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Intelligent robotic chapter 2 distribute

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Intelligent robotic chapter 2 distribute

  1. 1. 1
  2. 2.  The study of the position and orientation of a robot hand with respect to a reference coordinate system, given the joint variables and the arm parameters, OR  The analytical study of the geometry of motion of a robot arm with respect to a reference coordinate system.  Without regard the forces of moments that cause the robot motion.  It is the first step towards robotic control.2
  3. 3.  What you are given:  The length of each link  The angle of each joint  What you can find:  The position of any point (i.e. it’s (x, y, z) coordinates3
  4. 4.  Between two frames, the is a kinematic relationship either a translation, rotation or both. The relationship can be describe by a transformation matrix. {D} z2 z0 {C} Translation y2 and rotation {B} {A} Rotation Translation y0 x2 Rereference x0 frame Note: {D} = Frame D4
  5. 5. pu cos( ) sin( ) px A Puv RPxy B pv sin( ) cos( ) py A y Puv B RPxy v {A} A Puv Pxy B QPuv {B} Pxy u A A 1 A Q B BR BRT B A R x A Note: B R describes the rotations of {B} w.r.t. {A}5
  6. 6. px ix iu i x jv ix k w pu Pxyz py jy i u j y jv jy k w pv RPuvw pz k z iu k z jv kz kw pw z A Puvw B RPxyz Puvw Pxyz A Pxyz QP B uvw y A A 1 A BQ BR BRT B A R x6
  7. 7.  Rotation transformation matrices Rotation about x-axis by degrees - Yaw 1 0 0 z Rx ( ) 0 Cos Sin Roll 0 Sin Cos Rotation about y-axis by degrees - Pitch Pitch Cos 0 Sin y Ry ( ) 0 1 0 Sin 0 Cos x Yaw Rotation about z-axis by degrees - Roll Cos Sin 0 Rz ( ) Sin Cos 0 0 0 17
  8. 8.  Roll-pitch-yaw angles (Z-Y-X Euler angle-Relative axis) It provides a method to decompose a complex rotation into three consecutive fundamental rotations; roll, pitch, and yaw. Use post multiplication rule. Ruvw (mobile ) Rz ( ) Ry ( ) Rx ( ) Cos Sin 0 Cos 0 Sin 1 0 0 Sin Cos 0 0 1 0 0 Cos Sin 0 0 1 Sin 0 Cos 0 Sin Cos Ruvw is mobile with respect to the Rxyz8
  9. 9.  Yaw-pitch-roll angles (X-Y-Z fixed angle) Representation in yaw-pitch-roll angles allows complex rotation to be decomposed into a sequence of yaw, pitch and roll about the x, y and z axis. Use pre-multiplication rule. Ruvw ( fixed) RZ ( ) RY ( ) RX ( ) Cos Sin 0 Cos 0 Sin 1 0 0 Sin Cos 0 0 1 0 0 Cos Sin 0 0 1 Sin 0 Cos 0 Sin Cos Conclusion: Ruvw ( fixed) Ruvw (mobile )9
  10. 10.  Z-Y-Z Euler angle Read the Z-Y-Z Euler angles on page 30 (M. Zhihong)10
  11. 11.  Find the position of point P=[10 10] with respect to the global axis after it is transformed/rotated by [pi/3]  Find the position of point P=[10 10 10] with  respect to the global axis after it is transformed by [pi/4; pi/3; pi/6]11
  12. 12.  Homogeneous transformations •Transforms and translates. •The homogenous transformation matrix below is used to transform and translate. R is a 3x3 rotation matrix and P is a 3x1translation/position vector. R P H 0 0 0 1 Three fundamental rotation matrices of roll, pitch and yaw in the homogeneous coordinate system: C S 0 0 C 0 S 0 1 0 0 0 S C 0 0 0 1 0 0 0 C S 0Hz( ) Hy( ) Hx( ) 0 0 1 0 S 0 C 0 0 S C 0 0 0 0 1 0 0 0 1 0 0 0 112
  13. 13.  Homogeneous transformations Three fundamental rotation matrices of roll, pitch and yaw Hrpy in the homogeneous coordinate system: R CC CS S SC CS C SS Px SC SS S CC SS C CS Py H rpy S C S C C Pz 0 0 0 1 A point B’ can be found from the following relationship: B H rpy B13
  14. 14.  Homogeneous transformations O Translation without rotation Y 1 0 0 Px 0 1 0 Py N H P 0 0 1 Pz X A 0 0 0 1 Z Rotation without translation Y O nx ox ax 0 N ny oy ay 0 H nz oz az 0 X 0 0 0 1 Z14 A
  15. 15.  Example 1: Find a point B’ in {B} w.r.t to the reference frame {A} if the origin of {B} is (5,5,5) . Given B=(1,2,3). Given 0; 0; 0. B’ (5,5,5) B(1,2,3)15
  16. 16.  Solution 1 0 0 5 0 1 0 5 H 0 0 1 5 0 0 0 1 B HB 1 0 0 5 1 6 0 1 0 5 2 7 B 0 0 1 5 3 8 0 0 0 1 1 116
  17. 17.  Example 2: Find a point B’ in {N} w.r.t to the reference frame {M} if the origin of {N} is (3,5,4) . Given B=(3,2,1). {N} is rotated by ; 0; . 2 {N} B’ (3,5,4) {M} B(3,2,1)17
  18. 18. Solution: 1 0 0 3 3 0 0 0 1 5 2 6 B 0 1 0 4 1 6 0 0 0 1 1 118
  19. 19.  Example 3: Find a point P’ in {N} w.r.t to the reference frame {M} if the origin of {N} is (3,5,4) . Given B=(3,2,1). {N} is rotated by ; ; . 3 2 {N} B’ (3,5,4) {M} B (3,2,1) B’ =[ 0.7679 4.8660 1.0000 1.0000]19

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