system-level modeling with symbolics

© 2014 Maplesoft, a division of Waterloo Maple Inc. A CYBERNET group company

System Modelling with Symbolics

Chad Schmitke, Ph.D. Director, MapleSim Development


• System Modeling

• Numeric vs. Symbolic – Math -> Simulation

– Analysis

– Knowledge Capture

– Fast Simulation Code

• Case Study: Motion Platform

2

Overview


• 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

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System Modeling

© 2014 Maplesoft, a division of Waterloo Maple Inc. A CYBERNET group company 4

System Modeling


System Modeling


• 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

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Numerics vs. Symbolics


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


Math -> Simulation


Analysis

• Solve for the motion of the legs given the motion of the platform (Inverse Kinematics)


Analysis


Knowledge Capture


MapleSim Symbolic Formulation • A model’s chosen state variables directly impact the number and complexity of the resulting equations Coordinate Selection

Equation Generation


Code Optimization


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


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)


Equation Generation


Code Optimization


Model Definition

Simulation



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:



Code Optimization


Model Definition

Simulation

Equation Generation



MapleSim Symbolic Formulation • Expressions that are repeated within the equations are identified and isolated so they are only computed once



Code Optimization


Model Definition

Simulation

Equation Generation



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



Code Optimization


Model Definition

Simulation

Equation Generation



• 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)

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Case Study: Race Car Simulator


• Modeling:

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• Modeling:

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Platform



• Modeling:

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Controller



• Inverse Dynamics

• Inverse Kinematics 𝑃𝑜𝑠𝑖𝑡𝑖𝑜𝑛

𝐴𝑐𝑐𝑒𝑙𝑒𝑟𝑎𝑡𝑖𝑜𝑛

• Forward Kinematics 𝑃𝑜𝑠𝑖𝑡𝑖𝑜𝑛𝑉𝑒𝑙𝑜𝑐𝑖𝑡𝑦



• 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

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Summary

system-level modeling with symbolics

Engineering