机器人的位姿运动学

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】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, Orientation

4、, Normal directionsyznoaxaonF,运动坐标系zyxF,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 Body 0001xxxxyyyyobje

5、ctzzzznoapnoapFnoapzyxnoap【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 the following six c

6、onstraint 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 pre- or post-multipl

7、iedlEasy 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 pure translationlA pu

8、re 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 zyxpnoanoad1000100010001000100

9、01xxxxxxxxxxyyyyyyyyyynewzzzzzzzzzzdnoapnoapddnoapnoapdFdnoapnoapdFnew = 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( , )xyznoapRot xpcos

10、sin0sincos0001),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 zSCnRepresentation of Combined Transformations lExample: 1.Rotation of degrees about the x-axis,2.Followed by a translation of l1,l2,l3 (relative to the x-, y-, and z-axes respectively),

11、3.Followed by a rotation of degrees about the y-axis.lPre-multiply by each matrix:1, =( ,)xyznoapRot xp2,1231,123( , , )( , , )( ,)xyzxyznoapTrans l l lpTrans l l lRot xp3,2,123( ,)( ,)( , , )( ,)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 Re

12、lative 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 the 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

13、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 the determinant of the matrix.lTranspose the matrix.lReplace each element of the transposed matrix by its own minor (ad

14、joint matrix).lDivide the converted matrix by the determinant.4. 4. 变换矩阵的逆变换矩阵的逆nInverse of Rotation MatriceslThe inverse of a rotation 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 (

15、or a frame) matrix is the following:l1. Transpose the rotation portion of the matrix.l2. Take the negative of the dot-product of the 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

16、Inverse Kinematic EquationslForward kinematics includes substituting the known joint values into the equations to find the location 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

17、运动学Forward and Inverse Kinematics for PositioninglFour possibilities are common:a. Cartesian (gantry, rectangular) coordinatesb. Cylindrical coordinatesc. Spherical coordinatesd. Articulated (anthropomorphic or all-revolute) coordinates6. 6. 位置的正逆运动学方程位置的正逆运动学方程nCartesian CoordinateslThree linear mo

18、tions.100010(,)0010001xyRpcartxyzzppTTppppxyzppxpypzano【直角直角(台架台架) 坐标坐标】nCylindrical CoordinateslTwo linear and one revolute joints( , , )(0,0, )( , )( ,0,0)RpcylTTrlTransl Rot zTrans r00( , , )0010001RpcylCSrCSCrSTTrllxyzrlpaon【圆柱坐标圆柱坐标】Inverse SolutionlUse the position equations to find the joint

19、values.lThe application of ATAN2 function for correct determination of angles.nSpherical CoordinateslTwo revolute and one linear joints( , )( , )( ,)(0,0, )RPsphTTrRot zRot yTransr ( , )00001RPsphC CSS CrS CC SCS SrS STTrSCrC 【球坐标球坐标】Inverse SolutionlUse the position equations to determine the joint

20、 values.lCheck your answers for correct values.nArticulated CoordinateslWe will study later with the DenavitHartenberg methodology【链式链式 坐标坐标】Forward and Inverse Kinematics for OrientationlThree possibilities are common:a. Roll-pitch-yaw (RPY) anglesb. Euler anglesc. Articulated coordinates6. 6. 姿态的正

21、逆运动学方程姿态的正逆运动学方程在不改变位置的情况下，适当地旋转坐标系而使其达到所期望的姿态姿态。nRPY AngleslRotations relative to the current z-, y-, and x-axeszxyazanoznoanoxyxynoaRPY(,)( ,)( ,)( ,)0000001aonaonaoaonanaonanaoaonanaonanoononRot aRot oRot nCCCSSSCCSCSSSCSSSCCSSCCSSCSC C 【滚动角、俯仰角和偏航角滚动角、俯仰角和偏航角】Inverse SolutionlUse:2(,) and 2(,) a

22、yxayxATANn nATANnn2,()ozxayaATANnn Cn S2(),()nyaxayaxaATANa Ca So Co SnEuler AngleslRotations relative to the current z-, y-, and z-axes.ZXYaZanoZnoanoXYXYEuler( , ,)( , )( , ),( ,)0000001Rot aRot oRot aC C CS SC C SS CC SS C CC SS C SC CS SS CS SC 【欧拉角欧拉角】Inverse SolutionlUse:2(,)2(,)yxyxATANaaorAT

23、ANaa2(),()xyxyATANn Sn Co So C2(),)xyzATANa Ca SanArticulated AngleslWe will study later with the DenavitHartenberg methodology【链式关节链式关节】Denavit-Hartenberg (DH) Representation of Forward Kinematic Equations of Robots lMay be used for any configuration, whether specific coordinates or not.lCan includ

24、e joint offset, twist angles, multi-variable joints, and so on.lVery common.lMany other equations are based on this methodology8. 8. 机器人正运动学方程的机器人正运动学方程的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

25、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 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 b

26、oth.lIf z-axes are parallel, x-axes can be anywhere.D-H RepresentationlFour 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 perpendicu

27、lar to zn, and rotating zn an angle of will make 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 rota

28、ted) xn-axis a distance of an+1 to bring the origins 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

29、 we have transformed from one to the next.D-H Representation1n1n1nlA transformation matrix can be formed by:1111111111111110010001001000000100010000001000100100000100010001( ,)(0,0,)(,0,0)( ,)nnnnnnnnnnnnnnnnCSaSCCSdSTARot zTransdTrans aRot x 100001nCD-H RepresentationlAn A-matrix is:D-H Representat

30、ion11111111111111111100001nnnnnnnnnnnnnnnnnnCSCSSaCSCCCSaSASCdp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

31、 set of equations that allow determination of 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.I