Track planning method for assembling and calibrating parallel robot based on novel multi-target adaptive optimization algorithm

Abstract

The invention discloses an installation and calibration parallel robot trajectory planning method based on a novel multi-target adaptive optimization algorithm, and the method comprises the following steps: 1) constructing an installation and calibration parallel robot coordinate trajectory by taking a Clamped quintic B spline curve as an interpolation function; 2) based on time, impact and energy consumption dimensions, establishing an assembly and calibration platform multi-target trajectory optimization model under a multi-constraint condition; 3) performing iterative solution on the multi-target trajectory optimization model of the installation and calibration platform to obtain an optimized trajectory set of the installation and calibration parallel robot; and 4) evaluating the optimal track set by using a multi-target evaluation method to obtain an optimal track of a group of calibration parallel robots. The multi-target trajectory planning method is designed for the assembling and calibrating parallel robot, and a set of workpiece assembling and calibrating motion trajectories meeting the multi-target performance requirement are obtained.

Description

Technical Field

[0001] The present invention relates to the field of trajectory planning, and in particular to a trajectory planning method for an assembly and calibration parallel robot based on a novel multi-objective adaptive optimization algorithm. Background Art

[0002] During the assembly and calibration of large, heavy-loaded, precision workpieces, six-degree-of-freedom (6DOF) positional adjustments are required. This can be accomplished using either serial or parallel robots. Compared to serial robots, parallel robots offer superior load capacity and stiffness, eliminate cumulative errors, and offer improved motion accuracy. Therefore, parallel robots are often chosen for the assembly and calibration of large, heavy-loaded, precision workpieces.

[0003] The workpiece assembly and calibration task requires continuous adjustment of its six-degree-of-freedom (6DOF) position while avoiding obstacles along the path until docking and installation are complete. To achieve this, it is necessary to rationally plan the module's desired motion trajectory from the initial pose point to the target pose point, while also determining the desired motion trajectory of each drive joint of the parallel robot. This allows for effective motion control of the parallel robot in the future. Specifically, trajectory planning for the parallel robot primarily involves the workpiece's six-degree-of-freedom spatial motion trajectory, as well as the trajectories of motion parameters such as displacement, velocity, and acceleration of each drive joint.

[0004] The general robot trajectory planning method is joint space trajectory planning, that is, first the path pose points of the end effector are inversely solved into the displacement / rotation of each joint, and then the motion trajectory of each joint is planned based on the inverse solution. This process directly focuses on the displacement, velocity, acceleration and other motion parameters of the robot joint. The trajectory of the end effector needs to be obtained by performing a kinematic forward solution on the joint trajectory, so it lacks intuitiveness.

[0005] While trajectory planning is in progress, the generated trajectory needs to be optimized for performance. For the calibration motion of a workpiece, a single-objective optimization is usually performed on the total calibration time or the impact value corresponding to the motion trajectory to maximize the efficiency of the calibration motion time or optimize stability. However, there is a certain conflict between these two optimization objectives: the highest efficiency is often accompanied by a greater impact, while the lowest impact is often accompanied by a longer calibration time. Actual task requirements are often diverse, generally requiring multiple performance indicators. Therefore, single-objective trajectory optimization is difficult to meet actual task requirements, and multi-objective trajectory optimization is required to obtain a calibration motion trajectory curve that meets the actual task requirements.

[0006] Trajectory optimization requires iterative solution through intelligent optimization algorithms. Traditional particle swarm optimization algorithms leverage social and individual cognitive learning mechanisms, enabling them to more finely explore discovered high-quality areas, improving solution quality. However, this approach may lack solution diversity. The whale optimization algorithm, a relatively new intelligent optimization algorithm, is highly effective in guiding the algorithm to broadly explore the solution space due to its unique mechanism based on whale predation behavior. However, its convergence rate is relatively slow and it may require more computing resources to find the optimal solution. Summary of the Invention

[0007] The purpose of the present invention is to provide a trajectory planning method for a parallel robot based on a novel multi-objective adaptive optimization algorithm, comprising the following steps:

[0008] 1) Using the clamped quintic B-spline curve as the interpolation function, the coordinate trajectory of the parallel robot is constructed;

[0009] 2) Based on the dimensions of time, impact, and energy consumption, a multi-objective trajectory optimization model for the installation and calibration platform is established under multiple constraints;

[0010] 3) Iteratively solve the multi-objective trajectory optimization model of the assembly and calibration platform to obtain the optimized trajectory set of the assembly and calibration parallel robot;

[0011] 4) The optimal trajectory set is evaluated using a multi-objective evaluation method to obtain the optimal trajectory of a set of calibrated parallel robots.

[0012] Furthermore, the steps of constructing the coordinate trajectory of the parallel robot include:

[0013] a1) Determine the node vector u, that is:

[0014]

[0015] u i =t i-5 (i=6,···w+5)(2)

[0016] a2) Determine the control point vector Q. The steps are as follows:

[0017] a2.1) Let the domain [u k ,u n+1 ] the corresponding point P(u i )(i=5,6,…,w+5) and type value point P i-5 Overlap, we get:

[0018]

[0019] Where B j,5 (u i) is the basis function; i=5,6,...,w+5; Q i is the control point;

[0020] Basis function B i,k (u) is obtained by the following de Boor recursion formula:

[0021]

[0022] a2.2) Let B i,5 (u i )=0, simplifying formula (3) yields:

[0023]

[0024] a2.3) Convert equation (4) into matrix form to obtain:

[0025]

[0026] a2.4) Simplify formula (5) to obtain:

[0027]

[0028] a2.5) Construct the first-order and second-order derivative forms of the B-spline curve, namely:

[0029]

[0030] Where, P'(u) and P"(u) are the first-order and second-order derivative forms of the B-spline curve;

[0031] a2.6) Construct expressions for the velocity and acceleration at the start and end points of the trajectory based on the first- and second-order derivatives of the B-spline curve;

[0032] The velocity and acceleration expressions of the trajectory at the starting point are as follows:

[0033]

[0034] The velocity and acceleration expressions of the trajectory at the end point are as follows:

[0035]

[0036] a2.7) Combining equations (8) and (9), we obtain:

[0037] BQ=P * (10)

[0038]

[0039] Where: Q=[Q0 Q1 … Qw+4 ] T is the vector composed of the control points to be found, P * is the point vector containing the velocity and acceleration values ​​at the start and end points; P * =[0 0 P0 P1…P w 0 0] T ; The variables in the upper left and lower right corners of the B matrix represent the elements composed of the correlation coefficients in the expressions of the velocity and acceleration of the start and end points;

[0040] The expressions of the variables in the upper left and lower right corners of the B matrix are as follows:

[0041]

[0042] a2.8) Transform formula (10) to obtain:

[0043] Q=B -1 P * (13)

[0044] a2.9) Determine all control points using formula (13), substitute the node vector u and the control points into the general expression of the B-spline curve, and construct the coordinate trajectory of the parallel robot.

[0045] Furthermore, the objective functions of the multi-objective trajectory optimization model of the installation and calibration platform include an objective function based on time optimization, an objective function based on impact optimization, and an objective function based on energy consumption optimization.

[0046] Furthermore, the time-optimal objective function G1 is as follows:

[0047]

[0048] Where Δt i =t i -t i-1 (i=1,2,…,w) is the time interval between two adjacent type value points;

[0049] The objective function G2 based on impact optimization is as follows:

[0050]

[0051] Among them: jerk i represents the jerk of the i-driven joint during the motion process, and T represents the total time consumed by the calibration task.

[0052] The objective function G3 based on optimal energy consumption is as follows:

[0053]

[0054] Where: τ i and v i Respectively represent the driving force and speed of #i driving joint during the movement.

[0055] Furthermore, the constraints of the multi-objective trajectory optimization model of the calibration platform include joint displacement constraints, joint velocity constraints, joint acceleration constraints, and driving force constraints;

[0056] The joint displacement constraints are as follows:

[0057] s imin ≤s i ≤s imax (i=1,2,…,6) (17)

[0058] Where: s i ,s imax ,s imin are the displacement of the driven joint i and the upper and lower bounds of the allowed displacement range, respectively.

[0059] The joint velocity constraints are as follows:

[0060] |v i |≤v imax (i=1,2,…,6) (18)

[0061] Where: v i and v imax are the speed and maximum allowed speed of the driven joint i respectively.

[0062] The joint acceleration constraints are as follows:

[0063] |a i |≤a imax (i=1,2,…,6) (19)

[0064] Among them: a i and a imax are the acceleration and maximum allowed acceleration of #i driving joint respectively.

[0065] The driving force constraints are as follows:

[0066] |τ i |≤τ imax (i=1,2,…,6) (20)

[0067] Where: τ i and τ imax are the driving force and maximum allowable driving force of #i driving joint respectively.

[0068] Furthermore, the multi-objective trajectory optimization model G of the installation and calibration platform is as follows:

[0069]

[0070] Furthermore, the steps of iteratively solving the multi-objective trajectory optimization model of the installation and calibration platform include:

[0071] b1) Population POP and parameter initialization: Set the initial values ​​of each parameter in the algorithm according to the optimization requirements, and randomly initialize the population POP value;

[0072] b2) Target vector calculation and dominance comparison: Calculate the target function value corresponding to each individual in the initial population POP and determine the dominance relationship between all individuals;

[0073] b3) Generation of archive set A: Take all individuals that are not dominated by any other individuals as archive set A.

[0074] b4) Grid division and congestion assessment, the steps are as follows:

[0075] b4.1) First, the boundary of the target space is defined based on the target vector values ​​of all non-dominated particles in A and the expansion coefficient ε;

[0076] b4.2) Divide the target space into grid lines based on the quantile d and construct a grid system, such as Figure 3 As shown;

[0077] 4.3) According to the location of each particle in the archive set A in each dimension grid, assign its grid coordinate serial number, and according to the frequency of occurrence of all serial numbers φ k Complete the quantitative evaluation of the congestion degree of A; k = 1, 2, ..., η; η is the number of coordinate numbers;

[0078] b5) Determine whether the archive set A has reached the preset capacity. If so, use the roulette method to update the optimal individual, update the POP speed using formula (25), update the POP position using formula (26), and adjust to step b9). If not, proceed to step b6);

[0079] b6) Determine whether p ≥ 0.5. If so, update POP using formula (23) and proceed to step b8). If not, proceed to step b7); p is a probability value in the range [0, 1].

[0080] b7) Determine whether |A|>1 holds. If so, update POP using formula (24). Otherwise, update POP using formula (22).

[0081] b8) Parameter update: update the convergence factor υ, random value r, random value l, and probability value p;

[0082] b9) POP control comparison and archive set A update;

[0083] b10) Use the roulette wheel method to delete overflow individuals in the archive set A;

[0084] b11) Determine whether the current number of iterations iter is less than the maximum number of iterations. If so, set the number of iterations iter = iter + 1 and return to step b5). Otherwise, end the iteration.

[0085]

[0086] Where: L i represents the current position of particle i, L gbest represents the optimal particle position;

[0087] The coefficients A and C are as follows:

[0088]

[0089] Where: r is a random value in the interval [0,1], υ is the convergence factor, which decreases linearly from 2 to 0 during the entire iteration process. D is an intermediate parameter;

[0090]

[0091] Where: p is the probability value in the range of [0,1], b is the spiral shape parameter, and l is a random value in the interval of [-1,1]. * is the intermediate parameter;

[0092] The mathematical model of enveloping contraction is as follows:

[0093] L i (iter+1)=L gbest (iter)-A·D * (p<0.5)

[0094] L i (iter+1)=L rand -A|C·L rand -L i (iter)| (24)

[0095] Where: L rand Represents the position of a randomly selected particle

[0096]

[0097] Where: r1 and r2 are independent random variables in the interval [0,1]. i (iter+1), V i (iter) is the particle velocity at different iteration numbers; w0, c1, c2 are weights; is the coefficient.

[0098] Furthermore, the steps to update the optimal individual using the roulette method are:

[0099] c1) assigning a grid coordinate number to each particle in the archive set A according to its location in each dimension of the grid;

[0100] c2) Based on the grid number frequency φ k , select a grid number using the roulette wheel selection mechanism;

[0101] The probability of each grid number being selected is as follows:

[0102]

[0103] Where, φ i is the frequency; η is the number of grids with at least one particle inside; is the relational parameter.

[0104] c3) Randomly select a particle inside the selected grid as the global optimal particle L gbest .

[0105] Furthermore, the multi-objective evaluation method is as follows:

[0106]

[0107] Among them: G i,max and G i,min is the Pareto optimal solution set with respect to the objective function G i The maximum and minimum values ​​of λ i is the weight coefficient; is the multi-objective comprehensive optimal evaluation index; the individual with the smallest multi-objective comprehensive optimal evaluation index is the comprehensive optimal solution, and the motion trajectory corresponding to this individual is the trajectory obtained by trajectory planning based on the method proposed in this paper.

[0108] The technical benefits of this invention are undeniable. It employs Cartesian space trajectory planning for the trajectory planning of a parallel robot during assembly and calibration. This method directly focuses on the kinematic variables of the robot's end effector, is intuitive, and easily implements complex trajectory planning. Furthermore, solving the inverse kinematics of the parallel robot is relatively simple. The basic process of Cartesian space trajectory planning involves first planning the six-degree-of-freedom spatial motion trajectory of the workpiece to be assembled and calibrated. Then, through inverse kinematic modeling, the displacement, velocity, and acceleration curves of each driving joint are obtained for subsequent assembly and calibration motion control. BRIEF DESCRIPTION OF THE DRAWINGS

[0109] Figure 1 A framework diagram for implementing multi-target trajectory planning for a parallel robot;

[0110] Figure 2 Flowchart of the new multi-objective adaptive optimization algorithm;

[0111] Figure 3 Schematic diagram of the grid method principle;

[0112] Figure 4 The Pareto optimal frontier obtained by optimizing the new multi-objective adaptive optimization algorithm;

[0113] Figure 5 The Pareto optimal frontier obtained by solving the multi-objective particle swarm optimization algorithm;

[0114] Figure 6 The overall trajectory of the optimal lower position is balanced;

[0115] Figure 7 To balance the overall trajectory of the optimal posture;

[0116] Figure 8 This is a schematic diagram of the parallel robot mechanism;

[0117] Figure 9 To assemble and calibrate the parallel robot. DETAILED DESCRIPTION

[0118] The present invention will be further described below with reference to the following examples, but it should not be understood that the scope of the present invention is limited to the following examples. Without departing from the above technical ideas of the present invention, various substitutions and modifications can be made according to common technical knowledge and customary means in the art, and all should be included in the scope of protection of the present invention.

[0119] Example 1:

[0120] See also Figures 1 to 9 A trajectory planning method for a parallel robot based on a novel multi-objective adaptive optimization algorithm includes the following steps:

[0121] 1) Using the clamped quintic B-spline curve as the interpolation function, the coordinate trajectory of the parallel robot is constructed;

[0122] 2) Based on the dimensions of time, impact, and energy consumption, a multi-objective trajectory optimization model for the installation and calibration platform is established under multiple constraints;

[0123] 3) Iteratively solve the multi-objective trajectory optimization model of the assembly and calibration platform to obtain the optimized trajectory set of the assembly and calibration parallel robot;

[0124] 4) The optimal trajectory set is evaluated using a multi-objective evaluation method to obtain the optimal trajectory of a set of calibrated parallel robots.

[0125] Example 2:

[0126] A method for trajectory planning of a calibration parallel robot based on a novel multi-objective adaptive optimization algorithm, the technical content of which is the same as that of Example 1, further comprising the steps of constructing the coordinate trajectory of the calibration parallel robot:

[0127] a1) Determine the node vector u, that is:

[0128]

[0129] u i =t i-5 (i=6,···w+5) (2)

[0130] a2) Determine the control point vector Q. The steps are as follows:

[0131] a2.1) Let the domain [u k ,u n+1 ] the corresponding point P(u i )(i=5,6,…,w+5) and type value point P i-5 Overlap, we get:

[0132]

[0133] Where B j,5 (u i ) is the basis function; i=5,6,...,w+5; Q i is the control point;

[0134] Basis function B i,k (u) is obtained by the following de Boor recursion formula:

[0135]

[0136] a2.2) Let B i,5 (u i )=0, simplifying formula (3) yields:

[0137]

[0138] a2.3) Convert equation (4) into matrix form to obtain:

[0139]

[0140] a2.4) Simplify formula (5) to obtain:

[0141]

[0142] a2.5) Construct the first-order and second-order derivative forms of the B-spline curve, namely:

[0143]

[0144] Where, P'(u) and P"(u) are the first-order and second-order derivative forms of the B-spline curve;

[0145] a2.6) Construct expressions for the velocity and acceleration at the start and end points of the trajectory based on the first- and second-order derivatives of the B-spline curve;

[0146] The velocity and acceleration expressions of the trajectory at the starting point are as follows:

[0147]

[0148] The velocity and acceleration expressions of the trajectory at the end point are as follows:

[0149]

[0150] a2.7) Combining equations (8) and (9), we obtain:

[0151] BQ=P * (10)

[0152]

[0153] Where: Q=[Q0 Q1 … Q w+4 ] T is the vector composed of the control points to be found, P * is the point vector containing the velocity and acceleration values ​​at the start and end points; P * =[0 0 P0 P1 … P w 0 0] T ; The variables in the upper left and lower right corners of the B matrix represent the elements composed of the correlation coefficients in the expressions of the velocity and acceleration of the start and end points;

[0154] The expressions of the variables in the upper left and lower right corners of the B matrix are as follows:

[0155]

[0156] a2.8) Transform formula (10) to obtain:

[0157] Q=B -1 P * (13)

[0158] a2.9) Determine all control points using formula (13), substitute the node vector u and the control points into the general expression of the B-spline curve, and construct the coordinate trajectory of the parallel robot.

[0159] Example 3:

[0160] A trajectory planning method for a calibration parallel robot based on a novel multi-objective adaptive optimization algorithm, the technical content of which is the same as any one of Examples 1-2. Furthermore, the objective functions of the multi-objective trajectory optimization model of the calibration platform include an objective function based on time optimization, an objective function based on impact optimization, and an objective function based on energy consumption optimization.

[0161] Example 4:

[0162] A trajectory planning method for a parallel robot based on a novel multi-objective adaptive optimization algorithm, the technical content of which is the same as any one of Examples 1-3, and further, the objective function G1 based on time optimization is as follows:

[0163]

[0164] Where Δt i =t i -t i-1 (i=1,2,…,w) is the time interval between two adjacent type value points;

[0165] The objective function G2 based on impact optimization is as follows:

[0166]

[0167] Among them: jerk i represents the jerk of the i-driven joint during the motion process, and T represents the total time consumed by the calibration task.

[0168] The objective function G based on optimal energy consumption is as follows:

[0169]

[0170] Where: τ i and v i Respectively represent the driving force and speed of #i driving joint during the movement.

[0171] Example 5:

[0172] A method for trajectory planning of a calibration parallel robot based on a novel multi-objective adaptive optimization algorithm, the technical content of which is the same as any one of Examples 1-4, furthermore, the constraints of the multi-objective trajectory optimization model of the calibration platform include joint displacement constraints, joint velocity constraints, joint acceleration constraints, and driving force constraints;

[0173] The joint displacement constraints are as follows:

[0174] s imin ≤s i ≤s imax (i=1,2,…,6) (17)

[0175] Where: s i ,s imax ,s imin are the displacement of the driven joint i and the upper and lower bounds of the allowed displacement range, respectively.

[0176] The joint velocity constraints are as follows:

[0177] |v i |≤v imax (i=1,2,…,6) (18)

[0178] Where: v i and v imax are the speed and maximum allowed speed of the driven joint i respectively.

[0179] The joint acceleration constraints are as follows:

[0180] |a i |≤a imax (i=1,2,…,6) (19)

[0181] Among them: a i and a imax are the acceleration and maximum allowed acceleration of #i driving joint respectively.

[0182] The driving force constraints are as follows:

[0183] |τ i |≤τ imax (i=1,2,…,6) (20)

[0184] Where: τ i and τ imax are the driving force and maximum allowable driving force of #i driving joint respectively.

[0185] Example 6:

[0186] A trajectory planning method for a calibration parallel robot based on a novel multi-objective adaptive optimization algorithm, the technical content of which is the same as any one of Examples 1-5. Furthermore, the multi-objective trajectory optimization model G of the calibration platform is as follows:

[0187]

[0188] Example 7:

[0189] A method for trajectory planning of a calibration parallel robot based on a novel multi-objective adaptive optimization algorithm, the technical content of which is the same as any one of Examples 1-6, further comprising the step of iteratively solving the multi-objective trajectory optimization model of the calibration platform:

[0190] b1) Population POP and parameter initialization: Set the initial values ​​of each parameter in the algorithm according to the optimization requirements, and randomly initialize the population POP value;

[0191] b2) Target vector calculation and dominance comparison: Calculate the target function value corresponding to each individual in the initial population POP and determine the dominance relationship between all individuals;

[0192] b3) Generation of archive set A: Take all individuals that are not dominated by any other individuals as archive set A.

[0193] b4) Grid division and congestion assessment, the steps are as follows:

[0194] b4.1) First, the boundary of the target space is defined based on the target vector values ​​of all non-dominated particles in A and the expansion coefficient ε;

[0195] b4.2) Divide the target space into grid lines based on the quantile d and construct a grid system, such as Figure 3 As shown;

[0196] 4.3) According to the location of each particle in the archive set A in each dimension grid, assign its grid coordinate serial number, and according to the frequency of occurrence of all serial numbers φ k Complete the quantitative evaluation of the congestion degree of A; k = 1, 2, ..., η; η is the number of coordinate numbers;

[0197] b5) Determine whether the archive set A has reached the preset capacity. If so, use the roulette method to update the optimal individual, update the POP speed using formula (25), update the POP position using formula (26), and adjust to step b9). If not, proceed to step b6);

[0198] b6) Determine whether p ≥ 0.5. If so, update POP using formula (23) and proceed to step b8). If not, proceed to step b7); p is a probability value in the range [0, 1].

[0199] b7) Determine whether |A|>1 holds. If so, update POP using formula (24). Otherwise, update POP using formula (22).

[0200] b8) Parameter update: update the convergence factor υ, random value r, random value l, and probability value p;

[0201] b9) POP control comparison and archive set A update;

[0202] b10) Use the roulette wheel method to delete overflow individuals in the archive set A;

[0203] b11) Determine whether the current number of iterations iter is less than the maximum number of iterations. If so, set the number of iterations iter = iter + 1 and return to step b5). Otherwise, end the iteration.

[0204]

[0205] Where: L i represents the current position of particle i, L gbest represents the optimal particle position; D is the intermediate parameter;

[0206] The coefficients A and C are as follows:

[0207]

[0208] Where r is a random value in the interval [0,1], and υ is the convergence factor, which decreases linearly from 2 to 0 during the entire iteration process.

[0209]

[0210] Where: p is the probability value in the range of [0,1], b is the spiral shape parameter, and l is a random value in the interval of [-1,1]. * is an intermediate parameter;

[0211] The mathematical model of enveloping contraction is as follows:

[0212] L i (iter+1)=L gbest (iter)-A·D * (p<0.5)

[0213] L i (iter+1)=L rand -A|C·L rand -L i (iter)| (24)

[0214] Where: L rand Represents the position of a randomly selected particle

[0215]

[0216] L i (iter+1)=L i (iter)+V i (iter+1) (26)

[0217] Where: r1 and r2 are independent random variables in the interval [0,1]. i (iter+1), V i(iter) is the particle velocity at different iteration numbers; w0, c1, c2 are weights; is the coefficient.

[0218] Example 8:

[0219] A trajectory planning method for a parallel robot based on a novel multi-objective adaptive optimization algorithm, the technical content of which is the same as any one of Examples 1-7, and further, the steps of updating the optimal individual using the roulette wheel method are as follows:

[0220] c1) assigning a grid coordinate number to each particle in the archive set A according to its location in each dimension of the grid;

[0221] c2) Based on the grid number frequency φ k , select a grid number using the roulette wheel selection mechanism;

[0222] The probability of each grid number being selected is as follows:

[0223]

[0224] Where, φ i is the frequency; η is the number of grids with at least one particle inside; is the relational parameter.

[0225] c3) Randomly select a particle inside the selected grid as the global optimal particle L gbest .

[0226] Example 9:

[0227] A trajectory planning method for a parallel robot based on a novel multi-objective adaptive optimization algorithm, the technical content of which is the same as any one of Examples 1-8, and further, a multi-objective evaluation method is as follows:

[0228]

[0229] Among them: G i,max and G i,min is the Pareto optimal solution set with respect to the objective function G i The maximum and minimum values ​​of λ i is the weight coefficient; is the multi-objective comprehensive optimal evaluation index; the individual with the smallest multi-objective comprehensive optimal evaluation index is the comprehensive optimal solution, and the motion trajectory corresponding to this individual is the trajectory obtained by trajectory planning based on the method proposed in this paper.

[0230] Example 9:

[0231] A trajectory planning method for an assembly and calibration parallel robot based on a novel multi-objective adaptive optimization algorithm is presented as follows:

[0232] Considering the difficulty of solving the forward motion problem for parallel robots and to improve the intuitiveness of trajectory planning, we abandoned the traditional joint-space trajectory planning method and adopted Cartesian space trajectory planning for the trajectory planning of the parallel robot during assembly and calibration. This method directly focuses on the kinematic variables of the robot's end effector, is intuitive, and easily implements complex trajectory planning. Furthermore, solving the inverse kinematics of the parallel robot is relatively simple. The basic process of Cartesian space trajectory planning involves first planning the six-degree-of-freedom spatial motion trajectory of the workpiece to be assembled and calibrated. Then, through the inverse solution of the kinematic model, the displacement, velocity, and acceleration curves of each driving joint are obtained for subsequent assembly and calibration motion control.

[0233] The trajectory planning of the calibration parallel robot is divided into two key stages: trajectory generation and trajectory optimization. Trajectory generation refers to the parametric characterization of each trajectory using curves with good mathematical properties and high flexibility. Trajectory optimization refers to first building a multi-objective evaluation function related to trajectory optimization based on multiple performance requirements of the calibration task, then setting the kinematic and dynamic constraints of the calibration platform, and forming a multi-objective trajectory optimization model under multiple constraints. Finally, an intelligent optimization algorithm is used for iterative solution, and a set of optimal trajectories is determined through a multi-objective evaluation method. The implementation framework is as follows: Figure 1 shown.

[0234] The trajectory generation stage uses the clamped quintic B-spline curve as the interpolation function to ensure the smoothness, differentiability and local controllability of the generated trajectory. The main process of trajectory generation is:

[0235] (1) Determine the node vector u

[0236] In order to ensure the consistency of the start and end control points and the start and end type value points of the curve, according to the properties of the B-spline curve, the repetition degree of the node vector u at both ends is 6, that is:

[0237]

[0238] It is known that one of the generalized coordinate value point sequences is [t i ,P i ](i=0,1,…,w), then the remaining inner nodes of the node vector u can be obtained by corresponding to the relevant time nodes:

[0239] u i =t i-5 (i=6,···w+5)

[0240] Combining the above two equations, the node vector u can be determined.

[0241] (2) Determine the control point vector Q

[0242] According to the properties of B-spline curve, we can know the corresponding relationship between the type value points and the nodes. Let the domain [u k ,u n+1 ] the corresponding point P(u i )(i=5,6,…,w+5) and type value point P i-5 If we overlap, we get w+1 equations:

[0243]

[0244] Setting B i,5 (u i )=0, the above formula can be simplified to:

[0245]

[0246] Convert the above w+1 equations into matrix form and unify the subscripts of each element into a form related to w:

[0247]

[0248] The above formula can be simplified as:

[0249]

[0250] The number of control points to be solved is w+4, and there are a total of w constraint equations. Four additional equations are needed to find a unique solution for the control points. To ensure smoothness at the start and stop times, the velocity and acceleration at the start and end points of the trajectory are set to 0, and this setting is used as the additional four constraint equations. The first and second order derivatives of the B-spline curve are in the form of:

[0251]

[0252] The velocity and acceleration of the trajectory at the starting point are:

[0253]

[0254] The velocity and acceleration of the trajectory at the end point are:

[0255]

[0256] Combined, we can get:

[0257] BQ=P *

[0258]

[0259] Where: Q=[Q0 Q1 … Q w+4 ] T is the vector composed of the control points to be found, P* is the point vector containing the velocity and acceleration values ​​at the starting and ending points, that is, P * =[0 0 P0 P1 … P w 0 0] T , the variables in the upper left and lower right corners of the B matrix represent the elements composed of the correlation coefficients in the expressions of the velocity and acceleration of the start and end points, and their expressions are:

[0260]

[0261] The deformation can be obtained:

[0262] Q=B -1 P *

[0263] All control points can be determined using the above equation. Substituting the node vector u and the control points into the general expression of the B-spline curve completes the construction of the generalized coordinate trajectory of the clamped quintic B-spline curve for the module calibration motion. Furthermore, the displacement, velocity, and acceleration trajectories of each driven joint can be obtained by inversely solving the kinematic model.

[0264] The corresponding optimization objective function can be specified according to the actual workpiece assembly and calibration task requirements. Here, the assembly and calibration efficiency, stability, and energy consumption are used as three types of performance indicators, and a trajectory optimization objective function based on the three dimensions of time, impact, and energy consumption is established.

[0265] (1) Time-optimal objective function

[0266] The goal of time optimal trajectory optimization is to maximize the calibration efficiency while completing the calibration task, that is, to minimize the total calibration time. i =t i -t i-1 (i=1,2,…,w) is used as the optimization variable, and the objective function of trajectory optimization based on time optimization is:

[0267]

[0268] (2) Objective function based on impact optimization

[0269] During the calibration process, drastic changes in acceleration can cause significant impact on the calibration platform, affecting the calibration accuracy of the workpiece, accelerating the wear of the equipment's mechanical components, and potentially damaging the precision workpiece. Trajectory optimization with overall impact minimization as the optimization goal can improve calibration accuracy, stability, and reliability, extending the life of the equipment. The objective function for trajectory optimization based on impact optimization is:

[0270]

[0271] Among them: jerk i represents the jerk of the #i driven joint during the motion process, and T represents the total time consumed by the calibration task.

[0272] (3) Objective function based on optimal energy consumption

[0273] Trajectory optimization with minimum energy consumption as the optimization goal can reduce unnecessary energy loss, improve the energy utilization efficiency of the system, reduce operating costs, and mitigate negative impacts on the environment. The trajectory optimization objective function based on optimal energy consumption is:

[0274]

[0275] Where: τ i and v i Respectively represent the driving force and speed of #i driving joint during the movement.

[0276] When performing trajectory planning, in order to ensure the rationality and feasibility of the obtained trajectory, it must be based on a series of preset constraints to ensure that the planned trajectory not only optimizes the established goals but also meets the requirements of actual movement.

[0277] (1) Joint displacement constraint

[0278] In order to ensure the smooth progress of workpiece calibration, the displacement of each driving joint should be kept within its allowable range during the calibration process. Therefore, constraints must be imposed on the displacement of each joint during the trajectory planning stage, as shown in the following formula:

[0279] s imin ≤s i ≤s imax (i=1,2,…,6)

[0280] Where: s i ,s imax ,s imin They are respectively the displacement of the #i driven joint and the upper and lower limits of the allowed displacement range.

[0281] (2) Joint velocity constraints

[0282] The motion of the driven joints is driven by servo motors. Considering the rated speed limit of the motors, the speed of each driven joint should be constrained during trajectory planning, as shown in the following formula:

[0283] |v i |≤v imax (i=1,2,…,6)

[0284] Where: v i and v imaxare the speed of #i driving joint and the maximum allowed speed respectively.

[0285] (3) Joint acceleration constraints

[0286] In order to ensure the smoothness of the planned trajectory, the acceleration constraints of each driving joint are shown as follows:

[0287] |a i |≤a imax (i=1,2,…,6)

[0288] Among them: a i and a imax are the acceleration and maximum allowed acceleration of #i driving joint respectively.

[0289] (4) Driving force constraints

[0290] Similarly, given the rated torque limit of the servo motor, the output driving force of each driven joint must be constrained during the planning process to avoid exceeding the capability of the motor, as shown in the following formula:

[0291] |τ i |≤τ imax (i=1,2,…,6)

[0292] Where: τ i and τ imax are the driving force and maximum allowable driving force of #i driving joint respectively.

[0293] Combining the above optimization objective functions and constraints, the multi-objective trajectory optimization model of the calibration platform under multiple constraints is as follows:

[0294]

[0295] When solving multi-objective optimization problems, algorithms must strike a balance between solution quality and diversity. Each algorithm has its own unique advantages and limitations. The Whale Optimization Algorithm (WHA), due to its unique mechanism based on whale predation behavior, is very effective in guiding the algorithm to extensively explore the solution space. However, its convergence rate is relatively slow and it may require more computational resources to find the optimal solution. The Multi-Objective Particle Swarm Optimization (MPS) uses learning mechanisms based on social and individual cognition to enable more detailed exploration of discovered high-quality regions, thereby improving solution quality. However, this algorithm may lack solution diversity. Here, a novel multi-objective adaptive optimization algorithm is proposed that combines the advantages of the MPS and WHA, adaptively adjusting their strategies at different stages of the search process to achieve better performance. The core of this algorithm is to leverage the global search capabilities of the WHA to conduct extensive exploration while the external archive set A is not full. Once the archive set reaches a preset capacity, the algorithm switches to the PSO strategy to enhance the depth and precision of the local search. As the iterations progress, if the number of solutions in the archive set decreases due to the selection process, for example, if a small number of newly added solutions dominate a large number of original solutions in the archive set, the algorithm reactivates the global search mechanism of the whale optimization algorithm and expands the exploration of the solution space. This dynamic search strategy aims to leverage the strengths of both algorithms while avoiding their limitations.

[0296] Through this adaptive mechanism, the new multi-objective adaptive optimization algorithm can make intelligent decisions according to the dynamic characteristics of the target space and the current state of the archive set during the search process. The specific algorithm process is referred to Figure 2 The selection method of the global optimal particle adopts a grid selection method. The distribution of particles in the target space grid system is used to replace the traditional crowding measurement method. Specifically, firstly, the target vector value of all non-dominated particles in A is combined with the expansion coefficient ε to define the boundary of the target space. Then, the grid lines are divided in the target space according to the quantile d to construct a grid system, such as Figure 3 Finally, the grid coordinate serial number is assigned to each particle in the archive set A according to its location in each dimension grid, and the occurrence frequency of all serial numbers is φ k (k=1,2,…,η) can complete the quantitative evaluation of the congestion of A, where η is the number of grids with at least one particle inside. In each iteration, the algorithm will be based on the grid number frequency φ k A grid number is selected using the roulette wheel selection mechanism. The probability of each grid number being selected is as follows:

[0297]

[0298] Then randomly select a particle inside the selected grid as the global optimal particle L gbestThe roulette wheel selection mechanism helps incentivize the exploration of less crowded areas, i.e., areas that are currently less searched, thereby enhancing the diversity of A.

[0299] During the global search phase, the whale optimization strategy is mainly used, which can be abstracted into three mathematical models:

[0300] (1) In the optimization problem, due to the lack of prior knowledge to accurately locate the optimal solution, it is assumed that the best particle in the current iteration is the optimal solution, and other particles approach this optimal particle by updating their positions. The mathematical model of this process is:

[0301]

[0302] Where: L i represents the current position of particle i, L gbest represents the optimal particle position, and the coefficients A and C are obtained as follows:

[0303]

[0304] Where r is a random value in the interval [0,1], and υ is the convergence factor, which decreases linearly from 2 to 0 during the entire iteration process.

[0305] (2) Assuming that the particle performs spiral motion and encircling contraction with a probability of 50% each, the mathematical model is:

[0306]

[0307] Where: p is the probability value in the range [0,1], b is the spiral shape parameter, and l is a random value in the interval [-1,1].

[0308] The mathematical model of encirclement contraction is:

[0309] L i (iter+1)=L gbest (iter)-A·D * (p<0.5)

[0310] (3) In addition to moving toward the optimal particle, particles also perform random searches. Setting: When the parameter |A|>1, the process enters a random search state, randomly selecting a particle in the current population as a reference, and the remaining particles update their positions based on the position of the reference particle. The mathematical model of this process is:

[0311] L i (iter+1)=L rand -A|C·L rand -L i (iter)| (Formula 3)

[0312] Where: Lrand Represents the position of a randomly selected particle

[0313] In the local search phase, for each particle in the POP, its velocity and position are updated by the following formula. If the updated L i If the decision variable exceeds the limit, the position is reset to the boundary value and the speed is adjusted to move in the opposite direction. Calculate the new target vector value h(L i ), if L i <L best,i , then update L best,i =L i ; If L best,i <L i , then L best,i Remain unchanged; if L i With L best,i If they do not dominate each other, then update L with a probability of 50%. best,i =L i .

[0314]

[0315] L i (iter+1)=L i (iter)+V i (iter+1)(Formula 5)

[0316] Where: r1 and r2 are independent random variables in the interval [0,1].

[0317] A set of Pareto optimal solutions can be obtained by performing trajectory optimization through a new multi-objective adaptive optimization algorithm. In order to select a solution that balances various performance sub-goals as the trajectory of the installation and calibration operation, the present invention proposes a fuzzy evaluation method based on fuzzy set theory, namely

[0318]

[0319] Among them: G i,max and G i,min is the Pareto optimal solution set with respect to the objective function G i The maximum and minimum values ​​of λ i is the weight coefficient. This weight coefficient can be adjusted for different assembly and calibration conditions, allowing the trajectory that best suits the current condition to be selected from the Pareto optimal solution set. The currently obtained trajectory is the assembly and calibration motion trajectory for the workpiece. This trajectory must be converted into the motion trajectory of each driving joint using the inverse kinematic model of the parallel robot. This completes the assembly and calibration trajectory planning.

[0320] The 4-PPPS configuration parallel robot is used as the research object, and the schematic diagram of the mechanism is as follows: Figure 8 As shown, the three-dimensional model is Figure 9 As shown in the figure, a new multi-objective adaptive optimization algorithm is used to iteratively calculate the trajectory optimization model of the assembly and calibration parallel robot, and the obtained Pareto optimal frontier is shown in Figure 4 As a control experiment, the multi-objective particle swarm algorithm is used to iteratively calculate the trajectory optimization model, and the same parameters are kept the same as the previous settings. The Pareto optimal frontier is shown as follows. Figure 5 As shown. By comparing the two, it can be seen that under the same parameter configuration, the Pareto frontier obtained by the algorithm proposed in the present invention performs better in terms of uniformity and diversity, reflecting that the algorithm proposed in the present invention has excellent global search capabilities due to its adaptive switching mechanism. At the same time, it can be seen that the Pareto optimal solution set obtained by the new multi-objective adaptive optimization algorithm proposed in the present invention is superior to the multi-objective particle swarm algorithm in each optimization sub-objective. Among them, in terms of impact indicators, the algorithm proposed in the present invention is significantly superior to the multi-objective particle swarm algorithm with a gap of 20.5%, which means that the algorithm proposed in the present invention has certain superiority in solution quality, reflecting its excellent local search capabilities. Weight coefficient λ i Taking the values ​​of 3, 2, and 1 respectively, we can get the corresponding node vector and control point data, and further get the workpiece calibration motion trajectory with the best balance of various performance indicators. The overall trajectory of position and posture is as follows: Figure 6 and Figure 7 The resulting trajectory sequentially passes through each critical path value point and ultimately reaches the target pose. From the perspective of the task space, this demonstrates that the workpiece can be installed smoothly and without interference. This balanced optimal trajectory balances efficiency improvement with impact and energy reduction, achieving a comprehensive optimization of all three, validating the advanced trajectory planning method proposed in this paper.

Claims

1. A trajectory planning method for a parallel robot based on a novel multi-objective adaptive optimization algorithm, characterized in that: The following steps are involved: 1) Using the clamped quintic B-spline curve as the interpolation function, the coordinate trajectory of the parallel robot is constructed; 2) Based on the dimensions of time, impact, and energy consumption, a multi-objective trajectory optimization model for the installation and calibration platform is established under multiple constraints; 3) Iteratively solve the multi-objective trajectory optimization model of the assembly and calibration platform to obtain the optimized trajectory set of the assembly and calibration parallel robot; 4) The optimal trajectory set is evaluated using a multi-objective evaluation method to obtain the optimal trajectory of a set of calibrated parallel robots.

2. The trajectory planning method for a parallel robot based on a novel multi-objective adaptive optimization algorithm according to claim 1 is characterized in that: The steps for constructing and calibrating the parallel robot coordinate trajectory include: 1) Determine the node vector u, that is: u i =t i-5 (i=6,···w+5)(2) 2) Determine the control point vector Q. The steps are as follows: 2.1) Let the domain [u k ,u n+1 ] the corresponding point p(u i ) and type value point P i-5 Overlap, we get: Where B j,5 (u i ) is the basis function; i=5,6,...,w+5; Q i is the control point; Basis function B i,k (u) is obtained by the following de Boor recursion formula: 2.2) Let B i,5 (u i )=0, simplifying formula (3) yields: 2.3) Convert formula (4) into matrix form and get: 2.4) Simplify formula (5) and obtain: 2.5) Construct the first-order and second-order derivative forms of the B-spline curve, namely: Where, P'(u) and P"(u) are the first-order and second-order derivative forms of the B-spline curve; 2.6) Based on the first-order and second-order derivatives of the B-spline curve, construct the velocity and acceleration expressions at the starting and ending points of the trajectory; The velocity and acceleration expressions of the trajectory at the starting point are as follows: The velocity and acceleration expressions of the trajectory at the end point are as follows: 2.7) Combining formula (8) and formula (9), we get: BQ=P * (10) Where: Q=[Q0 Q1…Q w+4 ] T is the vector composed of the control points to be found, P * is the point vector containing the velocity and acceleration values ​​at the start and end points; P * =[0 0 P0 P1…P w 0 0] T ; The variables in the upper left and lower right corners of the B matrix represent the elements composed of the correlation coefficients in the expressions of the velocity and acceleration of the start and end points; The expressions of the variables in the upper left and lower right corners of the B matrix are as follows: 2.8) Transform formula (10) to obtain: Q=B -1 P * (13) 2.9) Determine all control points using formula (13), substitute the node vector u and the control points into the general expression of the B-spline curve, and construct the coordinate trajectory of the parallel robot.

3. The trajectory planning method for a parallel robot based on a novel multi-objective adaptive optimization algorithm according to claim 1 is characterized in that: The objective functions of the multi-objective trajectory optimization model of the installation and calibration platform include an objective function based on time optimization, an objective function based on impact optimization, and an objective function based on energy consumption optimization.

4. The trajectory planning method for a parallel robot based on a novel multi-objective adaptive optimization algorithm according to claim 3 is characterized in that: The time-optimal objective function G1 is as follows: Where Δt i =t i -t i-1 is the time interval between two adjacent type value points; i=1,2,...,w; The objective function G2 based on impact optimization is as follows: Among them: jerk i represents the jerk of the i-driven joint during the motion process, and T represents the total time consumed by the calibration task. The objective function G3 based on optimal energy consumption is as follows: Where: τ i and v i Respectively represent the driving force and speed of #i driving joint during the movement.

5. The trajectory planning method for a parallel robot based on a novel multi-objective adaptive optimization algorithm according to claim 1 is characterized in that: The constraints of the multi-objective trajectory optimization model of the calibration platform include joint displacement constraints, joint velocity constraints, joint acceleration constraints, and driving force constraints; The joint displacement constraints are as follows: s imin ≤s i ≤s imax (i=1,2,…,6)(17) Where: s i ,s imax ,s imin are the displacement of the driven joint i and the upper and lower bounds of the allowed displacement range, respectively. The joint velocity constraints are as follows: |v i |≤v imax (i=1,2,…,6)(18) Where: v i and v imax are the speed and maximum allowed speed of the driven joint i respectively. The joint acceleration constraints are as follows: |a i |≤a imax (i=1,2,…,6)(19) Among them: a i and a imax are the acceleration and maximum allowed acceleration of #i driving joint respectively. The driving force constraints are as follows: |t i |≤τ imax (i=1,2,…,6)(20) Where: τ i and τ imax are the driving force and maximum allowable driving force of #i driving joint respectively.

6. The trajectory planning method for a parallel robot based on a novel multi-objective adaptive optimization algorithm according to claim 1 is characterized in that: The multi-objective trajectory optimization model G of the installation and calibration platform is as follows:

7. The method for trajectory planning of a parallel robot based on a novel multi-objective adaptive optimization algorithm according to claim 1, characterized in that: The steps for iteratively solving the multi-objective trajectory optimization model of the calibration platform include: 1) Population POP and parameter initialization: Set the initial values ​​of each parameter in the algorithm according to the optimization requirements, and randomly initialize the population POP value; 2) Target vector calculation and dominance comparison: Calculate the target function value corresponding to each individual in the initial population POP and determine the dominance relationship between all individuals; 3) Generation of archive set A: Take all individuals that are not dominated by any other individuals as archive set A. 4) Grid division and congestion assessment, the steps are as follows: 4.1) First, based on the target vector values ​​of all non-dominated particles in A and combined with the expansion coefficient ε, the boundary of the target space is defined; 4.2) Divide the target space into grid lines according to the quantile d and construct a grid system; 4.3) According to the location of each particle in the archive set A in each dimension grid, assign its grid coordinate serial number, and according to the frequency of occurrence of all serial numbers φ k Complete the quantitative evaluation of the congestion degree of A; k = 1, 2, ..., η; η is the number of coordinate numbers; 5) Determine whether the archive set A has reached the preset capacity. If so, use the roulette method to update the optimal individual, and use formula (25) to update the POP speed, and use formula (26) to update the POP position, and adjust to step 9). If not, go to step 6); 6) Determine whether p ≥ 0.

5. If so, update POP using formula (23) and proceed to step 8). If not, proceed to step 7); p is a probability value in the range [0, 1]. 7) Determine whether |A|>1 holds. If so, update POP using formula (24). Otherwise, update POP using formula (22). 8) Parameter update: update the convergence factor υ, random value r, random value l, and probability value p; 9) POP control comparison and archive set A update; 10) Use the roulette wheel method to delete overflow individuals in the archive set A; 11) Determine whether the current number of iterations iter is less than the maximum number of iterations. If so, set the number of iterations iter = iter + 1 and return to step 5). Otherwise, end the iteration. Where: L i represents the current position of particle i, L gbest represents the optimal particle position; D is the intermediate parameter; The coefficients A and C are as follows: Where: r is a random value in the interval [0,1], υ is the convergence factor, which decreases linearly from 2 to 0 during the entire iteration process; Where: p is the probability value in the range of [0,1], b is the spiral shape parameter, l is a random value in the interval of [-1,1]; D * is an intermediate parameter; The mathematical model of enveloping contraction is as follows: L i (iter+1)=L gbest (iter)-A·D * (p<0.5) L i (iter+1)=L rand -A|C·L rand -L i (iter)|(24) Where: L rand Represents the position of a randomly selected particle 50 i (path+1)=L i (journey)+V i (path+1)(26) Where: r1 and r2 are independent random variables in the interval [0,1]; V i (iter+1), V i (iter) is the particle velocity at different iteration numbers; w0, c1, c2 are weights; is the coefficient.

8. The trajectory planning method for a parallel robot based on a novel multi-objective adaptive optimization algorithm according to claim 7 is characterized in that: The steps to update the optimal individual using the roulette method are: 1) Assign a grid coordinate number to each particle in the archive set A based on its location in each dimension of the grid; 2) Based on the grid number frequency φ k , select a grid number using the roulette wheel selection mechanism; The probability of each grid number being selected odds k As shown below Where, φ i is the frequency; η is the number of grids with at least one particle inside; is the relational parameter; 3) Randomly select a particle inside the selected grid as the global optimal particle L gbest ; 9. The method for trajectory planning of a parallel robot based on a novel multi-objective adaptive optimization algorithm according to claim 1, characterized in that: The multi-objective evaluation method is as follows: Among them: G i,max and G i,min is the Pareto optimal solution set with respect to the objective function G i The maximum and minimum values ​​of λ i is the weight coefficient; is the multi-objective comprehensive optimal evaluation index; the individual with the smallest multi-objective comprehensive optimal evaluation index is the comprehensive optimal solution.

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