A time-optimal trajectory planning method for rope-driven parallel robots based on GA-SQP

Abstract

The application relates to a GA-SQP-based time-optimal trajectory planning method for a rope-towing parallel robot, which comprises the following steps: establishing a robot kinematics model; transforming the positions of each key path point in the robot Cartesian space into corresponding rope lengths; establishing a time-optimal trajectory planning model based on cubic spline interpolation; pre-solving the time-optimal trajectory planning model by using a genetic algorithm (GA) to obtain a pre-solution result; taking the pre-solution result as an initial value for iterative solving of a sequential quadratic programming (SQP) algorithm, solving by using the SQP algorithm, and obtaining a time-optimal trajectory of the rope-towing parallel robot under a speed constraint condition. The application can obtain a robot motion trajectory with good smoothness, improves the motion stability of the robot, reduces the working time of the robot, thereby improving the working efficiency of the robot, has strong robustness, does not need a large amount of complex calculation, obtains a global optimal solution with high time efficiency, and has good real-time performance.

Description

Technical Field

[0001] This invention relates to the field of robot trajectory planning technology, and in particular to a time-optimal trajectory planning method for a rope-traction parallel robot based on GA-SQP. Background Technology

[0002] With the continuous development of science and technology, industrial robots possess advantages such as high work efficiency, stability, reliability, and good repeatability, playing an increasingly important role in industrial production. Time-optimal trajectory planning is a crucial research direction in industrial robotics. The quality of the planned trajectory significantly impacts the robot's work efficiency and motion smoothness; therefore, studying the time-optimal trajectory planning problem for robots is of great significance for improving robot operational efficiency.

[0003] Time-optimal trajectory planning has long been a challenging problem in the field of robotics, attracting widespread attention from major companies and research institutions. To further improve maneuverability while satisfying system speed, acceleration, or torque constraints, experts and scholars both domestically and internationally currently employ traditional exact optimization algorithms and modern heuristic algorithms to solve the time-optimal trajectory planning problem. However, exact optimization algorithms suffer from drawbacks such as long computation time and a tendency to get trapped in local optima; modern heuristic algorithms suffer from randomness and low computational efficiency. Therefore, how to avoid these problems and improve the performance of time-optimal trajectory planning algorithms for robots is a pressing technical issue that needs to be addressed in the field of robot trajectory planning. Summary of the Invention

[0004] To address the shortcomings of long computation time and low computation efficiency in existing technologies, the present invention aims to provide a time-optimal trajectory planning method for rope-traction parallel robots based on GA-SQP, which has low computational complexity and good real-time performance.

[0005] To achieve the above objectives, the present invention adopts the following technical solution: a time-optimal trajectory planning method for a rope-traction parallel robot based on GA-SQP, the method comprising the following sequential steps:

[0006] (1) Based on the model structure of the rope-traction parallel robot, establish the robot kinematic model based on vector theory;

[0007] (2) Based on the robot kinematics model, the robot kinematics is solved using the inverse kinematics solution method, and the positions of each key path point in the robot's Cartesian space are transformed into the corresponding rope lengths.

[0008] (3) Based on the rope length obtained in step (2), and with the total robot motion time as the optimization objective function, establish a time-optimal trajectory planning model based on cubic spline interpolation;

[0009] (4) Input the speed constraint conditions, and use the genetic algorithm (GA) to pre-solve the time-optimal trajectory planning model to obtain the pre-solution results;

[0010] (5) The pre-solved result is used as the initial value for the sequential quadratic programming algorithm, i.e., the SQP algorithm. The SQP algorithm is used to solve the problem and obtain the optimal time trajectory of the rope-traction parallel robot under the speed constraint.

[0011] In step (1), the robot kinematic model is:

[0012]

[0013] in, This represents the robot's coordinate rotation matrix. This represents the position vector of the centroid of the end-effector. Indicates the first The position vector of the root rope, Indicates the first The position vector of the point is indexed by the rope. This indicates the distance from the center of mass of the end-effector to the first point on the moving platform. The position vectors of the rope connection points , This indicates the number of ropes used in a rope-driven parallel robot.

[0014] Step (2) specifically refers to: for the robot's first A rope, , This indicates the number of ropes used in a rope-driven parallel robot. Given the sequence of key point positions in Cartesian space, i.e., the positions of critical path points, we use inverse kinematics to solve the robot's kinematics, thus obtaining the rope length, i.e., the sequence of rope length changes. .

[0015] Step (3) specifically refers to: the cubic spline interpolation function converting the interval Divided into small intervals , In each small interval The functions on the above are all cubic polynomials. ; Indicates the start time node. Indicates the end time point. This indicates the time node corresponding to each key point;

[0016] Assumption For the time series corresponding to each key point, Represents each small interval The robot passes through key points along the path sequentially according to the time intervals during operation, utilizing variables. The robot's first The first rope is planned based on the cubic spline interpolation method. The length of the rope corresponding to the segment trajectory; due to In the interval The expression is based on a cubic polynomial function, so its second derivative lies in the interval [0, 1]. The inner function is a linear function. If we use... The second derivative to indicate The resulting expression is:

[0017]

[0018] right Perform integration twice and utilize ,have to The expression is:

[0019]

[0020] right Taking the first derivative, we get:

[0021]

[0022] The above formula yields the result in the interval On and in the interval On for:

[0023]

[0024] Because the joint velocities are continuous at key points along the entire trajectory, utilizing... get:

[0025]

[0026] In the formula, , , Represented as:

[0027]

[0028] Assume the initial velocity is The velocity at the termination point is , get about The two equations are:

[0029]

[0030] Considering the actual working conditions, we assume that the first derivative values ​​at both ends are 0, which means that the velocity at the starting and ending points of the robot's motion trajectory is 0.

[0031] make , , , The above formula can be written in matrix form as follows:

[0032]

[0033] The chasing method is used to find That is, to obtain the rope length variation based on the cubic spline interpolation method;

[0034] Then, a time-optimal trajectory planning model is obtained based on cubic spline interpolation:

[0035]

[0036] Among them, due to the variable rope length A cubic polynomial function is used, and the corresponding velocity function is... It must be a quadratic polynomial, therefore the velocity trajectory is a parabola; Representing the two endpoints of the parabola The absolute value of the velocity at that point, Represents the vertex of the parabola The absolute value of the velocity at that point, This represents the time series between various key points. Indicates the robot's first The maximum speed allowed by the rope. This indicates the total time the robot takes to run. This represents the velocity constraints at three key points: the two endpoints and the vertex of the parabola. This represents the time constraint obtained under the maximum speed condition;

[0037] Step (4) specifically refers to: taking the velocity at the starting point as The velocity at the termination point is Set the robot's first Maximum speed allowed by the rope The genetic algorithm (GA) is used to pre-solve the time-optimal trajectory planning model, and the pre-solved results of the time-optimal trajectory planning model, i.e., the time series, are output. ;

[0038] Step (5) specifically refers to: solving the nonlinear optimization problem using the SQP algorithm based on the inequality constraints in the time-optimal trajectory planning model, with the initial values ​​of the iterations using the pre-solved results obtained from the genetic algorithm GA. Finally, the final solution of the time-optimal trajectory planning model, i.e., the time series, is obtained. According to time series Obtain the optimal trajectory in time, which corresponds to the rope length. .

[0039] As can be seen from the above technical solution, the beneficial effects of the present invention are as follows: First, the time-optimal trajectory planning model based on cubic spline interpolation designed in this invention effectively solves the time-optimal trajectory planning problem of rope-traction parallel robots, and can obtain a smooth robot motion trajectory, thereby improving the robot's motion stability; Second, the present invention can fully utilize the robot's speed performance, making the robot's working speed closer to the maximum allowable speed, reducing the robot's working time, thereby improving the robot's working efficiency, and also having strong robustness; Third, the hybrid algorithm combining the genetic algorithm GA and the sequential quadratic programming algorithm (SQP) adopted in this invention obtains the pre-solution result through the genetic algorithm GA, and gives the sequential quadratic programming algorithm a good iterative initial value, thereby ensuring the local superlinear convergence of the sequential quadratic programming algorithm, without a large amount of complex calculation, and obtains the global optimal solution with high time efficiency, thus having good real-time performance. Attached Figure Description

[0040] Figure 1 This is a flowchart of the method of the present invention. Detailed Implementation

[0041] like Figure 1 As shown, a time-optimal trajectory planning method for a rope-traction parallel robot based on GA-SQP is presented. This method includes the following sequential steps:

[0042] (1) Based on the model structure of the rope-traction parallel robot, establish the robot kinematic model based on vector theory;

[0043] (2) Based on the robot kinematics model, the robot kinematics is solved using the inverse kinematics solution method, and the positions of each key path point in the robot's Cartesian space are transformed into the corresponding rope lengths.

[0044] (3) Based on the rope length obtained in step (2), and with the total robot motion time as the optimization objective function, establish a time-optimal trajectory planning model based on cubic spline interpolation;

[0045] (4) Input the speed constraint conditions, and use the genetic algorithm (GA) to pre-solve the time-optimal trajectory planning model to obtain the pre-solution results;

[0046] (5) The pre-solved result is used as the initial value for the sequential quadratic programming algorithm, i.e., the SQP algorithm. The SQP algorithm is used to solve the problem and obtain the optimal time trajectory of the rope-traction parallel robot under the speed constraint.

[0047] In step (1), the robot kinematic model is:

[0048]

[0049] in, This represents the robot's coordinate rotation matrix. This represents the position vector of the centroid of the end-effector. Indicates the first The position vector of the root rope, Indicates the first The position vector of the point is indexed by the rope. This indicates the distance from the center of mass of the end-effector to the first point on the moving platform. The position vectors of the rope connection points , This indicates the number of ropes used in a rope-driven parallel robot.

[0050] Step (2) specifically refers to: for the robot's first A rope, , This indicates the number of ropes used in a rope-driven parallel robot. Given the sequence of key point positions in Cartesian space, i.e., the positions of critical path points, we use inverse kinematics to solve the robot's kinematics, thus obtaining the rope length, i.e., the sequence of rope length changes. .

[0051] Step (3) specifically refers to: the cubic spline interpolation function converting the interval Divided into small intervals , In each small interval The functions on the above are all cubic polynomials. ; Indicates the start time node. Indicates the end time point. This indicates the time node corresponding to each key point;

[0052] Assumption For the time series corresponding to each key point, Represents each small interval The robot passes through key points along the path sequentially according to the time intervals during operation, utilizing variables. The robot's first The first rope is planned based on the cubic spline interpolation method. The length of the rope corresponding to the segment trajectory; due to In the interval The expression is based on a cubic polynomial function, so its second derivative lies in the interval [0, 1]. The inner function is a linear function. If we use... The second derivative to indicate The resulting expression is:

[0053]

[0054] right Perform integration twice and utilize ,have to The expression is:

[0055]

[0056] right Taking the first derivative, we get:

[0057]

[0058] The above formula yields the result in the interval On and in the interval On for:

[0059]

[0060] Because the joint velocities are continuous at key points along the entire trajectory, utilizing... get:

[0061]

[0062] In the formula, , , Represented as:

[0063]

[0064] Because sufficient constraints are needed to solve the constrained problem, boundary conditions regarding the initial and final velocities can be added. Let's assume the initial velocity is... The velocity at the termination point is , get about The two equations are:

[0065]

[0066] Considering the actual working conditions, we assume that the first derivative values ​​at both ends are 0, which means that the velocity at the starting and ending points of the robot's motion trajectory is 0.

[0067] make , , , The above formula can be written in matrix form as follows:

[0068]

[0069] The chasing method is used to find That is, to obtain the rope length variation based on the cubic spline interpolation method;

[0070] Then, a time-optimal trajectory planning model is obtained based on cubic spline interpolation:

[0071]

[0072] Among them, due to the variable rope length A cubic polynomial function is used, and the corresponding velocity function is... It must be a quadratic polynomial, therefore the velocity trajectory is a parabola; Representing the two endpoints of the parabola The absolute value of the velocity at that point, Represents the vertex of the parabola The absolute value of the velocity at that point, This represents the time series between various key points. Indicates the robot's first The maximum speed allowed by the rope. This indicates the total time the robot takes to run. This represents the velocity constraints at three key points: the two endpoints and the vertex of the parabola. This represents the time constraint obtained under the maximum speed condition;

[0073] Step (4) specifically refers to: because sufficient constraints are needed to solve the constraint problem, boundary conditions regarding the initial and final velocities can be added, taking the velocity at the starting point as... The velocity at the termination point is Set the robot's first Maximum speed allowed by the rope The genetic algorithm (GA) is used to pre-solve the time-optimal trajectory planning model, and the pre-solved results of the time-optimal trajectory planning model, i.e., the time series, are output. .

[0074] Step (5) specifically refers to: solving the nonlinear optimization problem using the SQP algorithm based on the inequality constraints in the time-optimal trajectory planning model. Specifically, the fmincon function in the optimization toolbox of MATLAB software is used to solve the problem, where the initial value of the iteration is the pre-solved result obtained by the genetic algorithm GA. Finally, the final solution of the time-optimal trajectory planning model, i.e., the time series, is obtained. According to time series Obtain the optimal trajectory in time, which corresponds to the rope length. .

[0075] In summary, the time-optimal trajectory planning model based on cubic spline interpolation designed in this invention effectively solves the time-optimal trajectory planning problem for rope-traction parallel robots, yielding smooth robot motion trajectories and improving robot motion stability. This invention fully utilizes the robot's speed performance, making its working speed closer to the maximum permissible speed, reducing working time and thus improving robot efficiency, while also exhibiting strong robustness. The hybrid algorithm combining genetic algorithm and sequential quadratic programming algorithm used in this invention obtains pre-solution results through the genetic algorithm, providing the sequential quadratic programming algorithm with a good initial iterative value, thereby ensuring local superlinear convergence of the sequential quadratic programming algorithm. This eliminates the need for extensive and complex calculations, achieving the global optimal solution with high time efficiency and good real-time performance.

Claims

1. A time-optimal trajectory planning method for a rope-traction parallel robot based on GA-SQP, characterized in that: The method includes the following steps in sequence: (1) Based on the model structure of the rope-traction parallel robot, establish the robot kinematic model based on vector theory; (2) Based on the robot kinematics model, the robot kinematics is solved using the inverse kinematics solution method, and the positions of each key path point in the robot's Cartesian space are transformed into the corresponding rope lengths. (3) Based on the rope length obtained in step (2), and with the total robot motion time as the optimization objective function, establish a time-optimal trajectory planning model based on cubic spline interpolation; (4) Input the speed constraint conditions, and use the genetic algorithm (GA) to pre-solve the time-optimal trajectory planning model to obtain the pre-solution results; (5) The pre-solved result is used as the initial value for the sequential quadratic programming algorithm, i.e., the SQP algorithm, and the SQP algorithm is used to solve the problem to obtain the optimal time trajectory of the rope-traction parallel robot under the speed constraint. In step (1), the robot kinematic model is: ； in, This represents the robot's coordinate rotation matrix. This represents the position vector of the centroid of the end-effector. Indicates the first The position vector of the root rope, Indicates the first The position vector of the point is indexed by the rope. This indicates the distance from the center of mass of the end-effector platform to the first moving platform. The position vectors of the rope connection points , Indicates the number of ropes used in a rope-driven parallel robot; Step (2) specifically refers to: for the robot's first A rope, , This indicates the number of ropes used in a rope-driven parallel robot. Given the sequence of key point positions in Cartesian space, i.e., the positions of critical path points, we use inverse kinematics to solve the robot's kinematics, thus obtaining the rope length, i.e., the sequence of rope length changes. ; Step (3) specifically refers to: the cubic spline interpolation function converting the interval Divided into small intervals , In each small interval The functions on the above are all cubic polynomials. ; Indicates the start time node. Indicates the end time point. This indicates the time node corresponding to each key point; Assumption For the time series corresponding to each key point, Represents each small interval The robot passes through key points along the path sequentially according to the time intervals during operation, utilizing variables. The robot's first The first rope is planned based on the cubic spline interpolation method. The length of the rope corresponding to the segment trajectory; due to In the interval The expression is based on a cubic polynomial function, so its second derivative lies in the interval [0, 1]. The inner function is a linear function. If we use... The second derivative to indicate The resulting expression is: ； right Perform integration twice and utilize ,have to The expression is: ； right Taking the first derivative, we get: ； The above formula yields the result in the interval On and in the interval On for: ； Because the joint velocities at key points along the entire trajectory are continuous, utilizing... get: ； In the formula, , , Represented as: ； Assume the initial velocity is The velocity at the termination point is , get about The two equations are: ； Considering the actual working conditions, we assume that the first derivative values ​​at both ends are 0, which means that the velocity at the starting and ending points of the robot's motion trajectory is 0. make , , , The above formula can be written in matrix form as follows: ； The chasing method is used to find That is, to obtain the rope length variation based on the cubic spline interpolation method; Then, a time-optimal trajectory planning model is obtained based on cubic spline interpolation: ； Among them, due to the variable rope length A cubic polynomial function is used, and the corresponding velocity function is... It must be a quadratic polynomial, therefore the velocity trajectory is a parabola; Representing the two endpoints of the parabola The absolute value of the velocity at that point, Represents the vertex of the parabola The absolute value of the velocity at that point, This represents the time series between various key points. Indicates the robot's first The maximum speed allowed by the rope. This indicates the total time the robot takes to run. This indicates the velocity constraints at three key points: the two endpoints and the vertex of the parabola. This represents the time constraint obtained under the maximum speed condition; Step (4) specifically refers to: taking the velocity at the starting point as The velocity at the termination point is Set the robot's first Maximum speed allowed by the rope The genetic algorithm (GA) is used to pre-solve the time-optimal trajectory planning model, and the pre-solved results of the time-optimal trajectory planning model, i.e., the time series, are output. ; Step (5) specifically refers to: solving the nonlinear optimization problem using the SQP algorithm based on the inequality constraints in the time-optimal trajectory planning model, with the initial values ​​of the iterations using the pre-solved results obtained from the genetic algorithm GA. Finally, the final solution of the time-optimal trajectory planning model, i.e., the time series, is obtained. According to time series Obtain the optimal trajectory in time, which corresponds to the rope length. .

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