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INDIAN INSTITUTE OF TECHNOLOGY ROORKEE
Bond Graph Modelling of Robots
By Dr Pushparaj Mani Pathak
Associate Professor, Mechanical & Industrial Engineering Department,I.I.T. Roorkee
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Contents
• Introduction of Bond graph modelling• Modelling of planar 1 DOF robot• Modelling of planar 2 DOF robot• Modelling of Four legged walking robots
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Introduction
• Today we must develop capabilities to deal with variety. • Only way to deal with variety and diversity is to achieve
conceptual unification.• The requirement of a unified approach to modeling,
simulation and synthesis of physical systems residing in multi-energy domain may be stated as follows. – The language for a modeling should have concise lexicon valid over a
large variety of energy domain.– It should allow the modeler to portray the exchange with in and
across the domain.– The portraits so created should algorithmically lead to mathematical
or logical models. These models may then be subjected to predictive or deductive processes.
• In physical systems it is energy which plays the role of common currency of exchange between various domains and sustains the business of dynamics.
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• The idea of bond graph was proposed by H. M. Paynter (at MIT).
• He used a graphical language for representation of physical system in multi-energy domain through creating portrays of exchange of power.
• These portrays were further augmented by imposition of reference direction of power flow and causal relations or information exchange.
• In India Prof Amalendu Mukherjee of IIT Kharagpur worked extensively in bond graph modelling.
(1923-2002)
(1947-2015)
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• The significant value of these portrays is that one may arrive at mathematical or logical models in algorithmic manner.
• This brought both algorithmisation and unification.• For engineering analysis and synthesis, computers could be
deployed to be our deductive partners and also could be entrusted to perform simulation.
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Dynamic Systems
Electrical Mechanical Hydraulic Thermal Examples
Moving car Electric circuits Telescope positioning system.
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Steps in Design of Dynamic Systems
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An invariant nature of power exchange
• The direction of power flow at any moment is a system invariant.
• The force (effort) and the velocity (flow) are its factors.
)()()( tftetP =Power,
effort flow
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Power Variables
Power = F.v Power = T.w
Power = P.Q Power = V.iPower = Effort (e) x Flow (f)
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System variables for Different Energy Domains
Systems Effort (e) Flow (f)Mechanical Force (F) Velocity (v)
Torque (τ) Angular velocity (ω) Electrical Voltage (V) Current (i)Hydraulic Pressure (P) Volume flow rate (dQ/dt)Thermal Temperature (T) Entropy change rate (ds/dt)
Pressure (P) Volume change rate (dV/dt)Chemical Chemical potential (μ) Mass flow rate (dN/dt)
Enthalpy (h) Mass flow rate (dm/dt)Magnetic Magneto-motive force
(em)Magnetic flux rate (dφ/dt)
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Introductory Example: Electrical (RLC) Circuit
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≡
Reference power direction represented by half arrow on the bonds is given
It is as arbitrary as fixing coordinate system in classical analysis.
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Basic Elements of Bond Graph
Constraints
External Sources of input
Energy variables
Power variables e(t) and f(t)
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Basic Elements SE and SF
Source of effort (SE): maintains input effort, ex. Voltage sources, forces, pressure
Source of flow (SF): maintains input flow, ex. Velocity sources, current, flow sources
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Energy Storing Elements I
• Inertia/Inductance (Effort storage)• Idealisation of devices like mass,
inductance, inertia in mechanical, electrical, hydraulic systems respectively.
=
=
=
dtteGtf
dttFmtv
dtdvmtF
)()(
)()/1()(
/)(
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Energy Storing Elements C
• Compliance/Capacitor (Flow storage)• Idealisation of devices as spring,
capacitor, accumulators
dttfGte
dttfKtKQte
dttftQ
=
==
=
)()(
)()()(
)()(
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A Resistive Element (R)
• Idealization of devices like dampers, resistors, fluid carrying pipes.
• Dissipative element.• Removes energy and relates effort to flow.
eRforfRe )/1(==
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Transformer Element (TF)
• Two port elements altering magnitude of either flow or effort.
• Relates flow to flow or effort to effort by transformer modulus.
• Ratio b/a is transformer modulus.• Other examples are gear set,
pulleys, electric transformer.
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Gyrator Element (GY)
• Two port element which relates input effort to output flow or vice versa by a modulus.
• Ex. Electric motor, generator
Angular velocity output is proportional to applied voltage e
If the rotor spins rapidly and a small F1 will yield proportional velocity V2
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The (0) junction element
321
321
PPPfff
===+
321
321
eeeiii==
+=
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Examples of 0 junction
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The (1) Junction Element
Current through C and R is same.
Summation of voltage
Velocity is common. Summation of forces must
follow Newton’s law.
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Examples of 1 junction
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Constitutive laws of the 3-port junction elements
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e1 f1- e2 f2 + e3 f3 - e4 f4 = 0f1 = f2 = f3 = f4e1- e2 + e3 - e4 = 0
e1 f1+ e2 f2 + e3 f3 + e4 f4 = 01 junction being a flow equalizing junction
f1 = f2 = f3 = f4e1+ e2 + e3 + e4 = 0
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e1 f1+ e2 f2 + e3 f3 + e4 f4 = 00 junction being effort equalizing junction
e1 = e2 = e3 = e4
f1+ f2 + f3 + f4 = 0
f1 - f2 + f3 - f4 = 0
0
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Advantage of bond graphs
Satisfying kinematics implies force balance is automatically satisfied.
Satisfying force balance implies kinematics is automatically satisfied.
You can draw a bond graph from either perspective or a mixture of them.
A bond graph model is somewhere between a physical system and a mathematical model. You can look into either side from the model.
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Concept of causality
or F ma a F / m.= =Is it a causal equation?
Then what is the causal equation?
( ) ( ) ( ) ( )0
0t t
p t e d e d pτ τ τ τ−∞
= = +
Newton’s law:
( ) ( ) ( )0
1 0t
f t e d fm
τ τ= +
NO.
Effort on a mass causes flow is an integral causal form.
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Bond graph causality
• Causality establishes cause and the effect relationship between the factors of power of a bond. The history of cause signal must decide the present value of consequence signal.
• Causality: Indicates WHO causes WHAT to WHOM
• The information of effort is represented by putting a small transverse stroke (causal stroke) at the end of the bond
• The open end of the bond imparts the information of flowto the interacting element or junction
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Causality of BG elements
•Sources: SE Jef
SF Jef
Resistance: R-element:
J Ref
or V iR i V / R= =
J Ref
e Rf= f e / R=
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Causality of BG storage (I & C) elements
•I-element:
J IefJ e
f( ) ( ) ( )( )d
de t m t f t
t=( ) ( ) ( ) ( )
0
1 d 0t
f t e t t Pm t
= +
I
Integral causality Derivative causality
J ef
( ) ( ) ( ) ( )0
d 0t
e t K t f t t Q
= +
C
Integral causality Derivative causalityC-element:
J Cef
( ) ( ) ( )( )dd
f t e t / K tt
=
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• Storage elements are said to have desirable causality if it is integrating.
• Numerical routines are designed to integrate not differentiate.• Inertias: Effort causality. • Compliances: Flow causality
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Causality of junctions
0-junction
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2 3
4
e1 f1
e4 f4
e2f2
e3f3 0
12 3
4
e1 f1
e4 f4
e2f2
e3f3
Stong relations2 13 14 1
Weak relation1 1 2 2 3 3 4 4 0
1 2 3 4
f ff ff f
e f e f e f e fe e e e
===
− + − + == − +
1-junction
Strong bond
Strong bondStong relations1 32 34 3
Weak relation1 1 2 2 3 3 4 4 03 1 2 4
e ee ee e
e f e f e f e ff f f f
===
− + − + == − + +
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Causality of two-port (TF and GY) elements
Gyrator (GY)Transformer (TF)
2 11 2f fe e
μμ
==
TF
TF
GY
GY
e1f1
e2f2
e1f1
e2f2
e1f1
e2f2
e1f1
e2f2
μ
μ
μ
1 22 1
f f /e e /
μμ
==
2 11 2
e fe f
μμ
==
2 11 2
f e /f e /
μμ
==
μ
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Sequential Causality Assignment Procedure (SCAP)
• The following steps are to be followed for causality assignment
• Sources causality to be assigned as per sources: effort by effort sources, flow by flow sources. Extend the causal implications using all 0,1,TF and GY restrictions.
• Choose any C or I element and assign integral causality. Extend the causal implications of this action using 0,1, TF and GY restriction.
• Choose any R element that is unassigned and give it an arbitrary causality. In linear R elements, the causality is in principle indifferent, but indicates whether resistance or conductance need to be entered as parameter. In non linear R elements, equations are more comfortable in one direction according to equation form. Extend the causal implications, using 0,1, TF and GY restrictions.
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Analysis of models
• What information can you extract from a bond graph model?– Causality analysis shows dependencies.– State equations are derived directly from a causal bond graph.– Signal flow diagrams and Transfer functions can also be derived
from a bond graph using Mason’s rule.• Controllability/Observability.
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Examples of Bond graph modelling of mechanical systems
A spring- mass- damper system
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What is wrong with differential causality?
V C1SE C
0SE C
Energy given by cell:2
dE V i t VQ
QV Q / C EC
= =
= =
Energy stored:0 0
2
d
2
Qt
CqE V i dqC
QEC
τ= =
=
Where has half energy gone? Is it dissipated??
Neglecting resistance from the model is wrong.
1SE C
R
Correct BG
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Generation of System Equations
Selection of system variablesA system changes from one
state to another because ofabsorbed causes.
System variables are p’s and Q’s of the integrally causalled I and C storage elements.
∞−
=t
dcausecauseAbsorbed τ)(
∞−=
tdep τ
∞−=
tdfQ τ
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The system equations may be generated by answering the following twoquestions: What do the elements (all) give to the system (expressed in terms of
system variables)? What does the system give to storage elements with integral causality ?
The system variables are p3 and Q2
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(i) Answer• I3 gives flow f=P/m.• C2 gives effort e2=KQ• R4 gives e4=Rf=RP/m• SE gives e1=F(t)(ii) Answer• To I, e3 given, e3=e1-e2-e4• To C, f2 is given mPQf /==
•
mRPKQtFP /)( −−=•
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+
−−=
0)(
0/1/ tF
Qp
mKmR
Qp
From the second equation p=m dQ/dt which when replaced in the first leads to
)(2
2
tFKQdtdQR
dtQdm +−−=
)(2
2
tFKQdtdQR
dtQdm =++
This Eq. corresponds to that derived through traditional method as
)(2
2
tFKxdtdxR
dtxdm =++
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Systems with differential causality
• Differential causalities occurs in systems having such storage elements of which the outputs are determined by outputs of some other storage elements or sources.
• In such cases parameters of differentially causalled elements gets associated with other storage elements which have integral causality
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• In presence of differential causalities, the order of the set state equations is smaller than the order of the system, because storage elements can depend on each other.
• These kind of dependent storage elements each have their own initial value, but they together represent one state variable.
• Their input signals are equal, or related by a factor, which may not be necessarily constant.
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Activation
• Some bonds in a bond graph may be only information carriers.
• These bonds are not power bonds. Such bonds, where one of the factors of the power is masked are called Activated bonds.
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Observers
• Additional states can be added for measurement of any factor of power on a bond graph model using the Observerstorage elements.
• A flow activated C-element would observe the time integral of flow (and consequently flow), whereas an effort activated I-element would observe the generalized momentum (and consequently effort).
• Activated elements are perceived conceptual instrumentations on a model
• They don’t interfere in the dynamics of the system (i.e. their corresponding states never appear on the right-hand side of any state equation.
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Bond Graph Model of Robots
Example 1
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Creation of system equations
I) Identification of system variables. • System variables, in terms of
which the equations are derived, are the absorbed causes in storage elements with integral causality.
• Q and P represents the two system variables corresponding to the integrally causalled storage elements ‘C’ and ‘I’ found in a bond graph.
• Hence in the current example, P1 and Q3 are the system or state variables.
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• II) What element gives to system in term of system variables?
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In Equation we have term because I2 element is in differential causality.
What system gives to integrally causalled I and C element?
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Example 2
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Example 3
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State variables P3, P1, Q5 and Q6
What element gives to system in term of system variables?
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What system gives to integrally causalled I and C element?
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Modelling of four legged walking robots
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Development of Experimental set-up through solid work model
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Dynamic Modeling of a Quadruped Robot
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Dynamics of Body
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Body Translational Dynamics Body Angular Dynamics
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Dynamics of upper link
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Translational Velocity of frame {0}
Angular Velocity Propagation of leg link
Translational Velocity Propagation of leg link
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Dynamics of Prismatic link
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Velocity of the cylinder
Rate of change of contemporary length
Newton equation for translatory motion of the cylinder & piston
Euler equation for rotary motion of the cylinder & piston
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Dynamics of Prismatic link
Normal velocity at the contact pt. 4 & 5
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Bond graph of Prismatic link
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Multi bond graph model of quadruped control in joint space
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Joint position corresponding to time interval
Leg 1 Leg 2 Leg 3 Leg 4Joint
1Joint
2Joint
1Joint
2Joint
1Joint
2Joint
1Joint
2Initial joint angle -1.87 0.6 -1.67 0.53 -1.27 -0.6 -1.47 -0.53To <= t < T1/2 -1.97 1.01 -1.78 0.59 -1.36 -0.59 -1.18 -1.01T1/2 <= t < T1 -1.67 0.53 -1.87 0.6 -1.47 -0.53 -1.27 -0.6T1 <= t < T2/2 -1.78 0.59 -1.96 1.01 -1.18 -1.01 -1.36 -0.59T2/2 <= t < T2 -1.87 0.6 -1.67 0.53 -1.27 -0.6 -1.47 -0.53
Joint rotation for locomotion
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Simulation & Results
Animation
Experiment
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Performance measures
∈ of Compliance legged model < Rigid legged model
∈ = Specific resistanceP(v) = Power consumed Wm = Mass of the system kgg = Gravitational acceleration m/s2v = Velocity m/s
Trot gait
Rigid legged
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References
• R. Merzouki, A. K. Samantaray, P. M. Pathak, B.OuldBouamama, Intelligent Mechatronic Systems: Modeling, Control and Diagnosis, Springer-Verlag, London , 2012.
• Amalendu Mukherjee, Ranjit Karmakar, Arun Kumar Samantaray, Bond Graph In Modeling, Simulation And Fault Identification, I. K. International Pvt Ltd, 2006
• M. M. Gor, P. M. Pathak, A. K. Samantaray, J.-M. Yang and S. W. Kwak, Control oriented model-based simulation and experimental studies on a compliant legged quadruped robot, Robotics and Autonomous Systems, 72 June, (2015) 217–234.
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Acknowledgements
• My students• Anshul Pandey, Anurag Mittal, Himanshu Lohani (B.
Tech, Mechanical Engineering)• Balkrishna V. Jagadale (M. Tech)• Ganesh Kumar K. (M. Tech)• Rohit Khandare (M. Tech)• Rishit Chauhan (M. Tech)• V. L. Krishnan (PhD)• Mehul Gor (PhD)• Kamran Alam: B.Tech, Electronics & Communication• Piyush Kumar Vishwakarma, B.Tech, Electrical • Divya Anand, B.Tech, Production & Industrial Engineering
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Acknowledgment
The work of M.M. Gor, P. M. Pathak, and A. K. Samantaray has been funded by DST, India under Indo–Korea Joint Research in Science and Technology vide Grant No. INT/Korea/P–13.
The work of J.–M. Yang and S. W. Kwak was supported by the National Research Foundation of Korea grant funded by the Korea government (MEST) (No. NRF–2011–0027705).
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Thanks
