Abstract— In this paper, a simple trajectory generation method
for biped walking is proposed. The dynamic model of the five link
bipedal robot is first reduced using several biologically inspired
assumptions. A sinusoidal curve is then imposed to the ankle of
the swing leg's trajectory. The reduced model is finally obtained
and solved: it is an homogeneous 2nd order differential equations
with constant coefficients. The algebraic solution obtained
ensures a stable rhythmic gait for the bipedal robot. It’s
continuous in the defined time interval, easy to implement when
the boundary conditions are well defined.
Index Terms— Trajectory Generation, Biped Locomotion,
model reduction.
I. INTRODUCTION
Gait pattern generation [1] is one of key problems of
research devoted to bipedal robots. Two kinds of works
dedicated to the bipedal walking pattern generation can be
distinguished: studies assimilating robots as elementary
models and works considering all morphological data of the
robot, see [2] and references therein. For the first approach,
the linear inverted pendulum model concept is the mostly used
concept in order to generate the gait trajectory [3, 4, 5]. For
the second group of works attention is paid on the generation
of a trajectory tracking control using objective function
composed of one or more terms to minimize [2, 6, 7].
In this paper, a simple trajectory generation method is
proposed. The dynamic model of the five link bipedal robot is
first reduced using several biologically inspired assumptions.
A sinusoidal curve is then imposed to the ankle of the swing
leg's trajectory. The reduced model is finally obtained and
solved: it is an homogeneous 2nd order differential equations
with constant coefficients. The solution gives an algebraic
solution for joint desired trajectories that ensures a stable
rhythmic gait for the bipedal robot.
II. THE FIVE LINK BIPEDAL ROBOT
The planar bipedal robot prototype is composed of five links
associated to five DOF. Fig.1 shows the involved rotations for
each link. All physical parameters involved in the kinematic
and dynamic models are given by Table 1. They are inspired
from [8]. The parameters
i
i
i
k
,
L
,
m
and
iI , (
5
,..,
1
=
i
), design
mass, length, position of center of mass and inertia about
center of mass, respectively.
Figure 1. The five link bipedal robot
Table1. Physical parameters of the bipedal robot [8]
link
Right leg
Right
thigh
Pelvis
Left
thigh
Left leg
joint
Right
ankle
Right
knee
Right
hip
Left
hip
Left
knee
Link number
1
2
3
4
5
Mass (kg)
3.255
7.000
24.850
7.000
3.255
Lenght (m)
0.426
0.424
0.299
0.424
0.426
Center of
mass (kg.m)
0.164
0.366
1.530
0.366
0.164
Inertia about
centre of
mass (Kgm2)
0.184
0.184
0.206
0.184
0.184
Gait trajectory generation for a five link bipedal
robot based on a reduced dynamical model
1Yosra Arous & 2Olfa Boubaker
1,2 National Institute of Applied sciences and Technology,
Tunis, Tunisia
2olfa.boubaker@insat.rnu.tn
III. KINEMATIC AND DYNAMIC MODELING
The five-link bipedal robotic system with five degrees of
freedom can be described by the following direct kinematic
model:
)
(θ
h
X =
(1)
where
[
]
5
5
4
3
2
1
ℜ
∈
=
T
θ
θ
θ
θ
θ
θ
is
the
joint
displacement vector,
[
]
2
ℜ
∈
=
T
y
x
X
is the Cartesian
position vector and
2
)
(
ℜ
∈
θ
h
is a nonlinear function
described by:
( )
+
+
+
−
−
+
=
5
5
4
4
2
2
1
1
5
5
4
4
2
2
1
1
sin
sin
sin
sin
cos
cos
cos
cos
θ
l
θ
l
θ
l
θ
l
θ
l
θ
l
θ
l
θ
l
θ
h
The time derivative of the direct kinematic model (1) yields
the following differential kinematic model:
θ
θ
J
X
&
&
)
(
=
(2)
where
[
]T
y
x
X
&
&
& =
is
the
Cartesian
velocity
vector,
[
]
T
θ
θ
θ
θ
θ
θ
5
4
3
2
1
&
&
&
&
&
& =
is the vector of joint velocities and
)
(θ
J
is the so-called analytical Jacobian matrix given by:
( )
−
−
=
5
5
4
4
2
2
1
1
5
5
4
4
2
2
1
1
cos
cos
0
cos
cos
sin
sin
0
sin
sin
θ
l
θ
l
θ
l
θ
l
θ
l
θ
l
θ
l
θ
l
θ
J
A typical walking cycle may include three phases [1]: the
single support phase (SSP), the impact phase (IP) and the
double support phase (DSP), (see Fig.2). In this paper we
focus only on SSP.
Using Lagrange approach [9], the dynamical model of the
bipedal robot in SSP phase is described by:
DU
θ
G
θ
θ
H
θ
θ
M
=
+
+
)
(
)
,
(
)
(
&
&&
(3)
where
[
]
T
θ
θ
θ
θ
θ
θ
5
4
3
2
1
&&
&&
&&
&&
&&
&& =
is the vector of joint
accelerations,
[
]T
U
U
U
U
U
U
5
4
3
2
1
=
is the vector of
torque inputs,
)
(θ
M
is the symmetric positive definite inertia
matrix,
)
,
( θ
θ
H
& is the vector of centripetal and Coriolis torques
and
)
(θ
G
is the vector of gravitational torques given by
[
]
5
1
5
1
)
(
≤
≤
≤
≤
=
j
i
ij
M
θ
M
,
[ ]
5
1
5
1
)
,
(
≤
≤
≤
≤
=
j
i
ij
h
θ
θ
H
&
,
[ ]
5
1
)
(
≤
≤
=
i
i
G
θ
G
where:
)
cos(
)
(
)
cos(
))
(
(
)
cos(
)
cos(
)
(
)
(
5
1
5
5
1
5
15
5
1
5
1
5
4
4
1
4
14
3
1
3
1
3
13
2
1
2
1
5
2
1
4
2
1
3
2
1
2
12
2
1
5
4
3
2
1
2
1
1
11
θ
θ
k
l
l
m
M
θ
θ
ll
m
k
l
l
m
M
θ
θ
k
l
m
M
θ
θ
l
l
m
l
l
m
l
l
m
k
l
m
M
l
m
m
m
m
I
k
m
M
+
−
=
+
+
−
=
−
=
−
+
+
+
=
+
+
+
+
+
=
)
cos(
)
(
)
cos(
)
(
(
)
cos(
)
(
)
cos(
)
)
(
(
5
2
5
5
2
5
25
4
2
4
2
5
4
4
2
4
24
3
2
3
2
3
23
2
2
5
4
3
2
2
2
2
22
2
1
2
1
5
4
3
2
1
2
21
θ
θ
k
l
l
m
M
θ
θ
l
l
m
k
l
l
m
M
θ
θ
k
l
m
M
l
m
m
m
I
k
m
M
θ
θ
ll
m
m
m
k
l
m
M
+
−
=
+
+
−
=
−
=
+
+
+
+
=
−
+
+
+
=
Figure 2.The three phases of a typical walking cycle
0
)
cos(
)
cos(
35
34
3
2
3
3
33
3
2
3
2
3
32
3
1
3
1
3
31
=
=
+
=
−
=
−
=
J
M
I
k
m
M
θ
θ
k
l
m
M
θ
θ
k
l
m
M
)
cos(
))
(
(
)
(
0
)
cos(
)
)
(
(
)
cos(
)
)
(
(
5
4
5
5
4
5
45
4
2
4
5
2
4
4
4
44
43
4
2
4
2
5
4
4
2
4
42
4
1
4
1
5
4
4
1
4
41
θ
θ
k
l
l
m
M
I
l
m
k
l
m
M
M
θ
θ
l
l
m
k
l
l
m
M
θ
θ
l
l
m
k
l
l
m
M
−
−
=
+
+
−
=
=
+
+
−
=
+
+
−
=
5
2
5
5
5
55
5
4
5
5
4
5
54
53
5
2
5
5
2
5
52
5
1
5
5
1
5
51
)
(
)
cos(
)
(
0
)
cos(
)
(
)
cos(
)
(
I
k
l
m
M
θ
θ
k
l
l
m
M
M
θ
θ
k
l
l
m
M
θ
θ
k
l
l
m
M
+
−
=
−
−
=
=
+
−
=
+
−
=
)
sin(
))
(
(
)
sin(
)
)
(
(
)
sin(
)
(
)
sin(
)
(
0
5
1
5
5
1
5
15
4
1
4
1
5
4
4
1
4
14
3
1
3
1
5
13
2
1
2
1
5
2
1
4
2
1
3
2
1
2
12
11
θ
θ
k
l
l
m
h
θ
θ
ll
m
k
l
l
m
h
θ
θ
k
l
m
h
θ
θ
ll
m
ll
m
ll
m
k
l
m
h
h
+
−
−
=
+
+
−
−
=
−
=
−
+
+
+
=
=
)
sin(
)
(
)
sin(
)
)
(
(
)
sin(
0
)
sin(
)
)
(
(
5
2
5
5
2
5
25
4
2
4
2
5
4
4
2
4
24
3
2
3
2
3
23
22
2
1
2
1
5
4
3
2
1
2
21
θ
θ
k
l
l
m
h
θ
θ
l
l
m
k
l
l
m
h
θ
θ
k
l
m
h
h
θ
θ
ll
m
m
m
k
l
m
h
+
−
−
=
+
+
−
−
=
−
=
=
−
+
+
+
−
=
0
0
)
sin(
)
sin(
35
34
33
3
2
2
2
3
32
3
1
3
1
3
31
=
=
=
−
−
=
−
−
=
h
h
h
θ
θ
k
l
m
h
θ
θ
k
l
m
h
)
sin(
))
(
(
0
0
)
sin(
)
)
(
(
)
sin(
)
)
(
(
5
4
5
5
4
5
45
44
43
4
2
4
2
5
4
4
2
4
42
4
1
4
1
5
4
4
1
4
41
θ
θ
k
l
l
m
h
h
h
θ
θ
l
l
m
k
l
l
m
h
θ
θ
l
l
m
k
l
l
m
h
−
−
=
=
=
+
+
−
−
=
+
+
−
−
=
)
sin(
))
(
(
0
)
sin(
))
(
(
)
sin(
))
(
(
5
4
5
5
4
5
54
55
53
5
2
5
5
2
5
52
5
1
5
5
1
5
51
θ
θ
k
l
l
m
h
h
h
θ
θ
k
l
l
m
h
θ
θ
k
l
l
m
h
−
−
−
=
=
=
+
−
−
=
+
−
−
=
5
5
5
5
5
4
4
5
4
4
4
4
3
3
3
3
2
2
5
4
3
2
2
2
1
1
5
4
3
2
1
1
1
cos
))
(
(
cos
)
)
(
(
cos
cos
)
)
(
(
cos
)
)
(
(
θ
k
l
m
g
G
θ
l
m
k
l
m
g
G
θ
gk
m
G
θ
l
m
m
m
k
m
g
G
θ
l
m
m
m
m
k
m
g
G
−
=
+
−
=
=
+
+
+
=
+
+
+
+
=
−
−
−
−
=
1
1
0
1
0
0
0
0
0
0
0
1
1
0
0
0
0
1
1
0
0
0
0
1
1
D
IV. MODEL REDUCTION AND GAIT TRAJECTORY GENERATION
The purpose of this section is to reduce the nonlinear dynamic
model (3) into a solvable differential system in order to give
algebraic solutions of gait trajectories. The reduced model will
be obtained as follows: the robotic system (3) is first written
around an equilibrium point
eq
θ
as [9]:
v
B
x
A
x
+
=
&
(4)
∂
∂
−
=
×
−
×
×
5
5
1
5
5
5
5
0
0
eq
θ
θ
G
J
I
A
=
−
×
D
J
B
1
5
5
0
[
]T
eq
U
U
v
−
=
×5
1
0
[
]
T
eq
eq
θ
θ
θ
θ
x
&
& −
−
=
To generate joint desired joint trajectory vector:
( )
( )
( )
( )
( )
( )
[
]T
d
d
d
d
d
d
t
θ
t
θ
t
θ
t
θ
t
θ
t
θ
,5
,4
,3
,2
,1
=
we assume, for the bipedal robot, the following biologically
inspired assumptions (see Fig.3):
Assumption 1: ZMP stability [10] is imposed. This is can be
ensured by assuming that:
0
1 =
U
(5)
Assumption 2: the supporting leg is kept straight as:
( )
( )t
θ
t
θ
d
d
,2
,1
=
(6)
Assumption 3: The bipedal robot must have an upright posture
that is to say, it keeps its back straight such that:
( )
2
,3
π
t
θ
d
=
(7)
Assumption 4: the relationship between the right ankle joint
and the left hip joint is imposed such that:
( )
( )
α
t
θ
t
θ
d
d
+
=
,1
,4
(8)
where α is a given constant angle to be chosen.
Assumption 5: To generate a rhythmic stable movement, the
relationship between the right ankle joint and the left knee
joint is given by [11]:
( )
( )
)
(
sin2
,1
,5
t
T
π
t
θ
t
θ
d
d
−
=
(9)
From the assumptions (6)-(9), we can then easily deduce the
desired velocity vector and the desired acceleration vector
defined, respectively, by:
Figure 3. Biologically inspired assumptions
( )
( )
( )
( )
( )
( )
[
]
T
d
d
d
d
d
d
t
θ
t
θ
t
θ
t
θ
t
θ
t
θ
,5
,4
,3
,2
,1
&
&
&
&
&
&
=
( )
( )
( )
( )
( )
( )
[
]
T
d
d
d
d
d
d
t
θ
t
θ
t
θ
t
θ
t
θ
t
θ
,5
,4
,3
,2
,1
&&
&&
&&
&&
&&
&&
=
Particularly the following relations are deduced:
( )
( )t
θ
t
θ
d
d
,2
,1
&&
&&
=
(10)
( )
0
,3
=
t
θ
d
&&
(11)
( )
( )t
θ
t
θ
d
d
,1
,4
&&
&&
=
(12)
+
=
t
T
π
T
π
θ
θ
d
d
2
cos
2
2
,1
,5
&&
&&
(13)
By summing all the lines of the system (4) and using the
relation (5), (10)-(13), the following reduced homogeneous 2nd
order differential equations with constant coefficients is
obtained as:
( )
( )
)
(
)
(
,1
1
,1
1
t
s
t
h
K
t
θ
H
t
θ
M
d
d
−
−
−
=
+
&&
(14)
where:
(
)
eq
eq
θ
θ
M
M
M
M
M
M
M
M
M
M
M
)
(
2
45
25
15
14
12
23
55
44
22
11
1
+
+
+
+
+
+
+
+
+
=
eq
θ
θ
G
θ
G
θ
G
θ
G
H
∂
∂
+
∂
∂
+
∂
∂
+
∂
∂
=
5
5
4
4
2
2
1
1
1
eq
θ
θ
U
θ
G
θ
G
π
θ
G
α
θ
θ
G
θ
θ
G
θ
θ
G
θ
θ
G
θ
θ
G
K
eq
eq
1
5
5
3
3
4
4
5
5
5
4
4
4
3
3
3
2
2
2
1
1
1
2
1
2
+
∂
∂
+
∂
∂
+
∂
∂
+
∂
∂
+
∂
∂
+
∂
∂
+
∂
∂
+
∂
∂
−
=
∂
∂
−
=
t
T
π
θ
G
t
h
2
cos
2
)
(
5
5
=
t
T
π
M
T
π
t
s
2
cos
2
)
(
2
2
55
45
25
15
2
M
M
M
M
M
+
+
+
=
Solving (14) for the boundary conditions:
10
0
,1
)
(
θ
t
θ d
=
f
f
d
θ
t
θ
1
,1
)
(
=
and for the equilibrium point :
T
eq
π
π
π
π
π
θ
=
2
2
2
2
2
the analytical solution of the differential equation (14) is given
by [12]:
1
3
2
1
,1
2
cos
)
(
2
1
H
K
t
T
π
C
e
C
e
C
t
θ
t
r
t
r
d
−
+
+
=
(15)
where :
10
3
1
2
1
θ
C
H
K
C
C
−
+
−
=
f
f
f
t
r
t
r
f
t
r
f
e
e
H
K
t
T
π
C
e
H
K
θ
θ
C
C
2
1
1
1
3
1
10
1
3
2
2
cos
+
+
−
−
−
+
=
−
−
∂
∂
=
1
1
2
2
2
5
5
3
4
2
H
M
T
π
T
π
M
θ
G
C
1
1
1
2
,1
M
H
M
r
⋅
−
±
=
,
0
1
1
〈
H
M
To verify the algebraic solution (15) that asked a rather tedious
calculation two approaches are used: a symbolic computation
of the solution (15) using the symbolic toolbox of Matlab
software and a numerical integration of the differential
equation (14) using the ode45 function of the same software.
The three solutions were found superimposed for the physical
parameters given by Table 1.
Using the relations (15) and (6)-(9), the analytical expressions
of the joint trajectories are then deduced. We impose then to
the robotic model (3) the following second order linear input-
output behavior [8]:
( )
( )
(
)
( )
( )
(
)
( )
( )
(
)
0
=
−
+
−
+
−
t
θ
t
θ
K
t
θ
t
θ
K
t
θ
t
θ
d
p
d
v
d
&
&
&&
&&
(16)
where
5
5×
ℜ
∈
v
K
and
5
5×
ℜ
∈
p
K
are two positive definite
diagonal matrices chosen to guarantee global stability, desired
performances and decoupling proprieties for the controlled
system and such that the desired trajectories
d
θ ,
d
θ& and are
derived from the solution (15) and the relations (6)-(9). The
control law deduced from (3) and (16) is then given by:
( )
( )
( )
(
)
( )
( )
(
)
(
)
[
)]
(
)
,
(
)
(
1
θ
G
θ
θ
H
t
θ
t
θ
K
t
θ
t
θ
K
t
θ
θ
M
D
U
d
p
d
v
d
+
+
−
−
−
−
=
−
&
&
&
&&
(17)
Using the physical parameters given in table1, the joint
trajectories of the controlled biped robot are generated as see
by Fig. 4 where the desired joint position is designed by
( )
( )
( )
( )
( )
( )
[
]T
d
t
teta
t
teta
t
tata
t
teta
t
teta
t
5
4
3
2
1
θ
=
. The
walking cycle is shown by Fig.5.
Compared to previous works [8], this paper gives an algebraic
solution for desired joint trajectories that must be followed by
the robotic system. The result is a stable rhythmic movement
for the planar bipedal system.
Figure 4. Joint trajectories of the bipedal robot in the swing phase
Figure 5. The resulting walking cycle
V. CONCLUSION
A simple trajectory generation method is proposed for biped al
gait by imposing a sinusoidal curve to the ankle of the swing
leg's trajectory.
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