Method for generating workspace boundary of robot end effector based on linear programming

Abstract

The present invention relates to a method for generating the workspace boundary of a robot end effector based on linear programming, including the steps of: S1. Kinematic modeling of robot motion is constructed by parametrically representing joint motion according to the joint characteristics of the robot; S2. Linear equations for all constraints required to generate the workspace boundary of the robot end effector are respectively constructed; S3. A system of linear constraints that needs to be satisfied for generating the workspace boundary of the robot end effector is determined; S4. Robot joint motion parameters that satisfy the linear equations in the system of constraints are obtained; S5. Boundary joint motion parameters that simultaneously satisfy some of the constraints are found from the robot joint motion parameters that satisfy the linear equations in the system of linear constraints; S6. The workspace boundary points of the robot end effector are identified and determined. When arranging robots and industrial production lines, it is simplified to the work of overlapping the industrial production line and the robot workspace, greatly simplifying the robot layout work.

Description

Technical Field

[0001] The present invention relates to the technical field of automatic control, and particularly to a method for generating the workspace boundary of a robot. Background Art

[0002] With the continuous development of automatic control technology, robots are widely used in various industries. In various scenarios and tasks where robots are applied, there are different requirements for the positions and postures that the end effector of the robot needs to reach. When arranging robots and industrial production lines, the traditional method is to roughly estimate the activity range of the end effector of the robot based on the design drawings of the robot under the premise of meeting the task requirements. However, such an estimation method may result in the situation that the end effector of the robot cannot fully meet the task requirements. Since it is impossible to determine whether a robot can meet the requirements of a certain task, after the robot is deployed, technical workers need to conduct full-process tests in the form of manual point walking.

[0003] In the prior art, there is also a method of sampling in the joint space of the robot and then performing forward kinematics calculations to solve for the workspace of the end effector of the robot in the Cartesian space. Such a method is only applicable to robots with fewer joints. For multi-joint robots, a very large amount of calculations are required and there are a large number of redundant calculations. Since the six-dimensional pose of the end effector of the robot is sampled in the Cartesian space and the inverse kinematics equation is solved to verify the accessibility of the robot, there are problems such as the inability to ensure sufficient sampling density in the Cartesian space sampling, redundant calculations for a large number of points inside the workspace after sampling points, and the probability of failure in inverse kinematics calculations.

[0004] Therefore, there is an urgent need in the industry for a method to obtain the workspace of the end effector according to the joint characteristics of the robot, so as to determine whether the robot can reach a certain working point and meet the requirements of a certain task when arranging robots and industrial production lines. Summary of the Invention

[0005] To solve at least one of the above technical problems, the present invention proposes a method for generating the workspace boundary of a robot end effector based on linear programming.

[0006] The object of the present invention is achieved through the following technical solutions:

[0007] The present invention provides a method for generating the workspace boundary of a robot end effector based on linear programming, including the following steps:

[0008] S1. According to the joint characteristics of the robot, parameterize the joint movement and construct a kinematic model of the robot;

[0009] S2. Respectively construct linear equations for the robot assembly constraints, matrix rank deficiency constraints, constraints on the pose of the robot end effector, and constraints generated by introducing intermediate variables required to generate the workspace boundary of the robot end effector;

[0010] S3. Combine the joint motion parameters and the linear equations for the robot assembly constraints, matrix rank deficiency constraints, constraints on the pose of the robot end effector, and constraints generated by introducing intermediate variables to determine the linear constraint equation system that needs to be satisfied to generate the workspace boundary of the robot end effector;

[0011] S4. Obtain all the robot joint motion parameters that satisfy the linear equations in the constraint equation system according to the linear constraint equation system;

[0012] S5. Find the boundary joint motion parameters that simultaneously satisfy the robot assembly constraints, matrix rank deficiency constraints, and constraints on the pose of the robot end effector from all the robot joint motion parameters that satisfy the linear equations in the linear constraint equation system;

[0013] S6. Combine the robot assembly constraints to identify and determine the workspace boundary points of the robot end effector from the boundary joint motion parameters.

[0014] As a further improvement, in the step S1, kinematic modeling of the robot motion is constructed by parametrically representing the joint motion according to the joint characteristics of the robot, which specifically includes the following steps:

[0015] S11. Take a joint point on each joint of the robot and abstract all the joints of the robot into an abstract model composed of points and line segments;

[0016] S12. Parametrically represent the motion generated by the rotation pose or position change of the next joint point connected to the current joint on the current joint;

[0017] S13. Repeat the step S12 for each joint of the robot until the motion of each joint is parametrically represented to construct the kinematic modeling of the robot motion.

[0018] As a further improvement, in the step S12, parametrically representing the motion generated by the rotation pose or position change of the next joint point connected to the current joint on the current joint specifically includes the following steps:

[0019] S121. When the current joint is a rotational joint, the current joint motion parameter is represented by the rotation pose change of the next joint point using quaternion parameters;

[0020] S122. When the current joint is a telescopic joint, the current joint motion parameter is represented by the position change of the next joint point using three-dimensional vector parameters.

[0021] As a further improvement, in the step S2, the robot assembly constraints for constructing the workspace boundary are specifically as follows: For each joint of the robot, when the current joint is a rotational joint, the robot assembly constraint condition that the current joint needs to satisfy is that when the rotational posture of the next joint point changes, the rotational direction always remains consistent with the rotation axis of the current joint; when the current joint is a telescopic joint, the robot assembly constraint condition that the current joint needs to satisfy is that when the position of the next joint point changes, the two vectors that are not consistent with the positive telescopic direction of the current joint always remain unchanged.

[0022] As a further improvement, in the step S2, the linear equation with matrix rank deficiency constraint constructed is:

[0023]

[0024] Among them, represents the transpose matrix of the partial derivative with respect to , represents the linear equation of the robot assembly constraint condition, represents the joint motion parameters of the robot end effector, represents the joint motion parameters of other joints except the robot end effector, represents a random vector.

[0025] As a further improvement, in the step S3, the linear constraint equations that need to be satisfied for generating the workspace boundary of the robot end effector are as follows:

[0026]

[0027] Among them, represents the linear constraint equation for the pose of the robot end effector, represents the constraint linear equation generated by introducing intermediate variables, represents the intermediate variable introduced by the quadratic variable composed of joint motion parameters, represents the intermediate variable introduced by the bilinear variable composed of joint motion parameters.

[0028] As a further improvement, in the step S4, obtaining all the robot joint motion parameters that satisfy the linear equations in the linear constraint equations includes the following steps:

[0029] S41. Find the maximum and minimum values that the joint motion parameters can take on the premise of satisfying the linear constraint equations;

[0030] S42. Set a threshold value, and determine whether the maximum value in the maximum value range of all joint motion parameters is higher than the set threshold value. If so, divide the value range of this joint motion parameter into two equal parts from the midpoint, store the value ranges of the two groups of parameters obtained into the queue of joint motion parameters respectively, and keep the value ranges of other joint motion parameters unchanged;

[0031] S43. For the value range of the next group of parameters in the joint motion parameter queue, repeat steps S41 and S42 until the value ranges of all joint motion parameters are lower than the set threshold value.

[0032] As a further improvement, in step S5, among the robot joint motion parameters of all linear equations that satisfy the linear constraint equations, use the Newton iteration method to find the boundary joint motion parameters that simultaneously satisfy the robot assembly constraint, the matrix rank deficiency constraint, and the constraint on the pose of the robot end effector.

[0033] As a further improvement, in step S6, combine the robot assembly constraint to identify and determine the workspace boundary points of the robot end effector from the boundary joint motion parameters, including the following steps:

[0034] S61. Search for each group of boundary joint motion parameters, and respectively find the joint motion normal line of the high-dimensional geometric figure formed by the robot assembly constraint linear equations at the current boundary joint motion parameter position with respect to the joint motion parameters of the robot end effector. The specific formula is as follows:

[0035]

[0036] Among them, represents the joint motion normal line of the high-dimensional geometric figure formed by the robot assembly constraint linear equations at the current boundary joint motion parameter position with respect to the joint motion parameters of the robot end effector, is with respect to partial derivative, represents the random vector of the current boundary joint motion parameter;

[0037] S62. Combine the joint motion normal line to set a judgment function, and judge each joint motion parameter of the robot end effector through the judgment function to identify and determine the workspace boundary points of the end effector.

[0038] A method for generating the workspace boundary of a robot end effector based on linear programming provided by the present invention is characterized by comprising the following steps: S1. Kinematic modeling of robot motion is constructed by parametrically representing joint motion according to the joint characteristics of the robot; S2. Linear equations for robot assembly constraints, matrix rank deficiency constraints, constraints on the pose of the robot end effector, and constraints generated by introducing intermediate variables, which are required to generate the workspace boundary of the robot end effector, are respectively constructed; S3. Combining the joint motion parameters and the linear equations of the robot assembly constraints, matrix rank deficiency constraints, constraints on the pose of the robot end effector, and constraints generated by introducing intermediate variables, a system of linear constraints that needs to be satisfied to generate the workspace boundary of the robot end effector is determined; S4. All robot joint motion parameters that satisfy the linear equations in the system of constraint equations are obtained according to the system of linear constraints; S5. Boundary joint motion parameters that simultaneously satisfy the robot assembly constraints, matrix rank deficiency constraints, and constraints on the pose of the robot end effector are found from the robot joint motion parameters that satisfy the linear equations of the system of linear constraints; S6. Combining the robot assembly constraints, the workspace boundary points of the robot end effector are identified and determined from the boundary joint motion parameters. In the application process of the present invention, based on the structural dimensions, joint limits, and requirements of the task objective of the robot, for the boundary points of the workspace obtained by the robot end effector on the premise of meeting the task requirements, after connecting each workspace boundary point, the workspace boundary of the robot end effector is obtained. By introducing all the constraint conditions on the robot end effector, a large amount of redundant computation and storage space can be saved when calculating the workspace boundary points of the robot end effector according to the present invention. When arranging the robot and the industrial production line, the layout work is simplified to the work of overlapping the industrial production line and the robot workspace, so that the layout work of the automation scenario can be greatly simplified, and the accuracy of the layout can be greatly improved. BRIEF DESCRIPTION OF THE DRAWINGS

[0039] Figure 1 is a flowchart of the present invention;

[0040] Figure 2 is a schematic diagram of an abstract model of an embodiment of the present invention;

[0041] Figure 3 is a schematic diagram of the workspace boundary of the robot end effector of the present invention. DETAILED DESCRIPTION OF THE EMBODIMENTS

[0042] Combined with Figure 1 as shown, an embodiment of the present invention provides a method for generating the workspace boundary of a robot end effector based on linear programming, which specifically includes the following steps:

[0043] S1. According to the joint characteristics of the robot, parameterize the joint motion and construct a kinematic model of the robot. For example, the robot in this embodiment may be, but is not limited to, a joint robot for explanation, but not limited thereto. Specifically, the robot may include, but is not limited to, multiple joint axes, and the specific position, quantity, type, etc. of the joint axes can be arbitrarily set by those skilled in the art according to the application scope, field, working performance, etc. of the robot. Specifically, according to the functions that the robot can achieve, the joint axes may include, but are not limited to, rotary axes or / and telescopic axes, and two joint axes are connected by a joint arm (i.e., a joint link), and the joint characteristics of the robot include rotation or telescoping. More specifically, the end effector is an execution component installed at the end of the joint, and may include, but is not limited to, multi-fingered mechanical claws, paint spray guns, welding tools and other operation tools. More specifically, each joint of the robot may be, but is not limited to, mathematically represented by parameters to achieve mathematical modeling; more specifically, a mathematical modeling method of linear programming may be, but is not limited to, adopted, because linear programming (abbreviated as LP) is a basic mathematical theory and method for studying the extreme value problem of a linear objective function under linear constraint conditions, but not limited thereto. More specifically, step S1 may include, but is not limited to, the following content:

[0044] S11. As Figure 2 shown, take a joint point on each joint of the robot. In this embodiment, the joint points taken are the joint axes of the robot, and all joints of the robot are abstracted into an abstract model composed of points and line segments. A joint includes a joint point and a joint arm. The joint point, that is, the position where the joint axis is located, is represented by a point in the abstract model, and the joint arm is represented by a line in the abstract model;

[0045] S12. Represent the motion generated by the rotation attitude or position change of the next joint point connected to the current joint by parameters; specifically, it may include, but is not limited to, the following steps:

[0046] S121. When the current joint is a rotary joint, since the position of the next joint point connected to the current joint is determined by its rotation attitude, the motion parameters of the current joint are represented by the rotation attitude change of the next joint point using quaternion parameters. For example, in Figure 2 , the motion parameters of joint 1 are represented by the rotation attitude change of joint point P1 using quaternion parameters, and the motion parameters of joint 2 are represented by the rotation attitude change of joint point P2 using quaternion parameters, and so on for other joints.

[0047] S122. When the current joint is a telescopic joint, since the rotation attitude of the next joint point connected to the current joint does not change, the motion parameters of the current joint are represented by the position change of the next joint point using three-dimensional vector parameters.

[0048] S13. Repeat the steps of S12 for each joint of the robot until the motion of each joint is represented parametrically, and construct the kinematic model of the robot.

[0049] S2. Respectively construct linear equations for the robot assembly constraints, matrix rank deficiency constraints, constraints on the pose of the robot end effector, and constraints generated by introducing intermediate variables required to generate the boundary of the workspace of the robot end effector.

[0050] Construct the robot assembly constraints for generating the workspace boundary: Since the robot is formed by connecting different components with all joints, and the connections of each joint of the robot only allow limited degrees of freedom of motion between adjacent components, the parameters representing the joint motion need to satisfy the assembly constraint conditions introduced by the joint to limit the form of parameter changes. Specifically: For each joint of the robot, when it is determined that the current joint is a rotational joint, the robot assembly constraint condition that the current joint needs to satisfy is that when the rotational pose of the next joint point changes, the direction in the rotational pose is always consistent with the rotation axis of the current joint. Among them, the direction in the rotational pose is determined by the coordinate system definition of the next joint point. For example Figure 2 the rotational poses of the joint points P1 and P2 in 10 are (q 11 , q 12 , q 13 ) and (q 20 , q 21 , q 22 , q 23 ), then their corresponding rotation matrices are as follows:

[0051]

[0052]

[0053] Since joint 2 can only rotate around the Y-axis of P1, the constraint condition introduced by joint 2 is that the Y-axis direction in the rotational pose of P2 should always be consistent with the Y-axis of P1, that is

[0054] 2(q 11 q 12 -q 10 q 13 ) = 2(q 21 q 22 -q 20 q 23 )

[0055]

[0056] 2(q 10 q 11 +q12 q 13 ) = 2(q 20 q 21 +q 22 q 23 )

[0057] When the current joint is a telescopic joint, the robotic assembly constraint condition that the current joint needs to satisfy is that when the position (X, Y, Z) of the next joint point changes, the two vectors that are not in the same direction as the positive telescopic direction of the current joint always remain unchanged.

[0058] After deriving all the assembly constraint conditions introduced by the robotic joints, use to represent the linear equation of the robotic assembly constraint conditions, where represents the joint motion parameters of the robotic end effector, such as Figure 2 the quaternion parameters of joint point P6 in represents the joint motion parameters of other joints except the robotic end effector, such as Figure 2 the quaternion parameters of P1...P5 in Among all the joint motion parameters that satisfy

[0059] the constructed linear equation with matrix rank deficiency constraint is:

[0060]

[0061] where represents the transpose matrix of the partial derivative of with respect to represents the linear equation of the robotic assembly constraint conditions, represents a random vector.

[0062] The constructed linear equation for the pose constraint of the end effector is:

[0063] The constructed linear equation for the constraint introduced by the intermediate variables: Since linear programming requires that all constraint conditions are linear equations, quadratic variables, such as and bilinear variables, such as g i g j will be introduced when constructing the robotic assembly constraint, matrix rank deficiency constraint, and pose constraint of the robotic end effector. To linearize all the constraint equations, we introduce intermediate variables and some constraint conditions to define the relationship between the newly introduced intermediate variables and the original parameters.

[0064] For the quadratic variable we introduce an intermediate variable s i such that Suppose the value range of g i is [l i , u i . Three constraint equations need to be introduced to constrain the relationship between s i and g i , that is

[0065]

[0066] For the bilinear variable g i g j introduce an intermediate variable b ij such that b ij = g i g j . Suppose the value ranges of g i and g j are [l i , u i and [l j , u j . Four constraint equations need to be introduced to constrain b ij as well as the relationship between g i and g j , that is b ij = g i g j :

[0067]

[0068] Through the above description, the linear equations of all constraints generated by introducing intermediate variables are written as:

[0069]

[0070] Among them, represents the intermediate variable introduced by the quadratic variable composed of joint motion parameters (for example, ), represents the intermediate variable introduced by the bilinear variable composed of joint motion parameters (for example, b ij = g i g j ).

[0071] S3. Combine the joint motion parameters with the constructed linear equations of the robot assembly constraints, matrix rank deficiency constraints, constraints on the pose of the robot end effector, and constraints generated by introducing intermediate variables to determine the system of linear constraints that need to be satisfied to generate the workspace boundary of the robot end effector, and set an initial value range for each parameter. For example, in this embodiment, the joint motion parameter representing the rotational pose is a quaternion parameter, and the initial value range of each element can be set to [-1, 1].

[0072]

[0073] Among them, represents the linear constraint equation for the pose of the robot end effector,

[0074] represents the linear constraint equation generated by introducing intermediate variables.

[0075] S4. Obtain all the robot joint motion parameters that satisfy the linear equations in the system of linear constraints, including the following steps:

[0076] S41. Adopt the Pruning method to narrow the value range of each joint motion parameter through linear programming to achieve the purpose of pruning, and find the maximum and minimum values that the joint motion parameter can take on the premise of satisfying the system of linear constraints. Specifically: For each joint motion parameter, two linear programming problems need to be defined to find the minimum and maximum values that the current parameter can take on the premise of satisfying all the constraint conditions in the system of constraint equations. For example, when pruning the value range of a certain joint motion parameter g i we need to solve the following two linear programming problems:

[0077]

[0078] and

[0079]

[0080] By solving the above maximization and minimization problems, the upper and lower limits of the value range of g i are updated respectively.

[0081] S42. Set a threshold value and determine whether the maximum value within the value range of all joint motion parameters is higher than the set threshold. Specifically, after trimming all joint motion parameters, if there is a maximum value range among all joint motion parameters that is higher than the set threshold, we need to use the Branching method to divide the value range of this joint motion parameter into two equal parts from the midpoint, store the value ranges of the two sets of parameters obtained into the queue of joint motion parameters respectively, and keep the value ranges of other joint motion parameters unchanged;

[0082] S43. For the value range of the next set of parameters in the joint motion parameter queue, repeat steps S41 and S42 until the value ranges of all joint motion parameters are lower than the set threshold.

[0083] S5. After the operations of the trimming method and the branching method, many different value ranges of all joint motion parameters (i.e., and ) that satisfy the linear constraint equations are found. Use the Newton iteration method to find the boundary joint motion parameters that simultaneously satisfy the robot assembly constraints, the matrix rank deficiency constraints, and the constraints on the pose of the robot end effector (i.e., ) from the robot joint motion parameters that satisfy the linear constraint equations. The Newton iteration method is a method for approximately solving equations in the real number field and the complex number field. In this step, since after step S4, the value range of each joint motion parameter is less than the set threshold, the solution process using the Newton iteration method can quickly obtain a solution that satisfies all constraint conditions.

[0084] S6. Combine the robot assembly constraints to identify and determine the workspace boundary points of the robot end effector from the boundary joint motion parameters. Since the matrix rank deficiency constraint condition is only a necessary condition for a set of joint motion parameter values to be workspace boundary points, it is necessary to more accurately identify the values of each set of joint motion parameters in this step to determine whether it is a workspace boundary point of the robot end effector. The specific steps are as follows:

[0085] S61. Search for each set of boundary joint motion parameters. In this embodiment, taking a set of boundary joint motion parameters as an example, respectively find the joint motion normal formed by the linear equations of the robot assembly constraints at the position of the current boundary joint motion parameters with respect to the joint motion parameters of the robot end effector. The specific formula is as follows:

[0086]

[0087] Among them, is the partial derivative with respect to , denotes a random vector of current boundary joint motion parameters, obtained by the formula ;

[0088] S62. Set a judgment function in combination with the joint motion normal. In this embodiment, let be any set of joint motion parameters that satisfy the linear equation of the robot assembly constraints, adjacent to the joint point of . Judge each joint motion parameter of the robot end effector through the judgment function to identify and determine the workspace boundary points of the end effector. In this embodiment, determine whether is a workspace boundary point of a robot end effector through the judgment function . If is positive definite or negative definite, then is a workspace boundary point of a robot end effector; if the sign of cannot be determined, then is not a workspace boundary point of a robot end effector.

[0089] In the application process of the present invention, based on the structural dimensions, joint limits, and task objective requirements of the robot, for the boundary points of the workspace obtained by the robot end effector under the premise of meeting the task requirements, after connecting all the workspace boundary points, the workspace boundary of the robot end effector is obtained. By introducing all the constraint conditions for the robot end effector, a large amount of redundant calculation amount and storage space can be saved when calculating the workspace boundary points of the robot end effector through the present invention. When arranging the robot and the industrial production line, the layout work is simplified to the work of overlapping the industrial production line and the robot workspace, so that the layout work of the automated scenario can be greatly simplified, and the layout accuracy can be greatly improved.

[0090] The present invention is also applicable to any robot, including but not limited to industrial robotic arms, parallel robots, redundant robotic arms, dual-arm robots, or multi-fingered manipulators, etc.

[0091] The technical features of the above-described embodiments can be combined arbitrarily. For the sake of brevity of description, not all possible combinations of the technical features in the above embodiments are described. However, as long as there is no contradiction in the combination of these technical features, it should be considered as the scope described in this specification.

[0092] The above-described embodiments merely represent several implementation manners of the present invention. The description is relatively specific and detailed, but it should not be construed as a limitation on the scope of the invention patent. It should be noted that for those of ordinary skill in the art, without departing from the concept of the present invention, several modifications and improvements can still be made, and these all fall within the protection scope of the present invention. Therefore, the protection scope of the present invention patent shall be subject to the appended claims.

Claims

1. A method for generating the workspace boundary of a robot end effector based on linear programming, characterized in that It includes the following steps: S1. According to the joint characteristics of the robot, parameterize the joint motion and construct a robot dynamics model; S2. Respectively construct linear equations for the robot assembly constraints, matrix rank deficiency constraints, constraints on the pose of the robot end effector, and constraints generated by introducing intermediate variables required to generate the boundary of the robot end effector's workspace; S3. Combine the joint motion parameters and the linear equations of the robot assembly constraints, matrix rank deficiency constraints, constraints on the pose of the robot end effector, and constraints generated by introducing intermediate variables to determine the linear constraint equations that need to be satisfied to generate the boundary of the robot end effector's workspace; S4. Obtain the robot joint motion parameters that satisfy the linear equations in the constraint equations according to the linear constraint equations; S5. Find the boundary joint motion parameters that simultaneously satisfy the robot assembly constraints, matrix rank deficiency constraints, and constraints on the pose of the robot end effector from the robot joint motion parameters that satisfy the linear equations in the linear constraint equations; S6. Identify and determine the workspace boundary points of the robot end effector from the boundary joint motion parameters in combination with the robot assembly constraints.

2. The method for generating the workspace boundary of the robot end effector based on linear programming according to claim 1, wherein, In the step S1, parameterizing the joint motion according to the joint characteristics of the robot to construct the kinematic modeling of the robot motion specifically includes the following steps: S11. Take a joint point on each joint of the robot and abstract all the joints of the robot into an abstract model composed of points and line segments; S12. Parameterize the motion generated by the rotation attitude or position change of the next joint point connected to the current joint on the current joint; S13. Repeat the step S12 for each joint of the robot until the motion of each joint is parameterized to construct the kinematic modeling of the robot motion.

3. The method for generating the workspace boundary of the robot end effector based on linear programming according to claim 2, characterized in that In the step S12, parameterizing the motion generated by the rotation attitude or position change of the next joint point connected to the current joint on the current joint specifically includes the following steps: S121. When the current joint is a rotational joint, the current joint motion parameter is represented by the rotation attitude change of the next joint point using quaternion parameters; S122. When the current joint is a telescopic joint, the current joint motion parameter is represented by the position change of the next joint point using three-dimensional vector parameters.

4. The method for generating the workspace boundary of the robot end effector based on linear programming according to claim 3, wherein In the step S2, the robot assembly constraints for generating the workspace boundary are specifically: judge each joint of the robot. When the current joint is a rotational joint, the robot assembly constraint condition that the current joint needs to satisfy is that when the rotation attitude of the next joint point changes, the rotation direction always remains consistent with the rotation axis of the current joint; When the current joint is a telescopic joint, the robot assembly constraint condition that the current joint needs to satisfy is that when the position of the next joint point changes, the two vectors that are not in the positive telescopic direction of the current joint always remain unchanged.

5. The method for generating the workspace boundary of the robot end effector based on linear programming according to claim 1, wherein In the step S2, the linear equation of the constructed matrix rank deficiency constraint is: Among them, denotes the transpose matrix of the partial derivative with respect to , represents the linear equation of the assembly constraint conditions of the robot, represents the joint motion parameters of the robot end effector, represents the joint motion parameters of other parts except the robot end effector, represents a random vector.

6. The method for generating the workspace boundary of the robot end effector based on linear programming according to claim 1, characterized in that, In the step S3, the linear constraint equations that need to be satisfied to generate the boundary of the robot end effector's workspace are as follows: Among them, represents the constraint linear equation for the pose of the robot end effector, represents the constraint linear equation generated by introducing intermediate variables, represents the intermediate variable introduced by the quadratic variable composed of joint motion parameters, represents the intermediate variable introduced by the bilinear variable composed of joint motion parameters.

7. The method for generating the workspace boundary of the robot end effector based on linear programming according to claim 1, wherein In step S4, obtaining all the robot joint motion parameters that satisfy the linear equations in the constraint equation set according to the linear constraint equation set includes the following steps: S41. Find the maximum and minimum values that the joint motion parameters can take on the premise of satisfying the linear constraint equation set; S42. Set a threshold, and determine whether the maximum value in the maximum value range of all joint motion parameters is higher than the set threshold. If so, divide the value range of this joint motion parameter into two equal parts from the midpoint, store the value ranges of the two groups of parameters obtained into the queue of joint motion parameters respectively, and keep the value ranges of other joint motion parameters unchanged; S43. Repeat steps S41 and S42 for the value range of the next group of parameters in the joint motion parameter queue until the value ranges of all joint motion parameters are lower than the set threshold.

8. The method for generating the workspace boundary of the robot end effector based on linear programming according to claim 7, wherein In step S5, from all the robot joint motion parameters that satisfy the linear equations in the linear constraint equation set, find the boundary joint motion parameters that simultaneously satisfy the robot assembly constraint, the matrix rank deficiency constraint, and the constraint on the pose of the robot end effector through the Newton iteration method.

9. The method for generating the workspace boundary of the robot end effector based on linear programming according to claim 7, characterized in that, In step S6, identifying and determining the workspace boundary points of the robot end effector from the boundary joint motion parameters in combination with the robot assembly constraint includes the following steps: S61. Search for each group of boundary joint motion parameters, and respectively find the joint motion normal line of the high-dimensional geometric figure formed by the linear equations of the robot assembly constraint at the position of the current boundary joint motion parameter with respect to the joint motion parameters of the robot end effector. The specific formula is as follows: Among them, represents the joint motion normal of the high-dimensional geometric figure composed of the robot assembly constraint linear equations with respect to the joint motion parameters of the robot end effector at the current boundary joint motion parameter position, is the partial derivative with respect to ; represents the random vector of the current boundary joint motion parameters; S62. Set a judgment function in combination with the joint motion normal line, and judge each joint motion parameter of the robot end effector through the judgment function to identify and determine the workspace boundary points of the end effector.

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