# Two Degree Freedom Roboticic Manipulator Using Bondgraph Computer Science Essay

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This report is based on Modeling and controlling of two degree freedom robotic manipulator using Bondgraph and its subordinate software Symbols Shakti and controlling the robotic arm using PID controller.

Theory of Bondgraph is based on the concept that any dynamic system equation can be represented in two parameters basically effort and flow. It expresses any physical system in terms of power flow interactions. Based on this theory the Bondgraph model of the system equation shall be implemented by the software Symbols Shakti, software developed by renowned institute IIT Khargapur. A two degree freedom robotic arm is modeled on symbol Shakti which will be free to move in radial and angular directions.

The concept of PID controller is introduced in order to control the movements of robotic arm. The same will be simulated on symbols Shakti using all of its components used in the software including:-

Theory of Bondgraph

Symbols shakti

Introductory concept of mechanical handling of systems

PID controller

Taking into consideration of each and all concepts analytically and mathematically, this report will try to model and control the robotic arm using Bondgraph efficiently.

ABSTRACT 2

ACKNOWLEDGEMENT 3

INTRODUCTION 7

2. OBJECTIVE 8

3. PROPOSED PLAN 9

4. THEORETICAL CONCEPTS 10

4.1 Introduction of Bondgraph 10

4.1.1 History 10

23

4.3 Bondgraph for mechanical systems 23

Following methods are found to be effective in creating bondgraphs for mechanical systems:- 23

1. method of flow map(MFM) 23

2. method of effort map(MEP) 23

3 . mixed map(MM) 23

4.3.1 Method of flow map:- 23

this method is based on the concept that single port mechanical C or R element may be attached to a 1-junction where the relative velocity of ends of these elements are available. 23

Steps to create system Bondgraph by MFM 23

1.create 1-junctions depicting components of velocities of inertial points. 23

2 Create 1-junction depicting motions of end points of C and R elements. 23

3. relate and connect these junctions by TF if a lever relation exists between them. 23

4. create the relative velocities between the end points of C and R elements. 0-junctions may be used for adding or subtracting the velocities. 23

5. Attach C and R elements to 1-junctions where every relative velocities of their and points are created. 23

6. reduce the Bondgraph using reduction process. 23

4.3.2 MEM(method of effort map) 23

23

1.Create 0-junction for sources of efforts.such junctions may be treated as distributor of efforts. 23

2.Decide whether the tensile force in C is to be taken as positive or it is the compressive force which is to be taken positive. 23

3. for R elements decide what kind of relative velocities (compressive or stretching ) is to be taken as positive. 23

4.Addition of forces may be done using 1-junction. 23

5.linear forces may be converted to couples using TF elements. 24

6.reduce the bondgraph using appropriate forms. 24

24

4.3.3 MM(mixed map) 24

The 1-junctions on which the forces accelerating generalized inertial points are added corresponding to velocities of these inertial points in MEM. Now if a part of a system is modeled by MEM then the 1-junctions depicting these velocities may be used to create the bondgraph for the rest of the systems following MFM. 24

5.PID CONTROLLER 24

AÂ proportional-integral-derivative controllerÂ (PID controller)[2] is a genericÂ control loopÂ feedback mechanismÂ (controller) widely used in industrialÂ control systemsÂ - a PID is the most commonly used feedback controller. A PID controller calculates an "error" value as the difference between a measuredÂ process variableÂ and a desiredÂ setpoint. The controller attempts to minimize the error by adjusting the process control inputs. In the absence of knowledge of the underlying process, a PID controller is the best controller. 24

24

Fig 20. Block diagram of PID controller 24

5.1 Control strategy 24

1.force proportional to the error in position. 25

2.force proportional to the integral of error in position. 25

3.force proportional to the rate of change of position.(velocity) 25

The proportional, integral, and derivative terms are summed to calculate the output of the PID controller. DefiningÂ u(t)Â as the controller output, the final form of the PID algorithm is: 25

25

y 25

x 25

26

26

Fig 21: schematic diagram of a typical slider crank mechanism 26

The motion of the connecting rod (link 2) is resolved into motion of its centre of mass in the two principle x and y directions and its rotation about the centre of mass. The motion of link 2 and the slider are determined by the modulated transformers from crank rotation. crank position Î¸ is required for the transformer moduli. 26

6.2. 4-bar mechanism 26

26

Fig 22.4-bar mechanism 26

The mechanism is driven by a flexible drive shaft at constant speed. There is an elastic load at the output link-4. link -4 receives the rotational motion from crank rotation through link-3. This is represented by the TF connecting junctions. 27

We can deduce from the bondgraph model of 4-bar mechanism that all inertial elements, except I are with differential causalities indicating the links connected to the crank have dependent motions.these differential causalities can be eliminated by using pads. 27

7 Bond graph modeling 27

8. PROCEEDINGS TILL DATE 28

9. FUTURE WORK 29

REFERENCES 30

## INTRODUCTION

AÂ bond graphÂ is aÂ graphical representationÂ of a physicalÂ dynamic system. It is similar to the better knownÂ block diagramÂ andÂ signal-flow graph, with the major difference that the arcs in bond graphs representÂ bi-directionalÂ exchange of physicalÂ energy,

In 1959, Prof. H.M.Paynter gave the revolutionary idea of portraying systems in terms of

Power bonds, connecting the elements of the physical system to the so called junction

Structures which were manifestations of the constraints. This power exchange portray of a

System is called Bond Graph (some refer it as Bondgraph), which can be both power and

Information oriented.

By this approach, a physical system can be represented by symbols and lines, identifying

The power flow paths. The lumped parameter elements of resistance, capacitance and

Inertance are interconnected in an energy conserving way by bonds and junctions resulting

In a network structure. From the pictorial representation of the bond graph, the derivation of

System equations are so systematic that it can be algorithmized. The whole procedure of

Modeling and simulation of the system may be performed by some of the existing software

e.g., ENPORT, Camp-G, SYMBOLS, COSMO, LorSim etc. In our project we are going to use symbol Shakti software that was developed in India itself in IIT kharagpur.

## 2. OBJECTIVE

To understand the theory of Bondgraph, to establish a relationship between effort and flow sources of any dynamic system.

To implement the theories of Bondgraph on Symbols Shakti.

An introduction of mechanical handling of systems.

To be acquainted with the PID Controller.

An introductory knowledge of robotic.

Implementation of Bondgraph theory to modeling and controlling of a two degree freedom robotic manipulator using PID controller.

## 3. PROPOSED PLAN

Our objective in doing this project is to prepare a report on the modeling of two degree freedom robotic manipulator using theory of bondgraph and to control that robotic manipulator using PID controller. Here we aim to understand the theory of Bondgraph, using its theory to convert any complex system dynamic equation into effort flow diagram and the software Symbols Shakti will be used to implement the theory of Bondgraph. Using mechanisms and mechanical handling systems concepts the robotic arm will be modeled. We will then understand the PID controller and its functional approach and its mathematical modeling. After having the thorough knowledge of Bondgraph and PID controller we will model a robotic arm with enabled movement in radial and angular directions that will be controlled using the PID controller and control of that two degree freedom robotic arm will be done.

## 4.1.1 History

AÂ bond graph[1]Â is aÂ graphical representationÂ of a physicalÂ dynamic system. It is similar to the better knownÂ block diagramÂ andÂ signal-flow graph, with the major difference that the arcs in bond graphs representÂ bi-directionalÂ exchange of physicalÂ energy, while those in block diagrams and signal-flow graphs representÂ uni-directionalÂ flow of information.

In 1959, Prof. H.M. Paynter gave the revolutionary idea of portraying systems in terms of

Power bonds, connecting the elements of the physical system to the so called junction

Structures which were manifestations of the constraints. This power exchange portray of a

System is called Bond Graph (some refer it as Bondgraph), which can be both power and

Information oriented.

## 4.1.2 Power variables of bondgraph

The language of bond graphs aspires to express general class physical systems through

Power interactions. The factors of power i.e., Effort and Flow have different interpretations

In different physical domains. Yet, power can always be used as a generalized co-ordinate

to model coupled systems residing in several energy domains. One such system may be an

electrical motor driving a hydraulic pump or a thermal engine connected with a muffler;

where the form of energy varies within the system. Power variables of bond graph may not

be always realizable (viz. in bond graphs for economic systems); such factual power is

encountered mostly in non-physical domains and pseudo bond graphs.

## Mechanical

Force(F)

Torque(Ðƒ)

Velocity(V)

Angular velocity(ÏŽ)

Voltage(V)

Current(I)

## Hydraulic

Pressure(P)

Volume flow rate(dQ/dt)

## Thermal

Temperature(T)

Pressure(P)

Entropy change rate(dS/dt)

Volume change rate(dV/dt)

## Chemical

Chemical potential(Âµ)

Enthalpy (h)

Mole flow rate(dN/dt)

Mass flow rate (dm/dt)

## 4.1.3. Bond Graph Standard Elements

In bond graphs, one needs to recognize only four groups of basic symbols, i.e., three basic one port passive elements, two basic active elements, two basic two port elements and two basic junctions. The basic variables are effort (e), flow (f), time integral of effort (P) and the time integral of flow (Q).

## 4.1.4 Basic elements of Bondgraphs

The concept of energy ports:-

In discrete system modeling the basic elements are incorporated in a system through what may be called terminal of the element.

## Effort and Flow Sources:

The active ports are those, which give reaction to the source. For, example if we step on a

rigid body, our feet reacts with a force or source. For this reason, sources are called active ports. Force is considered as an effort source and the surface of a rigid body gives a

velocity source. They are represented as a half arrow pointing away from the source symbol as shown below.

## Fig 2. Representation of flow source

In electrical domain, an ideal shell would be represented as an effort source. Similarities

can be drawn for source representations in other domains.

## Basic 1-Port elements

A 1-port element is addressed through a single power port, and at the port a single pair of effort and flow variables exists. Ports are classified as passive ports and active ports.

The passive ports are idealized elements because they contain no sources of power. The inertia or inductor, compliance or capacitor, and resistor or dashpot are classified as passive elements.

## Fig 4 bondgraph representation of RLC circuit

'1' is used for common flow junction and '0' is used for common effort junction.

In bondgraph modeling the interconnections takes place through abstract entities called energy port. Energy port is represented by a bond having associated with it two conjugate factors of power.

## R-Elements:

The 1-port resistor is an element in which the effort and flow variables at the single port are

related by a static function. Usually, resistors dissipate energy. This must be true for simple

electrical resistors, mechanical dampers or dashpots, porous plugs in fluid lines, and other

analogous passive elements. The bond graph symbol for the resistive element is shown

below.

## Fig 5. Bondgraph symbol for resistive element

The half arrow pointing towards R means that the power i.e., product of F and V (or e * f) is

positive and flowing into R, where e, represents effort or force, and f, represents flow or

velocity. The constitutive relationship between e, f and R is given by:

## e = R * f

Power = e * f = R * f 2

## C-Elements:

Consider a 1-port device in which a static constitutive relation exists between an effort and a

displacement. Such a device stores and gives up energy without loss. In bond graph

terminology, an element that relates effort to the generalized displacement (or time integral

of flow) is called a one port capacitor. In the physical terms, a capacitor is an idealization of

devices like springs, torsion bars, electrical capacitors, gravity tanks, and accumulators, etc.

The bondgraphic symbol, defining constitutive relation for a C-element are shown below.

## Fig 6. Bondgraph symbol of C-element

Here, the current is the cause and the total charge (and hence voltage) is the consequence.

## , e= K

Here, flow is the cause and deformation (and hence effort) is the consequence. In a

capacitor, the charge accumulated on the plates (Q) or voltage (e) is given by,

## I-Elements:

A second energy storing 1-port arises if the momentum, P, is related by a static constitutive

law to the flow, f. Such an element is called an inertial element in bond graph terminology.

The inertial element is used to model inductance effects in electrical systems and mass or inertia effects in mechanical or fluid systems. The bond graph symbol for an inertial element is depicted in the figure given below...

## Fig 7. Bondgraph symbol for inertial element

Here, effort is the cause and velocity (and hence momentum) is the consequence. Similarly

the current in an inductor is given by;

i = L -1

## Basic 2-Port elements

There are only two kinds of two port elements, namely ``Transformer'' and ``Gyrator''. The

bond graph symbols for these elements are TF and GY, respectively. As the name

suggests, two bonds are attached to these elements.

## The Transformer:

The bondgraphic transformer can represent an ideal electrical transformer, a mass less

lever, etc. The transformer does not create, store or destroy energy. It conserves power and transmits the factors of power with proper scaling as defined by the transformer modulus

(discussed afterwards).

The meaning of a transformer may be better understood if we consider an example given

here. In this example, a mass less ideal lever is considered. Standard and bondgraphic

nomenclature of a lever are shown in the figure below. It is also assumed that the lever is

rigid, which means a linear relationship can be established between power variables at both

the ends of the lever.

## Fig 8. Power variables working on a lever and its representation

From the geometry, we have,

V2 = (b/a) V1

The power transmission implies

F2 = (a/b) F1 , so that V2 F2 = V1 F1.

In bondgraphs, such a situation may be represented as shown in the above figure.

The 'r' above the transformer denotes the modulus of the transformer, which may be a

constant or any expression (like 'b/a'). The small arrow represents the sense in which this

modulus is to be used.

fj = r fi , and ej = (1/r) ei.

Thus the following expression establishes the conservation of power,

ej fj = ei fi .

## The Gyrator :

A transformer relates flow-to-flow and effort-to-effort. Conversely, a gyrator establishes

relationship between flow to effort and effort to flow, again keeping the power on the ports

same. The simplest gyrator is a mechanical gyroscope, shown in the figure below.

## Fig 9. A mechanical Gyroscope and its bondgraph representation

A vertical force creates additional motion in horizontal direction and to maintain a vertical

motion, a horizontal force is needed. So the force is transformed into flow and flow is

transformed into force with some constant of proportionality. In this example, Izz stands for

moment of inertia about z axis. wx , wy and wz stand for angular velocities about respective

axes; Tx , Ty and Tz represent torque acting about the corresponding axis.

Tx = Izz wzwy .

The power transmission implies

Ty = Izz wz wx , so that Txwx = Ty wy .

Such relationship can be established by use of a Gyrator as shown in the figure above.

The Âµ above the gyrator denotes the gyrator modulus, where Âµï€ = Izz wz . This modulus does

not have a direction sense associated with it. This modulus is always defined from flow to

effort.

ej =Âµ ï€ fi , ei = Âµï€ fj .

Thus the following expression establishes conservation of power, ei fi = ej fj .

## The 3-Port junction elements

The name 3-port used for junctions is a misnomer. In fact, junctions can connect two or

more bonds. There are only two kinds of junctions, the 1 and the 0 junction. They conserve

power and are reversible. They simply represent system topology and hence the underlying

layer of junctions and two-port elements in a complete model (also termed the Junction

Structure) is power conserving.

1 junctions have equality of flows and the efforts sum up to zero with the same power

orientation. They are also designated by the letter S in some older literature. Such a

junction represents a common mass point in a mechanical system, a series connection (with same current flowing in all elements) in an electrical network and a hydraulic pipeline representing flow continuity, etc. Two such junctions with four bonds are shown in the figure below.)

## Fig 10.bondgraph representation of two junctions with four bonds

Using the inward power sign convention, the constitutive relation (for power conservation at

the junctions) for the figure in the left may be written as follows;

e1 f1 + e2 f2 + e3f3 + e4 f4 = 0.

As 1 junction is a flow equalizing junction,

f1 = f2 = f3 = f4 .

This leads to, e1 + e2 + e3 + e4 = 0.

Now consider the above bond graph shown on the right. In this case, the constitutive

relation becomes,

e1 f1 - e2 f2 + e3f3 - e4 f4 = 0 , and, f1 = f2 = f3 = f4 .

Thus, e1 - e2 + e3 - e4 = 0.

So, a 1 junction is governed by the following rules:

The flows on the bonds attached to a 1-junction are equal and the algebraic sum of the efforts is zero.The signs in the algebraic sum are determined by the half-arrow directions in a bond graph.0 junctions have equality of efforts while the flows sum up to zero, if power orientations are taken positive toward the junction. The junction can also be designated by the letter P. This junction represents a mechanical series, electrical node point and hydraulic pressure distribution point.

## 4. 1.5 Power directions on the bonds

When one analyses a simple problem of mechanics, say, the problem of a single mass and

spring system as shown in the figure below, one initially fixes a co-ordinate system.

equation(s) when a positive value of displacement and a negative value of force are seen.

## Fig 11.single mass and spring system

One may take positive displacement, x, towards right and all its time derivatives are

then taken positive towards right. The force acting on the mass may also be taken

positive towards right. One has to then create a view point which is general and any particular system interpretation should be easily derivable. This is done by assigning the bonds with Power directions.

J : junction,

E : element, J F

half arrow : direction of power Fig 12. Power direction representation

## 4.2. Causality

Causality establishes the cause and effect relationships between the factors of power. In bondgraphs, the inputs and the outputs are characterized by the causal stroke. The causal stroke indicates the direction in which the effort signal is directed (by implication, the end of the bond that does not have a causal stroke is the end towards which the flow signal is directed).

## Fig 13. Prime mover: an example of causality

The prime mover is driving the load i.e., the power is going from the prime mover to the load. Apart from sending the power, the prime mover also decides that the load should run at a particular speed depending on the setting of the governor.

## INERTANCE:-

For inertance I type storage elements, the flow (f) is proportional to the time integral of the effort.

## CAPACITANCE

For capacitive C type storage elements, the effort (e) is proportional to the time

integral of the flow (f).

## RESISTANCE

The resistive or dissipative elements do not have time integral form of constitutive

laws.

## e = R f or f= e/R

The flow and the effort at this port are algebraically related and can thus have any

type of causal structure, either with an open-ended bond (causal stroke is away from

the element, i.e., at the junction end) indicating a resistive causality or a stroke ended

bond indicating a conductive causality.

As per the above discussions, the causal strokes for I, C and R elements are shown in the figure below.

## Fig 15. Causal strokes of I,C and R elements

The elements I, C, R, SF, and SE are classified as single port, since they interact with the system through one bond only. However, the I, C and R elements can be connected to many bonds to represent tensorial nature, such as the spatial motion of a free body or the stress-strain relationships in a compressible material; in which case they are termed as field elements. The transformer, by its elemental relation, receives either flow or effort information in one bond and generates the same in its other bond. Thus, one of its port is open-ended with the other end stroked as shown in the figure below.

## Fig 16.bondgraph representation of tranformer

For the first case, the constitutive equations would be,

fj = r fi and ei = r ej ,

whereas; for the second, the relations are

fj = 1/r fi and ei = 1/r ej .

As mentioned earlier, the gyrator relates flow to effort and effort to flow, therefore, both of its ports have either open-ended or stroke-ended causality as shown below.

## Fig 17.bondgraph representation of gyrator

For the first case, the constitutive equations would be

ej = r fi and ei = r fj ,

whereas; for the second, the relations are

fj = r ei and fi = r ej .

## 4.2.1Strong bond and weak bond

At a 1 junction, only one bond should bring the information of flow; i.e., only one bond should be open ended and all others should be stroked as displayed in the

## Fig 18. Strong bond at junction 1

figure on the right. This uniquely causalled bond at a junction is termed as Strong bond. Similarly at a 0 junction stroked nearer to the junction.

## Fig 19. Strong bond at junction 0

This strong bond determines the effort at the junction, which the weak bonds (other bonds besides the strong bond. The proper causality, for a storage element (I or C), is called Integral Causality, where the cause is integrated to generate the effect. For example, in C element, a study of the constitutive equations reveal that flow is integrated and multiplied with the stiffness to generate effort.

Sometimes the causal strokes will have to be inverted, which means the constitutive

relationship for the corresponding element is written as a differential equation. For example, flow in a spring is the time derivative of the ratio of effort and stiffness. Such causality pattern is called Differential Causality.

## 4.2.2 Causality Assignment Procedure:

1. Assign fixed causalities to sources.

2. Propagate the causality through junctions, if possible, i.e. if any bond has got a

causality such that it has become the strong bond for a junction, the causality for all

other bonds (week bonds) is determined by laws for causality of junctions and if all

other bonds of junction are causalled, the last bond should be the strong bond.

Similarly, if any port of a two port is causalled (of TF and GY), the causality of the

other can be assigned.

3. Assign integral causality to one of the storage elements and propagate the causality through junctions. Continue the procedure with other storage elements. This should normally result in complete causalling of the graph.

4. If the graph is not completely causalled yet, start assigning a resistive causality to an R element and propagate it. Continue till the entire graph is causalled. In cases, where the model is determined through causalities of R-elements, there may be several possible causal models. It is always advisable to maximize resistive causalities and minimize the conductive causalities in R-elements.

5. If the system develops differential causalities in some storage elements, try minimizing its number of occurrence through assigning initial integral causalities to other storage elements than those selected before.

6. Try to avoid differential causalities by suitable changes to the model, such as

introducing some compliance or resistance or both.

7. Discard all models, which result in a causal structure, that violates junction causality rules.

## The proportional, integral, and derivative terms are summed to calculate the output of the PID controller. DefiningÂ u(t)Â as the controller output, the final form of the PID algorithm is:

where the tuning parameters[3] are:

## Proportional gain,Â Kp

Larger values typically mean faster response since the larger the error, the larger the proportional term compensation. An excessively large proportional gain will lead to process instability and oscillation.

## Integral gain,Â Ki

Larger values imply steady state errors are eliminated more quickly. The trade-off is larger overshoot: any negative error integrated during transient response must be integrated away by positive error before reaching steady state.

## Derivative gain,Â Kd

Larger values decrease overshoot, but slow down transient response and may lead to instability due to signal noise amplification in the differentiation of the error.

## 6.Modeling of mechanisms

The methodology of using bondgraph to simulate the dynamic behavior of mechanisms with the aid of two examples

Slider crank mechanism

4-bar mechanism.

## Slider- crank mechanism

the input crank has been driven by a constant velocity source. The slider is moving against the spring.

2

1

B

y

3

x

## 7 Bond graph modeling

A simple mechanical system shown in the figure below is considered here for explaining the modeling procedure.

## Fig 23. A simple mechanical system

The output from the pump would be flow and pressure, while the input would be torque and angular velocity applied to the pump shaft. The input power is provided by an electric motor. The power flow diagram is shown in the figure below.

## 8. PROCEEDINGS TILL DATE

During the semester we have so far have learnt concepts of

## Theory of bondgraphs:-

In the theory of Bondgraph we have learnt how to draw the effort flow graph of any system dynamic equation(e.g. electrical, mechanical etc.).using the power direction and we can assign the causality.Bondgraphs for mechanical systems can be created using flow map, method map and mixed map.

Introduction of Symbols Shakti: symbol Shakti is the next generation modeling tool with which we can easily model a system either in bondgraph form or by using its powerful object oriented encapsulation called capsules.

PID controller and its mathematical approach: Â The PID controller calculation (algorithm) involves three separate parameters, and is accordingly sometimes calledÂ three-term control: theÂ proportional, the integralÂ andÂ derivativeÂ values, denotedÂ P,Â I,Â andÂ D.Â TheÂ proportionalÂ value determines the reaction to the current error, theÂ integral value determines the reaction based on the sum of recent errors, and theÂ derivativeÂ value determines the reaction based on the rate at which the error has been changing.

Mechanisms and material handlings of the system using bondgraphs: this concept will be used with bondgraph modeling of mechanisms, simple load hoisting system and robotic manipulators. The models of the mechanisms are created using relations between the input speeds and velocity components of centre of masses of rigid links and angular velocities.

## 9. FUTURE WORK

After gaining knowledge of theory of bondgraphs and its usage to any system dynamic equation we will implement these theories on symbols Shakti. A model in symbol Shakti may be created using combination of bondgraphic elements in our project. The robotic arm modeled using symbol Shakti will have two degree freedom(r,Î¸) i.e radial and angular movements. The modeling of the robotic arm will be done using mechanical handling of the systems using bondgraph by 4-bar mechanism or slider crank mechanism. A PID controller Â calculates an "error" value as the difference between a measuredÂ process variableÂ and a desiredÂ set point. By tuning the three constants in the PID controller algorithm, the controller can provide control action designed for specific process requirements. Therefore, PID controller will be used for the control of robotic manipulator in our project.