1. COMP322/S2000/L91 Direct Kinematics- The Arm Equation Link Coordinates and Kinematics Parameters (Cont‘d) Another example: A 5-axis articulated robot (Rhino XR-3) (refer to class notes for details) Exercise: Determine the link coordinates and the kinematics parameters of the Motoman robot arm the SONY arm (LAB-2)

2. COMP322/S2000/L92 The Arm Equation For a n-axis robot arm, the arm equation is P 0 = T P n where P 0 is a point (vector) w.r.t. frame L 0, ie. The base frame; P n is a point (vector) w.r.t. frame L n, ie. The tool frame; T is the transformation matrix. Question: What is T? T is the composite transformation from one frame to another and is expressed in terms of the kinematics parameters.

3. COMP322/S2000/L93 Arm Equation Consider the transformation from one frame to another, say from frame L k-1 to L k. Idea is find the transformation that will bring L k-1 to align with L k. 4 fundamental motions: First two motions: Rotate L k-1 about z k-1 to bring x k-1 parallel to x k, ie. by an angle of  k => pure rotation: Rot(  k, z k-1 ) Translate L k-1 along z k-1 to bring x k-1 align with x k, ie. by a distance of d k => pure translation: Trans(d k, z k-1 ) => Screw (d k,  k, z k-1 )

4. COMP322/S2000/L94 Arm Equation Second two motions: Translate L k-1 along x k-1 (x k ) to bring L k-1 and L k (the two origins) to coincide, ie. by a distance of a k => pure translation: Trans(a k, x k-1 ) Rotate L k-1 about x k-1 (x k ) to bring z k-1 to align with z k, ie. by an angle of  k => pure rotation: Rot(  k, x k-1 ) => Screw (a k,  k, x k-1 ) Let T k-1 k denote the transformation from frame L k-1 to frame L k, i.e.P k-1 = T k-1 k P k

5. COMP322/S2000/L95 Arm Equation Question: T k-1 k = Screw (d k,  k, z k-1 ) Screw (a k,  k, x k-1 ) = A ? or T k-1 k = Screw (a k,  k, x k-1 ) Screw (d k,  k, z k-1 ) = B ? Note: Refer to class notes for the details to answer the above question.