Inverse Kinematics

Inverse Kinematics Set goal configuration of end effector calculate interior joint angles Analytic approach – when linkage is simple enough, directly calculate joint angles in configuration that satifies goal Numeric approach – complex linkages At each time slice, determine joint movements that take you in direction of goal position (and orientation)

Forward Kinematics - review Articulated linkage – hierarchy of joint-link pairs Pose – linkage is a specific configuration Pose Vector – vector of joint angles for linkage Degrees of Freedom (DoF) – of joint or of whole figure Types of joints: revolute, prismatic • Tree structure – arcs & nodes • Recursive traversal – concatenate arc matrices • Push current matrix leaving node downward • Pop current matrix traversing back up to node

Inverse Kinematics End Effector L1 q3 q2 L3 L2 q1 Goal

Inverse Kinematics Underconstrained – if fewer constraints than DoFs Many solutions Overconstrained – too many constraints No solution Reachable workspace – volume the end effector can reach Dextrous workspace – volume end effector can reach in any orientation

Inverse Kinematics - Analytic Given arm configuration (L1, L2, …) Given desired goal position (and orientation) of end effector: [x,y] or [x,y,z, y1,y2, y3] Analytically compute goal configuration (q1,q2) Interpolate pose vector from initial to goal

Analytic Inverse Kinematics q2 L2 Goal L1 (X,Y) q1

Analytic Inverse Kinematics Multiple solutions Goal (X,Y)

Analytic Inverse Kinematics L2 180- q2 L1 (X,Y) q1 qT

Analytic Inverse Kinematics L2 180- q2 L1 (X,Y) q1 Y qT X

Law of Cosines C A a B

Analytic Inverse Kinematics L2 (X,Y) L1 180- q2 q1 qT Y X

Analytic Inverse Kinematics

Iterative Inverse Kinematics When linkage is too complex for analytic methods At each time step, determine changes to joint angles that take the end effector toward goal position and orientation Need to recompute at each time step

Inverse Jacobian Method a2 d2=EF-J2 End Effector a2 x d2 q2 - Compute instantaneous effect of each joint - Linear approximation to curvilinear motion - Find linear combination to take end effector towards goal position

Inverse Jacobian Method Instantaneous linear change in end effector for ith joint = (EF - Ji) x ai

Inverse Jacobian Method What is the change in orientation of end effector induced by joint i that has axis of rotation ai and position Ji? Angular velocity

Inverse Jacobian Method Solution only valid for an instantaneous step Angular affect is really curved, not straight line Once a step is taken, need to recompute solution

Set up equations yi: state variable xi : system parameter fi : relate system parameters to state variable Inverse Jacobian Method - Mathematics

Matrix Form Inverse Jacobian Method - Mathematics

Inverse Jacobian Method - Mathematics Use chain rule to differentiate equations to relate changes in system parameters to changes in state variables

Inverse Jacobian Method - Mathematics Matrix Form

Change in position (and orientation) of end effector Change in joint angles Linear approximation that relates change in joint angle to change in end effector position (and orientation) Inverse Jacobian Method

Inverse Jacobian Method

= (S - J1) x a1 = w1 Inverse Jacobian Method

The Matrices

The Matrices N DoFs V – desired linear and angular velocities 3x1, 6x1 J – Jacobian Matrix of partials 3xN, 6xN N x 1 q – change to joint angles (unknowns)

Pseudo Inverse of the Jacobian

LU decomposition Solving using the Pseudo Inverse

A solution of this form When put into this formula Like this After some manipulation, you can show that it… …doesn’t affect the desired configuration Adding a Control Term But it can be used to bias The solution vector

Desired angles and corresponding gains are input ‘z’ is H differentiated Form of the Control Term Bias to desired angles (not the same as hard joint limits) Where the deviation is large, you bump up the solution vector in such a way that you don’t disturb the desired effect

Include this in equation Isolate vector of unknown Rearrange to isolate the inverse Some Algebraic Manipulation

LU decomp. Solving the Equations

Control Term Use to bias to desired mid-angle Does not enforce joint angles Does not address “human-like” or “natural” motion Only kinematic control – no forces involved

Other ways to numerically IK Jacobian transpose Alternate Jacobian – use goal position HAL – human arm linkage Damped Least Squares CCD

Jacobian Transpose Use projection of effect vector onto desired movement

Jacobian Transpose S

Jacobian Transpose …

Alternate Jacobian G Use the goal postion instead of the end-effector!!?? !? …

substitution Solve Damped Least Squares G

Hueristic Human-Like Linkage (HAL) 7 DoF linkage RA (q1,q2 ,q3) , RB ( q4), RC(q5,q6 ,q7) 3 DoF G Decompose into simpler subproblems 1 DoF Set hand position and rotation based on relative position of Goal to shoulder 3 DoF Fix wrist position – use as Goal

L L1 L2 Hueristic Human-Like Linkage (HAL) s Set q4 based on distance between shoulder and wrist w e Assume axis of elbow is perpendicular to plane defined by s, e, w use law of cosines

Hueristic Human-Like Linkage (HAL) s q4 w e Elbow lies on circle defined by w, s & q4 Determine elbow position based on heuristics For example: project forearm straight from hand orientation if arm intersects torso or a shoulder angle exceeds joint limit (or exceeds comfort zone) – Clamp to inside of limits

Hueristic Human-Like Linkage (HAL) s q4 w e From e and s, determine RA From e and w and hand orientation, determine RB

Cyclic-Coordinate Descent Traverse linkage from distal joint inwards Optimally set one joint at a time Update end effector with each joint change At each joint, minimize difference between end effector and goal Easy if only trying to match position; heuristic if orientation too Use weighted average of position and orientation.

Cyclic-Coordinate Descent .

Cyclic-Coordinate Descent Rotational joint: .

Inverse Kinematics