A method for solving the kinematics of a virtual ball joint

Abstract

This invention discloses a method for solving the kinematics of a virtual ball joint, belonging to the field of robot kinematics technology. To address the problem that the motion trajectory of existing virtual ball joint mechanisms is elliptical, making kinematic solving difficult and greatly limiting the application scope of virtual ball joints, this application effectively decomposes the forward and inverse motions of the virtual ball joint during operation, and models and calculates the forward and inverse motions of the virtual ball joint separately, obtaining relatively objective calculation results. Simulation verification is performed. To enable kinematic solving, approximate calculations are used in the solution process, resulting in some error between the calculated results and the true values. However, this invention is the first to propose a kinematic method for this structure, filling a gap in the field of kinematics. This application is mainly used for solving the kinematics of virtual ball joints.

Description

Technical Field

[0001] The invention relates to the technical field of robot kinematics, and in particular to a method for solving the kinematics of a virtual ball-joint joint. Background Art

[0002] The virtual ball joint is a highly flexible and compact joint structure, with three curved rods evenly distributed around the upper and lower platforms. The special shape of the curved rods allows the three curved rods to fit compactly in a small space. It is an ideal structure for robot joints. It is currently used for humanoid robot wrist joints and can achieve flexible bending in space. Some studies have proposed using rope drive transmission, with four rope drive anchor points evenly distributed around the circumference of the moving platform, to control the posture of the virtual ball joint moving platform, i.e., the terminal posture, by controlling the rope stretching.

[0003] With the development of robot application fields, robot structures have become more diversified, and more parallel mechanism joints have emerged for stiffness and precision. The forward kinematics of parallel mechanisms is often more difficult than the inverse kinematics. Virtual ball joints are a type of parallel mechanism, and because the motion trajectory of the mechanism is not a sphere but an ellipsoid, coupled with the complex spatial configuration of the structure, it is difficult to solve its kinematics. Due to this problem, the widespread application of virtual ball joints is greatly limited. Summary of the invention

[0004] In order to solve the problem that the motion trajectory of the existing virtual ball joint mechanism is an ellipse, it is difficult to solve its kinematics, which greatly limits the application scope of the virtual ball joint, and further provides a method for solving the kinematics of a virtual ball joint.

[0005] A method for solving kinematics of a virtual ball joint, wherein the simulation method is implemented by the following steps:

[0006] Step 1: Select the zero position of the robot parallel joint, simplify the parallel mechanism, set the design structural parameters used to determine the position of each kinematic pair of the parallel mechanism, and set the joint axis as S:

[0007] Step 2: Construct an inertial coordinate system;

[0008] Step 3: Split the parallel mechanism simplified in step 1 into several single-branch chains, and describe the kinematic pairs on each branch chain using the inertial reference system constructed by the kinematic screw in step 2;

[0009] Step 4: Perform forward kinematic solution on the kinematic pairs on each branch chain in step 3;

[0010] Step 5: Perform inverse kinematics solution on the kinematic pairs on each branch chain in step 3;

[0011] Furthermore, the specific method of constructing the inertial coordinate system in step 2 is as follows:

[0012] In the zero position state, select a convenient position on the static platform as the origin of the inertial reference coordinate system, and select the coordinate axis direction of the coordinate system reasonably according to the actual distribution of the kinematic pairs and the principle of simplified calculation; select a convenient position on the dynamic platform as the origin of the end tool coordinate system, and select the coordinate axis direction of the coordinate system reasonably according to the actual distribution of the kinematic pairs and the principle of simplified calculation;

[0013] Furthermore, the analysis process of the macroscopic simulation results in step 1 is as follows:

[0014] The stress cloud diagram of the simulation results is used to analyze the unit strain and stress state in different regions, and the units of interest (edge ​​regions) are extracted. The strain is analyzed to be elongation along the rolling direction (X axis) and compression along the normal direction (Y axis), which is used as the boundary condition for microstructure simulation.

[0015] Furthermore, the specific method for performing forward kinematic solution on the kinematic pairs on each branch chain in step 3 in step 4 is as follows:

[0016] Step 41: Use the transformation matrix M to describe the position and posture of the origin of the moving platform coordinate system in the inertial reference coordinate system;

[0017]

[0018] In the above matrix h c The connecting rod B in the virtual ball joint joint i C i The straight-line distance between them, h0 is the connecting rod A in the virtual ball joint kinematic pair i B i The straight-line distance between the connecting rod A i B i The straight line distance between them is equal to the connecting rod C i D i The straight-line distance between

[0019] in Where l c The connecting rod B in the virtual ball joint joint i C i The length of the connecting rod OA i Length;

[0020] Step 42: For the n kinematic pairs on the i-th single-spending chain, the end position and the zero position have the following transformation relationship:

[0021]

[0022] In the above formula, T i is the homogeneous transformation matrix at the end of the branch chain, S inrepresents the spinor coordinate of the nth joint of the i-th branch, θ in represents the rotation angle of the nth joint of the i-th branch;

[0023] At the same time, since the end effector of each branch chain is on the same platform, the end posture should be the same, so:

[0024] T1=T2=T3=…T i (2)

[0025] The Baker–Campbell–Hausdorff formula is one of the commonly used formulas in group theory and quantum mechanics. Since its complete form is relatively complicated, only the first few terms of its expansion are given as follows:

[0026]

[0027] In the above formula, A and B are matrices, and [] are Lie brackets;

[0028] Taking the first two terms in equation (3) as approximations, we get:

[0029] ln(e A e B )=A+B (4)

[0030] Combining equations (1) and (4), since the coordinate system transformation matrix requires the determinant to be non-zero, it is easy to know that M is a reversible matrix, and then there is a constraint equation:

[0031]

[0032] Step 43: Input the required structural design parameters and the numerical solution of the constraint equation of the driving pair to obtain the approximate solution of the terminal posture and complete the forward kinematics modeling and solution;

[0033] Furthermore, the specific method for performing inverse kinematics solution on the kinematic pairs on each branch chain in step 3 in step 5 is as follows:

[0034] Step 51: Space velocity ν s There are the following expressions:

[0035]

[0036] In the above formula:

[0037]

[0038]

[0039] T -1 represents the inverse of the terminal pose matrix, To express the derivative, represents the derivative of the terminal pose matrix;

[0040] Step 52: Calculate the spatial Jacobian matrix for the i-th single-spending chain:

[0041]

[0042] The i-th column of the Jacobian matrix describes the rotation of the i-th joint axis relative to the inertial coordinate system, and is also a function of the first several joint variables;

[0043] Step 53: Use the Newton-Raphson method to perform numerical calculations on the inverse kinematics.

[0044] g(θ)=x d -f(θ) (7)

[0045] Determine the initial value θ0 and list the first-order Taylor expansion of g(θ)

[0046]

[0047] The pseudo-inverse matrix optimization operation of Jacobian matrix is ​​introduced, and the error termination condition is set to iteratively complete the inverse kinematics solution;

[0048] The beneficial effects of this application compared to the prior art are as follows:

[0049] The present invention provides a method for solving the kinematics of a virtual ball-joint joint. By effectively decomposing the forward motion and inverse motion of the virtual ball-joint joint when it is working, and modeling and calculating the forward motion and inverse motion of the virtual ball-joint joint respectively, a relatively objective calculation result is obtained. Through simulation verification, in order to solve the kinematics, approximate calculations are used in the solution process. The result obtained has a certain error with the true value, but as a whole, a kinematic method of the structure is proposed for the first time, filling the gap of no kinematics. Moreover, the modeling and settlement process involved in this application is process-based and formulated, which is friendly to computer programming and easy to program. After programming, it is very effective for selecting structural design parameters, has high versatility, and is also conducive to promoting the practicality of virtual ball joints. BRIEF DESCRIPTION OF THE DRAWINGS

[0050] Figure 1 This is a schematic diagram of the structural axial side of the virtual ball joint described in this application;

[0051] Figure 2 This is a schematic diagram of the front view of the structure of the virtual ball joint described in this application;

[0052] Figure 3 This is a comparison chart of the forward kinematics theory calculation and simulation results in the X-axis position coordinates described in this application (the horizontal axis in the figure is time and the vertical axis is distance);

[0053] Figure 4 This is a comparison chart of the Y-axis position coordinates of the forward kinematics theory calculation and the simulation results described in this application (the horizontal axis in the figure is time and the vertical axis is distance);

[0054] Figure 5 This is a comparison chart of the Z-axis position coordinates between the forward kinematics theory calculation and the simulation results described in this application (the horizontal axis in the figure is time and the vertical axis is distance). DETAILED DESCRIPTION

[0055] Specific implementation method 1: Combination Figures 1 to 5 This embodiment describes a method for solving the kinematics of a virtual ball joint. The simulation method is implemented by the following steps:

[0056] Step 1: Select the zero position of the robot parallel joint, simplify the parallel mechanism, set the design structural parameters used to determine the position of each kinematic pair of the parallel mechanism, and set the joint axis as S:

[0057] Step 2: Construct an inertial coordinate system;

[0058] Step 3: Split the parallel mechanism simplified in step 1 into several single-branch chains, and describe the kinematic pairs on each branch chain using the inertial reference system constructed by the kinematic screw in step 2;

[0059] Step 4: Perform forward kinematic solution on the kinematic pairs on each branch chain in step 3;

[0060] Step 5: Perform inverse kinematics solution on the kinematic pairs on each branch chain in step 3.

[0061] Specific implementation method 2: Combination Figures 1 to 5 This embodiment is described. The difference between this embodiment and the first embodiment is that the specific method of constructing the inertial coordinate system in step 2 is as follows:

[0062] In the zero position state, a convenient position for calculation is selected on the static platform as the origin of the inertial reference coordinate system, and the direction of the coordinate axis of the coordinate system is reasonably selected according to the actual distribution of the kinematic pair and the principle of simplified calculation; a convenient position for calculation is selected on the dynamic platform as the origin of the end tool coordinate system, and the direction of the coordinate axis of the coordinate system is reasonably selected according to the actual distribution of the kinematic pair and the principle of simplified calculation. The other method steps are the same as those in the first specific implementation method.

[0063] Specific implementation method three: Combination Figures 1 to 5 This embodiment is described. The difference between this embodiment and the second embodiment is that the specific method of performing forward kinematic solution on the kinematic pairs on each branch chain in step 3 in step 4 is as follows:

[0064] Step 41: Use the transformation matrix M to describe the position and posture of the origin of the moving platform coordinate system in the inertial reference coordinate system;

[0065]

[0066] In the above matrix h c The connecting rod B in the virtual ball joint joint i C i The straight-line distance between them, h0 is the connecting rod A in the virtual ball joint kinematic pair i B i The straight-line distance between the connecting rod A i B i The straight line distance between them is equal to the connecting rod C i D i The straight-line distance between

[0067] in Where l c The connecting rod B in the virtual ball joint joint i C i The length of the connecting rod OA i Length;

[0068] Step 42: For the n kinematic pairs on the i-th single-spending chain, the end position and the zero position have the following transformation relationship:

[0069]

[0070] In the above formula, T i is the homogeneous transformation matrix at the end of the branch chain, S in represents the spinor coordinate of the nth joint of the i-th branch, θ in represents the rotation angle of the nth joint of the i-th branch;

[0071] At the same time, since the end effector of each branch chain is on the same platform, the end posture should be the same, so:

[0072] T1=T2=T3=…T i (2)

[0073] The Baker–Campbell–Hausdorff formula is one of the commonly used formulas in group theory and quantum mechanics. Since its complete form is relatively complicated, only the first few terms of its expansion are given as follows:

[0074]

[0075] In the above formula, A and B are matrices, and [] are Lie brackets;

[0076] Taking the first two terms in equation (3) as approximations, we get:

[0077] ln(e A e B )=A+B (4)

[0078] Combining equations (1) and (4), since the coordinate system transformation matrix requires the determinant to be non-zero, it is easy to know that M is a reversible matrix, and then there is a constraint equation:

[0079]

[0080] Step 43: Input the required structural design parameters and the numerical solution of the driving pair to obtain the approximate solution of the terminal posture and complete the forward kinematics modeling solution. The other components and connection methods are the same as those in the second specific implementation method.

[0081] Specific implementation method four: Combination Figures 1 to 5 This embodiment is described. The difference between this embodiment and the third embodiment is that the specific method for performing inverse kinematics solution on the kinematic pairs on each branch chain in step 3 in step 5 is as follows:

[0082] Step 51: Space velocity ν s There are the following expressions:

[0083]

[0084] In the above formula:

[0085]

[0086]

[0087] T -1 represents the inverse of the terminal pose matrix, To express the derivative, represents the derivative of the terminal pose matrix;

[0088] Step 52: Calculate the spatial Jacobian matrix for the i-th single-spending chain:

[0089]

[0090] The i-th column of the Jacobian matrix describes the rotation of the i-th joint axis relative to the inertial coordinate system, and is also a function of the first several joint variables;

[0091] Step 53: Use the Newton-Raphson method to perform numerical calculations on the inverse kinematics.

[0092] g(θ)=x d -f(θ) (7)

[0093] Determine the initial value θ0 and list the first-order Taylor expansion of g(θ)

[0094]

[0095] The pseudo-inverse matrix optimization operation of the Jacobian matrix is ​​introduced, and the error termination condition is set to iteratively complete the inverse kinematics solution. The other components and connection methods are the same as those of the third specific implementation method.

[0096] The present invention has been disclosed as above with preferred implementation cases, but it is not intended to limit the present invention. Any technician familiar with the profession can make some changes or modifications to equivalent implementation cases with equivalent changes by using the above-disclosed structures and technical contents without departing from the scope of the technical solution of the present invention. However, any simple modifications, equivalent changes and modifications made to the above implementation cases based on the technical essence of the present invention without departing from the content of the technical solution of the present invention are still within the scope of the technical solution of the present invention.

[0097] Example:

[0098] The virtual ball joint structure mainly consists of three identical OA i B i C i D i P branched chain. i The plane is the base platform (considered as a static platform in the analysis of single joint motion), and the radius of the circumscribed circle of the triangle formed by its vertices is r; PD i The plane is regarded as a moving platform, and the radius of the circumscribed circle of the triangle formed by its vertices is also r; the revolute pairs A1 and A2 are driving pairs, and the connecting rod A i B i =C i D i The length is h0, connecting rod B i C i The length is l c , all parameters are in mm. With O as the origin, OA1 as the positive direction of the y-axis, OA i The upward direction of the normal line of the plane is the positive direction of the z-axis to establish an inertial coordinate system s; take P as the origin, PD1 as the negative direction of the y-axis, PD i The object coordinate system b is established with the upward direction of the normal line of the plane as the positive direction of the z-axis.

[0099] It can be seen that the initial posture transformation matrix between the object coordinate system b and the inertial coordinate system s is

[0100]

[0101] in:

[0102] For the branched chain OA1B1C1D1P, Figure 2 It can be seen that the coordinates of the rotation units of each axis are as follows

[0103]

[0104] So the kinematic equation of the branch chain is

[0105]

[0106] Where T1 is the homogeneous transformation matrix of the branch end, S ji represents the i-th joint spinor coordinate of the j-th branch, θ ji represents the i-th joint rotation angle of the j-th branch;

[0107] Similarly, the kinematic equations of the second and third branches are obtained:

[0108]

[0109]

[0110] According to the Baker–Campbell–Hausdorff formula, we can get

[0111]

[0112] At the same time, since the end effector of each branch chain is on the same platform, the end posture should be the same, so there is

[0113] T1=T2=T3

[0114] It is easy to know that M is a reversible matrix, so we have

[0115]

[0116] This gives the system of equations

[0117]

[0118] Input design structure parameters r, h c , h0 and the driving pair rotation angle θ 11 ,θ 21 That is, the approximate position and posture of the terminal moving platform is obtained

[0119] For example, if the parameter is set to r=40, l c =100,h c =60, h0=20, input θ 11 =π / 12,θ 21 =π / 6, the remaining rotation angle radian array and the end position are obtained as

[0120] [-0.3023 -0.3023 0.5236 0.3023 0.3023 0.5236 0.0000 0.6046 0.60460.0000]

[0121]

[0122] Through simulation verification, it is shown that within the driving angle range of 30°, the dynamic platform can flip over 90°, and the maximum error of the end position does not exceed 5°.

[0123] Space velocity ν s have

[0124]

[0125] in

[0126]

[0127]

[0128] The spatial Jacobian matrix of the ith single-spending chain is calculated as

[0129]

[0130] Where J s That is, the Jacobian matrix under the fixed coordinate situation, Ad represents the adjoint matrix, Represents the angular velocity of the joint, and then the spatial Jacobian J s Converted to object Jacobian matrix J b ,as follows

[0131]

[0132] The coordinate vector x and the series forward kinematics equation x = f(θ) are used to represent the end coordinates, θ is the joint angle, and x is d is the target end coordinate system, Ad Tbs The adjoint matrix of Tbs satisfies the following according to the Newton-Raphson method:

[0133] g(θ)=x d -f(θ)

[0134] Where g(θ) represents the objective function of the iterative method

[0135] It is known that the initial estimate θ0 is close to the true solution θ d , write the above formula in Taylor expansion form, that is

[0136]

[0137] In the formula

[0138]

[0139] Δθ=θ d -θ 0

[0140] Substitute the Jacobian matrix of the series object, cut off the first term and simplify it to get

[0141] J(θ 0 )Δθ=x d -f(θ 0 )

[0142] θ i =θ i-1 +Δθ

[0143] where i=1,2,3,...,θ i represents the i-th iteration value of θ, θ i-1 The i-th iteration value of θ is the positive difference of the i-1-th iteration value, and Δθ represents the difference between the target joint angle and the initial joint angle;

[0144] After continuous iteration of the above two formulas, they finally converge to obtain the angles θ of each joint of the θ branch chain. d , the same goes for other branches, and finally the inverse kinematics of the parallel mechanism is obtained;

[0145] The iterative results of the rotation amplitude of each rotation pair of the branch chain obtained by inputting Example 1 are:

[0146] θ1=[0.5236 -0.3023 -0.3023 0.5236]

[0147] The calculation results are the same as those of forward kinematics, which verifies the feasibility of forward and inverse kinematics.

Claims

1. A method for solving the kinematics of a virtual ball joint, characterized by: The method is achieved by the following steps: Step 1: Select the zero position of the robot parallel joint, simplify the parallel mechanism, set the design structural parameters used to determine the position of each kinematic pair of the parallel mechanism, and set the joint axis as S: Step 2: Construct an inertial coordinate system; Step 3: Split the parallel mechanism simplified in step 1 into several single-branch chains, and describe the kinematic pairs on each branch chain using the inertial reference system constructed by the kinematic screw in step 2; Step 4: Solve the forward kinematics of the kinematic pairs on each branch chain in step 3; The specific method for performing forward kinematic solution on the kinematic pairs on each branch chain in step 3 in step 4 is as follows: Step 41: Use the transformation matrix M to describe the position and posture of the origin of the moving platform coordinate system in the inertial reference coordinate system; In the above matrix h c The connecting rod B in the virtual ball joint joint i C i The straight-line distance between them, h0 is the connecting rod A in the virtual ball joint kinematic pair i B i The straight-line distance between the connecting rod A i B i The straight line distance between them is equal to the connecting rod C i D i The straight-line distance between them, R is the rotation matrix of the rigid body rotation; in Where l c The connecting rod B in the virtual ball joint joint i C i The length of the connecting rod OA i Length; Step 42: For the n kinematic pairs on the i-th single-spending chain, the end position and the zero position have the following transformation relationship: In the above formula, T i is the homogeneous transformation matrix at the end of the branch chain, S in represents the spinor coordinate of the nth joint of the i-th branch, θ in represents the rotation angle of the nth joint of the i-th branch; At the same time, since the end effector of each branch chain is on the same platform, the end posture is the same, so: <h2 style=";text-align:left;direction:ltr">T1=T2=T3=…T<h2 style=";text-align:left;direction:ltr"> i <h2 style=";text-align:left;direction:ltr"> (2) The first few terms of the expansion of the Baker–Campbell–Hausdorff formula are as follows: In the above formula, A and B are matrices, and [] are Lie brackets; Taking the first two terms in equation (3) as approximations, we get: ln(ie A have been B )=A+B (4) Combining equations (1) and (4), since the coordinate system transformation matrix requires the determinant to be non-zero, it is easy to know that M is a reversible matrix, and then there is a constraint equation: Step 43: Input the required structural design parameters and the numerical solution of the constraint equation of the driving pair to obtain the approximate solution of the terminal posture and complete the forward kinematics modeling and solution; Step 5: Solve the inverse kinematics of the kinematic pairs on each branch chain in step 3; The specific method for performing inverse kinematics solution on the kinematic pairs on each branch chain in step 3 in step 5 is as follows: Step 51: Space velocity ν s There are the following expressions: In the above formula: T -1 represents the inverse operation of the matrix, To express the derivative; Step 52: Calculate the spatial Jacobian matrix for the i-th single-spending chain: The i-th column of the Jacobian matrix describes the rotation of the i-th joint axis relative to the inertial coordinate system, and is also a function of the first several joint variables; Step 53: Use the Newton-Raphson method to perform numerical calculations on the inverse kinematics. g(θ)=x d -f(θ) (7) Determine the initial value θ0 and list the first-order Taylor expansion of g(θ) The pseudo-inverse matrix optimization operation of the Jacobian matrix is ​​introduced, and the error termination condition is set to iteratively complete the inverse kinematics solution.

2. A method for solving the kinematics of a virtual ball joint according to claim 1, characterized in that: The specific method of constructing the inertial coordinate system in step 2 is as follows: In the zero-position state, a position that is convenient for calculation is selected on the static platform as the origin of the inertial reference coordinate system, and the direction of the coordinate system axis is reasonably selected based on the actual distribution of the kinematic pairs and the principle of simplified calculation; a position that is convenient for calculation is selected on the dynamic platform as the origin of the end tool coordinate system, and the direction of the coordinate system axis is reasonably selected based on the actual distribution of the kinematic pairs and the principle of simplified calculation.

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