© 2014 Maplesoft, a division of Waterloo Maple Inc.
A CYBERNET group company
System Modelling with Symbolics
Chad Schmitke...
© 2014 Maplesoft, a division of Waterloo Maple Inc.
A CYBERNET group company
• System Modeling
• Numeric vs. Symbolic
– Ma...
© 2014 Maplesoft, a division of Waterloo Maple Inc.
A CYBERNET group company
• What are they?
– Models of systems that inc...
© 2014 Maplesoft, a division of Waterloo Maple Inc.
A CYBERNET group company 4
System Modeling
© 2014 Maplesoft, a division of Waterloo Maple Inc.
A CYBERNET group company 5
System Modeling
© 2014 Maplesoft, a division of Waterloo Maple Inc.
A CYBERNET group company
• Talking about formulation approach, not
sim...
© 2014 Maplesoft, a division of Waterloo Maple Inc.
A CYBERNET group company
Coordinate Selection
Equation Generation
Symb...
© 2014 Maplesoft, a division of Waterloo Maple Inc.
A CYBERNET group company 8
Math -> Simulation
© 2014 Maplesoft, a division of Waterloo Maple Inc.
A CYBERNET group company 9
Analysis
• Solve for the motion
of the legs...
© 2014 Maplesoft, a division of Waterloo Maple Inc.
A CYBERNET group company 10
Analysis
© 2014 Maplesoft, a division of Waterloo Maple Inc.
A CYBERNET group company 11
Analysis
© 2014 Maplesoft, a division of Waterloo Maple Inc.
A CYBERNET group company 12
Analysis
© 2014 Maplesoft, a division of Waterloo Maple Inc.
A CYBERNET group company 13
Analysis
© 2014 Maplesoft, a division of Waterloo Maple Inc.
A CYBERNET group company 14
Knowledge Capture
© 2014 Maplesoft, a division of Waterloo Maple Inc.
A CYBERNET group company
MapleSim Symbolic Formulation • A model’s cho...
© 2014 Maplesoft, a division of Waterloo Maple Inc.
A CYBERNET group company
MapleSim Symbolic Formulation • Generated equ...
© 2014 Maplesoft, a division of Waterloo Maple Inc.
A CYBERNET group company
MapleSim Symbolic Formulation • Multiplicatio...
© 2014 Maplesoft, a division of Waterloo Maple Inc.
A CYBERNET group company
MapleSim Symbolic Formulation • Expressions t...
© 2014 Maplesoft, a division of Waterloo Maple Inc.
A CYBERNET group company
MapleSim Symbolic Formulation • Simulation pr...
© 2014 Maplesoft, a division of Waterloo Maple Inc.
A CYBERNET group company
• Highly accurate vehicle model
• 6-DoF motio...
© 2014 Maplesoft, a division of Waterloo Maple Inc.
A CYBERNET group company
• Modeling:
21
Case Study: Race Car Simulator
© 2014 Maplesoft, a division of Waterloo Maple Inc.
A CYBERNET group company
• Modeling:
22
Platform
Case Study: Race Car ...
© 2014 Maplesoft, a division of Waterloo Maple Inc.
A CYBERNET group company 23
Case Study: Race Car Simulator
© 2014 Maplesoft, a division of Waterloo Maple Inc.
A CYBERNET group company
• Modeling:
24
Controller
Case Study: Race Ca...
© 2014 Maplesoft, a division of Waterloo Maple Inc.
A CYBERNET group company 25
• Inverse Dynamics
• Inverse Kinematics 𝑃𝑜...
© 2014 Maplesoft, a division of Waterloo Maple Inc.
A CYBERNET group company 26
Case Study: Race Car Simulator
© 2014 Maplesoft, a division of Waterloo Maple Inc.
A CYBERNET group company 27
Case Study: Race Car Simulator
© 2014 Maplesoft, a division of Waterloo Maple Inc.
A CYBERNET group company
• System Modeling is important for capturing
...
© 2014 Maplesoft, a division of Waterloo Maple Inc.
A CYBERNET group company
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System-Level Modeling with Symbolics

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System-level models are essential for capturing the dynamic interactions of different domains. Their benefits, however, can be greatly extended when coupled with a symbolically capable environment. In these cases, the actual equations that govern the interactions can be extracted, viewed, manipulated, and modified – yielding insights and efficiencies that are simply not available with purely numeric tools. These environments can also provide significant speed improvements, leveraging their internal data structures to generate compact, highly-efficient c-code.

MapleSim, a modeling and simulation platform that is based on the Maple symbolic computation engine, and available through Altair’s HyperWorks Platform, provides such an environment. This presentation will discuss some of the above advantages in the context of MapleSim case studies.

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System-Level Modeling with Symbolics

  1. 1. © 2014 Maplesoft, a division of Waterloo Maple Inc. A CYBERNET group company System Modelling with Symbolics Chad Schmitke, Ph.D. Director, MapleSim Development
  2. 2. © 2014 Maplesoft, a division of Waterloo Maple Inc. A CYBERNET group company • System Modeling • Numeric vs. Symbolic – Math -> Simulation – Analysis – Knowledge Capture – Fast Simulation Code • Case Study: Motion Platform 2 Overview
  3. 3. © 2014 Maplesoft, a division of Waterloo Maple Inc. A CYBERNET group company • What are they? – Models of systems that incorporate multiple domians (electical, hydraulic, mechanical, thermal, etc.) • Why we care about them? – System models allow us to capture the dynamic interactions of different domains – capturing phenomena we might miss if the parts were treated independently 3 System Modeling
  4. 4. © 2014 Maplesoft, a division of Waterloo Maple Inc. A CYBERNET group company 4 System Modeling
  5. 5. © 2014 Maplesoft, a division of Waterloo Maple Inc. A CYBERNET group company 5 System Modeling
  6. 6. © 2014 Maplesoft, a division of Waterloo Maple Inc. A CYBERNET group company • Talking about formulation approach, not simulation approach – both simulate numerically • Numeric: System reconstructed at each step in the simulation by multiplying/adding numeric matrices • Symbolic: System constructed once during pre-processing phase and numerically evaluated at step in the simulation 6 Numerics vs. Symbolics
  7. 7. © 2014 Maplesoft, a division of Waterloo Maple Inc. A CYBERNET group company Coordinate Selection Equation Generation Symbolic Simplification Code Optimization Simulation Procedure Generation Model Definition Simulation MapleSim Symbolic Formulation Standard Numeric Formulation Model Definition Simulation Procedure Generation with Limited Optimization Simulation Numerics vs. Symbolics
  8. 8. © 2014 Maplesoft, a division of Waterloo Maple Inc. A CYBERNET group company 8 Math -> Simulation
  9. 9. © 2014 Maplesoft, a division of Waterloo Maple Inc. A CYBERNET group company 9 Analysis • Solve for the motion of the legs given the motion of the platform (Inverse Kinematics)
  10. 10. © 2014 Maplesoft, a division of Waterloo Maple Inc. A CYBERNET group company 10 Analysis
  11. 11. © 2014 Maplesoft, a division of Waterloo Maple Inc. A CYBERNET group company 11 Analysis
  12. 12. © 2014 Maplesoft, a division of Waterloo Maple Inc. A CYBERNET group company 12 Analysis
  13. 13. © 2014 Maplesoft, a division of Waterloo Maple Inc. A CYBERNET group company 13 Analysis
  14. 14. © 2014 Maplesoft, a division of Waterloo Maple Inc. A CYBERNET group company 14 Knowledge Capture
  15. 15. © 2014 Maplesoft, a division of Waterloo Maple Inc. A CYBERNET group company MapleSim Symbolic Formulation • A model’s chosen state variables directly impact the number and complexity of the resulting equations Coordinate Selection Equation Generation Symbolic Simplification Code Optimization Simulation Procedure Generation Model Definition Simulation 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 constraints = 42 equations Example: Stewart Platform Fast Simulation Code
  16. 16. © 2014 Maplesoft, a division of Waterloo Maple Inc. A CYBERNET group company MapleSim Symbolic Formulation • Generated equations are true for all time, using the previous example: -2 multiplications, 1 addition per step (versus original 6 and 4, respectively) Coordinate Selection Equation Generation Symbolic Simplification Code Optimization Simulation Procedure Generation Model Definition Simulation Fast Simulation Code
  17. 17. © 2014 Maplesoft, a division of Waterloo Maple Inc. A CYBERNET group company MapleSim Symbolic Formulation • Multiplications by 1’s, 0’s automatically removed (previous slide) • Simple equations directly solved, reducing the number of variables to integrate • Trigonometric simplifications: Coordinate Selection Symbolic Simplification Code Optimization Simulation Procedure Generation Model Definition Simulation Equation Generation Fast Simulation Code
  18. 18. © 2014 Maplesoft, a division of Waterloo Maple Inc. A CYBERNET group company MapleSim Symbolic Formulation • Expressions that are repeated within the equations are identified and isolated so they are only computed onceCoordinate Selection Symbolic Simplification Code Optimization Simulation Procedure Generation Model Definition Simulation Equation Generation Fast Simulation Code
  19. 19. © 2014 Maplesoft, a division of Waterloo Maple Inc. A CYBERNET group company MapleSim Symbolic Formulation • Simulation procedure generated as standalone C-code (no callbacks to MapleSim) • Portable • Fast • Easy to wrap for including in other toolchains • S-functions • FMUs Coordinate Selection Symbolic Simplification Code Optimization Simulation Procedure Generation Model Definition Simulation Equation Generation Fast Simulation Code
  20. 20. © 2014 Maplesoft, a division of Waterloo Maple Inc. A CYBERNET group company • Highly accurate vehicle model • 6-DoF motion platform • Patented kinematic layout • Maximum realism to the driver • Solutions for all joints of this simulator based on the error in the platform (Inverse Kinematics + Inverse Dynamics) 20 Case Study: Race Car Simulator
  21. 21. © 2014 Maplesoft, a division of Waterloo Maple Inc. A CYBERNET group company • Modeling: 21 Case Study: Race Car Simulator
  22. 22. © 2014 Maplesoft, a division of Waterloo Maple Inc. A CYBERNET group company • Modeling: 22 Platform Case Study: Race Car Simulator
  23. 23. © 2014 Maplesoft, a division of Waterloo Maple Inc. A CYBERNET group company 23 Case Study: Race Car Simulator
  24. 24. © 2014 Maplesoft, a division of Waterloo Maple Inc. A CYBERNET group company • Modeling: 24 Controller Case Study: Race Car Simulator
  25. 25. © 2014 Maplesoft, a division of Waterloo Maple Inc. A CYBERNET group company 25 • Inverse Dynamics • Inverse Kinematics 𝑃𝑜𝑠𝑖𝑡𝑖𝑜𝑛 𝐴𝑐𝑐𝑒𝑙𝑒𝑟𝑎𝑡𝑖𝑜𝑛 • Forward Kinematics 𝑃𝑜𝑠𝑖𝑡𝑖𝑜𝑛 𝑉𝑒𝑙𝑜𝑐𝑖𝑡𝑦 Case Study: Race Car Simulator
  26. 26. © 2014 Maplesoft, a division of Waterloo Maple Inc. A CYBERNET group company 26 Case Study: Race Car Simulator
  27. 27. © 2014 Maplesoft, a division of Waterloo Maple Inc. A CYBERNET group company 27 Case Study: Race Car Simulator
  28. 28. © 2014 Maplesoft, a division of Waterloo Maple Inc. A CYBERNET group company • System Modeling is important for capturing dynamics between different domains • Using symbolic technology offers a variety of benefits: – Math -> Simulation – Analysis – Knowledge Capture – Fast Simulation Code 28 Summary
  29. 29. © 2014 Maplesoft, a division of Waterloo Maple Inc. A CYBERNET group company
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