Layout algorithm for heterogeneous mirror image milling system

Abstract

The invention discloses a layout algorithm for a heterogeneous mirror image milling system. The layout algorithm comprises the following steps that A, robot geometric modeling and reachable space analysis are conducted on a hybrid robot and a series robot respectively; b, public area boundary points of the two reachable spaces are extracted, and then the area of an overlapping area of the reachable spaces is calculated; c, analyzing the operability and the bearing performance of the series robot according to the machining task; d, performing rule workspace calculation based on an optimization algorithm; in the reachable space overlapping area, the maximum regular geometry is defined as the maximum regular working space, and the machining track is located in the maximum regular working space; e, multi-index layout optimization based on principal component analysis; according to the boundary condition of the layout constraint condition, determining a layout parameter value range; and calculating a reachable space overlapping region, a maximum regular working space area, region operability and region bearing performance of each group of layout parameters by taking 1mm as a step length. According to the layout algorithm, the scientificity and the accuracy of layout optimization are improved.

Description

Technical Field

[0001] The present invention relates to the field of mirror milling of thin-walled components, and in particular to a layout algorithm for a heterogeneous mirror milling system. Background Art

[0002] Multi-robot systems are characterized by redundancy, robustness, scalability, and flexibility, enabling them to perform more complex collaborative tasks. In the study of multi-robot systems, system layout optimization is key to the placement design of robot work cells, affecting the system's collaborative work efficiency and workspace utilization.

[0003] The existing technology has a layout optimization method for a dual-robot machining center based on a genetic algorithm. The optimization goal is to maximize the workspace overlap rate. Compared with the original layout, the spatial overlap rate is increased by 14.6% and the machining cycle time is shortened by 1.2 minutes.

[0004] There is also a hybrid algorithm in the existing technology for multi-robot assembly system layout optimization, which combines five nature-inspired algorithms, including differential evolution algorithm (DE), artificial bee colony algorithm (ABC), charge search algorithm (CSS), particle swarm optimization algorithm (PSO) and genetic algorithm (GA), to simultaneously optimize the three indicators of layout area, operation time and operability to find the optimal layout design.

[0005] In the existing technology, the research objects are mostly homogeneous serial multi-robot systems, such as dual-tandem arm processing systems and homogeneous multi-robot assembly systems. They are mainly used for high-intensity, repetitive, and harsh environment operations, and are not suitable for scenarios with high requirements for precision or rigidity. For mirror milling scenarios, serial / hybrid heterogeneous multi-robot systems can better play the complementarity of multi-robot systems, where hybrid robots ensure the accuracy of milling processing and serial robots ensure the flexibility of support collaboration. Compared with homogeneous multi-robot systems, due to the asymmetry of the system structure, the layout indicators of heterogeneous multi-robot systems are difficult to quantify and evaluate, and there are many parameters and complex calculations. At present, there is a lack of layout optimization methods for multiple performance indicators of heterogeneous multi-robot systems. In addition, there is a correlation between the various indicators of system layout. The existing layout algorithms do not take into account the correlation of various indicators. The redundant features in the data will interfere with the construction of a comprehensive evaluation index model.

[0006] Therefore, a heterogeneous dual-robot system layout algorithm that comprehensively considers the four performance indicators of reachable space overlap area, maximum regular workspace, regional operability and regional carrying capacity has become a problem that the industry needs to solve. Summary of the Invention

[0007] In order to address the deficiencies in the prior art, the main purpose of the present invention is to provide a heterogeneous dual-robot system layout algorithm that comprehensively considers four performance indicators, namely, the overlapping area of ​​achievable space, the maximum regular workspace, regional operability, and regional carrying capacity. The algorithm quantifies the performance of the robots in the working area, and uses principal component analysis (PCA) to analyze the correlation between the four performance indicators to construct a comprehensive evaluation index function model.

[0008] To achieve the above main objectives, the present invention discloses a layout algorithm for a heterogeneous mirror milling system, wherein the heterogeneous mirror milling system includes a hybrid robot at the processing end and a serial robot at the support end. The layout algorithm includes the following steps:

[0009] A. Robot geometry modeling and reachable space analysis for hybrid robots and serial robots respectively;

[0010] B. Extract the common boundary points of the two reachable spaces and calculate the area of ​​the overlapping area of ​​the reachable spaces;

[0011] C. Analyze the operability and load-bearing performance of the serial robot for processing tasks;

[0012] D. Calculate the regular workspace based on the optimization algorithm; within the overlapping area of ​​the reachable space, the largest regular geometry is defined as the maximum regular workspace, and the machining trajectory is located within the maximum regular workspace;

[0013] E. Multi-index layout optimization based on principal component analysis; determine the range of layout parameter values ​​based on the boundary conditions of the layout constraints; calculate the reachable spatial overlap area, maximum regular workspace area, regional operability, and regional carrying capacity for each set of layout parameters with a step size of 1 mm.

[0014] In the present invention, the processing end uses a hybrid robot with high precision and good rigidity, which can meet the needs of milling processing. The support end uses a serial robot with good flexibility and large working space.

[0015] According to a specific embodiment of the present invention, in step A, a kinematic model of the hybrid robot is established by a closed-loop vector method, and a hierarchical search method is used to calculate the reachable space point cloud of the hybrid robot.

[0016] According to a specific embodiment of the present invention, in step A, a serial robot exponential product (POE) kinematic model is established, and the reachable space point cloud is calculated using Monte Carlo.

[0017] To simplify the calculation, only the accessible space of the symmetric plane is considered. The common accessible space area of ​​the two robots on the symmetric plane is one of the quantitative indicators for evaluating the layout plan. According to a specific embodiment of the present invention, in step B, the Alpha shapes algorithm is used to extract the boundary points of the common area of ​​the accessible space, and then the area K of the overlapping area of ​​the accessible space is calculated. v .

[0018] Analyze the operability and load-bearing performance of the support end serial robot for the processing task. The mirror milling support end needs to achieve flexible collaborative support within the operating space and have a high load-bearing capacity. Quantify the performance through operability indicators and load-bearing performance indicators, construct relative operability graphs and load-bearing performance graphs, eliminate areas with poor performance in the accessible space, and ensure that it has excellent operability and load-bearing performance. According to a specific embodiment of the present invention, step C includes the following steps:

[0019] C1. Define the determinant of the product of the Jacobian matrix and its transposed matrix as the operability index

[0020]

[0021] In the above formula, w is the maneuverability, J(q) is the robot Jacobian matrix, and ‖·‖ is the Euclidean norm. When w = 0, the robot is in a singular configuration. The larger w is, the further away from the singular configuration. The threshold method is used to eliminate areas with poor maneuverability in the reachable space to ensure that the working area has high maneuverability.

[0022] C2. The load-bearing capacity is defined as the extreme value of the modulus ‖F‖ of the generalized force at the end of the mechanism when the modulus ‖f‖ of the driving force is 1. When ‖f‖ is 1, the extreme value of the load-bearing capacity ‖F‖ max for

[0023]

[0024] where λ Fmax represents the largest eigenvalue of the matrix, is the maximum singular value of the matrix, H(q)=J -1 (q) is the force Jacobian matrix;

[0025] C3. Use the following method to quantify the performance of the working area

[0026]

[0027] Where k λ is the operability of a point in the working area, k η is the load-bearing performance of a point in the working area, and W is the working area.

[0028] According to a specific embodiment of the present invention, step D comprises the following steps:

[0029] D1, the maximum regular workspace is defined as the largest inscribed axis-aligned rectangle in the overlap region of the reachable space;

[0030] D2. Convert geometric problems into nonlinear programming problems

[0031] minf(x)

[0032] g i (x)≤0,i=1,2,...,m

[0033] h j (x)=0,j=1,2,...,r

[0034] Where f(x) is the objective function, g i (x)≤0 is the inequality constraint, h j (x) = 0 is an equality constraint. The nonlinear constraints for this problem include two points: a. All four vertices of the inscribed rectangle to be found must be inside or on the edge of the overlapping region; b. The intersection of the inscribed rectangle to be found and the overlapping region must be equal in size to the inscribed rectangle to be found.

[0035] D3. Set the optimization objective to the area of ​​the inscribed rectangle, and the optimization variables to be a vector group m = [m(1), m(2), m(3), m(4)] containing four elements, where m(1) and m(2) represent the horizontal and vertical coordinates of the lower left corner of the maximum inscribed rectangle to be found, and m(3) and m(4) represent the length and width of the rectangle, respectively.

[0036] D4. An axis-aligned rectangle in a two-dimensional plane is uniquely determined by the m vector. Use the interior point method (IPM) to solve the nonlinear optimization problem. Set the objective function f(x), the initial value m0, the upper bound supm of the variable, and the lower bound infm of the variable. Use the interior point method (IPM) to solve the nonlinear programming problem and terminate the solution when the change between two iterations is less than the tolerance. Use the maximum regular workspace area obtained as a quantitative indicator K of the layout. w .

[0037] Since the four performance indicators have a certain correlation, in order to eliminate data interference between different indicators, according to a specific embodiment of the present invention, in step E, principal component analysis (PCA) is used to analyze the indicator correlation and perform multi-objective optimization; a comprehensive evaluation indicator function model is established to search for the optimal layout parameters.

[0038] The present invention has the following beneficial effects:

[0039] Compared with existing algorithms, the present invention is aimed at serial / hybrid heterogeneous multi-robot systems, giving full play to the advantages of high precision of hybrid robots and high flexibility of serial robots, filling the gap in the layout optimization method of heterogeneous multi-robot systems. In terms of solving the maximum regular workspace, a solution algorithm based on nonlinear programming has been developed, which avoids complex mathematical derivation and has higher computational efficiency. The principal component analysis method (PCA) is used to perform correlation analysis on multiple performance indicators and construct a comprehensive evaluation index function model. The layout algorithm of the present invention can effectively eliminate data interference between different indicators, reduce information overlap, achieve multi-objective optimization, and improve the scientificity and accuracy of layout optimization.

[0040] In order to more clearly illustrate the purpose, technical solutions and advantages of the present invention, the present invention is further described in detail below with reference to the accompanying drawings and specific embodiments. BRIEF DESCRIPTION OF THE DRAWINGS

[0041] Figure 1 is a schematic diagram of a heterogeneous mirror milling system in Example 1;

[0042] Figure 2 This is a flowchart of the hierarchical search algorithm for the reachable space of the hybrid robot at the processing end in Example 1;

[0043] Figure 3 Schematic diagram of the reachable space of the hybrid robot at the processing end in Example 1;

[0044] Figure 4 Schematic diagram of the reachable space of the support end serial robot in Example 1;

[0045] Figure 5 Result diagram of the maneuverability analysis of the support end serial robot in Example 1;

[0046] Figure 6 Result diagram of workspace correction based on maneuverability of the support-end serial robot in Example 1;

[0047] Figure 7 1 is a graph showing the results of the load-bearing performance analysis of the support-end serial robot in Example 1;

[0048] Figure 8 This is a result diagram of the workspace correction of the support-end serial robot based on the load-bearing performance in Example 1;

[0049] Figure 9 is a schematic diagram of the spatial overlap area accessible by the heterogeneous mirror milling system in Example 1;

[0050] Figure 10 This is the result diagram of the maximum regular workspace solution in Example 1;

[0051] Figure 11A In Example 1, the reachable space overlap area K v The result graph;

[0052] Figure 11B In Example 1, the maximum regular workspace K w The result graph;

[0053] Figure 11C In Example 1, the regional operability K λ The result graph;

[0054] Figure 11D In Example 1, the regional carrying capacity K η The result graph;

[0055] Figure 12 This is a diagram showing the results of calculating the comprehensive evaluation index using principal component analysis in Example 1;

[0056] Figure 13 This is a schematic diagram of the optimal layout solution in Example 1. DETAILED DESCRIPTION

[0057] In the following description, many specific details are set forth in conjunction with the embodiments to facilitate a full understanding of the present invention. However, it should be understood that the following embodiments and detailed descriptions are only for illustrative purposes and do not limit the scope of protection of the present invention.

[0058] Example 1

[0059] This embodiment provides a layout algorithm for a heterogeneous mirror milling system. The heterogeneous mirror milling system includes a hybrid robot at the processing end and a serial robot at the support end. The layout algorithm includes the following steps:

[0060] A. Robot geometry modeling and reachable space analysis are performed for the hybrid robot and serial robot respectively. The kinematic model of the hybrid robot is established using the closed-loop vector method, and the hierarchical search method is used to calculate the reachable space point cloud of the hybrid robot. The product of exponential (POE) kinematic model of the serial robot is established, and the reachable space point cloud is calculated using Monte Carlo.

[0061] B. Extract the common boundary points of the two reachable spaces and calculate the area of ​​the overlapping area of ​​the reachable spaces. To simplify the calculation, only the reachable spaces on the symmetric plane are considered. The common reachable space area of ​​the two robots on the symmetric plane is one of the quantitative indicators for evaluating the layout plan. Use the Alpha shapes algorithm to extract the common boundary points of the reachable spaces and calculate the area K of the overlapping area of ​​the reachable spaces. v .

[0062] C. Analyze the operability and load-bearing performance of the serial robot for the processing task; analyze the operability and load-bearing performance of the support end serial robot for the processing task. The mirror milling support end needs to achieve flexible collaborative support within the operating space and have a high load-bearing capacity. Quantify the performance through operability indicators and load-bearing performance indicators, construct relative operability graphs and load-bearing performance graphs, and eliminate areas with poor performance in the accessible space to ensure it has excellent operability and load-bearing performance. Step C includes the following steps:

[0063] C1. Define the determinant of the product of the Jacobian matrix and its transposed matrix as the operability index

[0064]

[0065] In the above formula, w is the maneuverability, J(q) is the robot Jacobian matrix, and ‖·‖ is the Euclidean norm. When w = 0, the robot is in a singular configuration. The larger w is, the further away from the singular configuration. The threshold method is used to eliminate areas with poor maneuverability in the reachable space to ensure that the working area has high maneuverability.

[0066] C2. The load-bearing capacity is defined as the extreme value of the modulus ‖F‖ of the generalized force at the end of the mechanism when the modulus ‖f‖ of the driving force is 1. When ‖f‖ is 1, the extreme value of the load-bearing capacity ‖F‖ max for

[0067]

[0068] where λ Fmax represents the largest eigenvalue of the matrix, is the maximum singular value of the matrix, H(q)=J -1 (q) is the force Jacobian matrix;

[0069] C3. Use the following method to quantify the performance of the working area

[0070]

[0071] Where k λ is the operability of a point in the working area, k η is the load-bearing performance of a point in the working area, and W is the working area.

[0072] D. Calculate the regular workspace based on the optimization algorithm; within the overlapping area of ​​the reachable space, the largest regular geometric body is defined as the largest regular workspace, and the machining trajectory is located within the largest regular workspace; Step D includes the following steps:

[0073] D1, the maximum regular workspace is defined as the largest inscribed axis-aligned rectangle in the overlap region of the reachable space;

[0074] D2. Convert geometric problems into nonlinear programming problems

[0075] minf(x)

[0076] g i (x)≤0,i=1,2,...,m

[0077] h j (x)=0,j=1,2,...,r

[0078] Where f(x) is the objective function, g i (x)≤0 is the inequality constraint, h j (x) = 0 is an equality constraint. The nonlinear constraints for this problem include two points: a. All four vertices of the inscribed rectangle to be found must be inside or on the edge of the overlapping region; b. The intersection of the inscribed rectangle to be found and the overlapping region must be equal in size to the inscribed rectangle to be found.

[0079] D3. Set the optimization objective to the area of ​​the inscribed rectangle, and the optimization variables to be a vector group m = [m(1), m(2), m(3), m(4)] containing four elements, where m(1) and m(2) represent the horizontal and vertical coordinates of the lower left corner of the maximum inscribed rectangle to be found, and m(3) and m(4) represent the length and width of the rectangle, respectively.

[0080] D4. An axis-aligned rectangle in a two-dimensional plane is uniquely determined by the m vector. Use the interior point method (IPM) to solve the nonlinear optimization problem. Set the objective function f(x), the initial value m0, the upper bound supm of the variable, and the lower bound infm of the variable. Use the interior point method (IPM) to solve the nonlinear programming problem and terminate the solution when the change between two iterations is less than the tolerance. Use the maximum regular workspace area obtained as a quantitative indicator K of the layout. w .

[0081] E. Multi-index layout optimization based on principal component analysis: Determine the range of layout parameter values ​​based on the boundary conditions of the layout constraints. Calculate the reachable spatial overlap area, maximum regular workspace area, regional operability, and regional load-bearing capacity for each set of layout parameters with a step size of 1 mm. Because the four performance indicators have a certain correlation, principal component analysis (PCA) is used to analyze the correlation between indicators to eliminate data interference between different indicators. Multi-objective optimization is then performed. A comprehensive evaluation indicator function model is established to search for the optimal layout parameters.

[0082] Specifically, the system composition is first confirmed. The system uses hybrid robots to ensure the accuracy of milling processing and serial robots to ensure the flexibility of support collaboration, such as Figure 1The reachable space of the hybrid robot can be solved by using a hierarchical search method to estimate a search space that is larger than the actual reachable space of the mechanism. The X range in the length direction is [-600, 600]; the Y range in the width direction is [-800, 700]; and the Z range in the height direction is [400, 1100]. The estimated space is divided into several planes with an interval of 1mm along the Z direction. In each plane, the X and Y directions are discretized and searched with an interval of 1mm. When the search of this layer is completed, increase the Z value and repeat the above steps to search. The search process is as follows: Figure 2 As shown in . Since only the reachable space of the symmetric plane is considered, the reachable space of the hybrid robot on the symmetric plane is obtained as Figure 3 As shown. For the serial robot, based on the POE geometric model, the Monte Carlo method is used to calculate the reachable space. For each joint variable, a random value is taken within the joint variation range to generate a set of joint variable vectors. The program continuously calculates the TCP coordinates corresponding to each joint variable vector to obtain the reachable space point cloud of the serial robot, as shown in Figure 4 shown.

[0083] Next, we use the operability index to quantify the flexibility of the supporting robot in the operating space. The results are as follows: Figure 5 As shown in the figure, a threshold of 40% is defined, and any area with an operability lower than 40% of the maximum operability will be removed. The remaining area will have better operability. The result is as follows: Figure 6 The load-bearing capacity of the retained area is quantified, and the results are shown in Figure 7 Similar to the above operation, a threshold of 40% is defined, and any area with a load-bearing capacity lower than 40% of the maximum load-bearing capacity will be removed, and the remaining area will have a better load-bearing capacity. The result is shown in Figure 8 shown.

[0084] The developed algorithm is used below to solve the maximum regular workspace in the overlapping area of ​​the reachable space. Figure 9 As shown in the green area. Set the objective function f(x), initial value m0, upper bound supm and lower bound infm

[0085] f(x)=-m(3)×m(4)

[0086] m0=[(x min +x max ) / 2,(z min +z max ) / 2,(x max -x min ) / 2,(z max -z min ) / 2]

[0087] supm=[x max ,zmax ,x max -x min ,z max -z min ]

[0088] infm=[x min ,z min ,0,0]

[0089] The initial value m0 vector indicates that the search starts from the center of the maximum enclosing rectangle, and the upper bound of the variable supm represents the maximum enclosing rectangle. After determining the initial value, upper and lower bounds, objective function and constraints, use IPM to solve the nonlinear programming problem and set the iteration tolerance to 10 -5 , the iteration is terminated when the difference between two iterations is less than the tolerance. Taking the layout parameters (ΔX=1400mm, ΔZ=800mm) as an example, the output result is m=[689.9685,139.9928,180.0620,170.0131], and the maximum rule workspace area is S1=0.0306m 2 ,like Figure 10 As shown in the red box, the algorithm running time T1 = 5.37s. Compared with the widely used genetic algorithm, the genetic algorithm is Figure 10 As shown in the blue box, the maximum regular workspace area is S2 = 0.0293m 2 , the algorithm running time T2 = 79.46s. Compared with the genetic algorithm, the proposed algorithm has a solution result difference of only 4.4%, while the efficiency is improved by 93.2%.

[0090] The range of layout parameters is determined based on the boundary conditions corresponding to the layout constraints. The constraints of the layout parameters are: (1) there is an overlapping area in the reachable space of the two manipulators; (2) avoid interference between the manipulators. The calculation domain of the layout parameters is determined to be ΔX∈[1000mm,1750mm], ΔZ∈[-600mm,1350mm]. To improve efficiency, we first search for areas where extreme values ​​may appear in the entire calculation domain with a large step size, and only consider the overlapping area K of the reachable space. v , narrow the calculation domain. In the entire calculation domain process, the step size is selected as 1mm. When the layout parameters are (ΔX=1320mm,ΔZ=800mm), the reachable space overlap rate reaches the maximum value. According to the results of the global search, the calculation domain ΔX∈[1220mm,1420mm], ΔZ∈[700mm,900mm]. In the second search, the step size of 1mm is used to calculate the reachable space overlap area K for each set of layout parameters. v , maximum rule workspace K w , Regional operability K λ , Regional carrying capacity K η , the results are as follows Figures 11A-11Dshown.

[0091] Finally, the principal component analysis (PCA) method is used to analyze the correlation between the indicators, and the above four quantitative indicators are used as data samples for PCA analysis. Since the indicator data in the table are not at the same order of magnitude, the Z-Score method is used to standardize the data. By analyzing the eigenvalues ​​and eigenvectors of the correlation coefficient matrix, the principal components can be constructed through orthogonal transformation to replace the various performance indicators, thereby achieving data dimensionality reduction. By using the eigenvalues ​​of the correlation coefficient matrix as weights, a comprehensive evaluation index function for layout parameters can be constructed. The optimal layout solution is obtained by searching for the optimal value of the comprehensive evaluation index function. The optimal layout parameters are (ΔX = 1338 mm, ΔZ = 760 mm), the maximum value of the comprehensive index is 0.8158, and the optimal result is as follows. Figure 12 As shown in the middle blue dot. According to the layout parameters, the layout can be placed as follows Figure 13 shown.

[0092] Although the present invention has been described above through embodiments, the above embodiments are only used to exemplify the possible implementation schemes of the present invention and are not used to limit the scope of protection of the present invention. Any equivalent substitutions or changes made by those skilled in the art in accordance with the present invention should also be covered by the scope of protection defined by the claims of the present invention.

Claims

1. A layout algorithm for a heterogeneous mirror milling system, characterized in that: The heterogeneous mirror milling system includes a hybrid robot at the processing end and a serial robot at the support end. The layout algorithm includes the following steps: A. performing robot geometric modeling and reachable space analysis on the hybrid robot and the serial robot respectively; B. Extract the common boundary points of the two reachable spaces and calculate the area of ​​the overlapping area of ​​the reachable spaces; C. Analyze the operability and load-bearing performance of the serial robot for processing tasks; D. Calculate the regular workspace based on the optimization algorithm; within the overlapping area of ​​the reachable space, the largest regular geometric body is defined as the maximum regular workspace, and the machining trajectory is located within the maximum regular workspace; E. Multi-index layout optimization based on principal component analysis; determine the range of layout parameter values ​​based on the boundary conditions of the layout constraints; calculate the reachable spatial overlap area, maximum regular workspace area, regional operability, and regional carrying capacity for each set of layout parameters with a step size of 1 mm.

2. The layout algorithm according to claim 1, characterized in that In step A, a kinematic model of the hybrid robot is established by a closed-loop vector method, and a hierarchical search method is used to calculate the reachable space point cloud of the hybrid robot.

3. The layout algorithm according to claim 2, characterized in that In step A, an exponential product kinematic model of the serial robot is established, and a Monte Carlo method is used to calculate the reachable space point cloud.

4. The layout algorithm according to claim 1, wherein: In step B, the Alpha shapes algorithm is used to extract the boundary points of the common area of ​​the reachable space, and then the area K of the overlapping area of ​​the reachable space is calculated. v .

5. The layout algorithm according to claim 1, characterized in that Step C includes the following steps: C1. Define the determinant of the product of the Jacobian matrix and its transposed matrix as the operability index In the above formula, w is the maneuverability, J(q) is the robot Jacobian matrix, and ‖·‖ is the Euclidean norm. When w = 0, the robot is in a singular configuration. The larger w is, the further away from the singular configuration. The threshold method is used to eliminate areas with poor maneuverability in the reachable space. C2. The load-bearing capacity is defined as the extreme value of the modulus ‖F‖ of the generalized force at the end of the mechanism when the modulus ‖f‖ of the driving force is 1. When ‖f‖ is 1, the extreme value of the load-bearing capacity ‖F‖ max for where λ Fmax represents the largest eigenvalue of the matrix, is the maximum singular value of the matrix, H(q)=J -1 (q) is the force Jacobian matrix; C3. Use the following method to quantify the performance of the working area Where k λ is the operability of a point in the working area, k η is the load-bearing performance of a point in the working area, and W is the working area.

6. The layout algorithm according to claim 1, characterized in that Step D comprises the following steps: D1, the maximum regular workspace is defined as the largest inscribed axis-aligned rectangle in the overlap region of the reachable space; D2. Convert geometric problems into nonlinear programming problems minf(x) g i (x)≤0,i=1,2,...,m h j (x)=0,j=1,2,...,r Where f(x) is the objective function, g i (x)≤0 is the inequality constraint, h j (x) = 0 is an equality constraint. The nonlinear constraints for this problem include two points: a. All four vertices of the inscribed rectangle to be found must be inside or on the edge of the overlapping region; b. The intersection of the inscribed rectangle to be found and the overlapping region must be equal in size to the inscribed rectangle to be found. D3. Set the optimization objective to the area of ​​the inscribed rectangle, and the optimization variables to be a vector group m = [m(1), m(2), m(3), m(4)] containing four elements, where m(1) and m(2) represent the horizontal and vertical coordinates of the lower left corner of the maximum inscribed rectangle to be found, and m(3) and m(4) represent the length and width of the rectangle, respectively. D4, the axis-aligned rectangle in the two-dimensional plane is uniquely determined by the m vector; Use the interior point method (IPM) to solve the nonlinear optimization problem; set the objective function f(x), the initial value m0, the upper bound supm of the variable, and the lower bound infm of the variable; The interior point method (IPM) is used to solve the nonlinear programming problem. The solution is terminated when the change between two iterations is less than the tolerance. The maximum regular workspace area obtained by the solution is used as a quantitative indicator K of the layout. w .

7. The layout algorithm according to claim 1, characterized in that: In step E, principal component analysis (PCA) is used to analyze the correlation of indicators and perform multi-objective optimization; a comprehensive evaluation indicator function model is established to search for the optimal layout parameters.

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