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Kinematic Modelling in Robotics

Kinematic Modelling in Robotics. dr Dragan Kosti ć WTB Dynamics and Control October 22 th , 2010. Outline. Representing rotations and rotational transformations Parameterization of rotations Rigid motions and homogenous transformations DH convention for modeling of robot kinematics

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Kinematic Modelling in Robotics

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  1. Kinematic Modelling in Robotics dr Dragan KostićWTB Dynamics and Control October 22th, 2010

  2. Outline • Representing rotations and rotational transformations • Parameterization of rotations • Rigid motions and homogenous transformations • DH convention for modeling of robot kinematics • Forward kinematics • Case-study: kinematics of RRR-arm

  3. Representing rotations in coordinate frame 0 • Rotation matrix • xiand yi are the unit vectors in oixiyi

  4. Representing rotations in coordinate frame 1

  5. Representing rotations in 3D (1/4) • Each axis of the frame o1x1y1z1 is projected onto o0x0y0z0: R10 SO(3)

  6. Representing rotations in 3D (2/4) • Example: Frame o1x1y1z1 is obtained from frame o0x0y0z0 by rotation through an angle  about z0 axis. all other dot products are zero

  7. Representing rotations in 3D (3/4) • Basic rotation matrix about z-axis

  8. Representing rotations in 3D (4/4) • Similarly, basic rotation matrices about x- and y-axes:

  9. Rotational transformations • pi: coordinates of p in oixiyizi

  10. Parameterization of rotations (1/2) Euler angles ZYZEuler angle transformation:

  11. Parameterization of rotations (2/2) Roll, pitch, yaw angles XYZyaw-pitch-roll angle transformation:

  12. Rigid motions

  13. Note that • Consequently, rigid motion (d, R) can be described by matrix representing homogenous transformation: Homogenous transformations (1/2) • We have

  14. We augment vectors p0 and p1 to get their homogenous representations and achieve matrix representation of coordinate transformation Homogenous transformations (2/2) • Since R is orthogonal, we have

  15. Basic homogenous transformations

  16. Conventions (1/2) • 1. there are n joints and hence n + 1 links; joints 1, 2, , n; links 0, 1, , n, • 2. joint i connects link i − 1to link i, • 3. actuation of joint i causes link i to move, • 4. link 0 (the base) is fixed and does not move, • 5. each joint has a single degree-of-freedom (dof):

  17. Conventions (2/2) 6. frame oixiyizi is attached tolink i; regardless of motion of the robot, coordinates of each point on link iare constant when expressed in frame oixiyizi, 7. when joint i is actuated, link i and its attached frame oixiyiziexperience resulting motion.

  18. DH convention for homogenous transformations Position and orientation ofcoordinate frame i with respect to frame i-1 is specified by homogenous transformation matrix: where

  19. Physical meaning of DH parameters • Link length ai is distance from zi-1 to zi measured along xi. • Link twist i is angle between zi-1 and zi measured in plane normal to xi (right-hand rule). • Link offset di is distance from origin of frame i-1 to the intersection xi with zi-1, measured along zi-1. • Joint angle i is angle from xi-1 to xi measured in plane normal to zi-1 (right-hand rule).

  20. DH convention to assign coordinate frames • Assign zi to be the axis of actuation for joint i+1 (unless otherwise stated zn coincides with zn-1). • Choose x0 and y0 so that the base frame is right-handed. • Iterative procedure for choosing oixiyizi depending on oi-1xi-1yi-1zi-1 (i=1, 2, , n-1): • a) zi−1 and zi are not coplanar; there is an unique shortest line segment from zi−1 to zi, perpendicular to both; this line segment defines xi and the point where the line intersects zi is the origin oi; choose yi to form a right-handed frame, • b) zi−1is parallel to zi; there are infinitely many common normals; choose xi as the normal passes through oi−1; choose oi as the point at which this normal intersects zi; choose yito form a right-handed frame, • c) zi−1 intersects zi; axis xi is chosen normal to the plane formed by ziand zi−1; it’s positive direction is arbitrary; the most natural choice of oi is the intersection of zi and zi−1, however, any point along the zi suffices; choose yito form a right-handed frame.

  21. Forward kinematics (1/2) Homogenous transformation matrix relating the frame oixiyizito oi-1xi-1yi-1zi-1: Ai specifies position and orientation of oixiyiziw.r.t. oi-1xi-1yi-1zi-1. Homogenous transformation matrix Tji expresses position and orientation of ojxjyjzj with respect to oixiyizi:

  22. Forward kinematics (2/2) Forward kinematics of a serial manipulator with n joints can be represented by homogenous transformation matrix Hn0 which defines position and orientation of the end-effector’s (tip) frame onxnynzn relative to the base coordinate frame o0x0y0z0:

  23. Case-study: RRR robot manipulator

  24. DH parameters of RRR robot manipulator

  25. Forward kinematics of RRR robot manipulator (1/2) Coordinate frame o3x3y3z3 is related with the base frame o0x0y0z0via homogenous transformation matrix: where

  26. Position of end-effector: Forward kinematics of RRR robot manipulator (2/2) , , Orientation of end-effector:

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