Human body lower limb kinematics modeling method based on dual quaternion

Abstract

The invention discloses a human body lower limb kinematics modeling method based on dual quaternion, and relates to the technical field of biomechanical measurement and motion analysis, and the method comprises the following steps: building a lower limb Pluecker linear model by using binocular vision calibration data; on the basis of real-time visual data, dual quaternions of joint movement are constructed through point cloud registration and relative transformation calculation, and the poses of the limb segments are updated through chain multiplication of the dual quaternions; and performing differential operation on the dual quaternion time sequence to directly solve the dual velocity spinor so as to obtain accurate angular velocity and linear velocity. According to the method, binocular vision data and a unified mathematical framework are combined, the problems that a traditional method is not unified in model and complex in speed solution are solved, efficient and accurate conversion from three-dimensional coordinates to motion parameters is achieved, and the method is suitable for the fields of gait analysis, rehabilitation engineering, robot control and the like.

Description

Technical Field

[0001] This invention relates to the interdisciplinary field of advanced motion capture and biomechanical analysis technologies, and particularly to a method for modeling the kinematics of the human lower limbs based on dual quaternions. This invention integrates computer vision, dual quaternion mathematical theory, Plück coordinate geometry, and numerical computation methods and systems. More specifically, this invention provides a computer-based method for high-precision, singularity-free, real-time pose tracking and direct spatial velocity calculation of the human lower limb gait, based on binocular vision measurement and utilizing dual quaternions to perform rigid transformation and differential operations on the Plück line. This method can be widely applied in rehabilitation medicine, sports training, ergonomics, robotics, and other fields. Background Technology

[0002] Accurate modeling of human lower limb movement is crucial for understanding human movement mechanisms, assessing motor function, designing rehabilitation programs, and developing robotic lower limb motion systems. Traditional methods for modeling human lower limb kinematics typically describe rotation and translation separately, such as using Euler angles to describe rotation and considering translation components separately. This approach is prone to singularity problems when dealing with complex lower limb movements due to the coupling relationship between rotation and translation, and the accumulation of errors during computation is significant, leading to a decrease in model accuracy. Furthermore, joints such as the hip, knee, and ankle joints in the human lower limb not only have their own rotational and translational characteristics during movement, but their movements are also interconnected and mutually influential. Traditional methods struggle to accurately characterize these complex inter-joint motion relationships, especially when multi-joint coordinated movements are involved. To address the shortcomings of existing technologies, the purpose of this invention is to provide a human lower limb kinematics modeling method and system based on dual quaternions, to more accurately and efficiently describe human lower limb movement, providing a more reliable tool for research and applications in related fields. Summary of the Invention

[0003] The purpose of this invention is to overcome the shortcomings of the prior art and provide a method for modeling and analyzing lower limb gait motion based on binocular vision measurement, dual quaternions and Plück lines. This method can complete the seamless calculation from the original three-dimensional coordinates to the pose and then to the spatial velocity within a unified, singular, and efficient mathematical framework.

[0004] To achieve the above objectives, this invention employs a fundamental technical approach that integrates differential geometry and Lie group theory. Its core concept lies in: modeling each rigid segment of the human lower limb as a Plück line in a local coordinate system, which constitutes the geometric carrier of motion; utilizing the unit dual quaternion as a unified mathematical operator to simultaneously represent rotation and translation, achieving a rigid transformation of the Plück line; and finally, by performing time differentiation on this dual quaternion operator and utilizing the exponential mapping relationship between Lie groups and Lie algebras, directly and synchronously deriving the spatial angular velocity and linear velocity.

[0005] To achieve the above objectives, this invention discloses a method for kinematic modeling of the human lower limbs based on dual quaternions, the method comprising the following steps:

[0006] Model initialization: Using lower limb marker point cloud data acquired under static calibration posture using a binocular vision measurement system, and based on a predefined anatomical model and optimization algorithm, the instantaneous centers of the hip, knee, and ankle joints are calculated in three-dimensional space. Using the previous joint center as the origin and the direction pointing to the next adjacent joint center as the reference axis, a local coordinate system for each limb segment is established. The thigh and lower leg segments are represented as a Plück line in their respective local coordinate systems; wherein, the Plück line uses a six-dimensional coordinate pair. Define parameters. This is a unit direction vector from the center of the proximal joint of this limb segment to the center of the distal joint. The moment vector is the position vector of a point on the limb segment. With the direction vector The cross product is calculated as follows: .

[0007] Real-time motion tracking and transformation steps: Based on the 3D coordinate sequence of marker points acquired in real time by the binocular vision system during motion, for each frame of data, the singular value decomposition (SVD) algorithm is used to register the marker point cloud on each limb segment, solving for the optimal rigid body transformation of the limb segment from the static calibration posture to the current motion posture. This transformation includes a rotation matrix. and a translation vector The rotation matrix Convert to unit quaternion and using the formula Calculate dual part quaternions This allows us to construct a unit dual quaternion representing the pose of the limb segment in the global coordinate system. ,in As a dual unit, satisfying and Based on the parent-child connection relationship of limb segments, using the formula... Calculate the relative motion between adjacent limb segments, where For joint motion dual quaternions, and These are the global pose dual quaternions for the parent and child limb segments, respectively. Represents the multiplication of dual quaternions.

[0008] By sequentially multiplying and combining the joint motion dual quaternions along the kinetic chain, a composite dual quaternion transformation describing the distal limb segment (foot) relative to the root coordinate system (pelvis) is obtained. Using the aforementioned composite dual quaternions For the local Plück line of the distal limb segment Perform a rigid transformation to obtain its real-time pose in the global coordinate system. The transformation formula is: .

[0009] This invention employs a fundamental technical approach that integrates differential geometry and Lie group theory. Its core concept lies in: modeling each rigid segment of the human lower limb as a Plück line in a local coordinate system, which constitutes the geometric carrier of motion; utilizing the unit dual quaternion as a unified mathematical operator to simultaneously represent rotation and translation, achieving a rigid transformation of the Plück line; and finally, by performing time differentiation on this dual quaternion operator and utilizing the exponential mapping relationship between Lie groups and Lie algebras, directly and synchronously deriving the spatial angular velocity and linear velocity.

[0010] This invention provides a method for lower limb gait motion modeling and analysis based on binocular vision measurement, dual quaternions, and Plück lines. The method comprises the following three logically interconnected stages:

[0011] Phase 1: Initialization of the geometric model based on binocular vision and representation of the Plück line:

[0012] S101: System Calibration and Data Acquisition: Using a high-precision calibrated binocular vision system, the three-dimensional calibration coordinates of the marker points are acquired while the subject maintains a standard static posture.

[0013] S102: Establishment of joint centers and local coordinate system: Based on static calibration data, the functional centers of each joint are calculated using an algorithm based on the anatomical model.

[0014] S103: Plück Line Parametricization: Parametricizes each limb segment as a Plück line in its local coordinate system. Direction vector Defined as a unit vector pointing from the center of the proximal joint to the center of the distal joint. Moment vector. : Position vector from the center of the proximal joint With direction vector The cross product determines that, i.e. In this local coordinate system, due to the reference point The initial moment vector is usually taken as the zero vector. The value is zero. This notation provides an ideal mathematical basis for subsequent rigid transformations.

[0015] Phase Two: Dual Quaternion Motion Tracking and Transformation Based on Real-Time Visual Data

[0016] S201: Rigid Body Transformation Solution: For each frame of 3D coordinate data of marker points acquired in real time by the binocular vision system, perform the following operations for each limb segment: Input: The set of marker points for the limb segment under static calibration and the corresponding set of points in the current frame. Process: Calculate the centroid of the point set and the covariance matrix, and perform singular value decomposition (SVD) on the matrix. Output: Optimal rotation matrix. Translation vector Singular value decomposition provides the optimal solution in the least squares sense.

[0017] S202: Dual Quaternion Construction and Pose Representation: This step is one of the key innovations of this invention, unifying rotation and translation within a single mathematical entity. The rotation matrix... Convert to unit quaternion The conversion process is a technique known in the field, utilizing translation vectors. Construct the dual part of dual quaternions This formula originates from the mathematical derivation of representing rigid body transformations using dual quaternions. Therefore, a unit dual quaternion representing the pose of this limb segment in the global coordinate system is constructed. The dual quaternion automatically satisfies the unit constraint. and This constraint ensures that it represents a legal rigid body motion.

[0018] S203: Joint Motion Calculation: Based on the tree-like chain structure of the lower limbs, calculate the relative motion (i.e., joint motion) between adjacent limb segments. For joints connecting parent and child limb segments, their motion is given by the following formula. This operation is mathematically equivalent to calculating the relative pose of the child coordinate system in the parent coordinate system.

[0019] S204: Chain Transformation and Pose Update: To obtain the pose of the distal limb segment (foot) relative to the root coordinate system (pelvis), the dual quaternions of all joint transformations along the kinetic chain are sequentially multiplied and combined. .

[0020] Phase 3: Solving for the dual velocity spinor based on Lie group differential calculus:

[0021] This stage is the core innovation of this invention, as it realizes a paradigm shift in speed-based solution.

[0022] S301: Numerical Differential Calculation: Calculation of the dual quaternion time series representing the motion of the distal limb segment, obtained from binocular vision data sequences. Numerical differentiation is performed using the central difference method. This method has second-order accuracy and can effectively reduce errors.

[0023] S302: Dual Velocity Spinor Extraction – Lie Algebra Mapping: This is the theoretical core of velocity solving. In the Lie group represented by the dual quaternion, the velocity is directly related to the Lie algebra elements. By calculating the product of the time derivative and the inverse (conjugate) of the group element, it can be mapped to the Lie algebra space, yielding... ,in As a dual velocity spinor, this formula establishes a bridge from the differentiation of dual quaternions to the analysis of velocity in physical space, completely avoiding the construction and inversion of the traditional Jacobian matrix.

[0024] S303: Velocity Decoupling and Output: From Dual Velocity Rotor In the middle, its physical component is directly extracted: angular velocity. and linear velocity , here It is the absolute angular velocity of the end limb in the global coordinate system. It is the absolute linear velocity of the end limb corresponding to the origin of the coordinate system represented by the dual quaternion.

[0025] Compared with the prior art, the beneficial effects of the present invention are:

[0026] This invention is the first to integrate Plück's line, dual quaternions, and dual velocity spinors into a single mathematical framework. This unity fundamentally solves the problem of fragmented models in traditional methods and provides a solid mathematical foundation for human motion analysis.

[0027] This invention utilizes dual quaternion transformations of the Plück line, requiring only a few quaternion multiplications, with a computational complexity less than the equivalent 6x6 adjoint matrix multiplication. This invention simplifies the most cumbersome and error-prone spatial velocity calculation in traditional methods into a single dual quaternion multiplication and differential operation, which not only reduces computational complexity and improves real-time processing capabilities, but also achieves higher numerical accuracy and stability.

[0028] This invention provides a complete and non-redundant description of the instantaneous motion state of a limb using the final output dual velocity spinor, which is in a form that is completely consistent with the standard "motion spinor" in robotics. This allows the calculation results to be directly used for subsequent advanced applications such as dynamic analysis and robot trajectory tracking control without any conversion.

[0029] This invention combines binocular vision data with a unified mathematical framework, solving the problems of inconsistent models and complex speed calculations in traditional methods. It achieves efficient and accurate conversion from three-dimensional coordinates to motion parameters, and is applicable to fields such as gait analysis, rehabilitation engineering, and robot control. Attached Figure Description

[0030] Figure 1 This is a flowchart illustrating the kinematic modeling of the human lower limbs based on dual quaternions, as described in this invention. Figure 2 This is a schematic diagram of the rigid pose solution and chain transformation process based on singular value decomposition and dual quaternions of the present invention. Detailed Implementation

[0031] The technical solutions of the embodiments of the present invention will be clearly and completely described below with reference to the accompanying drawings. Obviously, the described embodiments are only some embodiments of the present invention, and not all embodiments. Based on the embodiments of the present invention, all other embodiments obtained by those skilled in the art without creative effort are within the scope of protection of the present invention.

[0032] Please see Figure 1-2 This invention provides a technical solution for a method of human lower limb kinematics modeling based on dual quaternions: the method includes the following steps:

[0033] Model initialization: Using lower limb marker point cloud data acquired under static calibration posture using a binocular vision measurement system, and based on a predefined anatomical model and optimization algorithm, the instantaneous centers of the hip, knee, and ankle joints are calculated in three-dimensional space. Using the previous joint center as the origin and the direction pointing to the next adjacent joint center as the reference axis, a local coordinate system for each limb segment is established. The thigh and lower leg segments are represented as a Plück line in their respective local coordinate systems; wherein, the Plück line uses a six-dimensional coordinate pair. Define parameters. This is a unit direction vector from the center of the proximal joint of this limb segment to the center of the distal joint. The moment vector is the position vector of a point on the limb segment. With the direction vector The cross product is calculated as follows: .

[0034] Real-time motion tracking and transformation steps: Based on the 3D coordinate sequence of marker points acquired in real time by the binocular vision system during motion, for each frame of data, the singular value decomposition (SVD) algorithm is used to register the marker point cloud on each limb segment, solving for the optimal rigid body transformation of the limb segment from the static calibration posture to the current motion posture. This transformation includes a rotation matrix. and a translation vector The rotation matrix Convert to unit quaternion and using the formula Calculate dual part quaternions This allows us to construct a unit dual quaternion representing the pose of the limb segment in the global coordinate system. ,in As a dual unit, satisfying and Based on the parent-child connection relationship of limb segments, using the formula... Calculate the relative motion between adjacent limb segments, where For joint motion dual quaternions, and These are the global pose dual quaternions for the parent and child limb segments, respectively. Represents the multiplication of dual quaternions.

[0035] By sequentially multiplying and combining the joint motion dual quaternions along the kinetic chain, a composite dual quaternion transformation describing the distal limb segment (foot) relative to the root coordinate system (pelvis) is obtained. Using the aforementioned composite dual quaternions For the local Plück line of the distal limb segment Perform a rigid transformation to obtain its real-time pose in the global coordinate system. The transformation formula is: .

[0036] The method for calculating joint centers based on marked point cloud data uses a sphere fitting algorithm to determine the center of the hip joint by using marked points on the pelvis. For the knee and ankle joints, a geometric constraint method based on an anatomical model or a combination thereof is used to determine the center.

[0037] Unit dual quaternion It strictly satisfies the following mathematical constraints to ensure that it represents a legal rigid body transformation, where the unit constraint satisfies Duality constraints are satisfied .

[0038] The method also includes a step for calculating the motion velocity:

[0039] Numerical molecular steps: For dual quaternion time series representing end-limb segment motion calculated from continuous frames of visual data. The central difference method is used for numerical differentiation to calculate its time derivative at time t. .

[0040] Dual velocity spinor extraction sub-step: Calculate the time derivative. and dual quaternions conjugate The product of and results in a dual vector of the form . ,in This is the dual velocity spinor.

[0041] Velocity decoupling step: from the dual velocity spinor In the middle, the real part of its pure quaternion form is directly extracted and multiplied by the scalar coefficient 2 to obtain the angular velocity vector of the end limb segment in the global coordinate system. Simultaneously, the dual part in its pure quaternion form is directly extracted and multiplied by a scalar coefficient of 2 to obtain the linear velocity vector on the distal limb segment corresponding to the origin of the dual quaternion. .

[0042] The method for modeling human lower limb kinematics based on dual quaternions includes the following three logically interconnected stages:

[0043] Step 1: System initialization and model establishment (corresponding to S101-S103).

[0044] S101: System Calibration and Data Acquisition: Using a high-precision calibrated binocular vision system, the three-dimensional calibration coordinates of the marker points are acquired while the subject maintains a standard static posture.

[0045] S102: Establishment of Joint Centers and Local Coordinate Systems: Based on static calibration data, an anatomical model-based algorithm is used to calculate the functional centers of each joint. Hip Joint Center: The estimated location of the femoral head center is calculated using a least-squares sphere fitting algorithm based on markers on the pelvis. Knee and Ankle Joint Centers: The longitudinal axis of the local coordinate system for each limb segment is determined using marker clusters on the thigh and lower leg, respectively. This axis intersects the anatomical transverse axis defined by markers on the medial and lateral epicondyles and the medial and lateral malleoli; the intersection point is the joint center. A local coordinate system is established for each limb segment, typically with the proximal joint center as the origin and the Z-axis pointing towards the distal joint center.

[0046] S103: Plück Line Parametricization: Parametricizes each limb segment as a Plück line in its local coordinate system. Direction vector Defined as a unit vector pointing from the center of the proximal joint to the center of the distal joint. Moment vector. : Position vector from the center of the proximal joint With direction vector The cross product determines that, i.e. In this local coordinate system, due to the reference point The initial moment vector is usually taken as the zero vector. The value is zero. This notation provides an ideal mathematical basis for subsequent rigid transformations.

[0047] Step 2: Real-time motion tracking and pose calculation (corresponding to S201-S204).

[0048] S201: Detailed Mathematical Expression of Global Pose Solution

[0049] Rigid body transformation solution: For each frame of 3D coordinate data of marker points acquired in real time by the binocular vision system, taking the right thigh as an example, perform the following operations: Input: The coordinate set of the right thigh marker points under static calibration in its local coordinate system. Input: The coordinates of the same group of markers in the global coordinate system at frame k. Procedure: Calculate the centroid of the point set. and , , Calculate the covariance matrix ,right Perform Singular Value Decomposition (SVD): ,in and It is an orthogonal matrix. It is a singular value matrix. Output: Calculate the rotation matrix. Translation vector The SVD method provides the optimal solution in the least squares sense, and this process is performed in parallel for each limb segment.

[0050] S202: The rigorous mathematical definition of the construction of dual quaternions

[0051] Construct the real part The obtained rotation matrix Convert to unit quaternion Ensure the actual units

[0052] Constructing dual parts : can be obtained Duality constraint is .

[0053] The pose dual quaternion of the right thigh in the global coordinate system can be obtained as follows: .

[0054] S203: Joint Motion Calculation

[0055] To calculate the relative motion between adjacent limb segments, we treat each dual quaternion as a coordinate system transformation operator. The flexion / extension angles of the knee joint can be obtained by calculating the relative transformation between the thigh and lower leg. The dual quaternion for calculating the knee joint transformation is... ,in Describe the transformation from the local coordinate system of the thigh to the world coordinate system. Describe the transformation from the local coordinate system of the lower leg to the world coordinate system. This describes the transformation from the local coordinate system of the lower leg to the local coordinate system of the thigh. real part The rotation angle around the local X-axis (internal and external axes) of the knee joint can be decomposed, i.e., the flexion / extension angle. Similarly, the ankle joint transformation can be calculated.

[0056] S204: Chain Transformation and Foot Pose Update

[0057] The pose composite transformation of the distal limb segment (right foot) relative to the root coordinate system (pelvis) is calculated as follows:

[0058] use Transforming the local Plück line of the foot to the pelvic coordinate system, this transformation formula reflects the geometric meaning of the dual quaternion acting on the Plück line: , where the direction vector Only the real part Rotation yields the moment vector. The transformation includes both rotation and translation, corresponding to the transformation law of Plück coordinates under rigid body motion, and obtains the real-time orientation and position of the toes in the pelvic coordinate system. This position data can be used to calculate spatiotemporal parameters such as stride length and stride width.

[0059] Step 3: Solving for spatial velocity based on Lie group differentials

[0060] S301: Numerical Differential Calculation: Calculation of the dual quaternion time series representing the motion of the distal limb segment, obtained from binocular vision data sequences. Numerical differentiation is performed using the central difference method. This method has second-order accuracy and can effectively reduce errors. The differential equation is: ,in This represents the sampling time interval.

[0061] S302: Dual Velocity Spinor Extraction – Lie Algebra Mapping: This is the theoretical core of velocity solving. In the Lie group represented by the dual quaternion, the velocity is directly related to the Lie algebra elements. By calculating the product of the time derivative and the inverse (conjugate) of the group element, it can be mapped to the Lie algebra space, yielding... ,in For the dual velocity spinor, expanding it yields... This formula establishes a bridge from the differentiation of dual quaternions to the analysis of physical space velocity, completely avoiding the construction and inversion of the traditional Jacobian matrix.

[0062] S303: The mathematical process of velocity decoupling: from dual velocity spinors In the middle, its physical component is directly extracted: angular velocity. for linear velocity for , here It is the instantaneous angular velocity of the foot in the global coordinate system. This value reflects the combined angular velocity of the foot around the ankle joint inversion / inversion and plantar flexion / dorsiflexion. It is the instantaneous linear velocity of the origin of the foot coordinate system (usually the center of the ankle joint) in the global coordinate system. It directly reflects the speed of foot movement in space and is a key indicator for calculating kinetic energy and evaluating gait symmetry.

[0063] Although embodiments of the invention have been shown and described, it will be understood by those skilled in the art that various changes, modifications, substitutions and alterations can be made to these embodiments without departing from the principles and spirit of the invention, the scope of which is defined by the appended claims and their equivalents.

Claims

1. A method for human lower limb kinematics modeling based on dual quaternion, characterized in that: The method comprises the following steps: Model initialization: the lower limb marker point cloud data collected by the binocular vision measurement system at the static calibration posture, based on the predefined anatomical model and optimization algorithm, the instantaneous center of the hip joint, knee joint and ankle joint in the three-dimensional space is calculated, the previous joint center is taken as the origin, the direction of the next adjacent joint center is taken as the reference axis, the local coordinate system of each limb segment is established, and the thigh and lower leg limb segments are represented as a Plucker straight line in their respective local coordinate system; wherein the Plucker straight line uses a six-dimensional coordinate pair Parameterized definition is performed It is a unit directional vector from the proximal joint center of the current limb segment to the distal joint center, It is a moment vector calculated by the cross product of the position vector of a point on the limb segment And the direction vector , that is ; Real-time motion tracking and transformation steps: Based on the 3D coordinate sequence of marker points acquired in real time by the binocular vision system during motion, for each frame of data, the singular value decomposition (SVD) algorithm is used to register the marker point cloud on each limb segment, solving for the optimal rigid body transformation of the limb segment from the static calibration posture to the current motion posture. This transformation includes a rotation matrix. and a translation vector ; the rotation matrix Convert to unit quaternion and using the formula Calculate dual part quaternions This allows us to construct a unit dual quaternion representing the pose of the limb segment in the global coordinate system. ,in As a dual unit, satisfying and Based on the parent-child connection relationship of limb segments, through the formula Calculate the relative motion between adjacent limb segments, where For joint motion dual quaternions, and These are the global pose dual quaternions for the parent and child limb segments, respectively. Represents the multiplication of dual quaternions; By sequentially multiplying the joint motion dual quaternions along the kinematic chain, a composite dual quaternion transformation describing the end limb segment (foot) relative to the root coordinate system (pelvis) is obtained , using the composite dual quaternion , a rigid transformation of the end limb segment local Plucker line is performed to obtain its real-time pose in the global coordinate system , the transformation formula is: .

2. The human lower limb kinematics modeling method based on dual quaternion according to claim 1, characterized in that: The method for calculating the center of the joint based on the marker point cloud data, for the hip joint, adopts a spherical fitting algorithm to determine the spherical center through the marker points on the pelvis; For the knee joint and the ankle joint, the geometric constraint method based on the anatomical model or the combination thereof is adopted to determine.

3. The human lower limb kinematics modeling method based on dual quaternion in claim 1, wherein: Unit dual quaternions The following mathematical constraints are strictly satisfied to ensure that it represents a legal rigid body transformation, where the unit constraint is satisfied The dual constraint is satisfied .

4. The human lower limb kinematics modeling method based on dual quaternion according to claim 1, characterized in that: The method further comprises a motion speed solving step: Numerical differentiation sub-step: for the dual quaternion time series representing the motion of the end limb segment calculated from the visual data of consecutive frames , the central difference method is used to numerically differentiate to calculate its time derivative at time t ; Dual velocity spinor extraction sub-step: calculating the time derivative the conjugate of the dual quaternion the product of which is a dual vector of the form where is the dual velocity spinor; Velocity decoupling sub-step: extracting directly the real part of its pure quaternion form from the dual velocity screw and multiplying it by the scalar factor 2, resulting in the angular velocity vector of the end limb segment in the global coordinate system ; while extracting directly the dual part of its pure quaternion form and multiplying it by the scalar factor 2, resulting in the linear velocity vector on the end limb segment corresponding to the origin of the dual quaternion .

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