1. COMP322/S2000/L111 Inverse Kinematics Given the tool configuration (orientation R w and position p w ) in the world coordinate within the work envelope, find the joint variables, (q). Formulation: R 0 n (q) | p 0 n (q) R w | p w T 0 n (q) = - - - - - - - - - - - - - = - - - - - - - - - 0 0 0 | 1 0 0 0 | 1 ==> 12 equations, n unknowns; r 1 (q) r 2 (q) r 3 (q) | p 0 n (q) Re-writing, T 0 n (q) = - - - - - - - - - - - - - - - - - - - - - 0 0 0 | 1

2. COMP322/S2000/L112 Indirect Kinematics Recall: r 1 (q), r 2 (q), r 3 (q) are orthonormal vectors, ==> r 1 (q) r 2 (q) = 0; r 2 (q) r 3 (q) = 0; r 1 (q) r 3 (q) = 0; and || r 1 (q) || = 1; || r 2 (q) || = 1; || r 3 (q) || = 1. ==> 6 independent constraints ==> solution exists for q if n >= 6 Note: Solution is NOT unique and may not exist. To reduce the number of equations, a tool configuration vector, W is defined.

3. COMP322/S2000/L113 Inverse Kinematics - Tool Configuration Vector Definition: The tool configuration vector, W, in R 6 is defined as p 0 n (q) W(q) = - - - - - - - - - - - - - - - - - exp( q n /  ) r 3 (q) where p 0 n (q) is the position of the tool frame; q n is the roll angle of the tool (  n ); and r 3 (q) is the unit vector giving the direction of the approach vector (z n ). Thus, the number of equations becomes 6.

4. COMP322/S2000/L114 Formulation of Direct & Inverse Kinematics Given a robot arm: Step 0: Determine the vector of joint variables, q. Step 1: Derive the link coordinates of the arm based upon the D-H representation and algorithm; Step 2: Derive kinematics parameters ( , d, a,  ); Step 3: Derive the arm equation, i.e. the transformation matrix, R 0 n (q) | p 0 n (q) T 0 n (q) = - - - - - - - - - - - - - 0 0 0 | 1 Step 4: Direct Kinematics: Given q, compute T 0 n (q) ; Step 5: Inverse Kinematics: Given the tool configuration vector, W, compute the vector of joint variables, q. Example: 5-axis spherical robot (details in class notes)