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Applied Mechanics and Materials Vols. 110-116 (2012) pp 2762-2767
Online available since 2011/Oct/24 at www.scientific.net
© (2012) Trans Tech Publications, Switzerland
doi:10.4028/www.scientific.net/AMM.110-116.2762
Running Model for a Compliant Wheel-Leg Hybrid Mobile Robot by
Using a Mass-Spring Model
Youngshik Kim 1, a and Dong-Hwan Shin 2, b
1 Department of Mechanical Engineering, Hanbat National University, Daejon, 305-719, Korea
2 Division of Robot System, DGIST, Daegu, Korea
a youngshik@hanbat.ac.kr, b sdh77@dgist.ac.kr
Keywords- mass-spring model; passive compliant joints; running; swing control; wheel-leg hybrid
mobile robot
Abstract—This research presents a running model for a compliant wheel-leg hybrid mobile robot.
A wheel-leg consists of three legs arranged by 120 degrees to each other and passive compliant
joints. For simplicity, each leg is treated as a linear spring such that a traditional mass-spring model
can be applied to model running of the wheel-leg hybrid robot. In order to achieve stable running,
we propose a simple controller that can converge a robot leg to a desired angle of attack
asymptotically during the swing phase. Hybrid wheel-leg running is then simulated and results are
discussed.
I. INTRODUCTION
For decades, many researchers have adopted legs and wheels to provide mobility for robots [1-5].
However, mobility of legged and wheeled robots are critically limited to ground conditions and
environment. For instance, wheeled robots can move efficiently on high traction surfaces whereas
legged robots provide good performance on rugged terrain. For these reasons, hybrid mobile robots
combining wheels and legs at the same time have drawn many researchers’ attention to maximize
robot mobility in all terrain.
A unicycle type model [6] is significantly used to describe wheeled mobile robot. In contrast,
several types of model are available in legged robots according locomotion and speed; minimal
model of walking, synthetic wheel, and mass-spring model of running [7]. Passive compliant joints
are also applied to reject disturbances, increase energy efficiency, and/or provide advanced
maneuverability in some robots . Traditionally, linear or torsional spring systems have been used to
describe these compliant joints [8, 9].
Furthermore, it is well known that the simple
mass-spring model, so called Spring Loaded
Inverted Pendulums (SLIPs) [8], can demonstrate
running dynamics of animals and humans
effectively. In the mass-spring model, a leg is an
ideal linear spring of negligible mass. The robot
body is then described by a point mass mounted
on the leg. Thus, kinetic and potential energy of
this system is conserved supposing a perfect
spring and no air resistance such that the transport
cost will be zero in an ideal world.
Recently, we are developing a novel wheel-leg
hybrid robot, Fig. 1, with passive compliant joints
inspired by animal and human legs [10]. Most
importantly, compliance in the leg is adopted to
provide energy-efficient running similar to
Fig. 1. Wheel-leg hybrid mobile robot with
passive compliant joints
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Applied Mechanics and Materials Vols. 110-116 2763
animals storing kinetic energy as elastic energy during passive motion. In this research we are
interested in simplified robot modeling and control for high speed locomotion similar to animals and
humans. Thus, we present how our novel wheel-leg with compliance can be modeled for running by
using the simple mass-spring model (SLIP) and a controller, which is the main contribution of this
research.
This paper is organized as follows: In Sec. II, we present a wheel-leg hybrid robot model
considering dynamics and kinematics. We then verify the proposed wheel-leg robot in simulation
and discuss simulation results in Sec. III. Finally, Sec. IV provides conclusions of this research and
future work.
II.
RUNNING MODEL
A hybrid wheel-leg is modeled by using a traditional spring-mass system. Applying the spring
loaded inverted pendulum model, inertia and mass of a leg is assumed to be negligible. The robot
body is then treated as the point mass, m. In this case, three identical compliant leg segments are
attached to the body and separated by 120 degrees as shown in Fig. 2.
Fig. 2. Hybrid wheel-leg modeling
Running of the wheel-leg system can be divided into swing and stance phases as shown in Fig. 3
similar to the SLIP model. It is, however, worthwhile to note that our wheel-leg rotates in a fixed
direction for forward motion similar to a wheel whereas a leg move back and forth periodically to
satisfy desired angles of attack.
First, considering the particle dynamics of mass, m, in Cartesian coordinates, (x 0 , y 0 ), during the
swing phase where only gravity is acting, we have,
x = 0
0
y
0
= −g
, (1)
where m represents the mass of the robot and g is the gravity. Similarly, considering the stance phase
when the leg i (i=1,2,3) rolls about a fixed point on the ground, we then have,
k
x0 = ( L0
− Li
)cos( θi
) ; i = 1,2,3
m
k
y0 = ( L0
− Li
)sin( θi
) − g
m
, (2)

2764 Mechanical and Aerospace Engineering, ICMAE2011
where coordinates (x i , y i ) represent the position of a foot point of the leg i, L 0 is the undeformed
length of the leg, L i is the length of the leg i, and θ i is an angle measured from the x-axis that is
parallel to the ground as defined by,
L = ( x − x ) + ( y − y )
θ
2 2
i i 0 i 0
tan
⎛ y
− y ⎞
−1 0 i
i
= ⎜ ⎟
x0
− xi
Foot positions of legs are then defined by,
⎝
⎠
. (3)
x = x + L cos( β )
i 0 i i
y = y + L sin( β )
i 0 i i
, (4)
where β 1 =β, β 2 =β +2π/3, β 3 =β −2π/3, and
β
β
− y y
tan
⎛ − ⎞
⎝ ⎠
1 1 0
=
1
= ⎜ ⎟
x1 − x0
. (5)
Furthermore, the angle of attack, θ*, Fig. 3, should be determined to provide stable motion. In this
case, robot motion is stable if the robot moves without falling on the ground. Toward this goal,
Rummel and Seyfarth [8] investigated self-stability and self-stable running as a function of robot
properties (leg length, mass, and joint stiffness), running speed, and angle of attack. As a result,
they illustrated several regions of these properties for stable running. If robot parameters are selected
in these stable regions, the robot is supposed to keep running without falling. Thus, we will choose
the angle of attack which satisfies the aforementioned regions of stable running for given robot
parameters for evaluation in Sec. III.
Now we discuss how we can achieve desired orientation of the wheel-leg for stable running.
During the swing phase, the wheel-leg must rotate one of its leg to the desired angle of attack before
this leg touches down the ground. Toward this goal, we need to control angular velocities of the
wheel-leg. For asymptotical stability [11], we apply the following controller during the swing phase,
θ = − tanh( θ −θ*)
, (6)
i
K θ
i
v y
Swing phase
Swing phase
Trajectory path of
the center of mass
v x
Stance phase
L

Applied Mechanics and Materials Vols. 110-116 2765
y(m)
1.5
0.5 1
t=0
0 5 10 15 20 25 30
x (m)
Vy
Vx
Fig. 4. Running trajectory path of the center of mass and sequential positions of the wheel-leg
y cos( θ ) − x sin( θ )
θ =
0 i 0 i
i
(7)
Li
Note that self-stability of legged robots physically depends on how angular momentum and
gravity of the center of mass are balanced during the stance phase to move without falling. Thus, it is
worthwhile to investigate robot energy during motion. The total energy of the robot consists of
kinetic, gravitational potential, and elastic potential energy,
E E E E
= + + , (8)
Kinetic Potential, Gravity Potential , Elastic
where
m
EKinetic
= x + y
2
E = mgy
2 2
(
0 0 )
Potential, Gravity 0
k
EPotential, Elastic
= L0
− Li
2
( )
2
. (9)
Note that the total energy (8) is conserved in the mass-spring model with an assumption of negligible
leg mass and no disturbance, which will be verified in the next section.
III. SIMULATION RESULTS AND DISCUSSIONS
A. Methods
First, we assume an ideal simulation environment with a perfect model and no disturbance as
mentioned in Sec. II. The robot parameters used in this research are as follows; m=80 kg, L 0 =1 m,
θ*=112 degrees, k=20000 N/m, K θ =50. As discussed in Sec. II, this angle of attack satisfies selfstability
for running according to [8] with the following initial condition for the center of mass;
position [x 0 (t=0), y 0 (t=0)]=[0, 1 m] and velocity [v x (t=0), y y (t=0)]=[5 m/sec, 0].
We simulate the proposed leg-wheel hybrid by using Matlab with tolerances of 1 ×10 -12 . Note that
running is divided into two phases for a single step whose governing dynamics are different, but
robot path trajectories are piecewise continuous.
B. Simulation Results
Now we present running simulation results, Fig. 4 and Fig. 5, applying the aforementioned robot
parameters. The robot trajectory path, Fig. 4, illustrates the motion of the center of robot mass. As
expected, this result demonstrates that the wheel-leg robot runs stably and smoothly similar to the
SLIP model. Note that the wheel-leg rotates clockwise to move forward (the positive x direction)
during the swing phase, which is different from the motion of a single leg used in the SLIP model as
discussed in Sec. II.

2766 Mechanical and Aerospace Engineering, ICMAE2011
L (m)
1
0.9
L1
L2
L3
Velocity (m/s)
Control input
(rad/sec)
Energy (Nm)
0.8
0 1 2 3 4 5 6 7
Time (sec)
(a) Leg length
6
Vx
4
2
0
-2
0 1 2 3 4 5 6 7
Time (sec)
(b) Robot velocity
0
-20
-40
-60
0 1 2 3 4 5 6 7
Time (sec)
(c) Control input
2000
1500
1000
500
Total
Kinetic
Potential(gravity)
Elastic potential
0
0 1 2 3 4 5 6 7
Time (sec)
(d) Energy
Fig. 5. Simulated wheel-leg hybrid running
Fig. 5 (a)-(d) show change in leg lengths (L 1 , L 2 , L 3 ), velocities of the center of mass, control
input for the swing phase, and system energy, respectively. A leg is compressed while contacting
with the ground. The amount of this leg deflection varies slightly for each step, but the maximum
deflection is estimated to be ~25% of the undeformed length. The average speed of the robot in the
x-direction is ~ 5 m/sec, which is almost same as the initial speed whereas vertical speed changes
similar to a bouncing ball. Our results also show that the control input bounded by K θ ca rotate the
leg to the desired angle of attack as designed. Finally, we find that total energy is conserved for
wheel-leg hybrid running as discussed in Sec. II. In this case, kinetic energy (49.8%~60.1%) and
gravitational potential energy (37.3%~44.4%) are significant when compared to total energy of
1784.8 Nm whereas elastic potential energy (0~12.9%) is relatively small.
Vy
IV. CONCLUSIONS
In this research we model a novel compliant wheel-leg hybrid robot based on a simple massspring
model. We also propose a controller that can converge the wheel-leg to a desired angle of
attack asymptotical during the swing phase. As a result, our simulation results verify that our hybrid
wheel-leg can run efficiently similar to the ideal SLIP model where leg mass and inertia are
negligible.
Future work will focus on extension of the proposed running model considering masses and
inertia effects of wheel-legs, which will allow us to investigate more realistic robot and ground
conditions.

Applied Mechanics and Materials Vols. 110-116 2767
ACKNOWLEDGMENT
This research was supported by the Hanbat National University, Daejon, Korea.
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Mechanical and Aerospace Engineering, ICMAE2011
10.4028/www.scientific.net/AMM.110-116
Running Model for a Compliant Wheel-Leg Hybrid Mobile Robot by Using a Mass-Spring Model
10.4028/www.scientific.net/AMM.110-116.2762