ECE 450 Introduction to Robotics

1. ECE 450 Introduction to Robotics Section: 50883 Instructor: Linda A. Gee 9/28/99 Lecture 08

2. Transformation Matrices for End Effectors Position Orientation 1 0 0 px 0 [n s a] or R 0 1 0 py Tposition = 0 0 0 1 pz 0 0 0 0 1 0 0 0 1

3. Euler Angle Representation of a Rotation Matrix • There are three Euler angle methods of representation • Eulerian Angles (System I) R,, =Rz, Ru, Rw, • Euler Angles (System II) R,, =Rz, Rv, Rw, • Roll, Pitch, Yaw:RPY(System III) R,, =Rz, Ry, Rx, z w v y u z x Roll  Pitch  Yaw  y x

4. Specifications of Hand Position • The position of the hand can be specified in other coordinates • Cylindrical • Spherical

5. 1 0 0 px 0 where 0R6 = Rotation matrix in Euler Angles, 0R6 [ n s a ], Roll, Pitch, Yaw 0 1 0 py 0T6 = 0 0 0 1 pz 0 0 0 0 1 0 0 0 1 Resultant Arm Transformation Matrix represents the arm transformation matrix and describes position and orientation of hand w.r.t. the base coordinate frame 0T6 (3x3 : orientation of hand; 3x1: location of end effector)

6. a s n d y r  0 1 0 0 rcos 0R6 x 0 0 1 0 rsin 0T6 = 0 0 0 1 d 0 0 0 1 0 0 0 1 End Effector Position using Cylindrical Coordinates z Tcylindrical = Tz,d Tz,Tx,r px = rcos py = rsin pz = d

7. 0 1 0 0 rcossin 0R6 0 0T6 = 0 1 0 rsinsin 0 0 0 1 rcos 0 0 0 1 0 0 0 1 End Effector Position using Spherical Coordinates z a Tspherical = Tz,Ty,Tz,r s px = rcos sin py = rsin sin pz = r cos r n  y  x