Kinematic representation! Iterative methods! Optimization methods

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Colleen Adams
5 years ago
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1 Human Kinematics

3 Kinematic representation! Iterative methods! Optimization methods

4 Kinematics Forward kinematics! given a joint configuration, what is the position of an end point on the structure?! Inverse kinematics! given the position for an end point on the structure, what angles do the joints need be to achieve that end point?

5 Quiz Which function is inverse kinematics? θ φ σ θ φ σ p θ, φ, σ = f(p) p p = f(θ, φ, σ)

6 Why inverse? More intuitive control! Maintain environment constraints! Calculate desired joint angles for control

7 Motion representation Global position and orientation are determined by 3 translational and 3 rotational DOFs Each joint can have up to 3 DOFs 1 DOF: knee 2 DOF: wrist 3 DOF: arm

8 Configurations q = x, y, z,,,, th, th, th, kn, x, y, z,,, θ thigh,φ thigh,σ thigh θ knee θ ankle,φ ankle

9 Constraints x, y, z,,, θ thigh,φ thigh,σ thigh d 0 Position constraint d 1 θ knee C(q) =h(q) p = 0 d 2 p θ ankle,φ ankle h(q) h(q) =T(x, y, z)r(,, )T(d 0 )R( th, th, th )T(d 1 )R( kn )T(d 2 )R( an, an )x a point on foot in world coordinates a point on foot in local coordinates

10 Solutions Closed form solutions can only be found for fairly simple mechanisms! Numerical solutions! No solution! Single solution! Multiple solution

11 Under-specified problem Multiple solutions! Mostly bad! How do we find the optimal solution?! Heuristics (move the outermost links first)! Closest to the current configuration! Energy minimization! Natural looking motion (whatever it means)

13 Kinematic representation! Iterative methods! Optimization methods

14 Iterative method Use inverse of Jacobian to iteratively step all the joint angles towards the goal! Girard and Maciejewski, Computational modeling for the computer animation of legged figures, SIGGRAPH 85

15 Jacobian matrix Constraint (m) i DOF (n) j C i q j Jacobian is a m by n matrix that relates differential changes of q to changes of C Jacobian maps the velocity in joint space to velocities in Cartesian space Jacobian depends on current state

16 IK and the Jacobian J = C q C = J q q = J 1 C q new = q + tj 1 C Linearize about current q

17 Invert Jacobian But Jacobian is most likely non-square Compute the pseudo inverse Jacobian J + C = J q J T C = J T J q (J T J) 1 J T C =(J T J) 1 J T J q J + C = q J + =(J T J) 1 J T

18 Under-constrained Where #DOFs is greater than #constraints, the problem is under-constrained and J T J is not invertible = = J T (JJ T )

19 Problems Pseudo inverse returns the least square minimum norm solution! Singularities cause deficient Jacobian! Linearization about current state causes error

20 Kinematic representation! Iterative methods! Optimization methods

21 Optimization method Find a solution that optimizes some numeric metric and satisfies constraints! Numeric metric! A function of q that measures the quantity to be minimized! Also called objective function

22 Optimization method Solve for joint configuration q min q subject to G(q) C(q) =0

23 Constraint derivatives We need a direction to move joints in such way that the handle moves towards the goal! The derivative of the constraint, i.e. Jacobian, gives directions where handles move if joints move h(q) p C(q) =h(q) p = 0 C(q) q = h(q) q

24 Constraint derivatives C(q) =h(q) p = 0 h(q) =T(x, y, z)r(θ 0,φ 0,σ 0 )TR(θ 1 )TR(θ 2,φ 2 )h k q =[x, y, z, θ 0,φ 0,σ 0,θ 1,θ 2,φ 2 ] C(q) q = h(q) q x, y, z, θ 0,φ 0,σ 0 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 p Need to know how to compute derivatives for each transformation θ 2,φ 2 h(q) h k : local coordinate of h

25 Quiz What is the derivative for the translation matrix?!!!! What is the derivative for a rotation matrix about z axis? cos sin 0 0 sin cos

26 Constraint derivatives h(q) What is the most efficient way to compute the? q Compute the gradient for the DOF on the outermost link first x, y, z, θ 0,φ 0,σ 0 h = T 0 (x, y, z)r 0 (θ 0,φ 0,σ 0 )T 1 R 1 (θ 1 )T 2 R 2 (θ 2,φ 2 )h k θ 1 p θ 2,φ 2 h(q) h k : local coordinate of h

27 Quiz What s the dimension of a system with n dofs and m position constraints?

28 Optimization method Solve for joint configuration q min q subject to G(q) C(q) =0

29 Objective functions Joint velocity! Power consumption! Similarity to the rest pose! Similarity to the natural pose

30 Unconstrained optimization Treat each constraint as a separate metric and minimize weighted sum of all metrics! Also called penalty methods! each spring pulls on constraint with force proportional to violation of the constraint

31 Unconstrained optimization Softening constraints by adding them to the objective function Minimize F (q) =G(q)+ i w i C i (q) 2

32 Gradient descent For each iteration, move in the direction of @q 2 X i w T C i Update the state q new = q α F q

33 Optimization loop while F (q) @q +2X i w i )T C i update q

34 Search step size The step size in each search iteration can be determined arbitrarily, through a binary search, or solved by another optimization process Step size: α where min α F (q + α q) q new = q + α q

35 Unconstrained optimization Pros:! simple and fast, no linear system to solve! near-singular configurations is less of a problem! Cons:! can t maintain constraints exactly! constraints fight against each other and the objective function

36 Constrained optimization Treat constraints as hard constraints! Use Lagrangian formulation to solve a nonlinear optimization L(q,λ)=G(q) λ C min q,λ L(q,λ)

37 Constrained method Pros! Enforce constraints exactly! Quadratic convergence! Cons! Large system of equations! Near-singular configurations cause instability

38 Leg example L0 hip T 0h T h1 R(q 0 )R(q 1 )R(q 2 ) L1 knee T 1k R(q 3 ) L2 T k2 ankle L3 R(q 4 )R(q 5 ) T 2a T a3 h 0

39 Leg example node0 hip q 0, q 1, q 2 q = {q 0, q 1, q 2, q 3, q 4, q 5 } node1 knee node2 q 3 To evaluate gradients of the obj function at each iteration, we need to evaluate constraint and compute Jacobian. C(q) = T 0h R 0 R 1 R 2 T h1 T 1k R 3 T k2 T 2a R 4 R 4 T a3 h 0 - p How to compute J(q)? q 4, q 5 node3 ankle h 0 p

40 Compute Jacobian T 0h T h1 R(q 0 )R(q 1 )R(q 2 ) Consider the derivative of C with respect to q 5 joint T 1k R(q 3 ) T k2 T 2a R(q 4 )R(q 5 ) T a3 h 0 = T 0h R(q 0 )R(q 1 )R(q 2 )T h1 T 1k R(q 3 )T k2 T 2a R(q 4 T a3 h 5 node->getparentbodynode()->gettransform() joint->gettransformfromparentbodynode() joint->gettransform(0) joint->gettransformderivative(1) joint->gettransformfromchildbodynode().inverse()

Reading. Realistic Character Animation. Modeling Realistic Motion. Two Approaches

Reading. Realistic Character Animation. Modeling Realistic Motion. Two Approaches Realistic Character Animation Reading Jessica Hodgins,,et.al,Animating Human Athletics,, SIGGRAPH 95 Zoran Popović, Changing Physics for Character Animation,, SIGGRAPH 00 2 Modeling Realistic Motion Model