Inverse Kinematics (part 2)Forward KinematicsForward & Inverse KinematicsWelman, 1993Gradient DescentGradient Descent for f(x)=gMinimizationTaking Safe StepsInverse of the DerivativeGradient Descent AlgorithmJacobiansSlide 12Jacobian Inverse KinematicsSlide 14Slide 15Slide 16Slide 17Jacobian for a 2D Robot ArmJacobian MatricesIncremental Change in PoseIncremental Change in EffectorIncremental Change in eIncremental ChangesChoosing ΔeBasic Jacobian IK TechniqueComputing the JacobianComputing the Jacobian Matrix1-DOF Rotational JointsSlide 29Rotational DOFsSlide 313-DOF Rotational JointsSlide 33Slide 34Slide 35Quaternion JointsSlide 37Slide 38Translational DOFsSlide 40Slide 41Slide 42Building the JacobianUnits & ScalingSlide 45End Effector OrientationSlide 47Scaled Rotation Axis6-DOF End EffectorDesired Change in OrientationEnd EffectorSlide 52Slide 53Slide 54Inverting the Jacobian MatrixInverting the JacobianUnderconstrained SystemsOverconstrained SystemsWell-Constrained SystemsPseudo-InverseDegenerate CasesSingle Value DecompositionJacobian TransposeSlide 64Iterating to the SolutionIterationWhen to StopFinding a Successful SolutionLocal MinimaTaking Too LongIteration SteppingOther IK IssuesJoint LimitsHigher Order ApproximationRepeatabilityMultiple End EffectorsMultiple ChainsGeometric ConstraintsOther IK TechniquesJacobian Method as a Black BoxInverse Kinematics (part 2)CSE169: Computer AnimationInstructor: Steve RotenbergUCSD, Winter 2004Forward KinematicsWe will use the vector:to represent the array of M joint DOF valuesWe will also use the vector:to represent an array of N DOFs that describe the end effector in world space. For example, if our end effector is a full joint with orientation, e would contain 6 DOFs: 3 translations and 3 rotations. If we were only concerned with the end effector position, e would just contain the 3 translations. M...21Φ Neee ...21eForward & Inverse KinematicsThe forward kinematic function computes the world space end effector DOFs from the joint DOFs:The goal of inverse kinematics is to compute the vector of joint DOFs that will cause the end effector to reach some desired goal state Φe f eΦ1 fWelman, 1993“Inverse Kinematics and Geometric Constraints for Articulated Figure Manipulation”, Chris Welman, 1993Masters thesis on IK algorithmsExamines Jacobian methods and Cyclic Coordinate Descent (CCD)Please read sections 1-4 (about 40 pages)Gradient DescentWe want to find the value of x that causes f(x) to equal some goal value gWe will start at some value x0 and keep taking small steps:xi+1 = xi + Δxuntil we find a value xN that satisfies f(xN)=gFor each step, we try to choose a value of Δx that will bring us closer to our goalWe can use the derivative to approximate the function nearby, and use this information to move ‘downhill’ towards the goalGradient Descent for f(x)=gf-axisx-axisxif(xi)df/dxgxi+1MinimizationIf f(xi)-g is not 0, the value of f(xi)-g can be thought of as an error. The goal of gradient descent is to minimize this error, and so we can refer to it as a minimization algorithmEach step Δx we take results in the function changing its value. We will call this change Δf.Ideally, we could have Δf = g-f(xi). In other words, we want to take a step Δx that causes Δf to cancel out the errorMore realistically, we will just hope that each step will bring us closer, and we can eventually stop when we get ‘close enough’This iterative process involving approximations is consistent with many numerical algorithmsTaking Safe StepsSometimes, we are dealing with non-smooth functions with varying derivativesTherefore, our simple linear approximation is not very reliable for large values of ΔxThere are many approaches to choosing a more appropriate (smaller) step sizeOne simple modification is to add a parameter β to scale our step (0≤ β ≤1)  1dxdfxfgxiInverse of the DerivativeBy the way, for scalar derivatives:dfdxdxdfdxdf11Gradient Descent Algorithm     } newat evaluate // along step // take1slope compute/ / { whileat evaluate // valuestarting initial11110000iiiiiiiiinxfxffxsfgxxxdxdfsgfxfxffxJacobiansA Jacobian is a vector derivative with respect to another vectorIf we have a vector valued function of a vector of variables f(x), the Jacobian is a matrix of partial derivatives- one partial derivative for each combination of components of the vectorsThe Jacobian matrix contains all of the information necessary to relate a change in any component of x to a change in any component of fThe Jacobian is usually written as J(f,x), but you can really just think of it as df/dxJacobians NMMNxfxfxfxfxfxfxfddJ...........................,1221212111xfxfJacobian Inverse KinematicsJacobiansLet’s say we have a simple 2D robot arm with two 1-DOF rotational joints:φ1φ2• e=[ex ey]JacobiansThe Jacobian matrix J(e,Φ) shows how each component of e varies with respect to each joint angle 2121,yyxxeeeeJ ΦeJacobiansConsider what would happen if we increased φ1 by a small amount. What would happen to e ?φ1•111yxeeeJacobiansWhat if we increased φ2 by a small amount?φ2•222yxeeeJacobian for a 2D Robot Armφ2•φ1 2121,yyxxeeeeJ ΦeJacobian MatricesJust as a scalar derivative df/dx of a function f(x) can vary over the domain of possible values for x, the Jacobian matrix J(e,Φ) varies over the domain of all possible poses for ΦFor any given joint pose vector Φ, we can explicitly compute the individual components of the Jacobian matrixIncremental Change in PoseLets say we have a vector ΔΦ that represents a small change in joint DOF valuesWe can approximate what the resulting change in e would be: ΦJΦΦeΦΦee  ,JddIncremental Change in EffectorWhat if we wanted to move the end effector by a