1. © 2012 Maplesoft, a division of Waterloo Maple Inc. A CYBERNET group company General Purpose Symbolic Computation Environment Equations are not simply text strings but objects that can be programmatically viewed, manipulated and solved GPSCEs provide commands for equation manipulation: – Symbolic differentiation – Trigonometric simplification – Analytical solving – Index reduction – Linearization – Etc. Presentation will use Maple as the example GPSCE What is a GPSCE

2. © 2012 Maplesoft, a division of Waterloo Maple Inc. A CYBERNET group company A model’s chosen state variables directly impact the number and complexity of the resulting equations Consider a Stewart Platform: Absolute coordinates: 78 coords (12 per leg, 6 for the platform), 78 dynamic equations +72 constraint equations = 150 equations Hybrid coordinates : 24 coords (3 per leg, 6 for the platform), 24 dynamic equations +18 constraint equations = 42 equations Coordinate Selection

3. © 2012 Maplesoft, a division of Waterloo Maple Inc. A CYBERNET group company Case Study: Inverse Dynamics of a Stewart Platform

4. © 2012 Maplesoft, a division of Waterloo Maple Inc. A CYBERNET group company Each inverse dynamics solution was exported to Simulink® as a MATLAB® sfunction using MapleSim’s Connector product Relative times were compared within Simulink® using a fixed-step solver – Simulation time: 100 s – Solver: Euler – Step size:.001 s Comparison of Approaches

5. © 2012 Maplesoft, a division of Waterloo Maple Inc. A CYBERNET group company Solving MethodRelative Speeds (x faster than real-time) Traditional (Two Models)4x Solved Inverse Kinematics8x Solved Inverse Kinematics and Solved Inverse Dynamics 66x Speed Comparison Machine: Intel Core i7, 1.7Ghz, 10GB RAM Traditional approach ran almost 4x faster than real-time (still takes advantage of the symbolic optimizations) Solved approach more than 16x faster than the traditional solution

6. © 2012 Maplesoft, a division of Waterloo Maple Inc. A CYBERNET group company 6 Fastest Model Code Simulation Failed