Chad Schmitke, Ph.D. Director of Maplesim Development, Maplesoft

© 2009 Maplesoft, a division of Waterloo Maple Inc.

Chad Schmitke, Ph.D.Director of Maplesim Development, Maplesoft

Physical Modelling


Goal: • To clearly define the concepts of:• Causal/Acausal modelling• Symbolic/Numeric formulation

• Highlight the differences between these modelling and formulation approaches

• Clearly communicate why these differences matter


1. Getting Context: Plants, Controllers, Causal, Acausal

2. Causal Modelling: How, Why, Why MapleSim

3. Acausal Modelling: How, Why, Why MapleSim

4. Simulation: What happens when you press “Simulate”

5. Symbolics: A key difference

6. Summary







6. Summary


Getting Context: Plant modelling

Problem it is trying to solve: Create a mathematical description of a system given:1. Mathematical models of the components included in the

system – for example:• Resistor: v = i*R• Mass: F = m*a

2. How the components are interconnected


Getting Context: Controller modelling

Problem it is trying to solve: Create a procedure to control how a target plant model will behave given:1. Desired behaviour of the system2. Plant model inputs that can be controlled (voltages, motions,

pressures, temperatures, etc.) 3. Outputs that can be measured (voltages, motions, pressures,

temperatures, etc.)

Control laws are more algorithmic than physical in nature

Controller Plant

modelinputs

measuredoutputs

desiredbehaviour


Getting Context: Causal vs. AcausalCasaul modelling:

• Based on the flow of a signal through a diagram• Well suited to Controller Modelling

Acasaul modelling: • Based on interconnection of components• Well suited to Plant Modelling

R Lv(t) J







6. Summary


Causal Modelling: How it’s done...Basic steps for building a causal model:

1. Formulate the differential equations and manipulate them to solve for the desired form (different depending on if you’re modelling a plant or a controller)

~

R L

V

∫ 𝑑2θ 𝑑 𝑡2

𝑑𝑡=𝑑θ 𝑑𝑡

∫ d i 𝑑𝑡 𝑑𝑡= i


Causal Modelling: How it’s done...Basic steps for building a causal model:

2. Transform equations into signal flow diagrams block by block using atomic operators (gains, adders, etc)


Causal Modelling: Why it’s used...This modelling approach is often used for the following

reasons:

• Controller Modelling: Intuitive way to model controllers

• Forced Insight: Process of formulating the equations can yield insight into how the system works

• Familiarity: Design tools of this nature have been around for decades

• Legacy Models: All of the previous models they have were built this way – they want to extend them or connect to them


Causal Modelling: Challenges...1. Complexity of equations does not scale linearly with the size of

the system• As complexity/size increases, so does the chance of errors• Prevents high fidelity modelling of larger systems,

particularly when applied to plant models

# of Links # of Additions # of Multiplications # of Acausal Blocks

1 2 7 5

2 21 82 9

3 135 660 13

4 669 3,974 17

5 2,726 19,224 21

* Cost of dynamic equations, joint coordinate formulation, basic symbolic simplify()

Example: 3D pendulum with increasing number of links:


Causal Modelling: Challenges...2. Generated model looks nothing like the formulated

equations or model diagram• Assumptions made during equation formulation lost• Hard to track errors• Hard to visually understand the purpose of the system

~R L

V ?

?


Causal Modelling: Challenges...3. Since these models have predefined inputs/outputs, it is difficult

to (properly) connect two causal models• This becomes more important as the scope of models

increases (i.e. connect powertrain model to chassis/tire model)

• In some cases this can require an equation re-formulation (to be done properly)

?

Engine/Powertrain

AngleInputs

Chassis/TireTorque Outputs


Causal Modelling: Why MapleSim...

1. MapleSim/Maple has tools for deriving, checking, and manipulating equations• Saves time and reduces errors during the 1st stage of causal

model creation (equation generation stage)

2. MapleSim/Maple can automatically convert equations to a component block (causal or acausal)• Automates the 2nd stage of causal model creation (saves

time, reduces error)

3. MapleSim/Maple provides a live document detailing the assumptions and equations that created the model• Saves time and reduces errors when modifying the model

equations in the future• Simplifies (and makes possible) the ability to properly link

two causal models together

MapleSim/Maple is an excellent causal modelling tool







6. Summary


Acausal Modelling: How it’s done...Basic steps for building an acausal model:

1. Use blocks or components to define the topology of your system

R Lv(t) J

~

R L

V


Acausal Modelling: Why it’s used...This modelling approach is often used for the following

reasons:

• Plant Modelling: Intuitive way to model plants – this approach is not used for controller modelling since controller models assume causal relationships

• Ease of Modifying Models: Changing the model only requires changing the way you’ve connected components

• Ease of Combining Systems: Combining separately created systems is trivial – details of connecting them automatically handled by the modelling tool

• Visual Clarity: Diagrams are easier to understand via visual inspection


Acausal Modelling: Challenges...

• Too Powerful, Too Easy: Some feel there is a loss of system insight because you don’t have to manually derive the equations

• Cannot Convert Legacy Models: There is no mapping between a legacy causal model and an acausal model…you can’t automatically convert/update your causal models


Acausal Modelling: Why MapleSim...

1. MapleSim/Maple can automatically generate the governing equations for any model and provides powerful tools for inspecting/manipulating those equations

2. MapleSim/Maple can automatically convert equations to a component block (causal or acausal)• Useful to researches who want to introduce a particular

physical phenomena• Ability to extend the MapleSim blockset without being a

programmer

3. As of MapleSim V2, 3D systems can be automatically visualized, providing additional insight into the behaviour/motion of a system

MapleSim/Maple is an excellent acausal modelling tool







6. Summary


Simulation: Two approaches...

User presses “Simulate”

Program generates the simulation procedure and integrates the system forward in time

Program displays results

Symbolic Formulation Numeric Formulation

User creates model (Causal, Acausal, or Both)

OR


Different formulation approaches

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


Standard Numeric Formulation

Model Definition

Simulation Procedure Generation with Limited

Optimization

Simulation

• Generated procedure is a set of routines that multiply/add numerical matrices to reformulate the equations at each time step

-6 multiplications, 4 additions per step

• Certain optimizations can be built into these routines but these are limited, and must be defined ahead of time


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


Equation Generation


Code Optimization


Model Definition

Simulation

Absolute coordinates (e.g. ADAMS):• 78 coords (12 per leg, 6 for the platform), • 78 dynamic equations, +72 constraint equations = 150 equations

Hybrid coordinates (MapleSim):•24 coords( 3 per leg, 6 for the platform)•24 dynamic equations+ 18 constraints = 42 equations

Example: Stewart Platform


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)

• Equations can be viewed, analyzed and manipulated in the Maple environment


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 onceCoordinate Selection


Code Optimization


Model Definition

Simulation

Equation Generation


MapleSim Symbolic Formulation • Using MapleSim’s Addons, optimized procedures can be exported to a variety of targets:• LabVIEW RT Toolchain• Simulink RTW Toolchain

• Alternatively, these procedures can be generated in Standalone C-code (no Connectivity Toolboxes required)



Code Optimization


Model Definition

Simulation

Equation Generation


Simulation: Why does this matter...MapleSim uses a symbolic formulation strategy.

Key Challenge when using a symbolic formulation:• Individuals who are used to numeric formulations will notice a

“slow-down” between when the press “simulate” and when they see their results

Key Advantage of using a symbolic formulation:• This is an enabling technology – it allows the generation of

real-time code for systems that numeric formulations cannot

PhysicalController

Code GeneratedPlant

Real-time code to test controller


Integrator

Desktop

• When simulating an entire system on an engineer’s desktop, a faster simulation procedure means quicker iterations through design modifications.

Plant ModelController Model

Simulation Procedure Result Visualization

Simulation: Why does this matter...


• When trying to simulate the plant model in real-time with external hardware, a faster simulation procedure is the difference between “possible” and “not possible”.

Integrator

Plant Model- Controller- User Feedback

Simulation Procedure

Real-time PlatformExternal Hardware



• Therefore, a better way to formulate the simulation procedure means that larger, higher-fidelity models can be tested in a real-time environment

• MapleSim uses the power of symbolics to generate extremely efficient simulation procedures

Integrator

Plant Model- Controller- User Feedback

Simulation Procedure

Real-time PlatformExternal Hardware








6. Summary


What about symbolics…The connection between MapleSim and it’s connection to the Maple

environtment is the key difference between MapleSim and other modelling tools.

1. If you’re deriving equations (plant or controller)…• Manage/manipulate/formulate equations• View/analyse equations• Automatically convert equations into component models

2. If you’re doing plant modelling…• Automatically generate equations for analysis• Tools to work with the equations (see 1.)

3. If you’re generating real-time code…• Enabling technology – fast, efficient code






5. How Symbolics fit in

6. Summary


Summary1. MapleSim is an excellent tool for causal modelling

1. Tools to naturally view/manipulate equations2. Equations can be automatically turned into components3. Knowledge capture

2. MapleSim is an exceptional tool for acausal modelling1. Equations can be viewed for added insight and analysis2. 3D visualization3. Knowledge capture

3. MapleSim can model both Acausal and Causal system in the same environment

4. The symbolic formulation strategies used by MapleSim allow the creation of real-time code for more complicated systems


Questions?


User

User

Hum

an e

ffort

Computer effort

Problem Analysis

Intuition & physics

Model equations

Execute numericalalgorithms

Numerical algorithms

General purposelanguages

e.g. FORTRAN

Specialized numericalmathematics

e.g. NAG, MATLAB

State-basedsimulation

e.g. Simulink

Acausal modelingenvironmentse.g. MapleSim

Simulation model

Problem Analysis Problem Analysis

Intuition & physics Intuition & physics

Model equations Model equations





Problem Analysis

Intuition & physics

Model equations



Simulation model Simulation model Simulation model

Numerical experts

Math experts

Modeling experts

Engineers

User

User

Math experts

Modeling experts

Engineers

User

UserModeling experts

Engineers

Engineers

The Evolution of Multi-Domain Modeling


What about symbolics…

Non-symbolic tools

Modelling Formulation

• Insight and Analysis• Equation Derivation Tools• Equations -> Components• more to come…

• Enabling Technology• Flexibility• more to come…

Current Requirements Desired/Perceived Requirements

MapleSim

Additional potential largely because of symbolic approach

Chad Schmitke, Ph.D. Director of Maplesim Development, Maplesoft

Documents

given plant model

given system

division of waterloo

mathematical models

building plant models

target plant model

maplesim acausal modelling

plant modellingproblem