Multi-objective optimization method and device for target ball mounting position of mechanical arm based on optimal control

Abstract

The application discloses a kind of mechanical arm target ball installation position multi-objective optimization method and device based on optimal control, the method first establishes the kinematic model of mechanical arm, the installation position parameterization of target ball in connecting rod local coordinate system is carried out, and the visibility function including occlusion geometric model and laser incidence angle constraint is constructed;Second, the target ball installation position optimization problem is modeled as the multi-objective optimization problem under the continuous time optimal control framework, and the performance indicators such as the maximum visibility coverage, the minimum incidence angle deviation and the uniformization of target ball distance are defined;Then, the Pontryagin maximum principle is used to derive the optimality necessary condition, and the optimal target ball configuration is solved based on HJB equation and adaptive dynamic programming technology.The application can significantly improve the visibility coverage of target ball when mechanical arm performs complex trajectory task and meet the incidence angle constraint, and provides theoretical support for high-precision industrial calibration system.

Description

Technical Field

[0001] This invention belongs to the field of industrial robot calibration and measurement technology, specifically involving a multi-objective optimization method for the installation position of a target ball in a robotic arm based on optimal control. It is particularly suitable for the target ball layout design in a high-precision calibration and online compensation system for a robotic arm assisted by a laser tracker. Background Technology

[0002] Industrial robotic arms play a crucial role in high-precision manufacturing tasks such as welding, drilling, and assembly. To achieve high-precision calibration and online compensation of robotic arms, laser trackers, as a high-precision spatial measurement device, are widely used for real-time measurement of the end-effector pose. Laser trackers acquire the spatial position information of the robotic arm by tracking a target ball (retroreflector) mounted on the arm body, thus providing data support for calibration, trajectory compensation, and accuracy evaluation.

[0003] In practical engineering applications, the selection of the target ball's installation position directly affects the availability and accuracy of the measurement system. When the robotic arm performs complex trajectory tasks, the target ball may be obstructed by the robotic arm's own linkage structure, causing the laser tracker to lose sight of the target and interrupting the measurement. Furthermore, an excessively high laser incident angle can significantly reduce measurement accuracy and may even prevent the laser tracker from locking onto the target ball. Therefore, how to rationally arrange multiple target balls on the robotic arm body so that at least one target ball remains visible to the laser tracker throughout the entire task execution process, while satisfying the incident angle constraint, becomes a critical engineering optimization problem.

[0004] Existing research often employs heuristic methods or simple geometric analysis to determine the target ball position, lacking systematic mathematical modeling and optimization theoretical support. This invention models the target ball installation position optimization problem as a multi-objective optimization problem within a continuous-time optimal control framework. It solves for the optimal target ball configuration using the Pontryagin maximum principle and the Hamilton-Jacobi-Bellman equation, providing a theoretical basis and systematic solution for the target ball layout design of a laser tracker-assisted robotic arm calibration system. Summary of the Invention

[0005] The purpose of this invention is to address the shortcomings of existing technologies by proposing a multi-objective optimization method for the installation position of a target ball in a robotic arm based on optimal control, in order to solve the problem that the target ball layout in existing technologies lacks systematic mathematical modeling and optimization theoretical support.

[0006] The objective of this invention is achieved through the following technical solution: a multi-objective optimization method for the installation position of a target ball in a robotic arm based on optimal control, specifically including the following steps:

[0007] S1: Obtain the angular trajectory of each joint of the robotic arm system and the pose of each link, define the position of the target ball with respect to the link and the decision variable vector, and define the normal vector of the target ball surface based on the laser tracker model;

[0008] S2: Establish a cylindrical geometric model of the robotic arm linkage, and based on the occlusion relationship, construct a system-wide visibility function for the probability that at least one target ball is visible;

[0009] S3: The target ball installation position optimization problem is modeled as a multi-objective optimization problem under a continuous-time optimal control framework. The objective function is constructed based on visibility, incident angle, constraints, and distance between target balls. The optimization problem is then transformed into an optimal control problem.

[0010] S4: Solve the optimal control problem based on the Pontryagin maximum principle, construct the Hamiltonian function, and obtain the necessary conditions for optimality;

[0011] S5: Solve the optimal control problem based on Hamilton-Jacobi-Bellman equations and adaptive dynamic programming. Use neural networks to approximate the value function, design an evaluation-execution network architecture, and use integral reinforcement learning and experience playback for the solution.

[0012] S6: The selection of the target ball mounting link is a discrete variable. A binary indicator variable is introduced to indicate whether the target ball is mounted on the corresponding link. Continuous relaxation and entropy regularization are applied to the binary variable, and the regularization coefficient is adjusted by annealing strategy. An alternating optimization algorithm and gradient projection method are designed to update the discrete variable and obtain the target ball mounting position.

[0013] Further, in step S1, a serial robotic arm system considering n degrees of freedom is established, the joint angular trajectory is defined, and the pose of each link is calculated through a homogeneous transformation matrix; the position of the target ball in the local coordinate system of the link is parameterized, and a decision variable vector containing the position of the target ball in the local coordinate system of the link and the link number to which it belongs is defined; a fixed laser tracker model is established, the normal vector of the target ball surface is defined, and the constraint condition that the laser incident angle must be less than the maximum allowable value is constructed.

[0014] Further, in step S2, based on the constructed cylindrical geometric model of the robotic arm link, the shortest distance between the ray and the link axis is defined, and a smooth approximate indicator function of the occlusion model is constructed; by combining the visibility function and the overall system visibility coverage, the overall system visibility function of the probability that at least one target ball is visible is obtained.

[0015] Furthermore, in step S3, four performance indicators are defined: maximizing visibility coverage, minimizing incident angle deviation, minimizing constraint violation penalty, and homogenizing the distance between target balls. The objective function is constructed through weighted synthesis. Physical constraints on target ball installation are introduced, including geometric feasibility constraints, minimum spacing constraints, workspace boundary constraints, and full-process visibility guarantee constraints. The optimization problem is transformed into an optimal control problem with path constraints. Virtual control inputs and virtual dynamic systems are introduced, and Bolza-type performance indicators are defined.

[0016] Further, in step S4, the necessary conditions for optimality include the state equation, costate equation, control optimality condition, and cross-section condition; expand the costate equation, calculate the gradient of the objective function with respect to the target ball position parameters, including the gradients of the visibility function, occlusion function, and incident angle term; and use the logarithmic obstacle function to handle inequality path constraints.

[0017] Furthermore, a system analysis is conducted to prove the monotonic convergence of the strategy iteration, the existence of the optimal solution, the error bound of the neural network approximation, the lower bound of the visibility coverage, the boundedness of the smooth approximation error, and the computational complexity.

[0018] Secondly, the present invention also provides a multi-objective optimization device for the installation position of a target ball on a robotic arm based on optimal control, comprising a memory and one or more processors, wherein the memory stores executable code, and when the processor executes the executable code, it implements the aforementioned multi-objective optimization method for the installation position of a target ball on a robotic arm based on optimal control.

[0019] Thirdly, the present invention also provides a computer-readable storage medium having a program stored thereon, which, when executed by a processor, implements the aforementioned multi-objective optimization method for the installation position of a target ball in a robotic arm based on optimal control.

[0020] Fourthly, the present invention also provides a computer program product, including a computer program, which, when executed by a processor, implements the aforementioned multi-objective optimization method for the installation position of a target ball in a robotic arm based on optimal control.

[0021] The beneficial effects of this invention are:

[0022] (1) The method of this invention rigorously models the target ball installation position optimization problem as a multi-objective optimization problem under a continuous-time optimal control framework. It formally describes the target ball visibility (anti-occlusion) and laser incident angle limitation during the entire motion process of the robotic arm through state-dependent inequality constraints, providing a theoretical guarantee for the continuity of the measurement chain. Compared with traditional heuristic or simple geometric analysis methods, the method of this invention can systematically handle dynamic occlusion relationships and incident angle constraints under complex trajectories, ensuring that at least one target ball can always be observed by the laser tracker during the entire task execution process, thereby ensuring the continuity and reliability of the high-precision calibration and online compensation data chain.

[0023] (2) The method of this invention unifies four mutually constraining objectives—maximizing visibility coverage, minimizing incident angle deviation, minimizing constraint violation penalties, and homogenizing the distance between target balls—within a continuous-time optimal control framework through a weighted strategy and constraint relaxation techniques, achieving multi-objective system collaborative optimization. By adjusting the weight coefficients, various indicators can be flexibly balanced according to actual engineering needs, avoiding the shortcomings of traditional methods where single-objective optimization leads to the deterioration of other performance aspects. Theoretical analysis provides a theoretical lower bound for visibility coverage, providing a clear performance benchmark for engineering acceptance.

[0024] (3) To address the difficulty of analytically solving the Hamilton-Jacobi-Bellman equation in optimal control problems, this invention employs adaptive dynamic programming and neural network approximation of the value function, designing a dual-network architecture of evaluation and execution. Combining integral reinforcement learning avoids complex gradient calculations, and acceleration of convergence is achieved through experience replay and batch updates, reducing computational complexity. This intelligent solution strategy can improve optimization efficiency while ensuring solution quality, making it suitable for rapid layout design of complex trajectories in practical industrial scenarios.

[0025] (4) To address the complex problem of hybrid optimization involving the target ball installation position (continuous variable) and the selection of the installation link (discrete variable), this invention employs an alternating optimization algorithm combining continuous relaxation with entropy regularization and annealing strategies. By introducing a binary indicator variable and relaxing it to a continuous variable, and adding an entropy regularization term and an integer penalty term, the regularization coefficient is gradually adjusted using the annealing strategy, thus achieving a joint and effective solution for both continuous and discrete variables. In the alternating optimization framework, the discrete variable update uses gradient projection and simplex projection, ensuring the feasibility and integer convergence of the decision variables, providing an operable optimization scheme for the installation position and link selection in practical engineering. Attached Figure Description

[0026] 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 some embodiments of the present invention. For those skilled in the art, other drawings can be obtained based on these drawings without creative effort.

[0027] Figure 1 This is a flowchart illustrating the overall system architecture of the method described in this invention.

[0028] Figure 2 This is a schematic diagram of the n-degree-of-freedom robotic arm system in an embodiment of the present invention.

[0029] Figure 3 This is a schematic diagram of the target ball mounting position and the robotic arm linkage structure system in an embodiment of the present invention.

[0030] Figure 4 This is a schematic diagram illustrating the geometric relationship between the laser tracker and the incident angle of the target ball in an embodiment of the present invention.

[0031] Figure 5 This is a schematic diagram of the cylindrical model of the robotic arm linkage and the determination of light occlusion in an embodiment of the present invention.

[0032] Figure 6 This is a schematic diagram of the evaluation-execution network structure based on adaptive dynamic programming in an embodiment of the present invention.

[0033] Figure 7 This is a schematic diagram of the convergence curve of the comprehensive objective function in an embodiment of the present invention.

[0034] Figure 8 This is a schematic diagram of the visibility coverage convergence curve in an embodiment of the present invention.

[0035] Figure 9 This is a schematic diagram of the Bellman error (HJB residual) in an embodiment of the present invention.

[0036] Figure 10 This is a schematic diagram illustrating the relationship between the approximation error of the value function and the network size in an embodiment of the present invention.

[0037] Figure 11 This is a schematic diagram showing the comparison results before and after the incident angle optimization in an embodiment of the present invention.

[0038] Figure 12 This is a schematic diagram of the strategy iteration convergence process in an embodiment of the present invention.

[0039] Figure 13 This is a schematic diagram of the smooth approximation error of the occlusion model in an embodiment of the present invention.

[0040] Figure 14This is a schematic diagram of the linkage selection relaxation variable annealing process in an embodiment of the present invention.

[0041] Figure 15 This is a diagram illustrating the overall performance comparison.

[0042] Figure 16 The present invention provides a structural diagram of a multi-objective optimization device for the installation position of a target ball in a robotic arm based on optimal control. Detailed Implementation

[0043] To make the objectives, technical solutions, and advantages of the embodiments of the present invention clearer, 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, 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.

[0044] This embodiment provides a multi-objective optimization method for the installation position of a target ball in a robotic arm based on optimal control, such as... Figure 1 As shown, the specific steps include:

[0045] S1: Establish the kinematic model of the robotic arm, where the robotic arm is as follows: Figure 2 The n-DOF robotic arm system structure is shown below. Joint angular trajectories are defined, and the pose of each link is calculated using a homogeneous transformation matrix. The target ball mounting position and the robotic arm link structure are as follows: Figure 3 As shown, the position of the target ball in the local coordinate system of the link is parameterized, and the position expression and velocity Jacobian matrix of the target ball in the base coordinate system are established; a laser tracker model is established, defining the direction vector, distance, and unit direction vector from the laser tracker to the target ball, defining the surface normal vector of the target ball, and constructing the laser incident angle constraint. The geometric relationship between the incident angle of the laser tracker and the target ball is as follows. Figure 4 As shown; the specific steps are as follows:

[0046] S11: Establish the dynamic model of the robotic arm system and define the joint angle vectors. , Let be the number of joints, and the trajectory of each joint over time be... , ,in The total execution time of the task.

[0047] The poses of each link are obtained recursively through the homogeneous transformation matrix:

[0048]

[0049] in For the first The DH transformation matrix of each joint:

[0050]

[0051] in The length of the link. The link twist angle, This is the link offset.

[0052] Extracting the rotation matrix from the transformation matrix and position vector :

[0053]

[0054] in, Denotes a special orthogonal group in three dimensions, defined as all groups whose determinant is 1. The set of orthogonal matrices, used in this paper to define the matrices representing the link attitude. Satisfy the geometric constraints for rotation of a three-dimensional rigid body.

[0055] S12: Install on the robotic arm body The position of the target ball in the local coordinate system of the link is parameterized. Let the first target ball be... The target ball is installed at the first On each link, its position in the local coordinate system of the link is: Define the decision variable vector:

[0056]

[0057] in, This represents the total number of links in the robotic arm (i.e., the number of degrees of freedom of the system). This represents the set of link numbers that can be optionally installed on the target ball.

[0058] No. The position of each target ball in the base coordinate system changes with the joint angle as follows:

[0059]

[0060] in Indicates at time , No. The rotation matrix of the local coordinate system of each link relative to the base coordinate system; Indicates at time , No. The position vector of the origin of the local coordinate system of each link in the base coordinate system.

[0061] The velocity of the target ball in the base coordinate system is:

[0062]

[0063] in For the first The velocity Jacobian matrix of the target ball relative to the joint angle:

[0064]

[0065] in For the first Each joint axis direction, For the first Standard basis vectors.

[0066] S13: Establish a laser tracker model, and fix the laser tracker in a fixed position in the workspace. Defined from laser tracker to the first The direction vectors of the target balls:

[0067]

[0068] The distance from the laser tracker to the target ball is:

[0069]

[0070] The unit direction vector is:

[0071]

[0072] Define the normal vector of the target sphere's surface. Let the first normal vector be... The installation normal vector of each target ball in the local coordinate system of the link is: Then its expression in the base coordinate system is:

[0073]

[0074] The laser incident angle is defined as the angle between the laser direction and the normal vector of the target sphere:

[0075]

[0076] The incident angle constraint requirement is as follows:

[0077]

[0078] in, This represents the maximum permissible angle of incidence. The angle of incidence constraint is equivalent to:

[0079]

[0080] S2: Establish a cylindrical geometric model of the robotic arm link, define the shortest distance between the light ray and the link axis, construct a smooth approximate indicator function for the occlusion model, and determine the occlusion of the robotic arm link cylindrical model and the light occlusion. Figure 5 As shown; by combining the overall visibility function and the overall system visibility coverage, the overall system visibility function is obtained, which represents the probability that at least one target ball is visible; the specific steps are as follows:

[0081] S21: Simplify each link into a cylindrical geometric model, the first... The cylindrical parameters of each link are based on the axis starting point. End point of axis and radius .

[0082] Define a parametric ray:

[0083]

[0084] Parametric representation of the connecting rod axis:

[0085]

[0086] The shortest distance between the ray and the axis of the connecting rod is:

[0087]

[0088] The occlusion conditions are ,in For safety margin.

[0089] S22: Define the first The target ball at time Occlusion indicator function:

[0090]

[0091] Introduce a smooth approximation function to replace the discontinuous indicator function:

[0092]

[0093] in For smoothing parameters.

[0094] Definition of the first The target ball at time Visibility function:

[0095]

[0096] in It is a smooth step function. For parameters.

[0097] The overall system visibility function is defined as the probability that at least one target ball is visible:

[0098]

[0099] The system's visibility coverage throughout the entire task is:

[0100]

[0101] S3: The target ball installation position optimization problem is modeled as a multi-objective optimization problem under a continuous-time optimal control framework. Four performance indicators are defined: maximizing visibility coverage, minimizing incident angle deviation, minimizing constraint violation penalty, and homogenizing the distance between target balls. The objective function is constructed through weighted synthesis. Physical constraints on target ball installation are introduced, including geometric feasibility constraints, minimum spacing constraints, workspace boundary constraints, and full-process visibility guarantee constraints. The optimization problem is transformed into an optimal control problem with path constraints. Virtual control inputs and a virtual dynamic system are introduced, and Bolza-type performance indicators are defined. The specific steps are as follows:

[0102] S31: Define three optimization objectives:

[0103] Objective 1: Maximize visibility coverage.

[0104]

[0105] Objective 2: Minimize the incident angle deviation.

[0106]

[0107] Objective 3: Minimize penalties for constraint violations.

[0108]

[0109] Introducing a regularization term for the distance between target balls to achieve uniformity:

[0110]

[0111] in, This indicates the initial moment when the robotic arm executes its task trajectory, and is used to assess the spatial distribution distance between target balls in the initial pose.

[0112] The weighted comprehensive objective function is:

[0113]

[0114] in The weighting coefficients satisfy the normalization condition. .

[0115] S32: The physical constraints for the target ball installation are as follows:

[0116] Geometric feasibility constraints:

[0117]

[0118] in For the first The center of the axis of each link This is the maximum permissible installation radius from the outer surface of the connecting rod to the axis.

[0119] Minimum spacing constraint between target balls:

[0120]

[0121] in This is the minimum allowable spacing.

[0122] Workspace boundary constraints for the target ball position:

[0123]

[0124] in, This indicates that the target ball moves along the local coordinate system of the connecting rod. Axis (corresponding to respectively) The position coordinate components of ) The corresponding coordinates in the local coordinate system of the link Maximum permissible installation offset distance in the axial direction.

[0125] Full-process visibility guarantee constraint:

[0126]

[0127] in The maximum allowable proportion of visibility discontinuities.

[0128] S33: Define the system state as the target ball configuration parameters. In addition, auxiliary state variables are introduced. Considering the need to handle the integral form of the objective function within the optimization framework, auxiliary states are introduced:

[0129]

[0130]

[0131]

[0132] Define extended state vector .Notice These are static decision variables, i.e. .

[0133] The state equation of the system is:

[0134]

[0135] in:

[0136]

[0137] S34: Introducing virtual control input This is applied to the target ball's position parameters, allowing the optimization process to be achieved through time evolution. Define a virtual dynamic system:

[0138]

[0139] in To optimize time for virtualization, To optimize the termination time.

[0140] Redefine the performance metrics as Bolza type:

[0141]

[0142] The terminal cost is:

[0143]

[0144] The operational cost is the control energy regularization term:

[0145]

[0146] in To control the weight matrix, This is the obstacle penalty function.

[0147] The optimal control problem can be formally expressed as:

[0148]

[0149] S4: Based on the Pontryagin maximum principle, derive the theoretical analytical solution to the optimal control problem, construct the Hamiltonian function, and derive the necessary conditions for optimality, including the state equation, co-state equation, control optimality condition, and cross-section condition; expand the co-state equation, calculate the gradient of the objective function with respect to the target ball position parameters, including the gradients of the visibility function, occlusion function, and incident angle term; and use the logarithmic obstacle function to handle inequality path constraints.

[0150] S4 specifically includes the following steps:

[0151] S41: Define costate variables Construct Hamiltonian functions:

[0152]

[0153] in, Costate variables are used to introduce the dynamic constraints of the system state equations into the optimal control framework, characterizing the sensitivity of the optimal cost to small changes in the system state.

[0154] Expanded to:

[0155]

[0156] According to Pontryagin's maximum principle, the optimal solution is... The following conditions must be met:

[0157] Equations of state:

[0158]

[0159] Costate equations:

[0160]

[0161] Control optimality conditions:

[0162]

[0163] This yields the optimal virtual control input:

[0164]

[0165] Cross section condition:

[0166]

[0167] S42: The gradient of the objective function with respect to the target ball's position parameters can be calculated using the chain rule. For the visibility coverage objective:

[0168]

[0169] in:

[0170]

[0171] The gradient of the visibility function for a single target ball is:

[0172]

[0173] in, Indicated by Let be a smooth step function of the independent variable. Let be the derivative of the smooth step function with respect to its independent variable. The cosine threshold corresponding to the target ball incident angle constraint.

[0174] The gradient of the occlusion function is:

[0175]

[0176] in, A smoothing parameter introduced in the occlusion model to control the smoothness of the approximate indicator function.

[0177] The partial derivative of the shortest distance between the ray and the connecting rod with respect to the target ball's position is:

[0178]

[0179] in The parameter values ​​are used to achieve the shortest distance.

[0180] The gradient of the incident angle term is:

[0181]

[0182] S43: For inequality path constraints The inequality constraints are transformed into an unconstrained problem using a logarithmic barrier function:

[0183]

[0184] in Here are the barrier parameters. The modified Hamiltonian function is as follows:

[0185]

[0186] S5: To address the difficulty of directly solving the optimal control problem derived above analytically, an approximate numerical solution for the optimal control problem is proposed based on the Hamilton-Jacobi-Bellman equations and adaptive dynamic programming. A neural network is used to approximate the value function, and a judge-execution network architecture is designed. Integral reinforcement learning and experience replay are employed for the solution. The specific steps are as follows:

[0187] S51: Establish the Hamilton-Jacobi-Bellman equations. Define the optimal value function. From state The minimum cost from departure to arrival can be expressed as:

[0188]

[0189] According to the principle of dynamic programming, the optimal value function satisfies the Hamilton-Jacobi-Bellman equation:

[0190]

[0191] Optimal control Substituting, we get:

[0192]

[0193] in This is an operational cost term that includes path constraint obstacle penalties and is independent of control.

[0194] The terminal boundary conditions are:

[0195]

[0196] For a sufficiently long optimization time Consider the steady-state value function Satisfies the steady-state Hamilton-Jacobi-Bellman equations:

[0197]

[0198] in This is the discount factor.

[0199] The corresponding steady-state optimal control law is:

[0200]

[0201] S52: Since the Hamilton-Jacobi-Bellman equations are generally difficult to solve analytically, an adaptive dynamic programming method is used for approximate solutions. The network structure is as follows: Figure 6 As shown. The value function is approximated using a neural network:

[0202]

[0203] in To output weights, For hidden layer weights, For bias, For activation function, To approximate the error, This represents the number of hidden layer neurons.

[0204] The gradient estimate of the value function is:

[0205]

[0206] in .

[0207] The approximate optimal control law is:

[0208]

[0209] S53: Design and implement a direct output control strategy for the network:

[0210]

[0211] in , , To execute network parameters, The number of hidden neurons in the execution network.

[0212] Define Hamilton-Jacobi-Bellman residuals (Bellman error):

[0213]

[0214] The gradient descent method is used to update the weights of the evaluation network and the execution network to minimize the Bellman error.

[0215]

[0216] in, To evaluate the learning rate of a network.

[0217] Hidden layer weight update:

[0218]

[0219] in .

[0220] Bias update:

[0221]

[0222] The policy error of the execution network is defined as:

[0223]

[0224] Perform network weight updates:

[0225]

[0226] in To implement the network learning rate, This is the weight decay coefficient.

[0227] S54: Employs integral reinforcement learning method, time interval The Hamilton-Jacobi-Bellman equation is integrated above:

[0228]

[0229] Define timing difference error:

[0230]

[0231] based on The parameters of the value function network are obtained by least squares optimization of the sampled data.

[0232]

[0233] in For regularization parameters, It is the Frobenius norm.

[0234] Given an initial feasible strategy According to the convergence criterion or Solve and improve the strategy evaluation steps:

[0235]

[0236]

[0237] in, To control the convergence tolerance threshold of the strategy, This is the convergence tolerance threshold for the value function.

[0238] An experience revisiting mechanism is introduced to accelerate convergence and improve learning stability. The experience pool is defined as follows:

[0239]

[0240] in, The total number of data samples stored. To accumulate costs, This indicates the state at the next moment.

[0241] During batch updates, random sampling is performed from the experience pool:

[0242]

[0243] in, The size of the sampling batch extracted during each batch update.

[0244] The information matrix is ​​constructed as follows:

[0245]

[0246] The target vector is constructed as follows:

[0247]

[0248] The weight update method is as follows:

[0249]

[0250] S6: Perform discrete optimization of link selection, introduce binary indicator variables, adopt continuous relaxation and entropy regularization, and use annealing strategy to adjust the regularization coefficient. Design an alternating optimization algorithm and gradient projection method to update discrete variables.

[0251] S6 specifically includes the following steps:

[0252] S61: Selection of target ball mounting linkage For discrete variables, introduce binary indicator variables. Indicates whether the k-th target ball is installed on the first target ball. On each link:

[0253]

[0254] The coupling relationship between continuous position parameters and discrete selection is as follows:

[0255]

[0256] S62: Relaxing a binary variable into a continuous variable Introducing entropy regularization promotes convergence to integer solutions:

[0257]

[0258] in, To avoid logarithmic singularity, a small constant is used.

[0259] The corrected objective function is:

[0260]

[0261] in and is the regularization coefficient. The third term is the integer penalty term, when... The value is zero when taken as an integer.

[0262] Use annealing strategy to gradually reduce and increase :

[0263]

[0264] S63: Employing an alternating direction optimization strategy: fixing discrete variables Using adaptive dynamic programming to solve continuous variables The optimal control problem; fixed continuous variables For discrete variables Relaxation optimization is performed using gradient projection to update the settings, as follows:

[0265]

[0266] in To reach the simplex The projection.

[0267] The simplex projection algorithm is used to ensure the sum is 1 and to apply non-negativity constraints. The simplex projection algorithm is as follows: [The algorithm is described in the original text, but the provided text is incomplete and cannot be translated accurately.] Arranged in descending order The calculation is as follows:

[0268]

[0269]

[0270]

[0271] S7: Perform system analysis to prove the monotonic convergence of strategy iteration, the existence of the optimal solution, the error bound of neural network approximation, the lower bound of visibility coverage, the boundedness of smooth approximation error, and the computational complexity.

[0272] S7 specifically includes the following steps:

[0273] S71: Definition From the strategy evaluation equation, we have:

[0274]

[0275]

[0276] From strategy improvement steps We can obtain:

[0277]

[0278] in , The strategy iteration algorithm has monotonically convergent properties, and the value function sequence satisfies And converges to the optimal value function .

[0279] Target ball installation area yes The non-empty compact set in the objective function exist Upper and lower semi-continuous and constraint functions Since the problem is continuous, according to Weierstrass's extremum theorem, the optimal control problem has an optimal solution under compact conditions.

[0280] S72: Setting Function In close proximity If it is continuous above, then for any There exist neural network parameters such that:

[0281]

[0282] set up If the approximation error is the value function, then the control strategy error is:

[0283]

[0284] The suboptimal bound of the objective function is:

[0285]

[0286] in Let be the Lipschitz constant of the objective function.

[0287] Taking into account the approximation error of the neural network Smoothing approximation error and discrete relaxation error The suboptimal bound of the optimal solution is:

[0288]

[0289] in The error terms are constants related to the problem size, and can be reduced by increasing the network size, increasing the smoothing parameter, and increasing the number of annealing iterations, respectively.

[0290] S73: Define the optimal solution as If the visibility weights in the objective function Constraints Feasibility conditions For all If true, then the lower bound of visibility coverage is given as:

[0291]

[0292] S74: When hour, The corresponding component is 0, and the smooth approximation is .when The same logic applies.

[0293] For any ,have:

[0294]

[0295] when hour, For all Summing the possible occlusion links yields a smooth approximation function. With the true indicator function The upper bound of the error between them.

[0296] S75: The computational complexity analysis of the algorithm is as follows:

[0297] The single-step update of value function approximation is The single-step update of the strategy evaluation is as follows: The single-step calculation for occlusion detection is as follows: Experience pool sampling and batch updates are The simplex projection of discrete variables is The complexity of each outer iteration is O(n). ,in The number of iterations for the inner strategy. This represents the number of discrete points in the trajectory over time.

[0298] Based on simulation and experimental results, Figure 7 The convergence curves of the combined objective function are shown, illustrating the changes in the objective function values ​​of the three optimization methods—OC-ADP, Genetic Algorithm (GA), and Particle Swarm Optimization (PSO)—with the number of iterations. The proposed method exhibits the fastest convergence speed and the lowest final convergence value (approximately 2.15), demonstrating superior optimization capability in multi-objective integrated optimization. Figure 8 The visibility coverage convergence curves are shown: the convergence process of the visibility coverage Γ(χ) of the three methods with the number of iterations is compared. The method in this paper finally achieves a coverage of 99.3%, which is significantly better than GA (96.2%) and PSO (97.5%), verifying the effectiveness of the method in target ball visibility optimization. Figure 9 The Bellman error (HJB residual) is shown using logarithmic coordinates, illustrating the convergence of the Bellman error during the ADP algorithm iteration process. The error exhibits an exponential decay trend, eventually converging to near zero, verifying the numerical stability of the value function iterative solution to the HJB equation. Figure 10 The relationship between the approximation error of the value function and the network size is shown: The relationship between the number of hidden neurons Nc in the neural network and the approximation error of the value function is illustrated. The error decreases at a rate of O(Nc^(-0.85)) as Nc increases, indicating that approximating the value function using a neural network has good convergence. Figure 11 Comparison of incident angles before and after optimization: The upper and lower sub-figures show the curves of the incident angles of the three target balls changing over time before and after optimization. Before optimization, the incident angles frequently exceeded the 30° constraint threshold, while after optimization, all incident angles met the constraint conditions, verifying the effectiveness of the constraint handling mechanism. Figure 12 The convergence process of the policy iteration algorithm is illustrated: the convergence process of the policy error and value function error with the number of iterations is shown. Both error curves decay exponentially and eventually fall below the convergence tolerance (0.05), verifying the convergence of the policy iteration algorithm. Figure 13 To address the smoothing approximation error of the occlusion model: the smoothing parameter β is displayed using double logarithmic coordinates. oThe relationship between the error and the smooth approximation error of the occlusion model. The error varies with β. o Increased to O(1 / β) o The rate decrease verifies the effectiveness of the smooth approximation method. Figure 14 Annealing process for selecting relaxation variables for the links: This demonstrates the selection of relaxation variables δ for the six links of target ball 1. 1j The evolution process with the number of annealing iterations. As the iterations proceed, the relaxation variables gradually converge from continuous values ​​to discrete values ​​(link 3 is selected), verifying the correctness of the discrete variable continuous relaxation method. Figure 15 To compare the overall performance of the three methods, a bar chart was used to compare their overall performance across five dimensions: objective function value, coverage, maximum incident angle, minimum spacing, and computation time. The method presented in this paper demonstrates the best performance in all three core metrics: objective function value, coverage, and maximum incident angle, while maintaining a moderate computation time, thus exhibiting the best overall performance.

[0299] This invention's method models the target ball layout optimization problem as an optimal control problem and solves it. This method ensures that the target ball system's visibility to the laser tracker meets preset requirements when the robotic arm executes a predetermined trajectory. Theoretical analysis and simulations show that this method can significantly improve the target ball visibility coverage in complex trajectory tasks and strictly satisfy the laser incident angle constraint, thereby guaranteeing the continuity and reliability of the high-precision calibration and online compensation data link. The occlusion model uses a smooth approximation and provides an upper bound on the error. When the smoothing parameter approaches infinity, it can approximately approach the true function, balancing computational feasibility and model accuracy.

[0300] This invention addresses path constraints (geometric feasibility, minimum spacing, workspace boundaries, and full-process visibility) through optimal control theory, derives necessary conditions for optimality using Pontryagin's maximum principle, improves solution efficiency through adaptive dynamic programming and integral reinforcement learning, and handles practical engineering choices through a mixed-variable optimization strategy. All aspects are supported by rigorous theoretical analysis: monotonically converging policy iterations, existence of optimal solutions (Weierstrass's extreme value theorem), neural network approximation error bounds, boundedness of smooth approximation errors, and computational complexity analysis. This results in a solution that combines high performance, high reliability, and a solid theoretical foundation, providing a complete theoretical-to-practice solution for industrial applications. Simulations and experiments verify its effectiveness in improving the usability and accuracy of measurement systems.

[0301] This invention addresses the optimization problem of multiple target spheres in practical engineering, considering practical factors such as the cylindrical geometry of the connecting rod, the calculation of the shortest distance between the ray and the axis, and the incident angle constraint, thus possessing direct engineering operability. Simulation and experimental results show that by introducing virtual control input and a virtual dynamic system, the static optimization problem is transformed into a dynamic optimal control problem. The numerical stability and convergence of the optimization process are guaranteed by utilizing the control energy regularization term and obstacle penalty function in the Bolza-type performance index. The effectiveness of this method has been verified on a typical industrial robotic arm model, providing a systematic theoretical framework and operable solution scheme for the target sphere layout design of a laser tracker-assisted robotic arm calibration system.

[0302] Corresponding to the aforementioned embodiment of a multi-objective optimization method for the installation position of a robotic arm target ball based on optimal control, the present invention also provides an embodiment of a multi-objective optimization device for the installation position of a robotic arm target ball based on optimal control.

[0303] See Figure 16 The present invention provides a multi-objective optimization device for the installation position of a target ball on a robotic arm based on optimal control, comprising a memory and one or more processors. The memory stores executable code, and when the processor executes the executable code, it is used to implement a multi-objective optimization method for the installation position of a target ball on a robotic arm based on optimal control as described in the above embodiment.

[0304] The embodiment of the multi-objective optimization device for the installation position of a target ball in a robotic arm based on optimal control provided by this invention can be applied to any device with data processing capabilities, such as a computer. The device embodiment can be implemented through software, hardware, or a combination of both. Taking software implementation as an example, as a logical device, it is formed by the processor of any data processing device loading the corresponding computer program instructions from non-volatile memory into memory for execution. From a hardware perspective, such as... Figure 16 The diagram shown is a hardware structure diagram of any device with data processing capabilities, which is the multi-objective optimization device for the installation position of a robotic arm target ball based on optimal control provided by the present invention. (Except for...) Figure 16 In addition to the processor, memory, network interface, and non-volatile memory shown, any data processing device in the embodiment may also include other hardware depending on the actual function of the data processing device, which will not be described in detail here.

[0305] The specific implementation process of the functions and roles of each unit in the above device can be found in the implementation process of the corresponding steps in the above method, and will not be repeated here.

[0306] For the device embodiments, since they basically correspond to the method embodiments, the relevant parts can be referred to in the description of the method embodiments. The device embodiments described above are merely illustrative. The units described as separate components may or may not be physically separate, and the components shown as units may or may not be physical units, that is, they may be located in one place or distributed across multiple network units. Some or all of the modules can be selected to achieve the purpose of the present invention according to actual needs. Those skilled in the art can understand and implement this without creative effort.

[0307] This invention also provides a computer-readable storage medium storing a program that, when executed by a processor, implements a multi-objective optimization method for the installation position of a target ball on a robotic arm based on optimal control, as described in the above embodiments.

[0308] The computer-readable storage medium can be an internal storage unit of any data processing device described in any of the foregoing embodiments, such as a hard disk or memory. The computer-readable storage medium can also be an external storage device of any data processing device, such as a plug-in hard disk, smart media card (SMC), SD card, flash card, etc., equipped on the device. Furthermore, the computer-readable storage medium can include both internal storage units and external storage devices of any data processing device. The computer-readable storage medium is used to store the computer program and other programs and data required by the data processing device, and can also be used to temporarily store data that has been output or will be output.

[0309] The present invention also provides a computer program product, including a computer program, which, when executed by a processor, implements the aforementioned multi-objective optimization method for the installation position of a target ball in a robotic arm based on optimal control.

[0310] Finally, it should be noted that the above embodiments are only used to illustrate the technical solutions of the present invention, and not to limit them; although the present invention has been described in detail with reference to the foregoing embodiments, those skilled in the art should understand that modifications can still be made to the technical solutions described in the foregoing embodiments, or equivalent substitutions can be made to some or all of the technical features; and these modifications or substitutions do not cause the essence of the corresponding technical solutions to deviate from the scope of the technical solutions of the embodiments of the present invention.

Claims

1. A multi-objective optimization method for the installation position of a target ball in a robotic arm based on optimal control, characterized in that, Specifically, the following steps are included: S1: Obtain the angular trajectory of each joint of the robotic arm system and the pose of each link, define the position of the target ball with respect to the link and the decision variable vector, and define the normal vector of the target ball surface based on the laser tracker model; S2: Establish a cylindrical geometric model of the robotic arm linkage, and based on the occlusion relationship, construct a system-wide visibility function for the probability that at least one target ball is visible; S3: The target ball installation position optimization problem is modeled as a multi-objective optimization problem under a continuous-time optimal control framework. An objective function is constructed based on visibility, incident angle, constraints, and the distance between target balls. The optimization problem is then transformed into an optimal control problem, specifically: defining four performance indices: maximizing visibility coverage, minimizing incident angle deviation, minimizing constraint violation penalties, and homogenizing the distance between target balls. These indices are then used to construct the objective function through weighted synthesis. Physical constraints on target ball installation are introduced, including geometric feasibility constraints, minimum spacing constraints, workspace boundary constraints, and full-process visibility guarantee constraints. The optimization problem is further transformed into an optimal control problem with path constraints, introducing virtual control inputs and a virtual dynamic system, and defining Bolza-type performance indices. S4: Solve the optimal control problem based on the Pontryagin maximum principle, construct the Hamiltonian function, and obtain the necessary conditions for optimality; the necessary conditions for optimality include the state equation, costate equation, control optimality condition, and cross-sectional condition; expand the costate equation, calculate the gradient of the objective function with respect to the target ball position parameters, including the gradients of the visibility function, occlusion function, and incident angle term; use the logarithmic obstacle function to handle inequality path constraints; S5: Solving the optimal control problem based on the Hamilton-Jacobi-Bellman equation and adaptive dynamic programming. Specifically, the Hamilton-Jacobi-Bellman equation is approximated using adaptive dynamic programming. A neural network is used to approximate the value function, and an evaluation-execution network architecture is designed. Integral reinforcement learning is used to integrate the Hamilton-Jacobi-Bellman equation. Based on the time-series difference error, the network parameters of the value function are optimized using least squares. An empirical backtracking mechanism is introduced to accelerate convergence and improve learning stability. The evaluation-execution network architecture first designs the execution network to directly output the control strategy based on the execution network parameters. The Hamilton-Jacobi-Bellman residual is defined as the Bellman error. The gradient descent method is used to update the weights of the evaluation network and the execution network to minimize the Bellman error. S6: The selection of the target ball mounting link is a discrete variable. A binary indicator variable is introduced to indicate whether the target ball is mounted on the corresponding link. Continuous relaxation and entropy regularization are applied to the binary variable, and the regularization coefficient is adjusted by annealing strategy. An alternating optimization algorithm and gradient projection method are designed to update the discrete variable and obtain the target ball mounting position.

2. The multi-objective optimization method for the installation position of a target ball in a robotic arm based on optimal control according to claim 1, characterized in that, In step S1, establish considerations n A serial robotic arm system with multiple degrees of freedom is constructed. The joint angular trajectories are defined, and the pose of each link is calculated through a homogeneous transformation matrix. The position of the target ball in the local coordinate system of the link is parameterized, and a decision variable vector containing the position of the target ball in the local coordinate system of the link and the link number to which it belongs is defined. A fixed laser tracker model is established, the normal vector of the target ball surface is defined, and the constraint condition that the laser incident angle must be less than the maximum allowable value is constructed.

3. The multi-objective optimization method for the installation position of a target ball in a robotic arm based on optimal control according to claim 1, characterized in that, In step S2, based on the cylindrical geometric model of the constructed robotic arm link, the shortest distance between the ray and the link axis is defined, and a smooth approximate indicator function of the occlusion model is constructed; by combining the visibility function and the overall system visibility coverage, the overall system visibility function of the probability that at least one target ball is visible is obtained.

4. A multi-objective optimization device for the installation position of a target ball in a robotic arm based on optimal control, comprising a memory and one or more processors, wherein the memory stores executable code, characterized in that, When the processor executes the executable code, it implements a multi-objective optimization method for the installation position of a robotic arm target ball based on optimal control, as described in any one of claims 1-3.

5. A computer-readable storage medium having a program stored thereon, characterized in that, When the program is executed by the processor, it implements a multi-objective optimization method for the installation position of a target ball in a robotic arm based on optimal control, as described in any one of claims 1-3.

6. A computer program product, comprising a computer program, characterized in that, When the computer program is executed by the processor, it implements a multi-objective optimization method for the installation position of a target ball in a robotic arm based on optimal control, as described in any one of claims 1-3.

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