Robotics: 3D Movements


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Robotics: 3D Movements

  1. 1. 3-D Movements
  2. 2. 3-D Spaces and Points <ul><li>Axes xyz </li></ul><ul><ul><li>right hand rule X  Y=Z </li></ul></ul>X Z Y v v 1 v 2 v 3 X Y Z
  3. 3. <ul><li>2 co-ordinate frames rotated about a common origin </li></ul><ul><li>i R j is the rotation matrix from frame j to i </li></ul>X 0 Y 0 Z 0 X 1 Y 1 Z 1
  4. 4. Rotation about the axes X Y Z
  5. 5. Direction Cosine Form <ul><li>View rotation as the dot product between each of the equations for 1 x 0 , 1 y 0 , and 1 z 0 and the axes 0 x 0 , 0 y 0 , 0 z 0 </li></ul>
  6. 6. Inverse of a rotation matrix
  7. 7. Composition of Rotations <ul><li>Consider three frames 0,1 and 2 rotated about a common origin with p a point relative to the common origin </li></ul><ul><li>Assume all frames coincide </li></ul><ul><ul><li>Rotate 1 and 2 about y 0 by  </li></ul></ul><ul><ul><li>Rotate 2 about z 1 by  </li></ul></ul><ul><li>Because rotations occurred in different frames </li></ul>
  8. 8. From origin P Left to right a From origin P Right to left a a a X Y Z X Y Z
  9. 9. Spatial Transformations <ul><li>Similar to 2-D case with and increase in dimensions </li></ul>Representation of Orientation <ul><li>Rotation matrix contains redundant information (9 terms) </li></ul><ul><ul><li>Only 3 terms independent </li></ul></ul><ul><li>3 term representations </li></ul><ul><ul><li>Euler angles </li></ul></ul><ul><ul><li>Rodrigues vectors </li></ul></ul><ul><li>4 term representations </li></ul><ul><ul><li>Quaternions </li></ul></ul>
  10. 10. Euler Angles <ul><li>Any non-redundant set of three successive rotations about principal axes </li></ul><ul><li>12 systems most common is ZYZ </li></ul><ul><li>+ Useful where they match the structure of the robot </li></ul><ul><li>- Can have two solutions </li></ul>
  11. 11. Roll, Pitch Yaw angles <ul><li>R xyz (  ) = R z (  ) R y (  ) R x (  ) </li></ul>X Y Z Roll  Yaw  Pitch 
  12. 12. Quaternions <ul><li>Proposed by Hamilton </li></ul><ul><li>Euler parameters make up the components of the quaternion </li></ul><ul><li>Quaternion Q has a scalar and a vector part </li></ul><ul><ul><li>Q=[w+v] </li></ul></ul><ul><li>If S =sin(  /2) and C=cos(  /2) then the rotation of  around axis k where u is a unit vector. </li></ul>
  13. 13. Quaternion Operations <ul><li>Addition: </li></ul><ul><li>Multiplication: </li></ul><ul><li>Norm (length): </li></ul><ul><li>Conjugate: </li></ul><ul><li>Inverse: </li></ul>
  14. 14. Quaternions as Rotations <ul><li>We define the representation of a (homogeneous) Cartesian point in quaternion space as: </li></ul><ul><li>Let q be a unit quaternion (i.e. N( q ) = 1 ). </li></ul><ul><li>The conjugation of p by q is defined as: </li></ul>
  15. 15. Quaternions as Rotations <ul><li>then </li></ul><ul><li>and if </li></ul><ul><li>then the effect on p is to rotate it anti-clockwise about axis v by  degrees. </li></ul><ul><li>Therefore every unit quaternion represents a rotation about an axis. </li></ul><ul><li>Note that the rotation q is the same as - q  redundancy </li></ul>
  16. 16. Quaternions as Rotations <ul><li>Example : rotate point P = [ x, y, z, w ] about vector v by  degrees. </li></ul><ul><ul><li>determine quaternion representation p of point P: </li></ul></ul><ul><ul><li>create quaternion q for rotation: </li></ul></ul><ul><ul><li>conjugate p by q : </li></ul></ul><ul><ul><li>recover new rotated coordinate: </li></ul></ul>
  17. 17. Rotation Matrix  Quaternion Conversion
  18. 18. Deconstructing the Quaternion <ul><li>Given arbitrary unit quaternion we can recover the associated rotation axis and angle: </li></ul>
  19. 19. Normalisation <ul><li>With successive rotations, errors can accumulate making the quaternions of non-unit length </li></ul><ul><li>Normalisation of Matrices is much harder </li></ul>
  20. 20. Quaternion Rotation Composition <ul><li>As with matrices, quaternion rotations may be composed of multiple basic rotations: </li></ul>note the order ( q 2 after q 1 ) property of conjugation
  21. 21. Advantages of Quaternions for Rotations <ul><li>Significantly less storage required (4 reals vs. 16 reals) </li></ul><ul><li>Much easier to construct arbitrary rotations about vectors. </li></ul><ul><li>Easily interpolated for smooth paths . </li></ul><ul><li>Transformation via quaternions more efficient. </li></ul>Operation Matrices Quaternions Storage 9 4 Transformation 9M 6A 15M 15A Composition 27M 18A 16M 12A Normalisation complicated 8M 3A 1sqrt