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Human running can be stabilized in a wide range of speeds by automatically adjusting muscular properties of leg and torso. It is known that fast locomotion dynamics can be approximated by a spring loaded inverted pendulum (SLIP) system, in which leg

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1
st
Reading
May 19, 2010 17:41 WSPC/S0219-8436 191-IJHR 00210
International Journal of Humanoid Robotics
1
Vol. 7, No. 2 (2010) 1–18c
World Scientiﬁc Publishing Company
3
DOI:10.1142/S0219843610002106
5
BIPED HOPPING CONTROL BASED ON SPRING LOADEDINVERTED PENDULUM MODEL
7
SEYED HOSSEIN TAMADDONI
∗
, FARID JAFARI
†
,ALI MEGHDARI
‡
and SAEED SOHRABPOUR
§
9
Center of Excellence in Design,Robotics, and Automation (CEDRA),
11
School of Mechanical Engineering,Sharif University of Technology, Tehran, Iran
13
∗
h.tamaddoni@gmail.com
†
jafari.farid@gmail.com
15
‡
meghdari@sharif.edu
§
sohrabpour@sharif.edu
17
ReceivedRevised
19
AcceptedHuman running can be stabilized in a wide range of speeds by automatically adjusting
21
muscular properties of leg and torso. It is known that fast locomotion dynamics canbe approximated by a spring loaded inverted pendulum (SLIP) system, in which leg is
23
replaced by a single spring connecting body mass to ground. Taking advantage of theinherent stability of SLIP model, a hybrid control strategy is developed that guarantees
25
a stable biped locomotion in sagittal plane. In the presented approach, nonlinear controlmethods are applied to synchronize the biped dynamics and the spring-mass dynamics.
27
As the biped center of mass follows the mass of the mass-spring model, the whole bipedperforms a stable locomotion corresponding to SLIP model. Simulations are done to
29
obtain a repeatable hopping for a three-link underactuated biped model. Results showthat periodic hopping gaits can be stabilized, and the presented control strategy provides
31
feasible gait trajectories for stance and swing phases.
Keywords
: Biped; hopping; jumping; running; SLIP; gait; control.
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1. Introduction
For decades, motion planning has been a crucial step in developing walking robots.
35
In particular, there is signiﬁcant interest on developing systematic methods of gaitgeneration for biped robots.
1
–
4
Among the existing methods, the objective function
37
oriented approach that regulates the motion of a planar link-segmental biped robotsuch that the cost is minimized has shown considerable advantages. Through this
39
method, biped locomotion can be characterized in terms of step length, progressionspeed, maximum step height and the stance knee bias angle. According to the objec-
41
tive function, joint angular displacement proﬁles can be then uniquely determinedbased on the initial condition of the system.
43
1
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2
S. H. Tamaddoni et al.
Despite diﬀerent gaits and speeds, human and legged animal fast locomotion
1
can be described by spring-like leg behavior
5
; hence, fast locomotion including run-ning, jumping, hopping, and etc. can be represented by a simple spring-mass sys-
3
tem, known as
Spring Loaded Inverted Pendulum
or SLIP.
6
,
7
This model is basedon observations which revealed that the energy level remains approximately con-
5
stant during running, hopping and jumping. Associated with the spring-like legbehavior,
leg stiﬀness
is deﬁned that is shown to remain almost constant during
7
fast locomotion.
8
Thus, the spring-mass representation seems to provide acceptableinsight to human fast locomotion.
9
,
10
It is also known that the spring-mass model
9
is repeatable and self-stabilizing for adequate leg parameter adjustments, includingangle of attack, leg stiﬀness, and leg length, and suﬃcient speeds.
8
,
11
11
In this paper, a new approach to biped trajectory planning and control in hop-ping locomotion is introduced based on synchronization of biped governing dynam-
13
ics and SLIP dynamics that corresponds to a particular hopping motion. Thismethod is biologically inspired by the idea that biped must follow human trajectory
15
which is considered an ideal self-stabilizing pattern in order to become capable of fast locomotion.
12
,
13
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A link-segmental model of the biped with three degrees of freedom and onedegree of underactuation in the ankle joint is developed in this paper. Using the
19
proposed control strategy, the biped joint torques are commanded such that thebiped dynamics follows its corresponding SLIP model. Later discussed, the initial
21
conditions are set in a way that a periodic hopping motion is achieved. Simulationresults show a periodic hopping locomotion can be achieved using the proposed
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synchronization control method.
2. Dynamics Model of Hopping
25
The robot is assumed to consist of three rigid links with mass, connected via rigid,frictionless, revolute joints to for a single open kinematic chain lying in sagittal
27
plane. The hopping occurs on a ﬂat horizontalsurface, thus without loss of generalityit is assumed that the ground height is zero with respect to the inertial frame.
29
A complete cycle of hopping consists of two consecutive phases:(1)
stance
in which leg end is in stationary contact with the ground and as an ideal
31
pivot supports the whole body. If the biped center of mass (CoM) reaches asuﬃcient upward velocity, the leg loses its contact to the ground and the phase
33
ends,(2)
ﬂight
in which there is no contact with the ground, and it ends as soon as a
35
sudden impact with the ground surface occurs.Figure 2 illustrates diﬀerent phases of running with coordinate conventions of this
37
paper labeled on a link-segmented model of biped robot in sagittal plane. The bipedmodel in this study has three links each of which has length, mass, and moment
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Biped Hopping Control Based on Spring Loaded Inverted Pendulum Model
3
of inertia. There is one degree of underactuation in ankle joint of the biped, so the
1
robot cannot apply any torque at its foothold.Assuming that
θ,
˙
θ,
¨
θ
, and
τ
represent the generalized coordinates, velocities,accelerations, and torques, respectively, the dynamic model describing the motionof the biped in stance phase can be written as,
14
D
(
θ
)¨
θ
+
C
(
θ,
˙
θ
)˙
θ
+
G
(
θ
) =
Bτ
(1)where
D
(
θ
) is the inertia matrix,
C
(
θ,
˙
θ
) is the Coriolis matrix,
G
(
θ
) is the gravity
3
vector, and
B
is the control matrix.For
i,j
= 1
,
2
,
3, the matrices
D,C,G
are given by:
D
ij
(
θ
) =
m
j
d
j
l
j
+
3
k
=
j
+1
m
k
l
i
l
j
cos(
θ
i
−
θ
j
)
C
ij
(
θ,
·
θ
) =
m
j
d
j
l
j
+
3
k
=
j
+1
m
k
l
i
l
j
sin(
θ
i
−
θ
j
)
·
θ
j
G
i
(
θ
) =
m
j
d
j
g
+
3
k
=
i
+1
m
k
l
i
g
sin
θ
i
.
(2)Since the biped is underactuated in the contact point, no torque can be exertedfrom the ground to the ﬁrst link. Therefore, the control matrix
B
corresponding tothe generalized torque vector
τ
= [
τ
2
τ
3
]
T
is given by,
B
=
−
1 01
−
10 1
(3)During ﬂight phase, ﬁve degrees of freedom including joint angles and ankle position
5
are deﬁned to describe the governing dynamics. The biped center of mass undergoesa ballistic trajectory, and the exerted torques in knee and hip deﬁnes the angle
7
trajectory during ﬂight. Given that the angle trajectories of ankle and knee jointsare known, the hip angle and ankle position is determinant by ﬂight dynamics.
9
When the ankle touches the ground at the end of ﬂight phase, an impact takesplace. Based on the existing impact models, the ground reaction forces at impact can
11
be represented by impulses with intensity
I
R
. In this paper, the impact is assumedinelastic, and the velocity of the ankle becomes zero instantaneously. As a result
13
of this model, the robot’s conﬁguration is assumed unchanged during impact andthere are only instantaneous changes in the velocities.
15
The Cartesian position of the ankle can be expressed in terms of Cartesianposition of biped center of mass as,
x
a
y
a
=
x
c
y
c
−
s
(
θ
) (4)
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S. H. Tamaddoni et al.
where
s
is determined from the in-between link parameters such as length, mass,
1
and position of center of mass.Let˙
θ
−
,
˙
θ
+
denote the velocity vector just before and after the impact, respec-tively. After impact, the ankle neither rebounds nor slides; therefore, the linearvelocity of the center of mass can be expressed in terms of the angular velocities just after impact as,
00
=
˙
x
+
c
˙
y
+
c
−
∂s
(
θ
)
∂θ
˙
θ
+
.
(5)The ground reaction impulse vector is given by,
I
R
=
m
∂s
(
θ
)
∂θ
˙
θ
+
−
˙
x
−
c
˙
y
−
c
(6)and the robot’s angular velocity vector after impact is obtained with respect to thevelocity before impact as,˙
θ
+
=
A
(˜
θ
) +
m∂s
T
(
θ
)
∂θ∂s
(
θ
)
∂θ
−
1
A
(˜
θ
)
m∂s
T
(
θ
)
∂θ
˙
θ
−
(7)where˜
θ
is the relative angles of the actuated joints with reference to the bipedbody, and
A
is the matrix that deﬁnes the total kinetic energy of the robot duringﬂight,
K
f
=12˙˜
θ
T
A
˙˜
θ
+12
m
(˙
x
2
c
+ ˙
y
2
c
)
.
(8)
3. Spring Loaded Inverted Pendulum Model
3
The planar spring model is characterized by alternating stance and ﬂight phases.Figure 1 illustrates the parameters of the spring mass model. The point mass
m
5
Fig. 1. Three-link model of biped robot.
1
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May 19, 2010 17:41 WSPC/S0219-8436 191-IJHR 00210
Biped Hopping Control Based on Spring Loaded Inverted Pendulum Model
5Fig. 2. Spring mass running model.
represents the eﬀective body mass, which is supported by a linear spring of stiﬀness
1
k
and rest length
l
0
. During stance phase, the spring is pinned to the ground withthe angle of attack
α
between the neutral axis and the point mass. During ﬂight
3
phase, the center of mass describes a ballistic curve, determined by the gravitationalforce acting downward.
5
The transition to stance phase occurs when the following landing condition issatisﬁed:
y
=
l
0
sin(
α
TD
) (9)where
α
TD
is the touch-down angle of attack.During stance phase, spring mass dynamics is governed by spring forces andgravity, and the equation of motion is:
m
¨
l
=
ml
˙
α
2
−
mg
cos
α
−
k
(
l
−
l
0
)
ml
¨
α
=
−
2
m
˙
l
˙
α
+
mg
sin
α
(10)where
l,l
0
are the spring current length and rest length, respectively, and
α
is the
7
angle of attack.Assuming small angle of attack,
α
, an approximated analytical solution can be
9
obtained for the spring mass system of Eq. (2).
9
However, since biped hops in awide range of angle of attacks, the equations of motion for spring mass system is
11
solved numerically in this work.During ﬂight phase, the center of mass describes a ballistic curve, determinedby the gravitational force acting downward:
¨
x
c
= 0¨
y
c
=
−
g.
(11)Combining equations (10) and (11), a hybrid model is formed that describes the
13
dynamics of spring mass in stance and ﬂight.Since, the horizontal component of center of mass velocity,
v
x
, remains
15
unchanged during ﬂight phase, its value should be the same at touchdown andtakeoﬀ to guarantee periodicity of hopping motion. This implies that symmetry of
17

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