A forward kinematics method for a general redundant parallel robot

Abstract

The present invention provides a forward kinematics method for a general redundant parallel robot, which relates to the technical field of industrial redundant parallel robot control. The method includes: establishing a branch coordinate system for the redundant parallel robot, where the branch coordinate system includes: a static platform coordinate system, a moving platform coordinate system, and a joint coordinate system; establishing a kinematic model of the redundant parallel robot to be solved according to the DH method and dual quaternions, where the kinematic model includes: a closed-chain structure kinematic sub-model and an open-chain structure kinematic sub-model; performing an elimination process on the kinematic model to obtain a corresponding constraint equation set; substituting the structural parameters corresponding to the kinematic model into the corresponding constraint equation set and determining the target pose according to the Gauss-Newton iteration method. The present invention solves two problems existing in the prior art: one is the error problem of compound hinges and other components during the manufacturing and assembly processes; the other is the problem of multiple solutions existing in the kinematically redundant parallel redundant robot.

Description

Technical Field

[0001] The present invention relates to the technical field of industrial redundant parallel robot control, and particularly to a forward kinematic solution method for a general redundant parallel robot. Background Art

[0002] In applications such as physical human - robot interaction, the working performance of parallel redundant robots is often restricted. The main reasons are that it is difficult to obtain the unique solution of its forward kinematics, and its directional workspace is relatively small. For example, in the famous Stewart platform, due to the existence of the second - type singularity, its maximum tilt angle is limited to about 45 degrees. By introducing kinematic redundancy, the directional workspace of the parallel redundant robot can be expanded, thereby effectively alleviating the limitations brought by the second - type singularity. 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 forward kinematic problem has not been significantly reduced. Similar to conventional parallel mechanisms, for a given driving joint variable, a parallel redundant robot with kinematic redundancy may also have multiple solutions.

[0003] Due to the mutual coupling relationship between each branch chain, the forward kinematic problem of the parallel mechanism is more complex than the inverse kinematic problem. There is a literature proposing a three - branch - chain spatial kinematic redundant parallel robot (3 - [R(RR - RRR)SR]) with (6 + 3) degrees of freedom. Figure 1 The schematic diagram of this mechanism is shown. There is a literature discussing its forward kinematics problem in the ideal state and analyzing adding different numbers of additional encoders to simplify the forward solution process. However, due to the errors in the manufacturing and assembly processes of compound hinges and other components, the existence of system errors is inevitable. Summary of the Invention

[0004] In order to overcome the deficiencies of the prior art, the purpose of the present invention is to provide a forward kinematic solution method for a general redundant parallel robot. The present invention solves two problems existing in the prior art: one is the error problem of compound hinges and other components in the manufacturing and assembly processes; the other is the problem that a parallel redundant robot with kinematic redundancy has multiple solutions.

[0005] To achieve the above - mentioned purpose, the present invention provides the following solutions:

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

[0007] Establishing the branch - chain coordinate systems of the redundant parallel robot; the branch - chain coordinate systems include: a static - platform coordinate system, a moving - platform coordinate system, and a joint coordinate system;

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

[0009] Perform an elimination process on the kinematic model to obtain the corresponding constraint equations;

[0010] Substitute the structural parameters corresponding to the kinematic model into the corresponding constraint equations and determine the target pose according to the Gauss-Newton iteration method.

[0011] Preferably, the expression of the closed-chain structure kinematic sub-model is:

[0012] ;

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

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

[0015] ;

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

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

[0018] ;

[0019] ;

[0020] ;

[0021] ;

[0022] ;

[0023] Among them, They are respectively the first link length parameter, the second link length parameter, the third link length parameter, the fourth link length parameter, the fifth link length parameter, the sixth link length parameter and the seventh link length parameter of the redundant parallel robot. denote , denote , denote .

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

[0025] ;

[0026] ;

[0027] ;

[0028] ;

[0029] ;

[0030] ;

[0031] .

[0032] Preferably, the expression of the constraint equation set corresponding to the closed-chain structure kinematic sub-model is:

[0033] .

[0034] Preferably, the expression of the constraint equation set corresponding to the open-chain structure kinematic sub-model is:

[0035] .

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

[0037] Using the Gauss-Newton iteration method to iterate the initial parameter values to obtain parameter estimation values, where the initial parameter values are the values at the starting position of the redundant parallel robot;

[0038] Determining all unknowns according to the parameter estimation values and the constraint equation set corresponding to the open-chain structure kinematic sub-model;

[0039] Substituting all unknowns 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 pose in space, and 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 pose in space, respectively.

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

[0044] The present invention provides a forward solution method for a general redundant parallel robot, including: establishing a branch chain coordinate system of the redundant parallel robot, where the branch chain coordinate system includes: a static platform coordinate system, a moving platform coordinate system, and a joint coordinate system; establishing a kinematic model of the redundant parallel robot to be solved according to the DH method and dual quaternions, where the kinematic model includes: a closed-chain structure kinematic sub-model and an open-chain structure kinematic sub-model; performing an elimination process on the kinematic model to obtain a corresponding constraint equation set; substituting the structural parameters corresponding to the kinematic model into the corresponding constraint equation set and determining the target pose according to the Gauss-Newton iteration method. In the case where there is no publicly available forward solution algorithm for 9-degree-of-freedom redundant parallel robots of the 3-[R(RR-RRR)SR], 3-[P(RR-RRR)URR], 3-[P(RR-RRR)RRRR], and 3-[P(RR-RRR)RRRP] configurations, the present invention first proposes a forward solution algorithm containing 158 geometric parameters, determines the unique forward solution of the redundant parallel robot. In high-precision scenarios such as industry and aerospace, the determined end pose is beneficial to realizing repeated positioning and precise operation. When the system knows that there is only a unique end pose, the situation where the same input causes the redundant parallel robot to "run" to different positions or postures will not occur, avoiding control ambiguity. BRIEF DESCRIPTION OF THE DRAWINGS

[0045] In order to more clearly illustrate the technical solutions in the embodiments of the present invention or the prior art, the following will briefly introduce the drawings required for use in the embodiments. Obviously, the drawings in the following description are only some embodiments of the present invention. For those of ordinary skill in the art, other drawings can be obtained based on these drawings without creative efforts.

[0046] Figure 1 It is a schematic structural diagram of a spatial kinematic redundant parallel robot with (6 + 3) degrees of freedom of three branch chains in the prior art provided by an embodiment of the present invention;

[0047] Figure 2 It is a flowchart of a forward solution method for a general redundant parallel robot provided by an embodiment of the present invention;

[0048] Figure 3 Schematic diagram of the closed-chain series structure provided by the embodiment of the present invention;

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

[0050] Figure 5 Schematic diagram of the connection method of the moving platform provided by the embodiment of the present invention;

[0051] Figure 6 Schematic diagram of the coordinate system of the first single branch chain provided by the embodiment of the present invention;

[0052] Figure 7 Schematic diagram of the coordinate system of the second single branch chain provided by the embodiment of the present invention;

[0053] Figure 8 Schematic diagram of the mechanism joint provided by the embodiment of the present invention;

[0054] Figure 9 is a schematic diagram of different poses of the parallel mechanism provided by the embodiment of the present invention. Among them, Figure 9(a) is the schematic diagram of the first pose, and Figure 9(b) is the schematic diagram of the second pose. Detailed implementation manners

[0055] Next, the technical solutions in the embodiments of the present invention will be clearly and completely described in conjunction with the accompanying drawings in the embodiments of the present invention. Obviously, the described embodiments are only a part of the embodiments of the present invention, rather than all the embodiments. All other embodiments obtained by those of ordinary skill in the art based on the embodiments of the present invention without creative efforts shall fall within the protection scope of the present invention.

[0056] To make the above objects, features, and advantages of the present invention more obvious and understandable, the present invention will be further described in detail below in conjunction with the accompanying drawings and specific implementation manners.

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

[0058] Step 100: Establish a branch chain coordinate system for the redundant parallel robot, where the branch chain coordinate system includes: a static platform coordinate system, a moving platform coordinate system, and a joint coordinate system;

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

[0060] Step 300: Eliminate elements from the kinematic model to obtain a corresponding constraint equation system;

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

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

[0063] The methods for describing the rigid body attitude include screw, Lie group as well as homogeneous transformation matrix, dual quaternion, and double quaternion, etc. Among them, the most commonly used is the 4×4 homogeneous transformation matrix.

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

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

[0066]

[0067] is the real part of the quaternion , is the imaginary part of the quaternion , , and are respectively , , the coefficients of , , is the imaginary unit, and ;

[0068] Its geometric meaning can be regarded as in three-dimensional space , , are the unit vectors 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 by matrix multiplication as:

[0070] ;

[0071] M(b)*c is the form of multiplying the quaternion b*c converted into the matrix M(b)*c. Here, only the multiplication rule of quaternions is described.

[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 anti-symmetric matrix and are extended to a four-dimensional form. The above formula completely describes the algebraic rule of quaternion multiplication. c0, c1, c2, and c3 are the ninth parameter, the tenth parameter, the eleventh parameter, and the twelfth parameter of the quaternion respectively.

[0073] In the formula , .

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

[0075] ;

[0076] Define the conjugate of a unit quaternion as: When and , we get .

[0077] The attitude of a rigid body can be represented by a unit quaternion, which is more efficient than using the traditional rotation matrix method. The unit quaternion can express three-dimensional rotation with fewer parameters, thus avoiding the singularity problem that may occur in the rotation matrix and reducing the storage and computation amount during the calculation process.

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

[0079] ;[[ID=4३]]

[0080] represents the element located at the i-th row and j-th column of the matrix in the matrix.

[0081] Clifford first proposed dual quaternions in 1878 based on quaternions to represent the motion group of a rigid body, that is: Among them, and are the real part and the dual part of the dual quaternion respectively. The real part of the dual quaternion function is the attitude of the current joint, is the dual symbol and .

[0082] H represents the quaternion algebra. D represents the dual number algebra. The tensor product of the quaternion algebra $\mathbb{H}$ and the dual number algebra $\mathbb{D}$ is used to describe the rigid body motion.

[0083] In the expression, it acts as: belongs to , that is , represents is a dual quaternion, which is the pose of a rigid body in space.

[0084] The multiplication of dual quaternions is still a dual quaternion, for example:

[0085] ;

[0086] Both $h_1$ and $h_2$ are general dual quaternions. Here it only illustrates the multiplication rule of dual quaternions.

[0087] In the formula .

[0088] The quaternion addition and multiplication defined above are used in this formula. The conjugate 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 its inverse: . The dual quaternion is used for the purpose of expressing pose transformation. The real part corresponds to the rotation matrix , while the dual part is defined as a function with the displacement vector :

[0092] ;

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

[0094]

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

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

[0097]

[0098] Therefore, the pose transformation can be expressed by dual quaternions as follows:

[0099] ;

[0100] This indicates that the orientation transformation can be represented as , and the position transformation can be represented as , where p1p2 are the thirteenth and fourteenth parameters of the fourth quaternion. In 1955, J. Denavit et al. proposed a method using four parameters to represent joint angle postures, namely the DH method, to describe the relative positions and postures of various components in a spatial mechanism, which is widely used in serial mechanisms. The relative relationship between link i and link i - 1 is established by combining dual quaternions as follows:

[0101] 1) Rotate around the axis by , that is

[0102] 2) Translate along the axis by , that is

[0103] 3) Rotate around the axis by That is

[0104] 4) Translate along the axis by , that is

[0105] is the first rotation angle, is the quaternion representation of rotating around the axis by , .

[0106] is the moving distance, is the quaternion representation of translating along the axis by , .

[0107] is the second rotation angle, is the quaternion representation of rotating around the axis by , .

[0108] is the moving distance, is the quaternion representation of translating along the axis by , 。

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

[0110]

[0111] In the formula 。

[0112] For the convenience of subsequent concise expression, the present invention represents the dual quaternion using 。

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

[0114] In practical applications, a compound hinge is often used to replace the traditional rotary pair with three axes intersecting at a point, that is, the 3R joint pair. In addition, the rotary pair connecting the moving platform can also be replaced by a prismatic 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, so as to maintain a consistent mathematical framework.

[0115] Furthermore, first establish the coordinate systems on the static platform and the tool of the moving platform, establish the base coordinate system at the center of the static platform, and establish the tool coordinate system at the center of the moving platform, which can be abbreviated as coordinate system {B} and coordinate system {P} respectively. Then, establish the joint coordinate systems on each branch in turn, and sort the joints from the base coordinate system to the tool coordinate system and call them 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, as Figure 6 shown. Among them 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, the parameter tables 1 and 2 of each structure on each branch chain can be obtained according to the joint coordinate system established by the DH method. Theoretically, the DH parameters of the three branch chains 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). Table 1 and Table 2 are shown as follows respectively:

[0116] Table 1 DH parameter table of the closed - chain structure

[0117]

[0118] Table 2 DH parameter table of the open - chain structure

[0119]

[0120] (RR - RRR) The closed - chain structure is a closed - loop series structure composed of rod 2, rod 3, rod 4, and rod 5. In the closed - chain mechanism, the kinematic mathematical model can be expressed using dual quaternions as:

[0121] (5);

[0122] Among them, represents the dual - quaternion form of the homogeneous transformation from the coordinate system {k} corresponding to the link DH parameters on the j - th branch chain to the coordinate system {i}, is the dual - quaternion form of the unit homogeneous transformation matrix. It represents that the coordinate system has not undergone any transformation, that is, the unit element in the dual - quaternion form. This equation expresses the kinematic constraints that satisfy the entire closed - chain structure, where the link DH parameters represent the rotation angle and the redundant degrees of freedom of the moving platform.

[0123] Taking the nominal parameters as an example, from Table 1 and formula (4), we can obtain the specific form between adjacent joints of the (RR - RRR) closed - chain structure (the expression of the link DH parameters on the branch chain corresponding to the sub - model of the closed - chain structure kinematics) as:

[0124]

[0125]

[0126]

[0127]

[0128]

[0129] Among them, are respectively the first link length parameter, the second link length parameter, the third link length parameter, the fourth link length parameter, the fifth link length parameter, the sixth link length parameter and the seventh link length parameter of the redundant parallel robot.

[0130] The open-chain serial mechanism can be regarded as a general 3-RRRSR mechanism. Using dual quaternions, its kinematic mathematical model can be expressed (the expression of the kinematic sub-model of the open-chain structure) as:

[0131] (6), where is the pose of the moving platform.

[0132] Furthermore, the expression of the link DH parameters on the branch chain corresponding to the kinematic sub-model of the open-chain structure is:

[0133]

[0134]

[0135]

[0136]

[0137]

[0138]

[0139]

[0140] In the expression the link length parameters of the redundant parallel robot are known quantities.

[0141] Taking the 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 represented in the form of dual quaternions.

[0142] 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}. Table 3 and Table 4 are shown as follows:

[0143] Table 3 Pose Table of Coordinate System {0} in Coordinate System {B}

[0144]

[0145] Table 4 Pose Table of Coordinate System {10} in Coordinate System {P}

[0146]

[0147] The pose transformation from coordinate system {B} to coordinate system {0} on link 1 can be denoted as , and the transformation from coordinate system {10} to coordinate system {P} can be denoted as .

[0148] The pose transformation from coordinate system {B} to coordinate system {0} on link 2 can be denoted as , and the transformation from coordinate system {10} to coordinate system {P} can be denoted as .

[0149] The pose transformation from coordinate system {B} to coordinate system {0} on link 3 can be denoted as , and the transformation from coordinate system {10} to coordinate system {P} can be denoted as .

[0150] The specific parameters are as follows:

[0151]

[0152]

[0153]

[0154]

[0155]

[0156]

[0157] They are the radii of the moving platform and the static platform respectively.

[0158] represents the dual quaternion form of the homogeneous transformation from the link coordinate system {k} to the coordinate system {i} on the j-th link. Equation (4) gives its specific transformation form, which contains a total of four parameters, including 1 variable and 3 constants. is the 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 respectively.

[0159] represents the dual quaternion form of the 4th-order identity matrix, and its physical meaning is that the object coordinate system has not rotated or translated.

[0160] The pose of the moving platform is , and in the forward 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 orientation of the tool coordinate system, that is, the moving platform.

[0161] is the rotation angle about the Z-axis of the i-th joint coordinate system in the j-th limb of the mechanism;

[0162] where is the redundant degree of freedom of the moving platform to be solved, that is, the rotation angle of the revolute pair connected to the moving platform is about Figure 7 in the rotated angle.

[0163] are the driving variables, namely the rotated angles about Figure 7 in the rotated angle.

[0164] are the constraint rotation angles, which are the rotated angles about Figure 7 in the rotated angle.

[0165] are the constraint rotation angles, which are the rotated angles about Figure 7 in the rotated angle.

[0166] are respectively the axis 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 limb.

[0167] After removing the same parameters in one limb, , contains a total of 40 parameters.

[0168] In the j-th limb, the pose transformation between the base coordinate system and the 0-th coordinate system is represented by 5 constant parameters, including 3 translation parameters and 2 rotation parameters. Similarly, the pose transformation from the 10-th coordinate system to the tool coordinate system in its j-th limb is also represented by 3 translation parameters and 2 rotation parameters.

[0169] Therefore, the complete redundant parallel mechanism has a total of 158 parameters, including 9 active driving parameters, 120 geometric structure constant parameters, and 29 unknown variables.

[0170] The active drives of configurations such as 3-[R(RR-RRR)SR], 3-[R(RRRRR)URR], 3-[R(RR-RRR)RRRR] and 3-[R(RR-RRR)RRRP] are all on , and when the drive is determined, the mathematical models of different configurations only differ in DH parameters, and their solution processes can be covered by the same mathematical model.

[0171] Furthermore, the forward kinematics solution can be expressed as: Given Find and . Each leg of this redundant parallel mechanism consists of two parts, including a closed-chain structure and an open-chain structure. All the unknown variables of each leg include the redundant joint angle of the moving platform and six passive joint angles , plus eight parameters to be determined in the tool coordinate system. Therefore, for the forward kinematics solution of this general redundant parallel mechanism, a total of 29 quantities to be determined must be calculated. The pose of the tool coordinate system is represented by a dual quaternion with eight variables as follows:

[0172]

[0173] The forward kinematics solution of this redundant parallel mechanism can be divided into two parts: 1) Solve the kinematic sub-model of the closed-chain structure. 2) Solve the pose of the moving platform and the redundant degrees of freedom .

[0174] Furthermore, this embodiment discloses the specific process of elimination:

[0175] First, multiply both sides of Equation (5) on the right by to obtain:

[0176] The expression of the constraint equation system corresponding to the kinematic sub-model of the closed-chain structure is:

[0177] (8)

[0178] The right side of this equation only includes one unknown variable , and the left side includes a total of two unknown variables. According to the product rule of dual quaternions, expand the left side of the equation to obtain:

[0179]

[0180] where , and is used to convert the dual quaternion into an 8-dimensional column vector. and are 4th-order invertible matrices related to the leg structure constants and the known conditions .

[0181] Then, expand the right side of Equation (8) according to the product of dual quaternions to obtain:

[0182]

[0183] where k ji is a first and second order polynomial function of θ j6 , K is an 8×1 matrix and each element of K is a first and second order polynomial function of θ j6 .

[0184] In the formula is a first and second order polynomial function of .

[0185] Let . Combining with equations (8)-(10), we can get:

[0186]

[0187] where is a second and second order polynomial function of . And from equations (9) and (11), we know that:

[0188]

[0189] It is easy to get from equation (12):

[0190]

[0191]

[0192]

[0193]

[0194] Substitute the quadratic polynomial in equation (11) into (13)-(16), we can get:

[0195] ; z 1ji is in the form of the quadratic polynomial function of equation (11).

[0196] Furthermore, multiply equation (6) on the left and right by simultaneously, we can obtain the constraint equation set corresponding to the kinematic sub-model of the open-chain structure, and the expression is:

[0197] (18).

[0198] The right side of this equation includes a total of Nine unknown variables. It can be easily deduced from the above that they can be directly obtained according to Equation (17). , so the left side of the equation can be regarded as only containing Three unknown variables.

[0199] According to the product principle of dual quaternions, expanding the left side of Equation (18) gives:

[0200]

[0201] is an 8th-order invertible matrix related to the constant parameters of the branched-chain structure. Similarly, according to the product principle of dual quaternions, expanding the right side of Equation (18) into an 8-dimensional vector:

[0202] Equation in is the first and second quadratic polynomial function of variables.

[0203] Similarly, let and from Equations (18)-(20), we can obtain:

[0204]

[0205] is the second and second quadratic polynomial function with

[0206] From Equation (19), we know that:

[0207]

[0208] The known trigonometric identity is:

[0209]

[0210] Calculate its Groebner basis from Equations (22) and (23). Using the gbasis command in MATLAB for symbolic operations, 46 bases can be obtained. Among them, there are 10 low-order bases without trigonometric functions. After removing 1 identity, there are 9 bases left, and the results are as follows:

[0211]

[0212]

[0213]

[0214]

[0215]

[0216]

[0217]

[0218]

[0219]

[0220] Substituting the quadratic polynomial function in (21) into the nine equations of (24)-(32), we get: the nine equations of (24)-(32), we get:

[0221]

[0222] Since both sides of the equation are divided by we can obtain:

[0223]

[0224] where At this point, each branch contains nine equations, so the three branches contain a total of 27 identities. Each equation contains nine variables, and the highest degree of the variables is 4.

[0225] Through two eliminations, five unknown variables are eliminated in each branch, and a total of 15 unknown variables are eliminated in the three branches. Now there are 14 unknowns left in the system of equations.

[0226] Furthermore, 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:

[0227] Using the Gauss-Newton iteration method to iterate the initial parameter values to obtain parameter estimation values;

[0228] Determining all unknowns according to the parameter estimation values and the constraint equations corresponding to the kinematic sub-model of the open-chain structure;

[0229] Substituting all unknowns into the position expression to obtain the target pose.

[0230] Furthermore, in the forward solution algorithm proposed in this embodiment, the pose of the tool coordinate system must follow the two constraint equations (1) and (2) of the dual quaternion, and at the same time, it needs to comply with the 12 constraint equations of Equation (17) and the 27 constraint equations of Equation (33). It can be seen from the derivation process that the 12 equations of Equation (17) and the 27 equations of Equation (33) are not a system of contradictory equations. Therefore, the closed-chain kinematic model (closed-chain structure kinematic sub-model) can be solved first using the least squares method and Equation (17), and Substituting it back into Equations (11) and (12) can solve .

[0231] Since the driving variable is known and can be obtained first . Therefore, the forward solution problem can be further transformed into solving the system of 29 11-variable polynomial equations composed of Equation (1), Equation (2), and Equation (33). It is expressed in the form of a 29-dimensional column vector as follows:

[0232]

[0233] Each term of which is a functional equation containing unknown variables, and the 11-dimensional column vector form of the unknown variables is as follows:

[0234] ;

[0235] The Gauss-Newton iteration method is often used to solve non-linear equations. In addition, a multi-variable polynomial equation generally has 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 iteration process. Among them, the Jacobi matrix needs to be calculated during the iteration process, each term of which is a polynomial function. Substitute a certain initial value into the following formula to start the iteration:

[0236]

[0237] This formula is the formula of the Gauss-Newton iteration method for solving the non-linear least squares problem

[0238] : The parameter estimation value of the k-th iteration.

[0239] : The parameter estimation value of the (k + 1)-th iteration.

[0240] : The Jacobi matrix calculated with as the input at the k-th iteration. For the input at the k-th iteration, the calculated Jacobi matrix.

[0241] : The 29-dimensional column vector calculated with as the input at the k-th iteration. For the input at the k-th iteration, the calculated 29-dimensional column vector.

[0242] Set the iteration termination condition as , then the solution of the 11-variable non-linear algebraic polynomial equation can be obtained. After obtaining and substituting it into equations (21) and (22) successively, and can be easily obtained. Find After that, substituting it into equations (21) and (22) successively, and can be easily obtained. Thus, all 29 unknowns can be obtained.

[0243] Finally, substituting the solved into equation (3), the form of the homogeneous transformation matrix representing the attitude can be obtained. Substituting the solved into equation (3), the form of the homogeneous transformation matrix representing the attitude can be obtained.

[0244] Furthermore, for a general redundant parallel single chain as shown in Figure 8 , different poses of the parallel mechanism are shown in Figure 9. Among them, 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, γ is the rotation angle of the moving platform coordinate system around the Z-axis of the static platform coordinate system, x Figure 8 p , y p , z p are all the positions of the moving platform coordinate system relative to the static platform coordinate system.

[0245] In this specification, each embodiment is described in a progressive manner. The key point of each embodiment is to illustrate the differences from other embodiments. For the same or similar parts among the embodiments, reference can be made to each other.

[0246] In the present invention, specific examples are used to elaborate on the principle and implementation manner of the present invention. The description of the above embodiments is only used to help understand the method and its core idea of the present invention; at the same time, for those of ordinary skill in the art, according to the idea of the present invention, there will be changes in the specific implementation manner and application scope. In summary, the content of this specification should not be construed as a limitation to the present invention.

Claims

1. A forward kinematics method for a general redundant parallel robot, characterized in that, Including: Establish the branch chain coordinate system of the redundant parallel robot; The branch chain coordinate system includes: a static platform coordinate system, a moving platform coordinate system, and a joint coordinate system; Establish the kinematic model of the redundant parallel robot to be solved according to the DH method and dual quaternions; the kinematic model includes: a closed-chain structure kinematic sub-model and an open-chain structure kinematic sub-model; Perform an elimination process on the kinematic model to obtain the corresponding constraint equations; Substitute the structural parameters corresponding to the kinematic model into the corresponding constraint equations and determine the target pose according to the Gauss-Newton iteration method; The expression of the closed-chain structure kinematic sub-model is: ； Among them, represents the dual quaternion form of the homogeneous transformation from the coordinate system { j} corresponding to the link DH parameters on the k -th branch chain to the coordinate system { i}, is the unit element in the form of dual quaternion, representing the kinematic constraint of the entire closed-chain structure, the j -th joint on the i -th branch chain, the rotation angle about the Z axis in the link DH parameters, where the link DH parameters represent the rotation angle and the redundant degrees of freedom of the moving platform, i ranges from 1 to 10; The expression of the open-chain structure kinematic sub-model is: ； in, is the posture of the moving platform, B Represents the base coordinate system of the center of the static platform of the mechanism, P The tool coordinate system representing the center of the moving platform, j For the j On the branch chain, the coordinate system { B }arrive{ 0 }’s pose transformation; The expression of the link DH parameters on the branch chain corresponding to the closed-chain structure kinematic sub-model is: ； ； ； ； ； Among them, 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, ； The expression of the link DH parameters on the branch chain corresponding to the open-chain structure kinematic sub-model is: ； ； ； ； ； ； 。 2. The direct kinematic method of a general redundant parallel robot according to claim 1, characterized in that The expression of the constraint equations corresponding to the closed-chain structure kinematic sub-model is: 。 3. A forward kinematics method for a general redundant parallel robot according to claim 1, characterized in that, The expression of the constraint equations corresponding to the open-chain structure kinematic sub-model is: 。 4. A forward kinematics method for a general redundant parallel robot according to claim 1, characterized in that, The 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: Using the Gauss-Newton iteration method to iterate the initial parameter values to obtain parameter estimation values, where the initial parameter values are the values at the starting position of the redundant parallel robot; Determine all unknowns according to the parameter estimation values and the constraint equations corresponding to the open-chain structure kinematic sub-model; Substitute all unknowns into the position expression to obtain the target pose.

5. A forward kinematic method for a general redundant parallel robot according to claim 4, characterized in that The position expression is: ； Among them, T is in the form of a homogeneous transformation matrix for the rigid body position and orientation in space, b 0 、b 1 、b 2 、b 3 、e 0 、e 1 、e 2 、e 3 are the first parameter, the second parameter, the third parameter, the fourth parameter, the fifth parameter, the sixth parameter, the seventh parameter, and the eighth parameter of the dual quaternion form of the rigid body position and orientation in space, respectively.

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