WO2020034403A1 - 基于轴不变量的多轴机器人正运动学计算方法 - Google Patents
基于轴不变量的多轴机器人正运动学计算方法 Download PDFInfo
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- B—PERFORMING OPERATIONS; TRANSPORTING
- B25—HAND TOOLS; PORTABLE POWER-DRIVEN TOOLS; MANIPULATORS
- B25J—MANIPULATORS; CHAMBERS PROVIDED WITH MANIPULATION DEVICES
- B25J9/00—Programme-controlled manipulators
- B25J9/16—Programme controls
- B25J9/1602—Programme controls characterised by the control system, structure, architecture
- B25J9/1607—Calculation of inertia, jacobian matrixes and inverses
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- the invention relates to a forward kinematics calculation method of a multi-axis robot, and belongs to the technical field of robots.
- Robots are a very hot area right now. A lot of scientific and engineering manpower has been invested in this field over the past few decades, and it has been studied for many years. However, once the number of axes and degrees of freedom are increased to a certain number, according to the existing textbooks and known methods of observation, modeling, calculation and control, they often fall into complex and out-of-control problems, and even cannot be solved.
- the technical problem to be solved by the present invention is to provide a forward kinematics calculation method of a multi-axis robot based on an axis invariant.
- the present invention adopts the following technical solutions:
- the multi-axis robot device includes a sequence of rods and a sequence of joints, and converts a sequence of joints in a tree chain into a corresponding sequence of axes and a sequence of parent axes thereof, and the axes of the sequence of axes are translation axes or rotation axes;
- the invariance of the axis invariant is used to establish an iterative kinematic equation based on the axis invariant, and the symbol of the iterative kinematic equation corresponds to a pseudo code, which reflects the topological relationship and the chain order relationship of the kinematic chain of the multi-axis machine device ;
- the inertial space is denoted by i, given the kinematic chain i l n from i to the rod n , the rod l, n, j ⁇ A , n> l, s is any point on the body l, and A is the sequence of axes; when Rotation vector, the iterative forward kinematics calculation steps of the motion chain i l n include:
- the Euler quaternion is used to represent fixed-axis rotation; the calculation of the rotation transformation matrix is equivalent to the matrix calculation of the chain quaternion, and the rotation transformation matrix sequence ⁇ i Q j
- Steps for iterative deceleration calculation based on axis invariants include:
- k is a member belonging to the kinematic chain i l n ;
- the method of the present invention proposes and proves an iterative real-time numerical modeling method based on axis invariant, including: iterative calculation of position vector, rotation vector, velocity vector, acceleration vector, and partial velocity vector based on axis invariant method. It has a concise motion chain symbol system, has the function of pseudo code, and has an iterative structure, which ensures the reliability of the system and mechanized calculations.
- Figure 3 is a schematic diagram of the meaning of the deflection speed.
- 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, l L is a subtree without l; recursive execution of l l, the computational complexity is
- ⁇ 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 of 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.
- the kinematic chain is a partially ordered chain; Both means The connection to the rod l, which also means from the rod l to the rod Connection Have total order;
- total order and partial order are attributes of an object itself.
- the corresponding symbol system has not yet appeared in mechanics and robot theory.
- i l i an empty or ordinary chain.
- the inertial space (environment) is denoted by i, and the ordinary chain i l i always exists.
- the reference index in the upper left corner of the four-dimensional complex number only indicates the relationship of motion, and the meaning of the projection reference system has been lost.
- the reference index is meaningless in a 4D complex number, it does not indicate that the index relationship is meaningless, because the multiplication and division of complex numbers is closely related to the order of action of complex numbers.
- Equation (14) is called a quaternion concatenation operation, which corresponds to a homogeneous transformation. Therefore, the sequence pose operation has concatenation of motion chains. Similar to the vector cross product operation, the quaternion product can be replaced by the corresponding conjugate matrix.
- Equation (18) contains only 16 multiplication operations and 12 addition operations. and Need to perform 27 multiplications and 18 additions.
- Is a 4 ⁇ 4 matrix which is composed as follows: the fourth column is a right-handed quaternion The fourth line is the left-handed quaternion which is Top left 3 ⁇ 3 contains among them: The upper right triangle is a vector of right hand order The lower left triangle is a left-handed vector which is The main diagonal is The 4th element.
- Equation (14) represents a position vector rotation operator, that is, rotation. Therefore, the Euler quaternion product operation corresponds to the product operation of the rotation transform matrix. Therefore, a rotation transformation chain is equivalent to a fixed-axis rotation chain, that is,
- the Euler quaternion can uniquely determine the rotation transformation matrix; the rotation transformation matrix can also uniquely determine the Euler quaternion, that is, Euler quaternion is equivalent to the rotation transformation matrix.
- the rotation vector has a one-to-one correspondence with the standard quaternion, that is, the quaternion represents the rotation of the fixed axis; the calculation of the rotation transformation matrix is equivalent to the matrix calculation of the chain quaternion.
- Equation (21) 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 (21) 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; Eq. (21) contains only one sine and cosine operation, six product operations, and six sum operations, and the calculation complexity is low; And joint variables The coordinate system and polarity are parameterized.
- ⁇ ′ means to find absolute derivative of projection coordinate system i; angular velocity Axis vector Equation (24) shows that absolute angular velocity and relative angular velocity are equivalent.
- Equation (28) the "absolute derivative of positive order" and the implication term Is the angular velocity cross-multiplication matrix from the projection reference system i to the measurement reference system l; The results are based on the projection coordinate system i, and the projection reference systems of all sum terms are consistent.
- the calculation method of the Jacobian matrix is usually used in the prior art, but the conclusions are not proved and the conclusions are not comprehensive.
- the Jacobian matrix In kinematics and dynamics analysis, it is more appropriate to refer to the Jacobian matrix as the deflection velocity.
- the Jacobian matrix generally refers to partial derivatives, it is not necessarily additive; in kinematics and dynamics, partial velocity refers to the partial derivative of the vector on joint variables, which is additive.
- the partial velocity is a transformation matrix corresponding to the velocity, and is a vector projection of a unit direction vector.
- yaw speed plays a key role. The calculation of yaw speed is the basic premise of dynamic system calculation.
- k is a member belonging to the kinematic chain i l l ;
- Equation (39) can be obtained from Equations (35), (33), (36), and (34).
- equation (40) is obtained. As i ⁇ n and versus with It has nothing to do with equation (41).
- Equations (35) to (38) have very important effects on kinematics and dynamics analysis. Not only do they have a clear physical meaning, they can also simplify the expression of kinematics and dynamics equations.
- the deflection velocity in equation (39) is the corresponding axis invariant
- equation (40) represents the first order of the position vector to the axis invariant, that is, the axis Vector With vector
- Equation (43) shows: Completed force Pair of axes The effect is the calculation of the moment.
- Equation (45) shows that for the axis invariant, 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.
- equation (47) the derivative of the bias velocity with respect to time t is still an iterative equation of the axis invariant.
- Axis invariant Is the coordinate vector of the basis e l , Essentially represents the projection of the basis e l onto the reference frame i. If formula (45) does not hold, the invariance of the reference basis e l as a reference is denied, that is, objectivity.
- the axis invariant of the MAS system is invariant to time, that is, the natural reference axis of the rigid body system is invariant. From equation (48), it can be known that the joint variables of the system are mapped one by one with the natural reference axis, and the number of joint variables of the body is determined by its independent motion dimension, but does not change the invariance of the natural reference axis to time differentiation.
- the derivative of a function's argument is called a derivative, and it is represented by d.
- the increment of the independent variable function is called variation, and it is expressed by ⁇ ; but variation does not consider the increment ⁇ t of time t, that is, ⁇ t ⁇ 0. It is precisely because the time increment ⁇ t is not taken into account that the variation of linear displacement and angular displacement is understood as the possible movement amount change at the same time t, that is, the virtual displacement.
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Abstract
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Claims (10)
- 一种基于轴不变量的多轴机器人正运动学计算方法,其特征是,多轴机器人装置包含杆件序列与关节序列,将树链中的关节序列转换成对应的轴序列及其父轴序列,所述轴序列的轴为平动轴或转动轴;使用轴集合来对应描述多轴机器装置,以自然坐标系为基础,利用轴集合的轴对应的轴不变量来计算多轴机器装置的控制参数;利用轴不变量的不变性建立基于轴不变量的迭代式运动学方程,并且所述迭代式运动学方程的符号对应到伪代码,反映所述多轴机器装置运动链的拓扑关系与链序关系;计算运动链的迭代式正运动学数值;计算基于轴不变量的迭代式偏速度。
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