Kinematics. Why inverse? The study of motion without regard to the forces that cause it. Forward kinematics. Inverse kinematics

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Sabina Reynolds
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1 Kinematics Inverse kinematics The study of motion without regard to the forces that cause it Forward kinematics Given a joint configuration, what is the osition of an end oint on the structure? Inverse kinematics Given the osition for an end oint on the structure, what angles do the joints need to be to achieve that end oint? Kinematics Why inverse? Which function is inverse kinematics? θ φ σ θ, φ, σ = f() θ φ σ = f(θ, φ, σ) Secify fewer degrees of freedom (DOFs) More intuitive control Maintain environment constraints Calculate desired joint angles for control

2 An interactive IK system Joint configurations q = x, y, z, θelvis, φelvis, σelvis, θthigh, φthigh, σthigh, θknee,... Create a handle on the character θelvis, φelvis, σelvis Interactively ull the handle around θthigh, φthigh, σthigh IK system figures out the aroriate the joint configurations θknee θankle, φankle Constraints Solutions Position constraint Closed form solutions can only be found for fairly simle mechanisms C(q) = h(q) = 0 Numerical solutions Orientation constraint h(q) No solution Single solution C(q) = d(q) v = 0 Multile solution

3 Under-secified roblem Ik methods Multile solutions Mostly bad How do we find the otimal solution? Heuristics (move the outermost links first) Closest to the current configuration Energy minimization Iterative methods Otimization methods Unconstrained otimization Constrained otimization Natural looking motion (whatever it means) Iterative method Jacobian matrix Use inverse of Jacobian to iteratively ste all the joint angles towards the goal Girard and Maciejewski, Comutational modeling for the comuter animation of legged figures, SIGGRAPH 85 Constraint (m) j DOF (n) i C i j Jacobian is a m by n matrix that relating differential changes of q to changes of C Jacobian mas the velocity in state sace to velocities in Cartesian sace Jacobian deends on current state

4 IK and the Jacobian Invert Jacobian J = C C = J But Jacobian is most likely non-square Comute the seudo inverse Jacobian J + C = J = J 1 C q new = q + tj 1 C Linearize about current q J T C = J T J (J T J) 1 J T C = (J T J) 1 J T J J + C = J + = (J T J) 1 J T = J T (JJ T ) 1 Otimization method Otimization method Find a solution that otimizes some numeric metric and satisfies constraints Numeric metric A function of q that measures the quantity to be minimized Solve for joint configuration q subject to min G(q) q C(q) = 0 Also called objective function

5 Objective functions Constraint Derivatives Joint velocity Power consumtion Similarity to the rest ose Similarity to the natural ose What do we want? A direction to move joints in such way that the constraint handles move towards the goal What do derivatives tell us? A direction constraint handles move if joints move Constraint derivatives Constraint derivatives C(q) = h(q) = 0 C(q) = h(q) q = [x, y, z, θ 0, φ 0, σ 0, θ 1, θ 2, φ 2 ] x, y, z, θ 0, φ 0, σ 0 What is the most efficient way to comute the? h(q) h(q) = T(x, y, z)r(θ 0, φ 0, σ 0 )TR(θ 1 )TR(θ 2, φ 2 )h k x, y, z, θ 0, φ 0, σ 0 h(q) = T(x, y, z)r(θ 0, φ 0, σ 0 )TR(θ 1 )TR(θ 2, φ 2 )h k h(q) = T(x, y, z)r(θ 0, φ 0, σ 0 )T R(θ 1) TR(θ 2, φ 2 )h k θ 1 θ 1 θ 1 θ 1 Need to know how to comute derivatives for each transformation θ 2, φ 2 h(q) h k: local coordinate of h θ 2, φ 2 h(q) h k: local coordinate of h

6 Unconstrained Otimization Unconstrained otimization Treat each constraint as a searate metric and minimize weighted sum of all metrics Also called enalty methods each sring ulls on constraint with force roortional to violation of the constraint Minimize F (q) = G(q) + i Move in the direction of F = G 2 i w i C i (q) 2 w i C i C i Udate the state q new = q α F Search and ste Unconstrained otimization Find direction and the ste size that move joints closer to constraints Direction: q = F Ste size: α where min F (q + α q) α q new = q + α q Pros: simle and fast, no linear system to solve near-singular configurations is less of a roblem Cons: constraints fight against each other and the objective function can t maintain constraints exactly linear convergence

7 Constrained otimization Constrained method Solve a linear system comrised of Jacobians and the quadratic metric Also called Lagrangian Multiliers L(q, λ) = G(q) λ C min L(q, λ) q,λ Pros Enforce constraints exactly Quadratic convergence Cons Large system of equations Near-singular configurations cause instability

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