机器人位姿运动学2017

1、KINEMATICS OF ROBOTS: POSITION ANALYSIS工业机器人是多自由度机构，必须知道每个关节变量才能确定机器人手机器人手的位置。Robot Reference FrameslWorld Reference FramelJoint Reference FramelTool Reference FramexyzWorld Reference FrameJoint Reference FramexyznoayznoaTool Reference Framex【机器人的参考坐标系机器人的参考坐标系】nRepresentation of a Point in SpacelA

2、point P in space can be represented by its three coordinates relative to a reference frame as: zxyaxczbyPxyzPabcijk【空间点的表示空间点的表示】1. 1. 机器人运动学的矩阵表示机器人运动学的矩阵表示nRepresentation of a Vector in SpaceA vector can be represented by three coordinates of its tail and its head. If the vector starts at point A

3、and ends at point B, then it can be represented by: ()()()ABxxyyzzBABABAPijkzxyaxczbyPxyzabcP【空间向量的表示空间向量的表示】-【矩阵矩阵】Application of a scale factorlMakes the matrix 4 by 1lAllows for introducing directional vectorsxyzPPPwP, yxxyPPabww为什么要引入比例因子？为什么要引入比例因子？nThe n-o-a Frame designationlaonApproach, Orie

4、ntation, Normal directionsyznoaxaonF,运动坐标系运动坐标系zyxF,全局参考坐标系全局参考坐标系zzzyyyxxxaonaonaonF【坐标系在参考坐标系原点的表示坐标系在参考坐标系原点的表示】方向余弦？方向余弦？nRepresentation of a Frame Relative to a Fixed Reference Framezyxnoap0001xxxxyyyyzzzznoapnoapFnoapaonF,运动坐标系zyxF,全局参考坐标系【坐标系在参考坐标系的表示坐标系在参考坐标系的表示】nRepresentation of a Rigid Bo

5、dy 0001xxxxyyyyobjectzzzznoapnoapFnoapzyxnoap【刚体的表示刚体的表示】1000zzzzyyyyxxxxpaonpaonpaonPaonTFrame representation Requirementslthe three unit vectors n, o, a are mutually perpendicularleach unit vectors length, represented by its directional cosines, must be equal to 1lThese constraints translate into

6、the following six constraint equations: (the dot-product of n and o vectors must be zero) (the magnitude of the length of the vector must be 1) and1n1o1a0on0an0oalThe same can be achieved by:n oa上式包含了正确的右手法则关系右手法则关系，所以一般使用这个等式判断3个向量之间的关系。Homogeneous Transformation Matricesl4 by 4 matrices:lCan be pr

7、e- or post-multipliedlEasy to find inverse of the matrixlRepresents both orientation and position information, including directional vectors0001xxxxyyyyzzzznoapnoapFnoap【齐次变换矩阵齐次变换矩阵】2. 2. 齐次齐次( (变换变换) )矩阵矩阵Representation of TransformationsA transformation may be in one of the following forms:lA pur

8、e translationlA pure rotation about an axislA combination of translations and/or rotations3. 3. 变换的表示变换的表示当空间的坐标系（向量、物体或运动坐标系）相对于固定的参考坐标系运动时，这一运动可以用类似于表示坐标系的方式来表示。nRepresentation of a Pure Translation zyxpnoanoad【纯平移变换的表示纯平移变换的表示】1000100010001zyxdddTnRepresentation of a Pure Translation zyxpnoanoad1

9、00010001000100010001xxxxxxxxxxyyyyyyyyyynewzzzzzzzzzzdnoapnoapddnoapnoapdFdnoapnoapdFnew = Trans (dx ,dy ,dz ) Fold 相对于固定坐标系，新坐标系位置可通过在原坐标系矩阵前面左乘左乘变换矩阵变换矩阵得到。nRepresentation of a Pure Rotation about an Axis yzoapl1l2l3l4pzpapopapopy1234cossinsincosxnyoazoapppllpppllpp1000cossin0sincosxnyozapppppp( ,

10、 )xyznoapRot xpcossin0sincos0001),Rot(x【绕轴纯旋转变换的表示绕轴纯旋转变换的表示】Rotation Matrices100( , )00Rot xCSSC0( , )0100CSRot ySC0( , )0001CSRot zSC1000010000cossin00sincos,10000cos0sin00100sin0cos,10000cossin00sincos00001,zRotyRotxRot齐次变换矩阵？齐次变换矩阵？nRepresentation of Combined Transformations lExample: lRotation

11、of degrees about the x-axis,lFollowed by a translation of l1,l2,l3 (relative to the x-, y-, and z-axes respectively),lFollowed by a rotation of degrees about the y-axis.1.Pre-multiply by each matrix:1, =( ,)xyznoapRot xp2,1231,123( , , )( , , )( , )xyzxyznoapTrans l l lpTrans l l lRot xp3,2,123( ,)(

12、 ,)( , , )( , )xyzxyzxyznoappRot ypRot yTrans l l lRot xp相对于固定的参考坐标系的每次变换，变换矩阵都是左乘的。【复合变换的表示复合变换的表示】nTransformations Relative to the Rotating (current) Frame lIn this case, matrices representing each transformation are post-multiplied.lIf transformations are relative to both the Universe frame and t

13、he current frame, each matrix is accordingly multiplied, either pre- or post-.当进行相对于运动坐标系或当前坐标系的轴的变换时：为计算当前坐标系中点的坐标相当于参考坐标系的变化，这时需要右乘变换矩阵右乘变换矩阵而不是左乘。【相对于旋转坐标系（当前坐标系相对于旋转坐标系（当前坐标系/运动坐标系）的变换运动坐标系）的变换】Inverse of MatriceslThe following steps must be taken to calculate the inverse of a matrix:lCalculate

14、the determinant of the matrix.lTranspose the matrix.lReplace each element of the transposed matrix by its own minor (adjoint matrix).lDivide the converted matrix by the determinant.4. 4. 变换矩阵的逆变换矩阵的逆所谓逆变换就是将被变换的坐标系返回到原来的坐标系。所谓逆变换就是将被变换的坐标系返回到原来的坐标系。nInverse of Rotation MatriceslThe inverse of a rota

15、tion matrix is its transpose because rotation matrices are “unitary”. Txx),(Rot),(Rot1【旋转矩阵的逆旋转矩阵的逆】nInverse of Transformation MatriceslThe inverse of a transformation (or a frame) matrix is the following:l1. Transpose the rotation portion of the matrix.l2. Take the negative of the dot-product of th

16、e P and n, P and o, and P and a vectors.lThe scale factors remain the same.1 and 00010001xxxxxyzyyyyxyzzzzzxyznoapnnnnoapoooTTnoapaaa p np op a【变换矩阵的逆变换矩阵的逆】Forward and Inverse Kinematic EquationslForward kinematics includes substituting the known joint values into the equations to find the location

17、 and orientationlInverse kinematics includes finding an equation that results in joint values if the desired position and orientation are specified.5. 5. 机器人的正逆运动学机器人的正逆运动学23关节与连杆l 在机器人中，通常有两类关节：转动关 节和移动关节l 自由度：物体能够相对于坐标系进行独立运动的数目l 不同于人类的关节，一般机器人关节为一个自由度的关节，其目的是为了简化力学、运动学和机器人的控制l 转动关节提供了一个转动自由度，移动关节

18、提供一个移动自由度，各关节间是以固定杆件相连接的24连杆参数 连杆长度：两个关节的关节轴线 Ji与 Ji+1 的公垂线距离为连杆长度，记为ai。 连杆扭转角：由Ji与公垂线组成平面P，Ji+1与平面P的夹角为连杆扭转角，记为i 。连杆偏移量：除第一和最后连杆外，中间的连杆的两个关节轴线Ji与Ji+1 都有一条公垂线ai，一个关节的相邻两条公垂线 ai与ai-1的距离为连杆偏移量，记为di。关节角：关节Ji的相邻两条公垂线ai与ai-1在以Ji为法线的平面上的投影的夹角为关节角，记为i。ai、i、di、i这组参数称为Denavit-Hartenberg(D-H)参数。Denavit-Harten

19、berg (DH) Representation of Forward Kinematic Equations of Robots lMay be used for any configuration, whether specific coordinates or not.lCan include joint offset, twist angles, multi-variable joints, and so on.lVery common.lMany other equations are based on this methodology6. 6. 机器人正运动学方程的机器人正运动学方

20、程的D-HD-H表示表示nD-H RepresentationlZ-axes along the joint motion. represents joint rotation. ld is joint linear displacement or distance between common normals.l is the twist angle between z-axes.la is the length of the common normal.D-H RepresentationlAssign z-axes to each joint along linear motion or

21、 revolute axis.lAssign x-axes along the common normal between successive z-axes.lNo need for y-axes.lIf z-axes coincide, x-axis is perpendicular to both.lIf z-axes are parallel, x-axes can be anywhere.D-H Representation连杆本身的参数连杆长度连杆两个轴的公垂线距离（x方向）连杆扭转角连杆两个轴的夹角（x轴的扭转角）连杆之间的参数连杆之间的距离相连两连杆公垂线距离（z方向平移距）连

22、杆之间的夹角相连两连杆公垂线的夹角（z轴旋转角）lFour transformations are necessary to go from one frame to the next:D-H RepresentationlRotate about the zn-axis an angle of . This will make xn and xn+1 parallel to each other. This is true because an and an+1 are both perpendicular to zn, and rotating zn an angle of will ma

23、ke them parallel (and thus, coplanar).lTranslate along the zn-axis a distance of dn+1 to make xn and xn+1 colinear. Since xn and xn+1 were already parallel and normal to zn, moving along zn will lay them over each other.lTranslate along the (already rotated) xn-axis a distance of an+1 to bring the o

24、rigins of xn and xn+1 together. At this point, the origins of the two reference frames will be at the same location.lRotate zn-axis about xn+1-axis an angle of to align zn-axis with zn+1-axis. At this point, frames n and n+1 will be exactly the same, and we have transformed from one to the next.D-H

25、Representation1n1n1nlA transformation matrix can be formed by:1111111111111110010001001000000100010000001000100100000100010001( ,)(0,0,)(,0,0)( ,)nnnnnnnnnnnnnnnnCSaSCCSdSTARot zTransdTrans aRot x 100001nCD-H RepresentationlAn A-matrix is:D-H Representation11111111111111111100001nnnnnnnnnnnnnnnnnnCS

26、CSSaCSCCCSaSASCdp在机器人的在机器人的基座基座与与手手之间的总变换为：之间的总变换为：yznoaTool Reference FramexnAAAA321n1 -n32211RHRTTTTTlA parameters table may look like:D-H RepresentationlExample: A simple 6-axis robot zyx123z0 x0z1x1z2x2z3x4x3z4z5x5z6x6a2a3a44D-H RepresentationlFind a set of equations that allow determination of

27、joint values from desired position and orientation information.lEach robot has a different solution.lIt may be necessary to use different approaches for each robot.lThis usually requires pre-multiplication of transformation matrices by inverse of individual A matrices, squaring of terms, divisions, and so on.Inverse Kinematic EquationslFor the shown example, the following may be found:where and In