![Composition of Rotation Matrices
• Recall for matrices
AB BA
(matrix multiplication is not commutative)
Rot[Z,90] Rot[Y,-90]
34
TEMPUS IV Project: 158644 – JPCR
Development of Regional Interdisciplinary Mechatronic Studies - DRIMS
ROBOTICS](https://image.slidesharecdn.com/46forwardkinematics-240219070740-0f1cb7f5/85/4_6-Forward-kinematics-ROBOTICS-ENGINNERING-ppt-34-320.jpg)
![Composition of Rotation Matrices
• Recall for matrices
AB BA
(matrix multiplication is not commutative)
Rot[Z,90]
Rot[Y,-90]
35
TEMPUS IV Project: 158644 – JPCR
Development of Regional Interdisciplinary Mechatronic Studies - DRIMS
ROBOTICS](https://image.slidesharecdn.com/46forwardkinematics-240219070740-0f1cb7f5/85/4_6-Forward-kinematics-ROBOTICS-ENGINNERING-ppt-35-320.jpg)
![Rot[Z,90]
Rot[Y,-90]
Rot[Z,90] Rot[Y,-90]
36
TEMPUS IV Project: 158644 – JPCR
Development of Regional Interdisciplinary Mechatronic Studies - DRIMS
ROBOTICS](https://image.slidesharecdn.com/46forwardkinematics-240219070740-0f1cb7f5/85/4_6-Forward-kinematics-ROBOTICS-ENGINNERING-ppt-36-320.jpg)
![Rot[z,90]Rot[y,-90] Rot[y,-90] Rot[z,90]
37
TEMPUS IV Project: 158644 – JPCR
Development of Regional Interdisciplinary Mechatronic Studies - DRIMS
ROBOTICS](https://image.slidesharecdn.com/46forwardkinematics-240219070740-0f1cb7f5/85/4_6-Forward-kinematics-ROBOTICS-ENGINNERING-ppt-37-320.jpg)
4_6 Forward kinematics ROBOTICS ENGINNERING.ppt
- 1. ROBOTICS Forward Kinematics TEMPUS IV Project: 158644 – JPCR Development of Regional Interdisciplinary Mechatronic Studies - DRIMS ROBOTICS
- 2. Forward Kinematics • Modeling assumptions • Review: – Spatial Coordinates • Pose = Position + Orientation – Rotation Matrices – Homogeneous Coordinates • Frame Assignment – Denavit Hartenberg Parameters • Robot Kinematics – End-effector Position, – Velocity, & – Acceleration 1st part 2nd part 2 TEMPUS IV Project: 158644 – JPCR Development of Regional Interdisciplinary Mechatronic Studies - DRIMS ROBOTICS
- 3. Industrial Robot sequence of rigid bodies (links) connected by means of articulations (joints) 3 TEMPUS IV Project: 158644 – JPCR Development of Regional Interdisciplinary Mechatronic Studies - DRIMS ROBOTICS
- 4. Robot Basics: Modeling • Kinematics: – Relationship between the joint angles, velocities & accelerations and the end-effector position, velocity, & acceleration 4 TEMPUS IV Project: 158644 – JPCR Development of Regional Interdisciplinary Mechatronic Studies - DRIMS ROBOTICS
- 5. Modeling Robot Manipulators • Open kinematic chain (in this course) • One sequence of links connecting the two ends of the chain (Closed kinematic chains form a loop) • Prismatic or revolute joints, each with a single degree of mobility • Prismatic: translational motion between links • Revolute: rotational motion between links • Degrees of mobility (joints) vs. degrees of freedom (task) • Positioning and orienting requires 6 DOF • Redundant: degrees of mobility > degrees of freedom • Workspace • Portion of environment where the end-effector can access 5 TEMPUS IV Project: 158644 – JPCR Development of Regional Interdisciplinary Mechatronic Studies - DRIMS ROBOTICS
- 6. Modeling Robot Manipulators • Open kinematic chain – sequence of links with one end constrained to the base, the other to the end-effector Base End-effector 6
- 7. Modeling Robot Manipulators • Motion is a composition of elementary motions Base End-effector Joint 1 Joint 2 Joint 3 7
- 8. Kinematic Modeling of Manipulators • Composition of elementary motion of each link • Use linear algebra + systematic approach • Obtain an expression for the pose of the end-effector as a function of joint variables qi (angles/displacements) and link geometry (link lengths and relative orientations) Pe = f(q1,q2,,qn ;l1,ln,1,n) 8 TEMPUS IV Project: 158644 – JPCR Development of Regional Interdisciplinary Mechatronic Studies - DRIMS ROBOTICS
- 9. Pose of a Rigid Body • Pose = Position + Orientation • Physical space, E3, has no natural coordinates. • In mathematical terms, a coordinate map is a homeomorphism (1-1, onto differentiable mapping with a differentiable inverse) of a subset of space to an open subset of R3. – A point, P, is assigned a 3-vector: AP = (x,y,z) where A denotes the frame of reference 9 TEMPUS IV Project: 158644 – JPCR Development of Regional Interdisciplinary Mechatronic Studies - DRIMS ROBOTICS
- 10. A B X X Y Y Z Z AP = (x,y,z) BP = (x,y,z) P 10 TEMPUS IV Project: 158644 – JPCR Development of Regional Interdisciplinary Mechatronic Studies - DRIMS ROBOTICS
- 11. Pose of a Rigid Body • Pose = Position + Orientation How do we do this? 11 TEMPUS IV Project: 158644 – JPCR Development of Regional Interdisciplinary Mechatronic Studies - DRIMS ROBOTICS
- 12. Pose of a Rigid Body • Pose = Position + Orientation • Orientation of the rigid body – Attach a orthonormal FRAME to the body – Express the unit vectors of this frame with respect to the reference frame XA YA ZA 12 TEMPUS IV Project: 158644 – JPCR Development of Regional Interdisciplinary Mechatronic Studies - DRIMS ROBOTICS
- 13. Pose of a Rigid Body • Pose = Position + Orientation • Orientation of the rigid body – Attach a orthonormal FRAME to the body – Express the unit vectors of this frame with respect to the reference frame XA YA ZA 13 TEMPUS IV Project: 158644 – JPCR Development of Regional Interdisciplinary Mechatronic Studies - DRIMS ROBOTICS
- 14. Rotation Matrices • OXYZ & OUVW have coincident origins at O – OUVW is fixed to the object – OXYZ has unit vectors in the directions of the three axes ix, jy,and kz – OUVW has unit vectors in the directions of the three axes iu, jv,and kw • Point P can be expressed in either frame: 14 TEMPUS IV Project: 158644 – JPCR Development of Regional Interdisciplinary Mechatronic Studies - DRIMS ROBOTICS
- 15. O X U V Y W Z AP = (x,y,z) BP = (u,v,w) P 15 TEMPUS IV Project: 158644 – JPCR Development of Regional Interdisciplinary Mechatronic Studies - DRIMS ROBOTICS
- 16. O X U V Y W Z AP = (x,y,z) P BP = (u,v,w) 16 TEMPUS IV Project: 158644 – JPCR Development of Regional Interdisciplinary Mechatronic Studies - DRIMS ROBOTICS
- 17. O X U V Y W Z AP = (x,y,z) P BP = (u,v,w) 17 TEMPUS IV Project: 158644 – JPCR Development of Regional Interdisciplinary Mechatronic Studies - DRIMS ROBOTICS
- 18. O X U V Y W Z AP = (x,y,z) BP = (u,v,w) P 18 TEMPUS IV Project: 158644 – JPCR Development of Regional Interdisciplinary Mechatronic Studies - DRIMS ROBOTICS
- 19. Rotation Matrices 19 TEMPUS IV Project: 158644 – JPCR Development of Regional Interdisciplinary Mechatronic Studies - DRIMS ROBOTICS
- 20. Rotation Matrices 1 X axis expressed wrt Ouvw 20
- 21. Rotation Matrices 1 Y axis expressed wrt Ouvw 21
- 22. Rotation Matrices 1 Z axis expressed wrt Ouvw 22
- 23. Rotation Matrices 23 TEMPUS IV Project: 158644 – JPCR Development of Regional Interdisciplinary Mechatronic Studies - DRIMS ROBOTICS
- 24. Rotation Matrices Z axis expressed wrt Ouvw X axis expressed wrt Ouvw Y axis expressed wrt Ouvw 24 TEMPUS IV Project: 158644 – JPCR Development of Regional Interdisciplinary Mechatronic Studies - DRIMS ROBOTICS
- 25. Rotation Matrices 1 U axis expressed wrt Oxyz 25
- 26. Rotation Matrices U axis expressed wrt Oxyz V axis expressed wrt Oxyz W axis expressed wrt Oxyz 26
- 27. Properties of Rotation Matrices • Column vectors are the unit vectors of the orthonormal frame – They are mutually orthogonal – They have unit length • The inverse relationship is: – Row vectors are also orthogonal unit vectors 27 TEMPUS IV Project: 158644 – JPCR Development of Regional Interdisciplinary Mechatronic Studies - DRIMS ROBOTICS
- 28. Properties of Rotation Matrices • Rotation matrices are orthogonal • The transpose is the inverse: • For right-handed systems – Determinant = -1(Left handed) • Eigenvectors of the matrix form the axis of rotation 28 TEMPUS IV Project: 158644 – JPCR Development of Regional Interdisciplinary Mechatronic Studies - DRIMS ROBOTICS
- 29. Elementary Rotations: X axis X Y Z 29 TEMPUS IV Project: 158644 – JPCR Development of Regional Interdisciplinary Mechatronic Studies - DRIMS ROBOTICS
- 30. Elementary Rotations: X axis X Y Z 30 TEMPUS IV Project: 158644 – JPCR Development of Regional Interdisciplinary Mechatronic Studies - DRIMS ROBOTICS
- 31. Elementary Rotations: Y axis X Y Z 31 TEMPUS IV Project: 158644 – JPCR Development of Regional Interdisciplinary Mechatronic Studies - DRIMS ROBOTICS
- 33. Composition of Rotation Matrices • Express P in 3 coincident rotated frames 33 TEMPUS IV Project: 158644 – JPCR Development of Regional Interdisciplinary Mechatronic Studies - DRIMS ROBOTICS
- 34. Composition of Rotation Matrices • Recall for matrices AB BA (matrix multiplication is not commutative) Rot[Z,90] Rot[Y,-90] 34 TEMPUS IV Project: 158644 – JPCR Development of Regional Interdisciplinary Mechatronic Studies - DRIMS ROBOTICS
- 35. Composition of Rotation Matrices • Recall for matrices AB BA (matrix multiplication is not commutative) Rot[Z,90] Rot[Y,-90] 35 TEMPUS IV Project: 158644 – JPCR Development of Regional Interdisciplinary Mechatronic Studies - DRIMS ROBOTICS
- 36. Rot[Z,90] Rot[Y,-90] Rot[Z,90] Rot[Y,-90] 36 TEMPUS IV Project: 158644 – JPCR Development of Regional Interdisciplinary Mechatronic Studies - DRIMS ROBOTICS
- 37. Rot[z,90]Rot[y,-90] Rot[y,-90] Rot[z,90] 37 TEMPUS IV Project: 158644 – JPCR Development of Regional Interdisciplinary Mechatronic Studies - DRIMS ROBOTICS
- 38. Decomposition of Rotation Matrices • Rotation Matrices contain 9 elements • Rotation matrices are orthogonal – (6 non-linear constraints) 3 parameters describe rotation • Decomposition is not unique 38 TEMPUS IV Project: 158644 – JPCR Development of Regional Interdisciplinary Mechatronic Studies - DRIMS ROBOTICS
- 39. Decomposition of Rotation Matrices • Euler Angles • Roll, Pitch, and Yaw 39 TEMPUS IV Project: 158644 – JPCR Development of Regional Interdisciplinary Mechatronic Studies - DRIMS ROBOTICS
- 40. Decomposition of Rotation Matrices • Angle Axis 40 TEMPUS IV Project: 158644 – JPCR Development of Regional Interdisciplinary Mechatronic Studies - DRIMS ROBOTICS
- 41. Decomposition of Rotation Matrices • Angle Axis • Elementary Rotations 41
- 42. Pose of a Rigid Body • Pose = Position + Orientation Ok. Now we know what to do about orientation…let’s get back to pose 42 TEMPUS IV Project: 158644 – JPCR Development of Regional Interdisciplinary Mechatronic Studies - DRIMS ROBOTICS
- 43. Spatial Description of Body • position of the origin with an orientation A X Y Z B 43 TEMPUS IV Project: 158644 – JPCR Development of Regional Interdisciplinary Mechatronic Studies - DRIMS ROBOTICS
- 44. Homogeneous Coordinates • Notational convenience 44 TEMPUS IV Project: 158644 – JPCR Development of Regional Interdisciplinary Mechatronic Studies - DRIMS ROBOTICS
- 45. Composition of Homogeneous Transformations • Before: • After 45 TEMPUS IV Project: 158644 – JPCR Development of Regional Interdisciplinary Mechatronic Studies - DRIMS ROBOTICS
- 46. Homogeneous Coordinates • Inverse Transformation 46 TEMPUS IV Project: 158644 – JPCR Development of Regional Interdisciplinary Mechatronic Studies - DRIMS ROBOTICS
- 47. Homogeneous Coordinates • Inverse Transformation Orthogonal: no matrix inversion! 47
- 48. Literature: Richard M. Murray, Zexiang Li, S. Shankar Sastry: A mathematical Introduction to Robotic Manipulation, University of California, Berkeley, 1994, CRC Press, pp.93-95. An electronic edition of the book is available from: http://www.cds.caltech.edu/~murray/mlswiki TEMPUS IV Project: 158644 – JPCR Development of Regional Interdisciplinary Mechatronic Studies - DRIMS ROBOTICS 48