Proceedings of 2015 IEEE
International Conference on Mechatronics and Automation
August 2 - 5, Beijing, China
Gait planning of biped robot based on feed-forward
compensation of gravity moment
ZHAO Jianghai and ZHANG Xiaojian
TANG Cheng
Institution of Advanced Manufacturing Technology
Hefei Institution of Physical Science, CAS
Changzhou, Jiangsu Province 213164, China
[email protected]
Research Center of Robotic Technique
Seaborne Cold Extrusion of Metal co., LTD
Zhangjiagang, Jiangsu Province 213164, China
[email protected]
different with the humanoid robot, the walking data cannot be
used again if a new robot is produced.
There are three models that is utilized to generate the
walking pattern: multi-links model[2], 3D-LIPM [3] and tablecar model[4]. The famous biped robot developed by Honda
adopts the multi-link model to generate the walking pattern
based on the off-line planning means. Base on table-car
model, Kajita constructs the simplified control model and use
a preview controller for the gait generation of the walking
mechanism of HRP-2L. The 3D-LIPM doesn’t consider the
influences of the swing and stance leg, and can’t adequately
depict dynamical properties of biped robot. So this model is
only applicable to a biped robot possessing a special structure.
Inspired by the energy consumption of the human
walking, some researchers design the walking trajectory,
which is applied to a walking robot with a small consumption
of energy. The artificial intelligent method plans the motion
traces of the each joint of the biped robot by the training and
the learning[5], but it is difficult to obtain a convergence
solution and waste a lot of time.
In this paper, the influence of the distributed mass on the
height of 3D-LIPM is analyzed, and the walking data
generated by 3D-LIPM is optimized. The walking trajectory of
the 3DLIPM is divided into two sections: single-stance phase
and double-stance phase. As a result, the end point velocity of
the swing leg is planned to a value of zero, and the landing
impact is efficiently reduced. After the trajectory of COM is
generated, the angular displacement and angular velocity of
joints of robot is calculated by numerical iterative method. the
walking gait is tested by the biped robot called intelligentpioneer.
Abstract - To improve the walking stability of biped robot,
the adjusting angle is added to the ankle joint for eliminating the
lateral disequilibrium. Three dimensional linear inverted
pendulum model(3D-LIPM) is applied for depicting the motion
of the biped robot. The motions of the stance and swing leg have
an influence on the height of center of mass(COM), which is
verified by the computer simulation. The model of the pendulum
is also established for analyzing key factors which lead to the
unbalance of the biped robot in the lateral plane, and analyzing
results can be used to optimize the gait data. The motion
trajectory of the each joint of the biped robot is generated by
using the cosine and cycloid functions. The landing velocity of the
swing leg is set to zero, and the impact is effectively reduced.
Based on the inertial measure unit(IMU), the horizontal posture
of the foot is kept by adjusting the ankle motors of the stance leg,
and the walk stability of the biped robot is greatly improved. The
inverse solution for discrete points of the walking trajectory is
obtained by a numerical iterative means, which can satisfy the
requirement of the control cycle of 10ms. The walking test is
implemented on the flat ground, and the biped robot can steady
walk with the single stance phase of 3s, the double-stance phase
of 2s and the step-length of 200mm. the experiment results prove
that compensation means of the gravity moment is available for
the walking gait of biped robot.
Index Terms - biped robot; walking gait; motion trajectory;
three dimensional linear inverted pendulum; numerical iterative
means.
I. INTRODUCTION
The planning of the motion trajectory is a key technique
about the research of the humanoid robot. On a flat ground,
the biped robot can also walks with an open-loop way
according to the given walking pattern which is calculated in
advance. Based on information acquired from the six
dimensional force sensors, the humanoid robot can adapt to
the complex terrain by using the criterion of zero moment
point. It is important that an efficient walking pattern can
provide a solid foundation for the walking control of
humanoid robot.
The planning method of walking pattern is categorized
into four major groups: bionic kinematics means, model-based
planning, optimum control-based realization and artificial
intelligence-based generated method. Data of the human
motion, captured by the professional analyzing equipment, can
be used to generate the complex motion trajectory[1]. This
means is simple and efficient, and the walking data can be
obtained without the complex calculation and analysis.
However, the physical parameters of the experiment human is
978-1-4799-7098-8/15/$31.00 ©2015 IEEE
II. 3DLIPM-BASED TRAJECTORY GENERATION OF COM
Figure 2 shows the simplified model depicted by 3D-LIPM
and coordinate system of Denavit-Hartenberg(D-H)
parameters. The initial position and posture of biped robot is
upright, and the basic coordinate system of the stance leg is in
accordance with the one of the swing leg.
The every link of 3D-LIPM from the ankle of stance leg to
the ankle of swing leg is defined as L1, L2, L3, L4, L5, L6 and L7
in sequence, and the conversion matrix of position and posture
i-1
is expressed as Ri between two adjacent links. Local
coordinate system of the each link conforms to the definition
shown in Fig.2. As the upper part of body and two arms keep
motionless during the robot walking, the upper body is
equivalent to a section of link L4 and the whole body of the
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robot is simplified as a model of seven links without regard to
the mass distribution.
c0 = ∑ (mi g ) × pt (i )
i =1
∑m | g |
i
(2)
i =1
Where mi and pt(i) are the mass of the each link and the
radius vector of COM in the reference coordinate system of
concentrated mass, respectively, g is the acceleration of
gravity in the reference coordinate system of concentrated
mass, g=[-9.8 0 0].
The trajectory change of center of gravity of robot is
shown in Fig.3. The walking process of the biped robot
consists of the three stages of start step, continuous
walking and stopping step, which are depicted by the
curve segment ab, be and eh, and hi shown in Fig.3,
respectively.
Referencing to the virtual prototype of the biped
robot constructed by the Solid Works, the mass of each
link and the initial coordinate values of the centre of mass
in the joint coordinate system, and the initial height of
centre of mass of 3D-LIPM is 572.6mm. Let the singlestance period and interpolation cycle of motion trajectory
of 3D-LIPM be 6s and 10ms, respectively. In addition, the
step length of robot walking is 200mm. As the motion
trajectory is similar in each waking-phase of biped robot,
we may only consider the change of COM in one of
section when robot walking forward. Replacing the joint
angle of the starting phase into the Eq.(1) and Eq.(2), the
trajectory of COM can be observed in Fig.4. According to
the model of 3D-LIPM, the height of COM should be
constant. It can be seen from the Fig.4 that the height of
COM changes with the distribution mass, and the height
of COM decreases from 572.6mm to 570.36mm during
the robot take a step forward. Thus the influence of the
swing leg may be omitted, and the walking model of robot
may be depicted by 3D-LIPM. The simulation results
about the walking trajectory in the starting phase of robot
along the x and y axis can be observed in Fig.5 and Fig.6,
respectively.
Figure 5 shows that the end-point velocity of the curve
segment be along x axis reach a maximum value of 0.4m/s,
and a strong landing impact can happen, which result in a
harmful influence on the stability of the biped robot.
Referencing to the trajectory of 3D-LIPM, the curve
segment of single-stance of right leg be can be replaced
by three sections of line segments bc, cd and de, and the
other stage trajectory can also be processed by this means.
In this process means, the line segment cd is taken as the
trajectory of the single-stance phase, and the line segment
de and the one of next phase ef can be synthesized by a
section of line segment df, which is taken as the trace of
the double-stance phase. The planned trajectory depicted
by dotted line with arrows is shown in Fig.3.
In the new planning trajectory, a transfer of COG from
rear stance-leg to front stance-leg is implemented with a
switch between the single-stance and the double-stance
phase, so the balance of next step can be guaranteed. Each
Fig.2 3D-LIPM of robot
The locomotion of the stance and swing leg induces the
change of the radius vector of each link when biped robot
walks forward. The reference coordinate system calculating
for the concentrated mass coincides with the basic
coordinate system of the stance leg. According to the
transformation matrix of forward kinematics of a given
time of walking cycle, the radius vector of each link Pt in
the reference coordinate system of the concentrated mass
can be expressed as
⎡c11s 1S1
⎤
⎢ s 1
⎥
⎢c22 S2
⎥
⎢c s 1S
⎥
⎢ 33 3
⎥
s 1
⎢
⎥
pt = c46 S6
⎢ w1 s ⎥
⎢c53 W3 Tw ⎥
⎢ w1 s ⎥
⎢c64 W4 Tw ⎥
⎢c w 1W sT ⎥
⎣ 76 6 w ⎦
7
(1)
Where sTw is the transfer matrix from the basic coordinate
of swing leg to the basic coordinate of stance leg, 1Wj is the
transfer matrix from the coordinate system of swing-leg
joint j to the basic coordinate system of swing leg, j = 1,
2,..., 7, 1Sj is the transfer matrix from the coordinate system
of joint of stance leg j to the basic coordinate system of
swing leg, j = 1, 2,...,7, cijs is the radius vector of stance-leg
link i in the coordinate system of the stance-leg joint j
under the initial position and posture, cijw is the radius vector
of stance-leg link i in the coordinate system of the stanceleg joint j under the initial position and posture.
The radius vector of the biped robot c0 in a given time t
can be calculated by
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of line segments is planned by a cycloid or a cosine
functions, and the velocity of start point and end point of
motion trajectory is set to zero. As a result, the landing
impact is eliminated, effectively.
trace of x axis motion
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x/mm
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t/ms
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trace of x axis velocity
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x(mm/ms)
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t/ms
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Fig.5 trajectory of centre of mass along x axis
trace of y axis motion
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Fig.3 Trajectory of othorcenter during robot walking
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t/ms
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trace of y axis velocity
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0
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t/ms
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y (m m / m s )
height of center of mass(mm)
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y /m m )
. pratical height of COG
+ idea height of COG
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Fig.4 height trajectory of centre of mass during starting phase
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t/ms
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Fig.6 trajectory of centre of mass along y axis
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III. OPTIMIZATION OF WALKING PATTERN
moment shown in Fig.8, compensated angle of feed-forward
exerting on the ankle joint can effectively eliminate the
unbalance along the side direction.
A. compensation of gravity moment for walking motion
We may reference to the balance control of human along
the side direction, which is shown in Fig.7. A posture swing
along the side direction happen by the action of gravity
moment. To sustain the equilibrium of body along the side
direction, an adjusting moment must be acted on the joint of
ankle for compensating the posture tilting. The whole body of
human revolves around the ankle to keep the balance of
walking. This regulatory mechanism can be illuminated by the
inverted pendulum model. The swing range of an adult is from
5mm to 7mm along the fore and aft direction, and the range of
swing along the side direction is between 3mm and 4mm.
trace of tilting angle
3
2.5
angle/degree)
2
1.5
1
0.5
0
0
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1000
1500
t/ms
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Fig.8 tilting angle induced by the gravity moment
trace of compensating angle
0
-0.01
angle/rad)
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-0.04
-0.05
Fig.7 posture swing induced by moment gravity
According to the model of inverted pendulum, the
dynamical equation of gravity moment inducing the body of
robot swing can be expressed as
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0
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1000
1500
t/ms
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Fig.9 compensating angle changes with a walking cycle
..
Mgl sin(θ ) = Ml θ
(3)
Where M is the concentrated mass of robot, l is the rod
length of inverted pendulum, d is the motion displacement
along the side direction acted by the gravity moment, θ is the
tilting angle induced by the gravity moment.
It can be observed from the Eq.(3), if the compensating
moment can’t be exerted on the ankle joint of biped robot , the
robot will fall down with the increase of tilting angle θ.
Equation.(3) is a transcendental equation, so it has no the
analytical solution. To deeply explain the influence of the
gravity moment on the stability of walking, the left side of
Eq.(3) can be simplified as a constant gravity moment.
Figure.8 shows the result of simulation about the relation
between a constant gravity moment and the tilting angle of
ankle joint.
The tilting angle along the side direction can reach three
degrees during a walking cycle of 3s. As the developed biped
robot controls the motor of the each joint by the mode of
position loop, a rotary angle planned by the cosine function is
added on the ankle joint for compensating the tilting angle
induced by the gravity moment. The added compensating
angle and the trajectory of ankle joint change with a walking
cycle are shown in the Fig.9 and Fig.10, respectively.
Compared with the tilting angle induced by the gravity
trace of rotary angle of ankle joint
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angle/rad)
2
-0.03
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t/ms
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Fig.10 trajectory of ankle joint with compensating angle
B. gait planning of biped robot
A walk cycle consists of the single-stance phase and
double-stance phase. The locomotive trajectory of the each
stage is planned by the cycloid function or the cosine
functions. Take the walking trajectory of start step for
example, the motion traces of the stance and swing leg are
shown in Fig.11 and Fig.12, respectively.
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An inertial measurement unit(IMU) is used for in-time
measuring the walking posture of robot. The landing impact is
retarded by adjusting the landing posture of swing leg to a
level posture. The control of landing posture is illuminated in
Fig.12. The controller of the landing posture employs the
digital PI algorithm, the roll and pitch angle of robot’s initial
posture is taken as the reference input of the controller.
During the robot walking, the posture error between the
reference values and the measuring ones is in-time calculated.
Then motors of ankle joint correct the tilting posture to keep a
horizontal landing-posture of the swing-leg.
trace of lateral motion
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displacement/mm)
displacement/mm)
trace of forward motion
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0
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2000
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1000
t/ms
2000
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t/ms
Fig.9 motion trajectory of stance leg
trace of forward walking
height of lifting and landing
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160
35
30
120
displacement/mm)
displacement/mm
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100
80
60
Fig.12 control for landing posture of robot
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IV. EXPERIMENT RUSULT
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The walking pattern is verified by the biped robot
developed by us. The gait data are transmitted to the joint
motors with a control cycle of 10ms. During the stage of
starting step, the main controller acquires the gait data of
biped robot, and the rotary angle and angular velocity change
with the walking period, which are shown in Fig.13 and
Fig.14, respectively. It can be concluded from the tesr results
that the velocity curve is continuous and smooth. In addition,
the landing velocity of the swing leg is planned with a value of
zero, so the biped robot strikes the ground gently. Compared
with the velocity of other joints, the hip joints’ velocity
changes, drastically. This phenomenon is accordance with the
temperature alarm of hip joints after the walking of a long
time.
Figure.15 shows the video snapshots of the biped robot. The
test results demonstrate that the robot can steady walk on a flat
ground with a step length of 200mm, and the single-stance and
double-stance phase is 3s and 2s, respectively.
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t/ms
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t/ms
Fig.10 motion trajectory of swing leg
C.
generation of gait data
To generate the walking pattern of the biped robot, the
trajectory of COG in the each stage is discretized with an
interpolation period of 10ms. The inverse solution of
kinematics of each discrete point of COG trajectory is
[6]
calculated by the method of Newton-Raphson , which is
suitable for solving the inverse solution of a section of
continuous trajectory. The joint angle of the discrete point
prior to the solving point is taken as the input of kinematics,
deviation of position and posture, adjusting angles is obtained
by replacing the deviation of the position and posture into the
Jacobian matrix of robot. The exact solution of inverse
kinematics for the discrete point is calculated by this timingcycle iterative method. As the displacement between two
adjacent points is very small, it spends less time to calculate
the solving point. This means is effective for solving the
inverse solution of kinematics. Fig.11 gives the course of
solving the inverse solution of discrete point.
V. Conclusion
A simple and effective means for the generation of gait
data is proposed in this paper. By analysing the influence of
distributed mass on the height of 3D-LIPM, the trajectory
depicting for the motion of COG walking cycle is synthesized
by the single-stance and double-stance phase. The velocity of
end point of the stance-leg and swing-leg is planned with a
zero value, and the landing impact can be effectively
eliminated. Moreover, an IMU is mounted on the body of
robot, and the land posture of the swing leg is adjusted to the
level status by measuring the posture of the upper body of
robot. In the future, the research on the walking control based
on the information acquired by the six dimension force
sensors will be carried out.
Fig.11 Newton-Raphson solution to inverse kinematics
D. control of landing ground
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angle(rad)
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1
1
2
3
4
5
6
0
-1
0
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time scale(10ms)
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2
x 10
velocity(rad/ms)
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3
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(d)swing of left leg
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(e)double-stance (f)swing of right leg
Fig.15 walking experiment of biped robots
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ACKNOWLEDGMENT
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1500
time scale(10ms)
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This work was supported by Natural Science Foundation of
Jiangsu Province(Grant NO.BK2012587) and supported by
Science and Technology Support Program of Jiangsu Province
(Grant No.BE2012057).
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Fig.13 position and velocity curve of each joint for stance leg
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2
3
4
5
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REFERENCES
angle(rad)
1
[1] S. Nakaoka, A. Nakazawa, K. Kaneko. “Task model of lower body
motion for a biped humanoid robot to imitate human dances”. IEEE/RSJ
International Conference on Intelligent Robots and Systems. Edmonton,
Canada: IEEE, 2005:3157-3162.
[2] T. Chayooth, T. Tomohito, O. Kenichi, et al. “Dynamic Rolling-Walk
Motion by the Limb Mechanism Robot ASTERISK”, Advanced Robotics,
25(1-2): 75-91, 2011.
[3] S. Kajita, T. Nagasaki, K. Kaneko, et al. “ZMP-based biped running
control-The HRP-2LR biped robot”. Robotics & Automation Magazine,
14(2): 63-72, 2007.
[4] S. Kajita, T. Yamaura, A. Kobayashi. “Dynamic walking control of a
biped robot a long a potential energy conserving oribit”. IEEE
Transactions on Robotics and Automation, 8(4): 431-438, 1992.
[5] X. Wang, T. Lu, Z. Xu, et al. “A state generator for humanoid robots
based on genetic algorithm”. Journal of Shanghai Jiaotong University,
43(3):503-507, 2009.
[6] S. Kajita. Humanoid robots[M]. Beijing: Tsinghua University Press,
2007., 1989.
0.5
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-0.5
-1
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time scale(10ms)
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1
x 10
v elocity(rad/m s)
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2
3
4
5
6
0.5
0
-0.5
-1
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time scale(10ms)
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Fig.14 position and velocity curve of each joint for swing leg
(a)lateral motion of COG (b)swing of right leg
(c)double-stance
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