Forward solution algorithm for parallel mechanical platform based on polynomial regression

Abstract

The present invention discloses a forward solution algorithm for a parallel mechanical platform based on polynomial regression, comprising the following steps: Step 1: constructing a six-variable cubic polynomial as follows: #imgabs0# wherein: is the parameter to be solved, A is the cubic coupling term parameter of 1, B is the quadratic coupling term parameter of 1, C is the linear term parameter of 1, and D is a constant parameter; Step 2: kinematically inversely solving the known posture parameters to obtain the corresponding robotic arm length, and then eliminating the remaining unknowns except for the parameters A, B, C, and D in the above formula; Step 3: establishing a set of 259 linear equations for solving A, B, C, and D, solving the linear equations, and obtaining the values of A, B, C, and D. The present invention solves the problems of non-globality and poor real-time performance of Newton iteration in kinematic forward solution problems, and can be used in a multivariate polynomial regression forward solution algorithm for a parallel mechanical platform. The polynomial regression method can converge globally and has stronger real-time performance.

Description

Technical Field

[0001] The present invention relates to a Stewart parallel platform, and in particular to a forward solution algorithm for a parallel mechanical platform based on polynomial regression. Background Art

[0002] The Stewart parallel platform has six sets of robotic arms, each of which connects the bottom static platform and the top dynamic platform through mechanical mechanisms such as universal joints or Hooke's joints. By adjusting the length of the six robotic arms, the dynamic platform can be controlled to achieve six degrees of freedom (x, y, z, α, β, γ) posture changes.

[0003] When the 6-DOF posture of the moving platform in the parallel mechanical platform is known, solving the length of each robotic arm when reaching that posture is called the inverse kinematic solution; when the lengths of the six robotic arms are known, solving the posture of the moving platform is called the direct kinematic solution.

[0004] The kinematic inverse solution can be solved by space vectors. Abstract the platform and establish a mathematical model such as Figure 1 .

[0005] The kinematic inverse solution algorithm is:

[0006]

[0007] in,

[0008]

[0009] is the rotation matrix of the parallel platform.

[0010] Due to the unique structural characteristics of the parallel mechanical platform, it is difficult to obtain a closed-form analytical solution for the kinematics. Numerical solutions are usually used.

[0011] Existing forward solutions for parallel platforms usually use the numerical solution method of Newton iteration.

[0012] Assume a given set of robot arm length data L tar =[l1,l2,l3,l4,l5,l6] T , and its corresponding pose parameters are X = [x, y, z, α, β, γ] T The mapping relationship between the posture parameters and the length of the robot arm is G, then,

[0013] L tar =G(X)

[0014] Given the initial pose parameter X0, the Newton formula for k-step iteration is:

[0015] X k+1 =X k +J -1(L tar -G(X k ))

[0016] Among them, (Φ-G(X k )) is the error between the robot arm length at the current position and the target robot arm length. J is the 6th-order Jacobian matrix of the motion platform.

[0017] The forward solution for the parallel platform described above requires multiple iterations involving a 6th-order matrix inversion, which is computationally intensive. Furthermore, the convergence of the Newton iteration is also affected by the initial value selection. Poor initial values can cause the iteration to fall into a local minimum, ultimately preventing convergence. Summary of the Invention

[0018] The present invention aims to provide a forward solution algorithm for a parallel mechanical platform based on polynomial regression, which can achieve global convergence and has stronger real-time performance.

[0019] To achieve the above object, the present invention is implemented by adopting the following technical solutions:

[0020] In order to improve the real-time and global performance of the system, the inverse kinematics formula Φ = G(X) can be converted into the forward formula X = G -1 (Φ), by solving G -1 To avoid multiple iterations. Due to the nonlinear nature of the parallel platform, it is impossible to find the inverse mapping by inverting G. According to the Weierstrass polynomial approximation theorem, any continuous function on a closed interval can be expressed as the limit of a polynomial sequence that converges uniformly. Based on this theory, the workspace is limited to a closed space near the initial posture. The length of the robotic arm obtained by inverse kinematics is also within a closed space, satisfying the conditions of the Weierstrass polynomial approximation theorem.

[0021] The polynomial regression-based forward solution algorithm for a parallel mechanical platform disclosed in the present invention comprises the following steps:

[0022] Step 1: Construct a six-variable cubic polynomial as follows:

[0023] The forward solution algorithm of the parallel mechanical platform based on polynomial regression is characterized by comprising the following steps:

[0024] Step 1: Construct a six-variable cubic polynomial as follows:

[0025]

[0026] Where: is the parameter to be solved, A is the cubic coupling term parameter of l, B is the quadratic coupling term parameter of l, C is the linear term parameter of l, and D is a constant parameter;

[0027] Step 2: Inversely solve the kinematics of the known pose parameters to obtain the corresponding robotic arm length, and then use the above formula to eliminate the remaining unknowns except parameters A, B, C, and D;

[0028] Step 3: Establish 259 linear equations to solve A, B, C, and D, and solve the linear equations to obtain the values of A, B, C, and D.

[0029] Preferably, in step 3, the Monte Carlo method is used to randomly select several groups of input and output data as training samples, establish a training equation group, and use the least squares method to solve the minimum norm least squares solution of the training equation group to establish the regression equation model of the six-variable cubic polynomial.

[0030] Further preferably, in step 3, an offline training and online solving method is adopted.

[0031] Preferably, the number of groups is 100,000 groups.

[0032] Preferably, in step 3, the input and output data are selected within a limited workspace.

[0033] The present invention can be used for a multivariate polynomial regression positive solution algorithm of a parallel mechanical platform. The polynomial regression method can globally converge and has stronger real-time performance.

[0034] The present invention solves the problems of non-globality and poor real-time performance of Newton iteration in kinematics forward solution problems. BRIEF DESCRIPTION OF THE DRAWINGS

[0035] Figure 1 It is the mathematical model of Stewart parallel platform. DETAILED DESCRIPTION

[0036] In order to make the objectives, technical solutions and advantages of the present invention more clearly understood, the present invention is described in further detail below.

[0037] Example 1

[0038] This embodiment discloses a forward solution algorithm for a parallel mechanical platform based on polynomial regression, which is as follows:

[0039] By constructing a six-variable cubic polynomial:

[0040]

[0041] Among them, A ijk , B mn , C p , D is the parameter to be solved, A is the cubic coupling term parameter of l, B is the quadratic coupling term parameter of l, C is the linear term parameter of l, and D is the constant parameter.

[0042] By inversely solving the kinematics of the known posture parameters to obtain the corresponding robot arm length, and then substituting it into the above formula, the remaining unknowns except parameters A, B, C, and D can be eliminated.

[0043] It can be seen from the constructed six-variable cubic polynomial that in order to solve the parameters A, B, C, and D, the number of linear equations that need to be established is 6 3 +6 2 +6+1=259. Then, the required parameters can be obtained by solving the linear equations, which is the "training" process.

[0044] However, since the constructed equation is the original unknown G -1 The cubic polynomial approximation of G is used, and the selected input manipulator length and output posture have uncertainty for the constructed equation. It is impossible to guarantee that the randomly selected 259 parameters can make the equation correct for G. -1 Therefore, in this embodiment, 100,000 sets of input and output data are randomly selected as training samples by the Monte Carlo method in a limited workspace. It can be seen that the training equations are overdetermined, and due to the -1 The number of approximations is less than ∞, making it impossible for all equations to hold simultaneously. However, the least squares method can be used to find the minimum norm least squares solution to this overdetermined equation. This completes the multivariate polynomial regression model. Using offline training and online solving can improve the system's real-time performance.

[0045] Example 2

[0046] Based on Example 1, this example compares and verifies the parallel mechanical platform forward solution algorithm based on polynomial regression disclosed in the present invention and the Newton iteration method, as follows:

[0047] The Newton iteration method and polynomial regression algorithm codes were written in MATLAB, and the Euclidean distance of their errors and the operation time were tested.

[0048] On the same hardware platform, after using 100,000 sets of training samples to train the polynomial coefficients, the Newton iteration and polynomial regression methods are compared for the same 10,000 sets of test samples, as shown in Table 1:

[0049] Table 1

[0050] Newton iteration Polynomial regression Mean Euclidean distance error 0.0020 0.0011 Average calculation time 89.1uS 26.2uS

[0051] As can be seen from Table 1, the polynomial regression is significantly better than the Newton iteration method.

[0052] Of course, the present invention may have many other embodiments. Without departing from the spirit and essence of the present invention, those skilled in the art may make various corresponding changes and modifications based on the present invention, but these corresponding changes and modifications should all fall within the scope of protection of the claims attached to the present invention.

Claims

1. A forward solution algorithm for a parallel mechanical platform based on polynomial regression, characterized by: The following steps are involved: Step 1: Construct a six-variable cubic polynomial as follows: in: Both represent the length of the robotic arm. , , , D is the parameter to be solved, for The cubic coupling term parameters, for The secondary coupling term parameters, for The linear term parameter, D is a constant parameter; Step 2: Inversely solve the kinematics of the known pose parameters to get the corresponding length of the robotic arm, and then eliminate the parameters in the above formula. , , , the remaining unknowns except D; Step 3: Establish the solution , , ,D's 259 linear equations, solving the linear equations, we get , , , the value of D.

2. The polynomial regression-based forward solution algorithm for parallel mechanical platforms according to claim 1, characterized in that: In step 3, the Monte Carlo method is used to randomly select several groups of input and output data as training samples, establish a training equation group, and use the least squares method to solve the minimum norm least squares solution of the training equation group to establish the regression equation model of the six-variable cubic polynomial.

3. The polynomial regression-based forward solution algorithm for parallel mechanical platforms according to claim 2, characterized in that: In step 3, an offline training and online solution method is adopted.

4. The polynomial regression-based forward solution algorithm for a parallel mechanical platform according to claim 2 or 3, characterized in that: The number of groups is 100,000 groups.

5. The polynomial regression-based forward solution algorithm for parallel mechanical platforms according to claim 2, characterized in that: In step 3, the input and output data are selected within a limited workspace.

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