Use of multilayer perceptrons as Inverse Kinematics solvers Nathan Mitchell University of Wisconsin, Madison December 14, 2010 1 of 12
Introduction 1. Scope 2. Background 3. Methodology 4. Expected Results 2 of 12
Project Scope The goal of this project is to examine the effectiveness of a multilayer perceptron as a function approxiator [?] for the inverse kinematics, or IK, problem. For the purposes of this project, inverse kinematics will have the following definition: Inverse Kinematics Problem Given a series of N rigid elements, E = {E 1, E 2,..., E N }, connected by rotational joints J = {J 12, J 23,..., J (N 1)(N) } with K degrees of freedom, where J uv connects element U to element V, and a target position t, determine angles J, such that the free tip of E N is located at t while the free tip of E 1 is fixed at the origin. For purposes of easier visualization in MatLab, we will set the value of K = 1 for the scope of this project. 3 of 12
Background and Purpose 1. IK important in many areas, including robotic control [?] and animation [?] [?]. 2. Animation often requires real-time or reasonable time computations, and robotics require robust algorithms to deal with mechanical flaws [?]. 3. Need fast numerical solutions to the complex nonlinear problem of IK. 4. Standard methods include the Jacobian and Cyclic Coordinate Descent, or CCD [?] [?] [?]. 4 of 12
Experimental Setup 1. The experiment will consist of a simple two dimensional linkage, with N links and variable link lengths. 2. A multilayer perceptron will then be constructed and trained on the linkage, whose output will be compared to the target point. The data points used will be chosen at random within the reachable sphere of the linkage. 3. The perceptron will accept the target as input and output N values as rotations. These rotations will be applied to the linkage and the error will be the difference between the resulting tip and the requested target. 4. Finally a different set of random points, along with a set of linear points, will be generated, and the perceptron will be tested and compared to the Jacobian and CCD methods. 5 of 12
Comparison criteria The effectiveness of the multilayer perceptron will be judged on three levels chosen for the concerns of the animation domain. 1. Correctness - Can the multilayer perceptron place the tip at the target as well as the Jacobian or CCD method? 2. Speed - Relative to the Jacobian and CCD, how fast is the multilayer perceptron at computing the answer? 3. Continuity - Do the joint positions from the multilayer perceptron move realistically when the target moves in a straight line? 6 of 12
Most Effective Perceptron Finally, these tests will be conducted with variations on the linkage with respect to the configuration of the multilayer perceptron. In particular, how do the following effect the performance of the approach: 1. Number of hidden layers 2. Activation Functions 7 of 12
Expected results While the experimental framework is still being coded, there are several key expected results: 1. A trained perceptron will only work for the linkage it was trained on. 2. The classical methods will provide higher quality results, but perhaps not the fastest. 8 of 12
Software in Use 1. The experiment framework will be written using MatLab code. 2. The multilayer perceptron code will be adapted from the demo code provided for the ECE539 class. 3. Based on earlier work in the semester, an existing implementation of the CCD method will be used. 4. The Jacobian method will be implemented from scratch in MatLab. 9 of 12
References (1) Simon Haykin. Neural Networks and Learning Machines. Prentice Hall, 3rd edition, 2009. Rick Parent. Computer Animation: Algorithms and Techniques. Morgan Kaufmann, 2nd edition, 2008. Peter Shirley and Steve Marschner. Fundamentals of Computer Graphics. A. K. Peters, 3rd edition, 2009. 10 of 12
References (2) L.C.T. Wang and C.C. Chen. A combined optimization method for solving the inverse kinematics problems of mechanical manipulators. Robotics and Automation, IEEE Transactions on, 7(4):489 499, August 1991. Chris Welman. Inverse kinematics and geometric constraints for articulated figure manipulation. Master s thesis, Simon Fraser University, 1993. 11 of 12
Questions Any Questions? 12 of 12