Inverse Kinematics Kris Hauser CS B 659 Principles

Inverse Kinematics Kris Hauser CS B 659: Principles of Intelligent Robot Motion Spring 2013

Articulated Robot q 2 q 1 • Robot: usually a rigid articulated structure • Geometric CAD models, relative to reference frames • A configuration specifies the placement of those frames (forward kinematics)

Inverse Kinematics • Problem: given desired position of point on robot (end effector), what joint angles reach it?

Inverse Kinematics • Problem: given desired position of point on robot (end effector), what joint angles reach it? q 4 q 2 q 3 q 1

Inverse Kinematics • Bring point xn on link n to target x. T • Find q s. t. x. T = Tn(q)xn q 4 q 2 q 3 q 1

Solving a general equation • Solve f(q) = 0 (vector valued nonlinear function) • Can include rotation constraints, multiple IK targets q 4 q 2 q 3 q 1

Two Approaches for IK • Analytical. Write out the equation in terms of q and “invert” it • Sometimes there are simple solutions for certain kinematic structures (e. g. , industrial robots). If so, computation is fast. • Numerical. Iteratively move q to a solution point f(q)=0 • General purpose • Can incorporate joint limits easily • May fall into local minima

Analytical endpoint positioning for a 2 R robot arm Without loss of generality, let joint 2 and E. E. position lie on x axis at the reference frame, joint 1 at origin x(q)-x. D = 0 L 2 L 1 q 2 q 1 x. D

Analytical endpoint positioning for a 2 R robot arm Without loss of generality (why? ), let joint 2 and E. E. position lie on x axis at the reference frame, joint 1 at origin x(q)-x. D = 0 L 2 L 1 q 2 q 1 x. D ||x(q)||2 = ||x. D||2 (lhs depends only on q 2) ||x(q)||2 = (L 1 + c 2 L 2)2 + (s 2 L 2)2 = L 12 + 2 c 2 L 2 L 1 + c 22 L 22 + s 22 L 22 = L 12 + L 22 + 2 c 2 L 2 L 1

Analytical endpoint positioning for a 2 R robot arm Without loss of generality, let joint 2 and E. E. position lie on x axis at the reference frame, joint 1 at origin x(q)-x. D = 0 L 2 L 1 q 2 q 1 x. D ||x(q)||2 = ||x. D||2 (lhs depends only on q 2) ||x(q)||2 = (L 1 + c 2 L 2)2 + (s 2 L 2)2 = L 12 + 2 c 2 L 2 L 1 + c 22 L 22 + s 22 L 22 = L 12 + L 22 + 2 c 2 L 2 L 1 c 2 = (||x. D||2 - L 12 - L 22)/(2 L 2 L 1) c 2(q 2) Elbow up Elbow down If rhs > 1, x. D is out of reach If rhs < -1, x. D is inside “inner circle”

Analytical endpoint positioning for a 2 R robot arm Without loss of generality, let joint 2 and E. E. position lie on x axis at the reference frame, joint 1 at origin x(q)-x. D = 0 q 2 up L 1 q 1 x. D q 2 down ||x(q)||2 = ||x. D||2 (lhs depends only on q 2) ||x(q)||2 = (L 1 + c 2 L 2)2 + (s 2 L 2)2 = L 12 + 2 c 2 L 2 L 1 + c 22 L 22 + s 22 L 22 = L 12 + L 22 + 2 c 2 L 2 L 1 c 2 = (||x. D||2 - L 12 - L 22)/(2 L 2 L 1) c 2(q 2) Elbow up Elbow down If rhs > 1, x. D is out of reach If rhs < -1, x. D is inside “inner circle”

Analytical endpoint positioning for a 2 R robot arm Without loss of generality, let joint 2 and E. E. position lie on x axis at the reference frame, joint 1 at origin x(q)-x. D = 0 q 2 L 1 q 1 x. D Once q 2 is found, consider angle θD of x. D w. r. t origin

Analytical endpoint positioning for a 2 R robot arm Without loss of generality, let joint 2 and E. E. position lie on x axis at the reference frame, joint 1 at origin x(q)-x. D = 0 q 2 L 1 q 1 x. D Once q 2 is found, consider angle θD of x. D w. r. t origin Compute angle θ of E. E. at q=(0, q 2)

Analytical endpoint positioning for a 2 R robot arm Without loss of generality, let joint 2 and E. E. position lie on x axis at the reference frame, joint 1 at origin x(q)-x. D = 0 L 1 q 1 x. D q 2 Once q 2 is found, consider angle θD of x. D w. r. t origin Compute angle θ of E. E. at q=(0, q 2) Let q 1 = θD-θ Done!

Analytical positioning and orienting for a 3 R robot arm • Left as an exercise • Hint: consider solution in prior slides for choosing the first two joint angles so that the third joint is located at the correct location. Then pick the third.

Drawbacks to Analytical IK • Most 6 DOF robots encountered in practice are designed to have convenient analytical IK solutions (e. g. , 3 intersecting orthogonal axes) • General methods for 6 DOF robots require solving a high-degree (16) polynomial, which is computationally expensive and suffers from numerical difficulties • What about a redundant manipulator (> 6 DOF)?

Papers

Multiplicity issue • Let n = # of DOFs, m = # of constraints • Roughly, in common cases • If n < m, there will be 0 IK solutions • If n = m, there may be 1 or more solution • If n > m, there will either be 0 or infinite number of solutions • Singularities: the uncommon case

Null Space • A motion from q->q’ that maintains f(q)=0 is known as a motion in the null space.

Null Space • Null space velocity dq must satisfy J(q)dq = 0 • => dq lies in the null space of J(q) • For any vector y, (I-J+J)y lies in the null space • Recall J+ is the pseudoinverse of J • A basis of the null space can be found by SVD • • • J = UWVT Let the last k diagonal entries of W be 0, first n-k nonzero WVTdq = 0 First n-k entries of VTdq must be zero Last k entries of VTdq may be non zero => Last k columns of V are a basis for null space

Reminder: Project Proposals • 2 -3 paragraphs, due 2/1 • Include: • Title, group members • Motivation, significance, and expected demonstration. • List of 3 -4 milestones • Milestone 1 must be completed, or else you will feel like a total failure. Usually complete this before spring break. • Milestone 2 should be completed, or else you will feel bad and will deserve a less than stellar grade. • Completing milestone 3 will make you (and me) feel pleased with your project. • Milestone 4 will make you (and me) very excited, yet is still possible. (Don‘t promise to cure cancer. ) • Any other details, e. g. , implementation, that may be relevant

Recap • Two general ways of solving inverse kinematics: analytical and numerical • With nonredundant manipulators, there a finite number of solutions (usually > 1, without joint limits) • With redundant manipulators, there an infinite number of solutions • Null space motions