Method, computer program, apparatus and system for quantifying and / or reducing measurement uncertainties of sensor units

Description

Method, computer program, device and system for quantifying and / or reducing measurement uncertainties of sensor units Description

[0001] The invention relates to a method for quantifying and / or reducing measurement uncertainties of sensor units. Furthermore, the invention relates to a method for generating and / or processing motion data of a freely moving body. The invention also relates to a method for capturing and / or evaluating motion sequences of a freely moving body. Furthermore, the invention relates to a method for detecting and / or evaluating dynamic instabilities of a freely moving body. The invention also relates to a method for generating and / or optimizing algorithms and / or models for calculating the dynamics of freely moving bodies. Furthermore, the invention relates to a computer program. Furthermore, the invention relates to a device for carrying out such methods. Furthermore, the invention relates to a system for acquiring and evaluating motion data.

[0002] The invention may also relate to: a standardized method for quantifying measurement inaccuracy and a calibration and adjustment routine for inertial sensors, in particular for angular acceleration sensors; a sensor for the precise kinematic measurement and robust assessment of balance disturbances during bipedal locomotion in humanoid systems. In particular, the invention may also include one or more of the following aspects: direct angular acceleration measurement for robust inverse dynamics to support balance control in humanoid robotics; robust inverse dynamics by evaluating the Newton-Euler equations with respect to a moving reference point and measuring the angular acceleration (also referred to as Research Paper I); estimation of an unexpected external force and its line of action occurring during bipedal locomotion Locomotion acting on a humanoid robot; wearable sensor for remote kinematic monitoring and assessment of balance loss during bipedal locomotion in humans; a wearable sensor and framework for precise remote monitoring of human movement (also referred to as Research Paper II); a framework for automatic detection of near falls using a wearable inertial measurement cluster (also referred to as Research Paper III); inertial measurement cluster for direct measurement of the angular acceleration vector in humanoid systems; direct angular acceleration measurements: a multi-method framework for standardized assessment of sensor uncertainty (also referred to as Research Paper IV); adaptation routine for reducing angular acceleration-related measurement uncertainty by optimizing model parameters.Therefore, the title of the patent application could also be: Sensor for precise kinematic measurements and robust assessment of balance disturbances during bipedal locomotion in humanoid systems.

[0003] Document DE 10 2023 120 648 B3 discloses a method for generating and / or processing motion data of a freely moving body, wherein at least two sensor units, each having at least one angular velocity sensor and at least one acceleration sensor, are arranged at a fixed distance from each other on the body, and an angular acceleration vector is directly determined on the basis of measurement signals from the at least two sensor units and by applying laws of general motion of rigid bodies. Furthermore, a device is known from document DE 10 2023 120 648 B3 which is designed to carry out such a method, wherein the device has at least two sensor units which can be arranged fixedly apart from each other on a freely movable body, each having at least one angular velocity sensor and at least one acceleration sensor.

[0004] For the definition of the state of the art or the technological background, in particular regarding paragraphs

[0053] until

[0082] The following references are made: [A1 ] WR Bussone, J. Olberding, and M. Prange. "Six-Degree-of-Freedom Accelerations: Linear Arrays Compared with Angular Rate Sensors in Impact Events". In: SAE International Journal of Transportation Safety 5.2 (2017), pp. 194-207. DOT: 10. 4271 / 2017-01-1465. [A2] A. Colome et al. " External force estimation during compliant robot manipulation". In: 2013 IEEE international conference on robotics and automation. IEEE, 2013, pp. 3535-3540. DOT: 10.1109 / ICRA.2013.6631072. [A3] S. Diaz, J. B. Stephenson, and M. A. Labrador. " Use of Wearable Sensor Technology in Gait, Balance, and Range of Motion Analysis". In: Applied Sciences 10.1 (2019). DOI: 10.3390 / app10010234. [A4] B. Fang, W. Chou, and L. Ding. " An optimal calibration method for a MEMS inertial measurement unit". In: International Journal of Advanced Robotic Systems 11.2 (2014). Publisher: SAGE Publications Sage UK: London, England. DOT: 10. 5772 / 57516. [A5] T. Feng et al. " Research on calibration method of MEMS gyroscope mounting error based on large-range autocollimator". In: IEEE Sensors Journal (2023). Publisher: IEEE. DOT: 10. 1109 / JSEN. 2023.3303254. [A6] M. Gießler and B. Waltersberger. " Robust inverse dynamics by evaluating Newton-Euler equations with respect to a moving reference and measuring angular acceleration". In: Autonomous Robots 47 (2023), pp. 465-481. DOT: 10.1007 / s10514-023-10092-x. [A7] M. Gießler et al. " A wearable sensor and framework for accurate remote monitoring of human motion". In: Communications Engineering 3.20 (2024), pp. 1 -15. DOT: 10. 1038 / s44172-024-00168-6. [A8] A. Harindranath and M. Arora. " A systematic review of user - conducted calibration methods for MEMS-based IMUs". In: Measurement 225 (2024). DOT: 10.1016 / j. measurement.2023.114001. [A9] C.-W. Ho and P.-C. Lin. " Design and implementation of a 12-axis accelerometer suite". In: 2009 IEEE / RSJ International Conference on Intelligent Robots and Systems. St. Louis, MO: IEEE, 2009, pp. 2197-2202. DOT: 10. 1109 / IROS. 2009. 5353892. [A10] Y.-S. Kang, K. Moorhouse, and J. H. Bolte. " Measurement of Six Degrees of Freedom Head Kinematics in Impact Conditions Employing Six Accelerometers and Three Angular Rate Sensors (6aw Configuration)". In: Journal of Biomechanical Engineering 133.11 (2011 ). DOT: 10.1115 / 1.4005427. [A11 ] Q. Leboutet et al. " Inertial Parameter Identification in Robotics: A Survey". In: Applied Sciences 11.9 (2021 ). DOT: 10.3390 / app11094303. [A12] Y. Lin, H. Zhao, and H. Ding. " External force estimation for industrial robots with flexible joints". In: IEEE Robotics and Automation Letters 5.2 (2020). Publisher: IEEE, pp. 1311-1318. DOT: 10.1109 / LRA.2020.2968058. [A13] P. G. Martin et al. " Measurement Techniques for Angular Velocity and Acceleration in an impact Environment". In: 1997. DOI: 10.4271 / 970575. [A14] P. G. Martin et al. " Measuring the Acceleration of a Rigid Body". In: Shock and Vibration 5.4 (1998), pp. 211 -224. DOI: 10.1155 / 1998 / 134562. [A15] A. J. Padgaonkar, K. W. Krieger, and A. I. King. " Measurement of Angular Acceleration of a Rigid Body Using Linear Accelerometers". In: Journal of Applied Mechanics 42.3 (1975), pp. 552-556. DOI: 10.1115 / 1.3423640. [A16] X. Ru et al. " MEMS Inertial Sensor Calibration Technology: Current Status and Future Trends". In: Micromachines 13.6 (2022). DOI: 10.3390 / M1_13060879. [A17] J. Wang and N. Liu. " MEMS-IMU automatic calibration system design and implementation". In: Journal of Physics: Conference Series. Vol. 2492. IOP Publishing, 2023. DOT: 10.1088 / 1742-6596 / 2492 / 1 / 012005. [A18] C. Woernle. Multibody Systems: An Introduction to the Kinematics and Dynamics of Rigid Body Systems. 2nd, expanded edition. Textbook. Berlin Heidelberg: Springer Vieweg, 2016. DOI: 10.1007 / 978-3-662-46687-2. [A19] Y. Xie et al. "A Review: Robust Locomotion for Biped Humanoid Robots". In: Journal of Physics: Conference Series 1487.1 (2020). DOI: 10.1088 / 1742-6596 / 1487 / 1 / 012048. [A20] C.-C. Yang and Y.-L. Hsu. "A Review of Accelerometry-Based Wearable Motion Detectors for Physical Activity Monitoring". In: Sensors 10.8 (2010), pp. 7772-7788. DOI: 10.3390 / sl 00807772. [A21 ] S. Yousefizadeh and T. Bak. "Unknown external force estimation and collision detection for a cooperative robot". In: Robotica 38.9 (2020). Publisher: Cambridge University Press, pp. 1665-1681. DOI: 10.1017 / S0263574719001681. For the definition of the state of the art or the technological background, in particular regarding paragraphs

[0083] bis

[0103] , wird außerdem auf die folgenden Referenzen verwiesen: [B1 ] Y. Adesida, E. Papi, and A. H. McGregor. " Exploring the Role of Wearable Technology in Sport Kinematics and Kinetics: A Systematic Review". In: Sensors 19.7 (2019). DOI: 10.3390 / sl 9071597. [B2] G. Bellusci et al. AN-5083: FIS1100 Attitude Engine Low Power Motion CoProcessor for High Accuracy Tracking Applications. Tech. rep. Fairchild Semiconductor Corporation, 2015. [B3] P. Bet, P. C. Castro, and M. A. Ponti. " Fall detection and fall risk assessment in older person using wearable sensors: A systematic review". In: International Journal of Medical Informatics 130 (2019), p. 103946. DOI: 10.1016 / j. ijmedinf.2019.08. 006. [B4] W. R. Bussone, J. Olberding, and M. Prange. " Six-Degree-of-Freedom Accelerations: Linear Arrays Compared with Angular Rate Sensors in Impact Events". In: SAE International Journal of Transportation Safety 5.2 (2017), pp. 194-207. DOI: 10. 4271 / 2017-01-1465. [B5] A. Colome et al. " External force estimation during compliant robot manipulation". In: 2013 IEEE international conference on robotics and automation. IEEE, 2013, pp. 3535-3540. DOI: 10.1109 / ICRA.2013.6631072. [B6] S. Diaz, J. B. Stephenson, and M. A. Labrador. " Use of Wearable Sensor Technology in Gait, Balance, and Range of Motion Analysis". In: Applied Sciences 10.1 (2019). DOI: 10.3390 / app10010234. [B7] F. Ferretti, A. Saltelli, and S. Tarantola. " Trends in sensitivity analysis practice in the last decade". In: Science of the Total Environment 568.1 (2016), pp. 666-670. DOI: 10.1016 / j. scitotenv. 2016.02.133. [B8] P. C. Fino, F. B. Horak, and C. Curtze. " Inertial Sensor-Based Centripetal Acceleration as a Correlate for Lateral Margin of Stability During Walking and Turning". In: IEEE Transactions on Neural Systems and Rehabilitation Engineering 28.3 (2020), pp. 629-636. DOI: 10. 1109 / TNSRE. 2020.2971905. [B9] M. Ghislieri et al. " Wearable Inertial Sensors to Assess Standing Balance: A Systematic Review". In: Sensors 19.19 (2019). DOI: 10.3390 / s19194075. [B10] M. Gießler and B. Waltersberger. " Robust inverse dynamics by evaluating Newton- Euler equations with respect to a moving reference and measuring angular acceleration". In: Autonomous Robots 47 (2023), pp. 465-481. DOI: 10.1007 / s10514-023-10092-x. [B 11 ] M. Gießler et al. " A wearable sensor and framework for accurate remote monitoring of human motion". In: Communications Engineering 3.20 (2024), pp. 1 -15. DOI: 10. 1038 / s44172-024-00168-6. [B12] F. A. Graybill and C.-M. Wang. " Confidence Intervals on Nonnegative Linear Combinations of Variances". In: Journal of the American Statistical Association 75.372 (1980), pp. 869-873. DOI: 10.2307 / 2287174. [B13] A. Harindranath and M. Arora. " A systematic review of user - conducted calibration methods for MEMS-based IMUs". In: Measurement 225 (2024). DOT: 10. 1016 / j. measurement.2023.114001. [B14] B. Henze, M. A. Roa, and C. Ott. " Passivity-based whole-body balancing for torquecontrolled humanoid robots in multi-contact scenarios". In: The International Journal of Robotics Research 35.12 (2016), pp. 1522-1543. DOT: 10.1177 / 027836491665381 E [B15] H. Herr and M. B. Popovic. " Angular momentum in human walking". In: Journal of Experimental Biology 211.4 (2008), pp. 467-481. DOT: 10.1242 / j eb.008573. [B16] C.-W. Ho and P.-C. Lin. " Design and implementation of a 12-axis accelerometer suite". In: 2009 IEEE / RSJ International Conference on Intelligent Robots and Systems. St. Louis, MO: IEEE, 2009, pp. 2197-2202. DOT: 10. 1109 / IROS. 2009. 5353892. [B17] J. Howcroft, J. Kofman, and E. D. Lemaire. " Review of fall risk assessment in geriatric populations using inertial sensors". In: Journal of NeuroEngineering and Rehabilitation 10.1 (2013). DOT: 10.1186 / 1743-0003-10-91. [B18] H. Hu et al. " Performance Evaluation of Optical Motion Capture Sensors for Assembly Motion Capturing". In: IEEE Access 9 (2021 ), pp. 61444-61454. DOT: 10. 1109 / ACCESS. 2021.3074260. [B19] Y. Huang et al. " Dynamic Parameter Identification of Serial Robots Using a Hybrid Approach". In: IEEE Transactions on Robotics 39.2 (2023), pp. 1607-1621. DOT: 10. 1109 / TRO.2022.3211194. [B20] International Organization for Standardization. ISO / IEC Guide 98-3:2008 - Uncertainty of measurement - Part 3: Guide to the expression of uncertainty in measurement (GUM:1995). 2023. URL: https: / / www.iso.org / standard / 50461.html. [B21 ] J. Jovic et al. " Humanoid and Human Inertia Parameter Identification Using Hierarchical Optimization". In: IEEE Transactions on Robotics 32.3 (2016), pp. 726-735. DOT: 10. 1109 / TR0.2016.2558190. [B22] Y.-S. Kang, K. Moorhouse, and J. H. Bolte. " Measurement of Six Degrees of Freedom Head Kinematics in Impact Conditions Employing Six Accelerometers and Three Angular Rate Sensors (6aw Configuration)". In: Journal of Biomechanical Engineering 133.11 (2011 ). DOT: 10.1115 / 1.4005427. [B23] Q. Leboutet et al. " Inertial Parameter Identification in Robotics: A Survey". In: Applied Sciences 11.9 (2021 ). DOT: 10.3390 / app11094303. [B24] Y. Lin, H. Zhao, and H. Ding. " External force estimation for industrial robots with flexible joints". In: IEEE Robotics and Automation Letters 5.2 (2020). Publisher: IEEE, pp. 1311-1318. DOT: 10.1109 / LRA.2020.2968058. [B25] P. G. Martin et al. " Measurement Techniques for Angular Velocity and Acceleration in an impact Environment". In: 1997. DOT: 10.4271 / 970575. [B26] P. G. Martin et al. " Measuring the Acceleration of a Rigid Body". In: Shock and Vibration 5.4 (1998), pp. 211 -224. DOT: 10.1155 / 1998 / 134562. [B27] M. Menolotto et al. " Motion Capture Technology in Industrial Applications: A Systematic Review". In: Sensors 20.19 (2020). DOT: 10.3390 / s20195687. [B28] G. Niemann et al. Maschinenelemente 1: Konstruktion and Berechnung von Verbindungen, Lagern, Wellen. Berlin, Heidelberg: Springer Berlin Heidelberg, 2019. DOT: 10. 1007 / 978-3-662-55482-1. [B29] S. J. Ovaska and S. Valiviita. " Angular acceleration measurement: a review". In: IMTC / 98 Conference Proceedings. IEEE Instrumentation and Measurement Technology Conference. Where Instrumentation is Going (Cat. No.98CH36222). Vol. 2. St. Paul, MN, USA: IEEE, 1998, pp. 875-880. DOT: 10.1109 / IMTC. 1998.676850. [B30] A. J. Padgaonkar, K. W. Krieger, and A. I. King. " Measurement of Angular Acceleration of a Rigid Body Using Linear Accelerometers". In: Journal of Applied Mechanics 42.3 (1975), pp. 552-556. DOT: 10.1115 / 1.3423640. [B31 ] F. Pianosi, F. Sarrazin, and T. Wagener. " A Matlab toolbox for Global Sensitivity Analysis". In: Environmental Modelling 6 Software 70.1 (2015), pp. 80-85. DOT: 10.1016 / j.envsoft.2015.04.009. [B32] X. Ru et al. " MEMS Inertial Sensor Calibration Technology: Current Status and Future Trends". In: Micromachines 13.6 (2022). DOT: 10.3390 / mi13060879. [B33] A. Saltelli. Global sensitivity analysis: the primer. John Wiley, Chichester, England, 2008. DOT: 10.1002 / 9780470725184. [B34] A. Saltelli et al. " Why so many published sensitivity analyses are false: A systematic review of sensitivity analysis practices". In: Environmental Modelling 6 Software 114.1 (2019), pp. 29-39. DOT: 10.1016 / j.envsoft.2019.01.012. [B35] M. Schepers et al. AN-5084: XKF3 - Low-Power, Optimal Estimation of 3D Orientation using Inertial and Magnetic Sensing. Tech. rep. Fairchild Semiconductor Corporation, 2015. [B36] R. Stribeck. " Die wesentlichen Eigenschaften der Gleit- und Rollenlager (German Text)". In: Zeitschrift des Vereins Deutscher Ingenieure 46 (1902), pp. 1341 -1348. [B37] Y. Wang et al. "Investigation on frictional characteristic of deep-groove ball bearings subjected to radial loads". In: Advances in Mechanical Engineering 7.7 (2015), pp. 1-12. DOT: 10.1177 / 1687814015586111. [B38] C. Woernle. Multibody Systems: An Introduction to the Kinematics and Dynamics of Rigid Body Systems. 2nd, expanded edition. Textbook. Berlin Heidelberg: Springer Vieweg, 2016. DOT: 10.1007 / 978-3-662-46687-2. [B39] Y. Xie et al. "A Review: Robust Locomotion for Biped Humanoid Robots". In: Journal of Physics: Conference Series 1487.1 (2020). DOT: 10. 1088 / 1742-6596 / 1487 / 1 / 012048. [B40] C.-C. Yang and Y.-L. Hsu. "A Review of Accelerometry-Based Wearable Motion Detectors for Physical Activity Monitoring". In: Sensors 10.8 (2010), pp. 7772-7788. DOT: 10.3390 / sl 00807772. [B41 ] S. Yousefizadeh and T. Bak. "Unknown external force estimation and collision detection for a cooperative robot". In: Robotica 38.9 (2020). Publisher: Cambridge University Press, pp. 1665-1681. DOT: 10.1017 / S0263574719001681.

[0005] The invention is based on the objective of structurally and / or functionally improving at least one of the aforementioned methods, computer program, device and / or system.

[0006] The problem is solved by a method having the features of claim 1. Furthermore, the problem is solved by a method having the features of claim 10. Furthermore, the problem is solved by a method having the features of claim 11. Furthermore, the problem is solved by a method having the features of claim 12. Furthermore, the problem is solved by a method having the features of claim 13. Furthermore, the problem is solved by a computer program having the features of claim 14. Furthermore, the problem is solved by a device having the features of claim 15. Furthermore, the problem is solved with a system having the features of claim 20. Advantageous embodiments and / or further developments are the subject of the dependent claims.

[0007] The method according to the invention serves to quantify and / or reduce measurement uncertainties of sensor units using an adjustment routine in order to cope with mechanical imperfections and thus to quantify and / or reduce the measurement uncertainty of at least one sensor unit.

[0008] The invention can aim at the combined modeling and optimization / compensation of deterministic error sources through parameter adjustment. A particular aspect of the invention can lie in the following: the holistic consideration and correction of all relevant error sources, from the individual components to the geometry of the cluster. Joint optimization of the parameters can be central. These parameters can explicitly describe both geometric imperfections of the cluster (relative vectors of the measurement points, orientation of the sensors) and sensor-internal corrections (of the SMD components themselves, such as axis skew, offsets, scale errors, etc.).

[0009] At least one sensor unit can be configured as an Inertial Measurement Unit (IMU). An IMU can include a three-axis accelerometer. An IMU can include a three-axis angular velocity sensor. The sensor unit can be configured as an Inertial Measurement Cluster (IMC). An IMC can include multiple spatially distributed IMUs. The number of IMUs can be four. The spatial distribution can be achieved by a rigid support. The rigid support can be made of polyvinyl chloride. The rigid support can have milled areas. The weight of the IMC can be approximately 0.8 kg. The geometry can be adapted to anthropometric dimensions. The geometry can be adapted to the dimensions of a human thorax. The sensor unit can be based on microelectromechanical systems (MEMS). MEMS sensors can have a compact design. Alternatively or additionally, fiber optic gyroscopes can be used. Alternatively or additionally to accelerometers, piezoelectric sensors can be used. Alternatively or additionally to triaxial sensors, uniaxial sensors in an orthogonal arrangement can be used.

[0010] The sensors can be integrated into a common housing. The housing can have a defined external geometry. The sensor unit can be portable. The sensor unit can be designed for several hours of battery operation. The sensor unit can be capable of operating independently.

[0011] Measurement uncertainty can include stochastic errors. Stochastic errors can be modeled as normally distributed noise. Stochastic errors can be modeled as additive white Gaussian noise. Measurement uncertainty can include deterministic errors. Deterministic errors can include bias. Deterministic errors can include scale factor errors. Deterministic errors can include non-orthogonality of the sensitivity axes. Deterministic errors can include crosstalk of the sensitivity axes. Deterministic errors can include temperature-dependent drift. Measurement uncertainty can be quantified as a standard deviation. The standard deviation can vary for different angular acceleration ranges. For angular accelerations up to 21 rad / s² 2 The standard deviation can average 0.3 rad / s 2The measurement uncertainty can be specified with 95% confidence intervals. The measurement uncertainty can decrease depending on the acceleration. For angular accelerations between 0 and 5 rad / s² 2 The standard deviation can be 0.458 rad / s 2 at angular accelerations above 20 rad / s². 2 The standard deviation can be 0.173 rad / s 2 be.

[0012] The fitting routine can be a calibration routine. The fitting routine can include an optimization routine. The optimization routine can be based on the least squares method or use robust regression. The optimization routine can use a Levenberg-Marquardt algorithm or Gauss-Newton. The fitting routine can estimate parameters of a model. The model can describe the sensor characteristics. The fitting routine can operate iteratively. The fitting routine can require initial values ​​for parameters. The initial values ​​can be extracted from CAD models. The initial values ​​can be obtained from sensitivity analyses. The fitting routine can be performed using a 3D coordinate measuring machine. The 3D coordinate measuring machine can determine error limits for mispositioning. The 3D coordinate measuring machine can determine error limits for misalignment. The fitting routine can include static calibration.Static calibration can take approximately 30 seconds. Static calibration can reduce constant offset errors. The fitting routine can use averaging across measurement samples. Averaging can reduce the influence of white noise.

[0013] Mechanical imperfections can be deterministic errors, such as non-orthogonality of measuring axes, stresses on a component that lead to offsets, scale errors (for example, it should measure 5 m / s^2, but measures 7 m / s^2), i.e., mechanical imperfections of at least one sensor unit itself. Mechanical imperfections can also be inherent to the design, since for at least one sensor unit no physical measurement point can be identified. However, a point can be modeled and the location in space can be found whose acceleration state is optimally represented by the measurement. Mechanical imperfections may only become relevant through use in a cluster, for example through the orientation of sensor units relative to each other.

[0014] Mechanical imperfections can include manufacturing tolerances. Manufacturing tolerances can occur during the production of the carrier. Mechanical imperfections can include assembly tolerances. Assembly tolerances can arise during the attachment of the sensors to the carrier. Imperfections can include misalignment between sensor coordinate systems. The misalignment can be described by rotation angles. The rotation angles can be parameterized as Euler angles. The rotation angles can be defined about the x, y, and z axes. The rotation angles can be less than 1°. Alternatively or additionally to Euler angles, quaternions can be used. Mechanical imperfections can include mispositioning. The mispositioning can be described by a three-dimensional offset vector. The offset vector can be added to the nominal position vector. The deviations can be in the millimeter range. The deviations can be in the sub-millimeter range. Mechanical imperfections can include an eccentric center of gravity. Mechanical imperfections can include a tilted axis of rotation. The tilt can be defined relative to the gravitational field.

[0015] Any accelerated motion of a pendulum can be applied / used. At least relative vectors between modeled, fictitious, or assumed accelerometer measurement points can be estimated. The alignment and / or orientation between at least two accelerometers and / or angular velocity sensors can be estimated.

[0016] The pendulum can be a physical pendulum. Alternatively or additionally, a torsion pendulum or a rotating table can be used. The pendulum can have one rotational degree of freedom. The pendulum can oscillate in a gravitational field. The oscillation can be damped. The damping can be viscous. The damping can be caused by bearing friction. The bearing friction can be caused by ball bearings. The bearing can be a rolling bearing. The pendulum can have a pendulum rod. The pendulum rod can have a defined mass distribution. The mass distribution can determine the center of gravity. The center of gravity can be eccentric. The pendulum can have a moment of inertia. The moment of inertia can be characterized by the radius of gyration. The reciprocal of the square of the radius of gyration can be 7.88 m. 2 The pendulum can carry reflective markers. The number of markers can be four. The markers can be in The markers must be positioned at defined intervals. These intervals can be 0.1 m, 0.2 m, 0.3 m, and 0.4 m. The markers can be attached to the pendulum rod. The pendulum motion can have different initial conditions. These initial conditions can include an initial angle. The initial angle can be 1.0234 rad or 0.4386 rad. The initial conditions can also include an initial angular velocity. The initial angular velocity can be 0.0053 rad / s or 0.0298 rad / s. Different initial conditions can cover different angular acceleration ranges.

[0017] The relative vectors can be position vectors between accelerometers. The position vectors can be expressed in a reference coordinate system. The reference coordinate system can be fixed to the reference sensor. The reference sensor can be designated with the number 0. The measurement point can be a physical point of the rigid body. The physical point can optimally represent the measured linear acceleration. The optimal representation can be defined in terms of minimum deviations. The measurement point can lie within the sensor volume. The sensor volume can be finite. The exact measurement point can be unknown. The relative vectors can have three coordinates. The coordinates can be defined in the x, y, and z directions. The vectors can be linearly independent. Linear independence can enable the solution of a system of equations. Linear independence can ensure a unique solution. The sensors can be located outside the same spatial plane. The relative vectors can be collinear with basis vectors.In an orthogonal arrangement, every relative vector can be parallel to a basis vector. The norms of the relative vectors can be 0.1375 m, 0.175 m, and 0.1 m.

[0018] The orientation can be described by rotation matrices. The rotation matrices can combine elementary rotations. The combination can be successive around three axes. The orientation can be defined relative to a reference sensor. The orientation can be assumed to be constant over time. The time-constant orientation can require a rigid connection. The orientation can be defined by three angles. The angles can be parameterized. They can be referred to as transformation angles. The transformation angles can be 0.0070 rad, 0.0054 rad, 0.0014 rad, and 0.0017 rad. The misalignment can result from the accelerometer's manufacturing process. The misalignment can be MEMS-specific. Unit quaternions can be used as an alternative or additional to rotation matrices. Geometric objects can be used as an alternative or additional to coordinate matrices.

[0019] A multi-method framework can be applied / used in combination with an optimization routine based on the least squares method.

[0020] The multi-method framework can combine various measurement techniques. These techniques can include MOCAP systems. Alternatively or additionally, a laser tracker or encoder can be used. The measurement techniques can include IMU measurements. The measurement techniques can include IMC measurements. The framework can include transformations between coordinate systems. These transformations can include rotations. These transformations can include translations. The framework can include post-processing methods. These post-processing methods can include filtering. These post-processing methods can include numerical differentiation. These post-processing methods can include numerical integration. The numerical differentiation can include backward differences (BD). The numerical differentiation can include central differences (CD). The backward differences method can be real-time capable. The central differences method can provide more accurate results.The central difference method may require a posteriori processing. The framework may include a validation component. Validation may be performed across all kinematic planes. The kinematic planes may include angles. The kinematic planes may include angular velocity. The kinematic planes may include angular acceleration.

[0021] The optimization routine can minimize an objective function. The objective function can describe the squared deviation between measurement and model. The objective function can be formulated as an L1 or L2 norm. The minimization can be performed iteratively. Minimization can use the Levenberg-Marquardt algorithm or the Gaussian-Newton algorithm. Optimization can consider constraints. Constraints can be parameter limits. The parameter limits can be derived from physical considerations. The parameter limits can be derived from sensitivity analysis. The parameter limits can include upper bounds. The parameter limits can include lower bounds. Optimization can provide a covariance matrix. The covariance matrix can quantify parameter uncertainties. The covariance matrix can provide 95% confidence intervals.

[0022] At least one sensor unit can have at least one angular velocity sensor and at least one acceleration sensor.

[0023] The angular velocity sensor can be a MEMS gyroscope. The gyroscope can have three orthogonal sensitivity axes. The sensitivity axes can be labeled x, y, and z. The gyroscope can have a measurement rate of 200 Hz. The gyroscope can have a measurement range of ±2000° / s. The angular velocity sensor can provide position-independent measurements. This position independence can be derived from the rigid body hypothesis. The sensor can exhibit noise. The noise can be modeled as white noise. The noise density can be taken from the datasheet. The sensor can have quaternion-based Kalman filter routines or Kalman filter routines. The filter routine can enable real-time integration at 200 Hz or higher.

[0024] The accelerometer can be a MEMS accelerometer. The accelerometer can have three orthogonal sensitivity axes. The accelerometer can have a sampling rate of 200 Hz. The accelerometer can have a measuring range of ±160 m / s². 2 exhibit. The measurement can be position-dependent. The position dependency can result from rigid body kinematics. Different linear accelerations can occur at different points on the rigid body. The acceleration sensors can be spatially distributed. The acceleration sensors can have a finite volume. The finite volume can leave an exact measurement point undefined. The measurement point can be a It may be a fictitious construct that approximates the behavior of a sensor, but does not necessarily have to be physically interpretable.

[0025] At least one of the following steps can be performed: using a pendulum, setting up an angular acceleration reference oscillating in a gravitational field; measuring the kinematics of the pendulum, in particular using a MOCAP system at a series of reflective markers on a pendulum rod (positions) and / or using IMU and IMC measurements of angular velocity and / or angular acceleration; deriving a mathematical model of the damped oscillation; estimating at least one complete set of parameters, in particular based on a least squares deviation of MOCAP data; performing a global sensitivity analysis, in particular during the estimation of the at least one set of parameters; comparing the measurement methods across all kinematic planes; determining parameter uncertainties from the optimization procedure.Feeding the parameter uncertainties into a Monte Carlo model simulation; Estimating model uncertainties from the model simulation; applying / using the estimated model uncertainties to quantify and / or reduce a measurement uncertainty of at least one sensor unit; validating the at least one sensor unit.

[0026] The angular acceleration reference can serve as a measurement standard. The measurement standard can have low measurement uncertainty. The measurement uncertainty of the reference can be two orders of magnitude smaller than that of the IMG. The reference can cover an angular acceleration range up to 21 rad / s². 2 The pendulum can oscillate in a plane. The plane of oscillation can be defined. The plane of oscillation can be characterized by a normal vector. The pendulum can be gimbal-mounted, hinged, supported, freely oscillating, or suspended at Euler angles.

[0027] The MOCAP system can include multiple infrared cameras. The number of cameras can be as high as twelve. The cameras can detect the positions of reflective markers. The markers can be attached to the pendulum pole. The number of markers can vary. The MOCAP system can have four markers. The markers can be positioned at defined intervals. The MOCAP system can have a measurement rate of 200 Hz. The MOCAP system can be configured as a Qualisys system. The system can output position vectors in its own coordinate system. The position measurements can have sub-millimeter accuracy. The MOCAP system can use Butterworth low-pass filters (BLPFs). The filter can be of the fourth order. The filter can reduce noise. Alternatively or additionally, a Kalman filter or extended Kalman filter can be used. Alternatively or additionally to a discrete-time filter, continuous-time filters can be used. Alternatively or additionally to low-pass filters, band-pass filters can be used.

[0028] The model can be an ordinary differential equation. The differential equation can describe pendulum motion. The model can take damping effects into account. The damping can be parameterized by coefficients of friction. The coefficient of friction can be 0.06. The coefficient of friction can be 0.02. The damping can depend on the bearing type. The damping can take Stribeck friction into account. The model can take experimental imperfections into account. Imperfections can include an eccentric center of gravity. The eccentricity can be described by a center of gravity vector. Imperfections can include a tilted axis of rotation. The tilt can be parameterized by angles α and β. The model can take gravitational influence into account. The gravitational constant can be location-dependent. The gravitational constant can be referenced to the PTB g-extractor. The gravitational constant can be determined for the latitude and altitude of the laboratory.

[0029] The parameter set can include physical parameters. Physical parameters can include the moment of inertia. Physical parameters can include the center-of-mass vector. Physical parameters can include the coefficient of friction. Physical parameters can include the gravitational constant. Physical parameters can include the lever arm of the frictional force. The lever arm can be taken from the bearing data sheet. The lever arm can be approximately 9.6 mm. The parameter set can include geometric parameters. Geometric parameters can include marker positions. Geometric parameters can Coordinate system orientation. Geometric parameters can include the offset of the MOCAP system. The parameter set can include 27 parameters. The parameters can have different physical units. The parameters can have different sensitivities. The sensitive parameters can be κ 2 , s_z and g include.

[0030] Sensitivity analysis can be global. Global analysis can use variance-based methods. Sensitivity analysis can quantify local effects. Sensitivity analysis can quantify total effects. Total effects can include higher-order interactions. Sensitivity analysis can include Fourier Amplitude Sensitivity Testing (FAST). FAST can compute time-dependent sensitivities. Sensitivity analysis can use the SAFE toolbox. Sensitivity indices can take values ​​between 0 and 1. High indices can identify important parameters. The indices can quantify main effects. The indices can quantify total effects. Sensitivity analysis can interact with optimization. The interaction can form a reciprocal feedback loop. The feedback loop can provide reliable initial values.

[0031] Monte Carlo simulation can propagate parameter uncertainties. Propagation can occur through repeated model evaluation. The number of simulations can reach 100,000. The simulations can use normally distributed parameter variations. The variations can be derived from a covariance matrix. The simulation can quantify model uncertainties. Quantification can be performed as standard deviations. The standard deviation can be acceleration-dependent. Confidence intervals can be calculated using the Satterthwaite method. The Satterthwaite method can be refined. The simulation can provide bias estimators. The bias can be IMC-specific. The bias can be within the interval [-0.1; 0.1] rad / s. 2 lay.

[0032] Validation can include a comparison across kinematic planes. The comparison can involve different measurement methods. Validation can use error measures. Error measures can be L1 standard. Error measures can be L2 standard. Error measures can be standard. Error measures can be L∞ standard. Error measures can include RMSE. Validation can include residual analysis. Residuals can be calculated between model and measurement. Validation can include graphical comparisons. Validation can include tables of error measures.

[0033] The method according to the invention serves to generate and / or process motion data of a freely moving body, wherein at least two sensor units, each having at least one angular velocity sensor and at least one acceleration sensor, are arranged at a fixed distance from one another on the body, and an angular acceleration vector is directly determined on the basis of measurement signals from the at least two sensor units and by applying laws of general motion of rigid bodies, wherein the measurement uncertainty of at least one sensor unit is quantified and / or reduced according to a method mentioned above.

[0034] The freely moving body can be a humanoid robot. The freely moving body can be a human. The freely moving body can be a humanoid system. The humanoid system can exhibit bipedal locomotion. The freely moving body can be modeled as a rigid multibody system. The segments can be assumed to be rigid bodies. The connections can be realized through joints.

[0035] The sensor units can be fixedly spaced. This fixed spacing can be achieved using a rigid support. The spacing can result in defined position vectors. These position vectors can be linearly independent. Linear independence can ensure uniqueness of the solution. The minimum number of sensor units can be two. Alternatively, three or four sensor units can be used. More sensor units can provide redundancy. Redundancy can reduce measurement uncertainty. Redundancy can enable outlier detection. The sensor units can be mounted on the torso. Mounting can be at the level of thoracic vertebrae 1-12. Attachment can be achieved using adjustable straps. The straps can provide stabilization at shoulder and waist level. This stabilization can compensate for relative movement between the IMG and... Minimize the torso. Alternatively or additionally, the head can be instrumented. Alternatively or additionally, extremities can be instrumented. Humans or robots can be instrumented. Quadruped systems can be instrumented as an alternative or additionally to bipedal systems. The freely moving body can be a car, vehicle, aircraft, mobile robot (drone, AGV, quadruped), quadrupedal system, six-legged system, eight-legged system, or n-legged system.

[0036] Direct determination can avoid numerical differentiation. Avoiding numerical differentiation can prevent noise amplification. The determination can be based on the rigid body equation. The rigid body equation can combine linear acceleration, angular velocity, and relative position vectors. The calculation can involve solving a system of linear equations. The system of equations can be formulated in dual vector space. The solution can be unique. Uniqueness can require linear independence. The solution can use circular permutation of the cross product. The determination can use sensor fusion. Sensor fusion can involve eight independent calculations per vector coordinate. Sensor fusion can use weighted averaging. The weight can be 0.8 for principal values. The weight can be (1 - 0.8) / 2 for outliers. The weight can be (1 - 0.8) for outliers. The weight can be alpha for principal values.The weighting can be (1 -alpha) for outliers.

[0037] Laws of general rigid body motion can include kinematic relationships. These relationships can link linear acceleration at different points. The relationship can be angular acceleration. The relationship can be angular velocity. The relationship can be relative position vectors. The laws can be formulated in arbitrary coordinate systems. The rigid body hypothesis can assume constant relative distances. The hypothesis can be valid for sufficiently stiff beams. The hypothesis can neglect soft-tissue artifacts. The laws can include Newton-Euler equations. The laws can be formulated according to Lagrange mechanics. The equations can be formulated for a moving reference point. The reference point can define a non-inertial frame of reference.

[0038] The method according to the invention serves to capture and / or evaluate the movement sequences of a freely moving body, wherein motion data of the body generated and / or processed according to a method mentioned above is used. Capture can include continuous recording. Recording can take place outside of laboratory conditions. Recording can take place during activities of daily living (ADLs). ADLs can include walking. ADLs can include running. ADLs can include climbing stairs. ADLs can include sitting-to-standing transitions. ADLs can include picking up and putting down objects. Evaluation can include comparison with reference patterns. Evaluation can include anomaly detection. Anomaly detection can be threshold-based. Anomaly detection can be AI-based or based on machine learning algorithms. The thresholds can be individualized.The threshold values ​​can be extracted from baseline measurements. The data acquisition can focus on trunk kinematics. The trunk kinematics can include angular acceleration about the transverse axis. The transverse axis can be the y-axis of the principal axis system.

[0039] The method according to the invention serves to detect and / or evaluate dynamic instabilities of a freely moving body, using motion data of the body generated and / or processed according to a method mentioned above. Dynamic instabilities can include loss of balance. The loss of balance can be caused by external disturbances. The disturbances can include stumbling. The disturbances can include slipping. The disturbances can include perturbations while standing. Detection can use thresholding methods. The thresholds can be individualized. The thresholds can be extracted from baseline curves. The thresholds can include local maxima. The thresholds can include global maxima. The thresholds can consider the frequency spectrum. The evaluation can include accumulated relative trunk torque (aTAM).The assessment may include the accumulated rate of change of hull torque (aRCTAM). Assessment can quantify recovery time. Assessment can resolve adaptation phenomena. Adaptation can include a 30% reduction. Detection can have a sensitivity of 100%. Detection can have a specificity of 98.4%. Detection can have a positive predictive value of 96.2%. Detection can have an F1 score of 98.1%. Classification can differentiate between stumbling events. Classification can differentiate between sliding events. Classification can differentiate between anterior and posterior instabilities. Classification can have an accuracy of 100%.

[0040] The method according to the invention serves to generate and / or optimize algorithms and / or models for calculating the dynamics of freely moving bodies, using motion data of the body generated and / or processed according to a method described above. The algorithms can include inverse dynamics. The inverse dynamics can calculate reaction forces. The inverse dynamics can calculate the zero-moment point (ZMP). The inverse dynamics can calculate the imaginary zero-moment point (IZMP). The inverse dynamics can estimate external forces. The estimation can include fusion with force-torque sensors. Models can describe multibody systems. The models can use a floating-base model. The models can use Newton-Euler equations. The models can use a moving reference point. The reference point can be fixed at the root segment. The algorithms can be real-time capable. The real-time capability can enable a fieldbus frequency of 125 Hz.The algorithms can operate without global position vectors. The algorithms can operate without global velocity vectors.

[0041] The method can serve to improve the measurement quality of an inertial sensor unit and may include at least one accelerometer and one angular velocity sensor, wherein the method may comprise at least one of the following steps: (a) calibrating the inertial sensor unit to identify and compensate for deterministic errors; (b) acquiring raw measurement data from the accelerometer and angular velocity sensor, which exhibit stochastic noise; (c) applying a machine learning model (ML model) to the measurement data compensated according to step (a), wherein the ML model is trained to detect and suppress the specific, sensor-characteristic stochastic noise (such as, in particular, thermal effects and electronic noise) of the sensors; (d) provide noise-reduced measurement data with significantly improved measurement quality compared to calibrated data only; (e) adapt the sensor geometry based on the achieved measurement quality, in particular reducing the distance between the accelerometers to enable a more compact inertial sensor unit while maintaining or exceeding the required measurement accuracy.

[0042] The computer program according to the invention comprises code sections that enable at least one of the above-mentioned methods to be executed when the computer program is run. The computer program can be written in a high-level programming language. The language can be Python. The language can be MATLAB. The language can be C++. The program can be modular. Modules can include data acquisition. Modules can include signal processing. Modules can include optimization. Modules can include visualization. Modules can include sensor fusion. The program can be executable on different operating systems. The program can be ROS2-compatible. The program can use the KDL library. The program can use the publisher-subscriber principle. The program can have a Webots, Gazebo, Isaac Gym, Isaac Sim, MuJoCo, Matlab, Recurdyn, and / or SimulationX interface. The interface can provide simulation results.The program can use the TF framework. The TF framework can provide coordinate transformations.

[0043] The device according to the invention serves to carry out a method according to the invention and comprises at least four acceleration sensors and at least one angular velocity sensor, each having at least one three-axis angular velocity sensor and at least one three-axis acceleration sensor, a rigid support on which the sensors are arranged such that three linear independent relative vectors are defined between a sensor serving as a reference and the other three sensors, and a computing unit for executing the adaptation routine.

[0044] The system according to the invention serves to capture and evaluate motion data and comprises the device according to the invention and the computer program according to the invention, wherein the system is designed to process the motion data in real time and to automatically quantify and reduce the measurement uncertainty of the sensor units.

[0045] The invention enables the systematic quantification and reduction of measurement uncertainties by compensating for mechanical imperfections such as misalignment and mispositioning of the sensor axes. Specialized calibration hardware is not required; instead, simple physical reference systems (e.g., pendulum motion) are used for parameter estimation. Through iterative optimization and statistical uncertainty analysis, a precise quantification of the measurement uncertainty with defined confidence intervals is achieved. This leads to a proven improvement in measurement accuracy and enables robust kinematic measurements for mobile and portable sensor systems.

[0046] The invention is described in more detail below with reference to the figures, which show schematically and by way of example: Figure 1 Schematic representation of a spatially distributed MEMS sensor configuration for direct measurement of the angular acceleration vector. In (a) the four spatially distributed 3D accelerometers and 3D gyroscopes mounted on a support are shown. In (b) the potential deterministic errors associated with the misorientation and relative vectors of the acceleration measurement points (mispositioning and model assumption of a dedicated measurement point) with respect to a direct angular acceleration measurement principle are visualized. Figure 2 Schematic representation of the specific deterministic errors of the IMC in (a) and an overview of the chain of reference frames for (b) the entire IMC sensor system (^V), the individual IMUs (^Sz), the 24 FILE COPY - TRANSLATION (RULE 26.3ter(e)) vibration plane >), the motion-capture system (^M) and the gravitational field (;). Figure 3 Overview of the chain of reference frames leading to (a) the entire IMC sensor (*^So) and the individual IMUs (^S / the plane of vibration ( O) and the body-fixed pendulum (*^P), (c) the motion capture system ( 'M) and the gravitational field ;) belong. Figure 4 Schematic curve of ideal a, ideal vs. measured acceleration coordinate ai, me ss, with scale error and offset. Figure 5 General oblique coordinate system K s compared to an orthogonal coordinate system Ksortho-

[0047] Sensory feedback based on precise kinematic measurements of body segments in humanoid systems plays a crucial role in applications aimed at improving the resilience of bipedal locomotion against balance disturbances. However, all measurements from inertial sensors inherently contain stochastic and potentially deterministic errors. The need for noise-amplifying numerical differentiation to indirectly measure highly dynamic kinematic quantities is a common limitation given the measurement uncertainty (i.e., the inherent stochastic noise) of the primary sensors used for humanoid systems, such as rotary encoders, inertial sensors, or optical motion detection systems.

[0048] We can model the measurement uncertainty introduced by the numerical time derivative using two principles. First, let the time derivative be numerically defined by the ÜJ (tf+At ) ~u> ( / ,■ ). real-time capable difference quotient approximated as Ar — Here, Z is the angular velocity vector, A is the angular acceleration vector, A is the time difference between two sampling points 6' and A'+l, and O(Az) is the error term due to linear approximation. Because of a finite Az corresponding to the sensor's sampling rate and a transient angular acceleration vector, an approximation error generally occurs with respect to the time derivative (O(A) ≠ 0). FILE COPY - TRANSLATION (RULE 26.3ter(e)) Secondly, the existing stochastic and deterministic errors in the measured signal must be taken into account. Deterministic errors play a subordinate role with regard to noise amplification. These are partly constant over time (e.g., offset / bias) or do not differ significantly between two sampling points (crosstalk effect of the sensitivity axes). Therefore, to investigate the primary source of noise amplification, we model the disturbed angular velocity measurement with = u; U) +nU), where ) represents the unknown angular velocity vector, possibly perturbed by deterministic errors, and This is the stochastic noise term (additive white Gaussian noise). Due to the underlying mathematical model of the normal distribution, the existing additive noise terms converge. ,li and il are not present in the measured signals when A approaches zero. Consequently, the stochastic noise is amplified. Furthermore, due to the normal distribution hypothesis of stochastic noise in the signals, unfavorable combinations of noise components can occur at neighboring sampling points of the This can cause non-physical artifacts. In summary, the noise-amplifying effect leads to a reduced signal-to-noise ratio and impractical results, for example, regarding the estimation of rollover stability indicators or externally applied forces.

[0049] To overcome this problem, various strategies have been employed to address the issue of noise amplification in simple numerical differentiation; these include both direct and indirect measurement methods. Other researchers have highlighted the real-time capability of angular acceleration measurement methods, specifically pointing out challenges related to noise amplification. They emphasized the need for tailored post-processing techniques for indirect approaches based on numerical differentiation. However, the effectiveness of filter techniques (e.g., finite impulse response filters) depends on the acceleration bandwidth to be resolved and the acceptable phase delay. In robotics, the use of filter techniques is limited due to the real-time requirements for sensor feedback. 26 FILE COPY - TRANSLATION (RULE 26.3ter(e)) Furthermore, state observer methods such as a Kalman filter can utilize direct joint position measurements 9 of an actuator to significantly improve the estimation of joint velocities 9 and joint accelerations 9 compared to simple backward differentiation. The advantage of the Kalman filter is that it optimally estimates the state variables (9 and 9), provided the noise of the joint position measurement is normally distributed, i.e., the signal is optimally filtered. However, using the joint position measurements 9 to indirectly determine the joint accelerations 9 is inherently subject to uncertainties, such as approximation errors caused by the finite sample size of the discrete-time measurement signals.

[0050] To mitigate the limitations of post-processing, direct measurement approaches for determining the angular acceleration vector have been proposed for biomechanical applications, such as analyzing the effects of car crashes on brain injuries using dummies. These approaches are based on the laws of rigid body motion and utilize piezoelectric accelerometers and gyroscopes. Other researchers have used a nine-accelerometer package (NAP) or twelve-accelerometer packages to measure angular acceleration. However, these approaches require either numerical integration or resolution of sign ambiguity in relative kinematic equations to determine the angular velocity vector, both of which rely on measurements of linear acceleration. Therefore, the estimation of the angular velocity vector is prone to error due to the noise inherent in the linear acceleration.Other researchers have used a sensor setup consisting of three three-dimensional accelerometers or, alternatively, six one-dimensional accelerometers, along with an additional gyroscope to measure the angular velocity vector separately. A significant limitation of the predominantly used direct measurement methods is the underlying assumption of orthogonal relative vectors between the multiple accelerometers. Mispositions and misorientations between the sensors and their mounting surface caused by mounting inaccuracies can lead to these limitations. 27 FILE COPY - TRANSLATION (RULE 26.3ter(e)) The respective coordinate frames on the accelerometer can lead to significant errors, see Fig. 1(b). To overcome the limitations caused by alignment errors and the assumption of orthogonal relative vectors, spatially distributed microelectromechanical systems (MEMS) such as 3D accelerometers and 3D gyroscopes can be combined with a more general model for angular acceleration measurement based on the laws of relative kinematics (Fig. 1(a)). This enables a direct measurement principle in which mispositioning and misorientation can be captured by descriptive model parameters.

[0051] MEMS devices are well-suited for realizing a portable sensor (a combination of multiple IMUs on a three-dimensional rigid substrate) that, due to its compact size, light weight, robustness, low power consumption, and cost-effectiveness, does not significantly restrict the range of motion or dynamics of the body segment. However, the kinematics measured with MEMS IMUs are affected by deterministic errors and stochastic noise. Therefore, a key aspect of MEMS IMU applications is to reduce or eliminate the effects of these errors. Stochastic noise and deterministic errors are typically assumed to be independent of each other; deterministic errors include bias (offset), scale factor errors, non-orthogonality and crosstalk of the sensitivity axes, alignment errors between the sensor and substrate, and bias due to thermal effects.Considering the underlying mechanical equations for direct angular acceleration measurements, potential deterministic errors can arise from manufacturing and assembly inaccuracies of the required three-dimensional support structure or from the mounting of the MEMS IMUs on it. Furthermore, when utilizing the laws of rigid body kinematics, knowledge of the relative vectors between the measurement points of the accelerometers becomes relevant, since the linear acceleration vector is generally not constant over the entire rigid body. Because MEMS accelerometers have a finite volume, the assumption that the measurement point is determined solely by the mounting position leads to... 28 FILE COPY - TRANSLATION (RULE 26.3ter(e)) This is known to contribute to additional measurement uncertainty. Ultimately, all these inherent stochastic and deterministic errors of the MEMS sensors, as well as additional deterministic errors in the direct measurement of the angular acceleration vector, contribute to the overall measurement uncertainty.

[0052] In general, a key difference between stochastic noise and deterministic errors is that deterministic errors can be eliminated by calibrating the sensors. Other researchers have investigated the proposed mathematical modeling of deterministic errors and calibration procedures for MEMS IMUs. For the proposed high-precision calibration and adjustment routines, motorized hardware, such as precision rotary tables, was used to provide reference data, including rotational position and angular velocity. The angular velocity of the rotary table served as the reference. For the MEMS accelerometer, the gravitational field can be used as a reference for the linear acceleration vector. Furthermore, both references can be used to determine the measurement uncertainty with respect to the sensitivity axes of the gyroscope and the accelerometer.However, regarding angular acceleration, no setup exists that provides an arbitrary accelerated motion, i.e., one suitable as a reference for angular acceleration. This significantly limits the quantification of measurement uncertainty and the adjustment of direct measurement principles.

[0053] Consequently, a crucial aspect of quantifying the influence of stochastic and deterministic errors in angular acceleration measurement methods is establishing an angular acceleration reference that exhibits a sufficiently low measurement uncertainty and can thus serve as a measurement standard. The requirements for the mechanical system are a general acceleration state and the condition that the angular acceleration can be derived without the use of indirect measurement techniques. Next, the measurement uncertainty of the reference itself must be taken into account in the empirical method for quantifying the sensor's measurement uncertainty. Furthermore, the angular acceleration reference can be used to implement extended calibration and adjustment routines that compensate for deterministic errors by adjusting descriptive model parameters accordingly. 29 FILE COPY - TRANSLATION (RULE 26.3ter(e)) It must be adapted to account for mispositioning and misorientation, for example. In summary, quantifying measurement uncertainty and implementing adjustment routines to compensate for the effects of deterministic errors are of great importance, as the precise determination of angular acceleration potentially has a significant impact on the robustness of bipedal locomotion. Such a reference would also play a pioneering role in comparing the measurement uncertainty of different sensor technologies.

[0054] The following paper elucidates the impact of model parameter optimization on mitigating IMC-related deterministic errors, with the aim of further reducing measurement uncertainty. Accurate kinematics measurement plays a central role in a variety of disciplines within robotics, including bipedal locomotion control and human motion capture. Therefore, quantifying and reducing the measurement uncertainty of inertial sensors is crucial and a major challenge in these fields. In this work, we propose an adjustment routine to address mechanical imperfections and thereby reduce the measurement uncertainty of the inertial measurement cluster (IMC), i.e., a sensor for directly measuring angular acceleration.Specifically, we use the arbitrarily accelerated motion of a pendulum to estimate the relative vectors between the measurement points of the modeled accelerometers. The results obtained show that the multi-method framework, in combination with a least-squares optimization-based adjustment routine, could enable the quantification and reduction of measurement uncertainties for a wide variety of sensors.

[0055] To monitor the kinematics of body segments in robotics or in studies of human motion analysis, wearable inertial sensors play a central role in diverse application scenarios, e.g. for the accurate evaluation of externally applied forces on collaborative manipulators [A2, A12, A21 ], for estimating inertial parameters of body segments in robotic systems [A11 ] or for remote sensing of human movements to gain insights into the complex 30 FILE COPY - TRANSLATION (RULE 26.3ter(e)) To provide insights into the interactions between active lifestyles and situations in which loss of balance occurs [A20, A3], particularly when used as sensory feedback for stability control algorithms to improve their resistance of bipedal locomotion to balance disturbances, high accuracy of kinematic sensing is required [A19]. Here, the primary inertial sensors used are based on micro-electro-mechanical systems (MEMS). A MEMS gyroscope and accelerometer directly measure the angular velocity and angular velocity, respectively. Acceleration vector in three orthogonal axes. However, the kinematic data measured by MEMS-IMUs are affected by deterministic errors and stochastic noise. Stochastic noise is modeled as a random error (normally distributed), while deterministic errors include offset errors, scaling factor errors, non-orthogonality, crosstalk effects of the sensitivity axes, and thermal effects [A16, A8]. In contrast to stochastic errors, deterministic errors are known to be mitigated by calibration and fitting of parameter models [A16, A8]. Calibration routines utilize high-precision motorized hardware such as precision rotary stages to provide reference data for rotation angle and angular velocity vector.In MEMS accelerometers, the gravitational field serves as a reference for the acceleration vector [A4, A5, A17]. These references allow for statistically determined measurement uncertainties for the sensitivity axes of the gyroscope and the accelerometer. In contrast, there is less information in the literature on approaches or parameter models proposed for the calibration and adaptation of direct measurement principles of the angular acceleration vector.

[0056] Furthermore, the angular acceleration vector is an important kinematic quantity, especially for assessing the dynamics of body segments. However, for quantifying the angular acceleration-dependent measurement uncertainty with respect to direct or indirect measurement principles, there were no or only inadequate approaches in the literature using an experimental setup [A14]. In most cases, various measurement principles or sensor technologies were used due to the 31 FILE COPY - TRANSLATION (RULE 26.3ter(e)) Lacking a reliable angular acceleration reference, comparisons were only qualitative [A10, A1, A13]. Another possibility was the use of numerical modeling. By solving the equations of motion, an ideal trajectory of the angular acceleration vector can be determined for a rigid body [A7]. However, numerical modeling has limitations regarding its accuracy in real-world applications. In particular, the accurate modeling of the inherent shortcomings of the measurement principle, such as stochastic noise or deterministic errors, is crucial. Consequently, the transfer of measurement uncertainty quantification based on simulation results to real-world applications critically depends on the accuracy of the error models used.

[0057] Regarding the direct measurement of angular acceleration, the proposed sensor concepts are mainly based on the use of spatially distributed inertial sensors [A15, A13, A9]. Our IMC concept uses MEMS inertial sensors, see [A7] and Research Paper IV. The spatial distribution can introduce additional deterministic errors. Manufacturing or assembly deficiencies can lead to misalignment between the coordinate systems of the sensors. Furthermore, the fundamental equation of the IMG shows that the principle of direct measurement depends on the relative vectors between the measurement points of the finite-sized accelerometers. The literature contains no information regarding the vector-based quantification of a spatial point whose acceleration vector trajectory is optimally represented by a measured but erroneous acceleration vector trajectory.This can be attributed to the fact that the relevance of modeling the estimated measurement point is negligible in the context of current accelerometer applications. Furthermore, the lack of an acceleration reference that establishes a general (not just translational) acceleration state makes estimating an accelerometer point impractical.

[0058] Using the multi-method framework proposed in research paper IV, we suggested angular acceleration using an oscillating pendulum in a gravitational field. Given the evidence-based, sufficient 32 FILE COPY - TRANSLATION (RULE 26.3ter(e)) Due to its small angular acceleration-dependent measurement uncertainty, it can be considered a reliable measurement standard. Consequently, it can be used as a calibration device. Because of the general acceleration state of the oscillating pendulum and the multi-method frame that provides a measurement standard, this setup offers a possibility for a calibration and adjustment routine for angular acceleration sensors. We use the angular acceleration reference of the multi-method frame to calibrate and adjust the IMC to reduce its measurement uncertainty. Therefore, we propose general parameter models to represent the additional potential deterministic errors of the IMC concept. Furthermore, we use least-squares optimization to estimate the parameters that represent the sensor misalignment and the relative vectors between the accelerometer points.

[0059] In Gießler et al. (2024) and in Research Paper IV, we introduced and validated the so-called Inertial Measurement Cluster (IMC), i.e., a sensor for the direct measurement of angular acceleration, and quantified its measurement uncertainty. In summary, the direct measurement principle of the IMC is based on angular velocity and spatially distributed acceleration measurements. Therefore, the IMC comprises a total of four Xsens MTi-20 VRU IMUs (Movella Inc., Nevada, USA) spatially distributed on a substrate assumed to be rigid (Fig. 2(b)). Each Xsens MTi-20 VRU consists of a 3D gyroscope and a 3D accelerometer based on MEMS technology. These MEMS inertial sensors are used to measure the linear acceleration vectors. a i and the angular velocity vectors to measure, which are based on the respective vector in the sensor package's own coordinate system reference. Any IMU, designated So, served as the reference, and its coordinate system ^*So as the reference frame for the IMC. The other three IMUs (Sj with 1 = 1? 2, 3) were connected with linearly independent vectors. r i,0 is positioned with respect to the reference. By using the same IMC setup, we can demonstrate the linear independence of r to utilize i,0 and their convectors to derive the well-known rigid body equation d i = EQ + W x Tj j + OJ X (w X Ff4o^ (7.1 ) 33 FILE COPY - TRANSLATION (RULE 26.3ter(e)) to solve for the angular acceleration vector W in the dual vector space, as shown in [7, Eq. (4.1)-(4.7)] and research paper I. To determine the coordinates a To extract V' from W for W — (t Cs,,,. k ', we interpret all vectors as their coordinate matrices in the non-inertial frame are referenced. The coordinate matrix of the angular acceleration is a, and its calculation ultimately yields (ri mx \ <«i - a0-w x (u> xr U) ),r 2<0 ) U'zj) x {a2- fl^i -wx (u> x r2,o),r3.o) (7.2) J r 3.Ü X TTj)) 7 ^ (a3- a0- wx (u > x r3.0), no) where ( » denotes the standard dot product. Note that c So, & the unit vectors of are, and the inverse matrix in Eq. (7.2) always exists due to the linear independence of the vectors. r 'd). Furthermore, the absolute Abi eitu ng ei n er Koo rd in aten matrix d ef in ie rt a Is d « = — v + wx (7.3) Ä where dr v the component-wise derivative of the coordinate matrix and W the angular velocity vector of is with which they relate to any d_ Inertial system rotates. Note that dr w the total derivative Q!= W of the angular velocity coordinate matrix, since with W rotates with respect to an arbit inertial frame

[0060] Parameter models for mapping the properties of finite-dimensional MEMS sensor components

[0061] To solve equation (7.2), the coordinates of W, and a i regarding required. As mentioned, the IMG uses MEMS-IMUs with its own orthonormal coordinate system for the sensor package. that defines the reference frame of the three orthogonal measurement axes of the geometrically finite extended gyroscope 34 FILE COPY - TRANSLATION (RULE 26.3ter(e)) and accelerometer. In real-world applications, potential deterministic errors are introduced due to inadequacies in the manufacturing of the IMC carrier or in the assembly process of the MEMS-IMUs. Furthermore, with respect to Eq. (7.2), the relative vectors of the accelerometer's measurement point (^1,0) also become relevant, since is position-dependent on a rigid body. Therefore, we propose additional parameter models to represent the properties of finite-dimensional sensor components.

[0062] In general, there are six degrees of freedom (DOF) for an IMU mounting on the support: three for its spatial position and three for its spatial orientation. To model the three DOFs with respect to orientation, we first define a general rotation matrix ' ' ' = ' ' ' ) (j reisuccessive elementary rotations, ordered around the z-, y- and x-axes) for Sf to determine its relative orientation with respect to ö to model 0 (misalignment), and thus to convert the measured vector coordinates from the i-th coordinate system to the reference frame of the to transform IMG, see

[0018]

[0063] Secondly, to model the three DOFs with respect to the relative vector between Si and So, we use the coordinates of r i,0, to account for potential mispositioning. Misposition refers to an unknown deviation in the positioning of the sensor circuit boards due to error-prone MEMS device assembly, tolerances, or manufacturing deficiencies in the carrier or the geometric positioning elements defined in the technical drawing, such as holes. Furthermore, the measured trajectory must a i * to be assigned to a single material point of the rigid body, whose actual acceleration trajectory, despite inherent stochastic and deterministic errors, is best described by a if) is approximated. Therefore, we do not only use ''ÄO to model mispositioning. We interpret r i,0 is more generally defined as the relative vector pointing from the measurement point of the sensor volume of accelerometer 0 to that of accelerometer i, and therefore takes into account both mispositionings and the generally unknown measurement point (Figure 2). Furthermore, we reference r i,0 in 35 FILE COPY - TRANSLATION (RULE 26.3ter(e)) According to the rigid body hypothesis, a measurement point model is not required for a gyroscope, since J is position-independent.

[0064] With these model assumptions, we introduce parametric models to describe the deterministic errors caused by the properties of the MEMS accelerometers and gyroscopes as the relative vector between the assumed measurement points and the misalignment. The implementation of the parametric models in Eq. (7.2) and the interpretation of all vectors as their coordinate matrices, which are in the non-inertial framework are referenced, results ( Sl, S1 jRai — ao - u? > < (w 3J) ). where C is defined as CJj) + S " S 'ÄWi + S " S '-'RÜJ2 + Stl S3 ÄüJ3 — — — — — — - - - — — _ z_ _ __ __ _ — (7.5) 4 where in Eq. (7.4) and (7.5) all vectors are interpreted as their coordinate matrices, which in their respective non-inertial frames *^So or are referenced.

[0065] To determine the coordinates of the angular acceleration vector, reference u? re f =(p eo x To compare those obtained through the measurement standard (research paper 4), ^IMC was placed within the framework the oscillatory (yz) plane (normal vector) e Ox) transformed. The transformation of the measured angular acceleration vector WING into ICO was performed by two rotations β and ξ about the y- and z-axes, respectively. The corresponding rotation matrix is ​​written as ''" R = R{ß< y ) In general, the overall parameter model is used to solve the problem of the relative orientations of the IMUs by the Cardan angles T ft» a i for i = 1, 2, 3 unc| through the relative vector coordinates between the assumed measurement points of ft and So with r to address i,0, 36 FILE COPY - TRANSLATION (RULE 26.3ter(e)) 9y. yi.jßi.ai,...,y3, A^«3»? T,o xJ Ti, Oy, JT, Oj ! • • •, n.tg •. n, u. j (7.6) It contained 20 parameters. It is noteworthy that the Cardan angles and the coordinates of r ' A in ^* H ? ~ were assumed to be constant over time.

[0066] ​As in Research Paper IV, a convention used here concerns the dependence of mathematical objects on the parameter vector ^P. In cases where we wish to emphasize this dependence, the entire vector is enclosed in brackets, e.g., ^IAIG Ä-, W for the IMC measurement result. To estimate the parameter values ​​and quantify their uncertainties, the measurement standard was used. w The angular acceleration reference obtained was adjusted by the IMC model equations, which are in were transformed. The optimal parameter vector ^P was obtained using the Isqnonlin routine – a confidence-space reflection algorithm – in MatLab (version 2023b, Mathworks, Natick, USA) to minimize the sum of the squared distances between ^ref at support points and the corresponding ^IMC, as determined from the raw measurements of and depends. The following equation writes the problem in ^O: ? P + = argmin ) - to| MC (U-: Wl-l (7 7 k F 2 Problem (7.7) is formulated as follows: that the discrete version of the L distance with / N, £2 ;= H | ( ] ; rt a( / A j_mode] (£: ') |“ V / is minimized, where N denotes the number of time points. Note that all 20 parameters were optimized simultaneously without intermediate steps. The determination of the initial values ​​픓₀ is explained in more detail in the following section.

[0067] To provide the optimizer with a suitable starting point ^Po, we define the identity matrix for prepared so that all initial Cardan angles were zero, and extracted the geometric dimensions from the CAD model as 37 FILE COPY - TRANSLATION (RULE 26.3ter(e)) r i,o = (-0.08m, Örn, Om)' r2.o = (O m, -0.08m, Om) 7 unc | T;{.n = (Om, O m, 0.05 m), Next, we fixed the IMG on the pendulum, approximately ~ rad around the z-axis of rotated to avoid an asymmetrical weight distribution. Consequently, the initial Cardan angles for n and each O rad and l rad : .

[0068] Regarding the least-squares optimization method for IMC adjustment, several angular acceleration reference trajectories are available. crucial. To ^ rel j UA) ZU To generate the results, we used the multi-method framework proposed in research paper 4. We repeated the procedure described there four times (J = 1. 2, 3, 4) mThe pendulum was subjected to various initial deflections. The optical motion capture system and the IMC, which was attached to the pendulum, simultaneously tracked the pendulum's movement at 200 Hz. The mounting position of the IMC relative to the pendulum remained unchanged. The results were four reference trajectories. with. / = 1» 2,3,4 unc j their corresponding raw measurement data of the IMC (^. and ai j^k ^). We then chained three of the four reference trajectories together. and also the raw data sets for. = 1, 2, 3, to generate a common dataset for the least-squares optimization procedure. This connection was possible because the ordinary differential equation of the pendulum (Research Paper IV, Eq. (6.9)) does not contain an explicit time dependence, such as a time-dependent excitation.

[0069] The model parameter was then defined with respect to the linked reference trajectory. (tk ) was adjusted using Eq. (7.7). We then obtained the optimized model parameter. , the residual R and the Jacobi matrix J. Using the nlpradci routine, we estimated the confidence intervals (Kl) of the 20 model parameters based on R and J.

[0070] To determine the effect of the optimized model parameter to identify, in particular regarding the parameters for mispositioning and the 38 FILE COPY - TRANSLATION (RULE 26.3ter(e)) Assuming a measurement point, we evaluated the model (Eq. (7.4)) with and the raw data of the IMG from the fourth attempt, which led to ) led to. Note that for this analysis we used an experiment that was not optimized.

[0071] Similarly, based on the raw data from the IMG of the fourth attempt, we evaluated • the model with reduced model parameters referred to as WeiP ed = rtl-g mill ZI kref ( a- )” , where , • the model with the initial model parameter W) corrected with an offset determined in a static position of the IMG ^IMC.corr(^ ) = W3 " as well as • the model with the initial model parameter is referred to as W).

[0072] Here, the reduced model parameters Wed only contain the aforementioned Cardan angles. All coordinates of ri,0 were interpreted as constants and set to the initial values ​​extracted from the CAD model. Since the relative vectors r Since i,0 are theoretically pairwise orthogonal, the IMG modeled in the CAD model could > W) ) the proposed approaches with the assumptions of strict orthogonality with respect to the position vectors of the sensors and exactly aligned measurement axes, as used in e.g. [15, 13, 9, 10].

[0073] The angular acceleration reference trajectories ( RA ) (Research work IV, Eq. (6.3)) for the four experiments E1, E2, E3 and E4 were obtained using the multi-method framework proposed in Research Paper IV. The resulting 2Distances (equivalent to RMSE) of the adapted mathematical pendulum model (Research Paper IV, Eqs. (6.12a) and (6.12b)) taking into account the mapping of the trajectories of the position vector of the pendulum markers for the four experiments were RMSEEI = 0.00061 m, RMSE E2 = 0.0014 m, 39 FILE COPY - TRANSLATION (RULE 26.3ter(e)) RMSE_E3 = 0.00045 m, as well as RMSE_E4 = 0.0015 m provided an indicator of the quality with respect to C..4., reff (tk) for j = 1, 2, 3, 4.

[0074] According to the parameter optimization procedure (Eq. (7.7)) and ^ rvf(ft \ we obtained the resulting optimal parameter vector The parameter vector values ​​are listed in Table 7.1 along with their 95% confidence intervals (Kl). All parameters relating to the misalignment of with reference to were found with comparable values ​​on the order of < 0.5°. Furthermore, the parameters for the coordinates of r ',() for = 1 * -, 3 referential in not significantly different from their initial estimates, with deviations of less than 10 mm. ß [rad] y fradj yi [rad] ft [rad] «i [rail] ya [rad] ft [rad] ay [rad] yj [rad] ft [rad] aj [rad] 0.0231 -0.7979 -0 0030 -0.0015 -0.0078 -0.0061, 0.0U2!l -0.0038 -0.0061 -0.0068 -0.0021t ±0.0739 ±0.0791 ±0.1511 ±0.161 ±0.0951 ±0.1377 ±0.0896 ±0.1673 ±0.0647 ±0.0042 ±0.0062 ro [in] r2,o M ft.o [m] -I) 0895 / 0.0080\ / ~0.0093\ / 0.01051 / -0.0023\ / 0.007ft -0.0095 | ± 0.0114 -0.0890 ± 0.00.82 -0.0016 ± 0.0061 -0.0044 \0.0049 / (-0.0016 / (0.0073 / \ 0.0503 / (0.0049 / Table 7.1: Optimized parameter vector for the linked experiment (El, E2 and E3) as calculated from Eq. (7.7). Point estimates are given with the radius of the corresponding 95% CI.

[0075] The comparison of the quantitative error measure E 2 The four different IMC model evaluations (Eq. (7.4)) based on the kinematic quantities of E4 are shown in Table 7.2. The reference used was the J taking into account the complete parameter vector Overall, the L ^distance for^IMC'f rtfH showed the smallest deviation with respect to ^ref l. ). Compared to the IMC model evaluations and ^lMCLoffset(ffc) resulted in absolute deviations of approximately 0.01 rad / s 2 or 0.02 rad / s 2 This resulted in relative deviations of 4% and 7%, respectively, taking into account the assumptions of pairwise orthogonal relative vectors.r i,0 and FILE COPY - TRANSLATION (RULE 26.3ter(e)) With precisely aligned measuring axes, the limitations of such models became clearly apparent. Compared to The L was used as a reference. 2 -Distance from • ^Po) um d as 2.44 times larger. Method error measure: £ 2 [rad / - [] 0.2923 1 W'lh: 0.303(i I- 04 0.3123 1.07 ^i. Mc(a: ^o> 0.7145 2.44 Table 7.2: Error measurements L 2 from ^IMC A ), obtained from different IMC model evaluations. All L 2 Distances are given with respect to the reference trajectory, which was chained together from the results of experiments E1, E2 and E3.

[0076] Well-known models for adjusting deterministic errors of MEMS inertial sensors to reduce measurement uncertainties are described in the literature [A16, A8]. However, for direct angular acceleration measurement with spatially distributed inertial sensors, further potential deterministic errors arise, such as misalignment or differing relative vectors of the acceleration measurement points between the MEMS sensors due to mechanical imperfections. The primary calibration and adjustment references used, such as high-precision rotary stages or the gravitational field, are not suitable for quantifying the measurement uncertainty and subsequent adjustment. The multi-method framework introduced in Research Paper IV addresses the problem of measurement uncertainty qualification for angular acceleration sensors.An adjustment routine is proposed to address mechanical imperfections and thereby reduce the measurement uncertainty of angular acceleration sensors by utilizing the angular acceleration reference of the multi-method framework.

[0077] From the model used for the IMG to calculate the angular acceleration from the measured a i and It follows that the coordinates of r i,0 between the acceleration measurement point coordinate frame and the orientation of the 41 FILE COPY - TRANSLATION (RULE 26.3ter(e)) Sensor coordinate frames (''■8,-) relative to the measurement coordinate frame of the IMC ''must be known. Due to mechanical imperfections in the fabrication of the support (assumed to be rigid), the mounting of the MEMS sensors on the support, or the finite geometric volume, and the underlying measurement principle of MEMS accelerometers, the actual relative vectors or the orientation of the sensor coordinate frames generally deviate from the ideal assumptions. This leads to the fact that the approaches proposed by other researchers offer only limited parameterizability for adjusting the measurement uncertainty due to the underlying assumptions of orthogonal relative vectors or ideally aligned sensor coordinate frames [A15, A13]. In contrast, the general model applied for the IMC (Eq. (7.4)) The relative vectors of the acceleration measurement points and the orientations of the sensor coordinate frames are mapped by 18 physically interpretable parameters. The selected sensor configuration (four 3D gyroscopes and 3D accelerometers) for the IMC includes redundant gyroscope and accelerometer measurement axes. The minimum sensor configuration with respect to the underlying model equation is represented by one 3D gyroscope and one 3D accelerometer, referred to as the reference, plus three spatially distributed 1D accelerometers, as shown in [A6]. It should be noted that the adjustment routine proposed here does not require redundancy in the IMC sensor configuration. However, redundancy was used to implement the sensor fusion concept proposed in [A7] to further reduce the measurement uncertainty of the IMC.It is hypothesized that by exploiting redundancy with sensor fusion, even lower measurement uncertainty values ​​could be achieved than when using the minimum sensor configuration, since each coordinate of the angular acceleration vector can be evaluated in eight different ways, thereby reducing the influence of stochastic noise (Bienayme identity for pairwise independent random variables).

[0078] With regard to our model, the rotation matrix dependent on Cardan angles with three parameters allows for three degrees of freedom for the respective spatial 42 FILE COPY - TRANSLATION (RULE 26.3ter(e)) To model the sensor's orientation, we introduced a minimo model that maps the trajectory of the generally error-prone linear acceleration vector to a material point of the rigid body, considering the positioning of the accelerometers, their finite geometric extent, and the measurement principle of MEMS accelerometers. The accuracy of this minimo model needs to be validated through further testing. However, an important aspect of the minimo model was that it led to a well-defined optimization problem where, for example, the Jacobian matrix was not badly conditioned.

[0079] The results obtained demonstrated the feasibility of IMC adjustment for reducing the distance between the teeth in relation to the angular acceleration vector reference. First, the L 2 -Distance by using the least-squares optimized ^P for the model evaluation ^IMU Ä. ■ ^P ) compared to the model evaluation based au f • ^Po) reduced by a factor of 2.44 (0.2928 rad / s 2 compared to 0.7145 rad / s 2 Secondly, the values ​​of *P were in physically realistic ranges and did not deviate significantly from the initial estimates of ^P(>). Additionally, the uncertainty of the values ​​was assessed using their 95% confidence intervals (Cis). For example, for T1, T2, or <%2, a 95% CI width was obtained that was two orders of magnitude larger than the estimated value itself. The low sensitivity of some model parameters was explained by their use on a 1D angular acceleration reference. w ref = a x e xO based (oscillation of the pendulum in a plane; see research paper IV) and on the identical IMC mounting orientation relative to the pendulum for all included trials. As a remedy for further evaluations, it is suggested to restrict the To circumvent 1D pendulum motion through various experiments with different mounting orientations of the IMG and the to estimate in a common optimization routine. Different assembly orientations of the IMG can be taken into account by creating an individual one for each. The Cardan angle combination βi and i is included to describe the actual orientation i. However, the 18 IMC-specific parameters for 43 FILE COPY - TRANSLATION (RULE 26.3ter(e)) All linked experiments remain constant. By selecting suitable mounting orientations so that each measurement axis of the IMC is adequately considered, the uncertainty of the parameters can be sufficiently analyzed.

[0080] From the 2.44-fold deviation of the L 2 -Distances in comparison between k-'l. XK 1 and ) i sThe need to adjust the IMC with respect to its mechanical imperfections is obvious. Since - ^Po) can be linked to the unadjusted approaches proposed in the literature, these have a significant limitation regarding measurement uncertainty. Furthermore, to compare the relevance of the introduced models, which compare the sensor coordinate frame orientation and the relative vectors of the acceleration measurement points, a reduced parameter set ^Pred was additionally introduced. The ratio of 1.04 between the model evaluation based on ^red and the reference ^P (0.3036 rad / s²) indicates 2 compared to 0.2928 rad / s 2 compared to 0.7145 rad / s 2We concluded that the calibration of the sensor coordinate frame alignment plays a significant role. On the other hand, we also confirmed that by adjusting the parameters (coordinates) with respect to the relative vectors of the acceleration measurement points of the L 2 The distance could be further reduced by 4%. This indicates the potential impact on measurement uncertainty when using the accelerometer's measurement point model. Finally, the comparison of the L distances from the adjustment routine based on the angular acceleration reference compared to ^Po corrected with an offset to reduce the constant bias in the IMC's rest position showed that _ 9 The L-distances were reduced by 7%.

[0081] The reliable angular acceleration vector reference provided by the multi-method framework can serve as a measurement standard and hence opens up new approaches to objectively quantify the sensor's measurement uncertainty, compare different concepts, or even adjust parametric models for error compensation. By exploiting the angular acceleration vector reference, we demonstrated the feasibility of a leastsquares optimization procedure to adjust the parametric model in order to reduce the measurement uncertainty of our IMC, which is caused by its specific deterministic 44 AKTENEXEMPLAR - ÜBERSETZUNG (REGEL 26.3ter(e)) errors. Especially, we exploited the arbitrary accelerated motion of the pendulum to estimate the relative vectors between the modeled accelerometers' measurement points. Crucial features were the more general model equation. Another advantage of our approach was that all parameters for misalignment and modeled accelerometers' measurement points were estimated simultaneously by least-squares optimization.

[0082] To validate the effect of adjustment routines or to refine the underlying parameter models, future work must include more diverse raw measurement data. For example, different mounting orientations of the IMC could be considered. If the results obtained can be confirmed, the multi-method framework, in combination with least-squares optimization-based adjustment routines, could pioneer the quantification and reduction of measurement uncertainty for a wide variety of sensors. / o

[0083] A bold symbol V' refers to a vector or a second-order tensor, generally speaking, if necessary, with a reference point P indicated by an upper-right index in parentheses. We agree that, without further reference to a coordinate frame, the symbol shall be understood as the 3 x 1 or 3 x 3 coordinate matrix of the vector or tensor with respect to the inertial frame that defines the oscillation plane. This is represented, see Fig. 3. Every vector has its origin at the origin of >. Another additional upper left index such as in V is used to find the coordinate matrix of the vector or tensor with respect to the frame. to indicate if this differs from differs.

[0084] Another convention used here concerns the dependence of mathematical objects on the parameter vector. In cases where we want to emphasize this dependency, specific parameters or the entire vector are added in parentheses, e.g. B. tb ) for di e Rotation matrix between the frames and *^O depending on the Cardan angles 0 and , or r ()(^P) for the displacement vector of regarding ^M, or for solving the model equation. 45 FILE COPY - TRANSLATION (RULE 26.3ter(e))

[0085] To obtain data that were sufficiently dynamic for non-trivial measurements and simultaneously easy enough to model, two variations of a simple pendulum experiment were conducted and their kinematics recorded. The pendulum itself consisted of a sufficiently stiff aluminum rod, so that it could be assumed to behave rigidly, equipped with four reflective markers mounted in pre-drilled holes (see Fig. 3c). Two hybrid ball bearings (HC 628 TN) were used to implement the bearing concept. The experiments themselves consisted of manually deflecting the rod, once by a smaller angle (~25°) and once by a larger angle (~58°). During the pendulum's motion, a twelve-camera motion capture system (infrared Miqus M3) (Qualisys AB, Gothenburg, Sweden) recorded the coordinates of the markers' position vectors at a sampling rate of 200 Hz.At the end of the rod, an IMU-based sensor, also operating at a 200 Hz sampling rate, measured the pendulum's kinematics as described in the following subsection. Data were recorded after the first complete oscillation to ensure the system was in a transitional state. Each measurement lasted approximately 25 s, resulting in NEI = 4402 data points for experiment E1 and NE2 = 4458 for E2.

[0086] In Gießler et al. (2024) we have the so-called Inertial Measurement Cluster (IMC), i.e. h. a sensor for the direct measurement of angular acceleration, was introduced and validated. Its setup and measurement principle can be briefly summarized as follows: A total of four Xsens MTi-20 VRU IMUs (Movella Inc., Nevada, USA) – each consisting of a 3D gyroscope plus a 3D accelerometer – were mounted on a custom-made base plate, as shown in Fig. 3a. One IMU, designated So, served as the reference, while the other three IMUs (Sf with 1 — 2, 3) m it linearly independent connecting vectors r i,0 were positioned relative to the reference. The linear independence of these three vectors and their respective covectors can be used to calculate the angular acceleration W from the linear acceleration. ai and the angular velocity >, extracted as the mean of the angular velocities measured by the four IMUs i = 0,..., 3, respectively, can be calculated. Specifically, as shown in [11, Eq. (4.1 )-(4.7)], the well-known rigid body equation can be used. 46 FILE COPY - TRANSLATION (RULE 26.3ter(e)) Of = tg + WX + WX (w XT o) (6.1 ) for iö i dual vector space can be solved and ultimately yields r (ri^ x ra.o) 1 ^ / <«1 - a0“ w X (w X r lt0 ), r 2t0 ) (7*2.0 x r 3 ( O) T (a a - a0- wx (wx r2,o) s r 3i o) (6.2) V(a3- a0- wx (wx r3,0), r lj0 ) Ws,oxr^o'T ) where k » > denotes the standard scalar product. Note that the inverse matrix in Eq. (6.2) is due to the linear independence of the vectors. r hÖ always exists.

[0087] The aim of this section is to motivate the model that was used as a reference to evaluate the obtained kinematic measurements. A damped plane oscillation can be described by a second-order differential equation of the form v = / (r. r; = vo, <(0> = <o r (6.3) described, whereby = $^(0 denotes the time evolution of the deflection angle of the pendulum with respect to its rest state within its oscillation plane, and the first or second time derivative,? € r der Parameter vector and. / • X Ä X E. R a suitable right side.

[0088] In our experimental setup, data outputs had to be combined with respect to different reference frames. Firstly, this concerned the linear accelerations. (f) and angular velocities (f) for ? = 0, 1, 3 von each of the four IMUs, provided in their own reference frames These outputs were combined within the reference frame of the reference IMU, i.e., ^So, to calculate the angular acceleration. W (O of the IMC to obtain. Secondly, a body-fixed reference frame was attached to the pendulum itself. assigned, which is located within the reference frame of the oscillation plane moved. Thirdly, the positions of the four reflective markers (^ ' with. / = L) were determined. captured by the motion-capture system within its reference frame *Ä*M and therefore considered 1 P designated to remove them 47 FILE COPY - TRANSLATION (RULE 26.3ter(e)) the coordinates of the markers (i.e., the parameters) in to distinguish, P«. where they are written J. Finally, the experiment took place in the coordinate system *G of the gravitational field.

[0089] To write Eq. (6.3) as simply as possible, all quantities were transformed into the reference frame " "o of the oscillation plane (yz-). While the problem of the relative orientations of the IMUs was addressed in the previous subsection, several additional assumptions were required for the remaining reference frames. To transform the direction of gravity (z-axis) in " KG in *, two rotations 9 and The rotation is performed around the x- and y-axes. The corresponding rotation matrix is ​​written as = O G Ä{$, <^) ( V cf.

[0038] , A similar transformation was performed for the motion capture frame M carried out, which was additionally shifted by a vector 'Po' to find the origin of to hit, and rotated around the y- and z-axes by angles fi and y respectively to hit the x-axes, i.e. Ijj _ R(ß„ y) Q er The pendulum reference frame Äp was also chosen such that the origin and the x-axis coincide with (Ko, so that the transformation between these two frames is achieved by an elementary rotation. 1? = ' ^P)) can be described. It is noteworthy that the coordinates of the reflecting markers within MCP were assumed to be constant over time.

[0090] To formulate model equation (6.3) in *Ko, we begin with the general equations of motion (EOM). ma s = mg + F T + F? (6.4) I (0) w + wx I (0) u> = r s x mg + M f (0) (6.5) where m denotes the mass of the pendulum, a s the linear acceleration of its center of mass, 9 the gravitational vector, -^r the reaction force, -^f the frictional force, T ((l) the inertia tensor with respect to the pendulum pivot 0 (the origin of r K( >), the vector that points from 0 to the center of mass of the pendulum, and the 48 FILE COPY - TRANSLATION (RULE 26.3ter(e)) Frictional torque with respect to 0. For the frictional force and the resulting “jf (0 ' Frictional torque of the bearing ■*'f and f, we assume that both are proportional to and thus I II ~ Ml l-^rl I as well as ~ with the coefficient of friction and its lever arm r b (approximately the outer ring radius). The effective line of action of is assumed to be orthogonal to -^r and the oscillation axis e °. of to be, so that F r = sign(^) / je O1 F r Equation (6.6) holds true. Substituting equation (6.6) into (6.4) yields F r + si” ii (<£) / / e o , x F r = ma, - ing (6.7) Furthermore, due to the positioning of *^O, it follows that w ~ That is, the angular velocity vector points towards the pendulum axis (x-). Solving equation (6.5) for the x-coordinate yields + = 0 (6.8) which expands in scalar notation to ^ + S, gll( ^)« a 7 — _ + — J— + r , = 0 (6.9) Here, the terms Q to <'4, ki, unc | kj defined as kl = >s( ^ ) +. sin( ) - signt (r s cost <p ) + r s hint ) ) k-> = r,.c< W) - r hi.siii(^) + sign (r s , uos(^) + r^sint < / ?)) Ci = (r^costsc) - r^ sin(^) 4- sign(^) / y (r Sv ci)s( s c) + r^sinC. / ;))) <p 2 <?2 - g ( COS( ^ )COH(Ö ) - sign (^) / / <‘os(t / dshi(^)) (6.10) c’3 = ^- / sj-ostsf ) + sign(< / ) / / (r, costal - r,, sin (( / ?))) •2 eg = g (cos(^)sin(ri) + signt > cos(W) c5=R~T? < c os( ) cos ( 0 ) ( r s cos ( g? ) + r_ „ sn i (<f )) + cos ( i / > i sii i ( (l ) ( ig eos ( ip ) - r s s sin ( ) ) ) 49 FILE COPY - TRANSLATION (RULE 26.3ter(e)) where s denotes the magnitude of the gravitational acceleration, and; s z the P - Components of in the yz plane. Since the component ' no torque component around the oscillation axis eOx caused it, so it was omitted from further analysis. Note three things: First, Eq. (6.9) / Atli normalized by normalizing the quotient m i f xx, i.e. the inverse of the squared radius of gyration of the pendulum about the x-axis, is replaced by the symbol e. Secondly, the sign function term sign ( ^) is intended to ensure that friction always opposes motion. We do not explicitly consider the case of sticking, as no sticking occurred during the observed period of the experiment. Thirdly, although equation (6.9) is formulated implicitly, it can be solved analytically by solving a quadratic equation, which yields the explicit equation (6.3).

[0091] When we assemble the final parameter vector, we obtain eight physical parameters that appear in the model equation. ' 1 s '' 1 h' ) occur, two initial conditions (^(h and 17 auxiliary geometric parameters ^ r ° v ' 1 ° v ' 1 f hv ' M y < 1 T- ' • ■ • ' r lv ' 1 A ' 1 A was j ns G esam t 27 unknowns yields: ft, r Sj “r #s: , r b , fi, g t tp0, 0 O , ro*, Ay, r 0? , y, ß, r lx , r ly , r lä£ ,..., r 4y , (6.11 ) where 0y and the components of " are, and F lx« » r lz ' • • • » r 4 e K om p onen t en von r j= l.2,3, 1 are.

[0092] To estimate the parameter values ​​and quantify their uncertainties, the position data from the MOCAP system were fitted using the pendulum's equations of motion in the '^O reference frame. The optimal parameter vector was obtained using the nlinf it routine—a Levenberg-Marquardt algorithm—in MATLAB (version 2023b, MathWorks, Natick, USA) to minimize the sum of the squared distances between the MOCAP measurements at the time increments tk and the corresponding model evaluations. The following equation is written FILE COPY - TRANSLATION (RULE 26.3ter(e)) the problem for the more general formulation in *^O (Eq. (6.12a)) and for the more specific formulation (Eq. (6.12b)): ^3* = arg uiin X, " (r n (1ß) + r / (^;^))||^ (6.12a) k, j — arg niin o p Ä(^Z A :? Pn - P r J ('P) (6.12b 'S k, j where the initial values ​​were determined using Eq. (6.9). Note that all 27 parameters were optimized simultaneously without intermediate steps. The determination of the initial values ​​for the Levenberg-Marquardt algorithm is explained in more detail in the following section.

[0093] Although problem (6.12) is formulated such that the discrete version of L 2 Since the distance is minimized, we also evaluated other measures to assess the deviation of the time-dependent data ^l^'^^' ^^^ from the model output at tk and for V. In particular, we evaluated ■= j |data(fi) - model (; ^)| • the total (Manhattan) distance =1 • the Euclidean distance as used in Eq. (6.12): / TV “\ J / L~:= [ V | <lala( / ) - miuld (^; ^' l |“ \ k = l / i as well as A'':= max [dal a( / < ) - inudd ( / / ,: ) 1 • the maximum (Chebyshev) distance k

[0094] where N denotes the number of time points. Note that these distances did not depend on the number of markers and therefore did not require summation over j.

[0095] One advantage of the nlinf it method described in the previous subsection is that the Jacobian matrix of the optimization problem with respect to all parameters—and thus the covariance matrix S, which is the mean squared value—is fixed. 51 FILE COPY - TRANSLATION (RULE 26.3ter(e)) The error is approximately determined using finite differences. In a subsequent step, £ was used to perform a Monte Carlo simulation to evaluate the uncertainty of the modeled acceleration as a function of the uncertainty of the parameters. The methodology was based on the ISO / BIPM “Guide to the Expression of Uncertainty in Measurement” [B20]. A total of M = 100,000 samples of the parameter vector were used. ( ^Py,, M aus gjner multivariate normal distribution with means V and covariance £ was calculated using the MATLAB routine mvnrnd. Equation (6.9) was applied to each parameter vector. solved numerically using the differential equation solver ode78 from MatLab, where the relative tolerance is set to 10 ' R and the absolute tolerance to 10 -9 was set.

[0096] Subsequently, the average acceleration uodel(f) of the M realizations A was calculated. ^inod ' ^Py ). To provide a reliable datasheet entry for the IMC and to check for a potentially acceleration-dependent uncertainty, the mean model accelerations were divided into classes (bins) with a width of 5 rad / s. 2 divided. Within each class, the following were determined for each corresponding time point: -differences modtd k ) ~ un d the •^biii * diffei eilens ^mudel ( fij ​​) calculated, where k = 1, Afjaiii and J ~ Both sets of differences were fitted to a normal distribution using the MatLab routine f itdist, from which estimators for the standard deviations were derived. CHM(' = — I 2 ( / imodcl(ffc ) - t ' or IN THE * A ( 1 = 1 j Af-i IS ( / -Anodd (ffc ) ~ 'ÄiKidel {^k ' ^P / ) | iy=i ;: won. n The 95% confidence interval (CI) for the sensor uncertainty, i.e. h. I^AlC.low, ^IMC.upI, whose width depended only on ^bin, was also determined by the routine f itdist. To obtain an estimator for the standard deviation over all k support points FILE COPY - TRANSLATION (RULE 26.3ter(e)) *. * 2 _ to obtain, a convex linear combination of — _ 1_ y X'hir,,, model. k.,. r , _...,,.,, > fc=l ; ' used (see refined Saterthwaite-Ivlle method). Likewise, for the CI of the model uncertainty mk=1, i.e. r niodel.lu, rt'model.up], ejneA refined Satterthwaite method [B 12] was used to calculate an average CI for each class, the width of which depended only on M. The value of 100,000 for the Monte Carlo samples was chosen such that the width of the cr The model class is less than 0.1% of the nominal value. Furthermore, the bias of the IMG relative to the reference is inevitably present in every class estimator. 2 MC is included. Therefore, we assume here that the corresponding random variables of the differences are and AC(0, < r niy( jp|) are normally distributed, independent, and in the case of the IMC, subject to an unknown bias^IC. Consequently, the resulting uncertainty ®res of the IMC is a superposition of the deviations, i.e. D res — DIMC “Ojiiodel ~ (^IMC ' < J res) ( whose standard deviation + rt.odd. ( 6J 3> can be estimated. Taking into account the different numbers of realizations underlying ^IMC and ^model, the resulting 95% confidence interval l is £ A'?s.1<> W , ' res, up] again approximately determined using the refined Satterthwaite method [12, GL (2.2)]. The estimator of the bias^lMC and its 95% confidence interval t ^IMCJow, ^IMC.up] were also won using the derf itdist routine.

[0097] To evaluate which parameters Since G ^P (Eq. (6.11)) of the ODE model (Eq. (6.9)) has the greatest influence – and should therefore be treated with higher accuracy or caution – a global sensitivity analysis is performed [B33], the basics of which are briefly described here. Let V ~ W the ODE model solution, obtained for the parameter vector ^P. The Cartesian product of the 95% confidence intervals (Cis) derived from the covariance matrix X forms a FILE COPY - TRANSLATION (RULE 26.3ter(e)) Hyperrectangle W around In this context, the symbol denotes any parameter as . that all parameters of within the limits of * / may vary, with the exception of US', which is fixed. As shown by Saltelli et al. [B33, Chapter 4], the state can be decomposed into conditional expectations (E) and variances (V) to obtain both a first-order sensitivity index (primary index) as well as to obtain a total sensitivity index ct. The former is defined by s ” = - W -. (6 ' 14and represents the effect resulting solely from the variation of the parameter W, while simultaneously taking into account known variations of the other parameters. Here, the indices on E and V indicate that all possible parameter values ​​in the respective subspace of Ä are considered for the calculation. The total sensitivity index is defined by (6.15) W) It represents the effect of varying the parameter ET, including interactions of arbitrary order with all other parameters. To efficiently calculate time-dependent sensitivity indices *^«(0), Fourier Amplitude Sensitivity Testing (FAST) is applied using the “sensitivity analysis for everybody” (SAFE) Toolbox [B31 ].

[0098] Initial parameter values

[0099] To provide the optimizer with a suitable starting point, several preparatory measures were taken. In particular, a reciprocal feedback loop consisting of sensitivity analysis (Eqs. (6.14) and (6.15)) and optimization procedures (Eq. (6.12)) was used to understand which model parameters are most sensitive and what parameter uncertainties were to be expected. The globally most sensitive parameters were S, r s z and g Therefore were FILE COPY - TRANSLATION (RULE 26.3ter(e)) For these three, the geometric dimensions were taken from the CAD model as precisely as possible, and in addition, the SCALTEC SBC 61 precision scale (CITRINE Solutions LTD., Oranit, West Bank) of accuracy class II was used to determine the mass with an accuracy of 0.1 g, which led to Rf = 1.94 kg / 0.246 kgm 2 = 7.88 m" 2 und r s== 0.339 m led From this procedure was also 1 — .00005111 g ewO It should be noted that (i) the mass of the pendulum also includes the moving part of the bearing and (ii) the mass distribution of the pendulum was designed to minimize -rt I in order to avoid an asymmetric bearing load with respect to the spatially separated rolling bearings. The gravitational constant was obtained from the The “S-Extractor” (www.ptb.de) provided by the “Physikalisch-Technische Bundesanstalt” for the location (latitude and altitude above sea level) of our laboratory yielded the value = 9.8087 m / s. Initial values ​​Vhn for the model ODE (Eq. (6.3)) as well as the Mi Offsets were taken from the MOCAP system. In particular, for experiment ET W = -1.0234 rad 0o = 0.0053 rad / s and for experiment E2 ^ () = -0.4386 rad o = 0.0298 rad / s, while To = (0.91 Dl, -1.40 ill, 1.47111) j n In both cases, the transformation angles β = 0.0070 rad, y = -0.0051 rad, = 0.0014 rad, and 0 = 0.0017 rad were chosen as the result of a preliminary optimization, starting from zero; for the lever arm of the acting frictional force, r b) The approximate outer diameter of the ball contact points was taken from the ball bearing's datasheet, where it is specified as 9.6 mm. No datasheet information was available for the coefficient of friction; the initial value was chosen based on experience, and z was = 0-0 for experiment E1 and Ä = 0.06 fQ r Experiment E2. Finally, the body-fixed positions of the markers relative to the pendulum were taken from the CAD model as ' = (, 0, —0.1), 'V, = (0, 0, -0.2) r F r3= (0.0.-0.3) 7 ' und rt4= (0, 0, -0.4) rjewei | s in Meters indicated.

[0100] Data post-processing FILE COPY - TRANSLATION (RULE 26.3ter(e))

[0101] As a reminder: The data output of our experiments includes (i) raw positional data from the MOCAP system, (ii) raw angular velocity data from the IMUs and (iii) calculated angular acceleration data (see above) ' )# from the IMC, see also Fig. 3. By transforming these quantities into the reference frame *^O#, the corresponding data for and its derivatives are named as follows: (0# (MOCAP), ' / 'S, U) # (| MU) unc ] <£s (1 (t)# ( | MC) In order for these measurements to be comparable, < i )# had to be numerically calculated twice and They can be differentiated once.

[0102] We further differentiate between possible applications, e.g., real-time versus a posteriori data availability. If real-time processing of the data is required, numerical differentiation can only be performed using backward differences (BD), whereas A-posteriori processing allows the advantages of central differences (CD) to be exploited. As an additional a-posteriori tool for noise reduction, a Butterworth low-pass filter 4 was applied to the MOCAP data. Order (BLPF) applied.

[0103] For the inverse comparison, i.e., for calculating angular positions from the angular velocity measurements ^3,- ( ), the IM U offers a quaternion-based Kalman filter routine (XKF3, see ) which enables real-time (200 Hz) numerical integration. For further noise reduction for the IMC-based states and the arithmetic mean of all four corresponding IMU states was calculated. An overview of all resulting quantities compared to the respective optimized model kinematics ( " and S^- ^ ) is given in Table 6.2. FILE COPY - TRANSLATION (RULE 26.3ter(e)) kinematics method error measure L¹ L² L∞ reference values ​​0.0038 0.0048 0.0097 factors for| • MOC'AP 1.16 3.71 3.03 • MOCAP(BLPF) 1- 1 1 · IMU(XKF3) 1.04 0.97 0.89 · IMC(XKF3, mean) 0.45 0.43 0.45 φ̇(t) [rad / s] reference values ​​0.0192 0.0218 0.0097 factors for · MOCAP (BD) 10.89 14.45 49.23 « MOC'AP (CD) 7.95 9.83 27.95 - MOC'AP (BLPF, BDl 3.88 4.58 7.54 • MOC'AP (BLPF A 'D) 3.1 4.21 7.33 · IMU 1 1 1 · IMC(mean) 0.43 0.44 0.46 [rad / s²] reference values ​​0.2159 0.297 1.3811 factors foi • MOC'AP (BD) 311.55 326.12 355.11 - MOC'AP (CD) 311.63 326.12 355.03 - MOC'AP (BLPF, BD) 29.03 29.16 35.3 • MOC'AP (BLPF, CD) 29.03 29.13 35.23 · IMU(BD) 7.18 6.46 4.53 · IMU (CD) 3.67 3.28 2.2 • IMC 1 1 1 Table 6.2: Error Measurement Quantities (^. A. ) the kinematic data ( ^(0, ^(0), obtained from different sources (MOCAP, IMU, IMC) in experiment E2. Reference values ​​of the residuals are given together with their associated prefactors, where 1 is used to denote the reference measurements. All deviations are compared with respect to the model (Eq. (6.3)) using the optimized parameter vector. The post-processing method is specified for the non-reference measurements, i.e., Butterworth low-pass filtering (BLPF), quaternion-based Kalman filtering (XKF3), center differentiation (CD), backward differentiation (BD), and averaging (mean).

[0104] Finally, we analyzed the influence of compensating for IMC-specific deterministic errors using a calibration and adjustment routine. In particular, the required spatial arrangement of the four accelerometers can be used to... 57 FILE COPY - TRANSLATION (RULE 26.3ter(e)) Misalignments between the MEMS sensors (mounting imperfections) can lead to errors. Furthermore, the coordinates of the relative vectors between the measurement points of the linear acceleration vectors are crucial geometric parameters (see Research Paper IV, Eq. (6.1)). For the current fundamental equations of the IMG, the linear acceleration aThe acceleration was measured at four defined material points. Since the MEMS accelerometers have a finite volume, the measured trajectory of a i can be assigned to a material point of the rigid body, which, despite the inherent stochastic and deterministic errors in (l i is optimally approximated.

[0105] Primary calibration and adjustment routines that utilize the gravitational field (static) or precise rotary tables can be used to correct misalignment between accelerometers and gyroscopes (see [B28]). However, the relative vector coordinates of the acceleration measurement point models cannot be adjusted by measurements in the gravitational field, since the same linear acceleration applies to every material point. Because the proposed multi-method framework employs an oscillating pendulum with rotational motion components, this experimental setup is, in principle, suitable for estimating the relative vector coordinates. Using the angular acceleration vector reference, we demonstrated a least-squares optimization concept to optimally estimate the descriptive parameters for misalignment and relative vector coordinates.The key results showed that the IMC model equations, evaluated with optimally fitted parameters, exhibited a 2.4-fold lower RMSE than the published direct measurement approaches with implicit model assumptions of correctly aligned and orthogonal relative vectors between the inertial sensors. Thus, the measurement standard is not only suitable for the empirical estimation of measurement uncertainty, but also represents a promising framework for adjusting parameter models to compensate for deterministic errors and thereby further reducing the measurement uncertainty of the IMG. At the same time, such parameter adjustment is only possible due to the underlying general IMC model. Especially with regard to the miniaturization of the IMG, it will be crucial to compensate for deterministic errors – particularly those caused by 58 FILE COPY - TRANSLATION (RULE 26.3ter(e)) Assembly imperfections are caused because reduced distances between acceleration measurement points amplify stochastic noise (see...). Research paper IV, Eq. (6.2)).

[0106] We validated the IMC-driven inverse dynamics framework using numerical modeling. Despite particular emphasis on the realistic noise behavior of the simulated sensors, we obtained a deviation from the experimental data (JMi r;ul / s versus .29 i rad / ). Furthermore, the impact of mechanical imperfections in the robot or high-frequency oscillations, and thus the validity of the results obtained from IMC-driven inverse dynamics, still need to be validated in real-world applications. Nevertheless, the quantification of measurement uncertainty based on the established measurement standard and the promising concept of a calibration and adjustment routine can jointly contribute to achieving comparable results in practical applications of humanoid robotics. Finally, an implicit constraint due to the underdetermined system of equations for calculating the point of application of the ground reaction force is the requirement that both feet must be in contact with a flat ground surface during the double-support phase. However, this constraint is met for most applications of a humanoid robot in a human environment. Consequently, inverse dynamics can be used to predict unactuated degrees of freedom (DOFs) in scenarios that approximately satisfy this assumption.When walking on two legs on uneven terrain, the precise kinematic measurements of the IMG for all body segments can be used for other promising concepts, such as estimating and controlling the total angular momentum of the body and its rate of change, in order to prevent the humanoid robot from tipping over ([B39]).

[0107] Regarding the remote assessment of balance adjustment reactions in everyday life, we validated the approach by using standardized stumble-and-slide-like perturbation paradigms in an anteroposterior direction and incorporating a wide range of simulated ADLs (activities of daily living) in controlled environments. However, everyday activities exhibit a broader range of movement patterns. Furthermore, other types of 59 FILE COPY - TRANSLATION (RULE 26.3ter(e)) Perturbations, such as missteps, can also occur in any spatial direction, for example, mediolaterally. Neither of these were considered in this work. Therefore, the achievable accuracy metrics and the applicability for remote assessment of balance correction reactions under everyday conditions still need to be evaluated through field tests. Nevertheless, the remote detection and differentiation of situational balance loss during everyday activities appears promising for long-term monitoring under less controlled conditions. Particularly due to the algebraic and mechanically interpretable structure of the framework, which offers high flexibility, we are confident that both the integration of trunk kinematic coordinates in the mediolateral direction for perturbation detection and the treatment of diverse ADLs will be feasible.Furthermore, the validation of the wearable sensor system framework was conducted using a homogeneous group of participants. Consequently, the results must be further validated through subsequent studies with more heterogeneous groups of participants, encompassing differences in age, physical condition, and health status. Future studies could, however, benefit from considering specific gait characteristics through personalized thresholds and from the framework's adaptability due to its algebraic structure and physically interpretable subroutines. This will also allow for further personalization of the entire framework by adapting its structure or the subroutines used, in order to more effectively manage the individual gait behavior of, for example, older adults or individuals with neuromusculoskeletal disorders.Taking into account the limitations of the proposed multi-method framework, which serves as the measurement standard for determining sensor uncertainty, the achievable amplitude was limited to approximately 1 'MI / S*"#, due to the underlying mechanical system of the pendulum oscillating in the gravitational field. This limitation was caused by the actual pendulum mass and mass distribution. and Ts#; inertia parameter). It should be noted that the range for quantifying the measurement uncertainty is directly related to the amplitude of the pendulum angle acceleration reference and is therefore limited. The in 60 FILE COPY - TRANSLATION (RULE 26.3ter(e)) The results documented in Research Paper II show that the angular acceleration reference is sufficient to resolve the trunk angular acceleration amplitude for undisturbed walking for both the humanoid robot and humans. However, the amplitude for disturbed locomotion in humans can be an order of magnitude higher than for undisturbed walking (see Research Paper II). Therefore, this range cannot be assessed with the pendulum used. Consequently, we cannot determine the empirical measurement uncertainty for these ranges. However, the observed decreasing measurement uncertainty at higher angular acceleration ranges suggests that this likely also applies to further increasing angular acceleration values. Therefore, the proposed pendulum angular acceleration reference covers the ranges with the highest measurement uncertainty.

[0108] For humanoid robots, loss of balance and falls represent a significant limitation, restricting their autonomous deployment in uncontrolled and changing environments. A similar vulnerability can be observed in humans with deficits in balance control. Despite the adaptability of the human musculoskeletal system and recent advances in robotics, this area remains a challenge. To gain new insights into the complex interactions between bipedal locomotion and situational balance loss, we focus on the precise measurement of body segment kinematics. This can be a crucial component in the detection and assessment of balance disorders and thus contribute to increasing the resilience of bipedal locomotion against balance loss.Therefore, in this dissertation, we developed and validated an inertial measurement cluster that avoids numerical differentiation and thus the strong amplification of signal noise. The dissertation comprises three sections, each dealing with humanoid robotics, human behavior, and the quantification of sensor measurement uncertainty. In the first theoretical study, we introduced the mathematical concept of the inertial measurement cluster and paid particular attention to its effects on… 61 FILE COPY - TRANSLATION (RULE 26.3ter(e)) Sensor-based detection of situational balance loss during bipedal locomotion in humanoid robots. To this end, we developed a robust inverse dynamics that replaces numerical differentiation with a direct measurement method for the angular acceleration vector. Subsequently, we outlined the theory of sensor data fusion to estimate both the magnitude and the line of action of external forces acting on the robot. In the second research area, we focused on the detection and evaluation of tripping and slipping events outside the laboratory. Specifically, we investigated the resolution accuracy and reproducibility of the portable sensor framework system (inertial measurement cluster and evaluation framework).It was demonstrated that this enables automated detection and evaluation of performance for restoring balance and that the known adaptation phenomenon to repeated stumble-like disturbances can be verified. Subsequently, we extended the functionality of the wearable sensor system and demonstrated high accuracy in the detection and classification of stumble and slip events during daily activities. In the third research area, we introduced an experimental angular acceleration reference for the objective quantification of the measurement uncertainty of the inertial measurement cluster. Furthermore, we confirmed that the reference source is accurate enough to serve as a measurement standard and the basis for an adjustment routine to compensate for the deterministic errors of the inertial measurement cluster.In summary, we have established a sensor applicable to both humans and humanoid robots that eliminates numerical differentiation and exhibits significantly reduced measurement uncertainty compared to current kinematic measurements. The inertial measurement cluster and the evaluation algorithms were key elements for the precise detection of balance disturbances in both humanoid robots and humans, as confirmed by comparisons with conventional sensors and numerical differentiation. Combined, the sensor and the frameworks have the potential to provide new insights into the causes of balance disturbances or factors contributing to inadequate reactive actions for fall prevention in both humans and humanoid robots.

[0109] Additions: Optimization problems: Adjustment of deterministic errors of the accelerometer; error terms relevant for MEMS sensors.

[0110] Zero Bias Error: An ideal signal, or an ideally measured signal, shows a value of zero at a defined time t. The measured value of a real sensor, however, is not zero. This error is primarily caused by manufacturing defects in the IC. In the case of the IMC, it can also be due to misalignment between the evaluation systems of the individual ICs. The zero bias error can exist as a statistical component, for example, due to misalignment, or as a dynamic component, invariant or variable due to environmental factors such as temperature influences. Scale factor error / sensitivity error: Can have a linear or non-linear form. Linear: ~ ' «i(t) »üt M: measured value. and I: ideal value Non-linear: = / («I) ■ ai(0 -> The function of the factor depends on the value of ai. Modeling: Scale error (linear) + zero bias Model: t», ideal ~ ßi.korr “ & ' T offset See Figure 4. Cross-axis sensitivity: The crosstalk effect is caused by the non-orthogonality of the measurement axes. If the measurement axis orientation is not orthogonal, there is a projective component onto one measurement axis, leading to cross-axis sensitivity. This is mainly due to This causes the measuring axes to be incompatible during manufacturing, while the evaluation coordinate system is assumed to be perpendicular. See Figure 5. Modeling this deterministic error: The evaluation system of the real inertial sensor is represented by a generally oblique coordinate system K. s described. The oblique coordinate system K can be transformed by a linear mapping. s on a fictitious orthogonal evaluation coordinate system to be mapped. We place the fictitious evaluation coordinate system. so that the z-axis of the z-axis of K s corresponds. The following applies: with Mn «T2 [ö: v ] ™ I «22 «23 \ 0 Ü 1 J These error terms typically arise from inaccuracies and manufacturing defects in the production process. During the application of MEMS units to carrier boards, soldering, or fixing the carrier board to other components, further deterministic errors can occur due to stresses. These can be modeled using the error terms described above. Optimization problem: with y, = g where g corresponds to the gravitational constant f (äi) = ~ ■ öi with / : ® with <fj SS (Ijj §*1 4. 4“ äj:> S'3:::: <4:44 '4 O-Sj / 7 ^ + 5 t ~ (fetig 4 ex)fi 4 (^< H SM 4 < 2) <fo 4 (fea-is« 4 c :i )ffy ä> ~ (&xöisrM 4 c x ) (<* i 14> 4 4 «i;Ä ) 4 (A4 &ij, M 4- 4 «22*4 4 «234:} 4 (A'. t oj 3M 4 C;?)^ argnnn X () / ■; — / '(a-. TV --> Min mit r & •”• / («<) ZT" with / (öj) ~ |^|[ ~ V^i ' *4 and Vf (at ™ Gradient vector according to the Parameters f (a,l ofJF

[0111] Not only are sensor-specific parameters such as misalignment or misposition between the SMD components optimized, but also the SMD components themselves. Even straight from the factory, the IMUs (SMD components) exhibit systematic errors, as described and explained in this document. These are not corrected by the manufacturer in standard products. By optimizing the parametric models that describe these systematic errors (caused by manufacturing inaccuracies or soldering), significant improvements in measurement accuracy are achieved. Here again, the same concept as above applies: optimization is performed using a reference. For the optimization described here, either a pendulum or a gravitational vector can be used. We are not specifically dependent on the pendulum here. Previously, we were!

[0112] Figures 1 to 3 illustrate an embodiment of the invention for quantifying and reducing measurement uncertainties of an inertial measurement cluster (IMG). Figure 1 shows the spatial arrangement of four MEMS sensors (accelerometers and angular velocity sensors) on a rigid polyvinyl chloride substrate, as well as the relevant deterministic errors such as misalignment and unknown measurement points (mispositioning). Figure 2 Figure 1 visualizes the IMC-specific error sources and the interlinking of the reference systems (IMC, individual IMUs, oscillation plane, motion detection system, and gravitational field). Figure 3 shows the interlinking of all reference systems (IMC, individual IMUs, oscillation plane, body-fixed pendulum system, MOCAP system, gravitational field) with the corresponding rotation matrices and measurement data. Figures 4 and 5 illustrate the deterministic errors: Figure 4 schematically shows the zero-bias error and scale error as the deviation between the ideal (dashed) and measured acceleration signal; Figure 5 visualizes the crosstalk error as a consequence of non-orthogonal measurement axes due to the mapping of an oblique coordinate system onto an orthogonal evaluation coordinate system.Overall, the embodiment demonstrates that the adaptation routine according to the invention transforms the measurement uncertainty of the IMC from initially unknown to quantified and parameter-optimized values ​​- without specialized calibration hardware (rotary tables, centrifuges), but by systematically using a simple pendulum motion as an angular acceleration reference with two orders of magnitude lower uncertainty than the IMC itself.

[0113] For further technical features of the present invention, reference is made to document DE 10 2023 120 648 B3 and the publication "M. Gießler et al. "A wearable sensor and framework for accurate remote monitoring of human motion". In: Communications Engineering 3.20 (2024), pp. 1-15. DOT: 10. 1038 / s44172-024-00168-6.", features of which also belong to the teaching of the present invention and are fully incorporated into the disclosure of the present invention.

[0114] For further relevant specialist knowledge, please refer to the paragraphs

[0004] and 0 referenced. List of abbreviations: COM focus EOM equation of motion NIF Non-Inertial Frame IZMP Imaginary zero moment point COP pressure center ZMP zero torque point CP detection point LIP Linear Inverted Pendulum WF Inertial system KDL Kinematics and Dynamics Library ROS2 Robot Operating System 2 BFF Body-Fixed Coordinate System ACD actor controller differentiation IR-MD Inertial Reference Pulse Differential MR-AD Mobile Reference - Analytically Derived DOF Degree of Freedom IMU (Inertial Measurement Unit) NAP Nine Accelerometer Package ARS angular velocity sensor cube ADL Activities of Daily Living IMC Inertial Measurement Cluster RMSE (root mean squared error) aTAM Accumulated relative hull angular momentum aRCTAM Accumulated rate of change of relative trunk angular momentum EF Evaluation framework ANOVA One-way repeated measures analysis of variance MOCAP Optical motion capture MEMS Micro-Electro-Mechanical Systems CI confidence interval BLPF Butterworth low pass filtered 67 FILE COPY - TRANSLATION (RULE 26.3ter(e)) XKF3 Quaternion-based extended Kalman filter routine (Xsens) BD Backward difference quotient CD Central ID difference quotient ODE Ordinary Differential Equation FAST Fourier Amplitude Sensitivity Test cf. compare e.g. for example That means 68 FILE COPY - TRANSLATION (RULE 26.3ter(e)) List of symbols total time derivative of the vector V Relative time derivative corresponding to the derivatives of the coordinates of the vector V mathematical model for damped oscillation (ODE) 0 = denotes the ODE model solution with respect to the Angular velocity denotes the ODE model solution with respect to the angle L . i: Angular momentum with respect to the reference point P M ,r 'Torque acting on a body with respect to the reference point P Position vector pointing from the origin of an inertial frame of reference to a non-inertial frame of reference. Inertia tensor of a rigid body with respect to point P, coordinate system F denotes the coordinate matrix of the vector or tensor with respect to the frame , if necessary, with reference point P E ( P ) r k coordinater k of the vector with respect to the frame ^1 iJ R or R ij Rotation matrix that transforms the coordinates from frame i to frame j ÜJ Angular acceleration vector i Joint position of the joint i <- Joint velocity of joint i < i Joint acceleration of the joint i F-statistic value M 'I®p Partial eta-squared (effect size) 69 FILE COPY - TRANSLATION (RULE 26. Sterte» f (2) Chi-squared value with two degrees of freedom (Category-1 ) I 1'0> 1 1 ' Moment of inertia of the subject's torso about the transverse axis (y-axis of the principal axes of the torso) with respect to point q Basis of three-dimensional Euclidean space ff Dual base of ff 9i Covariant basis vectors from ff 9 Contravariant basis vectors of Coordinates of the angular acceleration vector ei' for k = 1, 2, 3 or k = x, y, z $ Gender symbol for female d* gender symbol for male Sensitivity value with respect to a binary classification; spec specificity value with respect to a binary classification PI ' Positive predictive value, defined as the conditional probability of a failed trial under the condition of positive detection Fi Harmonic mean of the positive predictive value and sensitivity So, IMU 0, which serves as the reference for the IMC sensor, is denoted by S». IMU i, for i = 1, 2, 3, denotes the IMU i with respect to the IMC sensor. Parameter vector containing the physical parameters of the optimization model. Initial parameter vector r Optimal parameter vector (least squares optimization) i 1 Total Manhattan distance £ 2 Euclidean distance L°° Maximum Chebyshev distance ^method Systematic error of a normally distributed process FILE COPY - TRANSLATION (RULE 26.3ter(e)) ^method Standard deviation of a normally distributed process / net hod Empirical mean <r method>d Empirical standard deviation \l(' Potential systematic error of the IMC sensor a Cardan angle, predominantly associated with a rotation angle about the x-axis of an orthogonal coordinate system ß Cardan angle, predominantly associated with a rotation angle about the y-axis of an orthogonal coordinate system y Cardan angle, predominantly associated with a rotation angle about the z-axis of an orthogonal coordinate system 71 FILE COPY - TRANSLATION (RULE 26.3ter(e))

Claims

1. Patent claims 1. Method for quantifying and / or reducing measurement uncertainties of sensor units using an adjustment routine to cope with mechanical imperfections and thus quantify and / or reduce the measurement uncertainty of at least one sensor unit.

2. Method according to claim 1, characterized in that an arbitrarily accelerated motion of a pendulum is applied / used to estimate at least relative vectors between modeled, fictitious or assumed measurement points of accelerometers and / or to estimate an alignment and / or orientation between at least two accelerometers and / or angular velocity sensors.

3. Method according to at least one of claims 1 to 2, characterized in that a calibration and / or adjustment routine is used as the adaptation routine, in which a multi-method framework is applied / used in combination with an optimization routine based on the method of least squares.

4. Method according to at least one of claims 1 to 3, characterized in that the at least one sensor unit comprises at least one angular velocity sensor and at least one acceleration sensor, which are based in particular on microelectromechanical systems (MEMS), and / or are designed as an inertial measurement unit (IMU) or inertial measurement cluster (IMC) with several spatially distributed IMUs on a rigid support.

5. Method according to at least one of claims 1 to 4, characterized in that at least one of the following steps is carried out:

7. Using a pendulum, establish an angular acceleration reference oscillating in a gravitational field; measure the kinematics of the pendulum, in particular using a MOCAP system at a series of reflective markers on a pendulum rod (positions) and / or using IMU and IMC measurements of angular velocity and / or angular acceleration; 8. Deriving a mathematical model of damped pendulum oscillation; 9. Estimating at least one complete parameter set, in particular based on a least squares deviation of MOCAP data; 10. Performing a global sensitivity analysis, especially during the estimation of at least one set of parameters; 11. Comparing different measurement methods across all kinematic planes; 12. Determining parameter uncertainties from the optimization procedure; feeding the parameter uncertainties into a Monte Carlo model simulation; 13. Estimating model uncertainties from model simulation; 14. Applying / using the estimated model uncertainties to quantify and / or reduce a measurement uncertainty of at least one sensor unit; 15. Validate at least one sensor unit.

6. Method according to at least one of claims 1 to 5, characterized in that, 17. that in a parameterized model of at least one sensor unit, a misalignment of the sensor coordinate systems relative to a reference coordinate system is described by rotation matrices and / or Euler or Cardan angles and / or 18. a mispositioning and / or an unknown measurement point of an accelerometer is described by a three-dimensional offset vector, 19. and / or that these parameters are estimated within the framework of the fitting routine to compensate for deterministic errors.

7. Method according to claim 6, characterized in that the fitting routine provides a parameter vector with at least the components of a normalized moment of inertia, 20. of a center of mass vector, 21. one or more friction parameters, 22. of gravitational and transformation parameters for coordinate systems, as well as marker and sensor geometry coordinates 23. encompasses and all parameters together in a nonlinear 24. Least Squares method can be optimized.

8. Method according to at least one of claims 1 to 7, characterized in that a multivariate normal distribution is derived from a covariance matrix of the parameter vector determined by the optimization routine and model uncertainties of the angular acceleration reference and / or measurement uncertainties of the at least one sensor unit are determined on the basis of samples from this distribution in a Monte Carlo simulation.

9. Method according to claim 8, characterized in that the measurement uncertainty of the at least one sensor unit is specified as an acceleration-dependent standard deviation and / or as a 95% confidence interval of the angular acceleration vector, wherein the angular acceleration range is divided into several intervals and a respective standard deviation and a corresponding confidence interval are determined for each interval.

10. Method for generating and / or processing motion data of a freely moving body, wherein at least two sensor units, each comprising at least one angular velocity sensor and at least one acceleration sensor, are arranged at a fixed distance from one another on the body, and an angular acceleration vector is directly determined on the basis of measurement signals from the at least two sensor units and by applying the laws of general motion of rigid bodies, characterized in that the measurement uncertainty is at least a sensor unit is quantified and / or reduced according to a method according to at least one of claims 1 to 9.

11. Method for recording and / or evaluating the movement sequences of a freely moving body, characterized in that movement data of the body generated and / or processed according to a method according to claim 10 are used.

12. Method for detecting and / or evaluating dynamic instabilities of a freely moving body, characterized in that motion data of the body generated and / or processed according to a method according to claim 10 are used.

13. Method for generating and / or optimizing algorithms and / or models for calculating the dynamics of freely moving bodies, characterized in that motion data of the body generated and / or processed according to a method according to claim 10 are used.

14. Computer program, characterized in that the computer program comprises program code sections with which a method according to at least one of claims 1 to 13 can be executed when the computer program is executed on a computing unit.

15. Device for carrying out a method according to at least one of claims 1 to 13, characterized in that the device comprises at least four acceleration sensors and at least one angular velocity sensor, each comprising at least one triaxial angular velocity sensor and at least one triaxial acceleration sensor, 33. a rigid support on which the sensors are arranged such that three linear independent relative vectors are defined between a reference sensor and the other three sensors, and a computing unit for executing the adaptation routine 34. comprises.

16. Device according to claim 15, characterized in that the IMUs are based on microelectromechanical systems (MEMS) and have a measurement rate of 200 Hz and a measurement range of the angular velocity sensor of ±2000° / s and of the accelerometer of ±160 m / s 2 are executed.

17. Device according to at least one of claims 15 to 16, characterized in that the linear independent relative vectors between the IMUs have standards of 0.1375 m, 0.175 m and 0.1 m and / or that the rigid support is made of polyvinyl chloride and has a geometry adapted to anthropometric dimensions, in particular to the dimensions of a human thorax.

18. Device according to at least one of claims 15 to 17, characterized in that the device further comprises a pendulum which has an aluminium rod with defined reflective markings at intervals of 0.1 m, 0.2 m, 0.3 m and 0.4 m, 37. Hybrid ball bearings for realizing one degree of rotational freedom, and / or 38. a MOCAP system with twelve infrared cameras for capturing marker positions with a measurement rate of 200 Hz 39. exhibits, wherein the pendulum serves as an angular acceleration reference for the adjustment routine according to claim 1.

19. Device according to claim 18, characterized in that the computing unit of the device is designed to, 41. To implement a mathematical model of damped pendulum oscillation with a parameter vector that includes inertia parameters, center of mass coordinates, friction parameters, transformation parameters for coordinate systems and marker positions.

42. adapt the parameters to MOCAP measurement data using a least-squares optimization method, 43. to conduct a global sensitivity analysis and a Monte Carlo simulation to quantify uncertainty, and / or 44. To quantify and / or reduce the measurement uncertainty of the IMUs.

20. System for acquiring and evaluating motion data, comprising a device according to any one of claims 15 to 19.

45. and a computer program according to claim 14, 46. ​​the system is designed to process the motion data in real time and to automatically quantify and reduce the measurement uncertainty of the sensor units.

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