Lecture15-hierarchical-animation

4 1 1 4 2 1 θ ϖ θ ϖ θ θ θ θ z z y x x x v v v

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= 4 1 1 4 2 1 θ ϖ θ ϖ θ θ θ θ z z y x x x v v v v J Change in position Change in orientation Joint position velocities Relates a joint position change to end effector change Check dimensions: V = 6x1 J(θ) = 6x4 θ = 4x1

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Where we stand ( 29 θ θ J V = Can figure this out. Can figure this out. Want this. Instantaneous positional change vectors Desired change vector
Solve for θ ( 29 θ θ = - V J 1 Instantaneous positional change vectors Desired change vector Small problem: J(θ) may not be easy to invert.

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IK – Transpose of the Jacobian Compute how much the change vector contributes to the desired change vector: Project joint change vector onto desired change vector Dot product of joint change vector and desired change vector is what we want and it can be computed using the transpose of the Jacobian
IK – Transpose of the Jacobian θ = V J T = z y x z y x v v v V ϖ ϖ ϖ = 4 4 2 1 1 1 θ ϖ θ θ θ ϖ θ θ z x x z y x v v v v J

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Transpose of the Jacobian. Project change vector for joint i onto desired change. Those relative lengths determine degree of rotation applied to each joint. Joints who’s change vector are closer to parallel w/ desired change vector get rotated more. Then scale that by a constant It is just an approximation after all.
The big picture

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Computing Joint Positions Have: joint angles and segment lengths Have: location of base in world coordinates Need: joint position in world coordinates Traverse the arm computing positions in world coordinates as you go.
Computing Joint Positions (Example) Angle 1 = 92.95 degrees Seg. 1 is at (-0.05,0.998) Angle 2 = 68.5 deg. Seg 2 is at (-0.53, 1.16) Angle 3 = 37.0 Seg 3 is at (-1.00, 1.00) angle 1 angle 2 angle 3 Seg. 1 posn. x axis for joint 1 x a x i s f o r j o i n t 2 x axis for joint 3

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Computing Joint Positions Angle 1 = 92.95 degrees Seg. 1 is at (-0.05,0.998) Angle 2 = 68.5 deg.
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