WO2020034407A1 - 基于轴不变量的通用3r机械臂逆解建模与解算方法 - Google Patents
基于轴不变量的通用3r机械臂逆解建模与解算方法 Download PDFInfo
- Publication number
- WO2020034407A1 WO2020034407A1 PCT/CN2018/112705 CN2018112705W WO2020034407A1 WO 2020034407 A1 WO2020034407 A1 WO 2020034407A1 CN 2018112705 W CN2018112705 W CN 2018112705W WO 2020034407 A1 WO2020034407 A1 WO 2020034407A1
- Authority
- WO
- WIPO (PCT)
- Prior art keywords
- determinant
- matrix
- axis
- vector
- equation
- Prior art date
Links
Images
Classifications
-
- G—PHYSICS
- G06—COMPUTING; CALCULATING OR COUNTING
- G06F—ELECTRIC DIGITAL DATA PROCESSING
- G06F30/00—Computer-aided design [CAD]
- G06F30/20—Design optimisation, verification or simulation
Definitions
- the invention relates to a multi-axis robot 3R manipulator inverse solution modeling and calculation method, and belongs to the field of robot technology.
- a dynamic graph structure (Dynamic Graph Structure) can be used to dynamically build a directed Span tree based on the motion axis, which lays a foundation for the study of robot modeling and control of Variable Topology Structure.
- a dynamic graph structure (Dynamic Graph Structure) can be used to dynamically build a directed Span tree based on the motion axis, which lays a foundation for the study of robot modeling and control of Variable Topology Structure.
- the inverse solution of the position of the 3R manipulator refers to: given the structural parameters of the 3R manipulator and the expected position, calculate 3 joint variables to align the wrist position with the expected position.
- the existing 3D manipulator position inverse solution method based on DH parameters has the following disadvantages: the process of establishing the DH system and DH parameters is unnatural, and the application is cumbersome; the singularity problem caused by the calculation method needs to be dealt with; the application is easy to introduce into the system Measurement error.
- the principle of inverse solution of 3R manipulator based on D-H parameters is not universal, and it is difficult to generalize to solve the problem of inverse solution of general 6R manipulator.
- the technical problem to be solved by the present invention is to provide a universal 3R manipulator inverse solution modeling and calculation method based on axis invariants, which can improve the absolute positioning accuracy of the manipulator; compared with the DH parameter, the solution process is universal and can be obtained The system is completely reversed.
- the present invention adopts the following technical solutions:
- a general inverse modeling and solving method of 3R manipulator based on axis invariants which is characterized by:
- Ju-Gibbs or Ju-Gibbs gauge quaternion that is isomorphic to Euler quaternions is defined:
- step [1] In step [1],
- expression Represents the xth power of ⁇ ;
- Delimiter I axis invariant Cross product matrix, member For members Same time replacement; 1 is the three-dimensional identity matrix; i Q n represents the attitude; Along the vector axis Line position Zero point from origin The translation vector to the origin O l ;
- ⁇ is the projection symbol, and i
- ⁇ is the ⁇ projection vector in the geodetic coordinate system.
- step [2] the calculation formula of Dixon determinant based on the axis invariant is:
- I the rotation transformation matrix
- the coefficients of the combined variables are independent column vectors, so we choose Coefficients to form a square matrix
- the remaining column vectors must be The columns are related;
- I a Dixon matrix of size S ⁇ S, whose [i] [j] member is an N-th order polynomial of univariate ⁇ 1 .
- step [2] the determinant formula of the block matrix is:
- a square matrix of size (n + m) ⁇ (n + m) is M and a matrix of size n ⁇ n Is a sub-matrix consisting of the first n rows and any n columns of a square matrix M, a matrix of size m ⁇ m Is a sub-matrix consisting of the elements of m rows and the remaining m columns of the square matrix M;
- step [2] perform the row staircase calculation principle on the determinant:
- each term is an n-th order polynomial of ⁇ 1 ;
- the original determinant can be transformed into an upper triangular determinant by elementary row transformation, and then the non-zero diagonal Multiply the line elements to get the polynomial expression of the determinant; the formula is 0, and all solutions of ⁇ 1 are obtained;
- the specific method of row staging is to first sort the highest order of the first column of the determinant from high to low, and then perform a maximum of (S-1) ⁇ n elementary row transformation eliminations.
- the determinant whose elements are not 0; the elementary row transformation and elimination of the cofactors in the first row and the first column of the determinant are solved successively and iteratively.
- step [3] the steps of constructing Dixon polynomials for n “n-ary N-th order” polynomial systems are:
- a determinant consisting of a position vector or a rotation vector represents the volume of the vector expansion space; in different Cartesian spaces, it has a volume invariance. among them:
- Equation (28) n n-elements
- Equation (116) is a 16th-order mononomial equation for ⁇ 1
- Equation (5) is used to perform the determinant of quadratic block.
- the method of the present invention proposes a general 3R attitude inverse solution method based on axis invariants.
- Features are:
- the axis invariant helps to improve the absolute positioning accuracy of the robotic arm; the range of joint variables covers a full week, eliminating the singularity caused by the DH calculation principle; compared with the DH parameters, the solution process is universal , You can get all the inverse solutions of the system.
- Figure 3 is a schematic diagram of fixed axis rotation
- Figure 4 shows the derived invariants of the axis invariants.
- Natural coordinate axis The unit reference axis with a fixed origin that is coaxial with the motion axis or measurement axis is called the natural coordinate axis, also known as the natural reference axis.
- Natural coordinate system As shown in Figure 1, if the multi-axis system D is at zero position, all Cartesian body coordinate systems have the same direction, and the origin of the body coordinate system is on the axis of the motion axis, then the coordinate system is a natural coordinate system. , Referred to as the natural coordinate system.
- the advantages of the natural coordinate system are: (1) the coordinate system is easy to determine; (2) the joint variables at the zero position are zero; (3) the system attitude at the zero position is consistent; (4) it is not easy to introduce a measurement error.
- the coordinate axis is a reference direction with a zero position and a unit scale. It can describe the position of translation in that direction, but it cannot fully describe the rotation angle around the direction, because the coordinate axis itself does not have a radial reference direction, that is There is a zero position that characterizes rotation. In practical applications, the radial reference of this shaft needs to be supplemented. For example: in Cartesian F [l] , to rotate around lx, you need to use ly or lz as the reference zero.
- the coordinate axis itself is 1D, and three orthogonal 1D reference axes constitute a 3D Cartesian frame.
- the axis invariant is a 3D spatial unit reference axis, which is itself a frame. It has its own radial reference axis, the reference mark.
- the spatial coordinate axis and its own radial reference axis determine the Cartesian frame.
- the spatial coordinate axis can reflect the three basic reference attributes of the motion axis and the measurement axis.
- [2] is a 3D reference axis, which not only has an axial reference direction, but also has a radial reference zero position, which will be explained in section 3.3.1.
- the axis invariant For axis invariants, its absolute derivative is its relative derivative. Since the axis invariant is a natural reference axis with invariance, its absolute derivative is always a zero vector. Therefore, the axis invariant is invariant to time differentiation. Have:
- the basis vector e l is any vector consolidated with F [l] .
- the basis vector With Any vector of consolidation, Is F [l] and Common unit vector, so Is F [l] and Shared basis vector. So the axis is invariant Is F [l] and Common reference base.
- Axis invariants are parameterized natural coordinate bases and primitives for multi-axis systems. The translation and rotation of a fixed axis invariant are equivalent to the translation and rotation of a fixed coordinate system.
- optical measurement equipment such as laser trackers can be applied to achieve accurate measurement of invariants of fixed axes.
- the cylindrical pair can be applied to simplify the MAS kinematics and dynamics analysis.
- Invariant A quantity that does not depend on a set of coordinate systems for measurement is called an invariant.
- Natural coordinates take the natural coordinate axis vector as the reference direction, the angular position or line position relative to the zero position of the system, and record it as q l , which is called natural coordinates;
- natural motion vector will be determined by the natural axis vector And the vector determined by natural coordinates q l Called the natural motion vector. among them:
- the natural motion vector realizes the unified expression of axis translation and rotation.
- a vector to be determined by the natural axis vector and the joint such as Called the free motion vector, also known as the free spiral.
- the axis vector Is a specific free spiral.
- joint space The space represented by the joint natural coordinates q l is called joint space.
- the Cartesian space that expresses the position and pose (posture for short) is called a shape space, which is a double vector space or a 6D space.
- Natural joint space using the natural coordinate system as a reference, through joint variables Means that there must be when the system is zero The joint space is called natural joint space.
- Axis vector Is the natural reference axis of the natural coordinates of the joint. Because Is an axis invariant, so Is a fixed axis invariant, which characterizes the motion pair The structural relationship is determined by the natural coordinate axis. Fixed axis invariant Is a link Natural description of structural parameters.
- Natural coordinate axis space The fixed axis invariant is used as the natural reference axis, and the space represented by the corresponding natural coordinate is called the natural coordinate axis space, which is referred to as the natural axis space. It is a 3D space with 1 degree of freedom.
- any motion pair in the loop can be selected, and the stator and the mover constituting the motion pair can be separated; thus, a loop-free tree structure is obtained.
- the Span tree represents a span tree with directions to describe the topological relationship of the tree chain movement.
- I is a structural parameter; A is an axis sequence; F is a bar reference sequence; B is a bar body sequence; K is a motion pair type sequence; NT is a sequence of constrained axes, that is, a non-tree.
- Axis sequence a member of.
- Rotary pair R, prism pair P, spiral pair H, and contact pair O are special examples of cylindrical pair C.
- the motion chain is identified by a partial order set ().
- O () represents the number of operations in the calculation process, and usually refers to the number of floating-point multiplications and additions. It is very tedious to express the calculation complexity by the number of floating-point multiplication and addition. Therefore, the main operation times in the algorithm cycle are often used;
- l l k is the kinematic chain from axis l to axis k, and the output is expressed as And
- the cardinality is written as
- l l k execution process execution If Then execute Otherwise, end.
- the computational complexity of l l k is O (
- l L means to obtain a closed subtree composed of axis l and its subtrees, Is a subtree without l; recursively execute l l with a computational complexity of
- ⁇ means attribute placeholder; if the attribute p or P is about position, then Should be understood as a coordinate system To the origin of F [l] ; if the attribute p or P is about direction, then Should be understood as a coordinate system To F [l] .
- attribute variables or constants with partial order include the indicators indicating partial order in the name; either include the upper left corner and lower right corner indexes, or include the upper right corner and lower right corner indexes; Their directions are always from the upper left corner indicator to the lower right corner indicator, or from the upper right corner indicator to the lower right corner indicator.
- the description of the direction is sometimes omitted. Even if omitted, those skilled in the art can also use symbolic expressions. It is known that, for each parameter used in this application, for a certain attribute symbol, their directions are always from the upper left corner index to the lower right corner index of the partial order index, or from the upper right corner index to the lower right corner index.
- the symbol specifications and conventions in this application are determined based on the two principles of the partial order of the kinematic chain and the chain link being the basic unit of the kinematic chain, reflecting the essential characteristics of the kinematic chain.
- the chain indicator represents the connection relationship, and the upper right indicator represents the reference system.
- This symbolic expression is concise and accurate, which is convenient for communication and written expression.
- they are structured symbol systems that contain the elements and relationships that make up each attribute quantity, which is convenient for computer processing and lays the foundation for computer automatic modeling.
- the meaning of the indicator needs to be understood through the background of the attribute, that is, the context; for example: if the attribute is a translation type, the indicator at the upper left corner indicates the origin and direction of the coordinate system; if the attribute is a rotation type, the indicator at the top left The direction of the coordinate system.
- the coordinate vector in the natural coordinate system F [k] that is, the coordinate vector from k to l;
- rotation vector / angle vector I is a free vector, that is, the vector can be freely translated
- the angular position that is, the joint angle and joint variables, are scalars
- T means transpose of ⁇ , which means transpose the collection, and do not perform transpose on the members; for example:
- Projection symbol ⁇ represents vector or tensor of second order group reference projection vector or projection sequence, i.e. the vector of coordinates or coordinate array, that is, the dot product projection "*"; as: position vector
- the projection vector in the coordinate system F [k] is written as
- Is a cross multiplier for example: Is axis invariant Cross product matrix; given any vector
- the cross product matrix is
- the cross product matrix is a second-order tensor.
- i l j represents a kinematic chain from i to j
- l l k is a kinematic chain from axis l to k
- n represents the Cartesian Cartesian system, then Is a Cartesian axis chain; if n represents a natural reference axis, then For natural shaft chains.
- Position vector The projection vector on the three Cartesian axes is definition Since the index of the upper left corner of l r lS indicates the reference frame, l r lS both indirectly represents the displacement vector It also directly represents the displacement coordinate vector, that is, it has the dual functions of vector and coordinate vector.
- n> represents the full permutation of natural numbers [1: n], and there are n! Instances.
- I [i1, ... in] represents the number of reverse order of the arrangement ⁇ i1, ... in>.
- the calculation complexity of formula (1) is: n! Product of n times and n! Sub-addition has exponential complexity and can only be applied to determinants with smaller dimensions. For determinants with larger dimensions, Laplace formula is usually used for recursive operations. for Adjugate Matrix, then
- Simpler algorithms usually use Gaussian elimination or LU decomposition, first transform the matrix into a triangular matrix or a product of triangular matrices by elementary transformation, and then calculate the determinant.
- the above determinant calculation method for number fields is not applicable to high-dimensional polynomial matrices, and a determinant calculation method for block matrices needs to be introduced.
- Computing the determinant of Vector Polynomial is a specific block matrix determinant calculation problem. It expresses the inherent relationship between vectors and determinants at the vector level.
- the determinant calculation of the block matrix expresses the inherent laws of the block matrix and the determinant at the matrix level.
- Equations (3) and (4) can be generalized to n-dimensional space.
- a square matrix of size (n + m) ⁇ (n + m) is M and a matrix of size n ⁇ n Is a sub-matrix consisting of the first n rows and any n columns of a square matrix M, a matrix of size m ⁇ m Is a sub-matrix consisting of the elements of m rows and the remaining m columns of the square matrix M;
- each term is an n-th order polynomial with respect to ⁇ 1 .
- the original determinant can be changed to an upper triangular determinant through elementary row transformation, and then the nonzero diagonal elements are multiplied to obtain the determinant polynomial expression. This formula is 0, and all solutions of ⁇ 1 are obtained.
- the specific method of row staging is to first sort the highest order of the first column of the determinant from high to low, and then perform a maximum of (S-1) ⁇ n elementary row transformation eliminations. Determinants whose elements are not zero. Then the elementary row transformation and elimination of the cofactors of the first row and the first column of the determinant are solved successively and iteratively.
- N-th order polynomial system based on "N-carry word” N-th order polynomial system based on "N-carry word”.
- n "n-ary first-order” polynomial power products Independent variables appear N times repeatedly, then n “n-ary N-th order” polynomial systems are obtained "N-ary N-th Order Polynomial System” and "n-bit N Carry Word” Isomorphism.
- the first m of the auxiliary variable Y m are used to sequentially replace the m variables in the Original Variables X n , and "
- a determinant consisting of a position vector or a rotation vector represents the volume of the vector expansion space; in different Cartesian spaces, it has a volume invariance. among them:
- Equation (28) n n-elements
- S ⁇ S a polynomial equation of x 1 cannot be established; at this time, S ⁇ S is transformed into a row-echelon matrix Ech ( S ⁇ S ); obtain the square matrix after calculating the product of the Pivot of the matrix That is, S ′ independent column vectors are selected from S ⁇ S.
- n "n-ary N-th order" polynomial system The example (referred to as polynomial) is written as among them: And have according to Polynomials to determine Dixon matrices and separate variables and Select and Satisfy
- Equation (32) is a polynomial equation of univariate x 1 ; n-1 unknowns are eliminated; thus, a feasible solution of univariate x 1 can be obtained. If x 1 is also satisfied
- the steps are:
- the formula is a four-order 1st-order polynomial system that meets the Dixon elimination conditions. From formula (19) and formula (22), we get
- Axis vector Relative to rod And ⁇ l or natural coordinate system And F [l] is fixed, so this rotation is called fixed axis rotation.
- the projection vector is Zero vector after rotation
- the moment vector is
- the axial component is Rodrigues vector equation with chain index
- Equation (43) is about with Multivariable linear equation is an axis invariant Second-order polynomial. Given a natural zero vector As Zero reference, and Respectively the zero vector and the radial vector. Equation (43) is Symmetry Represents zero-axis tensor, antisymmetric part Represents the radial axial tensor, and the axial outer product tensor, respectively Orthogonal to determine the three-dimensional natural axis space; Equation (43) contains only one sine and cosine operation, 6 product operations, and 6 sum operations, and the computational complexity is low; And joint variables The coordinate system and polarity are parameterized.
- Equation (43) can be expressed as
- Ju-Gibbs quaternion a standardized Ju-Gibbs quaternion (referred to as the standard Ju-Gibbs quaternion, that is, a quaternion with a "label" of 1);
- Non-standard that is, its standard part is not 1. From equation (53), we can know that only given axis l and The canonical Ju-Gibbs quaternion, and the two axes are orthogonal, Is the canonical quaternion.
- I the rotation transformation matrix
- the basic properties of the Dixon determinant of the radial invariant and the kinematic chain are proposed to lay the foundation for the inverse kinematic analysis of the robot based on the invariant of the axis.
- the invariant of the axis is essentially different from the coordinate axis: the coordinate axis is a reference direction with a zero position and a unit scale. It does not have a radial reference direction, that is, there is no zero position that characterizes rotation. In actual application, the radial reference of the coordinate axis needs to be supplemented.
- the coordinate axis itself is 1D, and three orthogonal coordinate axes constitute a 3D Cartesian frame; the axis invariant is a 3D space unit reference axis (referred to as a 3D reference axis), which has a radial reference zero.
- the "3D reference axis" and its radial reference zero position can determine the corresponding Cartesian system.
- the axis invariant based on the natural coordinate system can accurately reflect the three basic attributes of "coaxiality", "polarity” and “zero position" of the motion axis and the measurement axis.
- the axis invariant is essentially different from the Euler axis: the directional cosine matrix (DCM) is a real matrix, the axis vector is the eigenvector corresponding to the eigenvalue 1 of the DCM, and is an invariant; the fixed axis invariant is the "3D reference axis ", Not only with the origin and axial direction, but also with the radial reference zero; in the natural coordinate system, the axis invariant does not depend on the adjacent consolidated natural coordinate system, that is, it has Variable natural coordinates; axis invariants have excellent mathematical operation functions such as nilpotency; in natural coordinate systems, DCM and reference polarities can be uniquely determined through axis invariants and joint coordinates; it is not necessary to establish for each member The respective systems can greatly simplify the modeling workload.
- DCM directional cosine matrix
- the fixed axis invariant is the "3D reference axis ", Not only with the origin and axial direction, but also with
- measuring the axis invariants can improve the measurement accuracy of structural parameters.
- iterative kinematics and dynamic equations including topological structure, coordinate system, polarity, structural parameters, and dynamic parameters can be established.
- NP problems All problems that are not solvable in definite polynomial time are called NP problems.
- the non-deterministic algorithm decomposes the problem into two stages: “guessing” and “verifying”: the “guessing” stage of the algorithm is non-deterministic, and the “verifying” stage of the algorithm is deterministic, and the correctness of the guessed solution is determined through verification. If it can be calculated in polynomial time, it is called a polynomial non-deterministic problem.
- the elimination of multivariate polynomials is generally considered to be an NP problem. Usually applied Based on the elimination of multiple polynomials, we have to resort to heuristic "guessing” and “verification” to solve the problem.
- Structural parameters and These are the structural parameters of the chain link l, which can be obtained by external measurement when the system is in the zero position. As shown in Fig. 4, the zero vector, the radial vector, and the axial vector are invariants independent of the rotation angle. The zero vector is a specific radial vector.
- Any vector can be decomposed into zero vector and axial vector, so
- Is the axis l and Common vertical line or common radial vector Is the axial vector of axis l. From equation (65), we can know that any structure parameter vector Can be decomposed into zero invariants independent of the coordinate system Axial invariant Their radial vectors are written as Structural parameter vector And axis invariants Uniquely determine the radial coordinate system, with 2 independent dimensions. If two axial invariants and Collinear
- the axial invariant and the zero invariant shown in equation (66) are the decomposition of the natural parameter by the structural parameter vector.
- the zero vector, radial vector, and axial vector derived from the axis invariant have the following relationships:
- the equation (71) is called the inversion formula of the zero vector; the formula (72) is the interchange formula between the zero vector and the radial vector; and the formula (73) is called the radial vector invariance formula. From (65), (71) to (73),
- (80) can be and Translates to about Of multiple linear types. Simultaneously, It has symmetry (rotation) for y l and ⁇ l . From equations (67), (74), and (75),
- Equation (81) is derived from three independent structural parameters And a motion variable ⁇ l . From equation (81),
- 0 solution.
- Equation (116) is a 16th-order mononomial equation for ⁇ 1 .
- Equation (4) transforms the determinant calculation of the vector polynomial into the determinant of three vectors, this step plays a decisive role; the axis invariant
- the derived invariants are structural parameters, and the system equation is a vector algebraic equation about the vector of the structural parameter and the joint variable (scalar).
Landscapes
- Engineering & Computer Science (AREA)
- Physics & Mathematics (AREA)
- Theoretical Computer Science (AREA)
- Computer Hardware Design (AREA)
- Evolutionary Computation (AREA)
- Geometry (AREA)
- General Engineering & Computer Science (AREA)
- General Physics & Mathematics (AREA)
- Manipulator (AREA)
- Management, Administration, Business Operations System, And Electronic Commerce (AREA)
- Complex Calculations (AREA)
Abstract
一种基于轴不变量的通用3R机械臂逆解建模与解算方法,应用n个n元N阶多项式的Dixon消元与求解原理,进行位姿逆解计算,根据机械臂n元3D矢量位姿方程,获得n个n元2阶多项式方程;应用基于轴不变量的Dixon行列式计算式及分块矩阵的行列式计算式简化行列式计算;应用n个n元N阶多项式的Dixon消元与求解原理完成位姿逆解计算,根据Dixon矩阵的行列式为0,得到一元高阶多项式方程,应用基于友阵的一元高阶多项式方程求解一元高阶多项式方程的解。上述方法可提高机械臂的绝对定位精度;与D-H参数相比,求解过程具有通用性,可以获得系统全部逆解。
Description
本发明涉及一种多轴机器人3R机械臂逆解建模与解算方法,属于机器人技术领域。
自主机器人研究的一个重要方面是需要解决变拓扑结构机器人的运动学建模问题。在MAS中,具有动态的图结构(Dynamic Graph Structure),可以动态地建立基于运动轴的有向Span树,为研究可变拓扑结构(Variable Topology Structure)的机器人建模与控制奠定了基础。为此,需要提出基于轴不变量的通用机械臂逆解原理,既要建立包含坐标系、极性、结构参数、关节变量的完全参数化的正运动学模型,又要实时地计算位姿方程;一方面,可以提高机器人的自主性,另一方面,可以提高机器人位姿控制的绝对精度。
3R机械臂位置逆解是指:给定3R机械臂结构参数及期望位置,计算3个关节变量,使腕心位置与期望位置对齐。现有的基于D-H参数的3R机械臂位置逆解方法存在如下缺点:建立D-H系及D-H参数的过程不自然,应用繁琐;需要处理由计算方法导致的奇异性问题;在应用时,易引入系统测量误差。基于D-H参数的3R机械臂逆解原理不具有普适性,难以推广来解决通用6R机械臂的逆解问题。
发明内容
本发明所要解决的技术问题是提供一种基于轴不变量的通用3R机械臂逆解建模与解算方法,可提高机械臂的绝对定位精度;与D-H参数相比,求解过程具有通用性,可以获得系统全部逆解。
为解决上述技术问题,本发明采用以下技术方案:
一种基于轴不变量的通用3R机械臂逆解建模与解算方法,其特征是,
应用n个“n元N阶”多项式的Dixon消元与求解原理,进行位姿逆解计算,主要包括以下步骤:
【1】根据机械臂n元3D矢量位姿方程,获得n个“n元2阶”多项式方程;
【2】应用基于轴不变量的Dixon行列式计算式、分块矩阵的行列式计算式或对行列式进行行阶梯化计算式简化行列式计算;
【3】应用n个“n元N阶”多项式的Dixon消元与求解原理完成位姿逆解计算,其中:根据Dixon矩阵的行列式为0,得到一元高阶多项式方程,应用基于友阵的一元高阶多项式方程求解一元高阶多项式方程的解。
其中:
式中,
为居-吉布斯规范四元数
模的平方;表达形式幂符
表示□的x次幂;右上角角标∧或
表示分隔符;轴不变量
是轴不变量
的叉乘矩阵;
是Gibbs矢量
的叉乘矩阵;若用“□”表示属性占位,则式中的表达形式□
[□]表示成员访问符。
步骤【1】中,
建立规范的姿态方程为:
建立规范的定位方程:
式中,
为任意杆件,表达形式
表示□的x次幂;右上角角标∧或
表示分隔符;
是轴不变量
的叉乘矩阵,杆件
为杆件
时同理替换;1为三维单位矩阵;
iQ
n表示姿态;
为沿矢量轴
的线位置;
为零位时由原点
至原点O
l的平动矢量;
|□为投影符,
i|□为□在大地坐标系的投影矢量。
步骤【2】中,基于轴不变量的Dixon行列式计算式为:
根据运动链Dixon行列式性质有:
并记:
由式(47)得3R运动学方程
由式(90)得
由(91)式得
记
则由式(51)及式得(93)
由式(92)及式(93)得
由式(95)得3R运动学多项式方程
多项式系统F
3(Y
2|T
2),根据双线性型行列式通式
则有
其中:
由式(80)及式(93)得
得简化的3元N阶Dixon行列式为
步骤【2】中,分块矩阵的行列式计算式为:
若记大小为(n+m)·(n+m)的方阵为M,大小为n·n的矩阵
是方阵M的前n行及任意n列元素构成的子矩阵,大小为m·m的矩阵
是方阵M后m行及剩余m列元素构成的子矩阵;由升序排列的矩阵列序号构成的序列cn及cm是序列[1:m+n]的子集,[cn,cm]∈<1:n+m>,且有cm∪cn=[1:m+n];则方阵M行列式与分块矩阵
及
的行列式关系为
步骤【2】中,对行列式进行行阶梯化计算原理:
对于S×S矩阵,其每一项是关于τ
1的n阶多项式;计算该矩阵的行列式时,可通过初等行变换将原行列式变为上三角行列式,再将非零的对角线元素相乘,得到行列式的多项式表达式;该式为0,得到τ
1的所有解;
行阶梯化的具体方法为,先对行列式第一列的最高阶次由高到低进行排序,再进行最多(S-1)×n次初等行变换消元,得到第一列只有第一个元素不为0的行列式;再对该行列式第1行及1列的余子式进行初等行变换消元,依次迭代求解。
步骤【3】中,n个“n元N阶”多项式系统的Dixon多项式构建步骤为:
引入辅助变量[y
2,y
3,…,y
n],且有
得
其中:
由式(17)得
考虑式(13)及式(18)得该多项式的Dixon行列式
在笛卡尔空间下,由位置矢量或转动矢量构成的行列式表示矢量张成空间的容积(Volume);在不同笛卡尔空间下具有容积的不变性。其中:
给定n个“n元N阶”多项式系统F
n(Y
n-1|X
n-1),n≥2;存在与消去变量x
2,…,x
n无关的Dixon矩阵
SΘ
S(x
1),其Dixon多项式
表示为分离变量
及
的双重线性型:
其中:
Det(
SΘ
S(x
1))=0; (28)
称式(28)中“n个n元”为Dixon消元的必要条件,从而获得可行解。
由式(28)、式(99)及式(100)得
式(116)是关于τ
1的16阶单项式方程,应用式(5)进行二次分块的行列式计算。
本发明所达到的有益效果:
本发明的方法提出了基于轴不变量的通用3R姿态逆解方法。特征在于:
具有简洁、优雅的运动链符号系统,具有伪代码的功能,具有迭代式结构,保证系统实现的可靠性及机械化演算。
具有基于轴不变量的迭代式,保证计算的实时性;实现坐标系、极性及系统结构参量的完全参数化,基于轴不变量的可逆解运动学具有统一的表达及简洁的结构化层次模型,保证位姿分析逆解(Analytical Inverse Solution to Position and Attitude)的通用性。
直接应用激光跟踪仪精密测量获得的基于固定轴不变量的结构参数,保证位姿逆解的准确性;从而,使系统的绝对定位与定姿精度接近重复精度。
由于轴不变量可以精确测量,有助于提高机械臂的绝对定位精度;由于关节变量的范围覆盖完整的一周,消除了D-H计算原理导致的奇异性;与D-H参数相比,求解过程具有通用性,可以获得系统全部逆解。
图1自然坐标系与轴链;
图2固定轴不变量;
图3为定轴转动示意图;
图4为轴不变量的导出不变量。
下面对本发明作进一步描述。以下实施例仅用于更加清楚地说明本发明的技术方案,而不能以此来限制本发明的保护范围。
定义1自然坐标轴:称与运动轴或测量轴共轴的,具有固定原点的单位参考轴为自然坐标轴,亦称为自然参考轴。
定义2自然坐标系:如图1所示,若多轴系统D处于零位,所有笛卡尔体坐标系方向一致,且体坐标系原点位于运动轴的轴线上,则该坐标系统为自然坐标系统,简称自然坐标系。
自然坐标系优点在于:(1)坐标系统易确定;(2)零位时的关节变量为零;(3)零位时的系统姿态一致;(4)不易引入测量累积误差。
由定义2可知,在系统处于零位时,所有杆件的自然坐标系与底座或世界系的方向一致。系统处于零位即
时,自然坐标系
绕轴矢量
转动角度
将
转至F
[l];
在
下的坐标矢量与
在F
[l]下的坐标矢量
恒等,即有
由上式知,
或
不依赖于相邻的坐标系
及F
[l];故称
或
为轴不变量。在不强调不变性时,可以称之为坐标轴矢量(简称轴矢量)。
或
表征的是体
与体l共有的参考单位坐标矢量,与参考点
及O
l无关。体
与体l即为杆件或轴。
轴不变量与坐标轴具有本质区别:
(1)坐标轴是具有零位及单位刻度的参考方向,可以描述沿该方向平动的位置,但不能完整描述绕该方向的转动角度,因为坐标轴自身不具有径向参考方向,即不存在表征转动的零位。在实际应用时,需要补充该轴的径向参考。例如:在笛卡尔系F
[l]中,绕lx转动,需以ly或lz为参考零位。坐标轴自身是1D的,3个正交的1D参考轴构成3D的笛卡尔标架。
(2)轴不变量是3D的空间单位参考轴,其自身就是一个标架。其自身具有径向参考轴,即参考零位。空间坐标轴及其自身的径向参考轴可以确定笛卡尔标架。空间坐标轴可以反映运动轴及测量轴的三个基本参考属性。
【1】给定旋转变换阵
因其是实矩阵,其模是单位的,故其有一个实特征值λ
1及两个互为共轭的复特征值λ
2=e
iφ及λ
3=e
-iφ;其中:i为纯虚数。因此,|λ
1|·||λ
2||·||λ
3||=1,得λ
1=1。轴矢量
是实特征值λ
1=1对应的特征矢量,是不变量;
【2】是3D参考轴,不仅具有轴向参考方向,而且具有径向参考零位,将在3.3.1节予以阐述。
对轴不变量而言,其绝对导数就是其相对导数。因轴不变量是具有不变性的自然参考轴,故其绝对导数恒为零矢量。因此,轴不变量具有对时间微分的不变性。有:
因基矢量e
l是与F
[l]固结的任一矢量,基矢量
是与
固结的任一矢量,又
是F
[l]及
共有的单位矢量,故
是F
[l]及
共有的基矢量。因此,轴不变量
是F
[l]及
共有的参考基。轴不变量是参数化的自然坐标基,是多轴系统的基元。固定轴不变量的平动与转动与其固结的坐标系的平动与转动等价。
因此,应用自然坐标系统,当系统处于零位时,只需确定一个公共的参考系,而不必为系统中每一杆件确定各自的体坐标系,因为它们由轴不变量及自然坐标唯一确定。当进行系统分析时,除底座系外,与杆件固结的其它自然坐标系只发生在概念上,而与实际的测量无关。自然坐标系统对于多轴系统(MAS)理论分析及工程作用在于:
(1)系统的结构参数测量需要以统一的参考系测量;否则,不仅工程测量过程烦琐,而且引入不同的体系会引入更大的测量误差。
(2)应用自然坐标系统,除根杆件外,其它杆件的自然坐标系统由结构参量及关节变量自然确定,有助于MAS系统的运动学与动力学分析。
(3)在工程上,可以应用激光跟踪仪等光学测量设备,实现对固定轴不变量的精确测量。
(4)由于运动副R及P、螺旋副H、接触副O是圆柱副C的特例,可以应用圆柱副简化MAS运动学及动力学分析。
定义3不变量:称不依赖于一组坐标系进行度量的量为不变量。
定义6自然坐标:以自然坐标轴矢量为参考方向,相对系统零位的角位置或线位置,记为q
l,称为自然坐标;称与自然坐标一一映射的量为关节变量;其中:
定义9关节空间:以关节自然坐标q
l表示的空间称为关节空间。
定义10位形空间:称表达位置及姿态(简称位姿)的笛卡尔空间为位形空间,是双矢量空间或6D空间。
定义12自然坐标轴空间:以固定轴不变量作为自然参考轴,以对应的自然坐标表示的空间称为自然坐标轴空间,简称自然轴空间。它是具有1个自由度的3D空间。
如图2所示,
及
不因杆件Ω
l的运动而改变,是不变的结构参考量。
确定了轴l相对于轴
的五个结构参数;与关节变量q
l一起,完整地表达了杆件Ω
l的6D位形。给定
时,杆件固结的自然坐标系可由结构参数
及关节变量
唯一确定。称轴不变量
固定轴不变量
关节变量
及
为自然不变量。显然,由固定轴不变量
及关节变量
构成的关节自然不变量
与由坐标系
至F
[l]确定的空间位形
具有一一映射关系,即
给定多轴系统D={T,A,B,K,F,NT},在系统零位时,只要建立底座系或惯性系,以及各轴上的参考点O
l,其它杆件坐标系也自然确定。本质上,只需要确定底座系或惯性系。
给定一个由运动副连接的具有闭链的结构简图,可以选定回路中任一个运动副,将组成该运动副的定子与动子分割开来;从而,获得一个无回路的树型结构,称之为Span树。T表示带方向的span树,以描述树链运动的拓扑关系。
描述运动链的基本拓扑符号及操作是构成运动链拓扑符号系统的基础,定义如下:
【1】运动链由偏序集合(]标识。
【2】A
[l]为取轴序列A的成员;因轴名l具有唯一的编号对应于A
[l]的序号,故A
[l]计算复杂度为O(1)。
【3】
为取轴l的父轴;轴
的计算复杂度为O(1)。计算复杂度O()表示计算过程的操作次数,通常指浮点乘与加的次数。以浮点乘与加的次数表达计算复杂度非常烦琐,故常采用算法循环过程中的主要操作次数;比如:关节位姿、速度、加速度等操作的次数。
定义以下表达式或表达形式:
本申请中约定:在运动链符号演算系统中,具有偏序的属性变量或常量,在名称上包含表示偏序的指标;要么包含左上角及右下角指标,要么包含右上角及右下角指标;它们的方向总是由左上角指标至右下角指标,或由右上角指标至右下角指标,本申请中为叙述简便,有时省略方向的描述,即使省略,本领域技术人员通过符号表达式也可以知道,本申请中采用的各参数,对于某种属性符,它们的方向总是由偏序指标的左上角指标至右下角指标,或由右上角指标至右下角指标。例如:
可简述为(表示由k至l)平动矢量;
表示(由k至l的)线位置;
kr
l表示(由k至l的)平动矢量;其中:r表示“平动”属性符,其余属性符对应为:属性符φ表示“转动”;属性符Q表示“旋转变换矩阵”;属性符l表示“运动链”;属性符u表示“单位矢量”;属性符w表示“角速度”;角标为i表示惯性坐标系或大地坐标系;其他角标可以为其他字母,也可以为数字。
本申请的符号规范与约定是根据运动链的偏序性、链节是运动链的基本单位这两个原则 确定的,反映了运动链的本质特征。链指标表示的是连接关系,右上指标表征参考系。采用这种符号表达简洁、准确,便于交流与书面表达。同时,它们是结构化的符号系统,包含了组成各属性量的要素及关系,便于计算机处理,为计算机自动建模奠定基础。指标的含义需要通过属性符的背景即上下文进行理解;比如:若属性符是平动类型的,则左上角指标表示坐标系的原点及方向;若属性符是转动类型的,则左上角指标表示坐标系的方向。
(1)l
S-杆件l中的点S;而S表示空间中的一点S。
(7)左下角指标为0时,表示机械零位;如:
(8)0-三维零矩阵;1-三维单位矩阵;
(9)约定:“\”表示续行符;“□”表示属性占位;则
(12)
iQ
l,相对绝对空间的旋转变换阵;
(14)对于一给定有序的集合r=[1,4,3,2]
T,记r
[x]表示取集合r的第x行元素。常记[x]、[y]、[z]及[w]表示取第1、2、3及4列元素。
(15)
il
j表示由i到j的运动链;
ll
k为取由轴l至轴k的运动链;
分块矩阵的高维行列式计算:
其中:I[i1,…in]表示排列<i1,…in>的逆序个数。式(1)计算复杂度为:n!次n个数积及n!次加法,具有指数计算复杂度,只能适用于维度较小的行列式。对于维度较大的行列式,通常应用Laplace公式进行递规运算,记
为
的伴随矩阵(Adjugate Matrix),则有
更简单的算法通常应用高斯消去法或LU分解法,先通过初等变换将矩阵变为三角阵或三角阵的乘积,后计算行列式。上述针对数域的行列式计算方法不适用于高维度的多项式矩阵,需要引入分块矩阵的行列式计算方法。计算矢量多项式(Vector Polynomial)的行列式是一个特定的分块矩阵行列式的计算问题,它在矢量层次上表达了矢量与行列式的内在联系。而分块矩阵行列式计算则从矩阵层次上表达分块矩阵与行列式的内在规律。
则有
式(3)及式(4)可以推广至n维空间。
实施例1
另一方面,
上面的结果验证了式(4)的正确性。
给出分块矩阵的行列式计算定理:
若记大小为(n+m)·(n+m)的方阵为M,大小为n·n的矩阵
是方阵M的前n行及任意n列元素构成的子矩阵,大小为m·m的矩阵
是方阵M后m行及剩余m列元素构成的子矩阵;由升序排列的矩阵列序号构成的序列cn及cm是序列[1:m+n]的子集,[cn,cm]∈<1:n+m>,且有cm∪cn=[1:m+n];则方阵M行列式与分块矩阵
及
的行列式关系为
对行列式进行行阶梯化计算原理:
对于S×S矩阵,其每一项是关于τ
1的n阶多项式。计算该矩阵的行列式时,可通过初等行变换将原行列式变为上三角行列式,再将非零的对角线元素相乘,得到行列式的多项式表达式。该式为0,得到τ
1的所有解。
行阶梯化的具体方法为,先对行列式第一列的最高阶次由高到低进行排序,再进行最多(S-1)×n次初等行变换消元,得到第一列只有第一个元素不为0的行列式。再对该行列式第1行及1列的余子式进行初等行变换消元,依次迭代求解。
实施例2
步骤为:rk代表第k行。得
基于“N进位字”的N阶多项式系统:
n个“n元N阶”多项式系统的Dixon多项式:
引入辅助变量[y
2,y
3,…,y
n],且有
在多元多重多项式(8)中,用辅助变量Y
m的前m个依次替换原变量(Original Variables)X
n中的m个变量,记“|”为替换操作符,得到增广的(Extended)多项式
由式(6)及式(12)得
其中:
由式(17)得
式(15)中分离变量与文献不同:原变量X
n-1被辅助变量Y
n-1替换的次序不同,Dixon多项式也不同。考虑式(13)及式(18)得该多项式的Dixon行列式
在笛卡尔空间下,由位置矢量或转动矢量构成的行列式表示矢量张成空间的容积(Volume);在不同笛卡尔空间下具有容积的不变性。其中:
n个“n元N阶”多项式的Dixon行列式的阶次及替换变量项数分别为:
n个“n元N阶”Dixon矩阵:
给定n个“n元N阶”多项式系统F
n(Y
n-1|X
n-1),n≥2;存在与消去变量x
2,…,x
n无关的Dixon矩阵
SΘ
S(x
1),其Dixon多项式
表示为分离变量
及
的双重线性型:
其中:
若
则有
Det(
SΘ
S(x
1))=0。 (28)
称式(28)中“n个n元”为Dixon消元的必要条件,从而获得可行解。若
SΘ
S存在零行或零列向量,则无法建立x
1的多项式方程;此时,通过除标量积之外的初等变换,将
SΘ
S变为行阶梯(Row Echelon)矩阵Ech(
SΘ
S);在计算该矩阵的杻轴(Pivot)的积之后得方阵
即在
SΘ
S中选取S′个独立的列向量。
确定双线性型
称其为结式或消去式。式(32)是单变量x
1的多项式方程;消去了n-1个未知量;从而,可以获得单变量x
1的可行解。若x
1同时满足
即有
若有必要条件
若同时满足
①确定系统结构。方程数及独立变量数记为n;独立变量记为X
n;多项式复合变量记为
替换变量记为
替换变量数为n-1;大小为S·S的Dixon矩阵记为
其成员系数如式(24)所示,其中:S由式(32)确定;待消去变量为x
1。
④Dixon矩阵成员如式(32)所示,由式(32)计算Dixon矩阵
SΘ
S的(n+1)·S
2个系数。
⑤当满足式(37)及式(38)直接解判别准则时,由式(34)及式(35)得全部数值解。
实施例3
对多项式系统(39)进行Dixon消元。
步骤为:该式是4个“4元1阶”多项式系统,满足Dixon消元条件。由式(19)及式(22),得
其中:
由式(34)及式(40)得5个解:
解得:τ
3=1,τ
4=-2。将
τ
3及τ
4代入式(39)得τ
2=1。同样,可得其他三组解。显然,因变量不满足式(26),式(40)所示的Dixon矩阵不对称。该例表明Dixon行列式为零对于多重线性多项式系统是充分的。
基于轴不变量的定轴转动
轴矢量
相对于杆件
及Ω
l或自然坐标系
及F
[l]是固定不变的,故称该转动为定轴转动。单位矢量
绕轴
转动
后,转动后的零位矢量
对系统零位轴
的投影矢量为
转动后的零位矢量
对系统径向轴
的矩矢量为
轴向分量为
故得具有链指标的Rodrigues矢量方程
若
由式(42),得
若
即坐标系
与F
[l]的方向一致,由 式(42)可知:反对称部分
必有
因此,系统零位是自然坐标系
与F
[l]重合的充分必要条件,即初始时刻的自然坐标系方向一致是系统零位定义的前提条件。利用自然坐标系可以很方便地分析多轴系统运动学和动力学。
式(43)是关于
和
的多重线性方程,是轴不变量
的二阶多项式。给定自然零位矢量
作为
的零位参考,则
及
分别表示零位矢量及径向矢量。式(43)即为
对称部分
表示零位轴张量,反对称部分
表示径向轴张量,分别与轴向外积张量
正交,从而确定三维自然轴空间;式(43)仅含一个正弦及余弦运算、6个积运算及6个和运算,计算复杂度低;同时,通过轴不变量
及关节变量
实现了坐标系及极性的参数化。
称(45)为改进的Cayley变换。即有
由式(46)得规范的位置方程
“居-吉布斯”四元数的确定:
其中:
其中:
由式(52)得
习惯上,单关节及运动链的期望姿态以规范的Ju-Gibbs四元数(简称规范Ju-Gibbs四元数,即“标部”为1的四元数)表示;但是它们积运算通常是不规范的,即其标部不为1。由式(53)可知:只有给定轴l及
的规范Ju-Gibbs四元数,且两轴正交,
才为规范四元数。
由式(53)得
由四维复数性质得
由式(48)至式(50)及式(55)得
由式(50)、式(54)及式(57)得
类DCM及性质:
其中:
显然,类DCM可以通过Ju-Gibbs四元数表达。因此,式(59)姿态方程及式(47)位置方程是关于Ju-Gibbs四元数的表达式。
分块方阵的逆:
若给定可逆方阵K、B及C,其中B及C分别为l×l、c×c的方阵;A、D分别为l×c、c×l的矩阵,且
则有
基于轴不变量的Dixon行列式计算原理:
下面基于轴不变量,提出径向不变量及运动链的Dixon行列式基本性质,为基于轴不变量的机器人逆运动学分析奠定基础。
【1】轴不变量
首先,轴不变量与坐标轴具有本质区别:坐标轴是具有零位及单位刻度的参考方向,可以描述沿轴向平动的线位置,但不能完整描述绕轴向的角位置,因为坐标轴自身不具有径向 参考方向,即不存在表征转动的零位。在实际应用时,需要补充坐标轴的径向参考。坐标轴自身是1D的,3个正交的坐标轴构成3D的笛卡尔标架;轴不变量是3D空间单位参考轴(简称3D参考轴),具有径向参考零位。“3D参考轴”及其径向参考零位可以确定对应的笛卡尔系。以自然坐标系为基础的轴不变量可以准确地反映运动轴及测量轴的“共轴性”、“极性”与“零位”三个基本属性。
其次,轴不变量与欧拉轴具有本质的区别:方向余弦矩阵(DCM)是实矩阵,轴矢量是DCM的特征值1对应的特征矢量,是不变量;固定轴不变量是“3D参考轴”,不仅具有原点及轴向,也有径向参考零位;在自然坐标系下,轴不变量不依赖于相邻固结的自然坐标系,即在相邻固结的自然坐标系下具有不变的自然坐标;轴不变量具有幂零特性等优良的数学操作功能;在自然坐标系统中,通过轴不变量及关节坐标,可以唯一确定DCM及参考极性;没有必要为每一个杆件建立各自的体系,可以极大地简化建模的工作量。
同时,以唯一需要定义的笛卡尔直角坐标系为参考,测量轴不变量,可以提高结构参数的测量精度。基于轴不变量的优良操作及属性,可以建立包含拓扑结构、坐标系、极性、结构参量及动力学参量的迭代式的运动学及动力学方程。
由式(59)及式(47)可知:多轴系统的姿态及位置方程本质上是多元二阶多项式方程,其逆解本质上归结于多元二阶多项式的消元问题,包含Dixon矩阵及Dixon行列式计算的两个子问题。用式(47)的表达3R机械臂位置方程,是3个“3元2阶”多项式,应用Dixon消元方法计算逆解,有两个替换变量,在计算8×8的Dixon行列式时,最大可能的阶次为16。由式(4)可知:行列式计算是一个排列过程,面临着“组合爆炸”的难题。
所有的不在确定的多项式时间内可解的问题称为NP问题。非确定性算法将问题分解为“猜测”与“验证”两个阶段:算法的“猜测”阶段具有非确定性,算法的“验证”阶段具有确定性,通过验证来确定猜测的解是否正确。假如可以在多项式时间内计算出来,就称为多项式非确定性问题。多元多项式的消元通常被认为是NP问题。通常应用
基进行多元多项式的消元,不得不求助于启发式的“猜测”与“验证”来解决问题。
【2】径向不变量
任一个矢量可以分解为零位矢量及轴向矢量,故有
其中:
显然,
是轴l及
的公垂线或公共径向矢量,
是轴l的轴向矢量。由式(65)可知:任一个结构参数矢量
可分解为与坐标系为无关的零位不变量
及轴向不变量
它们的径向矢量记为
结构参数矢量
及轴不变量
唯一确定径向坐标系,具有2个独立维度。若两个轴向不变量
及
共线,则记为
因此,称式(66)所示的轴向不变量及零位不变量是结构参数矢量对自然轴的分解。
由式(69)及式(70)可知:同一个轴的三个径向矢量的行列式为零;同一个轴的任意两个轴向矢量的行列式为零。可以用轴不变量及其导出的不变量来简化Dixon行列式计算。
由轴不变量导出的零位矢量、径向矢量及轴向矢量具有以下关系:
称式(71)为零位矢量的反转公式;称式(72)为零位矢量与径向矢量的互换公式;称式(73)为径向矢量不变性公式。由式(65)、式(71)至式(73)得
由式(74)得
【3】运动链Dixon行列式性质
定义
由式(52)得
其中:
由式(62)及式(66)得
由式(79)证得
由式(80)及式(83)得
由式(80)及式(84)得
基于轴不变量的Cayley变换
由式(86)得
定义
故有
基于轴不变量的3R机械臂位置逆解方法
【1】根据机械臂n元3D矢量位姿方程,获得n个“n元2阶”多项式方程。
由式(47)得3R运动学方程
由式(90)得
由式(91)得
若记
则由式(61)及式得(93)
由式(92)及式(93)得
下面,阐述3R机械臂运动学方程的Dixon行列式的结构模型及特点。
由式(95)得3R运动学多项式方程
多项式系统F
3(Y
2|T
2),根据双线性型行列式通式
则有
其中:
由式(18)、式(95)及式(96)得
由式(22)及式(101)可知式(99)成立。由式(80)及式(93)得
由式(93)、式(102)及式(103)得
显然,式(104)中的y
2阶次β2∈[0:3]及y
3阶次β3∈[0:1]。考虑式(101)后三项:
中的y
2阶次β2∈[0:3]及y
3阶次β3∈[0:1];
中的y
2阶次β2∈[0:2]及y
3阶次β3∈[0:1];
中的y
2的阶次β2∈[0:3]及y
3的阶次β3∈[0:1]。由上可知:式(101)中的y
2阶次β2∈[0:3]及y
3的阶次β3∈[0:1]。故有S=8。
【2】应用“基于轴不变量的Dixon行列式计算”方法,“分块矩阵的高维行列式计算”方法或者“对行列式进行行阶梯化计算”方法简化行列式计算。
根据运动链Dixon行列式性质,由式(80)及式(93)得
由式(105)得
由式(106)得
由式(107)得
由式(101)得
将式(108)至式(110)代入式(111)得
【3】应用n个“n元N阶”多项式的Dixon消元与求解原理完成位姿逆解计算,其中:根据Dixon矩阵的行列式为0,得到一元高阶多项式方程,应用基于友阵的一元高阶多项式方程求解一元高阶多项式方程的解。
一元n阶多项式p(x)=a
0+a
1x+…a
n-1x
n-1+x
n具有n个解。若能找到一个矩阵A,满足|A-λ
l·1
n|·v
l=0,其中:l∈[1:n],λ
l为该矩阵的特征值,v
l为对应的特征矢量。若矩阵A的特征方程为
则称该矩阵为多项式p(x)的友矩阵(Companion Matrix,简称友阵),因此,多项式方程p(λ
l)=0的解为其友阵A的特征方程|A-λ
l·1
n|=0的解。
若多项式p(x)的友阵为
则由矩阵A的特征向量构成的矩阵为范德蒙德(Vandermonde)矩阵为
且有
p(λ
l)=|A-λ
l·1
n|=0。 (115)
由式(28)、式(99)及式(100)得
因S=8,应用式(1)计算
的复杂度为8·8!=322560;而应用式(5)进行二次分块的行列式计算,其中:2·2分块矩阵计算复杂度为4!(2·2!+2·2!+1)/(2!2!)=30,4·4分块分矩阵计算复杂度为8!(30+30+1)/(4!4!)=4270。一般情况下,式(116)是关于τ
1的16阶单项式方程。
该方法的过程表明:整体与局部、复杂与简单是对立统一的;式(4)将矢量多项式的行列式计算转化为三个矢量的行列式,这一步骤起到了决定性的作用;轴不变量及其导出的不变量都是结构参量,系统方程是关于结构参数的矢量与关节变量(标量)的矢量代数方程。
以上所述仅是本发明的优选实施方式,应当指出,对于本技术领域的普通技术人员来说,在不脱离本发明技术原理的前提下,还可以做出若干改进和变形,这些改进和变形也应视为本发明的保护范围。
Claims (8)
- 一种基于轴不变量的通用3R机械臂逆解建模与解算方法,其特征是,应用n个“n元N阶”多项式的Dixon消元与求解原理,进行位姿逆解计算,主要包括以下步骤:【1】根据机械臂n元3D矢量位姿方程,获得n个“n元2阶”多项式方程;【2】应用基于轴不变量的Dixon行列式计算式、分块矩阵的行列式计算式或对行列式进行行阶梯化计算式简化行列式计算;【3】应用n个“n元N阶”多项式的Dixon消元与求解原理完成位姿逆解计算,其中:根据Dixon矩阵的行列式为0,得到一元高阶多项式方程,应用基于友阵的一元高阶多项式方程求解一元高阶多项式方程的解。
- 根据权利要求3所述的基于轴不变量的通用3R机械臂逆解建模与解算方法,其特征是,步骤【2】中,基于轴不变量的Dixon行列式计算式为:根据运动链Dixon行列式性质有:并记:由式(47)得3R运动学方程由式(90)得由(91)式得记则由式(51)及式得(93)由式(92)及式(93)得由式(95)得3R运动学多项式方程多项式系统F 3(Y 2|T 2),根据双线性型行列式通式则有其中:由式(80)及式(93)得得简化的3元N阶Dixon行列式为
- 根据权利要求4所述的基于轴不变量的通用3R机械臂逆解建模与解算方法,其特征是,步骤【2】中,对行列式进行行阶梯化计算原理:对于S×S矩阵,其每一项是关于τ 1的n阶多项式;计算该矩阵的行列式时,可通过初等行变换将原行列式变为上三角行列式,再将非零的对角线元素相乘,得到行列式的多项式表达式;该式为0,得到τ 1的所有解;行阶梯化的具体方法为,先对行列式第一列的最高阶次由高到低进行排序,再进行最多(S-1)×n次初等行变换消元,得到第一列只有第一个元素不为0的行列式;再对该行列式第1行及1列的余子式进行初等行变换消元,依次迭代求解。
- 根据权利要求3所述的基于轴不变量的通用3R机械臂逆解建模与解算方法,其特征是,步骤【3】中,n个“n元N阶”多项式系统的Dixon多项式构建步骤为:引入辅助变量[y 2,y 3,…,y n],且有得其中:由式(17)得考虑式(13)及式(18)得该多项式的Dixon行列式在笛卡尔空间下,由位置矢量或转动矢量构成的行列式表示矢量张成空间的容积(Volume);在不同笛卡尔空间下具有容积的不变性;其中:给定n个“n元N阶”多项式系统F n(Y n-1|X n-1),n≥2;存在与消去变量x 2,…,x n无关的Dixon矩阵 SΘ S(x 1),其Dixon多项式 表示为分离变量 及 的双重线性型:α[l]∈[0,N·(n-l+1)-1],l∈[2:n]; (23)其中:称式(28)中“n个n元”为Dixon消元的必要条件,从而获得可行解。
Applications Claiming Priority (2)
Application Number | Priority Date | Filing Date | Title |
---|---|---|---|
CN201810933468.4 | 2018-08-16 | ||
CN201810933468.4A CN108959828B (zh) | 2018-08-16 | 2018-08-16 | 基于轴不变量的通用3r机械臂逆解建模与解算方法 |
Publications (1)
Publication Number | Publication Date |
---|---|
WO2020034407A1 true WO2020034407A1 (zh) | 2020-02-20 |
Family
ID=64469134
Family Applications (1)
Application Number | Title | Priority Date | Filing Date |
---|---|---|---|
PCT/CN2018/112705 WO2020034407A1 (zh) | 2018-08-16 | 2018-10-30 | 基于轴不变量的通用3r机械臂逆解建模与解算方法 |
Country Status (2)
Country | Link |
---|---|
CN (1) | CN108959828B (zh) |
WO (1) | WO2020034407A1 (zh) |
Families Citing this family (1)
Publication number | Priority date | Publication date | Assignee | Title |
---|---|---|---|---|
CN111967099B (zh) * | 2020-07-20 | 2021-04-27 | 居鹤华 | 多自由度机械臂矢量多项式系统最优求解方法 |
Citations (3)
Publication number | Priority date | Publication date | Assignee | Title |
---|---|---|---|---|
CN101648376A (zh) * | 2009-09-11 | 2010-02-17 | 北京理工大学 | 控制机器人操作的方法和装置 |
US20120072022A1 (en) * | 2010-09-20 | 2012-03-22 | Samsung Electronics Co., Ltd. | Robot and control method thereof |
CN104772773A (zh) * | 2015-05-08 | 2015-07-15 | 首都师范大学 | 一种机械臂运动学形式化分析方法 |
Family Cites Families (3)
Publication number | Priority date | Publication date | Assignee | Title |
---|---|---|---|---|
CN101244561A (zh) * | 2008-03-21 | 2008-08-20 | 哈尔滨工程大学 | 蒸汽发生器检修机械臂安装过程运动学逆解方法 |
CN103010491B (zh) * | 2012-11-30 | 2015-04-22 | 北京控制工程研究所 | 一种气浮台抓捕试验机械臂控制方法 |
CN106055810A (zh) * | 2016-06-07 | 2016-10-26 | 中国人民解放军国防科学技术大学 | 用于在轨快速抓捕的姿轨臂一体化运动规划方法 |
-
2018
- 2018-08-16 CN CN201810933468.4A patent/CN108959828B/zh active Active
- 2018-10-30 WO PCT/CN2018/112705 patent/WO2020034407A1/zh active Application Filing
Patent Citations (3)
Publication number | Priority date | Publication date | Assignee | Title |
---|---|---|---|---|
CN101648376A (zh) * | 2009-09-11 | 2010-02-17 | 北京理工大学 | 控制机器人操作的方法和装置 |
US20120072022A1 (en) * | 2010-09-20 | 2012-03-22 | Samsung Electronics Co., Ltd. | Robot and control method thereof |
CN104772773A (zh) * | 2015-05-08 | 2015-07-15 | 首都师范大学 | 一种机械臂运动学形式化分析方法 |
Non-Patent Citations (1)
Title |
---|
NI, ZHENSONG ET AL.: "General 6R Robot Inverse Solution Algorithm based on Quaternion Matrix and a Groebner Base", JOURNAL OF TSINGHUA UNIVERSITY - SCIENCE AND TECHNOLOGY, vol. 53, no. 5, 31 May 2013 (2013-05-31), pages 683 - 687, XP055685712, DOI: 10.4236/alamt.2018.81004 * |
Also Published As
Publication number | Publication date |
---|---|
CN108959828B (zh) | 2019-12-06 |
CN108959828A (zh) | 2018-12-07 |
Similar Documents
Publication | Publication Date | Title |
---|---|---|
CN108972558B (zh) | 一种基于轴不变量的多轴机器人动力学建模方法 | |
CN108942942B (zh) | 一种基于轴不变量的多轴机器人逆运动学建模与解算方法 | |
CN108983705B (zh) | 一种基于轴不变量的多轴机器人系统正运动学建模与解算方法 | |
Kukelova et al. | A clever elimination strategy for efficient minimal solvers | |
WO2020034421A1 (zh) | 一种基于轴不变量的多轴机器人系统建模与控制方法 | |
Orekhov et al. | Solving cosserat rod models via collocation and the magnus expansion | |
Wise et al. | Certifiably optimal monocular hand-eye calibration | |
WO2020034415A1 (zh) | 基于轴不变量的通用6r机械臂逆解建模与解算方法 | |
WO2020034407A1 (zh) | 基于轴不变量的通用3r机械臂逆解建模与解算方法 | |
WO2020034416A1 (zh) | 基于轴不变量的通用7r机械臂逆解建模与解算方法 | |
WO2020034417A1 (zh) | 基于轴不变量多轴机器人d-h系及d-h参数确定方法 | |
CN108942943B (zh) | 基于轴不变量的多轴机器人正运动学计算方法 | |
WO2020034418A1 (zh) | 基于轴不变量及dh参数1r/2r/3r逆解建模方法 | |
WO2020034405A1 (zh) | 基于轴不变量的树链机器人动力学建模与解算方法 | |
WO2020034402A1 (zh) | 基于轴不变量的多轴机器人结构参数精测方法 | |
CN108959829B (zh) | 基于轴不变量的非理想关节机器人动力学建模方法 | |
Stölzle et al. | Modelling handed shearing auxetics: Selective piecewise constant strain kinematics and dynamic simulation | |
CN111967099B (zh) | 多自由度机械臂矢量多项式系统最优求解方法 | |
CN109086544B (zh) | 基于轴不变量的闭链机器人动力学建模与解算方法 | |
Bobev et al. | Wrapped M5-branes and AdS5 black holes | |
CN108803350B (zh) | 基于轴不变量的动基座多轴机器人动力学建模与解算方法 | |
Daney et al. | Calibration of parallel robots: on the Elimination of Pose–Dependent Parameters | |
Boiadjiev et al. | A Novel, Oriented to Graphs Model of Robot Arm Dynamics. Robotics 2021, 10, 128 | |
Bayro-Corrochano et al. | Analysis and computation of the intrinsic camera parameters | |
Bandyopadhyay et al. | Analytical Determination of Principal Twists and Singular Directions in Robot Manipulators |
Legal Events
Date | Code | Title | Description |
---|---|---|---|
NENP | Non-entry into the national phase |
Ref country code: DE |
|
121 | Ep: the epo has been informed by wipo that ep was designated in this application |
Ref document number: 18930160 Country of ref document: EP Kind code of ref document: A1 |
|
122 | Ep: pct application non-entry in european phase |
Ref document number: 18930160 Country of ref document: EP Kind code of ref document: A1 |