What is a GPSCE
• 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:
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Symbolic differentiation
Trigonometric simplification
Analytical solving
Index reduction
Linearization
Etc.
• Presentation will use Maple as the example GPSCE
© 2012 Maplesoft, a division of Waterloo Maple Inc.
A CYBERNET group company
Coordinate Selection
• A model’s chosen state variables directly impact the number and
complexity of the resulting equations
• Consider a Stewart Platform:
• Absolute coordinates:
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78 coords (12 per leg, 6 for the platform),
78 dynamic equations
+72 constraint equations = 150 equations
• Hybrid coordinates :
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24 coords (3 per leg, 6 for the platform),
24 dynamic equations
+18 constraint equations = 42 equations
© 2012 Maplesoft, a division of Waterloo Maple Inc.
A CYBERNET group company
Case Study: Inverse Dynamics of a Stewart Platform
© 2012 Maplesoft, a division of Waterloo Maple Inc.
A CYBERNET group company
Comparison of Approaches
• 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
© 2012 Maplesoft, a division of Waterloo Maple Inc.
A CYBERNET group company
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
Solving Method
Traditional (Two Models)
Solved Inverse Kinematics
Solved Inverse Kinematics and
Solved Inverse Dynamics
Relative Speeds
(x faster than real-time)
4x
8x
66x
© 2012 Maplesoft, a division of Waterloo Maple Inc.
A CYBERNET group company
Fastest Model Code
Simulation Time per Step (μs) - Smaller is Better
1200
Simulation Failed
1000
800
600
400
MapleSim
Dymola
SimMechanics
SimulationX
200
0
6-Link Pendulum
6
Stewart Platform Controlled Stewart
© 2012 Maplesoft, a division of Waterloo Maple Inc.
A CYBERNET group company