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Forward Kinematics

Forward Kinematics. Professor Nicola Ferrier ME Room 2246, 265-8793 ferrier@engr.wisc.edu. Forward Kinematics. Modeling assumptions Review: Spatial Coordinates Pose = Position + Orientation Rotation Matrices Homogeneous Coordinates Frame Assignment Denavit Hartenberg Parameters

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Forward Kinematics

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  1. Forward Kinematics Professor Nicola Ferrier ME Room 2246, 265-8793 ferrier@engr.wisc.edu Professor N. J. Ferrier

  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 Today Next Lecture Professor N. J. Ferrier

  3. Industrial Robot sequence of rigid bodies (links) connected by means of articulations (joints) Professor N. J. Ferrier

  4. Robot Basics: Modeling • Kinematics: • Relationship between the joint angles, velocities & accelerations and the end-effector position, velocity, & acceleration    Professor N. J. Ferrier

  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 Professor N. J. Ferrier

  6. Modeling Robot Manipulators • Open kinematic chain • sequence of links with one end constrained to the base, the other to the end-effector End-effector Base Professor N. J. Ferrier

  7. Modeling Robot Manipulators • Motion is a composition of elementary motions End-effector Joint 2 Joint 1 Joint 3 Base Professor N. J. Ferrier

  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,q2,¼,qn ;l1¼,ln,1¼,n) Professor N. J. Ferrier

  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 Professor N. J. Ferrier

  10. P Z BP = (x,y,z) Z AP = (x,y,z) Y B Y A X X Professor N. J. Ferrier

  11. Pose of a Rigid Body • Pose = Position + Orientation How do we do this? Professor N. J. Ferrier

  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 YA XA ZA Professor N. J. Ferrier

  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 YA XA ZA Professor N. J. Ferrier

  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: Professor N. J. Ferrier

  15. P Z AP = (x,y,z) W BP = (u,v,w) V Y O X U Professor N. J. Ferrier

  16. P Z AP = (x,y,z) W BP = (u,v,w) V Y O X U Professor N. J. Ferrier

  17. P Z AP = (x,y,z) W BP = (u,v,w) V Y O X U Professor N. J. Ferrier

  18. P Z AP = (x,y,z) W BP = (u,v,w) V Y O X U Professor N. J. Ferrier

  19. Rotation Matrices Professor N. J. Ferrier

  20. Rotation Matrices 1 X axis expressed wrt Ouvw Professor N. J. Ferrier

  21. Rotation Matrices 1 Y axis expressed wrt Ouvw Professor N. J. Ferrier

  22. Rotation Matrices 1 Z axis expressed wrt Ouvw Professor N. J. Ferrier

  23. Rotation Matrices Professor N. J. Ferrier

  24. Rotation Matrices X axis expressed wrt Ouvw Y axis expressed wrt Ouvw Z axis expressed wrt Ouvw Professor N. J. Ferrier

  25. Rotation Matrices 1 U axis expressed wrt Oxyz Professor N. J. Ferrier

  26. Rotation Matrices U axis expressed wrt Oxyz V axis expressed wrt Oxyz W axis expressed wrt Oxyz Professor N. J. Ferrier

  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 Professor N. J. Ferrier

  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 Professor N. J. Ferrier

  29. Elementary Rotations: X axis Z Y X Professor N. J. Ferrier

  30. Elementary Rotations: X axis Z Y X Professor N. J. Ferrier

  31. Elementary Rotations: Y axis Z Y X Professor N. J. Ferrier

  32. Elementary Rotations: Z-axis Z Y X Professor N. J. Ferrier

  33. Composition of Rotation Matrices • Express P in 3 coincident rotated frames Professor N. J. Ferrier

  34. Composition of Rotation Matrices • Recall for matrices AB ¹ BA (matrix multiplication is not commutative) Rot[Z,90] Rot[Y,-90] Professor N. J. Ferrier

  35. Composition of Rotation Matrices • Recall for matrices AB ¹ BA (matrix multiplication is not commutative) Rot[Z,90] Rot[Y,-90] Professor N. J. Ferrier

  36. Rot[Z,90] Rot[Y,-90] Rot[Z,90] Rot[Y,-90] Professor N. J. Ferrier

  37. Rot[z,90]Rot[y,-90] ¹ Rot[y,-90] Rot[z,90] Professor N. J. Ferrier

  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 Professor N. J. Ferrier

  39. Decomposition of Rotation Matrices • Euler Angles • Roll, Pitch, and Yaw Professor N. J. Ferrier

  40. Decomposition of Rotation Matrices • Angle Axis Professor N. J. Ferrier

  41. Decomposition of Rotation Matrices • Angle Axis • Elementary Rotations Professor N. J. Ferrier

  42. Pose of a Rigid Body • Pose = Position + Orientation Ok. Now we know what to do about orientation…let’s get back to pose Professor N. J. Ferrier

  43. Spatial Description of Body • position of the origin with an orientation Z B Y A X Professor N. J. Ferrier

  44. Homogeneous Coordinates • Notational convenience Professor N. J. Ferrier

  45. Composition of Homogeneous Transformations • Before: • After Professor N. J. Ferrier

  46. Homogeneous Coordinates • Inverse Transformation Professor N. J. Ferrier

  47. Homogeneous Coordinates • Inverse Transformation Orthogonal: no matrix inversion! Professor N. J. Ferrier

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