A spatial posture evaluation method for the conceptual configuration of a branch-driven parallel robot

Abstract

The present invention discloses a method for evaluating the spatial posture of a conceptual configuration of a branched-chain driven parallel robot. First, a kinematic analysis method based on singular values ​​is used to judge the reachability of all the terminal postures of the branched-chain driven parallel mechanism, and the terminal posture of the branched-chain driven parallel mechanism in the reachable workspace is obtained; then, based on the screw theory and the principle of virtual work, the complete Jacobian matrix and the full stiffness matrix of the mechanism are obtained, and the stiffness performance parameters of all the terminal postures of the branched-chain driven parallel mechanism are calculated, and the stiffness performance parameters corresponding to the terminal postures of all the mechanisms are obtained; finally, according to the terminal posture in the reachable workspace, combined with the stiffness performance parameters corresponding to the terminal postures of all the mechanisms, the optimal posture set is screened and obtained. The present invention can not only quantitatively evaluate the optimal posture set, but also realize the innovative mechanism concept design that meets the needs by comparing the optimal posture sets of mechanisms with different configurations.

Description

Technical Field

[0001] The present invention relates to a parallel mechanism design method in the field of robotics for high-end equipment, and in particular to a spatial posture evaluation method for a conceptual configuration of a branch-driven parallel robot. Background Art

[0002] Parallel mechanisms, unlike serial mechanisms, feature short transmission chains, fewer members, high specific stiffness, and manageable control. Consequently, they are widely used in robotics and support structures. Because branch-driven parallel mechanism robots can actively control their position in relatively confined spaces, they are widely used in aerospace, industrial assembly lines, modern logistics, and rehabilitation medicine.

[0003] Taking the field of rehabilitation medicine as an example, the number of people with joint disabilities is increasing year by year around the world, which places higher demands on the design of human joint enabling mechanisms. The application of parallel mechanisms in joint rehabilitation can produce joint rehabilitation mechanisms with high precision and relatively high stiffness, such as branch-driven parallel mechanism products. However, current designs are usually reverse-engineered and analogically designed based on existing products, while the forward design of product lines based on design requirements is still somewhat difficult. The main reason is the lack of accurate quantitative evaluation methods. For complex mechanical motion systems, how to positively evaluate the performance of conceptual design mechanisms based on requirements, screen out structural types with excellent kinematic and dynamic performance, and then quantitatively evaluate and select more suitable posture configurations, and then use mechanical impedance transmission control to make them universal and help patients with rehabilitation training, etc., is of great research significance.

[0004] Researchers at Purdue University in the United States (Bi Z M. Design of a spherical parallel kinematic machine for ankle rehabilitation [J]. Advanced Robotics, 2013, 27 (2): 121-132) calculated the degrees of freedom and performed forward and inverse kinematic analysis on the spherical parallel mechanism PKM to determine the workspace of the mechanism. They also verified the correctness of the kinematic solution by deriving the Jacobian matrix for evaluating dexterity.

[0005] Researchers from the University of Nottingham in the UK (Russo M, Ceccarelli M. Analysis of a wearable robotic system for ankle rehabilitation[J]. Machines, 2020, 8(3): 48-63) conducted kinematic and static analysis on the designed S-4SPS parallel mechanism to evaluate the feasibility of the mechanism's range of motion and the load on the user's ankle joint, and verified the correctness of the analysis results through numerical simulation models.

[0006] Therefore, a mechanism with excellent performance can effectively improve the range of posture selection during motion work, and there is also coupling between different postures and mechanism performance. Therefore, it is necessary to select a mechanism with better performance and a better posture to bring out the optimal performance of the branch-driven parallel mechanism and improve the final effect of the robot equipment. Summary of the Invention

[0007] In order to solve the problems existing in the background technology, the purpose of the present invention is to provide a spatial posture evaluation method for the conceptual configuration of a branched-chain driven parallel robot.

[0008] To achieve the above object, the technical solution of the present invention is as follows:

[0009] Step 1: Use the kinematic analysis method based on singular values ​​to judge the reachability of all the end positions of the branch-driven parallel mechanism, and obtain the end position of the branch-driven parallel mechanism in the reachable workspace;

[0010] Step 2: Calculate the stiffness performance parameters of all end-positions of the branch-driven parallel mechanism to obtain the stiffness performance parameters corresponding to all end-positions of the mechanism;

[0011] Step 3: Based on the end-position in the reachable workspace, combined with the stiffness performance parameters corresponding to the end-positions of all mechanisms, the singularity and stiffness are comprehensively quantified to screen and obtain the optimal pose set.

[0012] The step 1 is specifically as follows:

[0013] S11: Perform continuous inverse kinematics analysis on all end-positions of the branch-driven parallel mechanism to obtain the mapping relationship between the end-position and the motion parameters of the branch.

[0014] S12: Perform continuous forward kinematic analysis on the mechanism branch motion parameters to obtain the mapping relationship between the mechanism branch motion parameters and the mechanism end pose;

[0015] S13: According to the mapping relationship between the end-position of the mechanism and the motion parameters of the mechanism branch, and the mapping relationship between the motion parameters of the mechanism branch and the end-position of the mechanism, the reachability judgment based on the singular value is performed on all the end-positions of the mechanism to obtain the end-position of the branch-driven parallel mechanism in the reachable workspace.

[0016] The S13 is specifically:

[0017] S131: taking one of the multiple angles at the end of the mechanism as a reference angle;

[0018] S132: Calculating the condition number corresponding to each mechanism end posture at the same reference angle based on the mapping relationship between the mechanism end posture and the mechanism branch motion parameters and the mapping relationship between the mechanism branch motion parameters and the mechanism end posture, and performing reachability judgment on each mechanism end posture at the current value of the reference angle based on the condition number to obtain the reachable end posture at the current value of the reference angle;

[0019] S133: Change the value of the reference angle, repeat S132, and obtain the achievable end-positions at different values ​​of the reference angle, until all achievable end-positions within the value range of the reference angle are obtained and used as the end-positions of the branch-driven parallel mechanism in the achievable workspace.

[0020] In S132, the condition number C of each end position of the mechanism is calculated by the following formula:

[0021]

[0022] Among them, σ max (J) is the maximum singular value of the mechanism Jacobian matrix J, σ min (J) is the minimum singular value of the Jacobian matrix J of the mechanism, R + represents the set of real numbers.

[0023] In the above S132, after eliminating the end-position postures of the mechanism with abnormal condition numbers according to the "3σ accurate measurement", the median value of the condition numbers of the end-positions of each mechanism under the current value of the reference angle is determined and used as the judgment threshold. The end-positions of the mechanism with condition numbers less than the judgment threshold are recorded as reachable end-positions and placed in the reachable workspace.

[0024] The step 2 is specifically as follows:

[0025] S21: Determine the drive Jacobian matrix and constraint Jacobian matrix of each end-position of the mechanism based on the screw theory, and then obtain the complete Jacobian matrix;

[0026] S22: Calculate the stiffness of each branch in the branch-driven parallel mechanism, then perform mechanical analysis on the branch-driven parallel mechanism based on the stiffness of each branch and the complete Jacobian matrix. Combined with the principle of virtual work, obtain the full stiffness matrix of the branch-driven parallel mechanism at the current end position of the mechanism.

[0027] S23: obtaining stiffness performance parameters of the current end position of the mechanism according to the full stiffness matrix of the branch-driven parallel mechanism at the current end position of the mechanism;

[0028] S24: Repeat S21-S23 to calculate and obtain the stiffness performance parameters corresponding to the end positions of all mechanisms.

[0029] In said S23, the stiffness performance parameters of the current end position of the mechanism include the minimum eigenvalue K of the full stiffness matrix of the branch-driven parallel mechanism at the current end position of the mechanism. min , maximum eigenvalue K max And the stiffness specificity KSI, where the calculation formula of stiffness specificity KSI is:

[0030]

[0031] The beneficial effects of the present invention are:

[0032] (1) The present invention uses a parallel mechanism with redundant drive degrees of freedom. Compared with conventional parallel mechanisms, the redundant drive degrees of freedom improve the reliability, accuracy, and response speed of the system, increase the flexibility of the system, and meet the actual motion posture adjustment requirements.

[0033] (2) The screw theory is introduced for stiffness calculation. The screw method can obtain the stiffness calculation results of the drive end mechanical chain more concisely and accurately, and can be used for the quantitative evaluation and optimization of the stiffness of the local mechanical chain.

[0034] (3) The present invention quantitatively determines the motion and stiffness performance of each posture in the working space of the mechanism, which can lay the foundation for the rhythmic control of continuous motion.

[0035] The present invention can effectively screen postures with excellent motion performance and stiffness performance. It can also select a mechanism configuration that meets the requirements by comparing the optimal posture sets of mechanisms with different configurations, thereby improving the effect of actual multi-axis linkage spatial posture adjustment. BRIEF DESCRIPTION OF THE DRAWINGS

[0036] Figure 1 This is a three-dimensional assembly diagram of the branch chain driven parallel mechanism selected in the embodiment of the present invention;

[0037] Figure 2 A schematic diagram of an unreachable workspace according to an embodiment of the present invention;

[0038] Figure 3Schematic diagram of the inaccessible working space of the 3-UPS / S parallel mechanism;

[0039] Figure 4 The singularity rate curve diagram of the embodiment selected for the present invention in the working space;

[0040] Figure 5 is the singularity rate curve of the 3-UPS / S parallel mechanism in the working space;

[0041] Figure 6 The maximum eigenvalue K in the working space of the embodiment selected for the present invention max Result graph;

[0042] Figure 7 is the maximum eigenvalue K of the 3-UPS / S parallel mechanism in the working space max Result graph;

[0043] Figure 8 The minimum eigenvalue K in the working space of the embodiment selected for the present invention min Result graph;

[0044] Figure 9 is the minimum eigenvalue K of the 3-UPS / S parallel mechanism in the working space min Result graph;

[0045] Figure 10 The figure is a graph of the stiffness specificity KSI result of the embodiment selected in the present invention in the working space;

[0046] Figure 11 The stiffness specificity KSI result diagram of the 3-UPS / S parallel mechanism in the workspace;

[0047] Figure 12 Flow chart of the method of the present invention.

[0048] In the figure: static platform 1, dynamic platform 2, supporting ball joint 3, RUPS branch chain 4, servo motor 41, rotary joint 42, universal joint 43, first sleeve 431, universal joint fork 432, second sleeve 433, thrust ball bearing 434, deep groove ball bearing 435, moving joint outer cylinder 44, moving joint inner rod 45, first ball joint 46, ball ring 461, fisheye bearing 462. DETAILED DESCRIPTION

[0049] The present invention will be further described below with reference to the embodiments and the accompanying drawings.

[0050] See also Figure 1The present invention selects a 3-RUPS / S type parallel mechanism with redundant drive degrees of freedom. The mechanism consists of three RUPS branches 4, a middle supporting ball chain, an upper moving platform 2 and a lower static platform 1. The lower rotary joint is fixedly connected to the static platform through a bearing, so that the RUPS branch can only rotate along the static platform, and the supporting ball chain is fixedly connected to the moving platform. The middle supporting hinge has only one supporting ball joint 3, which is used to constrain the three moving degrees of freedom of the moving platform so that it can rotate along three axes. The static coordinate system A-XYZ and the moving coordinate system B-XYZ are respectively fixedly connected to the center of the static platform and the center of the supporting ball joint. Three servo motors are fixed on the static platform, and the output shafts of the servo motors are respectively connected to the rotary joints 42 on the three branches through bearings. The rotation of the output shaft drives the rotary joint 42 to rotate, thereby driving the branch chain to move and changing the posture of the moving platform.

[0051] The existence of component 42 determines that the conceptual configuration of the mechanism is 3-RUPS / S type. Without this component, the conceptual configuration of the mechanism is 3-UPS / S type.

[0052] Three identical fully symmetrical RUPS drive branches are connected in series from bottom to top by a rotary joint 42, a universal joint 43, a mobile joint and a first ball joint 46. The universal joint includes a first sleeve 431, a second sleeve 433, a universal joint fork 432, a thrust ball bearing 434 and a deep groove ball bearing 435; the first ball joint 46 includes a ball ring 461 and a fisheye bearing 462; the servo motor 41 is fixed on the static platform 1, and the servo motor output shaft is connected to the bottom of the rotary joint 42 through a bearing; four threaded holes are provided above the rotary joint 42 for connecting the sleeve 431 of the universal joint 43; the mobile joint consists of two parts: a mobile joint outer cylinder 44 and a mobile joint inner rod 45. A servo motor and a screw are provided in the cavity between the mobile joint outer cylinder 44 and the mobile joint inner rod 45. The rotational motion of the servo motor output shaft will be converted into linear motion of the screw (moving pair). The second sleeve 433 above the universal joint 43 and the first sleeve 431 below are both provided with four threaded holes, which are respectively connected to the rotating joint 42 and the movable joint outer tube 44; the movable joint inner rod 45 is connected to the first ball joint 46 by a set screw; the first ball joint 46 is composed of a fisheye bearing 462, a ball ring 461, etc., and the square boss extending from the ball ring 461 of the first ball joint 46 is provided with four threaded holes for connecting to the moving platform 2; compared with the first ball joint 46, the supporting ball joint 3 has different sleeve shapes of its fisheye bearing and the extended shaft shape of its ball ring.

[0053] The calculation formula for spatial freedom is:

[0054]

[0055] Among them, n is the total number of components of the mechanism, g is the number of kinematic pairs, f iis the number of degrees of freedom of the i-th kinematic pair. In the proposed 3-RUPS / S mechanism, the total number of mechanism components is 11, and the number of kinematic pairs is 13, including 3 rotational pairs (with 1 degree of freedom), 3 universal joints (with 2 degrees of freedom), 3 translation pairs (with 1 degree of freedom), and 4 ball joints (with 3 degrees of freedom). There are also 3 local rotational degrees of freedom (at the universal joints). Therefore, the number of degrees of freedom of this mechanism is:

[0056] m=6×(11-13-1)+(3×1+3×2+3×1+4×3-1×3)=3 (2)

[0057] According to the calculation results of the degrees of freedom, the parallel mechanism has 3 degrees of freedom. Each branch chain has two drives, a rotating joint 42 and a moving joint, for a total of 6 drives. The number of drives is greater than the number of degrees of freedom, making it a redundant drive mechanism.

[0058] The number of drives and degrees of freedom of the 3-UPS / S parallel mechanism are both 3, so the 3-RUPS / S mechanism selected in the present invention is compared and analyzed with the 3-UPS / S mechanism to illustrate the improvement of the kinematics and stiffness performance of the mechanism by the additional redundant drive degrees of freedom.

[0059] like Figure 12 As shown, the present invention includes the following steps:

[0060] Step 1: Use the singular value-based kinematic analysis method to determine the reachability of all the end-positions of the branch-driven parallel mechanism, and obtain the end-position of the branch-driven parallel mechanism in the reachable workspace (i.e., the reachable end-position);

[0061] Step 1 is as follows:

[0062] S11: Perform continuous inverse kinematics analysis on all end-positions of the branch-driven parallel mechanism to obtain the mechanism branch motion parameters (specifically, the distance between the head and the end of each branch) and the mapping relationship between the mechanism end-position and the mechanism branch motion parameters;

[0063] In this embodiment, for the 3-RUPS / S parallel mechanism, since it has three rotational degrees of freedom, the end position is taken as the final rotational position of the moving platform, and the branch chain motion parameter is taken as the distance between the beginning and the end of the branch chain.

[0064] The rotational position of the mechanism is described using T&T (title and torsion) Euler angles, where α is the precession angle, ranging from [0° to 360°), β is the nutation angle, ranging from [0° to 45°), and γ is the rotation angle of the moving platform around the z-axis, ranging from [-60° to 60°]. The rotation matrix from the static platform A-XYZ to the moving platform B-XYZ is:

[0065]

[0066] in, It represents the rotation matrix from the static platform A-XYZ to the moving platform B-XYZ. u, v, and w represent the three columns of the rotation matrix respectively. Rot() represents the rotation matrix around a certain coordinate axis.

[0067] In terms of the initial design values, the radius of the moving platform is set to 50mm, the radius of the static platform is 70mm, and the distances between the support ball joint and the upper and lower platforms are 85mm and 365mm respectively. The translation matrix from the static platform A-XYZ to the moving platform B-XYZ is for:

[0068]

[0069] Where T represents the matrix transpose.

[0070] Three points D on the moving platform i The position vectors in the static coordinate system A-XYZ and the moving coordinate system B-XYZ are and The two have the following transformation relationship:

[0071]

[0072] in, Known, respectively:

[0073]

[0074] Substituting into (5) we can get D i The coordinates of the three points in the static coordinate system A-XYZ, C1, C2, C3 are fixed, so the distance between the head and the end of each branch chain under a given posture can be obtained. On this basis, the distance C between the head and the end of each branch chain can be obtained by calculating the posture of each angle. i D i The range of changes is:

[0075]

[0076] The length parameters of each joint of the branch chain can be calculated based on the results of continuous inverse kinematics to meet the needs of length change.

[0077] S12: Perform continuous forward kinematic analysis on the mechanism branch motion parameters to obtain the mapping relationship between the mechanism branch motion parameters and the mechanism end pose;

[0078] Each branch chain actually has two straight lengths, R and UPS. Select the first and last ends of the branch chain, C. i D iThe length of the branch chain is used as the motion parameter of the branch chain. Compared with the series mechanism, the continuous forward kinematics analysis of the parallel mechanism is more complicated. In this embodiment, the continuous inverse kinematics analysis results of the parallel mechanism are combined and solved by a numerical solution.

[0079] First, given an initial rotation posture (α0, β0, γ0), the length of each branch at the beginning and end can be obtained through continuous inverse kinematics analysis. The target length of the branch chain is known to be L i , the difference between the two Recorded as

[0080]

[0081] L i Expressed as a function of three rotation angles α, β, and γ, that is, L i (α, β, γ), then the length of the three branches is L i The relationship between the angular velocity of the moving platform and the rotational speed is as follows:

[0082]

[0083]

[0084] Where, They represent the linear speeds of the three branches along the line connecting the head and the end, They represent the angular velocity of the moving platform, and the 3×3 coefficient matrix is ​​the inverse matrix of the Jacobian matrix of the mechanism (the Jacobian matrix describes the mapping relationship between the parameters of the mechanism branches and the end posture, so the coefficient matrix is ​​the inverse matrix of the Jacobian matrix constructed according to the definition). Multiplying both sides of Equation (9) by dt, Equation (9) becomes the length increment ΔL of the three branches at the beginning and end. i The relationship between the incremental position angle of the moving platform is:

[0085]

[0086] Substituting (8) into (11) we can get the first rotation angle increment (Δα (1) ,Δβ (1) ,Δγ (1) ), we can further get the first corrected moving platform angle (α1, β1, γ1) (α1=α0+Δα (1) ,β1=β0+Δβ (1) ,γ1=γ0+Δγ (1) ), according to the corrected rotation angle, the length of each branch chain is obtained through continuous inverse kinematics analysis Calculate again the target length L of the branch chain i Deviation Right now

[0087]

[0088] Substituting (12) into (11), we can get the second rotation angle increment (Δα (2) ,Δβ (2) ,Δγ (2) ), repeat the above steps to get the length of each branch chain The new length obtained will be closer to the target length Li, and the above iterative process will be repeated until the new head and end lengths are With target length L i When the deviation is within the allowable error range, the continuous forward kinematics analysis is completed. At this time, the dynamic platform angle (α n , β n , γ n ) is the actual rotational position of the moving platform that needs to be solved.

[0089] S13: According to the mapping relationship between the end posture of the mechanism and the motion parameters of the mechanism branch, and the mapping relationship between the motion parameters of the mechanism branch and the end posture of the mechanism, the reachability judgment based on the singular value is performed on all the end postures of the mechanism to obtain the end posture of the branch-driven parallel mechanism in the reachable workspace (i.e., the reachable end posture).

[0090] S13 is specifically:

[0091] S131: taking one of the multiple angles at the end of the mechanism as a reference angle;

[0092] S132: Calculating the condition number corresponding to each mechanism end posture at the same reference angle based on the mapping relationship between the mechanism end posture and the mechanism branch motion parameters and the mapping relationship between the mechanism branch motion parameters and the mechanism end posture, and performing reachability judgment on each mechanism end posture at the current value of the reference angle based on the condition number to obtain the reachable end posture at the current value of the reference angle;

[0093] Among them, the condition number C of each end posture of the mechanism is calculated by the following formula:

[0094]

[0095] Among them, σ max (J) is the maximum singular value of the Jacobian matrix J of the mechanism. The Jacobian matrix J of the mechanism is determined by the mapping relationship between the end posture of the mechanism and the motion parameters of the branch chain of the mechanism, and the mapping relationship between the motion parameters of the branch chain of the mechanism and the end posture of the mechanism. min (J) is the minimum singular value of the Jacobian matrix J of the mechanism, R + represents the set of positive real numbers.

[0096] In S132, after eliminating the end-position postures of the mechanism with abnormal condition numbers according to the "3σ accurate measurement", the median of the condition numbers of the end-position postures of each mechanism under the current value of the reference angle is determined and used as the judgment threshold. The end-position postures of the mechanism with condition numbers greater than the judgment threshold are recorded as reachable end-positions and placed in the reachable workspace. Among them, using the "3σ accurate measurement" to eliminate abnormal data means that the difference between the data and the array average value is greater than 3 times the standard deviation of the array data, which is regarded as a singular value and discarded, and the corresponding posture is also regarded as a singular posture. The present invention uses the median as the screening criterion. When the condition number is greater than this value, the parallel mechanism has strong singularity and poor motion performance under the corresponding posture. It is not suitable as a point in the reachable workspace and is regarded as a singular point. On the contrary, the mechanism has excellent motion performance under this posture, and this posture should be selected into the reachable workspace.

[0097] S133: Change the value of the reference angle, repeat S132, and obtain the achievable end-positions at different values ​​of the reference angle, until all achievable end-positions within the value range of the reference angle are obtained and used as the end-positions of the branch-driven parallel mechanism in the achievable workspace.

[0098] Step 1 is more specific:

[0099] Step 2: Calculate the stiffness performance parameters of all end-positions of the branch-driven parallel mechanism to obtain the stiffness performance parameters corresponding to all end-positions of the mechanism;

[0100] Step 2 is as follows:

[0101] S21: Determine the drive Jacobian matrix and constraint Jacobian matrix of each end-position of the mechanism based on the screw theory, and then obtain the complete Jacobian matrix;

[0102] In this embodiment, the axis of the rotary joint 42 is S 1，i , the direction vector of moving joint 4 is S 4，i The universal joint 43 is equivalent to two intersecting non-coplanar revolute pairs, and the corresponding axis is S 2，i 、S 3，i The first ball joint 46 is equivalent to three intersecting non-coplanar revolute pairs, and the corresponding axis is S 5，i 、S 6，i 、S 7，i (i=1~3)

[0103] Then the helical system on the branch chain is:

[0104]

[0105] Among them, $ 1,i is the spiral vector of the revolute joint on branch i, S 1,i is the axis of the revolute joint on branch chain i, is the direction vector from the center B of the moving platform to the revolute joint on branch chain i, $ j,i are the two spiral vectors of the universal joint on branch i, is the direction vector from the center B of the moving platform to the universal joint on branch chain i, S j,i are the two axes of the universal joint on branch chain i, $ 4,i is the spiral vector of the moving joint on branch i, S 4,i is the direction vector of the moving joint on branch i, $ k,i are the three helical vectors of the spherical joint on branch chain i, is the direction vector from the center B of the moving platform to the spherical joint on the branch chain i, S k,i are the three axes of the spherical joint on branch chain i.

[0106] Therefore, the instantaneous motion spiral of the moving platform is c It can be expressed as a linear combination of the helical vectors of each branch motion pair, where ω c represents the angular velocity of the moving platform relative to the static coordinate system, v c represents the linear velocity of the moving platform relative to the static coordinate system, Expresses the velocity (angular velocity or linear velocity) of the j-th kinematic pair of the i-th branch:

[0107]

[0108] in, is the velocity (angular velocity or linear velocity) of the j-th moving pair of the i-th branch, ω c is the angular velocity of the moving platform relative to the static coordinate system, v c is the linear velocity of the moving platform relative to the static coordinate system.

[0109] Next, find an antihelix for each branched helix. s r,i,1 is the anti-helical direction vector of the helical system on branch i, For s r,i,1 For the moment of the moving platform center B, the anti-helix is ​​reciprocally multiplied with both sides of equation (17) and written in matrix form, we can obtain:

[0110]

[0111] J c is called the constraint Jacobian matrix of the parallel mechanism. Therefore, the key is to find the anti-helix of each branched spiral system. It is not difficult to prove that two coplanar line vectors are anti-helices to each other. Therefore, in order to simultaneously rotate the two rotational pairs of the universal joint 43 and the three rotational pairs of the first ball joint 46, the reciprocal product is 0. Then the direction vector of the anti-helix must pass through points S and U. Therefore, the anti-helix $ r,i,1 It can be expressed as:

[0112]

[0113] Among them, Unit(·) is the vector unitization, is the direction vector from the spherical joint S on branch chain i to the universal joint U, Normalize the direction vector.

[0114] In fact, the i-th branch mobile pair P i The motion spiral is [0, S 4，i ], S 4，i and are collinear vectors, and the reciprocal product of the two helices is not 0. Therefore, there is no such anti-helix on each branch of this mechanism, and the present invention does not consider the constrained Jacobian matrix of the parallel mechanism.

[0115] Next, constrain the driving component R on each branch chain. i and P i , find the antihelix of the remaining five helices$ r,i,2 , same as above, $ r,i,2 It can be expressed as:

[0116]

[0117] The anti-helix $ r,i,2 The reciprocal product of both sides of equation (17) is obtained (it is related to the revolute joint 42 i The motion spiral 1,i The reciprocal product is 0 only when the RUPS are collinear):

[0118]

[0119] Write the above formula in matrix form:

[0120]

[0121] in,

[0122]

[0123] Among them, J x $ r,i,2 The matrix formed, J q is the coefficient matrix of the hinge drive pair velocity, The speed of the three hinge drive pairs.

[0124] J a It is called the driving Jacobian matrix of the parallel mechanism. Since the mechanism has no constraint Jacobian matrix, the present invention selects the driving Jacobian matrix J a as the complete Jacobian matrix J of the mechanism.

[0125] S22: Calculate the stiffness of each branch in the branch-driven parallel mechanism, then perform mechanical analysis on the branch-driven parallel mechanism based on the stiffness of each branch and the complete Jacobian matrix. Combined with the principle of virtual work, obtain the full stiffness matrix of the branch-driven parallel mechanism at the current end position of the mechanism.

[0126] In this embodiment, it is assumed that the dynamic and static platforms of the parallel mechanism, the revolving pairs, universal joints, translation pairs, and ball joints in each branch chain are all rigid bodies. Each RUPS branch chain is subject to a driving force and a driving force couple along the translation pair P. The UPS segment of each branch chain is taken as the analysis object. represents the flexibility of the i-th branch along the rod direction, represents the flexibility of the i-th branch in the vertical direction of the rod (i = 1 to 3). Then, the elastic deformation of the parallel mechanism branch under the action of the driving force is:

[0127]

[0128] in, is the deformation of the i-th branch along the rod direction, is the deformation of the i-th branch in the direction perpendicular to the rod, is the driving force of the i-th branch along the rod direction, is the driving couple of the i-th branch in the direction perpendicular to the rod, L i and A i Denote the connecting rod length and cross-sectional area of ​​the i-th branch UPS segment, E and I respectively. z They represent the elastic modulus and moment of inertia of the UPS segment of the i-th branch respectively. By calculating the above two formulas, we can directly obtain and The value of .

[0129] Since stiffness and flexibility are reciprocals of each other, and when the RUPS are collinear, there is no driving force along the moving pair direction and no flexibility in the vertical rod direction, so the stiffness K can be obtained. i for:

[0130]

[0131] Now assume that the 3-RUPS / S parallel mechanism dynamic platform is subjected to an external load, which is expressed as ω=[f T ,m T ] T , where f and m represent force and torque, respectively. Let τ be the driving force provided by each branch chain, and ignore the effect of deadweight. The dynamic platform maintains equilibrium under the action of external load and driving force, which can be expressed as:

[0132]

[0133] Δq represents the displacement of each branch chain under the action of the driving force, k = diag[k1, k2, k3], k is the stiffness matrix of the branch chain, diag[] represents the diagonal matrix, k1, k2, k3 are the stiffness of the first, second, and third branches respectively, assuming that the small displacement of the dynamic platform in the static coordinate system is Δx = [Δx, Δy, Δz] T and Δθ=[Δθ x ,Δθ y ,Δθ z ] T , Δx represents the small linear displacement of the moving platform, Δθ represents the small angular displacement of the moving platform, and the deadweight is neglected. According to the principle of virtual work:

[0134] ω T ΔX=τ T Δq (27)

[0135] Where ΔX=[Δx T ,Δθ T ] T represents the deformation of the moving platform. Substituting Equation (27) and Δq = JΔX into Equation (27) and simplifying it, we can obtain:

[0136] ω=J T kJΔX (28)

[0137] Among them, J T kJ is K, which is defined as the full stiffness matrix of the parallel mechanism.

[0138] S23: obtaining stiffness performance parameters of the current end position of the mechanism according to the full stiffness matrix of the branch-driven parallel mechanism at the current end position of the mechanism;

[0139] In S23, the stiffness performance parameters of the current end-position of the mechanism include the minimum eigenvalue K of the full stiffness matrix of the branch-driven parallel mechanism at the current end-position of the mechanism. min , maximum eigenvalue K max And the stiffness specificity KSI, where the calculation formula of stiffness specificity KSI is:

[0140]

[0141] S24: Repeat S21-S23 to calculate and obtain the stiffness performance parameters corresponding to the end positions of all mechanisms.

[0142] Step 3: Based on the end-position in the reachable workspace, combined with the stiffness performance parameters corresponding to the end-position of all mechanisms, the singularity and stiffness are comprehensively quantified to screen out the optimal pose set. The branch-driven parallel mechanism performs continuous pose adjustment according to the optimal pose set. In the specific implementation, you can first screen the poses with good stiffness specificity KSI and motion performance (if there is no such pose, some performance can be sacrificed according to design requirements), and then calculate the K under these poses. min , K max In the case of , the poses that are not extreme values ​​and have a small gradient near the pose are selected as the optimal pose set.

[0143] After performing discrete point fitting on the terminal positions of each mechanism and the corresponding condition number C obtained in step S13, the obtained result is shown in the figure below: Figure 2-Figure 5 In order to better reflect the circularity and symmetry of the results, the present invention defines the horizontal coordinate f1(α, β) as:

[0144] f1(α,β)=β×cos(α) (30)

[0145] The vertical coordinate f2(α,β) is defined as:

[0146] f2(α,β)=β×sin(α) (31)

[0147] The vertical coordinate is defined as the γ angle.

[0148] In order to quantitatively characterize the motion performance of the mechanism under different rotation angles γ, the present invention calculates the singularity rate r under different γ angles. sing , the singularity rate r at each γ angle sing The ratio of the singular posture under the rotation angle γ is used to calculate, and the result is drawn as follows Figure 4 and Figure 5 shown.

[0149] Figure 4 Among them, the maximum value is 4.155459e-01 (indicated by the ▲ symbol, the same below), the minimum value is 3.859889e-02 (indicated by the ▇ symbol, the same below), the average value is 0.0613, the standard deviation is 0.0466, the variance is 0.0022, and the second-order central moment is 0.0022.

[0150] Figure 5 The maximum value is 1, the minimum value is 4.878049e-02, the mean is 0.1031, the standard deviation is 0.2016, the variance is 0.0406, and the second-order central moment is 0.0402.

[0151] By comparing the above parameter quantification result graphs, we can evaluate the difference in motion performance within the workspace between the 3-RUPS / S mechanism and the 3-UPS / S mechanism:

[0152] (1) The 3-RUPS / S parallel mechanism has fewer singular poses in the set workspace.

[0153] (2) The 3-RUPS / S has a more uniform distribution of singular postures. Both mechanisms have more singular postures at γ = 0°, but the 3-UPS / S has a singularity rate of 100% at γ = 0°, indicating that the mechanism cannot use the posture at γ = 0° during motion. The singularity rate of the 3-RUPS / S parallel mechanism when γ is less than -20° and greater than 20° is similar to that of the 3-UPS / S parallel mechanism, both around 5%. Although the singularity rate of the 3-RUPS / S is slightly higher than that of the 3-UPS / S in the range of -20° to +20° (excluding 0°), the singularity rate is only 41.55% at γ = 0°, less than 50%, indicating that the distribution of singular postures is more uniform, and there are more posture options during motion.

[0154] (3) Considering the actual multi-axis linkage situation, if the maximum rotation angle of the joint in each direction does not exceed 40°, the set posture space of the parallel mechanism will be reduced to no more than 40° in each direction. In this case, it can be found that the 3-UPS / S mechanism still has a very large singularity rate near γ = 0°, and there are many singular postures, while the singularity rate of the 3-RUPS / S mechanism is further reduced near γ = 0°, and singular postures will only appear when the β angle is close to 0°. The available multi-axis linkage postures are significantly higher than the former.

[0155] like Figures 6 to 11 As shown in the figure, the stiffness characteristic distribution of 3-RUPS / S parallel mechanism and 3-UPS / S parallel mechanism in the workspace is calculated respectively. Figures 2 to 5 As a result, all the postures when γ=-1.2° are selected to draw the stiffness characteristic curve. The definition of the horizontal and vertical coordinates of the curve is the same as Figure 2 、 3 The vertical axis is the corresponding stiffness characteristic index.

[0156] Figure 6 The maximum value is 5.209522e+00, the minimum value is 5.979950e-01, the mean is 2.1129, the standard deviation is 1.0251, the variance is 1.0507, and the second-order central moment is 1.0484. 1GN=10^9N.

[0157] Figure 7 The maximum value is 1.4997, the minimum value is 0.4860, the mean value is 0.8299, the standard deviation is 0.2303, the variance is 0.0530, and the second-order central moment is 0.0529.

[0158] Figure 8The maximum value is 2.3208, the minimum value is -0.7006, the average value is 0.8146, the standard deviation is 0.6544, the variance is 0.4282, and the second-order central moment is 0.4272. 1MN=10^6N.

[0159] Figure 9 The maximum value is 2.4263, the minimum value is -0.0002, the mean is 0.8947, the standard deviation is 0.6257, the variance is 0.3914, and the second-order central moment is 0.3906.

[0160] Figure 10 The maximum value is 3.7842, the minimum value is -0.4780, the mean is 0.6600, the standard deviation is 0.8370, the variance is 0.7006, and the second-order central moment is 0.6990.

[0161] Figure 11 The maximum value is 3.0671, the minimum value is -0.0369, the mean is 1.1976, the standard deviation is 0.8782, the variance is 0.7713, and the second-order central moment is 0.7696.

[0162] Combining several quantitative results of indicators, we can evaluate some differences in the stiffness characteristic distribution of the two mechanisms:

[0163] (1) The optimal stiffness performance of the 3-RUPS / S mechanism is better than that of the 3-UPS / S mechanism. Figure 10 、 Figure 11 It can be seen that the optimal stiffness performance of the two mechanisms appears when the angles α and β are close to 0° (the KSI index takes the maximum value at this time), and the maximum KSI of the 3-RUPS / S mechanism is 3.7977×10 -3 , compared with 3.0675×10 -3 The maximum eigenvalue K of the stiffness matrix of the 3-RUPS / S mechanism is increased by 19.23%. max and the minimum eigenvalue K min Both are improved compared to the 3-UPS / S mechanism. This indicates that after the introduction of the redundant drive joint R, the stiffness performance of the parallel mechanism is further improved near the posture with excellent stiffness performance (the posture when the α and β angles are close to 0°).

[0164] (2) The stiffness performance distribution of the 3-UPS / S mechanism within the working range is more uniform than that of the 3-RUPS / S mechanism. Figure 6 、 Figure 7 It can be seen that the K of the two institutions max The overall distribution of indicators is similar, but Figures 8 to 11 It can be seen that the minimum eigenvalue K of the stiffness matrix of the 3-RUPS / S mechanism in the boundary area ismin It is very small, even close to 0, which leads to its KSI index being very small in the boundary area, the stiffness characteristics being very unevenly distributed, and the differences under different postures being large; while the minimum eigenvalue K of the stiffness matrix of the 3-UPS / S mechanism in the boundary area is min There are three maximum values ​​that are 120° apart from each other. Although its KSI index is close to 0, it is much more averaged than the 3-RUPS / S mechanism, and the stiffness characteristics of the mechanism configuration are more evenly distributed in the entire workspace.

[0165] (3) Combination Figures 2 to 5 , in the position where the angles α and β are close to 0°, the singularity of the 3-RUPS / S mechanism is large, and the result of the stiffness analysis shows that the stiffness performance of the mechanism is optimal in these positions. This shows that for the 3-RUPS / S mechanism, there is a certain mutual constraint between stiffness and singularity, and it is impossible to obtain the optimal value at the same time. Fortunately, by comparing Figure 10 and Figure 2 It can be seen that in the postures where the angles α and β are close to 0°, the gradient of the singularity decrease is significantly greater than the gradient of the stiffness evaluation index KSI decrease, indicating that in postures close to 0° but not 0° such as β = 2.4°, the mechanism has weak singularity and good stiffness performance, and can be selected as the configuration posture of multi-axis linkage.

[0166] Therefore, based on the kinematic singularity performance and full stiffness performance analysis method proposed in the present invention, the performance quantitative evaluation of the selected 3-RUPS / S, a parallel mechanism with redundant drive degrees of freedom, was completed, and postures with excellent performance were screened out to achieve an innovative mechanism concept design that meets the needs.

[0167] The above embodiments are intended only to illustrate the technical concepts and features of the present invention. Their purpose is to enable those skilled in the art to understand the contents of the present invention and implement them accordingly. They are not intended to limit the scope of protection of the present invention. Any equivalent changes or modifications made in accordance with the spirit of the present invention are intended to be covered by the scope of protection of the present invention.

Claims

1. A method for evaluating the spatial posture of a conceptual configuration of a branched-chain driven parallel robot, characterized in that: The following steps are involved: Step 1: Use the kinematic analysis method based on singular values ​​to judge the reachability of all the end positions of the branch-driven parallel mechanism, and obtain the end position of the branch-driven parallel mechanism in the reachable workspace; Step 2: Calculate the stiffness performance parameters of all end-positions of the branch-driven parallel mechanism to obtain the stiffness performance parameters corresponding to all end-positions of the mechanism; The step 2 is specifically as follows: S21: Determine the drive Jacobian matrix and constraint Jacobian matrix of each end-position of the mechanism based on the screw theory, and then obtain the complete Jacobian matrix; S22: Calculate the stiffness of each branch in the branch-driven parallel mechanism, then perform mechanical analysis on the branch-driven parallel mechanism based on the stiffness of each branch and the complete Jacobian matrix. Combined with the principle of virtual work, obtain the full stiffness matrix of the branch-driven parallel mechanism at the current end position of the mechanism. S23: obtaining stiffness performance parameters of the current end position of the mechanism according to the full stiffness matrix of the branch-driven parallel mechanism at the current end position of the mechanism; In said S23, the stiffness performance parameters of the current end position of the mechanism include the minimum eigenvalue of the full stiffness matrix of the branch-driven parallel mechanism at the current end position of the mechanism K min , the maximum eigenvalue K max And the stiffness specificity KSI, where the calculation formula of stiffness specificity KSI is: ； S24: Repeat S21-S23 to calculate and obtain the stiffness performance parameters corresponding to the end positions of all mechanisms; Step 3: Based on the end-position in the reachable workspace, combined with the stiffness performance parameters corresponding to the end-positions of all mechanisms, the singularity and stiffness are comprehensively quantified to screen and obtain the optimal pose set.

2. The spatial posture evaluation method of the conceptual configuration of a branched-chain driven parallel robot according to claim 1 is characterized in that: The step 1 is specifically as follows: S11: Perform continuous inverse kinematics analysis on all end-positions of the branch-driven parallel mechanism to obtain the mapping relationship between the end-position and the motion parameters of the branch. S12: Perform continuous forward kinematic analysis on the mechanism branch motion parameters to obtain the mapping relationship between the mechanism branch motion parameters and the mechanism end pose; S13: According to the mapping relationship between the end-position of the mechanism and the motion parameters of the mechanism branch, and the mapping relationship between the motion parameters of the mechanism branch and the end-position of the mechanism, the reachability judgment based on the singular value is performed on all the end-positions of the mechanism to obtain the end-position of the branch-driven parallel mechanism in the reachable workspace.

3. The spatial posture evaluation method of the conceptual configuration of a branched-chain driven parallel robot according to claim 2 is characterized in that: The S13 is specifically: S131: taking one of the multiple angles at the end of the mechanism as a reference angle; S132: Calculating the condition number corresponding to each mechanism end posture at the same reference angle based on the mapping relationship between the mechanism end posture and the mechanism branch motion parameters and the mapping relationship between the mechanism branch motion parameters and the mechanism end posture, and performing reachability judgment on each mechanism end posture at the current value of the reference angle based on the condition number to obtain the reachable end posture at the current value of the reference angle; S133: Change the value of the reference angle, repeat S132, and obtain the achievable end-positions at different values ​​of the reference angle, until all achievable end-positions within the value range of the reference angle are obtained and used as the end-positions of the branch-driven parallel mechanism in the achievable workspace.

4. The method for evaluating the spatial posture of a conceptual configuration of a branched-chain driven parallel robot according to claim 3, characterized in that: In S132, the condition number C of each end position of the mechanism is calculated by the following formula: in, is the maximum singular value of the mechanism Jacobian matrix J, is the minimum singular value of the mechanism Jacobian matrix J, represents the set of real numbers.

5. The method for evaluating the spatial posture of a conceptual configuration of a branched-chain driven parallel robot according to claim 3, characterized in that: In said S132, according to "3 After eliminating the end-positions of the mechanism with abnormal condition numbers, the median of the condition numbers of each end-position of the mechanism under the current value of the reference angle is determined and used as the judgment threshold. The end-positions of the mechanism with condition numbers less than the judgment threshold are recorded as reachable end-positions and placed in the reachable workspace.

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