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6CCE3ROS/7CCSMROB ROBOTIC SYSTEM King’s College London Dr Shan Luo TUTORIAL 6 1. Get the Jacobian of the three-link manipulator below. Write in terms of a frame {4} located at the tip of the hand and having the same orientation as frame {3}. Fig. 1. A 3R nonplanar arm Ans: Solution 1: First, we calculate 4 0 , and then differentiating 0 . Finally, (4 Θ) can be calculated as 4 0 (0 Θ). 1 1 0 0 0 90° 1 0 0 2 0 0 3 0 1 0 = [ 1 1 0 0 1 1 0 0 0 0 1 0 0 0 0 1 ] 2 1 = [ 2 2 0 1 0 0 1 0 2 2 0 0 0 0 0 1 ] 3 2 = [ 3 3 0 2 3 3 0 0 0 0 1 0 0 0 0 1 ] 4 3 = [ 1 0 0 3 0 1 0 0 0 0 1 0 0 0 0 1 ] 4 0 = 1 0 2 1 4 3 3 2 = [ 123 123 1 11 + 212 + 3 123 123 1 11 + 212 23 23 0 22 0 0 0 1 ] Solution 2: Velocity propagation. 1 1 = [ 0 0 1 ] 1 1 = [ 0 0 0 ] 2 2 = 1 2 1 1 + [ 0 0 2 ] = [ 2 0 2 2 0 2 0 1 0 ] [ 0 0 1 ] + [ 0 0 2 ] = [ 2 1 2 1 2 ]

2 2 = 1 2 ( 1 1 + 1 1 × 1 2) = [ 2 0 2 2 0 2 0 1 0 ] ([ 0 0 0 ] + [ 0 1 1 0 ]) = [ 0 0 1 1 ] 3 3 = 2 3 2 2 + [ 0 0 3 ] = [ 3 3 0 3 3 0 0 0 1 ] [ 2 1 2 1 2 ] + [ 0 0 3 ] = [ 23 1 23 1 2 + 3 ] 3 3 = 2 3 ( 2 2 + 2 2 × 2 3) = [ 3 3 0 3 3 0 0 0 1 ] ([ 0 0 1 1 ] + [ 0 2 2 22 1 ]) = [ 32 2 32 2 1 1 22 1 ] 4 4 = 3 3 4 4 = 3 4 ( 3 3 + 3 3 × 3 ) = 3 3 + 3 3 × 3 4 = [ 32 2 32 2 1 1 22 1 ] + [ 0 3( 2 + 3) 323 1 ] = [ 32 2 32 2+3( 2 + 3) 1 1 22 1 323 1 ] (4 Θ) = [ 0 32 0 0 32 3 1 22 323 0 0 ] 2. A certain two-link manipulator has the following Jacobian: (Θ) = [ 11 212 212 11 + 212 212 ] .0 Ignoring gravity, what are the joint torques required in order that the manipulator will apply a static force vector = 10 0 0 Ans: = (Θ) 00 = [ 11 212 11 + 212 212 212 ] [ 10 0 ] 1 = 1011 10212 2 = 10212