6CCE3ROS/7CCSMROB ROBOTIC SYSTEM King’s College London
Dr Shan UTORIAL 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}.
Solution 1:
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First, we calculate 0𝑇, and then differentiating 0𝑃
. Finally, 4𝐽(Θ) can be calculated as 𝑑𝑖
0𝑅 0𝐽(Θ). 4
Fig. 1. A 3R nonplanar arm
0 𝐿3 0 𝐶1 −𝑆1 00
0𝑇 = [𝑆1 𝐶1 0 0 ] 10010 0001
𝐶2 −𝑆2 0 𝐿1
1𝑇=[0 0 −1 0]
2 𝑆2 𝐶2 0 0
0001 𝐶3 −𝑆3 0 𝐿2
2𝑇 = [𝑆3 𝐶3 0 0 ] 30010 0001
3𝑇=[0 1 0 0] 4 0010 0001
𝐶1𝐶23 −𝐶1𝑆23 𝑆1 𝐿1𝐶1 + 𝐿2𝐶1𝐶2 + 𝐿3
0𝑇 = 0𝑇1𝑇2𝑇3𝑇 = [𝑆1𝐶23 −𝑆1𝑆23 −𝐶1 𝐿1𝑆1 + 𝐿2𝑆1𝐶2 4 1234 𝑆23 𝐶23 0 𝐿2𝑆2
Solution 2:
Velocity propagation.
1𝑊 =[0] 1𝑣 =[0] 1̇1
̇ 0 𝐶2 0 𝑆2 0 0 𝑆2𝜃1
2𝑊 = 2𝑅 1𝑊 + [ 0 ] = [−𝑆 0 𝐶 ] [ 0 ] + [ 0 ] = [𝐶 𝜃̇ ] 211̇22̇̇21
𝜃2 0−10𝜃1 𝜃2 ̇ 𝜃2
2 2 1 1 1 𝐶2 0 𝑆2 0 0 0 𝑉=𝑅(𝑉+𝑊×𝑃)=[−𝑆 0 𝐶]([0]+[𝐿𝜃̇])=[0]
0 −1 0 0 0 −𝐿1𝜃1
̇̇ 0 𝐶3 𝑆3 0 𝑆2𝜃1 0 𝑆23𝜃1
3𝑊=3𝑅2𝑊+[0]=[−𝑆 𝐶 0][𝐶𝜃̇]+[0]=[𝐶 𝜃̇ ] 3 2 2 ̇ 3 3 21 ̇ 231
𝜃3 0 0 1 ̇ 𝜃3 ̇ ̇ 𝜃2 𝜃2 + 𝜃3
̇ 3 3 2 2 2 𝐶3 𝑆3 0 0 0̇ 𝑆3𝐿2𝜃2
𝑉=𝑅(𝑉+𝑊×𝑃)=[−𝑆 𝐶 0]([0]+[𝐿2𝜃2 ])=[ 𝐶𝐿𝜃̇ 3222333̇322
4𝑊 = 3𝑊 43
001−𝐿1𝜃1−𝐿𝐶𝜃̇ ̇ ̇ 2 2 1 −𝐿1𝜃1−𝐿2𝐶2𝜃1
44333333 ̇̇
𝑉=𝑅(𝑉+ 𝑊× 𝑃)= 𝑉+ 𝑊× 𝑃=[ 𝐶𝐿𝜃̇ ]+[−𝐿3(𝜃2+𝜃3)] 4333𝑦334322
0 𝐶3𝐿2 𝐿3 ] −𝐿1 −𝐿2𝐶2 −𝐿3𝐶23 0 0
̇̇̇ 𝐶3𝐿2𝜃2+𝐿3(𝜃2 + 𝜃3) ]
−𝐿1𝜃1 − 𝐿2𝐶2𝜃1 − 𝐿3𝐶23𝜃1
−𝐿1𝜃1−𝐿2𝐶2𝜃1
2. A certain two-link manipulator has the following Jacobian:
0𝐽(Θ) = [−𝑙1𝑠1 − 𝑙2𝑠12 −𝑙2𝑠12] . 𝑙1𝑐1 + 𝑙2𝑐12 𝑙2𝑐12
Ignoring gravity, what are the joint torques required in order that the manipulator will apply a static
0̂ force vector 𝐹 = 10𝑋0?
𝜏 = 0𝐽𝑇(Θ) 0𝐹
𝜏 = [−𝑙1𝑠1 − 𝑙2𝑠12 𝑙1𝑐1 + 𝑙2𝑐12] [10]
𝜏1 = −10𝑙1𝑠1 − 10𝑙2𝑠12
𝜏2 = −10𝑙2𝑠12
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