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STUDY OF COUPLED DYNAMICS BETWEEN BODY AND LEGS OF A FOUR LEGGED
WALKING ROBOT
V. L. Krishnan(a), P. M. Pathak(b), S. C. Jain(c)
Robotics and Control Laboratory
Mechanical and Industrial Engineering Department
Indian Institute of Technology, Roorkee, 247667, India
(a) vlk08dme@iitr.ernet.in, (b) pushpfme@iitr.ernet.in, (c) sjainfme@iitr.ernet.in
ABSTRACT
The paper presents a study of coupled dynamics
between body and legs of a four legged robot with two
articulate joints per leg. This study brings out the
influence produced by the ground reaction forces
through robot legs on the posture of robot body i.e.
body rotations about X, Y and Z axis. An object oriented
approach has been used for the bond graph modeling of
locomotion dynamics of the four legged robot while
taking into consideration the robot-ground interaction
forces. Detailed kinematics analysis of a single leg has
been carried out. Sub-model created for a single leg is
repeatedly used for developing the bond graph model of
the four legged walking robot. A multi bond graph is
used to represent the system. A dynamical gait is
proposed and implemented through joint control. Joint
control has been achieved using a proportional
derivative control law for each joint of the four legged
robot. The robot locomotion has been successfully
demonstrated through simulation and experiments on a
robot prototype.
Keywords: Four legged walking robot, coupled
dynamics between body and legs, bond graph modeling
1. INTRODUCTION
Legged robots have evolved as a better alternative
compared to their wheeled counterparts for field
applications such as military combat or transport
operations, material handling and rescue operations,
hazardous site inspection; and for extraterrestrial
applications viz. Mars or space exploration etc. Legged
robots are more suitable for such applications as they
have greater agility and also they can move well on all
kind of terrain whereas wheeled robots require only
paved paths for better performance. Among the legged
robots, six or more legged robots are suitable from the
locomotion stability perspective. However, four legged
robots offer a good compromise between locomotion
stability and speed.
Research in rigid legged robots started almost in
early 1980’s with an attempt to realize rigid legged
locomotion mainly on flat terrain. Waldron and
McGhee (1986) presented the design of Adaptive
Suspension Vehicle (ASV), which had morphology
analogous to a six legged insect. Hartikainen et al.
(1992) developed MECANT-I, a hexapod walking
machine for forests applications. These robots mainly
used statically stable gaits for their locomotion. Zhang
and Song (1993) presented a study of the stability of
generalized wave gaits. Gonzalez de Santos and
Jimenez (1995) introduced discontinuous gaits and
Estremera and Gonzalez de Santos (2002) proposed free
gaits for locomotion of quadruped robots on irregular
terrain. Later on researchers got motivated to explore
dynamically stable gaits for realizing faster locomotion.
Furusho et al. (1995) realized bounce gaits on
SCAMPER, a rigid legged quadruped with its design
similar to mammals. Estremera and Waldron (2006)
proposed a leg thrust control method for the
stabilization of dynamic gaits in rigid legged quadruped
robot KOLT. Garcia et al. (2003) suggested that in
addition to developing suitable gaits for legged
locomotion, consideration of actuator dynamics and
friction is essential for getting the real legged
locomotion behaviour. Bowling (2005) examined the
robot’s ability to use ground contact to accelerate its
body. Yoneda and Hirose (1992) employed biologically
inspired approach to realize smooth transition from
static to dynamic gait in the quadruped walking robot
‘TITAN-IV’. Biological inspired approach refers to an
extensive use of sensory feedback and reflex
mechanisms (similar to that found in animals) for
locomotion control. Further Kurazume et al. (2001)
accomplished dynamic trot gait control for ‘TITAN-
VIII’. Inagaki et al. (2006) proposed a method for the
gait generation and walking speed control of an
autonomous decentralized multi-legged robot by using a
wave Central Pattern Generator (CPG) model. Wyfells
et al. (2010) has presented a design and realization of
quadruped robot locomotion using Central pattern
generators.
Legged robot is a multi-body dynamic system.
Realization of various locomotion behaviors viz.
walking, running etc. requires a precise understanding
of the coupled dynamics between the body and legs of a
legged robot. Also influence of ground reaction forces
on the robot body is an important matter of
investigation because it is quite intuitive that
instantaneous leg tip velocity of a legged robot depends
upon the body state variables. Hence with the objective
to investigate the coupled dynamics between the body
and legs of a four legged walking robot, the present
work has been carried out. This paper presents a three
dimensional study of coupled dynamics between body
and legs and generation of stable walking in a four
legged robot. The coupled dynamics and robot
locomotion has been demonstrated through simulation
results and experiments on a robot prototype.
Bond graph technique (2006) has been used for
object oriented modelling of the four legged robot.
Bond graph is an explicit graphical tool for capturing
the common energy structure of systems. It gives power
exchange portray of a system. It provides a tool not only
for the formulation of system equations, but also for
intuition based discussion of system behavior viz.
controllability, observability, fault diagnosis, etc. 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 element to model
coupled systems residing in different energy domains.
In order to avoid repetitive modeling of same type of
structure and to express a very large system in modular
form, objects are created and then joined together to
create an integrated system model. In the present work
bond graph model of the four legged robot is created
and simulated in SYMBOLS Shakti (2006), a bond
graph modelling software.
2. BOND GRAPH MODELLING OF FOUR
LEGGED ROBOT
Modelling of four legged robot consists of modelling of
translational and angular dynamics of robot legs and
body. Figure 1 shows the schematic diagram of a four
legged robot model. In Fig. 1, {A} is inertial frame of
reference and {V} is body frame. Each leg of the robot
has two links. The joint between links i and i+1 is
numbered as i+1. A coordinate frame {i+1} is attached
to (i+1) joint.
The linear dynamics of a body is governed by
Euler’s first law and angular dynamics by Euler’s
second law. Linear dynamics of a body can be given by
()
V
A
AA
G
VV
FM V
•
= (1)
Figure 1: Schematic diagram of four legged robot
Figure 2: Multi bond graph of four legged robot
Where, A
V
F
is resultant of forces at joints ends and
external forces acting at center of gravity (CG),
expressed with respect to frame {A}; ()
V
A
AG
V
•
is
acceleration of CG of body with respect to frame {A};
MV is the mass of the body.
Angular dynamics of the body can be given by
()()()
A
V V VA VA
V
VVV
Nh h
ω
•
=+×
(2)
Where, V
V
N is the moment acting on the body; ()
VA
V
h
and ()
VA
V
ω
are respectively the angular momentum
and angular velocity of body with reference to inertial
reference frame {A}.
These fundamental equations of motion (1) and (2)
along with linear velocity and angular velocity
propagation relations for leg links (presented later in
this section) guides the bond graph modeling of a four
legged robot. Figure 2 shows the multi bond graph
model of a four legged walking robot. The multi bond
graph consists of sub model for the robot body and the
four legs. Figure 3(a) and (b) respectively presents the
bond graph of the ‘LEG’ and the ‘Joint actuator’ sub-
models.
Robot body sub model represents its translation and
angular dynamics. The velocity of the body CG frame
{V} is obtained from the linear inertia of the body.
Euler junction structure (EJS) can be used to represent
the angular dynamics of a body (Mukherjee et al. 2006).
Hence, it has been used to represent the rotational
dynamics of the robot body as well as that of the links
of robot legs in the present work.
The Euler equations used in the creation of the EJS
sub-model of body are deduced from Eq. (2) and are
given by Eq. (3) as
()
x
xx z y yz
NI II
ωωω
=+−
, (3a)
()
yyy xzzx
NI II
ωωω
=+−
, (3b)
()
z
zz y x xy
NI II
ωωω
=+−
. (3c)
Where x
N,y
N,z
N are the torques and ,
x
y
ωω
and
z
ω
are angular velocities acting about the principal axes
of the corresponding body fixed frame. Linear velocity
of the {0} frame of each leg is given as
00
() () [()( )]
AA AA A VV V A
VV V
VVRP
ω
=+−× (4)
In Eq. 4, A
VR represent the transformation from body
frame {V} to inertial frame {A} and can be expressed
as,
A
V
cc ssc cs csc ss
Rcsssscccsssc
ssc cc
θ
φψ
θ
φψφψ
θ
φψφ
θ
φψ
θ
φψφψ
θ
φψφ
θψθ ψθ
−+
=+−
−
ψ, θ and
φ
are the Euler angles representing robot body
rotation about X, Y, Z axis of the body fixed frame {V}.
0
()
VV
i
P represent the position vector of frame {0} of ith
Figure 4: Bond graph of four legged robot locomotion
dynamics
(a)
(b)
Figure 3(a) LEG sub-model (b) Joint actuator sub-
model
leg with respect to body CG frame {V}. It can be
expressed as 0
()[ ]
VV T
iixiyiz
PRRR=, where ‘i’
denotes leg 1 to 4. Value of Rix, Riy and Riz
corresponding to leg 1 to 4 is listed in Table 1 in
appendix.
Linear Velocity Propagation (LVP) sub model
shown in the ‘Body’ part of the multi bond graph in Fig.
2 takes the angular velocity from body ()
VA
V
ω
(obtained from EJS) and linear velocity ()
AA
V
V
(decided by body mass) as input and gives out the
velocity of {0} frame to the link 1 of each leg. Frame
{0} and {1} are coincident for each leg. Hence, the
velocity of frame {1} is same as frame {0} i.e.
10
()( )
AA AA
VV=.
‘Leg’ sub model, shown in detail in Fig. 3(a),
represents a two DOF leg. ‘Leg’ takes angular and
linear velocity of body and joint torques about X-axis as
input. ‘LEG’ sub model uses Angular Velocity
Propagation (AVP) and LVP sub models of links 1 and
2 and gives out leg tip velocity as output. Link lengths l1
and l
2 are taken along the principal Y-axis of the links
and hence represented in vector form as,
[]
0
1000
T
P=,
[]
1
21
00
T
Pl=,
[]
2
32
00
T
Pl=.
Thus ‘LEG’ furnishes complete dynamics of a two link
leg. The various sub models shown in Fig. 2 for leg ‘1’
can also be used to model leg 2, 3 and 4.
LVP and AVP sub models for leg links can be used
to find the velocity of tip of link1 and link 2 of a leg i.e.
frame {2} and frame {3} respectively. Governing
equation for AVP of links of a leg can be given as per
theory (Craig 2006),
11 1
11
() ()()
iA i iA ii
ii i i
R
ωωω
++ +
++
=+ (5)
Where, 1
1
()
ii
i
ω
+
+ is the angular velocity of (i+1) link as
observed from ith link and expressed in (i+1)th frame.
The term can be expressed for link 1 and 2 respectively
as,
10 1
1
()[ 00]
T
ωθ
•
=,21 2
2
()[ 00]
T
ωθ
•
=.
()
iA
i
ω
is the angular velocity of the ith link with respect
to inertial frame {A} and expressed in i
th frame.
1
1
()
iA
i
ω
+
+is the angular velocity of (i+1) link with
respect to inertial frame and expressed in (i+1)th frame.
The above equation (5) is represented by the sub model
‘AVP of Link’ in each leg of the robot.
Governing equation for the link tip velocity and
link CG velocity are given as,
11
()() [()()]
AA AA A iA ii
iiiii
VVR P
ω
++
=+ × (6)
This can be simplified as,
11
[ ( )] [ ( )] [ ][ ( )][ ( )]
AA AA A ii iA
iiiii
VVRP
ω
++
=+−× (7)
For position of a link CG, ()[00]
i
ii T
GGi
Pl=
[ ( )] [ ( )] [ ][ ( )][ ( )]
ii
AA AA A ii i A
GiiGi
VVRP
ω
=+−× (8)
Equations (6), (7), (8) represent the LVP of Link in each
leg of the robot. CG velocity of links depends on link
inertia. In bond graph model ‘I’ elements (representing
mass of a link), are attached at flow junctions. They
yield the CG velocities of links. The starting point of
the current link is same as the previous link tip. Hence,
the tip velocity of the previous link and the angular
velocity of the current link are used to find the tip
velocity and CG velocity of the current link. )( i
Ai
ω
in
above equations can be obtained from the AVP for the
current link.
The leg tip sub-model in Fig. 2 represents the
modelling of leg tip-ground interaction. The robot is
assumed to be walking on a hard surface with no
slipping of legs. An ‘R’ element is appended to ‘1’
junction of each leg in the X and Y direction, to model
the frictional resistance offered by ground. Similarly,
‘C’ and ‘R’ elements are attached in Z-direction to
model the normal reaction force from the ground. Leg
tip position detectors in each direction yields the leg tip
position coordinates.
The integrated bond graph model representing the
four legged robot locomotion dynamics is presented in
Fig. 4. Parametric values assumed for the purpose of
simulation of the four legged robot model are shown in
Table 2 in appendix.
3. GAIT PATTERN
Gait pattern represents the sequence of leg movements
required for realizing locomotion of a robot while
maintaining body stability. In the present work, a
bounding walk gait pattern has been implemented for
achieving robot locomotion. The gait has been used by
Lasa and Buehler (2000) for their single-link legged
quadruped robot SCOUT-II. In this gait pattern, either
the front or rear legs of the quadruped robot are
simultaneously lifted up or brought down to the ground
in a particular phase of the gait. Figure 5(i-viii)
represents schematically the eight phases of the gait
pattern of a locomotion cycle.
To implement the gait pattern, position of the joints
of each leg must be controllable. The voltage supplied
for controlling the position of joints can be given as
()
()
iPdii Vdii
VK K
θθ θθ
=−+−
(9)
Where, i
V is the input voltage supplied at the ith joint of
a leg. The voltage supply to a joint actuator is
implemented in bond graph, through an ‘SE’ element of
joint actuator sub-model shown in Fig. 3(b). KP and KV
are respectively the proportional and derivative gains;
θdi is the desired value of rotation, θi is the actual value
of rotation, di
θ
is the desired joint velocity and i
θ
is the
actual joint velocity.
The joint reference command used for moving a
particular joint to the desired position ‘θi+1’ at the end of
a certain time interval Δti+1 i.e. (ti+1-ti), can be expressed
by the following equation,
1
()
1(1 )
ii
tt
ii
ke
λ
θθ
+
−−
+=± − (10)
In Eq. (14), i
θ
and 1i
θ
+ respectively represents the
joint angular displacement values at the beginning and
end of a time interval Δti+1; k is a factor by which the
joint angle is to be increased or decreased, λ is an
integer. It can be noted from Eq. (14) if i
tt=, then
i
θθ
= and when 1i
tt
+
= then ik
θθ
=±, for very large
values of λ which leads the exponential term to a zero
value.
4. SIMULATION RESULTS AND DISCUSSION
For the selected gait pattern and robot parameters,
simulation has been carried out for 3 cycles. A
locomotion cycle takes 3.2 seconds. Simulation results
are presented in Fig. 6, 7, 8 and 9. Fig. 6(a) and (b)
respectively shows front leg joint 1 displacement (θ1F)
and front leg joint 2 displacements (θ2F) versus time.
Similarly, rear leg joint 1 (θ1R) and rear leg joint 2 (θ2R)
displacement versus time is presented in Fig. 7(a) and
(b). It can be noted that the leg joint angular
displacement plots corresponds with the specified gait
pattern.
Figure 8(a), (b) and (c) respectively presents the
variation of robot body Euler angles ψ, θ and
φ
versus
time. ψ, θ and
φ
represents the robot body rotation about
X, Y and Z axis. As a consequence of the selected gait
pattern for locomotion (in which the two front or rear
legs are simultaneously lifted or brought down
simultaneously), the body angular displacement ‘ψ’ is
oscillatory, as shown in Fig. 8(a). Variation of ‘ψ’ with
respect to time signifies the coupled dynamics between
the legs and the body. In the figure, second cycle of
locomotion has been further split into eight phases (of
the specified gait pattern) to explain the coupled
dynamics. It can be noted that a correspondence
between the various phases of the gait pattern and the
body pitching motion ‘ψ’ exists. For instance, in the
first two phases of the gait pattern (i.e. Fig. 5(i) and (ii))
the rear leg links rotation should result in body pitching
about +X-axis. The reason being the rear leg hip joint
will be at lesser height as compared to its front leg
(i)
(ii)
(iii)
(iv)
(v)
(vi)
(vii) (viii)
Figure 5: Phases of gait pattern
(a)
(b)
Figure 6: Simulation results of rigid legged quadruped
robot locomotion: Front leg joint angular displacements
(a) θ1F (radians) versus time (s) (b) θ2F (radians) versus
time (s)
counterpart due to the specified rotation. Simulation
result in Fig. 8(a) validates this observation. Similarly
corresponding to the gait pattern phase shown in Fig.
5(iv), pitching about X axis varies from positive to
negative values due to the variation of the difference in
the vertical height of front and rear leg hip joints.
Similarly it can be observed that ‘ψ’ is almost equal to
zero radians i.e. robot body is absolutely horizontal, at
the beginning of a locomotion cycle, at the end of the
fourth and eighth phase of a locomotion cycle. The
reason for the fact is that the front and rear leg hip joints
in the respective phases are at almost equal elevation.
Figure 8(b) indicates that there is no significant
angular displacement ‘θ’ about Y-axis (as expected),
because the joint torque is supplied only about X-axis.
Figure 8(c) shows that the body is turning about +Z-
axis. Figure 8(d) indicates quadruped robot progression,
as since the robot body CG displacement occurs along
positive Y-axis i.e. the direction of locomotion.
Figure 9(a), (b), (c) and (d) respectively shows the
displacement of tip of legs 1, 2, 3 and 4, along positive
Y-axis. It can be noted from the simulation results that
the leg tip displacement values for leg 2 and 4 are
greater than that of leg 1 and leg 3. Thus the simulation
(a)
(b)
Figure 7: Simulation results of rigid legged quadruped
robot locomotion: Rear leg joint angular
displacements (a) θ1R (radians) versus time (s) (b) θ2R
(radians) versus time (s)
(a)
(b)
(c)
(d)
Figure 8: Simulation results of rigid legged quadruped
robot locomotion (a) ψ (radians) versus time (s) (b) θ
(radians) versus time (s) (c)
φ
(radians) versus time (s)
(d) YCG (m) versus time (s)
results indicate that the robot is turning about positive
Z-axis. The turning about Z-axis can be attributed to the
modelling of leg tip-ground interaction.
The frictional resistance offered by the ground to
legged robot locomotion has been modeled, by
appending a dissipative R-element (bond graph
element) at the 1-junction representing leg tip velocity.
The value of the parameter ‘R’ representing frictional
resistance has been assigned greater value in X-direction
compared to that in the locomotion direction i.e. RgX >
RgY.
In the next section, the experimental realization of
locomotion of quadruped robot prototype is presented.
5. EXPERIMENTAL RESULTS
Experimental set-up designed for realizing robot
locomotion is presented in Fig. 10. The quadruped robot
comprises of four legs and a body over which the
controller CM5+ is mounted. The front and rear legs
have been designed identical in all respects. Each leg of
the quadruped robot has two rigid links connected
through revolute joints, one at the hip and second one at
the knee of a leg. The links are rotated through
‘Dynamixel’ series AX-12+ actuators, deployed at the
joints.
Bioloid control behavior interface of the Robotis Inc.
is used for the purpose of controlling robot locomotion.
Control algorithm is fed to the CM5+ controller through
a personal computer. A serial to USB data cable is used
for the communication between the PC and the CM5+
controller. SMPS is used for supplying required power
to the actuators and electronic circuitry. CM5+
controller uses ATMega 128 (128Kbyte flash memory)
as the main processor in it. It operates in the voltage
range of 7V-12V.
A program corresponding to the specified gait pattern
is communicated to the controller CM5+. Consequently
the robot locomotion is accomplished. Figure 11
presents the snapshots of locomotion of the quadruped
robot. Snapshots indicate a clear progression of the
robot. Figure 12 presents the leg tip and body CG
trajectory in XY plane, plotted using the experimental
data. The body CG trajectory and the leg tip
(a)
(b)
(c)
(d)
Figure 9: Simulation results of rigid legged quadruped
robot locomotion: (a) Leg 1 tip Y displacement (m)
versus time (s) (b) Leg 2 tip Y displacement (m) versus
time (s) (c) Leg 3 tip Y displacement (m) versus time
(s) (d) Leg 4 tip Y displacement (m) versus time (s)
Figure 10: Experimental prototype of four legged robot
displacement plots indicate that the robot is almost
progressing as desired, along a straight line in the
locomotion direction. Slight deviation of the body CG
and leg tip trajectories from the straight line path can be
attributed to the unpredictable and arbitrary lag in
command signals to the front or rear leg joint actuators.
Figure 13(a), (b), (c) and (d) respectively presents the
actual and commanded joint angle displacement
trajectories of joint 1 and 2 of the front and rear legs.
There is a slight deviation in the commanded and actual
joint trajectory along with some arbitrary lag in the
front and rear joint trajectory.
6. CONCLUSIONS
The paper presents the coupled dynamics between the
legs and body during locomotion. This study has been
carried out through simulation as well as experiment on
a four legged robot design with two articulate joints per
leg. Modeling and simulation of the four legged robot
has been carried out using bond graph technique. For
the specified gait pattern, the body angular displacement
trajectory for ψ and θ demonstrates the influence of
ground reaction forces, transmitted through legs, on the
robot body. The gait pattern has been tested on a
quadruped robot experimental model and locomotion
has been successfully realized.
This dynamic model of the four legged robot
locomotion can be further extended for simulating
running behavior, obstacle avoidance etc. Flexible leg
concept may be incorporated to enable the robot to walk
on uneven terrain. In fact the motivation for this work
originated with a desire to build a model of four legged
walking robot with flexible legs. Flexible legged robot
model can be used to study the effect of flexible leg
dynamics on the stability and impact tolerance of the
robot.
APPENDIX
Table 1:Position of Frame {0} with respect to Body CG
Leg ‘i’ Rix R
iy R
iz
Leg 1 -0.042 0.068 -0.045
Leg 2 0.042 0.068 -0.045
Leg 3 -0.042 -0.068 -0.045
Leg 4 0.042 -0.068 -0.045
(a)
(b)
(c)
(d)
(e)
(f)
Figure 11: Snapshots of rigid legged quadruped robot locomotion
Figure 12: Experimental results of rigid legged
quadruped robot locomotion: Leg tip and body CG
displacement along X-axis (m) versus Y-axis (m)
Table 2: Four Legged Robot Parameters
Parameters Value
Robot body mass Mb
= 0.43Kg;
Moment of Inertia
(M. I.) of body
IBXX = 0.007Kg-m2,
IBYY = 0.004Kg-m2,
I
B
ZZ = 0.002Kg-m2
Leg link lengths l1 = 0.068m, l2 = 0.025m
Leg link mass Ml1= 0.075Kg; Ml2= 0.015Kg
M. I. of link ‘1’ Ixx1 = 0.0002Kg-m2,
Iyy1 = 0.00025Kg-m2,
I
zz
1= 0.00001Kg-m2
M. I. of link ‘2’ Ixx2 = 0.00001Kg-m2,
Iyy2 = 0.000003Kg-m2,
I
zz
2= 0.000004Kg-m2
Joint actuator
parameters
Inductance: Lm =0.001H;
Resistance: Rm= 0.1Ohms;
Motor constant: Kt = 0.2 N-m/A;
Gear ratio: n = 254
Leg tip-ground
interaction
Stiffness: Kg = 100000N/m;
Damping: RgZ = 1000N-s/m;
Frictional resistance:
RgX = 800N-s/m,
R
g
Y = 400N-s/m
Gain Values Proportional gain:
KP = 2500V/rad;
Derivative gain: KV =25V/rad/s
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