MECHANICAL SYSTEM MODELLING OF ROBOT DYNAMICS
USING A MASS/PULLEY MODEL
L. J. Stocco and M. J. Yedlin
The Department of Electrical and Computer Engineering, The University of British Columbia
2332 Main Mall, Vancouver, BC, Canada, V6T 1Z4
Keywords: Mass matrix, inertia matrix, MP model, pulley, differential transmission, mechanical system representation,
robot dynamics, impedance, equivalent electric circuit.
Abstract: The well-known electro-mechanical analogy that equates current, voltage, resistance, inductance and capacitance
to force, velocity, damping, spring constant and mass has a shortcoming in that mass can only be used to simulate
a capacitor which has one terminal connected to ground. A new model that was previously proposed by the
authors that combines a mass with a pulley (MP) is shown to simulate a capacitor in the general case. This new
MP model is used to model the off-diagonal elements of a mass matrix so that devices whose effective mass is
coupled between more than one actuator can be represented by a mechanical system diagram that is
topographically parallel to its equivalent electric circuit model. Specific examples of this technique are
presented to demonstrate how a mechanical model can be derived for both a serial and a parallel robot with both
two and three degrees of freedom. The technique, however, is extensible to any number of degrees of freedom.
1 INTRODUCTION
The concept of impedance and its generalization
reactance, has been used to define equivalent
circuits of mechanical and electro-mechanical
systems since the development of the Maxwell
model of solids. The idea that driving point
impedances could be decomposed into terms that
parallel electrical elements was initiated by (Foster,
1924) who showed that the frequency response of
any system is determined by the poles and zeros of
its transfer function. The conditions for network
synthesis are described by (Brune, 1931) and later
applied by (Paynter 1961) who introduced bond
graphs to distinguish and represent effort and flow
variables in a graphical setting. Examples of
electro-mechanical system simulations are
numerous and include magnetic circuits (Hamill,
1993), mechatronics and electromechanical
transducers (Tilmans, 1996), (van Amerongen &
Breedveld, 2003), (Sass et al., 2004).
Mechanical block diagrams are routinely used
to model robot dynamics although some (Eppinger
& Seering, 1992) limit them to a single axis while
others (Yamakita et al., 1992) rely entirely on
equivalent electric circuits to avoid the inherent
difficulties of creating mechanical models of multi-
axis devices, transmission systems or other systems
with coupled dynamics.
Section 2 of this paper describes the
conventional electro-mechanical analogy and points
out a limitation of the mass model. It goes on to
describe a new mass/pulley (MP) model which
overcomes the inherent deficiency in the
conventional mass model. In Section 3, it is shown
how the new MP model can be used to model the
dynamics of devices which have coupled effective
masses. Examples are provided which include both
2-DOF and 3-DOF serial and parallel manipulators.
Lastly, concluding remarks are made in Section 4.
2 ELECTRO-MECHANICAL
ANALOGIES
The ability to define an electro-mechanical
equivalent circuit stems from the parallelism in the
differential equations that describe electrical and
mechanical systems, each of which involve an
across variable, a through variable and an
impedance or admittance variable. In electrical
circuits, voltage
E(s) is the across variable and
current
I(s) is the through variable. In mechanical
systems, velocity
V(s) is the across variable and
force
F(s) is the through variable (i.e. flow variable
25
J. Stocco L. and J. Yedlin M. (2007).
MECHANICAL SYSTEM MODELLING OF ROBOT DYNAMICS USING A MASS/PULLEY MODEL.
In Proceedings of the Fourth International Conference on Informatics in Control, Automation and Robotics, pages 25-32
DOI: 10.5220/0001614400250032
Copyright
c
SciTePress
(Fairlie-Clarke, 1999)). This results in a
correspondence between resistance
R and damping
B, inductance L and spring constant K, and
capacitance C and mass M shown in (1-3). An
alternate approach treats force as the across variable
and velocity as the through variable but that approach
is not used here. By (1-3), the electromechanical
equivalents shown in Figure 1 can be substituted for
one another to model a mechanical system as an
electrical circuit and vise versa.
Figure 1: Admittance of electro-mechanical equivalents.
2.1 Classical Mass Model Limitation
Each of the components in Figure 1 has two
terminals except for the mass which has only one.
This is due to the fact that the dynamic equation of a
mass (3) does not accommodate an arbitrary
reference. Acceleration is always taken with respect
to the global reference, or ground. Consider the two
systems in Figure 2 which are well known to be
analogous.
Figure 2: LC circuit and mechanical equivalent.
In Figure 2, the voltage across the capacitor
e
c
corresponds to the velocity of the mass v. Both of
these are relative measurements that only correspond
to one another because both are taken with respect to
ground. Consider, on the other hand, the circuit in
Figure 3 which contains a capacitor with one
terminal open circuited.
Figure 3: RC circuit and mechanical equivalent.
In Figure 3, the capacitor carries no current
and therefore, has no effect on the output voltage e
o
.
In other words, the voltages at n
1
and n
2
are equal so
the capacitor behaves like a short circuit. In the
mechanical “equivalent”, it is not possible to
connect a non-zero mass M to node n
1
without
affecting the output velocity v
o
. This is due to the
implicit ground reference of the mass (shown by a
dotted line) which prevents it from ever behaving
like a mechanical short circuit. Note that this same
limitation does not apply to the spring or damper
since they both act as a mechanical short circuit
(infinitely stiff connection) if one terminal is left
unconnected, just like their electrical counterparts,
the inductor and resistor.
2.2 The Mass/Pulley (MP) Model
Because of the above limitation, there are
mechanical systems which can not be modelled
using a mechanical system diagram. Elaborate
transmission systems such as robotic manipulators
may contain mass elements that are only present
when relative motion occurs between individual
motion stages. Currently, systems such as these can
only be modelled using electric circuits since
capacitors can be used to model this type of
behaviour but masses cannot.
It would be useful to have a mechanical
model which simulates the behaviour of a capacitor
without an implicit ground connection so that any
mechanism (or electric circuit) could be modelled by
a mechanical system diagram. This new model
should have two symmetric terminals (i.e. flipping
the device over should not affect its response), obey
Ohm’s Law, and be able to accommodate non-zero
velocities at both terminals simultaneously. A model
proposed by the authors (Stocco & Yedlin, 2006)
combines a mass with the pulley-based differential
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26
transmission shown in Figure 4. The pulley system
obeys the differential position / velocity relationship
shown in (4,5).
Figure 4: Pulley based differential transmission.
Note from (5) that although the pulley
provides the desired differential velocity input, it
also introduces an undesired 2:1 reduction ratio.
However, setting v
1
to 0 (i.e. connecting n
1
to
ground) results in (6). Therefore, a similar pulley
system with one input tied to ground could be used
to scale up velocity by an equivalent ratio.
The double pulley system shown in Figure 5
is a differential transmission with a unity gear ratio.
The primary pulley provides the differential input
while the secondary pulley cancels the reduction
ratio to achieve unity gain. A mass connected to the
secondary pulley is accelerated by a rate equal to the
difference between the acceleration of the two
inputs, n
1
and n
2
. This system simulates the
behaviour of a capacitor that may or may not be
connected to ground (Figure 5). Voltage E
1
corresponds to velocity V
1
, voltage E
2
corresponds to
velocity V
2
, current I corresponds to tension F and
capacitance C corresponds to mass M as shown by
(7,8). Note that the free-body diagram of the centre
pulley shows that the tension F in the primary cable
is equal to the tension F in the secondary cable. The
system must be balanced because any net force on
the massless centre pulley would result in infinite
acceleration of the pulley and therefore, the mass as
well.
The MP model uses ideal cables with zero
mass and infinite length and stiffness. The ideal
cables travel through the system of massless,
frictionless pulleys without any loss of energy. The
MP model operates in zero gravity so the mass is
only accelerated as a result of cable tension and/or
compression. Unlike practical cables, the ideal
cables never become slack. When an attractive force
is applied between n
1
and n
2
, F<0 and the mass is
accelerated downward. A block diagram of the MP
model is presented in Figure 6 where P has the same
value as M in Figure 5. Note that, unlike a pure
mass, the MP model has two terminals, n
1
and n
2
which correspond to the two ends of the primary
cable.
Figure 5: Mass / pulley equivalent of a capacitor.
Figure 6: Block diagram of MP model.
Consider Figure 7 which is the mechanical
system from Figure 3 with the mass replaced by an
MP model. With terminal n
2
left unconnected, the
primary cable of the MP model travels freely
through the primary pulley without accelerating the
mass or consuming energy. The MP model behaves
like a mechanical short circuit, just like the capacitor
in Figure 3. Also note the topological similarity
between the electrical circuit in Figure 3 and its true
mechanical equivalent in Figure 7. This is a direct
result of the topological consistency between the
capacitor and the MP model, both of which have two
symmetric terminals. As pointed out in (Stocco &
Yedlin, 2006), this consistency allows one to
analyze mechanical systems using electric circuit
analysis techniques once all masses have been
replaced by MP models.
MECHANICAL SYSTEM MODELLING OF ROBOT DYNAMICS USING A MASS/PULLEY MODEL
27
Figure 7: Mechanical equivalent using MP model.
3 ROBOT MASS MATRIX
Consider the simplified dynamics of a 2-DOF robot
(9) where M is the mass matrix, B is the damping
matrix, F is a vector of joint forces/torques (10), R is
a vector of joint rates r
1
and r
2
(10), and s is the
Laplace operator. Spring constants, gravitational and
coriolis effects are assumed to be negligible for the
purpose of this example. If the damping in the
system is dominated by the actuator damping
coefficients, B is a diagonal matrix (10). M, on the
other hand, represents the effective mass perceived
by each joint and is not diagonal or otherwise easily
simplified in general.
For simple kinematic arrangements such as
the redundant actuators shown in Figure 8 which
only have a single axis of motion, M is shown in
(11). The system responses are modeled by the
mechanical system diagram shown in Figure 9 and
the dynamic equation shown in (10). Using the
electromechanical transformation described in
Section 2, this system can also be represented by the
electrical circuit analogy shown in Figure 9.
Performing nodal analysis on the circuit in
Figure 9 results in (12) by inspection. Note however,
that (12) contains the term i
1
-i
2
as well as v
2
which
corresponds to the end-point velocity in the
mechanical system or, in other words, the sum of the
joint rates r
1
+r
2
. To obtain a correspondence between
electrical and mechanical component values, the
dynamic equation (10) is rearranged in (13) where
the associated damping
B' and mass M' matrices are
shown in (14,15). From (14), the resistor
admittances g
1
and g
2
and capacitor values c
1
and c
2
correspond to the equivalent damping and mass
values
b'
1
, b'
2
, m'
1
and m'
2
(16) respectively.
Figure 8: Redundant rotary & prismatic actuators.
Figure 9: System models of redundant actuators.
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In this simple example, masses are sufficient
to model the system behaviour but only because the
device has a single degree of freedom so
M' is
diagonal and there is no cross-coupling between
actuators. In general, however, effective mass is not
always decoupled and the off-diagonal elements of
M' can be expected to be non-zero. When M' is not
diagonal, conventional single-terminal masses are
unable to model the entire effective mass of the
system. They can not model the off-diagonal terms
that describe inertial effects resulting from relative
motion of the actuators.
3.1 Serial 2-DOF Robot
Consider the 2-DOF serial robot shown in Figure 10.
The mass matrix for this mechanism is approximated
in (Craig, 1989) by two point masses d
1
and d
2
placed
at the distal actuator and end-effector as indicated
below. The resulting mass matrix (17) has the terms
shown in (18-20) where q
1
and q
2
are the joint angles
and l
1
and l
2
are the link lengths. Just as in the
previous example, actuator damping coefficients b
1
and b
2
are taken to dominate the total system
damping.
Figure 10: 2-DOF serial robot.
The equivalent circuit model of this system is
shown in Figure 11. It is similar to Figure 9 except
that the capacitor values are configuration dependent
and a third capacitor c
12
is included to model the
coupled mass terms that are present. Performing
nodal analysis results in (21) and the corresponding
M' matrix in (22) which can be rearranged to solve
for the mechanical model parameters in terms of the
physical mass values in (23).
B' is the same diagonal
matrix as in (14).
Figure 11: Electrical model of 2-DOF serial robot.
Note from (22) that
M' is diagonal (i.e. p'
12
=0)
when
p'
12
=0. From (19,20), this is merely the special
case when q
2
= ±π2. Therefore, it is not possible to
model this system using only masses due to their
implicit ground reference, as described in Section
2.1. The off-diagonal terms can, however, be
modelled using the MP model proposed in Section
2.2. It results in a mechanical system model that is
topologically identical to the equivalent circuit in
Figure 11 where each grounded capacitor (c
1
,c
2
) is
replaced by a regular mass and each ungrounded
capacitor (c
12
) is replaced by an MP model since the
MP model is able to accommodate a non-zero
reference acceleration. The resulting mechanical
system is shown in Figure 12.
Although
p'
12
has a negative value when π2<q
2
<π2,
the net mass perceived by each actuator is always
positive because M is positive definite. When
p'
12
is
negative, it simply means that the motion of actuator
1 reduces the net mass perceived by actuator 2, but
the net mass perceived by actuator 2 is always
greater than zero.
3.2 Parallel 2-DOF Robot
The same technique can be applied to parallel
manipulators such as the 2-DOF 5-bar linkage used
MECHANICAL SYSTEM MODELLING OF ROBOT DYNAMICS USING A MASS/PULLEY MODEL
29
by (Hayward et al., 1994). In the case of parallel
manipulators, each actuator is referenced to ground
but there remains a coupling between the effective
mass perceived by each actuator which, like a serial
manipulator, is configuration dependent. This
coupling is modelled by c
12
and p'
12
in the equivalent
electrical and mechanical models shown in Figure
13. Typically, parallel manipulators also have
coupled damping terms due to their passive joints
which would be modelled by a conductance g
12
added between nodes 1 and 2 (i.e. in parallel with
c
12
). However, for the sake of simplicity, the
damping of the passive joints are neglected here.
Figure 12: Mechanical model of a 2-DOF serial robot.
Figure 13: Model of a 2-DOF parallel robot.
Performing nodal analysis on the circuit in
Figure 13 results in (24) by inspection. For a parallel
robot, currents and voltages correspond directly to
joint forces and joint rates so
B'= B and M'= M . For a
mass matrix of the form shown in (17), the elements
of the
M' matrix, and therefore the parameter values
associated with the masses and MP models of Figure
13, are shown in (26).
3.3 Multiple DOF Robots
This technique is easily extended to devices with
any number n of degrees of freedom. With serial
manipulators, the compliance and damping is often
mainly in the actuators and the damping B and
spring K matrices are diagonal (27,28). With parallel
manipulators, the B and K matrices typically contain
off-diagonal terms but they are easily modelled
using conventional techniques since springs and
dampers are 2-terminal devices which can be placed
at any two nodes in a system diagram.
To account for inertial cross-coupling, the
model must contain a capacitor and/or MP model
between every pair of actuators. For example, the
electric circuit model and corresponding mechanical
system model of a serial 3-DOF manipulator are
shown in Figure 14. The capacitance C matrix
resulting from the nodal analysis (29) of the circuit
in Figure 14 is shown in (30).
Just as in the previous examples, the 3x3 mass
matrix
M' (32) is rearranged into the form shown in
(31) to parallel the current/voltage relationship of
(29). For the mass matrix
M of the form shown in
(33), the entries of the
M' matrix are solved for in
(34).
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Figure 14: Model of a 3-DOF serial robot.
Similarly, for a parallel 3-DOF robot, the electric
circuit model and corresponding mechanical system
model are shown in Figure 15. For a mass matrix of
the form shown in (33), the elements of
M' are
shown in (35).
Figure 15: Model of a 3-DOF parallel robot.
4 CONCLUSION
It is argued that a plain mass is not a complete and
general model of a capacitor since a mass only has
MECHANICAL SYSTEM MODELLING OF ROBOT DYNAMICS USING A MASS/PULLEY MODEL
31
one terminal whereas a capacitor has two. The
response of a mass corresponds to its acceleration
with respect to ground and, therefore, can only be
used to simulate a capacitor which has one terminal
connected to ground. It cannot be used to simulate a
capacitor which has a non-zero reference voltage. A
new model described here that consists of a mass
and a pulley correctly simulates the response of a
capacitor in the general case.
It is shown that the MP model can be used to
model systems with cross-coupled effective masses
which are otherwise, impossible to model with pure
masses alone. This includes both serial and parallel
manipulators with any number of degrees of
freedom. The mechanical system model that is
obtained fully describes the dynamic response of the
system and is topologically identical to its electric
circuit equivalent. As shown in (Stocco & Yedlin,
2006), this makes it possible to apply electric circuit
analysis techniques to mechanical systems, directly.
ACKNOWLEDGEMENTS
The authors gratefully acknowledge Tim Salcudean
for his valuable comments during the preparation of
this manuscript.
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