A two-stage multi-constraint excitation trajectory design method for robotic arms

Abstract

This invention provides a two-stage, multi-constraint excitation trajectory design method for a robotic arm, belonging to the field of robotic arm dynamic parameter identification and trajectory optimization technology. The method includes: establishing an identification observation matrix containing inertial and friction parameters based on a robotic arm model; designing joint excitation trajectories using a finite-term Fourier series; generating observability evaluation indices based on the observation matrix; further generating a first-stage excitation optimization model based on joint motion constraints, spatial safety constraints, and self-collision detection constraints to obtain safe and executable candidate trajectories; generating a second-stage observability lower bound based on the first-stage observability optimization results; and comprehensively optimizing dynamic excitation, velocity reversal, spectral richness, and joint balance indices under the constraints of the observability lower bound to generate a second-stage hierarchical trajectory optimization model, obtaining the optimal excitation trajectory for dynamic parameter identification experiments. This invention can improve trajectory safety, excitation sufficiency, and overall identification accuracy.

Description

Technical Field

[0001] This invention relates to the field of robot dynamics parameter identification and trajectory optimization technology, and in particular to a two-stage multi-constraint excitation trajectory design method for identifying the dynamics parameters of a robotic arm. Background Technology

[0002] With the rapid development of industrial automation and robotics, robotic arms have been widely used in assembly, handling, welding, precision operation, and human-robot collaboration. Currently, much research focuses on achieving high-precision control of robotic arms, and identifying the dynamic parameters of robotic arms based on experimental data is an important means to improve the accuracy and control performance of robotic arm models.

[0003] In the process of identifying the dynamic parameters of a robotic arm, the design of the excitation trajectory directly affects the amount of information in the collected data and the reliability of the parameter identification results. Existing methods typically use Fourier series, polynomials, etc., to parameterize the joint trajectory and optimize the trajectory based on observability evaluation indicators such as the condition number of the observation matrix, minimum singular value, determinant index, or D-optimal index. However, existing methods often focus on improving the information quality of the observation matrix, neglecting the constraints of the actual execution process of the robotic arm, and are prone to generating excitation trajectories that are difficult to execute directly. If motion constraints are satisfied through post-processing methods such as scaling or manual correction, the acceleration and frequency characteristics of the trajectory may be altered, resulting in the loss of identification information.

[0004] Furthermore, in practical dynamic parameter identification, the excitation trajectory not only needs to ensure the observation matrix has good numerical properties, but also needs to fully excite velocity, acceleration, and different frequency components within the allowable motion range of the robotic arm to fully stimulate its dynamic characteristics. If the trajectory motion amplitude is too small, the velocity variation is insufficient, or the joint motion is concentrated in a local area for a long time, although its observation indicators may meet certain requirements, the excitation of nonlinear dynamic and frictional characteristics may still be insufficient, thus affecting the stability and generalization effect of parameter estimation. Therefore, it is necessary to design an excitation trajectory design method that can simultaneously consider the information quality of the observation matrix and the sufficiency of excitation while meeting the actual motion safety constraints of the robotic arm, to solve the above problems. Summary of the Invention

[0005] In view of this, the present invention provides a two-stage multi-constraint excitation trajectory design method for robotic arms, in order to solve the problem that existing methods for identifying excitation trajectories based on the dynamic parameters of robotic arms are difficult to simultaneously consider the quality of observation matrix information, actual motion safety constraints, and the sufficiency of experimental excitation.

[0006] To achieve the above objectives, this invention provides a two-stage multi-constraint excitation trajectory design method for a robotic arm, specifically including:

[0007] Step S1 linearizes the dynamic parameters based on the dynamic equations of the robotic arm to obtain the minimum set of inertial parameters of the robotic arm, and establishes a linear observation equation that includes inertial parameters and friction parameters.

[0008] Step S2 parameterizes the excitation trajectory of each joint of the robotic arm based on a finite-term Fourier series, and obtains the joint angle, joint velocity and joint acceleration of each joint of the robotic arm.

[0009] Step S3 constructs an observation matrix based on the linear observation equation and generates observability evaluation indexes. Under the conditions of satisfying joint motion constraints, spatial safety constraints, and self-collision detection constraints, the first-stage trajectory optimization is performed with the observability evaluation indexes as the optimization objective to obtain the optimal observability evaluation index values ​​and safe and executable candidate excitation trajectories for the first stage.

[0010] Step S4 generates the second-stage observability lower limit constraint based on the first-stage optimal observability evaluation index, and performs second-stage trajectory optimization by comprehensively considering dynamic incentive index, velocity reversal index, spectrum richness index, and joint balance index to obtain the optimal incentive trajectory.

[0011] Step S5: Based on the optimal excitation trajectory, obtain the motion data and joint torque data of the robotic arm, identify the dynamic parameters, and evaluate the safety and executability of the optimal excitation trajectory, the sufficiency of the excitation, and the accuracy of the identification of the robotic arm's dynamic parameters.

[0012] Further, in step S1, a linear observation equation containing inertial parameters and friction parameters is established based on the robot arm dynamics equation; wherein, the inertial parameters include the minimum inertial parameters obtained by linearizing the robot arm dynamics equation, and the friction parameters include viscous friction parameters, Coulomb friction parameters, and nonlinear velocity friction parameters.

[0013] Furthermore, in step S2, the sine and cosine coefficients in the finite Fourier series are used as trajectory optimization variables, and the joint angles, joint velocities, and joint accelerations of each joint of the robotic arm are generated based on the trajectory optimization variables.

[0014] Furthermore, in step S3, a total observation matrix is ​​constructed based on the dynamic parameter identification observation matrix corresponding to the candidate excitation trajectory at multiple sampling times. After normalizing the total observation matrix, an information matrix is ​​constructed, and a D-observability evaluation index is generated based on the eigenvalues ​​of the information matrix.

[0015] Further, in step S3, the joint motion constraints include joint angle constraints, joint velocity constraints, and joint acceleration constraints; the spatial safety constraints include height constraints of the robotic arm links and joints relative to a preset safety plane; the self-collision detection constraints include a self-collision detection constraint penalty term based on the distance between non-adjacent links; under the conditions of satisfying the joint motion constraints, spatial safety constraints, and self-collision detection constraints, the optimal observability evaluation index value for the first stage and a safe and executable candidate excitation trajectory are obtained.

[0016] Furthermore, in step S4, the second stage observational lower limit constraint is generated based on the optimal observability evaluation index of the first stage and the preset proportional coefficient. This enables the second stage trajectory optimization process to comprehensively optimize the dynamic excitation index, velocity reversal index, spectral richness index, and joint balance index while maintaining the observability of the candidate excitation trajectory at a preset level, thereby obtaining the optimal excitation trajectory for identifying the dynamic parameters of the robotic arm.

[0017] Further, in step S5, before sending the optimal excitation trajectory to the robotic arm controller, a safety and executability check is performed on the joint motion restrictions, spatial safety distance, and self-collision risk of the optimal excitation trajectory, and the excitation sufficiency is also checked. After the optimal excitation trajectory meets the requirements of safety and executability and excitation sufficiency, the robotic arm is driven to execute the optimal excitation trajectory, joint motion data and joint torque data are collected, the data are processed by zero-phase filtering, an experimental observation matrix and an experimental torque vector are constructed, the least squares method is used to identify the dynamic parameters of the robotic arm, and the identification accuracy of the dynamic parameters corresponding to the optimal excitation trajectory is evaluated based on the normalized root mean square error.

[0018] Furthermore, the two-stage multi-constraint optimization model is solved using a heuristic optimization algorithm, which includes one of the following: genetic algorithm, particle swarm optimization algorithm, differential evolution algorithm, or Bayesian optimization algorithm.

[0019] The present invention provides a two-stage multi-constraint excitation trajectory design method for robotic arms, which effectively solves the problems of insufficient consideration of actual motion safety constraints and difficulty in ensuring sufficient experimental excitation by single observability optimization in the traditional robotic arm excitation trajectory design process. It significantly improves the safety and executability of the excitation trajectory, the observability maintenance capability, and the overall dynamic parameter identification accuracy. Attached Figure Description

[0020] To more clearly illustrate the technical solutions of the embodiments of the present invention, the accompanying drawings used in the embodiments are briefly described below. It should be understood that the following drawings are only used to illustrate some embodiments of the present invention and do not constitute a limitation on the scope of protection of the present invention.

[0021] Figure 1 The flowchart illustrates the steps of a two-stage multi-constraint excitation trajectory design method for a robotic arm, as provided in an embodiment of the present invention.

[0022] Figure 2 The flowchart is shown in the embodiment of the present invention to illustrate a two-stage multi-constraint excitation trajectory design method for a robotic arm.

[0023] Figure 3 This is a schematic diagram of the optimal excitation trajectory in an embodiment of the present invention.

[0024] Figure 4 This is a schematic diagram of the optimal excitation trajectory safety constraint check in an embodiment of the present invention.

[0025] Figure 5 This is a diagram showing the filtered processing of joint angular acceleration and joint torque in an embodiment of the present invention. Detailed Implementation

[0026] refer to Figure 1 and Figure 2 The specific steps of the two-stage multi-constraint excitation trajectory design method for a robotic arm according to the present invention are as follows:

[0027] Step S1 linearizes the dynamic parameters based on the dynamic equations of the robotic arm to obtain the minimum set of inertial parameters of the robotic arm, and establishes a linear observation equation that includes inertial parameters and friction parameters.

[0028] In this embodiment, a dynamic parameter identification model is established based on the rigid body dynamics equations of the robotic arm. For those with... The dynamic equation of a robotic arm with multiple joints can be expressed as:

[0029]

[0030] in, These are the joint angles, angular velocities, and angular acceleration vectors of the robotic arm in generalized coordinates. The inertia matrix; This is the matrix of Coriolis force and centrifugal force. The gravity vector For joint friction torque, This is the driving torque of the joint.

[0031] The rigid body dynamics equations of the robotic arm are linearized to obtain the linear regression forms of the inertial parameters of each robotic arm joint:

[0032]

[0033] in, Indicates the first The standard inertial parameter vector of a robotic arm link, They represent the first A link revolves around its link coordinate system Moment of inertia of the shaft They represent the first The product of inertia of each link, Indicates the first The mass of each link. They represent the first The center of mass of the connecting rod lies along its coordinate system. Directional coordinates They represent the first The first-order mass moment of each link; redundant parameters in the inertial parameters are eliminated according to the robot arm configuration, and the eliminated parameters are combined to obtain the minimum inertial parameter set. Establish the minimum inertial parameter observation matrix:

[0034]

[0035] Furthermore, considering the influence of actual robotic arm joint friction on the driving torque, joint friction parameters are incorporated into the dynamic parameter identification model. This embodiment uses viscous friction, Coulomb friction, and nonlinear velocity-dependent friction terms for joint friction modeling. The joint friction torque is:

[0036]

[0037] in, For viscous friction, For Coulomb friction term, For nonlinear velocity-dependent friction terms, This represents the joint friction torque.

[0038] The friction equation can be further written in a linearized form with respect to the coefficients:

[0039]

[0040] Therefore, the linearized model of the frictional torque of all joints of the robot is as follows:

[0041]

[0042] in, Represents the friction parameter observation matrix. This represents the vector of friction parameters to be identified.

[0043] Therefore, by combining the minimum inertial parameter observation matrix and the friction parameter observation matrix, a linear observation equation containing both inertial and friction parameters can be established:

[0044]

[0045] The linear observation equation is used to construct the dynamic parameters identification observation matrix based on the candidate excitation trajectory and to calculate the trajectory observability evaluation index.

[0046] Step S2 parameterizes the excitation trajectory of each joint of the robotic arm based on a finite-term Fourier series, and obtains the joint angle, joint velocity and joint acceleration of each joint of the robotic arm.

[0047] In this embodiment, to facilitate the generation of joint excitation trajectories with periodicity, continuity, and multiple frequency components, a finite-term Fourier series is used to parameterize the trajectories of each joint of the robotic arm. Assume the robotic arm has a total of... Each joint, in the time domain is , For the total runtime of the experimental trajectory, the first... Joint trajectory:

[0048]

[0049] in, The first The angle, angular velocity, and angular acceleration of each joint. This is the initial angle of the joint. These are the coefficients of the sine term and the coefficients of the cosine term, respectively. The harmonic order is... For the fundamental frequency, For the first Angular frequency. Using The cosine term ensures that the initial position of the joint is not deviated from the initial position at the initial moment, thus facilitating the satisfaction of the initial position constraints of the joint trajectory. For all joints of the robotic arm, the Fourier coefficients of each joint can be combined into trajectory optimization variables. By changing these trajectory optimization variables, different candidate excitation trajectories can be generated.

[0050] In this embodiment, the first-stage trajectory optimization model and the second-stage trajectory optimization model are solved using a genetic algorithm. It should be noted that the genetic algorithm is only one specific implementation of a heuristic optimization algorithm. In other embodiments, other heuristic optimization algorithms capable of globally searching trajectory parameters, such as particle swarm optimization or Bayesian optimization, can also be used to solve the trajectory optimization model.

[0051] Step S3 constructs an observation matrix based on the linear observation equation, generates observability evaluation indexes, and performs first-stage trajectory optimization with the observability evaluation indexes as the optimization objective under the conditions of joint motion constraints, spatial safety constraints, and self-collision detection constraints, to obtain the optimal observability evaluation index values ​​and safe and executable candidate excitation trajectories in the first stage.

[0052] Specifically, at the sampling time At each location, calculate the corresponding observation matrix:

[0053]

[0054] By stacking the observation matrices corresponding to multiple sampling times along the row direction, the total observation matrix corresponding to the candidate excitation trajectory over the entire sampling time range is obtained:

[0055]

[0056] in, This represents the total observation matrix. The total observation matrix is ​​used to characterize the overall excitation information of the candidate excitation trajectory over the entire motion cycle for identifying the dynamic parameters to be determined.

[0057] In this embodiment, the total observation matrix is ​​column normalized to obtain a normalized observation matrix:

[0058]

[0059] in, Represents the normalized observation matrix. This represents a diagonal matrix composed of the scale factors of each column of the total observation matrix. The The diagonal elements are:

[0060]

[0061] in, This indicates the number of rows in the total observation matrix. Represents the total observation matrix. Line number Column elements.

[0062] Construct an information matrix based on the normalized observation matrix:

[0063]

[0064] in, The information matrix reflects the overall excitation degree of the candidate excitation trajectory in the parameter space to be identified, and its eigenvalue distribution can be used to evaluate the numerical stability and information richness of the dynamic parameter identification.

[0065] In this embodiment, an observability evaluation index is generated based on the eigenvalues ​​of the information matrix:

[0066]

[0067] in, Indicates observability evaluation index, Information matrix The 1 eigenvalue, This indicates the number of parameters to be identified.

[0068] After obtaining the observability evaluation index, a first-stage trajectory optimization model is established. The first-stage trajectory optimization model aims to improve the observability of candidate excitation trajectories, while introducing joint motion constraints, spatial safety constraints, and self-collision detection constraints during the actual execution of the robotic arm, so that the candidate excitation trajectories output in the first stage are safe and executable.

[0069] Specifically, the joint motion constraint is expressed as follows:

[0070]

[0071] in, These represent the lower limit and upper limit of the joint angle, respectively. These represent the lower limit and upper limit of joint velocity, respectively. These represent the lower limit and upper limit of joint acceleration, respectively.

[0072] Furthermore, based on the forward kinematics of the robotic arm, the spatial positions of each link and joint in the base coordinate system are calculated, and spatial safety constraints are established according to these spatial positions. Specifically, the spatial safety constraints are the link height constraints above the ground.

[0073]

[0074] in, Indicates the first Each link and joint at time height, Indicates the preset safety plane height. Indicates the first The equivalent envelope radius of each link is used to prevent accidents such as links or joints colliding with the experimental platform when the robotic arm executes the excitation trajectory.

[0075] Furthermore, to avoid self-collisions during the execution of candidate excitation trajectories, this embodiment establishes self-collision detection constraints based on the forward kinematics of the robotic arm. Specifically, non-adjacent links of the robotic arm are represented as capsules with equivalent envelope radii, and the surface distances between non-adjacent links are calculated. The robotic arm comprises... A rigid link, defined at time 1 No. The central axis segment of each link serves as the origin connecting adjacent joints. and line segments:

[0076]

[0077] For any set of non-adjacent links participating in self-collision detection The shortest distance between the centerline segments of two connecting rods is defined as:

[0078]

[0079] in, and They respectively represent the positions located at the th The central axis segment of each link and the The central axis segment of each link Any point in space is a three-dimensional position vector. This represents the Euclidean distance between two points in space.

[0080] To account for the geometry of the connecting rod, this embodiment uses a capsule approximation, and the first... root link and the first Each link is assigned an envelope radius. and Then the two connecting rods are The surface distance at time is:

[0081]

[0082] in, This indicates that there is still a surface gap between the two connecting rods. It means that we just made contact. Indicates geometric overlap.

[0083] Generate a self-collision penalty term based on the proximity of the objects:

[0084]

[0085] in, This indicates the self-collision penalty. This indicates the number of sampled items involved in the penalty calculation. Indicates the number of sampling times. This represents the set of non-adjacent links that participate in self-collision detection. This indicates the preset self-collision safety distance.

[0086] Based on the aforementioned observability evaluation index, joint motion constraints, spatial safety constraints, and self-collision penalty term, a first-stage trajectory optimization model is established:

[0087]

[0088] in, The vector representing the optimization variables of the candidate excitation trajectories. This represents the optimal candidate excitation trajectory output by the first-stage trajectory optimization model. This is the fitness function for the first-stage trajectory optimization model. This represents the observability evaluation index corresponding to the candidate incentive trajectories in the first stage. This represents the self-collision penalty coefficient. It should be noted that joint motion constraints and spatial safety constraints are hard constraints that candidate excitation trajectories must satisfy, used to limit the feasible region of the first-stage trajectory optimization; while self-collision detection constraints involve determining the minimum distance between non-adjacent links. Their constraint boundaries have non-smoothness and discrete triggering characteristics. If directly treated as hard constraints, it can easily lead to the elimination of a large number of candidate trajectories and reduce the search efficiency of the heuristic optimization algorithm. Therefore, self-collision constraints appear in the fitness function as a penalty term. This represents the first-stage feasible region, which consists of joint motion constraints, spatial safety constraints, and self-collision detection constraints. satisfy:

[0089]

[0090] in, They represent the optimization variables of the candidate trajectory, respectively. The generated first Each joint at any time Joint angle, joint angular velocity, joint angular acceleration, They represent the first The upper and lower limits of the angle, angular velocity, and angular acceleration of each joint. Indicates the first Each link at the sampling time Spatial safety distance.

[0091] In the feasible region Solve the first-stage trajectory optimization model internally to obtain the candidate excitation trajectories output by the first-stage trajectory optimization model and their corresponding optimal observability evaluation index:

[0092]

[0093] in, This is the optimal observability evaluation index obtained by the first-stage trajectory optimization model.

[0094] Step S4 generates the second-stage observability lower limit constraint based on the first-stage optimal observability evaluation index, and performs second-stage trajectory optimization by comprehensively considering dynamic incentive index, velocity reversal index, spectrum richness index, and joint balance index to obtain the optimal incentive trajectory.

[0095] The second-stage lower limit of observation is generated based on the optimal observability evaluation index and the preset proportional coefficient:

[0096]

[0097] in, This indicates the lower limit of observation for the second stage. The second stage observability lower limit is used to limit the minimum observability level of candidate excitation trajectories during the second stage trajectory optimization process, so that the second stage optimization can improve the excitation performance without excessively reducing the amount of observation information required for dynamic parameter identification.

[0098] In this embodiment, the candidate excitation trajectories obtained from the first-stage trajectory optimization model are used as the initial trajectories for the second-stage trajectory optimization model, and an initial solution set for the second-stage trajectory optimization is generated based on the initial trajectories. As one implementation method, a preset amplitude perturbation can be applied near the Fourier coefficients corresponding to the optimal trajectory in the first stage to generate several hot-start individuals; simultaneously, some randomly generated trajectory parameter individuals are retained to improve population diversity during the second-stage optimization process.

[0099] The following explains the quality indicators of the auxiliary excitations used in the second-stage trajectory optimization model:

[0100] 1) Dynamic incentive indicators:

[0101] To evaluate the dynamic utilization of candidate excitation trajectories at the velocity and acceleration levels, a dynamic excitation index is constructed. , for the Each joint at discrete time The angular velocity and angular acceleration of the relative motion constraints are defined as follows: The normalized utilization rate of the joint velocity and acceleration is:

[0102]

[0103] in, These represent the upper limits of joint velocity and acceleration, respectively. The normalized utilization rate is used to characterize the degree to which the candidate excitation trajectory utilizes the range of joint velocity and acceleration of the robotic arm.

[0104] To reduce the impact of a few instantaneous peak values ​​on the dynamic excitation evaluation results, this embodiment uses quantile statistics to characterize the typical dynamic level of the trajectory over most of the time. The 0.9 quantile operator, If is the number of sampling points, then the th The velocity and acceleration of each joint are expressed horizontally as follows:

[0105]

[0106] Furthermore, to characterize the persistence of the high dynamic state throughout the entire trajectory, thresholds for velocity and acceleration utilization are set as follows: Calculate the percentage of time when the normalized utilization rate exceeds the corresponding threshold:

[0107]

[0108] in, This is an indicator function that takes the value 1 if the condition within the parentheses is true, and 0 otherwise. , These represent the percentage of time velocity and acceleration utilization of the joint throughout the entire trajectory that are not lower than the threshold, reflecting the continuity of the dynamic state.

[0109] The final dynamic incentive index adopts a four-factor average method:

[0110]

[0111] The overline indicates all. The arithmetic mean of the joints is taken. The larger the value, the more fully the candidate excitation trajectory excites the dynamic characteristics of the robotic arm in terms of velocity and acceleration.

[0112] 2) Speed ​​reversal index:

[0113] To evaluate the sufficiency of joint velocity direction changes in candidate excitation trajectories, a velocity reversal index is constructed. Effective reversal of joint velocity direction enables experimental data to cover dynamic and friction terms under different velocity signs, which helps to improve the excitation diversity of the dynamic regression matrix.

[0114] Since the joint speed may fluctuate near zero, this embodiment focuses on the first... At each joint Time setting dead zone threshold And construct a stable symbol sequence as follows:

[0115]

[0116] According to the statistics of the stable symbol sequence, the first The effective number of rotations of each joint is To avoid excessively frequent reversals due to short-term jitter, a preset minimum interval can be set between two adjacent effective reversals. When the interval between two adjacent changes in speed direction is less than the preset minimum interval, it will not be counted in the number of effective reversals.

[0117] The effective rotation count of all joints is normalized to obtain the velocity reversal index:

[0118]

[0119] in, For the number of joints, This indicates the preset number of normalization commutations. The larger the value, the more fully the candidate trajectory changes in velocity direction.

[0120] 3) Spectral richness index:

[0121] To evaluate the frequency component richness of candidate excitation trajectories, a spectral richness index is constructed based on the energy distribution of harmonic coefficients in a finite-term Fourier series. This embodiment constructs a spectral richness index based on Fourier coefficients and uses energy entropy to measure the uniformity of energy distribution across harmonic orders. Let the Fourier order be... , No. The first joint The sine and cosine coefficients of the first harmonic are respectively , Then the first The first joint The energy of a first harmonic is defined as:

[0122]

[0123] The harmonic energy is normalized to an energy percentage:

[0124]

[0125] in, This represents the regularization coefficient used to avoid division by zero.

[0126] Based on the normalized energy proportions, normalized Shannon entropy is introduced to evaluate the uniformity of energy distribution for each harmonic:

[0127]

[0128] in, Indicates the first The spectral richness evaluation value for each joint. If the energy is almost entirely concentrated in a few harmonic orders, the energy distribution is sharp. Smaller; if the energy is more evenly distributed across all orders, then A larger value indicates that the frequency components of the joint trajectory are richer.

[0129] The spectral richness index is the average of the spectral richness evaluation values ​​for each joint:

[0130]

[0131] The larger the value, the more uniform the energy distribution of the candidate excitation trajectory is in the multi-harmonics, and the richer the frequency components, which is beneficial for covering more motion timescales within a limited experimental time.

[0132] 4) Joint balance index:

[0133] To evaluate whether the candidate excitation trajectories provide balanced excitation to different joints, a joint balance index is constructed. For the first The velocity utilization sequence of each joint during the entire trajectory cycle is as follows: ,in For discrete sampling points, The total number of discrete sampling points during the entire trajectory period is represented by the root mean square of the velocity utilization rate, which characterizes the overall excitation intensity of the joint during the entire trajectory period.

[0134]

[0135] The overall excitation intensity of all joints is denoted as set. , For the number of joints, this embodiment uses the coefficient of variation to represent the degree of uneven excitation between joints:

[0136]

[0137] in, This represents the mean. This represents the standard deviation. Further define the joint balance index:

[0138]

[0139] in, This represents the equilibrium penalty adjustment coefficient.

[0140] Based on the aforementioned dynamic incentive index, velocity reversal index, spectral richness index, and joint balance index, an auxiliary incentive quality index is constructed:

[0141]

[0142] in, This indicates the quality indicators of auxiliary incentives. These represent the weight coefficients of the corresponding indicators.

[0143] After establishing the quality indicators for auxiliary incentives, based on the second-stage observational lower limit... A second-stage trajectory optimization model is constructed, which comprehensively optimizes the observability evaluation index and the auxiliary incentive quality index. The second-stage trajectory optimization model is expressed as follows:

[0144]

[0145] in, This represents the optimal incentive trajectory for the second stage of optimization. This is the fitness function for the second-stage trajectory optimization model. These represent the weighting coefficients of the observability evaluation index and the auxiliary incentive quality index, respectively. This represents the observability evaluation index for the candidate trajectories in the second-stage optimization model. Through the hierarchical objective function described above, when the observability of the candidate excitation trajectories is insufficient, the second-stage optimization process prioritizes raising the observability evaluation index to above the lower limit of observability. Once the observability evaluation index meets the lower limit requirement, the optimization process further improves the dynamic excitation intensity, velocity reversal sufficiency, spectral richness, and joint excitation balance. Therefore, the second-stage trajectory optimization can enhance the excitation sufficiency of the candidate excitation trajectories for the actual dynamic characteristics of the robotic arm while maintaining the amount of information for dynamic parameter identification, ultimately obtaining the optimal excitation trajectory for the robotic arm dynamic parameter identification experiment. .

[0146] In this embodiment, a six-degree-of-freedom robotic arm is used to conduct a dynamic parameter identification experiment, and a genetic algorithm is employed to solve the two-stage trajectory optimization model. Specifically, the coefficients of the sine and cosine terms in a finite-term Fourier series are used as optimization variables in the genetic algorithm. In the first stage, the genetic algorithm searches for candidate excitation trajectories with better observability within the safe and feasible region. In the second stage, the genetic algorithm uses the aforementioned candidate excitation trajectories as the initial trajectory or a hot-start trajectory, and further optimizes the auxiliary excitation quality index under the lower bound of observability constraints. Through genetic operations such as selection, crossover, mutation, and elite retention, the trajectory parameters are updated generation by generation until a preset number of iterations is reached or the convergence condition is met. The population size, maximum number of iterations, crossover probability, mutation probability, and termination condition of the genetic algorithm can be set according to the degrees of freedom of the robotic arm, the order of the Fourier series, and computational resources. In a specific embodiment, the population size can be set to 100, the crossover probability can be set to 0.8, and the maximum number of iterations in both the first and second stages can be set to 800. The above parameters are merely an example and do not constitute a limitation on the scope of protection of this invention.

[0147] Step S5 involves acquiring robotic arm motion data and joint torque data based on the optimal excitation trajectory to identify dynamic parameters, evaluating the safety, sufficiency, and accuracy of the optimal excitation trajectory in identifying robotic arm dynamic parameters. Optimal Fourier coefficients are obtained, and optimal excitation trajectories for each joint of the robotic arm are generated based on these coefficients. The optimal excitation trajectory includes reference data for joint angles, joint velocities, and joint accelerations, which can be used as the input trajectory for the robotic arm dynamic parameter identification experiment. Figure 3 A schematic diagram of the optimal excitation trajectory obtained based on two-stage multi-constraint optimization in this embodiment is given. Figure 3It can be seen that the optimal excitation trajectory has continuous and smooth variation characteristics, and can form multi-joint joint motion within the experimental time range, providing effective experimental excitation data for dynamic parameter identification. To further illustrate the comprehensive excitation effect of the optimal excitation trajectory, the two-stage multi-constraint optimized trajectory of this invention is compared with the genetic algorithm trajectory considering only safety constraints, while setting all parameters to be consistent. The results are shown in Table 1. It should be noted that the observation index is used to characterize the information quality of the observation matrix for dynamic parameter identification, but it cannot determine the identification effect of the excitation trajectory alone. For actual robotic arm dynamic parameter identification experiments, the trajectory also needs to have sufficient velocity and acceleration excitation and a rich spectral distribution. Therefore, in the second-stage optimization, this invention does not simply pursue the maximization of the observation index, but rather improves the dynamic excitation and velocity reversal of the trajectory while ensuring that the observation index is not lower than the preset lower limit, thereby obtaining a trajectory with better comprehensive excitation performance.

[0148] Table 1 Comparison of comprehensive incentive quality indicators for the two incentive trajectories

[0149] Incentive Trajectory Observability indicators Dynamic incentive indicators Speed ​​reversal index Spectrum abundance index Joint balance index Genetic algorithm trajectory considering only safety constraints 0.4539 0.3849 0.5714 0.5532 0.3900 This invention provides a two-stage, multi-constraint trajectory optimization method. 0.4426 0.4982 0.7738 0.5789 0.5022 rate of change Maintain 97.51% +29.44% +35.42% +4.65% +28.77%

[0150] As shown in Table 1, compared with the genetic algorithm trajectory considering only safety constraints, the observability index of the two-stage multi-constraint optimized trajectory of this invention is slightly reduced, but still remains at a high level; at the same time, its dynamic excitation index, velocity reversal index, spectral richness, and joint balance index are all improved. These results demonstrate that the method of this invention can improve the comprehensive excitation performance of the trajectory while maintaining the quality of observation information, avoiding the problem of insufficient dynamic excitation of the trajectory due to simply pursuing observability indexes.

[0151] To verify the actual executability of the optimal excitation trajectory, this embodiment... Figure 3 The optimal excitation trajectory shown is subjected to safety constraint checks, including joint angle constraints, joint velocity constraints, joint acceleration constraints, link height constraints, and self-collision detection constraints. Specifically, each link of the robotic arm is represented as a capsule envelope, with the envelope radii of links 1 through 6 set to 50mm, 45mm, 40mm, 40mm, 40mm, and 40mm, respectively. Figure 4 Table 1 presents the variations in joint angles, joint velocities, joint accelerations, link height above ground, and minimum safe distance between links for the optimal excitation trajectory within the experimental time range. Table 2 further provides the preset boundaries of each safety constraint, the actual extreme values ​​of the trajectory, and the number of boundary violations. Figure 4As shown in Table 2, the actual variation ranges of joint angles, velocities, and accelerations are all within the corresponding preset motion limits, with zero out-of-bounds points. During movement, each link of the robotic arm satisfies the ground clearance constraint, with a minimum ground clearance margin not less than the preset safety boundary. The minimum distance between non-adjacent links is always greater than the preset safety distance, eliminating any risk of self-collision. Therefore, the optimal excitation trajectory satisfies the preset safety constraints throughout the entire experimental time and can serve as an executable input trajectory for the robotic arm dynamics parameter identification experiment.

[0152] Table 2. Results of Safety Constraint Check for Optimal Excitation Trajectory

[0153] Inspection Items Constraints Check value Outbound points Inspection results Maximum normalized utilization rate of joint angle ≤1 0.6689 0 satisfy Maximum normalized utilization rate of joint angular velocity ≤1 0.9865 0 satisfy Maximum normalized utilization rate of joint angular acceleration ≤1 0.6362 0 satisfy Space link height above ground ≥0mm 2.8mm 0 satisfy Self-collision linkage safety distance ≥0mm 2mm 0 satisfy

[0154] After obtaining the optimal excitation trajectory that satisfies safety constraints, the robotic arm is driven to execute the optimal excitation trajectory, and data from each joint of the robotic arm is collected. Since joint angular acceleration is usually obtained by differential velocity signal analysis, and joint torque data is easily affected by sensor noise, current fluctuations, etc., this embodiment performs filtering processing on the collected joint angular acceleration and joint torque data. The filtering result is as follows: Figure 5 As shown.

[0155] Based on the filtered experimental data, an experimental observation matrix and joint torque vector are constructed for dynamic parameter identification, and the least squares method is used for dynamic parameter identification. Based on the experimental observation matrix and experimental torque vector, the least squares method is used to identify the dynamic parameter vector:

[0156]

[0157] in, This represents the identified dynamic parameter vector. Represents the experimental observation matrix. This represents the vector of dynamic parameters to be identified. This represents the experimental torque vector. Further, substituting the identified dynamic parameter vector into the linear observation equation yields the predicted joint torque:

[0158]

[0159] To further verify the effectiveness of the two-stage multi-constraint excitation trajectory design method described in this invention, this embodiment uses the normalized root mean square error (RMSE) between the predicted joint torque and the experimentally acquired joint torque as a quantitative evaluation index to evaluate the identification effect of the dynamic parameters corresponding to the optimal excitation trajectory. For comparison, this embodiment selects the genetic algorithm-optimized trajectory considering only safety constraints as the comparison trajectory. Under the same robotic arm model, the same sampling conditions, the same filtering method, and the same least squares identification method, the torque prediction errors corresponding to the two excitation trajectories are calculated respectively. This embodiment uses a six-degree-of-freedom robotic arm for experiments and uses the normalized root mean square error as the evaluation index. The normalized root mean square error is expressed as:

[0160]

[0161] in, This indicates the number of data points involved in the error calculation. Indicates the first The experimental acquisition torque at each sampling point Indicates the first Predicted torque at each sampling point and These represent the maximum and minimum values ​​of the torque collected in the experiment, respectively.

[0162] Table 3 shows the comparison of the average NRMSE of the two excitation trajectories under different speed ranges. As can be seen from Table 3, under the three speed ranges of 0-120° / s, 0-150° / s, and 0-180° / s, the average NRMSE corresponding to the method of this invention are 6.92%, 7.28%, and 6.62%, respectively, all lower than the 7.47%, 10.85%, and 6.94% corresponding to the genetic algorithm trajectory considering only safety constraints. The average NRMSE reduction rates are 7.36%, 32.90%, and 4.61%, respectively. The results indicate that the method of this invention can reduce the overall dynamic parameter identification error and improve the overall identification accuracy of the excitation trajectory for parameter identification.

[0163] Table 3. Comparison of average NRMSE of the two excitation trajectories at different speed ranges

[0164] Speed ​​range (° / s) Average NRMSE of genetic algorithm trajectories considering only safety constraints / % This invention provides a two-stage, multi-constraint optimization trajectory average NRMSE / % Reduction rate / % 0-120 7.47 6.92 7.36 0-150 10.85 7.28 32.90 0-180 6.94 6.62 4.61

[0165] The technical solutions of the embodiments of the present invention have the following beneficial effects:

[0166] In the technical solution of this invention, a finite-term Fourier series is used to parameterize the excitation trajectory of each joint of the robotic arm. In the first stage of trajectory optimization, the observability evaluation index is used as the optimization target, and combined with joint motion constraints, spatial safety constraints, and self-collision detection constraints, safe and executable candidate excitation trajectories are obtained. In the second stage of trajectory optimization, an observability lower limit constraint is set based on the optimization results of the first stage, and dynamic excitation, velocity reversal, spectral richness, and joint balance indices are comprehensively optimized while maintaining the quality of observation information to obtain the optimal excitation trajectory. The technical solution provided by this invention effectively solves the problems of insufficient consideration of actual motion safety constraints and difficulty in ensuring sufficient experimental excitation by single observability optimization in the traditional robotic arm excitation trajectory design process, and significantly improves the safety and executability of the excitation trajectory, the observability maintenance capability, and the overall dynamic parameter identification accuracy.

[0167] The above description is only a preferred embodiment of the present invention and is not intended to limit the present invention. Any modifications, equivalent substitutions, improvements, etc., made within the spirit and principles of the present invention should be included within the scope of protection of the present invention.

[0168] The contents not described in detail in this specification are common knowledge to those skilled in the art.

Claims

1. A method for designing a two-stage, multi-constraint excitation trajectory for a robotic arm, characterized in that, The method includes: Step S1 linearizes the dynamic parameters based on the dynamic equations of the robotic arm to obtain the minimum set of inertial parameters of the robotic arm, and establishes a linear observation equation that includes inertial parameters and friction parameters. Step S2 parameterizes the excitation trajectory of each joint of the robotic arm based on a finite-term Fourier series to obtain the joint angle, joint velocity and joint acceleration of each joint of the robotic arm. Step S3 constructs an observation matrix based on the linear observation equation and generates observability evaluation index; under the conditions of satisfying joint motion constraints, spatial safety constraints and self-collision detection constraints, the first stage trajectory optimization is performed with the observability evaluation index as the optimization objective to obtain the first stage optimal observability evaluation index value and safe and executable candidate excitation trajectory. Step S4 generates the second-stage observability lower limit constraint based on the first-stage optimal observability evaluation index, and performs second-stage trajectory optimization by comprehensively considering dynamic incentive index, velocity reversal index, spectrum richness index and joint balance index to obtain the optimal incentive trajectory; Step S5 involves acquiring motion data and joint torque data of the robotic arm based on the optimal excitation trajectory, identifying dynamic parameters, and evaluating the safety, sufficiency, and accuracy of the identification of the robotic arm's dynamic parameters based on the optimal excitation trajectory.

2. The method according to claim 1, characterized in that, In step S1, the dynamic parameters are linearized based on the robotic arm's dynamic equations to obtain the minimum set of inertial parameters of the robotic arm, and a linear observation equation containing inertial and frictional parameters is established, including: Based on the derivation of the Newton-Euler formula, the rigid body dynamics equation of the robotic arm is obtained: in These represent the joint angle, angular velocity, and angular acceleration vectors of the robotic arm, respectively. The inertia matrix; The matrix represents the Coriolis force and the centrifugal force. Represents the gravity term. For joint friction torque, This refers to the driving torque of the robotic arm joints. The rigid body dynamics equations of the robotic arm are linearized to obtain the linear regression forms of the inertial parameters of each robotic arm joint: in, Indicates the first The standard inertial parameter vector of a robotic arm link, They represent the first A link revolves around its link coordinate system Moment of inertia of the shaft, They represent the first The product of inertia of each link, Indicates the first The mass of each link. They represent the first The center of mass of the connecting rod lies along its coordinate system. Directional coordinates They represent the first The first-order mass moment of each link; redundant parameters in the inertial parameters are eliminated according to the robot arm configuration, and the eliminated parameters are combined to obtain the minimum inertial parameter set. Establish the minimum inertial parameter observation matrix: A friction parameter observation matrix was constructed based on joint velocity. The friction parameter observation matrix includes viscous friction terms, Coulomb friction terms, and nonlinear velocity friction terms; combining the minimum inertial parameter observation matrix and the friction parameter observation matrix yields a linear observation equation containing inertial and friction parameters: in, This represents the vector of friction parameters to be identified.

3. The method according to claim 1, characterized in that, In step S2, the excitation trajectory of each joint of the robotic arm is parameterized based on a finite-term Fourier series to obtain the joint angle, joint velocity, and joint acceleration of each joint of the robotic arm, including: Assume the robotic arm has a total of One joint, experimental time domain is , For the total runtime of the experimental trajectory, the first... For joints, a periodic trajectory is constructed using truncated Fourier series: in, For the first Joint angle, For the first Joint angular velocity, For the first Joint angular acceleration, This is the initial angle of the joint. , For the first First harmonic coefficient, The harmonic order is... For the fundamental frequency, For the first Angular frequency.

4. The method according to claim 1, characterized in that, Step S3, which involves constructing an observation matrix based on the linear observation equation and generating observability evaluation indicators, includes: At sampling time At each location, calculate the corresponding observation matrix: By stacking the observation matrices corresponding to multiple sampling times along the row direction, the total observation matrix corresponding to the candidate excitation trajectory over the entire sampling time range is obtained: in, The total observation matrix is ​​used to characterize the overall excitation information of the dynamic parameters to be identified for the candidate excitation trajectory throughout the entire motion cycle. The total observation matrix is ​​column-normalized to obtain the normalized observation matrix: in, Represents the normalized observation matrix. This represents a diagonal matrix composed of the normalized coefficients of each column of the total observation matrix; the diagonal matrix The The diagonal elements are: in, Represents a diagonal matrix The diagonal elements, This indicates the number of rows in the total observation matrix. Represents the total observation matrix. Line 1 Column elements; Construct an information matrix based on the normalized observation matrix: in, The information matrix reflects the overall excitation degree of the candidate excitation trajectory in the parameter space to be identified, and its eigenvalue distribution can be used to evaluate the numerical stability and information richness of the dynamic parameter identification. Generate observability evaluation indicators based on the eigenvalues ​​of the information matrix: in, Indicates observability evaluation index, Information matrix The 1 eigenvalue, This indicates the number of parameters to be identified.

5. The method according to claim 4, characterized in that, In step S3, under the conditions of satisfying joint motion constraints, spatial safety constraints, and self-collision detection constraints, the first stage of trajectory optimization is performed with the observability evaluation index as the optimization objective to obtain the optimal observability evaluation index value and a safe and executable candidate excitation trajectory, including: Using the observability evaluation index as the optimization objective of the first-stage trajectory optimization model, joint motion constraints are established based on the joint limitations of the robotic arm. These joint motion constraints are expressed as follows: in, , These represent the lower limit and upper limit of the joint angle, respectively. , These represent the lower limit and upper limit of joint velocity, respectively. , These represent the lower limit and upper limit of joint acceleration, respectively. The spatial positions of each link and joint in the base coordinate system are calculated based on the forward kinematics of the robotic arm, and spatial safety constraints are established based on these spatial positions. in, Indicates the first Each link and joint at time height, Indicates the preset safety plane height. Indicates the first The equivalent envelope radius of each link; A self-collision detection model is established based on the forward kinematics of the robotic arm. Non-adjacent links of the robotic arm are represented as capsules with equivalent envelope radii. The surface distances between non-adjacent links are calculated. The robotic arm contains a total of [missing information - likely related to collision detection]. A rigid link, defined at time 1 No. The central axis segment of each link is the origin connecting adjacent joints. and line segments The link is represented as having an equivalent envelope radius. The capsule body, for any non-adjacent link participating in self-collision detection and Calculate the shortest distance between the centerline segments of the two connecting rods: in, and They respectively represent the positions located at the th The central axis segment of each link and the The central axis segment of each link Any point in space is a three-dimensional position vector. Represents the Euclidean distance between two points in space; The surface distance between the two links is obtained based on the shortest distance. in, Indicates the first The first link and the first Each link at time Surface distance, , Indicates the first root link and the first The radius of the root link's envelope. This indicates that there is still a surface gap between the two connecting rods. It means that we just made contact. Indicates geometric overlap; A self-collision constraint is established based on the surface distance. When the surface distance is less than zero, the candidate excitation trajectory is determined to have a self-collision risk. When the surface distance is greater than zero and less than a preset safety distance, a self-collision penalty term is generated. in, Indicates the number of sampling times. Indicates the self-collision penalty. This represents the set of non-adjacent links that participate in self-collision detection. This indicates the preset self-collision safety distance. Indicates the number of sampled items involved in the penalty calculation; Based on the aforementioned observability evaluation index, joint motion constraints, spatial safety constraints, and self-collision penalty term, a first-stage trajectory optimization model is established: in, The vector representing the optimization variables of the candidate excitation trajectories. This represents the optimal candidate excitation trajectory output by the first-stage trajectory optimization model. This is the fitness function for the first-stage trajectory optimization model. This represents the observability evaluation index corresponding to the candidate incentive trajectories in the first stage. Represent the self-collision penalty coefficient; establish the first-stage feasible region. : in, They represent the optimization variables of the candidate trajectory, respectively. The generated first Each joint at any time Joint angle, joint angular velocity, joint angular acceleration, They represent the first The upper and lower limits of the angle, angular velocity, and angular acceleration of each joint. Indicates the first Each link at the sampling time Spatial safety distance; In the feasible region Solve the first-stage trajectory optimization model internally to obtain the candidate excitation trajectories output by the first-stage trajectory optimization model and their corresponding optimal observability evaluation index: in, This is the optimal observability evaluation index obtained by the first-stage trajectory optimization model.

6. The method according to claim 1, characterized in that, Step S4, which generates the second-stage observational lower bound constraint based on the optimal observability evaluation index described in the first stage, includes: According to the optimal observability evaluation index The second-stage observational lower limit is generated using a preset scaling factor: in, This indicates the second-stage observational lower bound constraint. This represents a preset scaling factor, which uses the second-stage observability lower bound constraint as a threshold constraint in the second-stage trajectory optimization process to limit the minimum observability level of candidate excitation trajectories during the second-stage trajectory optimization process. in, This represents the observability evaluation index corresponding to the candidate stimulus trajectories in the second stage; the candidate stimulus trajectories obtained from the trajectory optimization model in the first stage... The initial trajectory is used as the initial trajectory for the second-stage trajectory optimization model, and the initial solution set for the second-stage trajectory optimization is generated based on the initial trajectory of the second-stage trajectory optimization model.

7. The method according to claim 6, characterized in that, In step S4, the second-stage trajectory optimization is performed by integrating dynamic excitation indicators, velocity reversal indicators, spectral richness indicators, and joint balance indicators to obtain the optimal excitation trajectory, including: A dynamic excitation index is constructed based on the joint velocity utilization, joint acceleration utilization, velocity coverage, and acceleration coverage of the candidate excitation trajectory. The dynamic excitation index is expressed as follows: in, This represents a dynamic incentive indicator. Indicates joint speed utilization rate. Indicates the utilization rate of joint acceleration. Indicates speed coverage. Indicates acceleration coverage. , , , This represents the corresponding weight coefficient; A velocity reversal index is constructed based on the number of velocity direction changes at each joint in the candidate excitation trajectory: in, Indicates the speed reversal index, Indicates the number of joints in the robotic arm. Indicates the first The number of effective changes in velocity direction of each joint in the candidate excitation trajectory. Indicates the preset normalization commutation number; A spectral richness index is constructed based on the energy distribution of harmonic coefficients of each order in a finite Fourier series: in, This represents a spectral richness index. Denotes the order of the Fourier series. Indicates the first The first joint The proportion of first harmonic energy They represent the first The first joint The coefficients of the first-order sine and cosine terms, Represents the regularization coefficient; A joint balance index is constructed based on the dispersion of joint motion utilization in candidate excitation trajectories. For the first... The velocity utilization sequence of each joint during the entire trajectory cycle is as follows: ,in For discrete sampling points, The total number of discrete sampling points during the entire trajectory period is represented by the root mean square of the velocity utilization rate, which characterizes the overall excitation intensity of the joint during the entire trajectory period. in, Indicates the first The overall excitation intensity of each joint during the entire trajectory period is denoted as set . , The number of joints is represented by the coefficient of variation, indicating the degree of uneven excitation between joints. in, This represents the mean. To represent the standard deviation, we further define the joint balance index: in, This represents the equilibrium penalty adjustment coefficient. ; Based on the aforementioned dynamic incentive index, velocity reversal index, spectral richness index, and joint balance index, an auxiliary incentive quality index is constructed: in, This indicates the quality indicators of auxiliary incentives. , , , These represent the weighting coefficients of the corresponding indicators; After establishing the quality indicators for auxiliary incentives, the second-stage observational lower limit constraint is applied. A second-stage trajectory optimization model is constructed, which comprehensively optimizes the observability evaluation index and the auxiliary incentive quality index. The second-stage trajectory optimization model is expressed as follows: in, This represents the optimal incentive trajectory for the second stage of optimization. This is the fitness function for the second-stage trajectory optimization model. Let represent the weight coefficients of the second-stage observability evaluation index and the auxiliary excitation quality index, respectively. Solving the second-stage trajectory optimization model, we can improve the dynamic excitation, velocity reversal, spectral richness, and joint excitation balance of the candidate excitation trajectories while satisfying the second-stage observability lower limit, thereby obtaining the optimal excitation trajectory. .

8. The method according to claim 1, characterized in that, In step S5, based on the optimal excitation trajectory, the motion data and joint torque data of the robotic arm are obtained for dynamic parameter identification. The safety and executability of the optimal excitation trajectory, the sufficiency of the excitation, and the accuracy of the robotic arm dynamic parameter identification are evaluated, including: Before the optimal excitation trajectory is deployed to the robotic arm controller, a safety and executability check is performed on the excitation trajectory to determine whether the optimal excitation trajectory exceeds the joint movement limit, whether it will collide with the experimental platform, and whether it will self-collision. After the check is passed, the sufficiency of the excitation trajectory is further determined. If the requirements are met, the optimal excitation trajectory is deployed to the robotic arm controller, the robotic arm is driven to execute the optimal excitation trajectory, and the joint motion data and joint torque data of the robotic arm are collected and saved. At the same time, the saved experimental data is subjected to zero-phase filtering to obtain filtered experimental data. Based on the filtered experimental data, an experimental observation matrix and joint torque vector are constructed for dynamic parameter identification, and the least squares method is used for dynamic parameter identification: in, This represents the identified dynamic parameter vector. Represents the experimental observation matrix. This represents the vector of dynamic parameters to be identified. The experimental torque vector is represented by this vector. Further, the identified dynamic parameter vector is substituted into the linear observation equation to obtain the predicted joint torque. The normalized root mean square error (RMSE) between the predicted joint torque and the experimentally acquired joint torque is used as a quantitative evaluation index to evaluate the identification effect of the dynamic parameters corresponding to the optimal excitation trajectory. The normalized root mean square error is expressed as: in, This indicates the number of data points involved in the error calculation. Indicates the first The experimental acquisition torque at each sampling point Indicates the first Predicted torque at each sampling point and These represent the maximum and minimum values ​​of the torque collected in the experiment, respectively.

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