A Trajectory Planning Method for Parallel Mechanisms Considering Joint Friction

Abstract

The present invention discloses a trajectory planning method for a parallel mechanism considering joint friction. The present invention comprehensively optimizes the objective by improving the particle swarm algorithm, the joint friction model, and time optimality, so as to achieve trajectory planning to minimize the wear of the mechanical structure and the energy consumption during operation. The specific steps are as follows: Step 1, build a 3-PPR parallel platform with redundant branches; Step 2, establish the kinematic equation for the 3-PPR parallel platform and establish the joint friction model of the parallel platform; Step 3, optimize the motion trajectory of the parallel platform based on the joint friction model. The present invention introduces the friction energy consumption characterized by the joint friction model into the trajectory optimization of the parallel platform based on the weighted coefficient method, and forms a comprehensive optimization objective function with time optimality, so that the planned trajectory of the parallel platform minimizes the energy consumption caused by joint friction to the greatest extent and at the same time minimizes the wear of the mechanical structure to the greatest extent.

Description

Technical Field

[0001] The present invention belongs to the technical field of robot trajectory planning, and particularly relates to a method for trajectory planning of a parallel mechanism based on an improved optimization algorithm and a joint friction model. Background Technique

[0002] With the progress and innovation of technology, robot technology has become the standard of productivity and production level in the industry. It is the result of the development of machines and one of the important symbols of a country's automation level and productivity level. High-precision alignment technology is a highly representative high-end technology in the current field of robot technology. High-precision alignment technology integrates technologies such as precision machinery, electronics, servo drive, motion control, intelligent communication, and image processing, involving the intersection and integration of many disciplines, and has high theoretical research and practical application value.

[0003] Domestic and foreign research starts from the layout of the kinematic chain of the parallel mechanism and deeply studies the motion performance, motion control, and error analysis of the platform. Due to the characteristics of the parallel mechanism such as non-linearity, strong coupling, multiple inputs, and multiple outputs, starting from the control system software and hardware, the robustness, intelligence, self-adaptability, and control accuracy of the system are improved. However, most of the research focuses on improving the alignment accuracy of the alignment platform, and there are few studies on the trajectory planning of the parallel platform and the accuracy error caused by joint friction. During the long-term movement of the parallel platform, the joint friction caused by a large load will cause wear of the mechanical structure and reduce the alignment accuracy of the parallel platform. On the one hand, it is difficult to establish a friction model that conforms to the actual situation. On the other hand, the friction model generally has complex non-linearity, and it is difficult to optimize the joint friction model.

[0004] From the perspective of practical application, for the trajectory planning aiming at joint friction, in addition to ensuring the alignment accuracy within a long time, it can also reduce the energy consumption brought by the operation of the parallel platform to the greatest extent and extend the mechanical life of the parallel platform. Therefore, the trajectory planning method of the parallel mechanism considering joint friction is very important for practical applications. Summary of the Invention

[0005] The present invention aims at the problems that during the movement of a high-precision parallel alignment platform, the overall movement energy consumption caused by joint friction is too large and the mechanical structure wear caused by long-term movement reduces the alignment accuracy of the parallel platform. Through reasonable trajectory planning, the wear of the mechanical structure and the energy consumption during operation are reduced to the greatest extent.

[0006] In order to achieve the above object, a trajectory planning method for a parallel mechanism considering joint friction is proposed. A trajectory planning method for a parallel mechanism based on an improved particle swarm algorithm, a joint friction model, and a comprehensive optimization goal of time optimality.

[0007] The technical solution adopted by the present invention to solve its technical problems includes the following steps:

[0008] Step 1: Build a 3-PPR parallel platform with redundant branches;

[0009] Step 2: Establish the kinematic equation of the 3-PPR parallel platform and establish the joint friction model of the parallel platform;

[0010] Step 3: Optimize the motion trajectory of the parallel platform based on the joint friction model.

[0011] Further, the building of the 3-PPR parallel platform with redundant branches in Step 1 is specifically implemented as follows: The 3-PPR parallel robot consists of a moving platform, a fixed platform, and 4 branches connecting the moving platform and the fixed platform. One of the 4 branch groups is set as a redundant branch. Each branch from bottom to top is respectively a prismatic pair P1, a prismatic pair P2, and a revolute pair R. The mechanism power input end is the prismatic pair P1 of the other 3 branches (the other 3 branches excluding the redundant branch); the branches are arranged in an equilateral rectangle, specifically as Figure 1 shown.

[0012] Further, the establishment of the joint friction model of the parallel platform in Step 2 is specifically implemented as follows:

[0013] 2-1. Construct the forward and inverse kinematic equations of the 3-PPR parallel platform with redundant branches through the homogeneous coordinate transformation method.

[0014] 2-1-1. Establish a fixed coordinate system O0-X0Y0Z0 with the center O0 of the rectangle formed by the plane branches of the fixed platform as the origin, and establish a moving coordinate system O1-X1Y1Z1 with the geometric center point O1 of the moving platform as the origin. According to the spatial constraints of the parallel platform, determine the coordinates of the slider center points M i , N i in the fixed coordinate system and the position coordinates of the revolute pair rotation center L i in the moving coordinate system. According to the mapping relationship between any position vector T p in the moving coordinate system and the fixed coordinate system, obtain the position vector L i of the revolute pair rotation center L io relative to the fixed coordinate system.

[0015] L io = Rot·T p + P(1)

[0016] where Rot is the attitude matrix of the moving platform relative to the fixed platform, and P = [x, y, z] T represents the position vector of the moving coordinate system relative to the fixed coordinate system.

[0017] 2-1-2. Calculate the slider center point Mi , N i to the position vector of the rotation center L of the revolute pair i

[0018] 2-1-3. Due to the particularity of the 3-PPR parallel platform, the displacement of the rotation centers L1 and L2 of the revolute pairs in the x0-axis direction is the same as that of the center points M1 and M2 of the sliders, and the displacement of the rotation centers L1 and L2 of the revolute pairs in the y0-axis direction is the same as that of the center points N1 and N2 of the sliders. The displacement of the rotation centers L3 and L4 of the revolute pairs in the x0-axis direction is the same as that of the center points N3 and N4 of the sliders, and the displacement of the rotation centers L3 and L4 of the revolute pairs in the y0-axis direction is the same as that of the center points M3 and M4 of the sliders. Let the displacements of the center points M1 and M2 of the lower sliders in the y0 direction be S1 and S2, the displacements of the center points M3 and M4 of the lower sliders in the x0 direction be S3 and S4, the displacements of the center points N1 and N2 of the upper sliders in the x0 direction be S5 and S6, and the displacements of the center points N3 and N4 of the upper sliders in the y0 direction be S7 and S8. Then, according to and the positional relationship of each slider relative to the moving platform can be obtained:

[0019] S k = f k (x, y, θ) k = 1, 2,..., 8 (2)

[0020] where x represents the offset of the moving platform relative to the fixed coordinate system in the x0-axis direction, y represents the offset of the moving platform relative to the fixed coordinate system in the y0-axis direction, and θ represents the offset of the moving platform relative to the fixed coordinate system rotating in the positive direction around the z0 axis; f k (x, y, θ) represents the functional relationship of the positions of each slider relative to the moving platform.

[0021] 2-2. Based on the positional relationship between the sliders and the moving platform and the Coulomb-viscous friction model, a joint friction model of the parallel platform is established, and its loss during the movement is characterized in the form of energy.

[0022] 2-2-1. For the Coulomb friction of the branch chain joints, the normal pressures received by the prismatic pairs P1 and P2 are f n1 and f n2 , respectively. Then, the Coulomb friction received by the moving branch chain of the parallel platform is

[0023]

[0024] where μ represents the Coulomb friction coefficient; f c represents the total Coulomb friction received by the parallel platform, which is composed of the Coulomb friction f ci between the sliders; sgn() represents the sign function; ​represents the velocity of each slider relative to the fixed coordinate system, i.e., related to S k is a derivative relationship.

[0025] 2-2-2. For the viscous friction of the branched-chain joints, since the motion trajectories of the driving joints need to be planned, it is necessary to derive the velocity mapping of each slider group relative to the driving joints:

[0026]

[0027] Among them, J is the Jacobian matrix, that is, the relationship between the pose of the moving platform of the parallel platform and the moving speed of the driving joints.

[0028] The viscous friction suffered by the motion branches of the parallel platform is:

[0029]

[0030] Among them, η represents the viscous friction coefficient; f v represents the total viscous friction suffered by the parallel platform, which is composed of the viscous friction f vk between each slider.

[0031] The friction loss of the parallel platform joints characterized by energy consumption can be expressed as:

[0032]

[0033] Based on the weighted coefficient method, a comprehensive optimization objective function composed of the minimum joint friction energy consumption and the time optimization is established:

[0034] min J = ξ1σE + ξ2(t f - t0) (7)

[0035] In the formula, ξ1 and ξ2 are optimization weight coefficients, and ξ1 + ξ2 = 1. σ is the elastic coefficient, which balances the influence caused by the difference in the order of magnitude between time and energy consumption. According to the actual operation situation, the trajectory equality constraints and inequality constraints of the parallel platform are written:

[0036]

[0037] In the formula, s(t), are the displacement function, velocity function, and acceleration function of the driving joints of the parallel platform respectively; s0, represent the displacement, velocity, and acceleration values at the initial time t0 respectively; s f , represent the displacement, velocity, and acceleration values at the end time t f respectively.

[0038] Further optimize the joint friction model described in step 3. Using the kinematic parameters of the mechanism as constraints, solve it with the improved particle swarm algorithm. Introduce a particle mutation process based on chaotic mapping to increase particle diversity and prevent particles from falling into local optimal points. Use the value of the iteration number to adaptively adjust the inertia weight ω, mutation rate r ch and the mutation iteration number g ch to increase the coordination of global search and local search. The specific optimization implementation steps are as follows:

[0039] 3-1. Set the initial inertia weight ω, learning factors c1, c2, the number of particle swarms num, the maximum iteration number ger, and the range limits of particle velocity and position, and set the initial mutation rate r of particle chaotic mapping ch and the initial mutation iteration number g ch .

[0040] 3-2. Select an interpolation function, such as a polynomial interpolation function, B-spline interpolation function, etc. as the trajectory curve of the parallel platform, that is, the displacement curve s(t) of the driving joint. The unknown parameters in the interpolation function constitute the multi-dimensional position parameters of the particle. Randomly initialize the velocity and position of the particle. Calculate the fitness function value of the particle through the comprehensive optimization objective function, and take the fitness function value of each particle as the individual optimal position pbest, and take the optimal fitness function value in the particle swarm as the global optimal position gbest;

[0041] 3-3. After each iteration, update the velocity and position of the particle, and calculate the corresponding fitness function value. If the fitness function value is better than the particle's individual optimal position pbest, update the optimal position of the particle. If the fitness function value of the particle is better than the global optimal gbest, update the global optimal position. The position and velocity update formulas are as follows:

[0042]

[0043] Among them, the inertia weight ω determines the optimization ability of the particle optimization algorithm. Increasing ω enhances the global search ability of the particle, and decreasing ω enhances the local search ability of the particle. Use the adaptive inertia weight particle algorithm:

[0044]

[0045] Among them, f avg is the average objective function value of all current particles, f min is the minimum objective function value of all current particles, ω max , ω min are the maximum inertia weight and minimum inertia weight respectively.

[0046] 3-4. The particle mutation counter increases with the number of iterations. When the particle mutation count reaches the set mutation iteration number, to prevent the particles from falling into local optimal solutions, the particles are sorted according to their individual optimal positions, and the particles with better performance are retained. The number of such particles is determined by the current mutation rate. At the same time, the remaining particles with poorer performance generate a chaotic sequence using the chaotic mapping method and replace the original particle sequence to ensure that the mapping results in a relatively uniform distribution of the chaotic sequence.

[0047] 3-5. A relatively large initial value can be set for the mutation rate and the initial mutation iteration number to improve the diversity of particles in the initial stage of optimization. During the iteration process, a linear differential decreasing strategy is adopted for the mutation rate, and a linear increasing strategy is adopted for the mutation iteration number:

[0048]

[0049] In the formula, r chmax and r chmin are the upper and lower limits of the set mutation rate, g′ ch is the mutation iteration number during the previous mutation, and rat is the set linear increasing factor. Iter represents the i-th iteration; the total number of iterations is ger.

[0050] The beneficial effects of the present invention are as follows:

[0051] 1. Based on the weighted coefficient method, the frictional energy consumption is used to characterize the joint friction and introduced into the trajectory optimization of the parallel platform, forming a comprehensive optimization objective function with the time optimal, so that the planned trajectory of the parallel platform reduces the energy consumption caused by joint friction to the greatest extent and reduces the wear of the mechanical structure to the greatest extent.

[0052] 2. For the optimization of complex optimization functions, the improved particle swarm algorithm introduces adaptive particle mutation, which has good global search ability and local search ability while ensuring global search ability, and has a fast convergence speed. Brief Description of the Drawings

[0053] Figure 1 Schematic diagram of the 3-PPR parallel platform mechanism.

[0054] Figure 2 Algorithm flow of the improved particle swarm algorithm. Detailed Embodiments

[0055] The present invention will be further described below in conjunction with the drawings and embodiments.

[0056] As Figure 1 and 2 shown, the technical solutions adopted by the present invention to solve its technical problems include the following steps:

[0057] Step 1, build a 3-PPR parallel platform with redundant branches;

[0058] Step 2: Establish the kinematic equations for the 3-PPR parallel platform and establish the joint friction model of the parallel platform;

[0059] Step 3: Optimize the motion trajectory of the parallel platform based on the joint friction model.

[0060] For the construction of the 3-PPR parallel platform with redundant legs described in Step 1, the specific implementation is as follows: The 3-PPR parallel robot consists of a moving platform, a fixed platform, and 4 legs connecting the moving platform and the fixed platform. One of the 4 leg groups is set as a redundant leg. Each leg consists of a prismatic pair P1, a prismatic pair P2, and a revolute pair R from bottom to top. The mechanism power input end is the prismatic pair P1 of the other 3 legs (the 3 legs excluding the redundant leg); the legs are arranged in an equilateral rectangle, specifically as Figure 1 shown.

[0061] For the establishment of the joint friction model of the parallel platform described in Step 2, the specific implementation is as follows:

[0062] 2-1. Construct the forward and inverse kinematic equations of the 3-PPR parallel platform with redundant legs through homogeneous coordinate transformation method.

[0063] 2-1-1. Establish a fixed coordinate system O0-X0Y0Z0 with the center O0 of the rectangle formed by the plane legs of the fixed platform as the origin, and establish a moving coordinate system O1-X1Y1Z1 with the geometric center point O1 of the moving platform as the origin. According to the spatial constraints of the parallel platform, determine the coordinates of the slider center points M i , N i in the fixed coordinate system and the position coordinates of the revolute pair rotation center L i in the moving coordinate system. According to the mapping relationship between any position vector T p in the moving coordinate system and the fixed coordinate system, obtain the position vector L i of the revolute pair rotation center L io relative to the fixed coordinate system.

[0064] L io = Rot·T p + P

[0065] where Rot is the attitude matrix of the moving platform relative to the fixed platform, and P = [x, y, z] T represents the position vector of the moving coordinate system relative to the fixed coordinate system.

[0066] 2-1-2. Calculate the position vectors i of the slider center points M i , N i to the revolute pair rotation center L

[0067] 2-1-3. Due to the particularity of the 3-PPR parallel platform, the displacement of the rotation centers L1 and L2 of the revolute pairs in the x0-axis direction is the same as that of the center points M1 and M2 of the sliders, and the displacement of the rotation centers L1 and L2 of the revolute pairs in the y0-axis direction is the same as that of the center points N1 and N2 of the sliders. The displacement of the rotation centers L3 and L4 of the revolute pairs in the x0-axis direction is the same as that of the center points N3 and N4 of the sliders, and the displacement of the rotation centers L3 and L4 of the revolute pairs in the y0-axis direction is the same as that of the center points M3 and M4 of the sliders. Let the displacements of the center points M1 and M2 of the lower sliders in the y0 direction be S1 and S2, the displacements of the center points M3 and M4 of the lower sliders in the x0 direction be S3 and S4, the displacements of the center points N1 and N2 of the upper sliders in the x0 direction be S5 and S6, and the displacements of the center points N3 and N4 of the upper sliders in the y0 direction be S7 and S8. Then, according to and the positional relationship of each slider relative to the moving platform can be obtained:

[0068] S k = f k (x, y, θ) k = 1, 2,..., 8

[0069] where x represents the offset of the moving platform relative to the fixed coordinate system in the x0-axis direction, y represents the offset of the moving platform relative to the fixed coordinate system in the y0-axis direction, θ represents the offset of the moving platform relative to the fixed coordinate system rotating in the positive direction around the z0-axis; f k (x, y, θ) represents the functional relationship of the positions of each slider relative to the moving platform.

[0070] 2-2. Based on the positional relationship between the sliders and the moving platform and the Coulomb-viscous friction model, a joint friction model of the parallel platform is established, and its loss during the movement is characterized in the form of energy.

[0071] 2-2-1. For the Coulomb friction of the branch chain joints, the normal pressures received by the prismatic pairs P1 and P2 are f n1 and f n2 , respectively. Then, the Coulomb friction received by the moving branch chain of the parallel platform is

[0072]

[0073] where μ represents the Coulomb friction coefficient; f c represents the total Coulomb friction received by the parallel platform, which is composed of the Coulomb friction f ci between each slider; sgn() represents the sign function; represents the velocity of each slider relative to the fixed coordinate system, that is, it has a derivative relationship with S k .

[0074] 2-2-2. For the viscous friction of the branched chain joints, since it is necessary to plan the motion trajectory of the driving joints, it is necessary to derive the velocity mapping of each sliding group relative to the driving joints:

[0075]

[0076] Among them, J is the Jacobian matrix, that is, the relationship between the pose of the moving platform of the parallel platform and the moving speed of the driving joints.

[0077] The viscous friction suffered by the moving branch of the parallel platform is

[0078]

[0079] Among them, η represents the viscous friction coefficient; f v represents the total viscous friction suffered by the parallel platform, which is composed of the viscous friction f vk between each slider.

[0080] The friction loss of the parallel platform joints characterized by energy consumption can be expressed as:

[0081]

[0082] Based on the weighted coefficient method, a comprehensive optimization objective function composed of the minimum joint friction energy consumption and the optimal time is established:

[0083] min J = ξ1σE + ξ2(t f - t0)

[0084] In the formula, ξ1 and ξ2 are optimization weight coefficients, and ξ1 + ξ2 = 1. σ is the elastic coefficient, which balances the influence of the difference in the order of magnitude between time and energy consumption through the elastic coefficient. According to the actual operation situation, write the trajectory equality constraints and inequality constraints of the parallel platform:

[0085]

[0086] In the formula, s(t), are the displacement function, velocity function, and acceleration function of the driving joints of the parallel platform; s0, represent the displacement, velocity, and acceleration values at the initial time t0; s f , represent the displacement, velocity, and acceleration values at the end time t f .

[0087] Furthermore, the joint friction model described in step 3 is optimized. Taking the kinematic parameters of the mechanism as the constraint conditions, the improved particle swarm algorithm is used for solution. Introduce the particle mutation process based on chaotic mapping to increase particle diversity and avoid particles falling into local optimal points; use the value of the number of iterations to calculate the inertia weight ω and the mutation rate rch and the mutation iteration number g ch Conduct adaptive adjustment to increase the coordination of global search and local search.

[0088] The specific optimization implementation steps of Step 3 are as follows:

[0089] 3-1. Set the initial inertia weight ω, learning factors c1, c2, the number of particle swarms num, the maximum iteration number ger, and the range limits of particle velocity and position, and set the initial mutation rate r of particle chaotic mapping ch and the initial mutation iteration number g ch .

[0090] 3-2. Select a suitable interpolation function, such as a polynomial interpolation function, a B-spline interpolation function, etc. as the trajectory curve of the parallel platform, that is, the displacement curve s(t) of the driving joint. The unknown parameters in the interpolation function constitute the multi-dimensional position parameters of the particle. Randomly initialize the velocity and position Calculate the fitness function value of the particle through the comprehensive optimization objective function, and take the fitness function value of each particle as the individual optimal position pbest, and take the optimal fitness function value in the particle swarm as the global optimal position gbest;

[0091] 3-3. After each iteration, update the velocity and position of the particle, and calculate the corresponding fitness function value. If the fitness function value is better than the individual optimal position pbest of the particle, update the optimal position of the particle. If the fitness function value of the particle is better than the global optimal gbest, update the global optimal position. The position and velocity update formulas are as follows:

[0092] v j = ω × v j + c1 × rand() × (pbest - p j ) + c2 × rand() × (gbest - p j )

[0093] p j = p j + v j

[0094] Among them, the inertia weight ω determines the optimization ability of the particle optimization algorithm. Increasing ω enhances the global search ability of the particle, and decreasing ω enhances the local search ability of the particle. The adaptive inertia weight particle algorithm is adopted:

[0095]

[0096] Among them, f avg is the average objective function value of all current particles, f minis the minimum objective function value of all current particles, ω max , ω min are the maximum inertia weight and the minimum inertia weight respectively.

[0097] 3-4. The particle mutation counter increases with the number of iterations. When the particle mutation count reaches the set mutation iteration number, to avoid the particles falling into the local optimal solution, the particles are sorted according to their individual optimal positions, and the particles with better performance are retained. The number of these particles is determined by the current mutation rate. At the same time, the remaining particles with poorer performance generate a chaotic sequence using the chaotic mapping method and replace the original particle sequence to ensure that the mapping makes the chaotic sequence relatively evenly distributed.

[0098] 3-5. The mutation rate and the initial mutation iteration number can be set with larger initial values to improve the diversity of particles in the initial stage of optimization. During the iteration process, a linear differential decreasing strategy is adopted for the mutation rate, and a linear increasing strategy is adopted for the mutation iteration number:

[0099]

[0100] where r chmax , r chmin are the upper and lower limits of the set mutation rate, g' ch is the mutation iteration number at the previous mutation, and rat is the set linear increasing factor.

Claims

1. A trajectory planning method for a parallel mechanism considering joint friction, characterized in that By improving the particle swarm optimization algorithm, the joint friction model, and time optimality to comprehensively optimize the objective, the trajectory planning is realized to minimize the wear of the mechanical structure and the energy consumption during operation. The specific steps are as follows: Step 1: Build a 3-PPR parallel platform with redundant branches; Step 2: Establish the kinematic equation for the 3-PPR parallel platform and establish the joint friction model of the parallel platform; Step 3: Optimize the motion trajectory of the parallel platform based on the joint friction model; The construction of the 3-PPR parallel platform with redundant branches described in step 1 is specifically implemented as follows: the 3-PPR parallel robot consists of a moving platform, a fixed platform, and four branches connecting the moving platform and the fixed platform, one of the four branches is a redundant branch; each branch from bottom to top is a moving branch , mobile vice , rotating pair R, the power input end of the mechanism is the moving pair of the remaining three branches except the redundant branch ; Each branch chain is arranged in an equilateral rectangle; Optimize the motion trajectory of the parallel platform based on the joint friction model described in step 3, use the improved particle swarm algorithm to solve with the mechanism kinematic parameters as the constraint conditions; introduce a particle mutation process based on chaotic mapping to increase particle diversity and avoid particles falling into local optimal points; use the value of the number of iterations to adaptively adjust the inertia weight , mutation rate and the mutation iteration number to increase the coordination of global search and local search. The specific optimization implementation steps are as follows: 3-1. Set the initial inertia weight , learning factor , the number of particle swarms num, the maximum number of iterations ger, and the range limits of the particle velocity and position, and set the initial mutation rate of the particle chaotic mapping and the initial mutation iteration number ; 3-2. Select the interpolation function, i.e., the displacement curve of the driving joint , the unknown parameters in the interpolation function constitute the multi-dimensional position parameters of the particle; randomly initialize the velocity and position of the particle. Calculate the fitness function value of the particle by comprehensively optimizing the objective function, and take the fitness function value of each particle as the individual optimal position pbest, and take the optimal fitness function value in the particle swarm as the global optimal position gbest; 3-3. After each iteration, update the velocity and position of the particle and calculate the corresponding fitness function value. If the fitness function value is better than the particle's individual optimal position pbest, then update the optimal position of the particle. If the fitness function value of the particle is better than the global optimal gbest, then update the global optimal position. The position and velocity update formulas are as follows: (9) Among them, the inertia weight determines the optimization ability of the particle optimization algorithm. Increasing enhances the global search ability of the particle, and decreasing enhances the local search ability of the particle. An adaptive inertia weight particle algorithm is adopted: (10) where is the average objective function value of all current particles, is the minimum objective function value of all current particles, are the maximum inertia weight and the minimum inertia weight, respectively; 3-4. The particle mutation counter increases with the number of iterations. When the particle mutation count reaches the set mutation iteration number, to avoid the particle falling into the local optimal solution, sort according to the particle's individual optimal position, retain the particles with better performance, and the number of which is determined by the current mutation rate. At the same time, the remaining particles with poor performance generate a chaotic sequence using the chaotic mapping method and replace the original particle sequence to ensure that the mapping makes the chaotic sequence relatively evenly distributed; 3-5. Set relatively large initial values for the mutation rate and the initial mutation iteration number to improve the diversity of particles in the initial stage of optimization. During the iteration process, adopt a linear differential decreasing strategy for the mutation rate and a linear increasing strategy for the mutation iteration number: (11) where are the upper and lower limits of the set mutation rate, is the number of mutation iterations during the previous mutation, is the set linear increment factor; represents the i-th iteration; the total number of iterations is .

2. A trajectory planning method for a parallel mechanism considering joint friction according to claim 1, characterized in that The establishment of the joint friction model of the parallel platform described in Step 2 is specifically implemented as follows: 2-1. Construct the forward and inverse kinematic equations of the 3-PPR parallel platform with redundant branches through the homogeneous coordinate transformation method; 2-2. Based on the position relationship between the slider and the moving platform and the Coulomb-viscous friction model, establish the joint friction model of the parallel platform and characterize its loss during motion in the form of energy.

3. A trajectory planning method for a parallel mechanism considering joint friction according to claim 2, characterized in that The specific implementation of Step 2-1 is as follows: The center of the rectangle formed by the fixed platform plane linkages is used as the origin to establish a fixed coordinate system . The geometric center point of the moving platform is used as the origin to establish a moving coordinate system ; According to the spatial constraints of the parallel platform, determine the coordinates of the slider center points , in the fixed coordinate system and the position coordinates of the rotation center of the revolute pair in the moving coordinate system. According to the mapping relationship between any position vector in the moving coordinate system and the fixed coordinate system, obtain the position vector of the rotation center of the revolute pair relative to the fixed coordinate system ; (1) Among them, is the attitude matrix of the moving platform relative to the fixed platform, represents the position vector of the moving coordinate system relative to the fixed coordinate system; 2-1-2. Calculate the center point of the slider , to the rotation center of the revolute pair position vector , ; 2-1-3. Due to the special nature of the 3-PPR parallel platform, the rotation centers of the revolute pairs , have the same displacement as the center points of the sliders in the -axis direction. The rotation centers of the revolute pairs have the same displacement as the center points of the sliders , in the -axis direction. The rotation centers of the revolute pairs , have the same displacement as the center points of the sliders , in the -axis direction. The rotation centers of the revolute pairs , have the same displacement as the center points of the sliders , in the -axis direction. Let the displacements of the center points of the lower sliders , in the direction be , respectively. Let the displacements of the center points of the lower sliders , in the direction be respectively. Let the displacements of the center points of the upper sliders , respectively. Let the displacements of the center points of the upper sliders , in the direction be , respectively. Let the displacements of the center points of the upper sliders , in the direction be , respectively. Then, based on and , the relative position relationships of each slider with respect to the moving platform can be obtained:​ (2) Among them, x represents the offset of the moving platform relative to the fixed coordinate system in the axis direction, y represents the offset of the moving platform relative to the fixed coordinate system in the axis direction, represents the offset of the moving platform relative to the fixed coordinate system rotating around the positive direction of the axis; represents the functional relationship of the positions of the respective sliders relative to the moving platform.

4. A trajectory planning method for a parallel mechanism considering joint friction according to claim 3, characterized in that The specific implementation of Step 2-2 is as follows: 2-2-1. For the Coulomb friction of the branched chain joints, the prismatic pairs and the prismatic pairs are subjected to the normal pressures of and respectively. Then, the Coulomb friction suffered by the moving branches of the parallel platform is as follows: (3) Among them, represents the Coulomb friction coefficient; represents the total Coulomb friction on the parallel platform, which is composed of the Coulomb friction among the sliders; represents the sign function; represents the velocity of each slider relative to the fixed coordinate system, that is, it has a derivative relationship with ; 2-2-2. For the viscous friction of the branch joint, since the motion trajectory of the driving joint needs to be planned, it is necessary to derive the velocity mapping of each slider relative to the driving joint: (4) where J is the Jacobian matrix, that is, the relationship between the pose of the moving platform of the parallel platform and the moving speed of the driving joint; The viscous friction received by the motion branch of the parallel platform is: (5) Among them, represents the viscous friction coefficient; represents the total viscous friction suffered by the parallel platform, which is composed of the viscous friction among the sliders; The joint friction loss of the parallel platform characterized by energy consumption is expressed as: (6) Among them, represents the end moment, represents the initial moment; Establish a comprehensive optimization objective function composed of the minimum joint friction energy consumption and time optimality based on the weighted coefficient method: (7) In the formula, is the optimized weight coefficient, and , is the elastic coefficient, which balances the influence caused by the difference in the order of magnitude between time and energy consumption through the elastic coefficient; according to the actual operating conditions, write the trajectory equality constraints and inequality constraints of the parallel platform: (8) where are the displacement function, velocity function, and acceleration function of the driving joints of the parallel platform, respectively; respectively represent the initial moment displacement, velocity, and acceleration values; respectively represent the end moment displacement, velocity, and acceleration values.

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