General redundant parallel robot positive solution method

Abstract

The invention provides a general redundant parallel robot positive solution method, and relates to the technical field of industrial redundant parallel robot control. The method comprises the following steps: establishing a branch chain coordinate system of the redundant parallel robot, wherein the branch chain coordinate system comprises a static platform coordinate system, a movable platform coordinate system and a joint coordinate system; according to a DH method and dual quaternion, a kinematics model of the redundant parallel robot to be solved is established, and the kinematics model comprises a closed chain structure kinematics sub-model and an open chain structure kinematics sub-model; performing elimination processing on the kinematic model to obtain a corresponding constraint equation set; and substituting the structure parameters corresponding to the kinematic model into the corresponding constraint equation set, and determining a target pose according to a Gauss-Newton iteration method. According to the invention, two problems in the prior art are solved: 1, the error problem of the composite hinge and other parts in the manufacturing and assembling process is solved; and secondly, the parallel redundant parallel robot with kinematic redundancy has the problem of multiple solutions.

Description

Technical Field

[0001] The invention relates to the technical field of industrial redundant parallel robot control, and in particular to a general redundant parallel robot positive solution method. Background Art

[0002] In applications such as physical human-machine interaction, the working performance of parallel redundant parallel robots is often subject to some limitations. The main reason is that the unique solution of its kinematic positive solution is difficult to obtain and its directional workspace is relatively small. For example, the maximum tilt angle of the famous Stewart platform is limited to about 45 degrees due to the existence of second-class singularities. By introducing kinematic redundancy, the directional workspace of the parallel redundant parallel robot can be expanded, thereby effectively alleviating the limitations brought by the second-class singularities. And different fixtures can be designed according to the redundant degrees of freedom of the moving platform to meet different usage scenarios. However, even with kinematic redundancy, the complexity of the kinematic positive solution problem has not been significantly reduced. Similar to conventional parallel mechanisms, for given drive joint variables, parallel redundant parallel robots with kinematic redundancy may also have multiple solutions.

[0003] Due to the mutual coupling relationship between the branches, the direct solution of the parallel mechanism is more complicated than the inverse solution. A three-branch spatial kinematic redundant parallel robot with (6+3) degrees of freedom (3-[R(RR-RRR)SR]) was proposed in the literature. Figure 1 The schematic diagram of the mechanism is shown. Some literature discusses its forward kinematics problem under ideal conditions and analyzes the addition of different numbers of additional encoders to simplify the forward solution process. However, due to the errors of the composite hinge and other components during the manufacturing and assembly process, the existence of system errors is inevitable. Summary of the invention

[0004] In order to overcome the shortcomings of the prior art, the purpose of the present invention is to provide a general redundant parallel robot positive solution method, which solves two problems existing in the prior art: one is the error problem of compound hinges and other components during the manufacturing and assembly process; the other is the problem of multiple solutions for kinematically redundant parallel robots.

[0005] To achieve the above object, the present invention provides the following solutions:

[0006] A general redundant parallel robot forward solution method, comprising:

[0007] Establishing a branch chain coordinate system of a redundant parallel robot; the branch chain coordinate system includes: a static platform coordinate system, a dynamic platform coordinate system and a joint coordinate system;

[0008] A kinematic model of the redundant parallel robot to be solved is established according to the DH method and the dual quaternion; the kinematic model includes: a closed-chain structure kinematic sub-model and an open-chain structure kinematic sub-model;

[0009] Performing elimination processing on the kinematic model to obtain a corresponding set of constraint equations;

[0010] The structural parameters corresponding to the kinematic model are substituted into the corresponding constraint equations and the target posture is determined according to the Gauss-Newton iteration method.

[0011] Preferably, the closed-chain structure kinematic sub-model is expressed as:

[0012] ;

[0013] in, The dual quaternion form representing the homogeneous transformation from coordinate system {k} to coordinate system {i} corresponding to the DH parameters of the link on the j-th branch chain, is the unit element in the form of dual quaternion, which represents the kinematic constraints of the entire closed chain structure. It is the rotation angle around the Z axis in the DH parameters of the connecting rod of the i-th joint on the j-th branch, where the DH parameters of the connecting rod represent the redundant degrees of freedom of the rotation angle and the moving platform, and i is 1-10.

[0014] Preferably, the expression of the open-chain structure kinematic sub-model is:

[0015] ;

[0016] in, is the position and posture of the moving platform, B represents the base coordinate system of the static platform center of the mechanism, and P represents the tool coordinate system of the moving platform center. j is the pose transformation from the label system {B} to {0} on the j-th branch.

[0017] Preferably, the expression of the DH parameter of the link on the branch chain corresponding to the closed chain structure kinematic sub-model is:

[0018] ;

[0019] ;

[0020] ;

[0021] ;

[0022] ;

[0023] in, are respectively the first rod length parameter, the second rod length parameter, the third rod length parameter, the fourth rod length parameter, the fifth rod length parameter, the sixth rod length parameter and the seventh rod length parameter of the redundant parallel robot, express , express , express .

[0024] Preferably, the expression of the DH parameter of the link on the branch chain corresponding to the kinematic sub-model of the open chain structure is:

[0025] ;

[0026] ;

[0027] ;

[0028] ;

[0029] ;

[0030] ;

[0031] .

[0032] Preferably, the constraint equation group corresponding to the closed-chain structural kinematic sub-model is expressed as:

[0033] .

[0034] Preferably, the constraint equation group corresponding to the open-chain structure kinematic sub-model is expressed as:

[0035] .

[0036] Preferably, substituting the structural parameters corresponding to the kinematic model into the corresponding constraint equations and determining the target pose according to the Gauss-Newton iteration method includes:

[0037] The initial parameter value is iterated by using the Gauss-Newton iteration method to obtain the parameter estimation value, wherein the initial parameter value is the value of the starting position of the redundant parallel robot;

[0038] Determine all unknown quantities based on parameter estimates and the constraint equations corresponding to the open-chain structural kinematic submodel;

[0039] Substitute all unknown quantities into the position expression to obtain the target pose.

[0040] Preferably, the position expression is:

[0041] ;

[0042] Among them, T is the homogeneous transformation matrix form of the rigid body posture in space, b0, b1, b2, b3, e0, e1, e2, e3 are the first parameter, second parameter, third parameter, fourth parameter, fifth parameter, sixth parameter, seventh parameter and eighth parameter of the dual quaternion form of the rigid body posture in space respectively.

[0043] The present invention discloses the following technical effects:

[0044] The present invention provides a general redundant parallel robot forward solution method, comprising: establishing a branched chain coordinate system of the redundant parallel robot, wherein the branched chain coordinate system comprises: a static platform coordinate system, a dynamic platform coordinate system and a joint coordinate system; establishing a kinematic model of the redundant parallel robot to be solved according to a DH method and dual quaternions, wherein the kinematic model comprises: a closed-chain structure kinematic sub-model and an open-chain structure kinematic sub-model; performing elimination processing on the kinematic model to obtain a corresponding set of constraint equations; substituting the structural parameters corresponding to the kinematic model into the corresponding set of constraint equations and determining the target posture according to a Gauss-Newton iteration method. The present invention is aimed at 9-DOF redundant parallel robots of 3-[R(RR-RRR)SR], 3-[P(RR-RRR)URR], 3-[P(RR-RRR)RRRR] and 3-[P(RR-RRR)RRRP] configurations, where there is no public correct solution algorithm. The present invention first proposes a correct solution algorithm containing 158 geometric parameters, and determines the unique correct solution of the redundant parallel robot. In high-precision scenarios such as industry and aerospace, the determined end position and posture are conducive to repeated positioning and precise operation. When the system knows that there is only one end position and posture, the same input will not cause the redundant parallel robot to "run" to different positions or postures, thereby avoiding control ambiguity. BRIEF DESCRIPTION OF THE DRAWINGS

[0045] In order to more clearly illustrate the embodiments of the present invention or the technical solutions in the prior art, the drawings required for use in the embodiments will be briefly introduced below. Obviously, the drawings described below are only some embodiments of the present invention. For ordinary technicians in this field, other drawings can be obtained based on these drawings without paying creative labor.

[0046] Figure 1 A schematic diagram of the structure of a spatial kinematic redundant parallel robot with three branches and (6+3) degrees of freedom in the prior art provided by an embodiment of the present invention;

[0047] Figure 2 A flow chart of a general redundant parallel robot forward solution method provided by an embodiment of the present invention;

[0048] Figure 3 A schematic diagram of a closed-chain series structure provided by an embodiment of the present invention;

[0049] Figure 4 A schematic diagram of an open-chain series structure provided by an embodiment of the present invention;

[0050] Figure 5 A schematic diagram of a connection method of a moving platform provided in an embodiment of the present invention;

[0051] Figure 6 A schematic diagram of a first single-branched chain coordinate system provided in an embodiment of the present invention;

[0052] Figure 7 A schematic diagram of a second single-branched chain coordinate system provided in an embodiment of the present invention;

[0053] Figure 8 A schematic diagram of a mechanism joint provided by an embodiment of the present invention;

[0054] FIG9 is a schematic diagram of different postures of a parallel mechanism provided by an embodiment of the present invention, wherein FIG9( a ) is a schematic diagram of a first posture, and FIG9( b ) is a schematic diagram of a second posture. DETAILED DESCRIPTION

[0055] The following will be combined with the drawings in the embodiments of the present invention to clearly and completely describe the technical solutions in the embodiments of the present invention. Obviously, the described embodiments are only part of the embodiments of the present invention, not all of the embodiments. Based on the embodiments of the present invention, all other embodiments obtained by ordinary technicians in this field without creative work are within the scope of protection of the present invention.

[0056] In order to make the above-mentioned objects, features and advantages of the present invention more obvious and easy to understand, the present invention is further described in detail below with reference to the accompanying drawings and specific embodiments.

[0057] like Figure 2 As shown, the present invention provides a general redundant parallel robot forward solution method, comprising:

[0058] Step 100: Establishing a branch chain coordinate system of a redundant parallel robot, wherein the branch chain coordinate system includes: a static platform coordinate system, a dynamic platform coordinate system and a joint coordinate system;

[0059] Step 200: establishing a kinematic model of the redundant parallel robot to be solved according to the DH method and the dual quaternion, wherein the kinematic model includes: a closed-chain structure kinematic sub-model and an open-chain structure kinematic sub-model;

[0060] Step 300: performing elimination processing on the kinematic model to obtain a corresponding set of constraint equations;

[0061] Step 400: Substitute the structural parameters corresponding to the kinematic model into the corresponding constraint equations and determine the target posture according to the Gauss-Newton iteration method.

[0062] Specifically, this embodiment discloses the theoretical basis of the redundant parallel mechanism:

[0063] Methods for describing the posture of a rigid body include spinors, Lie groups, As well as homogeneous transformation matrices, dual quaternions and multiple quaternions, among which the most commonly used is the 4×4 homogeneous transformation matrix.

[0064] This embodiment uses dual quaternions to describe the posture and position of the rigid body in space. Compared with the homogeneous transformation matrix which requires 16 numbers, the dual quaternion is expressed by 8 numbers, which effectively improves the compactness of the representation. This method not only simplifies mathematical processing and numerical calculations, but also significantly reduces the complexity of the problem.

[0065] Hamilton first proposed quaternions in 1843. in, , , and are all real numbers, H represents quaternion algebra, and the unit quaternion satisfies the normalization condition:

[0066]

[0067] is a quaternion The real part of is a quaternion The imaginary part of , and They are , , The coefficient of , , is an imaginary unit, and ;

[0068] Its geometric meaning can be seen as in three-dimensional space , , is the unit vector corresponding to the three orthogonal coordinate axes, that is is along the X-axis direction, is along the Y-axis direction, is along the Z-axis direction.

[0069] The product of quaternions is also a quaternion. The result of multiplying two quaternions can be expressed using matrix multiplication as follows:

[0070] ;

[0071] M(b)*c is the form of converting the quaternion b*c into the matrix M(b)*c multiplication. Here we only explain the multiplication rule of quaternion.

[0072] Among them, the matrix M(b) is the left multiplication matrix of the quaternion b, and its construction rule is: the first parameter b0 of the quaternion is located on the diagonal, and the second parameter b1, the third parameter b2, and the fourth parameter b3 of the quaternion form an antisymmetric matrix and are expanded to a four-dimensional form. The above formula fully describes the algebraic law of quaternion multiplication, and c0, c1, c2, and c3 are the ninth parameter, tenth parameter, eleventh parameter, and twelfth parameter of the quaternion.

[0073] In the formula , .

[0074] The addition of two quaternions can be expressed as:

[0075] ;

[0076] The conjugate of a unit quaternion is defined as: when and When we get .

[0077] Unit quaternions can be used to represent the posture of a rigid body, which is more efficient than using the traditional rotation matrix method. Unit quaternions can express three-dimensional rotations with fewer parameters, thus avoiding the singularity problem that may appear in the rotation matrix and reducing the storage and computational complexity during the calculation process.

[0078] The transformation relationship between the unit quaternion and the rotation matrix can be defined as follows:

[0079] ;

[0080] Indicates that the matrix The element in row i and column j.

[0081] Clifford first proposed dual quaternions in 1878 based on quaternions to describe the motion group of rigid bodies, namely: in, and are the real part and the dual part of the dual quaternion respectively. The real part of the dual quaternion function is the posture of the current joint. is the dual symbol and .

[0082] H stands for quaternion algebra. D stands for dual number algebra. The tensor product of the quaternion algebra H and the dual number algebra D is used to describe rigid body motion.

[0083] In the expression, it acts as: belong ,Right now ,express It is a dual quaternion, which is the position of the rigid body in space.

[0084] The multiplication of dual quaternions is still dual quaternions, such as:

[0085] ;

[0086] h1 and h2 are both general dual quaternions. Here we only explain the multiplication rule of dual quaternions.

[0087] In the formula .

[0088] The quaternion addition and multiplication defined above are used in this formula. The conjugation of a quaternion is defined as follows:

[0089]

[0090] This is the conjugate of the real part and the dual part of the quaternion. The unit dual quaternion satisfies equation (1) and the following conditions:

[0091] therefore , This shows that the conjugate of the unit dual quaternion is the inverse: . Dual Quaternion Used to express the purpose of posture transformation. Real part Corresponding rotation matrix , and the dual part is defined as with displacement vector Function:

[0092] ;

[0093] Therefore, the position expressed as a dual quaternion is:

[0094]

[0095] Where p is the position of the origin of the joint coordinate system in space.

[0096] The relationship between the dual quaternion and the homogeneous transformation matrix can be expressed as:

[0097]

[0098] Therefore, the posture transformation can be expressed using dual quaternions as:

[0099] ;

[0100] This shows that the direction change can be expressed as , the position transformation can be expressed as , where p1p2 are the thirteenth and fourteenth parameters of the quaternion. In 1955, J.Denavit et al. proposed a method using four parameters to represent the joint angle posture, namely the DH method, to describe the relative position and posture between the components in the spatial mechanism, which is widely used in series mechanisms. The relative relationship of link i to link i-1 is established in combination with the dual quaternion as follows:

[0101] 1) Winding Axis rotation ,Right now

[0102] 2) Along Axis Translation ,Right now

[0103] 3) Winding Axis rotation Right now

[0104] 4) Along Axis Translation ,Right now

[0105] is the first rotation angle, It's around Axis rotation The quaternion representation of .

[0106] is the moving distance, It is along Axis Translation The quaternion representation of .

[0107] is the second rotation angle, It's around Axis rotation The quaternion representation of .

[0108] is the moving distance, It is along Axis Translation The quaternion representation of .

[0109] The dual quaternion method can be used to express the DH method as follows:

[0110]

[0111] In the formula .

[0112] In order to facilitate the subsequent expression, the dual quaternion is expressed by the present invention. .

[0113] Furthermore, the kinematics of the parallel mechanism mainly includes the kinematics forward solution and the kinematics inverse solution. The redundant parallel mechanism studied in the present invention comprises three branches, and the mechanism diagram of a single branch is as follows: Figure 3-5 As shown. And each branch chain contains a closed chain series structure and an open chain series structure. In the kinematic inverse solution, each branch chain is independent of each other and there is no coupling relationship. Therefore, the kinematic inverse solution of the redundant parallel mechanism studied in this embodiment can be converted into the inverse solution of a general six-degree-of-freedom open chain series mechanism and the solution of a closed chain structure.

[0114] In practical applications, compound hinges are often used to replace the traditional revolute pair consisting of three axes intersecting at one point, namely the 3R joint pair. In addition, the revolute pair connecting the moving platform can also be replaced by a translation pair. Therefore, the kinematic models of 3-[R(RR-RRR)-SR], 3-[R(RR-RRR)URR] and 3-[R(RR-RRR)RRRR] can be uniformly described by the same mathematical model. And 3-[R(RR-RRR)RRRP] can replace the corresponding joint pair variables and make appropriate adjustments to maintain a consistent mathematical framework.

[0115] Furthermore, first establish the coordinate system on the static platform and the dynamic platform tool, and establish the base coordinate system at the center of the static platform , establish the tool coordinate system at the center of the moving platform They can be referred to as coordinate system {B} and coordinate system {P} respectively. Then, the joint coordinate system on each branch chain is established. , sort each joint in turn from the base coordinate system to the tool coordinate system and call it coordinate {i} (i=1,2,...,10). Finally, the joint coordinate system of a single branch of the redundant parallel mechanism can be established by the DH rule, such as Figure 6 As shown. and are the X-axis direction and Z-axis direction of its coordinate system {i} respectively. After these two directions are determined, the Y-axis direction is also determined, where x i Includes x1-x 10 , zi Including z1-z 10 . Therefore, according to the joint coordinate system established by the DH method, the parameter table 1 and parameter table 2 of each structure on each branch chain can be obtained respectively. Theoretically, the DH parameters of the three branches are the same, but the actual parameters are different due to factors such as manufacturing errors and assembly errors. Among them, Table 1 is the DH parameter table of the closed chain structure, and Table 2 is the DH parameter table of the open chain structure (the first DH parameter, the second DH parameter, the third DH parameter and the fourth DH parameter). Tables 1 and 2 are as follows:

[0116] Table 1 DH parameters of closed chain structure

[0117] Table 2 DH parameters of open chain structure

[0118] The (RR-RRR) closed chain structure is a closed series structure consisting of rod 2, rod 3, rod 4, and rod 5. In the closed chain mechanism, the kinematic mathematical model can be expressed as follows using dual quaternions:

[0119] (5);

[0120] in, The dual quaternion form representing the homogeneous transformation from coordinate system {k} to coordinate system {i} corresponding to the DH parameters of the link on the j-th branch chain, It is the dual quaternion form of the unit homogeneous transformation matrix. It indicates that the coordinate system has not undergone any transformation, that is, it is the unit element in the dual quaternion form. This equation expresses the kinematic constraints that satisfy the entire closed-chain structure, where the connecting rod DH parameters represent the redundant degrees of freedom of the rotation angle and the moving platform.

[0121] Taking the nominal parameters as an example, from Table 1 and formula (4), we can obtain the relationship between adjacent joints of the (RR-RRR) closed chain structure: The specific form of (the expression of the DH parameters of the link on the branch chain corresponding to the closed chain structure kinematic sub-model) is:

[0122]

[0123]

[0124]

[0125]

[0126]

[0127] in, They are respectively the first rod length parameter, the second rod length parameter, the third rod length parameter, the fourth rod length parameter, the fifth rod length parameter, the sixth rod length parameter and the seventh rod length parameter of the redundant parallel robot.

[0128] The open-chain serial mechanism can be regarded as a general 3-RRRSR mechanism, and its kinematic mathematical model can be expressed using dual quaternions (the expression of the kinematic sub-model of the open-chain structure) as follows:

[0129] (6), where is the position of the moving platform.

[0130] Furthermore, the expression of the DH parameter of the link on the branch chain corresponding to the open chain structure kinematic sub-model is:

[0131]

[0132]

[0133]

[0134]

[0135]

[0136]

[0137]

[0138] In the expression The rod length parameters of the redundant parallel robot are known.

[0139] Taking nominal parameters as an example, according to the parameters in Table 3 and Table 4, the pose transformation between the fixed platform and the moving platform and the joint coordinate system can be expressed in the form of dual quaternions.

[0140] Table 3 is the pose table of coordinate system {0} in coordinate system {B}, and Table 4 is the pose table of coordinate system {10} in coordinate system {P}. Tables 3 and 4 are as follows:

[0141] Table 3 The position and posture of coordinate system {0} in coordinate system {B}

[0142] Table 4 The pose of coordinate system {10} in coordinate system {P}

[0143] The pose transformation from coordinate system {B} to coordinate system {0} on branch 1 can be written as , the transformation from coordinate system {10} to coordinate system {P} can be expressed as .

[0144] The pose transformation from coordinate system {B} to coordinate system {0} on branch 2 can be written as , the transformation from coordinate system {10} to coordinate system {P} can be expressed as .

[0145] The pose transformation from coordinate system {B} to coordinate system {0} on branch 3 can be written as , the transformation from coordinate system {10} to coordinate system {P} can be expressed as .

[0146] Its specific parameters are as follows:

[0147]

[0148]

[0149]

[0150]

[0151]

[0152]

[0153] are the radii of the moving platform and the static platform respectively.

[0154] The dual quaternion form represents the homogeneous transformation from the link coordinate system {k} to the coordinate system {i} on the j-th branch. Formula (4) gives its specific transformation form, which contains four parameters, including 1 variable and 3 constants. is a variable, which is the rotation angle around the Z axis in the DH parameters. The three constants are the translation distance along the Z axis, the rotation angle around the X axis, and the translation distance along the X axis.

[0155] It represents the dual quaternion form of the 4th-order unit matrix. Its physical meaning is that the object coordinate system does not undergo rotation and translation.

[0156] The position of the moving platform is In the correct solution algorithm, it can be expressed by 8 unknown variables and used as the result to be solved. Its physical meaning represents the position and posture of the tool coordinate system, that is, the moving platform.

[0157] is the rotation angle of the j-th branch of the mechanism with the Z direction of the i-th joint coordinate system as the rotation axis;

[0158] in is the redundant degree of freedom of the moving platform to be determined, that is, the rotation angle of the revolute pair connected to the moving platform is about Figure 7 middle The angle of rotation.

[0159] The driving variables are Figure 7 middle The angle of rotation.

[0160] The rotation constraints are Figure 7 middle The angle of rotation.

[0161] The rotation constraints are Figure 7 middle The angle of rotation.

[0162] They are the axial directions of joint 1, joint 2, joint 3, joint 4, joint 5, joint 6, joint 7, joint 8, joint 9 and joint 10 on the i-th branch respectively.

[0163] After removing the same parameters in a branch chain, , There are 40 parameters in total.

[0164] In the jth branch, the pose transformation between the base coordinate system and the 0th coordinate system is It is represented by 5 constant parameters, including 3 translation parameters and 2 rotation parameters. The pose transformation from the 10th coordinate system in its j-branch chain to the tool coordinate system is also represented by 3 translation parameters and 2 rotation parameters.

[0165] Therefore, the complete redundant parallel mechanism has a total of 158 parameters, including 9 active drive parameters, 120 geometric structure constant parameters, and 29 unknown variables.

[0166] The active drives of the 3-[R(RR-RRR)SR], 3-[R(RRRRR)URR], 3-[R(RR-RRR)RRRR] and 3-[R(RR-RRR)RRRP] configurations are In general, when the drive is determined, the mathematical models of different configurations only differ in the DH parameters, and their solution processes can be included in the same mathematical model.

[0167] Furthermore, the kinematic solution can be expressed as: beg and Each branch of the redundant parallel mechanism consists of two parts, including a closed chain structure and an open chain structure. All unknown variables of each branch include the redundant joint angle of the moving platform and 6 passive joint rotations , plus the 8 parameters to be determined of the tool coordinate system, a total of 29 quantities to be determined must be calculated for the positive solution of this general redundant parallel mechanism. The position and posture of the tool coordinate system are expressed as follows using the dual quaternion of 8 variables:

[0168]

[0169] The positive solution of the redundant parallel mechanism can be divided into two parts: 1) solving the closed-chain structure kinematic sub-model. 2) solving the position and orientation of the moving platform. and redundant degrees of freedom .

[0170] Furthermore, this embodiment discloses the specific process of elimination:

[0171] First, multiply both sides of equation (5) by get:

[0172] The constraint equations corresponding to the closed-chain structural kinematics submodel are expressed as follows:

[0173] (8)

[0174] The right side of the equation contains only one unknown variable , the left side includes 2 unknown variables. According to the product rule of dual quaternions, expanding the left side of the equation, we can get:

[0175]

[0176] In the formula ,in The role of is to convert the dual quaternion into an 8-dimensional column vector. and is the branched chain structure constant and the known conditions The related 4th-order reversible matrix.

[0177] Then expand the right side of equation (8) according to the dual quaternion product to obtain:

[0178]

[0179] Among them, k ji is about θ j6The first quadratic polynomial function of , K is a matrix with 8 rows and 1 column and each element of K is related to θ j6 The first quadratic polynomial function of .

[0180] In the formula About The first quadratic polynomial function of .

[0181] make By connecting equations (8)-(10) in parallel, we can obtain:

[0182]

[0183] in About The second quadratic polynomial function of . And from equations (9) and (11), we can know that:

[0184]

[0185] From formula (12), it is easy to get:

[0186]

[0187]

[0188]

[0189]

[0190] The quadratic polynomial in equation (11) Substituting into (13)-(16) we get:

[0191] ;z 1ji It is the quadratic polynomial function form of formula (11).

[0192] Furthermore, we multiply the left and right sides of equation (6) by The constraint equations corresponding to the open-chain structure kinematic sub-model can be obtained, and the expression is:

[0193] (18).

[0194] The right side of the equation includes 9 unknown variables, it is easy to know from the above derivation that according to formula (17) we can directly find , so the left side of the equation can be regarded as containing only 3 unknown variables.

[0195] According to the product principle of dual quaternions, expanding the left side of equation (18) yields:

[0196]

[0197] is an 8th-order reversible matrix related to the constant parameters of the branched chain structure. Similarly, according to the product principle of dual quaternions, the right side of equation (18) is expanded into an 8-dimensional vector:

[0198] Mode middle About First and second degree polynomial functions of a variable.

[0199] Similar order And from equations (18)-(20) we can get:

[0200]

[0201] yes is a quadratic polynomial function of the variables.

[0202] From formula (19), we can know that:

[0203]

[0204] The known trigonometric identities are:

[0205]

[0206] The Groebner basis is calculated by equations (22) and (23). Using the gbasis command in MATLAB for symbolic operations, 46 bases can be obtained, of which 10 are low-order bases without trigonometric functions. After removing one identity, there are 9 bases left. The results are shown below:

[0207]

[0208]

[0209]

[0210]

[0211]

[0212]

[0213]

[0214]

[0215]

[0216] (21) Substituting the quadratic polynomial function into the nine equations (24)-(32) yields:

[0217]

[0218] because So divide both sides of the equation by You can get:

[0219]

[0220] in So far, we can get that each branch contains 9 equations, so the three branches contain a total of 27 identities. Each equation contains 9 variables, and the highest degree of the variables is 4.

[0221] Through two eliminations, 5 unknown variables in each branch are eliminated, and a total of 15 unknown variables in the three branches are eliminated. Now there are 14 unknown variables left in the system of equations.

[0222] Furthermore, the structural parameters corresponding to the kinematic model are substituted into the corresponding constraint equations and the target posture is determined according to the Gauss-Newton iteration method, including:

[0223] The initial parameter values ​​are iterated using the Gauss-Newton iteration method to obtain parameter estimates;

[0224] Determine all unknown quantities based on parameter estimates and the constraint equations corresponding to the open-chain structural kinematic submodel;

[0225] Substitute all unknown quantities into the position expression to obtain the target pose.

[0226] Furthermore, in the correct solution algorithm proposed in this embodiment, the tool coordinate system posture must comply with the two constraint equations (1) and constraint equation (2) of the dual quaternion, and at the same time must comply with the 12 constraint equations of equation (17) and the 27 constraint equations of equation (33). From the derivation process, it can be seen that the 12 equations of equation (17) and the 27 equations of equation (33) are not contradictory equations. Therefore, the least squares method and equation (17) can be used to solve the closed chain kinematic model (closed chain structural kinematic submodel), and the solution can be obtained. Substituting it back into equation (11) and equation (12) we can solve .

[0227] Due to the driving variable It is known and can be solved first Therefore, the positive solution problem can be further transformed into solving 29 11-variable polynomial equations composed of equations (1), (2) and (33). It is expressed as a 29-dimensional column vector as follows:

[0228]

[0229] Each of these is a function equation containing unknown variables, and the 11-dimensional column vector form of the unknown variables is as follows:

[0230] ;

[0231] The Gauss-Newton iteration method is often used to solve nonlinear equations. In addition, multivariate polynomial equations generally have many solutions, but in engineering applications, only one feasible solution is generally required. Therefore, the Gauss-Newton iteration method can be used to start the iteration from the initial value in the workspace (the value of X at the starting position of the redundant parallel robot), and finally obtain a stable solution through the iterative process. The Jacobi matrix needs to be calculated during the iteration process, Each term of is a polynomial function. Substitute the following formula to start the iteration:

[0232]

[0233] This formula is the formula of the Gauss-Newton iteration method, which is used to solve nonlinear least squares problems.

[0234] : Parameter estimates for the kth iteration.

[0235] : Parameter estimate for the k+1th iteration.

[0236] : At the kth iteration, The Jacobi matrix calculated for the input.

[0237] : At the kth iteration, The 29-dimensional column vector calculated for the input.

[0238] Set the iteration termination condition to The solution of the 11-variable nonlinear algebraic polynomial equation can be obtained . Find Then, we can easily find . At this point, all 29 unknown quantities can be solved.

[0239] Finally, the solution Substituting into equation (3) we can find the form of the homogeneous transformation matrix representing the posture.

[0240] Furthermore, generally redundant parallel single branch chain such as Figure 8 As shown in Figure 9, different postures of the parallel mechanism are shown, where α is the rotation angle of the moving platform coordinate system around the X-axis of the static platform coordinate system, β is the rotation angle of the moving platform coordinate system around the Y-axis of the static platform coordinate system, and γ is the rotation angle of the moving platform coordinate system around the Z-axis of the static platform coordinate system. p ,y p , z p They are all the positions of the moving platform coordinate system relative to the static platform coordinate system.

[0241] The various embodiments in this specification are described in a progressive manner, and each embodiment focuses on the differences from other embodiments. The same or similar parts between the various embodiments can be referenced to each other.

[0242] The present invention uses specific examples to illustrate the principles and implementation methods of the present invention. The above examples are only used to help understand the method and core ideas of the present invention. At the same time, for those skilled in the art, according to the ideas of the present invention, there will be changes in the specific implementation methods and application scope. In summary, the content of this specification should not be understood as limiting the present invention.

Claims

1. A general redundant parallel robot forward solution method, characterized in that: include: Establish the branch chain coordinate system of the redundant parallel robot; The branch chain coordinate system includes: a static platform coordinate system, a dynamic platform coordinate system and a joint coordinate system; A kinematic model of the redundant parallel robot to be solved is established according to the DH method and the dual quaternion; the kinematic model includes: a closed-chain structure kinematic sub-model and an open-chain structure kinematic sub-model; Performing elimination processing on the kinematic model to obtain a corresponding set of constraint equations; The structural parameters corresponding to the kinematic model are substituted into the corresponding constraint equations and the target posture is determined according to the Gauss-Newton iteration method.

2. A general redundant parallel robot forward solution method according to claim 1, characterized in that: The expression of the closed-chain structure kinematic submodel is: ； in, The dual quaternion form representing the homogeneous transformation from coordinate system {k} to coordinate system {i} corresponding to the DH parameters of the link on the j-th branch chain, is the unit element in the form of dual quaternion, which represents the kinematic constraints of the entire closed chain structure. It is the rotation angle around the Z axis in the DH parameters of the connecting rod of the i-th joint on the j-th branch, where the DH parameters of the connecting rod represent the redundant degrees of freedom of the rotation angle and the moving platform, and i is 1-10.

3. A general redundant parallel robot forward solution method according to claim 2, characterized in that: The expression of the open-chain structure kinematic submodel is: ； in, is the position and posture of the moving platform, B represents the base coordinate system of the static platform center of the mechanism, and P represents the tool coordinate system of the moving platform center. j is the pose transformation from the label system {B} to {0} on the j-th branch.

4. A general redundant parallel robot forward solution method according to claim 3, characterized in that: The expression of the DH parameter of the link on the branch chain corresponding to the closed chain structure kinematic sub-model is: ； ； ； ； ； in, are respectively the first rod length parameter, the second rod length parameter, the third rod length parameter, the fourth rod length parameter, the fifth rod length parameter, the sixth rod length parameter and the seventh rod length parameter of the redundant parallel robot, express , express , express .

5. A general redundant parallel robot forward solution method according to claim 4, characterized in that: The expression of the DH parameter of the link on the branch chain corresponding to the open chain structure kinematic sub-model is: ； ； ； ； ； ； 。 6. A general redundant parallel robot forward solution method according to claim 3, characterized in that: The constraint equations corresponding to the closed-chain structural kinematics submodel are expressed as follows: 。 7. A general redundant parallel robot forward solution method according to claim 3, characterized in that: The constraint equations corresponding to the open-chain structure kinematic sub-model are expressed as follows: 。 8. A general redundant parallel robot forward solution method according to claim 1, characterized in that: Substituting the structural parameters corresponding to the kinematic model into the corresponding constraint equations and determining the target posture according to the Gauss-Newton iteration method includes: The initial parameter value is iterated by using the Gauss-Newton iteration method to obtain the parameter estimation value, wherein the initial parameter value is the value of the starting position of the redundant parallel robot; Determine all unknown quantities based on parameter estimates and the constraint equations corresponding to the open-chain structural kinematic submodel; Substitute all unknown quantities into the position expression to obtain the target pose.

9. A general redundant parallel robot forward solution method according to claim 8, characterized in that: The position expression is: ； Among them, T is the homogeneous transformation matrix form of the rigid body posture in space, b0, b1, b2, b3, e0, e1, e2, e3 are the first parameter, second parameter, third parameter, fourth parameter, fifth parameter, sixth parameter, seventh parameter and eighth parameter of the dual quaternion form of the rigid body posture in space respectively.

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