A method and system for estimating the state of a flexible-joint robotic arm based on a polyhedron

Abstract

The application discloses a kind of based on polyhedral flexible joint mechanical arm state estimation method and system, it is related to mechanical arm control technical field, including the following steps: by actual system state acquisition estimation error, by estimation error to Luenberger state observer performance quantification, generate objective function;The gain matrix of Luenberger state observer is obtained by solving target function;Based on the estimation state of state variable obtained by Luenberger state observer after solving, error polyhedron for describing estimation error is constructed, and the recursive formula of error polyhedron is obtained by Minkowski sum operation;The state estimation interval of state variable is obtained by estimation state and the recursive formula of error polyhedron.The application solves the problem that traditional Kalman filtering cannot provide credible state boundary, and real-time gives credible interval containing real state.

Description

Technical Field

[0001] This invention relates to the field of robotic arm control technology, and in particular to a method and system for estimating the state of a flexible joint robotic arm based on multicellular structures. Background Technology

[0002] With the increasing demand for flexible control and human-computer interaction in various fields of life, flexible joint robotic arms have begun to attract much attention in the field of control research. Compared with traditional rigid robotic arms, flexible joint robotic arms have significant advantages such as high compliance and safe interaction: their joint materials are lighter and lower in cost, their operation is safe and reliable, and they also possess the dual characteristics of low power consumption and high performance. These types of robotic arms are widely used in aerospace robotics, medical surgery and rehabilitation assistance, and underwater robots.

[0003] In the control of rotary flexible joints, accurately tracking the setpoint is the primary goal, and accurate state estimation is a prerequisite for precise control. State estimation of a flexible robotic arm refers to the process of real-time estimation of key variables such as joint angles, angular velocities, end-effector positions, and deformation states using sensor information, system models, and estimation algorithms. Flexible robotic arm systems typically exhibit high-order dynamic characteristics, complex dynamic models, and strong coupling. System sensor information is limited; it can usually only measure motor end angles, partial strain, or acceleration, and cannot directly obtain end-effector positions or all modal information. Furthermore, external disturbances to the system, such as friction, external loads, and sensor noise, can introduce modeling errors, all of which affect estimation accuracy. Finally, the system has high real-time requirements, especially in high-speed motion or interactive control, where state estimation must be fast and accurate, with high demands on safety and predictability. Therefore, how to achieve both flexibility and precise control of the robotic arm while maintaining high accuracy has become a major challenge in current scientific research.

[0004] Existing state estimation methods aim to improve the accuracy of pose estimation for flexible joints of spatial robotic arms under complex disturbances. The technical solution is developed in three steps, with an overall logical chain of "modeling-disturbance processing-optimized estimation." The specific characteristics of this method are as follows: Flexible joint filtering modeling that aligns with real-world system characteristics: Breaking away from the limitations of traditional models that neglect joint flexibility, this model fully considers the multi-source interference faced by the space robot arm on track, such as external disturbance torque, joint friction torque, actuator noise, and sensor noise. A filtering model is constructed based on the dynamics principles of flexible joints. Simultaneously, the model is linearized and discretized to better suit actual sensor measurement methods in engineering, such as data acquisition at fixed time intervals, laying the foundation for subsequent accurate estimation.

[0005] Targeted solutions for estimating the impact of multi-source interference: Addressing the issues of unknown interference characteristics and lack of prior models, this method rapidly calculates the interference magnitude using only the difference between real-time measurement data from the joint encoder and the estimated measurement values. Furthermore, it employs the Gauss-Markov theorem to design the interference estimation gain, ensuring that the obtained interference results are unbiased under the minimum variance principle, providing a reliable basis for subsequent interference mitigation.

[0006] The accuracy of state estimation is optimized by improving the Kalman filter: Based on the traditional extended Kalman filter, the above-mentioned disturbance estimation results are incorporated for feedforward compensation. This is achieved through two processes: "time update" (estimating the current state based on the previous state and disturbance) and "measurement update" (correcting the estimated state based on real-time measurement data), dynamically adjusting the filter gain matrix. Ultimately, this ensures that the estimation errors of joint angles and angular velocities reach their optimal level with minimum variance, achieving high-precision pose estimation and providing support for end-effector positioning and motion control in on-orbit operations of space robotic arms.

[0007] Existing solutions explicitly model and estimate multi-source disturbances such as external perturbations, joint friction, actuator noise, and sensor noise, and suppress them through feedforward compensation. A minimum variance unbiased estimator is designed using the Gauss-Markov theorem to optimize the estimation error covariance. Based on the extended Kalman filter framework, the computational complexity is low, making it suitable for real-time online estimation.

[0008] Kalman filtering categorizes uncertainties in a system into process noise and measurement noise, corresponding to "unpredictability of internal system dynamics" and "errors in external observations," respectively. Process noise is the fundamental input parameter of the filtering model, describing the "unpredictable deviations" caused by external disturbances, model simplification, or inherent randomness during the system's state evolution. Mathematically, it is typically modeled as zero-mean Gaussian white noise, with its intensity quantified by the process noise covariance matrix. Measurement noise is the "deviation between observed values ​​and the true state" caused by sensor accuracy limitations, environmental interference, or signal transmission errors. The ideal assumption of Kalman filtering is that "noise is zero-mean Gaussian white noise," but in real-world scenarios, colored noise, impulse noise, and non-Gaussian noise are common, thus existing Kalman filtering methods have weak robustness. Furthermore, this approach outputs state point estimates and cannot provide confidence intervals or boundaries for possible states. Summary of the Invention

[0009] In view of the defects of the existing technology, the present invention provides a method and system for state estimation of flexible joint robotic arms based on multicellular structures, which solves the existing problems.

[0010] The present invention adopts the following technical solution: In a first aspect, the present invention provides a method for state estimation of a flexible joint robotic arm based on multiple cells, comprising the following steps: The dynamic model of the flexible joint robotic arm system is transformed into a spatial state equation; the spatial state equation is discretized to obtain a discrete state space model that includes external disturbances. The estimation error is obtained by using the actual system state. The performance of the Luneburg state observer is quantified by the estimation error to generate the objective function. The objective function is solved to obtain the gain matrix of the Luneburg state observer. The system state equation is obtained by placing closed-loop poles in the discrete state space model. Based on the system state equation, a Luneburg state observer for predicting the system state is designed. Based on the solved Luneburg state observer, the estimated state of the state variable is obtained, and an error polycell is constructed to describe the estimation error. The recursive formula of the error polycell is obtained through Minkowski summation. The state estimation interval of the state variable is obtained through the estimated state and the recursive formula of the error polycell.

[0011] Preferably, the discrete state-space model is as follows: ; ; In the formula, and They are respectively Time and The system state at any given moment. For the discrete system matrix, For discrete input matrices, The discrete perturbation input matrix, For discrete output matrices, For discrete measurement noise matrix, For system disturbance, for Real-time output volume for System input at any given time.

[0012] Preferably, the system state equations are as follows: ; ; In the formula, A For the intermediate matrix, To stabilize the perturbation input matrix, To stabilize the output matrix, To stabilize the measurement noise matrix; The system status includes the deflection angle of the servo motor, the deflection angle of the connecting rod, the angular velocity of the motor, and the relative angular velocity of the connecting rod. The control input is the servo motor voltage.

[0013] Preferably, the Luneburg state observer is specifically as follows: ; In the formula, and for Time and State estimate at time 10:00 This is the gain matrix of the Luneburg observer.

[0014] Preferably, the performance of the Romberg state observer is quantified by estimating the error to generate an objective function, specifically including the following steps: The estimation error is input into the Romberg state observer to obtain the error recursion formula; Based on the estimation error and the error recursion formula, a Lyapunov function is constructed. Through stability analysis of the Lyapunov function, a method is derived that satisfies the following condition for the estimation error: Matrix inequality constraints on performance metrics; The matrix inequality is transformed into an objective function by applying Schur's complement lemma.

[0015] Preferably, the objective function is as follows: ; In the formula, It is a positive definite matrix. The exponential decay rate, T For transpose, For matrix variables, This refers to the disturbance suppression performance index; The gain matrix of the Luneburg state observer is shown below: .

[0016] Preferably, the error multicell is specifically as follows: ; In the formula, It is an estimation error. yes k The estimated value of the time error, yes k The radius of time.

[0017] In a second aspect, the present invention provides a state estimation system for a flexible joint robotic arm based on multicellular structures, comprising: A module is established to transform the dynamic model of the flexible joint robotic arm system into a spatial state equation; the spatial state equation is discretized to obtain a discrete state space model that includes external disturbances. The solution module is used to obtain the estimation error from the actual system state, quantify the performance of the Luneburg state observer using the estimation error, and generate the objective function; solve the objective function to obtain the gain matrix of the Luneburg state observer; perform closed-loop pole placement on the discrete state-space model to obtain the system state equation; and design a Luneburg state observer for predicting the system state based on the system state equation. The estimation module is used to obtain the estimated state of the state variables based on the solved Luneburg state observer, construct an error polynomial to describe the estimation error, obtain the recursive formula of the error polynomial through Minkowski sum operation, and obtain the state estimation interval of the state variables through the estimated state and the recursive formula of the error polynomial.

[0018] Compared with the prior art, the above-mentioned at least one technical solution adopted by the present invention can achieve the following beneficial effects: This invention addresses the state estimation problem of flexible articulated robotic arms under complex disturbance environments by combining multicell estimation with an LMI-based Luneburger state observer. Firstly, the method optimizes the observer gain to meet performance specifications through pole placement and LMI design, effectively suppressing the impact of external disturbances and modeling errors on state estimation. Secondly, it uses multicells to geometrically enclose the estimation error and obtains a compact interval estimate of the state variables through Minkowski and recursive methods, solving the problems of traditional Kalman filtering's inability to provide reliable state boundaries and its weak robustness to non-Gaussian noise. This invention provides a reliable interval containing the true state in real time, offering clear boundary information for the safe control and constraint management of flexible robotic arms under conditions of model uncertainty, sensor noise, and external load variations, significantly improving the system's robustness and engineering practicality. Attached Figure Description

[0019] To more clearly illustrate the technical solutions in the embodiments of the present invention or the prior art, the drawings used in the description of the embodiments or the prior art will be briefly introduced below. Obviously, the drawings described below are only some embodiments of the present invention. For those skilled in the art, other drawings can be obtained based on these drawings without creative effort.

[0020] Figure 1 This is a flowchart of a method for estimating the state of a flexible joint robotic arm based on multicellular structures according to the present invention; Figure 2 This is a physical model of the flexible joint robotic arm of the present invention; Figure 3 The structure of the flexible joint system of the present invention; Figure 4 This is a structural diagram of the closed-loop system of the flexible joint robotic arm of the present invention; Figure 5 State variables of a flexible joint robotic arm A comparison chart of the estimated curves; Figure 6 State variables of a flexible joint robotic arm A comparison chart of the estimated curves; Figure 7 State variables of a flexible joint robotic arm A comparison chart of the estimated curves; Figure 8 State variables of a flexible joint robotic arm Comparison of estimated curves. Detailed Implementation

[0021] 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.

[0022] This invention addresses the control problem of flexible joint robotic arms by proposing a research method based on multi-cell estimation. The design aims to achieve high-precision state tracking and effectively suppress disturbances generated by the robotic arm.

[0023] Multiple body estimation is a membership estimation method specifically designed for uncertain systems where the noise boundary is known but the statistical properties are unknown. It uses a multiple body geometric structure to wrap the feasible set of states of the system, achieving a bounded estimate of the system state through recursive propagation and updates. It can be used to estimate the modal coordinates or end-effector deformation of flexible arms, providing state boundaries for robust controllers such as model predictive control or membership feedback. In distributed flexible systems, multiple body estimation can also achieve state fusion between subsystems. For example, in a multi-segment flexible arm, each segment can be considered a subsystem, and overall state estimation can be achieved through multiple body fusion.

[0024] A multicellular body is a centrally symmetric convex polyhedron, which can be represented as:<c,M> Here, c is the center vector, and M is the generating matrix, which determines the shape and volume of the polytope. Polytopes, in relation to the Minkowski equations, are suitable for state propagation in linear systems, and their complexity can be controlled through dimensionality reduction. The core of polytope estimation lies in representing the set of state uncertainties using polytopes and predicting and updating them using system models and observation information. The initial state is wrapped in a polytope, and then the state polytope is propagated to the next time step according to the system dynamics model. When new observation data is obtained, the observation information is fused using the observation equations, shrinking the state polytope to obtain a more accurate state estimate. To prevent the complexity of the polytope from increasing over time, dimensionality reduction is also required. In distributed systems, multiple subsystems may have overlapping states, requiring a fusion strategy to handle polytope intersections. While the box wrapping method is conservative, it is simple to implement, while strip intersection optimization reduces conservatism by minimizing the volume of the fused polytope. Simultaneously, convexity programming is used to estimate the linearized error boundary, which is then wrapped in a polytope. When dealing with complex and variable systems, the P-norm is introduced as a "compactness" index for multiple cells, transforming the state estimation problem into an optimization problem of the multiple cell expansion coefficient. The optimal generating matrix is ​​then solved using the Linear Matrix Inequality (LMI), i.e., the objective function. This invention uses... Norms and compactness emphasize constraints on extreme dimensions, and multicellular structures... The smaller the circumscribed radius, the more limited its maximum extension in all dimensions. It is suitable for scenarios that require no significant divergence in any direction, such as the multi-dimensional constraints of joint angles in flexible robotic arms.

[0025] To better achieve high-precision state tracking and effectively suppress interference generated by the robotic arm, refer to Figure 1 The technical solution adopted in this invention specifically includes the following steps: S1: Model the rotary flexible joint robotic arm. The detailed steps can be broken down into S11~S14 as follows.

[0026] S11: Introduces the physical model of the rotary flexible joint robotic arm, referring to... Figure 2 and Figure 3 The module base is mounted on the load gear of the SRV02 system. When the servo angle... When rotated counterclockwise, its value increases in the positive direction. This occurs when the control voltage is positive. When the speed is >0, the servo motor and connecting rod will rotate counterclockwise. The lower connecting rod connected to the base has a length of... The quality is The length and mass of the upper connecting rod are respectively... and The total length of the connecting rod can be changed by adjusting its mounting position. The distance between the servo motor's central axis and the midpoint of the upper connecting rod is a variable. It is indicated that the moment of inertia of the entire connecting rod is... The value of the link's deflection angle changes with the position of the upper link. This indicates that its value increases positively when rotated counterclockwise.

[0027] The structure of a flexible joint system is as follows Figure 2 As shown, the control variable is the input servo motor voltage. This voltage generates torque at the load gear of the servo mechanism. This drives the connecting rod and base to rotate. The coefficient of viscous friction of the servo mechanism. This represents the resistance acting on the load gear, while the frictional force on the connecting rod is determined by the viscous damping coefficient. Characterization. Ultimately, the flexible joint is modeled as having stiffness. A linear spring.

[0028] S12: Mathematical modeling of the dynamic characteristics of the rotary flexible joint robotic arm system using the Euler-Lagrange equations.

[0029] The Euler-Lagrange function is defined as the total kinetic energy of the system minus the spring potential energy: (1); In the formula, L The total kinetic energy generated by the rotation of the flexible joint robotic arm system. It is the equivalent moment of inertia on the servo motor side. and These represent the angular velocity of the motor and the relative angular velocity of the connecting rod, respectively.

[0030] Among them, the sum of the total kinetic energy generated by the rotation of SRV02 and the total kinetic energy of the deflection of the connecting rod. for: (2); Elastic potential energy stored in a spring for: (3); The moment of inertia of the upper connecting rod about its axis of rotation is defined as: (4); The moment of inertia of the lower connecting rod about its center of mass is defined as: (5); The expression for the overall rotational inertia of the connecting rod is: (6); The moment of inertia of the upper connecting rod about the servo pivot is: (7); For equation (1) Differentiate: (8); (9); Substituting the above equation into equation (2), and assuming that the viscous damping of the connecting rod is negligible, i.e. =0, which gives us the first equation of motion: (10); In the formula Torque is generated at the load gear of the servo motor.

[0031] For equation (1) Differentiate: (11); (12); The second equation of motion is obtained: (13); S13: Convert the two motion equations (10) and (13) into spatial state expressions.

[0032] The servo motor of the rotary flexible joint robotic arm deflects the angle. Deflection angle of the connecting rod 、 servo motor deflection angular velocity and the deflection angular velocity of the connecting rod Treat it as a state variable, output quantity y For the servo motor deflection angle and the deflection angle of the connecting rod The standard space state expression is constructed as follows: (14); in, ; In the formula, The derivative of the state variable. This is the system input, i.e., the servo motor voltage. .

[0033] The servo angular acceleration needs to be calculated from the mathematical model of the rotary flexible joint robotic arm. From equation (10) minus equation (13), we get: (15); The servo angular acceleration was obtained by sorting. equation: (16); Substituting (16) into (13), we obtain the angular acceleration of the flexible deformation of the connecting rod. equation: (17); S14: Rearranged into a space state equation: (18); (19); Torque is generated at the load gear of the servo motor. Substitute (18): (20); (twenty one); In the formula, It has a high overall gear ratio. It is the back electromotive force constant of the motor. It is the motor current torque constant. It's the motor efficiency. It's gearbox efficiency. It is the armature resistance of the motor.

[0034] Depend on The linearized system matrix is ​​obtained by rearranging the state variables. and linearized input matrix : ; ; S2: An LMI-based design was developed. Disturbance estimation is performed using a disturbance-resistant Luneburg state observer, such as Figure 4 As shown, this includes modeling errors, nonlinearities, and external disturbances. The detailed steps can be broken down into S21~S27 as follows.

[0035] S21: System discretization.

[0036] Using a ZOH zero-order hold to construct discrete state-space equations, disturbances are unavoidable in actual system operation; therefore, using... Characterizes disturbances in the external system.

[0037] (twenty two); In the formula, and They are respectively Time and The system state at any given moment. For the discrete system matrix, For discrete input matrices, The discrete perturbation input matrix, For discrete output matrices, For discrete measurement noise matrix, This is a system disturbance.

[0038] S22: Extreme Point Configuration: Will Substituting into discrete form, Given the controller gain matrix, and using closed-loop pole placement to ensure system stability, the system state equation is: (twenty three); in The stable system after pole placement is simplified and redefined as follows: (twenty four); In the formula, A For the intermediate matrix, To stabilize the perturbation input matrix, To stabilize the output matrix, To stabilize the measurement noise matrix.

[0039] S23: Construct the Luneburg state observer.

[0040] (25); in, This is the state estimate. for The gain matrix of the perturbation-resistant Luneburg observer needs to be solved using LMI. Therefore It is a prediction term based on the system model, using the current state to estimate and predict the state at the next time step. It is a correction term that uses the output error to correct the prediction result, so that the state estimate converges to the true state.

[0041] Define estimation error ,but: (26); S24: Stability analysis of Lyapunov.

[0042] Define candidate quadratic Lyapunov functions: (27); in Let be the positive definite matrix to be determined.

[0043] S25: Performance analysis.

[0044] remember ,at this time The derivation is as follows: Lyapunov stability analysis.

[0045] (28); In the formula, It is a disturbance suppression performance indicator. It is the exponential decay rate of the Lyapunov function.

[0046] From the above, we can conclude that: (29); when hour: (30); If there is no interference, the system performance function The system will not exceed its initial value, indicating that its state will not diverge and that it is stable. Without external torque disturbances or load disturbances, the system's vibration amplitude and tracking error will not increase.

[0047] Along the error system (24) Performing forward difference operations, we obtain: (31); when hour, The norm (infinite norm) describes the maximum amplitude of a signal: (32); In the formula, It is a positive subset of the set of integers Z, representing the discrete-time index set, i.e., all non-negative integers {0,1,2,3,…}, which here represents the sampling time sequence number of the signal.

[0048] The goal of control is to address disturbances. ∈ To ensure error And it satisfies the gain constraint: (33); Expressing the infinity norm in mathematical form: (34); Based on equation (29), the instantaneous constraint is extended to the period from the initial moment to... k The cumulative effect at time −1 is derived from the Lyapunov function, resulting in the exponentially weighted cumulative perturbation suppression inequality: (35); Ultimately, under interference conditions, the system can achieve stability by meeting the following performance indicators: (36); Therefore, combining get: (37); S26: LMI Derivation and Schur Supplement Lemma.

[0049] The constraint (37) is transformed into matrix form using Schur's complement lemma: (38); The gain constraint is guaranteed by using the negative definite condition of the matrix: (39); Applying Schur's complement lemma, the quadratic terms of L in the matrix, i.e. PL elimination: (40); Applying variable substitution Eliminating the coupling between L and P, we finally obtain the standard convex optimized LMI form: (41); S27: Solving for controller gain.

[0050] The optimal solution is obtained by solving the LMI using MATLAB's LMI toolbox.

[0051] right Right multiplication Transpose both sides of the equation and find the positive definite matrix to be found. The final observer gain matrix is: (42); S3: Introducing a cluster estimation approach: Treat model residuals, friction, and load disturbances as unknown but bounded, and enclose the true state within a multi-cell structure. The prediction step uses Minkowski summation, and the update step uses strip intersection to cut the observed information into new boundaries. This results in a compact region that can be proven to contain the true state, rather than a point estimate. The controller uses the region center as feedforward and calculates the safety margin using the outer radius, allowing collision risks to be eliminated in advance.

[0052] S31: The k The true state of a moment Located within an interval, with the estimated state at the center of the interval. The radius of the interval is determined by the matrix describe: (43); No. kThe radius of the interval at time +1 By the k Moment It is obtained by recursion. This represents the observer gain, used to correct the estimate; The perturbation matrix and the range of the perturbation are used to describe the uncertainty of the system. The core update formula for the interval observer is derived by combining these formulas. (44); S32: Real State It is an estimated state With error term The sum, here It is the bias in state estimation: (45); error It itself is located within an interval, and the center of the interval is , yes k The estimated value of the time error; the radius is determined by... The description, combined with equation (43), is equivalent to the actual state. exist Nearby, the error range is from limit: (46); The estimated state obtained by the observer The error range is from describe: (47); S33: No. k Error range at time +1 It is obtained by performing interval operations on two parts: (48); The error range resulting from the state update; Represents the multiplication operation between a matrix and an interval; Characterizing perturbation The resulting error range; yes k Perturbation value at any time; This represents the addition operation of the intervals, that is, the expansion of the two intervals. The radius of the final error interval corresponds exactly to the interval in equation (46), thus ensuring the consistency of the recursion.

[0053] S34: Generation The update formula for the error matrix is: [The original generator matrix is ​​then updated using the formula]. After linear transformation, and with the noise generation matrix Horizontal splicing, implemented using the Minkowski sum method: (49); It is the sum of the error estimate and the disturbance value, and is also regarded as the center vector of the multiple cell estimate.

[0054] Error range The propagation equation for the error multicell is: the error multicell at the next moment, derived from the current error multicell through the system dynamics. Propagation, coupled with noise interference. : (50); (51); The standard form of multicellular propagation is finally obtained: (52); S35: Based on the above theorem, the interval design is as follows: (53); in, , yes The One element, yes The One element, It is a matrix The Line 1 The elements of the column. The inferred condition. It doesn't have to be a Metzler type, and coordinate transformation is unnecessary. In discrete-time scenarios, it's difficult to find suitable time-varying coordinate transformations. Therefore, applying multicellular set membership estimation to the state observation of a rotary flexible joint manipulator can greatly reduce computational complexity and ensure real-time state estimation.

[0055] Example The following description, in conjunction with the accompanying figures, further illustrates this method. This implementation case relies on a flexible joint robotic arm system. Based on its discretized dynamics and pole configuration parameters, an LMI observer and a multicellular state estimation simulation model are built. The implementation process is shown in detail, and the robustness of the observer and the state envelope effect are demonstrated to better understand the actual performance of this method.

[0056] S1: Parameter settings for the flexible joint robotic arm system, as shown in Table 1.

[0057] Table 1 Specific parameters of the flexible rotary joint robotic arm After substituting the parameter values ​​into the system's mathematical model and performing calculations, the specific input-output matrix values ​​were obtained. The perturbation matrix values ​​were also randomly set. The simulation data are as follows: ; ; ; ; Simulation results: The optimal observer gain matrix value was calculated to be: ; S2: Selection of Reference State and Initial Conditions S2.1: Initial state settings: The actual initial state of the system: ; Observer initial state: ; Initial error: e (:,1)= x (:,1)- xobser (:,1); S2.2: Initial parameters of multicellular organisms: Initial multicell matrix Heo 0= diag (0.3*( Ones ( nx ,1))), used for the uncertainty of the initial state of the envelope.

[0058] S3: LMI Observer and Multicellular Design: A robust observer is designed using the LMI optimization method, and perturbation uncertainties are handled by combining multiple cells.

[0059] S3.1: Observer Gain Design: Define decision variables: symmetric positive definite matrices P (4×4) Auxiliary matrix Y (4×2) Performance indicators ; Construct LMI constraints (including attenuation coefficient) ): ; The L, error system matrix, is obtained by solving using the LMI toolbox. The pole moduli are all less than 1.

[0060] S3.2: Multicellular state envelope: The initial multicellular body was formed by During initialization and iteration, the error propagation matrix and the multi-cell matrix are used. And fuse the multicell matrix corresponding to the perturbation. Ultimately, it is the union of multiple cells. Envelope of the true state; envelope boundaries of each state dimension The upper and lower bounds of the observed values ​​are obtained. 、 S4: External Disturbance and Simulation Settings: The perturbation model is selected as sinusoidal random perturbation. omega (:, k )=1* sin ( k ); (For random angles, simulate the uncertainty disturbances of the actual system); Simulation parameter settings: sampling time. Total simulation time .

[0061] Figures 5 to 8 These correspond to the state variables of the flexible joint robotic arm. and The estimated curves are compared, and each figure is presented in discrete time intervals. k The horizontal axis represents the values ​​of the state variables, and the vertical axis represents the estimated values ​​of the corresponding state variables. Figure 5 State variables of a flexible joint robotic arm The curve comparison diagram of the ellipsoidal bundle method is used to compare the true value with the estimation result of this invention, and to show the coverage of the upper and lower bounds of the interval calculated by the ellipsoidal bundle estimation set on the true trajectory and the change of estimation error over time. Figure 6 State variables of a flexible joint robotic arm The comparison of the estimated curves obtained by the ellipsoidal beam method is used to illustrate the tracking effect of the estimated curves obtained by using ellipsoidal beam propagation and order reduction merging on the real state under uncertainty conditions, as well as the convergence and compactness of the interval boundary. Figure 7 State variables of a flexible joint robotic arm A comparison chart of curves estimated using the ellipsoidal bundle method is used to illustrate... The estimation results are consistent with the true values ​​and reflect the ability of the upper and lower bounds derived from the ellipsoidal bundle estimation set to cover the true state. Figure 8 State variables of a flexible joint robotic arm A comparison of curves estimated using the ellipsoidal bundle method is provided to illustrate... The comparison between the estimated curve and the true curve is shown, and it is explained that the interval boundary calculated based on the ellipsoidal bundle estimation set can maintain a small interval width while ensuring that the true state is included, thereby achieving deterministic interval state estimation.

[0062] The core advantage of this invention lies in its bounded robust estimation of the state set of the flexible arm using convex multicells. This not only adapts to the time-varying characteristics of the flexible robotic arm but also meets the requirements for real-time control and safe operation. In practical applications, it offers the following advantages: It exhibits greater robustness and higher model error compatibility in complex and perturbed environments.

[0063] The polytope method directly characterizes the "boundary range" of uncertainty through zonotopes, requiring only known upper bounds of the perturbation, such as the maximum amplitude of sensor noise or the error range of model parameters, without any probabilistic assumptions. The polytope method can accurately characterize the feasible region of structured uncertainty through the geometry of polyhedron vertices, edges, and faces. This invention constrains the elastic deformation of the flexible arm within a polytope defined by the stiffness matrix error, ensuring that the estimation results always fall within the physically feasible range.

[0064] Integrating multi-source constraints improves estimation accuracy and feasibility.

[0065] State estimation for flexible robotic arms needs to satisfy multiple physical constraints, such as joint angle limits, angular velocity limits, deformation thresholds, and observation constraints. The multi-cell method has a natural advantage in constraint fusion. The multi-cell method can achieve the fusion of multi-sensor information through multi-cell intersection, union, and Minkowski sum operations. This invention estimates the intersection of multi-cells to obtain a more compact feasible state region; when a sensor fails, only its corresponding constraint multi-cell needs to be removed, without affecting the overall estimation stability.

[0066] It is highly adaptable to real-time control, balancing accuracy and speed.

[0067] Although the geometric operations of polytopes may seem complex, they are characterized by low computational complexity due to the use of special polytope representations such as zonotopes, defined by a generator matrix. Their computational efficiency can meet the real-time control requirements of robotic arms, typically in the millisecond range. They can be directly deployed in the embedded controllers of robotic arms (such as STM32 and DSP), requiring no high-performance computing resources and meeting the time constraints of real-time control, such as a 1ms control cycle.

[0068] Explicitly ensure system security and constraint compliance.

[0069] The operation of a flexible robotic arm must satisfy multiple constraints: joint rotation limits, elastic deformation limits, end-effector obstacle avoidance boundaries, and load-bearing thresholds. Multi-cell estimation can directly transform these physical constraints into boundary constraints of the multi-cell, actively eliminating states exceeding the constraints during state estimation to ensure the estimation results always remain within the feasible region. When faced with disturbances such as sudden load changes or external collisions, the multi-cell can rapidly expand or contract to update the state boundaries, maintaining system stability and operational safety. This is particularly suitable for scenarios with stringent safety requirements, such as human-robot collaboration and operations in confined spaces.

[0070] Based on the same concept, the present invention also provides a state estimation system for a flexible joint robotic arm based on multicellular structures, including an establishment module, a solution module, and an estimation module.

[0071] The module is used to transform the dynamic model of the flexible joint robotic arm system into a spatial state equation; the spatial state equation is discretized to obtain a discrete state-space model that includes external disturbances.

[0072] The solution module is used to obtain the estimation error through the actual system state, and to quantify the performance of the Luneburg state observer by using the estimation error to generate the objective function; the objective function is solved to obtain the gain matrix of the Luneburg state observer; the system state equation is obtained by performing closed-loop pole placement on the discrete state space model; and a Luneburg state observer for predicting the system state is designed based on the system state equation.

[0073] The estimation module is used to obtain the estimated state of the state variables based on the solved Luneburg state observer, construct an error polycell to describe the estimation error, obtain the recursive formula of the error polycell through Minkowski sum operation, and obtain the state estimation interval of the state variables through the estimated state and the recursive formula of the error polycell.

[0074] Although preferred embodiments of the invention have been described, those skilled in the art, upon learning the basic inventive concept, can make other changes and modifications to these embodiments. Therefore, the appended claims are intended to be interpreted as including both the preferred embodiments and all changes and modifications falling within the scope of the invention.

[0075] Obviously, those skilled in the art can make various modifications and variations to this invention without departing from its spirit and scope. Therefore, if these modifications and variations fall within the scope of the claims of this invention and their equivalents, this invention also intends to include these modifications and variations.

Claims

1. A method for state estimation of a flexible joint robotic arm based on multicellular structures, characterized in that, Includes the following steps: The dynamic model of the flexible joint robotic arm system is transformed into a spatial state equation. Discretize the spatial state equation to obtain a discrete state-space model that includes external disturbances; The estimation error is obtained from the actual system state, and the performance of the Romberg state observer is quantified using the estimation error to generate the objective function. The objective function is solved to obtain the gain matrix of the Luneburg state observer; the system state equation is obtained by performing closed-loop pole placement on the discrete state-space model; and a Luneburg state observer for predicting the system state is designed based on the system state equation. Based on the solved Luneburg state observer, the estimated state of the state variable is obtained, and an error polycell is constructed to describe the estimation error. The recursive formula of the error polycell is obtained through Minkowski summation. The state estimation interval of the state variable is obtained through the estimated state and the recursive formula of the error polycell.

2. The method for state estimation of a flexible joint robotic arm based on multicellular structures as described in claim 1, characterized in that, The discrete state-space model is shown below: ； ； In the formula, and They are respectively Time and The system state at any given moment. For the discrete system matrix, For discrete input matrices, The discrete perturbation input matrix, For discrete output matrices, For discrete measurement noise matrix, For system disturbance, for Real-time output volume for System input at any given time.

3. The state estimation method for a flexible joint robotic arm based on multicellular structures as described in claim 2, characterized in that, The specific system state equations are as follows: ； ； In the formula, A For the intermediate matrix, To stabilize the perturbation input matrix, To stabilize the output matrix, To stabilize the measurement noise matrix; The system status includes the deflection angle of the servo motor, the deflection angle of the connecting rod, the angular velocity of the motor, and the relative angular velocity of the connecting rod. The control input is the servo motor voltage.

4. The state estimation method for a flexible joint robotic arm based on multicellular structures as described in claim 3, characterized in that, The Luneburg state observer is specifically shown below: ； In the formula, and for Time and State estimate at time 10:00 This is the gain matrix of the Luneburg observer.

5. The method for state estimation of a flexible joint robotic arm based on multicellular structures as described in claim 1, characterized in that, The performance of the Romberg state observer is quantified by estimating the error, and an objective function is generated. This process includes the following steps: The estimation error is input into the Romberg state observer to obtain the error recursion formula; Based on the estimation error and the error recursion formula, a Lyapunov function is constructed. Through stability analysis of the Lyapunov function, a method is derived that satisfies the following condition for the estimation error: Matrix inequality constraints on performance metrics; The matrix inequality is transformed into an objective function by applying Schur's complement lemma.

6. The method for state estimation of a flexible joint robotic arm based on multicellular structures as described in claim 4, characterized in that, The objective function is as follows: ； In the formula, It is a positive definite matrix. The exponential decay rate, T For transpose, For matrix variables, This refers to the disturbance suppression performance index; The gain matrix of the Luneburg state observer is shown below: 。 7. The state estimation method for a flexible joint robotic arm based on multicellular structures as described in claim 1, characterized in that, The error multicell is specifically shown below: ； In the formula, It is an estimation error. yes k The estimated value of the time error, yes k The radius of time.

8. A state estimation system for a flexible joint robotic arm based on multicellular structures, characterized in that, include: A module is established to transform the dynamic model of a flexible joint robotic arm system into a spatial state equation; Discretize the spatial state equation to obtain a discrete state-space model that includes external disturbances; The solution module is used to obtain the estimation error from the actual system state, quantify the performance of the Romberg state observer using the estimation error, and generate the objective function. The objective function is solved to obtain the gain matrix of the Luneburg state observer; the system state equation is obtained by performing closed-loop pole placement on the discrete state-space model; and a Luneburg state observer for predicting the system state is designed based on the system state equation. The estimation module is used to obtain the estimated state of the state variables based on the solved Luneburg state observer, construct an error polynomial to describe the estimation error, obtain the recursive formula of the error polynomial through Minkowski sum operation, and obtain the state estimation interval of the state variables through the estimated state and the recursive formula of the error polynomial.

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