An Approach to Implementation

of Physical Simulation Models

Shpakov Vladimir

St.Petersburg Institute for Informatics and Automation of the Russian Academy of Sciences

39, 14th line, St. Petersburg, Russian Federation

Keywords: Process-Oriented, Hybrid Simulation, Model Design, Transitive Model, Rule-based Approach.

Abstract: Advantages of a rule-based approach to dynamic system physical models specification and implementation

are discussed. Much need for the approach at the modern state of system engineering is pointed out. In par-

ticular, such an approach may be useful for simulation of control and self-organizing systems. A rule-based

situation formalism of an interacting hybrid processes specification is briefly stated and some ways of its

use for physical simulation model implementation are shown. Facilities of the considered methods are illus-

trated by examples of some simple dynamic system models implementations. Specifications of these physi-

cal models and some results of simulation are presented.

1 INTRODUCTION

A mathematical approach to dynamic system simu-

lation serves now as accepted. It consists in follow-

ing. At first, mathematic model of the system works

out on base of corresponding dynamic laws (me-

chanics, thermodynamics, electrodynamics and oth-

ers). As a rule, the model is a set of differential and

algebraic equations about abstract variables. Here

system physical parameters form equations coeffi-

cients. Once, the coefficients constitute rather com-

plicated dependences upon parameters. Model ex-

periments carry out on computers using numerical

methods of the equations solution.

An imperfection of such an approach is mani-

fested already when system parameters modifying in

course of model experiments. It is caused by neces-

sity of equations coefficients recalculation. Initial

system model modification or extension causes al-

teration of the equations and the software for their

processing.

An alternative of this abstract mathematical ap-

proach is using of physical models of dynamic sys-

tems. In this case the model variables are real system

process states, such as distances, velocities, forces,

angles, temperatures, pressures, levels and so on.

The states evolutions are executed with procedures,

which realize corresponding dynamic laws, subject

to satisfying of existing constraints and relations

between system elements. Among physical models

advantages, it is worth to note less labor expendi-

tures on the model development, its modification

and expansion. The physical approach is rather con-

venient for synergetic systems simulation. It enables

comparatively easy to realize systems interacting

with each other and with environment.

There are a lot of computer means for dynamic

system simulation including that intended for physi-

cal simulation models development (Modelica, Sim-

ulink, SimMechanics, SimElectronics and others)

(Avvizano, 2008). Available computer means for

physical model realization (physics engines) are

oriented substantially on carrying out model experi-

ments, system analysis and computer games devel-

opment (Boeing, 2007). Their usage for creation of

the models embedded for example in control sys-

tems, training or teaching equipment is rather awk-

ward. At the same time, high processing power of

modern computers enables to develop relatively

simple methods of such models specification and

implementation.

Hereinafter a short description of a rule-based

formalism is presented and capability of its using for

implementation of the physical simulation models is

discussed. In the third section an example of simple

dynamic system specification and some results of its

simulation are presented.

173

Vladimir S..

An Approach to Implementation of Physical Simulation Models.

DOI: 10.5220/0003981501730176

In Proceedings of the 2nd International Conference on Simulation and Modeling Methodologies, Technologies and Applications (SIMULTECH-2012),

pages 173-176

ISBN: 978-989-8565-20-4

 2012 SCITEPRESS (Science and Technology Publications, Lda.)

2 PROCESSES REALIZATION IN

TERMS OF USING A

RULE-BASED SITUATION

FORMALISM OF PROCESSES

SPECIFICATION

Real physical interacting processes almost always

are hybrid ones that is they have discrete and contin-

uous components. A base of the formalism is an

abstract model of hybrid automaton (Henzinger,

1996), in which discrete states are represented by

logical variables. A set of logical variables contains

also a subset of predicates of the continuous states.

The predicates are used to specify interactions be-

tween continuous and discrete-event components of

the processes. The logic variables can determine

both discrete changes of continuous component and

changes of their evolution dynamic. Thus a state of

the processes model may be represented by a set of

real variables

for the continuous components and

a set of logic variables

W for the discrete compo-

nents. The last consists of subset

Q for the discrete

states and subset

G of predicates, that is GQW U

To specify processes it is necessary to define transi-

tion functions of the following types:

},{: TrueFalseQW ×→σ — function of dis-

crete state transitions;

XXW →×δ : — function of continuous state

transitions;

},{: TrueFalseGX ×→

— predicate value de-

pendence of continuous process states.

The formalism expressiveness and effectiveness

of the model implementation essentially depend up-

on specific forms of these functions. A definitional

domain of the transition function σ may be repre-

sented with help of logic formulae that makes the

representation rather obvious. In the general case,

the model specification may need whatever logic

formulae. To our mind as the condition it is conven-

ient to use an elementary conjunction of the logic

variables. Such conjunctions may be understandably

interpreted as logic-dynamic situations (Shpakov,

2004). The situation

S may be defined as

following:

jjjj

sssS ,...,,...

= , where

ws = , or

ws ¬=

∈

, ||,...1 WNNn

== .

Designating the set of situations

S the type of

the transition function of discrete states may now be

defined as following:

},{: TrueFalseQS ×→

This function may be assigned with help of a collec-

tion of production rules “condition → action”. In

the rules the logic-dynamic situation is used as a

condition and the procedure, which assigns defined

values to logic states, is used as an action. At that the

rule takes the following form:

'''

,...,,...,

1 mi

jjjj

rrrS →

(1)

where

qr = or

qr ¬= , Qq

∈

To implement an arbitrary logic formula it is

necessary to use several rules with the different con-

ditions and the same executive parts. It is a specifi-

cation of a normal disjunction form and it enables to

realize any logic formula.

The function

must for every mode determine

new values of the process states which correspond

with a current process state, the system dynamic and

also with current states of some processes interacting

with this process. The hybrid process mode may be

naturally represented by a logic situation considered

above. It was proposed to find the new values of the

continuous process states by calculation of their

transitive relations with the current states (Alur,

2000). Algorithms of the transitive relations calcula-

tions for some elementary processes (integration,

differentiation, smoothing and others) are presented

in (Shpakov, 2006). Complicated processes may be

implemented by different connections of the elemen-

tary ones and using arithmetic operations and func-

tional procedures. In this case the transitive function

may be represented with help of a collection of

following rules:

),('

ikkkj

xxxS τ=→

(2)

where

∈

— the situation, corresponding to the

mode,

— transitive relation, Xxx

∈, —

states of the continuous processes.

Since the hybrid process modes very often de-

pend on their states being at some assigned ranges it

is convenient to represent the function

by a col-

lection of following rules:

kjkjjkj

gxbxxax →+

≤

∧

≥ ))()((

4321

(3)

where

GgXxxxx

kjjjj

∈∈ ,,,,

4321

a and

b -

constants, corresponding to the range. More compli-

cated predicates may be received by using simple

ones and rules (1).

A computer implementation of the processes

specified by rules (1, 2, 3) is executed with help of

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an interpreter of these rules. The sets of process

states variables (

and W ) are represented in the

interpreter with arrays of records. All rules are im-

plemented with help of condition operators

“

if…then…”. The interpreter base is an executing

procedure which scanning lists of rules in a cycle.

The processing algorithm calculates the value of the

rule condition part. If this value is equal to

True

then a procedure of the executive part of the rule is

triggered.

3 AN EXAMPLE OF PHYSICAL

MODEL IMPLEMENTATION

The approach described was used when developing

in SPIIRAS a prototype of computer environment

oriented on a collection of interacting hybrid pro-

cesses simulation. The environment has quite intui-

tive and obvious interface for editing rules (1, 2, 3)

and processes visualization. Later, specifications in

the environment of a springy hoop dropping on a

plane are presented.

Physical parameters of the hoop are mass

)(m , a

radius

)(R and an elastic stiffness )(k . The process

states are vertical acceleration

)(

A , velocity )(

V ,

and a center of mass height over the plane

)(h and

the hoop elastic deformation

)( hR − , which origi-

nates when the hoop getting in contact with the

plane. It is necessary to simulate the hoop moving in

two situations: a free fall when

)( Rh > , and an elas-

tic contact interaction when

)( Rh >¬ . The hoop

free fall is occurred under the action of the weight

)(P , and at the elastic impact its movement is de-

termined by the simultaneous actions of the weight

and the elastic reaction force

)(

. At the situation

)( Rh > the hoop acceleration is equal to the gravita-

tional acceleration

)(g , the velocity and the height

is determined by integrals of the acceleration and the

velocity, respectively. The elastic reaction force is

proportional to the elastic deformation

)( hR

−

. In

figure 1 there is a copy of part of the environment

editor where the rules of the hoop state transition are

presented.

Figure 1: State transition rules of a spring hoop dropping

on a plane.

Variable names in the rules correspond to ones

pointed above. The situations with names (braking)

and (speed-up) are determined as follows

)()0(),( brakingVRh

→

)()0(),( upspeedVRh

−→><

The rules number 2 and 3 calculate current sizes

of elastic reaction forces on sections of braking and

speeding up. The fourth rule calculates a sum of the

forces and the fifth one does the hoop acceleration.

The rules number 6 and 7 calculate the hoop current

velocity and the height correspondingly. And the

eighth rule calculates the size of the elastic hoop

deformation.

A chart of the hoop mass center height changing

is presented in figure 2.

Figure 2: A chart of the hoop mass center changing.

A graphical chart of the elastic reaction force

changing is presented in figure 3. The force differs

from zero only when

)( Rh

. It increases on the

section of braking and decreases on the section of

speeding up.

Now we show how to extend the model to the

case when there is a horizontal component of the

velocity. In this case in the presence of friction the

hoop begins to rotate as a result of its contact with

the plane. At the same time the hoop horizontal ve-

locity decreases.

AnApproachtoImplementationofPhysicalSimulationModels

175

Figure 3: Change of the hoop spring force during its con-

tact with the plane.

It is necessary to assign a hoop moment of inertia

and include in the state vector variables for represen-

tation of new process states: a horizontal coordinate

)(X , a horizontal acceleration )(

A , a velocity

)(

V , a friction force )(

F , a moment of the couple

)(

tng

, an angle, an angle acceleration, an angle

rate and a velocity of the band in regard to the plane

)(

tng

. The friction force originates in a situation

defined

slideVRh

tng

→>∧< )0||()(

Figure 4: The transition rules for simulation of the hoop

moving after its contact with the plane.

Figure 5: Changes of the hoop height, its horizontal and

angle velocities during its contact with the plane.

The rules for the process states evolution specifi-

cation are presented in figure 4. An addition of these

12 rules to the existing 8 ones enables to simulate an

elastic hoop dropping in the presence of the horizon-

tal component of the velocity and the hoop rotation.

Curves of the height, the horizontal and angle ve-

locities found in result of such hoop dropping simu-

lation are presented in figure 5.

4 CONCLUSIONS

A programming of dynamic system computer mod-

els based on use of transition rules (1, 2, 3) has a

number of advantages in comparison with using of

universal programming languages. A usage of the

rules enables to shorten the time of the model devel-

opment. The program is turned out more obvious.

The model modification, extension and validation

are simplified essentially. A transmission of

knowledge from technicians to programmers is facil-

itated.

ACKNOWLEDGEMENTS

The present research was supported partly by project

funded by grants 12-01-00015 of the Russian Foun-

dation for Basic Research.

REFERENCES

Alur R., Henzinger T.A., Lafferriere G., Pappas G. J.,

2000. Discrete Abstractions of Hybrid Sys-

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Adrian Boeing, Thomas Bräunl, 2007. Evaluation of real-

time physics simulation sys-

tems, http://www.adrianboeing.com/pal/papers/p281-

boeing.pdf.

Carlo Alberto Avvizano, 2008 Review of existing simula-

tion tools, http://sirslab.dii.unisi.it/I-RAS/wp-

content/uploads/2008/05/simulinkreviewx.pdf.

Henzinger T.A., 1996. The Theory of Hybrid Automa-

ta, Proceedings of the 11

Annual IEEE Symposium

on Logic in Computer Science (LICS 96).

Shpakov V.M., 2004. Executable specifications of produc-

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tomatization, control. No. 3, (in Russian).

Shpakov V. M., 2006. Dynamic knowledge specification

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Applications

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