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Modeling, stability and walking pattern generators of biped robots: A review

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Modeling, stability and walking pattern generators of biped robots: a

review

Hayder F. N. Al-Shuka, F. Allmendinger, B. Corves and Wen-Hong Zhu

Robotica / Volume 32 / Issue 06 / September 2014, pp 907 - 934

DOI: 10.1017/S0263574713001124, Published online: 05 December 2013

Link to this article: http://journals.cambridge.org/abstract_S0263574713001124

How to cite this article:

Hayder F. N. Al-Shuka, F. Allmendinger, B. Corves and Wen-Hong Zhu (2014). Modeling, stability and walking pattern
generators of biped robots: a review. Robotica, 32, pp 907-934 doi:10.1017/S0263574713001124

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 Robotica (2014) volume 32, pp. 907–934. © Cambridge University Press 2013
doi:10.1017/S0263574713001124

Modeling, stability and walking pattern generators of

biped robots: a review
Hayder F. N. Al-Shuka†∗ , F. Allmendinger†, B. Corves†
and Wen-Hong Zhu‡
†Department of Mechanism and Machine Dynamics, RWTH Aachen University, Aachen, Germany
‡Canadian Space Agency, 6767, Route de l’Aéroport, Longueuil (St-Hubert), QC, Canada, J3Y 8Y9
(Accepted October 29, 2013. First published online: December 5, 2013)

SUMMARY
Biped robots have gained much attention for decades. A variety of researches have been conducted
to make them able to assist or even substitute for humans in performing special tasks. In addition,
studying biped robots is important in order to understand human locomotion and to develop and
improve control strategies for prosthetic and orthotic limbs. This paper discusses the main challenges
encountered in the design of biped robots, such as modeling, stability and their walking patterns. The
subject is difficult to deal with because the biped mechanism intervenes with mechanics, control,
electronics and artificial intelligence. In this paper, we collect and introduce a systematic discussion
of modeling, walking pattern generators and stability for a biped robot.

KEYWORDS: Biped robot; Stability; Zero moment point; Orbital stability; Walking pattern
generators.

1. Introduction
Industrial robots are often rigidly attached to the ground. In contrast, mobile robots are able to move
in specific environments depending on the tasks they are required to perform. Much attention has
been drawn to the design and control of mobile robots due to their significant characteristics such as
their versatile mobility, sensing and reacting in their specific environments.1 Mobile robots can be
classified as legged robots, wheeled robots and tracks.
Legged robots offer significant advantages over wheeled robots and tracks in view of the workable
environments, the energy consumption and the adaptability.2 Although wheeled robots are simple
and lightweight structures, they need regular terrains for motion and lack, e.g. the ability to climb
stairs. Tracks help to overcome this drawback, but they consume a lot of energy due to the high
friction between the chains and the ground. In contrast, legged robots are able to move in regular and
irregular terrains with versatile mobility. With configuration changes, they can easily adapt to irregular
environment. Due to the small contact areas of their feet, legged robots can be efficiently operated.2
The biped robot is designed to imitate human-like locomotion and perform certain tasks such as
activities in danger environments, assistances to the elderly and entertainment.3 The biped robot can
consist of a trunk and legs with/without feet or even entire human-like mechanism depending on the
desired application (see Section 3).
The biped robots offer advantages over multi-legged robots. The biped robots have considerably
higher adaptability enabling them to easily surpass obstacles like narrow paths or stairs. This
adaptability is particularly important when the robot is required to perform human-centered tasks.
Due to the smaller ground contact area and fewer actuators used, their energy consumption can
be lower than multi-legged robots. This is consistent with the conclusion that two-legged animals
have higher efficiency and adaptability than multi-legged animals.4 For a historical review of legged
machines, we refer readers to Bekey3 and Raibert.5

* Corresponding author. Email: al-shuka@igm.rwth-aachen.de

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 908 A review on walking pattern generators of biped robots

Fig. 1. Important design issues of a biped robot.

Some challenges encountered in the design of biped robots are follows:

r Biped robots have unstable structures due to the passive joint located at the unilateral foot–ground
contact.6–9
r Due to the unilateral foot–ground contact and the varying configurations throughout the gait cycle,
their mechanical description is highly nonlinear. During the single-support phase, the robot is
under-actuated, while turning into an over-actuated system during the double-support phase.10
Consequently, the dynamic description and control laws change during transition from one phase
to another.7 It is noted that the biped robot could be fully actuated during the swing phase but with
some limitations (see Sections 2 and 3.2).
r Biped robots have many degrees of freedom (DOFs). A humanoid robot may have more than
30-DOF, making their mechanical behavior and control difficult.11–13 To avoid this difficulty,
most researchers have used simple models based on approximations and assumptions. A trade-off
between simplicity and accuracy becomes necessary.
r Biped robots interact with different unknown environments. This requires robust algorithms for
the generation of reference trajectories and control. These algorithms should be insensitive to the
possible disturbances and noises. However, stabilization and online adaptive control schemes could
solve this dilemma.
These challenges are associated with mechanics, control, electronics, artificial intelligence (AI)
and human anatomy so that studying biped robots is interdisciplinary. Unified solutions are therefore
difficult to develop. The possible issues to be encountered in the design of biped robots are illustrated
in Fig. 1.
In this paper, we focus on the mechanical aspects of the biped robots. The gait cycle of biped
locomotion is presented in Section 2. Section 3 discusses the modeling of biped mechanism, while
Section 4 considers the stability of biped robots. Section 5 introduces classification of the methods
used in the generation of walking patterns. The conclusions are given in Section 6.

2. Gait cycle
The complete gait cycle of human walking consists of two main successive phases: the double support
phase (DSP) and the single support phase (SSP) with intermediate sub-phases.14–16 The DSP arises
when both feet contact the ground resulting in a closed chain mechanism, while the SSP starts when
the rear foot is not supported by the ground with the front foot flatting on the ground. One should
note that the percentage of DSP is about 20% of time during one stride of the gait cycle whereas SSP
is about 80% of time.8,16
Due to the complexity of biped mechanisms, most researchers have simplified the gait cycle of
biped walking to understand the kinematics, biomechanics and control schemes of bipeds. Studies
have shown that there are three essential patterns used for the generation of periodic biped walking.10
Fig. 2 illustrates the third pattern grading from simple to complex configurations.
Pattern 17 : It consists of successive DSP and SSP without sub-phases, as shown in Fig. 2(a).
The swing feet are always level to the ground during leaving and striking the ground. The biped
mechanism that adopts this pattern may undergo under-actuation during the SSP if one considers the
ankle joints as passive joints. In contrast, if the ankle joints are active (powered), the system will be
fully actuated with allowable ankle torque to keep the zero moment point (ZMP) within the contact
area of the stance foot.12 However, the legs constitute over-actuated system during the DSP. In this
pattern, maximum ground support is provided during each step resulting in a slow or moderate biped

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 A review on walking pattern generators of biped robots 909

Fig. 2. Types of biped walking patterns.

walking.14 Huang and co-workers17,18 have stated that this type of pattern could result in unstable
walking due to the sudden landing of whole sole on the ground at the beginning of the DSP. This
drawback can be overcome by pattern 2.
Pattern 214,17,18 : This is analogous to the first pattern with an exception that the swing leg will leave
and land the ground with specified angle as shown in Fig. 2(b). This results in a smooth transition of
the striking foot from the heel to the whole sole at the beginning of the DSP. This pattern consists of
two sub-phases of DSP and one phase of SSP.
Pattern 310 : This pattern is close to the human walking consisting of two sub-phases of DSP and
SSP each. The first sub-phase of DSP starts when the rear foot begins to rotate about its front edge
during a small rotation of the front foot about the heel. Its second sub-phase starts when the front foot
is level to the ground while the rear foot continues to rotate about its front edge. The first sub-phase of
the SSP arises in the moment at which the rear foot is lifted above the ground meanwhile the stance
foot is level to the ground. Finally, the second sub-phase of the SSP occurs when the stance foot starts
to rotate about its front edge. Fig. 2(c) shows the walking stages of this pattern.
Additional DOF is gained during the second sub-phase of the SSP. Consequently, the system is
under-actuated during this sub-phase. This motivates the researchers to investigate the stability issue
of biped mechanism without foot or with a point foot because the robot coinciding with this walking
pattern is a system of under-actuation.19

Remark. It is possible to modify the mentioned patterns to generate the desired motion. For example,
pattern 2 can be performed in one phase of DSP instead of two sub-phases such that the front and

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 910 A review on walking pattern generators of biped robots

rear feet can rotate simultaneously at the same time. Another modification to pattern 3 can be seen in
Sato et al.20

3. Modeling
Humans and animals have matchless mobility due to their versatile reconfigurations. Consequently,
these creatures can walk easily and smoothly using their amazing control systems. The functionality
of human and animal locomotion is still not entirely analyzed.21 During motion, humans utilize only
about 20-DOF, although the human musculoskeletal system provides more than 300-DOF.8,22,23 This
fact allows humans to use a large number of DOFs that may be locked or released in order to adapt
their motion patterns to the environment and the desired tasks. Most researchers have attempted
to understand human walking principle using different simplified models of biped robots. Possible
choices are as follows:
r 2-link model: It uses one leg consisting of thigh, knee and shin without foot.5,24
r 3-link model: It consists of a trunk (torso) and two links for thigh and shin (shank) with locked
knees and without feet. This sagittal plane model has 5-DOF.25
r 5-link model: A sagittal plane model consisting of a trunk and two legs with knees, but without
feet. This model is frequently used, since it is considered a reasonable trade-off between simplicity
and leg anatomy.10,26,12 The leg motion can be extended from planar to spherical if a joint with
more than 1-DOF is chosen for the hip joint such as a universal joint or a ball-and-socket joint.
r 7-link model: It is a 5-link model with additional feet.
r 9-link model: It is analogous to the latter model adding two links connecting the ankle joints to the
sole plates. This is the closest model that most researchers proposed.10,17,18
A humanoid robot may comprise a total of more than 30-DOF.12,13

3.1. Kinematics
After a suitable model is selected, equations are derived to mathematically describe the kinematic
and dynamic behavior during walking. Direct kinematics can be written when the joint coordinates
are known and the position and orientation of robot’s links are formulated as a function of these
coordinates.
Define the rotation matrix R(t) ∈ SO(3) that transforms a 3 × 1 velocity expressed in a body
frame to the same vector expressed in the inertial frame, then

Ṙ (t) = (w×) R(t) (1)

with
⎡ ⎤
0 −wz wy
(w×) = ⎣ wz 0 −wx ⎦ (2)
−wy wx 0
where wx , wy and wz denote the components of the instantaneous angular velocity of the frame.
The relationship between the vector of the joint velocity of the biped mechanism and the velocity
of the end effector (swing foot in case of biped robot) can be described in Eq. (3),

ξ̇ = Jq q̇, (3)

where ξ̇ ∈ Rm represents the augmented linear/angular velocity vector of the end effector such that
1 ≤ m ≤ 6, Jq ∈ Rm×nq is the Jacobian matrix that relates the vector of the joint velocity of biped
mechanism to the body velocity of the end effector, nq is the number of generalized coordinates
of biped mechanism and q ∈ Rnq represents the generalized angular displacement vector of biped
mechanism.27
In contrast, the inverse kinematics is used to obtain the time history of joint coordinates as a
function of position and orientation of the robot’s links in Cartesian space. Therefore, the inverse
kinematics is very useful in determining the actuator motion necessarily needed throughout the entire

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 A review on walking pattern generators of biped robots 911

gait cycle of biped robots. The output of the inverse kinematic calculation provides the reference
motion of actuators for control purposes.28 Thus, the angular displacements of the biped mechanism
can be found as follows:

q̇ = Jq# ξ̇ , (4)
where
 −1
Jq# = JqT Jq JqT . (5)
Appropriate methods for the derivation of kinematic equations of biped robots can be found in the
refs. [29–31].
Remark. In effect, the optimal control theory can be used alternatively for determining the optimal
reference trajectory of joint displacements of biped mechanism. In addition, it can be used successfully
to control the biped mechanism with free reference trajectory (see Section 5.1.4).

3.2. Dynamics
Two approaches are commonly used to obtain the differential equations of the motion describing
system dynamics: Lagrange’s equation and the Newton–Euler equation. The dynamic equations can
govern the dynamic responses of the robot system to the input joint torques generated by actuators
and other active forces.31 Inverse dynamic calculations provide actuator torques as a function of the
desired motion. In forward dynamics, the differential equation of the motion is solved for the system
motion with active force/torque vector as an input. In control problem, we select appropriate control
law to generate input controls (torques/forces) that track the desired reference trajectory of the robot
mechanism.
In the following, the dynamic equations will be described assuming that the biped mechanism
walking consists of three successive phases (SSP, contact phase (impact phase), and DSP).
During the SSP, the biped robot behaves as an open-chain mechanism, therefore the governed
Lagrangian (or Newton–Euler32 ) dynamic equation can be written as

M q̈ + C q̇ + g = Aτ, (6)

where M ∈ Rnq ×nq is the mass robot matrix, q, q̇ and q̈· ∈ Rnq are the absolute angular displacement,
velocity and acceleration of the robot links, C ∈ Rnq ×nq represents the Coriolis and centripetal robot
matrix, g ∈ Rnq is the gravity vector, A ∈ Rnq ×nτ is the mapping matrix derived by the principle of
virtual work,33,34 τ ∈ Rnτ is the actuating torque vector and nτ represents the number of actuators.
Remark. It is known that the biped robot during the SSP behaves as an open-chain mechanism with
fully actuated- (nq = n = nτ ) or under-actuated-chain mechanism (n > nτ ) depending on the desired
walking patterns, where n represents the number of DOFs.
During the impact phase, the robot configuration remains unchanged with abrupt change in joint
velocity. Thus, the resulting dynamic equation can be written as

to +t
M(q)(q̇ + − q̇ − ) = J T ∫ λdt, (7)
to

J T (q̇ + − q̇ − ) = 0, (8)

where q̇ + and q̇ − refer to the angular velocity vector of the biped mechanism after and before impact
respectively, J ∈ Rnλ ×nq represents the Jacobian matrix resulting from the constrained motion of the
biped robot, λ ∈ Rnλ is the constrained force/moment vector at the foot–ground interaction with the
dimension of nλ and t is the time. Equation (7) expresses the theorem of conservation of momentum,
while Eq. (8) denotes the constraint of the motionless stance foot.
During DSP, the configuration of the biped mechanism changes, therefore the constrained
Lagrangian dynamic equation during this phase is

M q̈ + C q̇ + g = Aτ + J T λ, (9)

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 912 A review on walking pattern generators of biped robots

φ (q) = 0, (10)

where φ(.) represents the constraint equation with the same notations mentioned earlier.
The difficulties encountered in the analysis of the dynamics and control of the biped robots are the
over-actuation during DSP and the under-actuation during SSP. The inverse dynamic problem of an
over-actuated system is undetermined, and hence infinite combinations of torques can be applied to
achieve the desired output trajectory. There are two schemes that are used commonly for the control
of the over-actuated biped mechanism. The first one includes minimization of joint torques using the
algebraic optimization,12,35–37 whereas the second control method is represented by the kinematic
Jacobian.7,38 However, this strategy cannot guarantee the generation of continuous dynamic response
(actuating torques/ground reaction forces) at transition instances (from SSP to DSP and vice versa).
The under-actuation during the swing phase poses particular difficulty for the control of biped robot.
Four state-of-the-art control methods are used to solve this problem: the time scaling method,12,39
hybrid zero dynamics,40 the differentially flatness-based approach25 and the port-Hamiltonian
method.41 Further elaboration of these methods is left for subsequent papers.
For biped robots, moderate path deviations are acceptable in the sense of stability. Therefore,
elastic joints are preferred over stiff joints as they absorb shock and store energy.7 Adding elastic
joints means increasing the DOF of the system, and flexible joints further complicate the dynamics
of the robots. In addition, it is found that the compact biped model is insufficient to describe some
aspects of human walking; therefore, compliant legs are employed to improve the behavior of biped
mechanism. The field of compliant legs has been widely investigated by Seyfarth and co-workers.42–46
The foot–ground contact poses another particular challenge to the dynamic behavior of biped
robots. The consequences of these unilateral contacts with impulsive forces were well discussed in
refs. [10, 47].

4. Stability
The biped mechanism is unstable during the SSP. One of the challenges in the design and control
of biped robots is to keep its balance during walking in different kinds of environments. The reason
for instability is the under-actuation due to the passive joint of the foot–ground contact. This means
that the control of the feet is dependent on the control of the mechanism that is above the feet.6,48
Common stability theories, such as analysis of eigenvalues, gain and phase margins and the Lyapunov
stability can be applied to particular modes of biped robot gait. However, these approaches cannot
guarantee the biped stability for all modes of motion.49
In general, there are two types of stability criteria on which the trajectories of a biped mechanism
depend: static stability and dynamic stability. Static stability restricts the vertical projection of the
center of mass of the biped to remain inside the support polygon. The support polygon is defined as
the area represented by the stance foot during the SSP and the bounded area between the supported
feet during the DSP.10 This type of stability leads to slow gait and biped robot with large feet.50 Thus,
the position of the center of mass, pcom , can be calculated as
nl
m i pi
pcom = i=1 nl , (11)
i=1 mi

where nl is the number of biped links, mi is the mass of link (i) and pi is the position of the center of
mass of link (i). The ground projection of pcom can be easily found by determining its components.
Dynamic stability provides more freedom than static stability since the projected center of mass
of the biped may leave the support polygon and thus allows for faster gait.50 Following are the four
specific techniques to analyze dynamic stability49 :
(a) ZMP, (b) the periodicity-based gait, (c) theory of capture points51,52 and (d) the foot placement
estimator.53 In this paper we discuss the first two strategies.

4.1. Zero-moment point (ZMP)

The notion of ZMP was proposed by Vukobratovic and Stepaneko,54 exploiting the passive joint
of the foot–ground contact. It is applied to the biped mechanism in designing walking patterns and
control schemes. The ZMP is the point on the ground where the net moment vector of the inertial

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 A review on walking pattern generators of biped robots 913

Fig. 3. Relationship between ZMP, FZMP and COP. (a) Dynamically stable, (b) dynamically unstable (this
figure is cited from Vukobratovic and Borovac6 ).

and gravitational forces of the entire body has zero components in horizontal planes.6,14,48 In brief, if
ZMP is located inside the support polygon, then the system is stable and the center of pressure (COP)
of the foot coincides with ZMP. If the ZMP is outside the support polygon, the system is unstable and
the ZMP will be outside the stability margin comprising fictitious ZMP (FZMP)6 as shown in Fig. 3.
Point P in the mentioned figure represents the location of the zero components of the net moments
affecting the foot at horizontal planes. It is clear that in the stable case, one can determine the position
of ZMP by calculating the position of COP. This is conducted by using force sensors at the sole plate
of the foot.14 In effect, the theoretical calculation of ZMP can be performed by the following two
formulations:
(a) Formulation 1 (more computational formulation)
By knowing the inertial and gravitational forces, the ZMP coordinate can be found as in Eqs. (12)
and (13)55
n n 
mi (z̈i + g) xi − mi ẍi zi − ni=1 Iiy q̈iy
xzmp = i=1
n i=1
, (12)
i=1 (z̈i + g)
n n n
i=1 mi (z̈i + g) yi − i=1 mi ÿi zi + i=1 Iix q̈ix
yzmp = n , (13)
i=1 (z̈i + g)

where x and y axes are parallel to the horizontal plane for the biped motion, z-axis is pointing
upwards, (Iix , Iiy )T is the inertial vector of link i, q̈ix and q̈iy are the angular acceleration of link (i)
about x and y respectively, g is the gravitational acceleration, (xzmp , yzmp , 0) is the coordinate of ZMP
and (xi , yi , zi ) is the coordinate of the mass center of link (i). The above equations can be used to
investigate the stability of biped mechanism during SSP and DSP. It has been noted that Eqs. (12)
and (13) were described wrongly in most references as shown in Table I.
(b) Formulation 2 (less computational formulation)
Equations (10) and (11) are highly nonlinear; therefore, most researchers have used Eqs. (14) and
(15) alternatively during the SSP using the static equations of fixed stance foot,7

xZMP = −τy /Fz , (14)

yZMP = τx /Fz , (15)

where τy and τx are the ankle joint torques about the referred axes and Fz is the normal component
of the ground reaction force.
Thus, the necessary associated constraint is

−lf ≤ xZMP (t) ≤ 0, (for sagittal plane) (16)

where lf is the length of the stance foot assuming the origin of the Cartesian coordinate system placed
at the tip of the stance foot.

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 914 A review on walking pattern generators of biped robots
Table I. Wrong ZMP equations used in the literature.

References The ZMP equation

n n n
i=1 mi (z̈i + g) xi − i=1 mi ẍi zi − i=1 Iiy q̈iy
[17] xzmp = n
(z̈
i=1 i + g)
n n n
mi (z̈i + g) yi − mi ÿi zi + i=1 Iiy q̈iy
yzmp = i=1
n i=1

i=1 (z̈i + g)

n n n
mi (z̈i + g) xi − mi ẍi zi − i=1 Iiy q̈iy
[18] xzmp = i=1
n i=1
(z̈
i=1 i + g)
n n n
mi (z̈i + g) yi − mi ÿi zi − i=1 Iix q̈ix
yzmp = i=1
n i=1

i=1 (z̈i + g)

n n n
mi (z̈i − g) xi − mi ẍi zi + i=1 Iiy q̈iy
[56] xzmp = i=1
n i=1

i=1 (z̈i − g)
n n n
mi (z̈i − g) xi − mi ẍi zi − i=1 Iiy q̈iy
[57] xzmp = i=1
n i=1
(z̈
i=1 i − g)
n n n
mi (z̈i + g) yi − mi ÿi zi + i=1 Iix q̈ix
yzmp = i=1
n i=1

i=1 (z̈i + g)

The authors have considered g = −9.8m/s 2 .

n 
mi (z̈i + gz ) xi − ni=1 mi (ẍi + gx )zi
[21] xzmp = i=1
n
i=1 (z̈i + gz )

n 
mi (z̈i + gz ) yi − ni=1 mi (ÿi + gy )zi
yzmp = i=1
n
i=1 (z̈i + gz )

where gz = −g for level ground

Since ZMP coincides with COP, the ZMP coordinate can be found during DSP according to Eq.
(17),58

Fzf Fzr
ZMP = COP = copf + copr , (17)
Fzf + Fzr Fzf + Fzr

where COP, copf and copr represent the center of pressure for the biped robot during DSP, the front
foot COP and the rear foot COP respectively, Fzf and Fzr are normal components of ground reaction
forces for front and rear feet respectively.
In general, the ZMP-based biped mechanisms are characterized by the following points.
r The stance foot of the biped should remain in full contact all the time.59
r All the joints of the biped mechanism are actuated and rigidly controlled to track predetermined
trajectories that can simplify the control task during the SSP.
r Unnatural motion with high-energy consumption.
r Most conventional humanoid biped robots are always driven by electric motors via gears, and their
walking patterns are generated based on the linear inverted pendulum mode (see Section 5.1.3 and
Table II).

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 Table II. The most distinctive biped/humanoid robots built in different countries with their characteristics.

A review on walking pattern generators of biped robots

Year Country Structure Gait speed Stability Walking pattern generat-
(m/s) criterion ors

1973 Japan/Waseda University WABOT-1 is a hydraulically powered biped walking robot with two 5-DOF 0.0043 Static Model-based gaita
legs, a 1-DOF trunk and artificial hands.163
1980 Japan/Waseda University Wl-9DR is a hydraulically powered biped walking robot with total 10 0.045 Static/quasi- Model-based gaita
DOFs.5,163 dynamic
walking
1980– Japan/University of Tokyo A series of BIPER biped robots have been developed. BIPER-3 is a biped robot - Periodic COG-based gait
1984 with four actuators at the hip and point feet. While BIPER-4 is equipped with
eight actuators (4 actuators for each leg), BIPER-5 is similar to BIPER-3.128
1982– USA/CMU Leg A series of running legged (one-legged, biped and quadruped) prototypes have 4.3 Periodic Natural dynamics-based
1985 Laboratory been developed. The planar biped robot has 3 DOFs with tether boom for gait
constraining the biped motion in 2.5-m radius circle.5
1985 Japan/Waseda University WL-10RD is a 12-DOF hydraulic-powered biped walking robot with 1.3 s/step ZMP Model-based gaitb
microcomputers installed on the upper body.63
1990 Canada/T. McGeer A 2D passive (gravity-based powered on a shallow incline) walker with two 0.46 Periodic Natural dynamics-based
straight legs (without knees) and semicircular feet.78,164 In addition, the gait
knee-jointed legs have been designed.79
1996 Japan/Honda R&D Co. A series of Honda humanoid robots have been developed. In 2000, the updated Walking: ZMP COG-based
version called ASIMO was developed with a height of 1.2 (m), a weight of 0.75, gait/model-based gait
43 (kg) and electrically actuated (DC servomotors with harmonic drive running:
gears) 34 DOFs (3 DOFs for the head, 7 DOFs for each arm, 6 DOFs for 2.5
each leg and 1 DOF for each hand). In 2011 the updated version of ASIMO
was announced with 57 DOFs.81,95,165–167
1996 Japan/Waseda University Wabian family has been created. The updated version is called Kobian with 0.5 ZMP Model-based gaitb
about 1.4 (m) tall, 62 (kg) in weight and 48 electrically actuated (DC
motors/harmonic drive gears) DOFs (12-DOF legs, 3-DOF waist, 14-DOF
arms, 8-DOF hands, 4-DOF neck and 7-DOF face).168–173
1996– USA/MIT Spring Flamingo is a planar biped with 13.5 (kg) weight and 0.9 (m) height, 0.75–1.2 Capture Natural dynamics-based
2000 and six electrically actuated (series elastic actuator) DOFs (3 DOFs for each point/ gait
leg).160–162,174–177 periodic
1997 Francec Rabbit is a planar five-link biped robot with 4 degrees of actuation (two Walking: Periodic Optimization-based
actuators at the hip and two at the knees) and point feet.100,178,179 0.75,d gait/model-based gait
running:
1.25

915
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 916
Table II. Continued.

Year Country Structure Gait speed Stability cri- Walking pattern generat-
(m/s) terion ors

1998 USAe /Cornell University Tinker toy walking model is a two-leg toy with no control system and 0.0277 Periodic Natural dynamics-based
gravity-based power.80 gait
2001– Japanf /Sony Co. A series of SDR humanoid biped robots have been developed. The latest 0.23 ZMP COG-based
2006 version is QRIO (SDR-XII) with small-scale size (height of 0.58 (m) and gait/model-based gait
weight of 6.5 (kg)) and 38 electrically actuated (servo motor with gear unit)
DOFs (4 DOFs for the head, 2 DOFs for the torso, 5 DOFs for each arm, 6
DOFs for each leg and 10 DOFs for the fingers).180–186
2001 USA/Cornell University A 3D-passive dynamic walker with two legs, knees, curved feet, compliant 0.51 Periodic Natural dynamics-based
heels and two swinging arms for stable gait.155 gaitg
2002 Japan/AIST and HRP family (HRP-2P, HRP-2, HRP-3P, HRP-3, HRP-4C and HRP-4) has been HRP-4C: ZMP COG-based
KAWADA Industries developed. HRP-4C is a humanoid biped robot with Japan female 0.5, gait/model-based gait

A review on walking pattern generators of biped robots

Inc. appearance, weight of 43 (kg), length of 1.58 (m) and electrically actuated HRP-2:
(DC servomotors with harmonic drive gears) 42 DOFs (6 DOFs for each 0.694
arm, 2 DOFs for each hand, 6 DOFs for each leg, 3 DOFs for the waist, 3
DOFs for the neck and 8 DOFs for the face). While the updated version
HRP-4 is characterized by weight of 39 (kg), length of 1.51 (m) and 34
DOFs (2 DOFs for the neck, 7 DOFs for each arm, 2 DOFs for each hand, 2
DOFs for the waist and 6 DOFs for each leg).187–197
2002 Japan/Waseda University WL-15, WL-16, WL-16R and WL-16RII have been developed with 0.2500 ZMP Model-based gaitb
electric-powered 6-DOF parallel mechanisms, two legs and a waist for
multi-purpose use.198,199
2003 The Netherlandsh /Delft Mike is a 2D passive dynamic walker with weight of 7 (kg), length of 0.75 (m) 0.4 Periodic Natural dynamics-based
University and McKibben muscles (pneumatic actuators) at the hip and knee joints for gait
active control. It is similar to the close copy of McGeer by Garcia et al.200,201
2003 Germanyi /Technisce Johnnie is a humanoid biped robot with length of 1.8 (m), weight of 40 (kg) 0.6667 ZMP COG-based
Universität München and electrically actuated (DC brush motor with harmonic drive gears) 17 gait/model-based gait
DOFs (2 × 3 DOFs for the hip, 2 × 1 DOFs for the knees and 2 × 2 DOFs
for the ankles).98,99

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 A review on walking pattern generators of biped robots
Table II. Continued.

Year Country Structure Gait speed Stability cri- Walking pattern generat-
(m/s) terion ors

2004 USA/MIT Toddler is a 3D passive dynamic walker with a single passive joint at the hip, 0.2238 Periodic Natural
two active joints at each ankle (pitch and roll) and large curved feet.202 dynamics-based202
gait/reinforcement
learning control based
on optimization
strategy203
2004 The Netherlands/Delft Denise is an autonomous 3D biped robot with 8 (kg) weight, 1.5 (m) tall, five 0.4 Periodic Natural dynamics-based
University internal DOFs (1 × 2 DOFs for ankles which are equipped with spring and gait
damper, 1 × 2 DOFs for passive knees and 1 DOF for pneumatically
actuated (McKibben muscles) hip).204,205
2004 Republic of A series of KHR humanoid biped robots have been developed. The updated 0.3472 ZMP COG-based
Koreaj /KAIST version is HUBO with height of 1.25 (m), weight of 55 (kg) and 41 gait/model-based gait
electrically actuated (servo motor + harmonic speed reducer + drive unit)
DOFs (6 DOFs for each leg, 4 DOFs for each arm, 6 DOFs for the head, 7
DOFs for hand and 1 DOF for the trunk).206–214
2004 China/Beijing Institute of BHR humanoid biped robot family has been developed. The updated version is 0.1389 ZMP COG-based
Technology called BHR-2 with 1.6 (m) length and 63 (kg) weight and 32 electrically gait/model-based gait
actuated (DC motors) DOFs (2 × 3 DOFs for hips, 2 × 1 DOFs for the
knees, 2 × 2 DOFs for the ankles, 2 × 3 DOFs for the shoulder 2 × 1DOFs
for the elbows, 2 × 2 DOFs for the wrists, 2 × 3 DOFs for the hands and 2
DOFs for the head).215,216
2005 USA/Cornell University Cornell’s biped is a 3D passive dynamic walker with 12.7 (kg) in weight, two 0.44 Periodic Natural dynamics-based
arms, two 0.81-(m) long legs, a small upright torso with a leg-angle-bisecting gait
mechanism and wide curved feet. There are 5 internal DOFs: one free hip
joint, two periodically locked knees and two controlled ankle joints.102,103
2006 Germany/Technische LOLA is a humanoid biped robot with 1.8-(m) height and equipped with 0.9278 ZMP COG-based
Universität München electrically actuated (permanent magnet synchronous motors PMSM) 22 gait/model-based gait
DOFs (2 × 7 DOFs for the legs, 2 × 3 DOFs for the arms and 2 DOFs for
torso).217
2006 Japan/Toyota Central A humanoid biped robot has been developed with 1.30-(m) height, 50 (kg) Running: ZMP COG-based
R&D Labs. Inc. and weight and 15 DOFs (7 DOFs per leg and 1 DOF for the waist).96,97 1.9444 gait/model-based gait
Toyota Motor Corp. Inc.

917
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 918
Table II. Continued..

Year Country Structure Gait speed Stability cri- Walking pattern generat-
(m/s) terion ors

2009 Spaink /University Carlos The RH-1 is a humanoid biped robot with 1.50-(m) height, 50 (kg) weight and 0.194 ZMP COG-based
III of Madrid 21 electrically actuated (servo motors) DOFs (6 DOFs for each leg, 4 DOFs gait/model-based gait
for each arm and 1 DOF for the chest).218–220
a
The walking patterns of the biped robot have been performed by constraining the COM of biped to be within the stability margin. The interpolation technique could be used

A review on walking pattern generators of biped robots

for generation stable swing foot trajectory.
b
To find appropriate mathematical relationship between the COG of the biped robot and the ZMP trajectory, Eqs. (10) and (11) were approximated using FFT. The interpolation
technique was used for generating stable swing foot trajectory.
c
There are other prototypes built in France, such as BIP2000,221–228 SHERPA biped robot229 and the amazing small-scale humanoid robot NAO.230,231
d
RABBIT has been designed to be able to walk with average forward speed of at least 5 (km/h) and to run more than 12 (km/h).
e
For more details on the prototypes built by Ruina and colleagues, we refer to the website Andy Ruina-Biorobotics and Locomotion Lab – Cornell University: http://
ruina.tam.cornell.edu/research/index.php
f
HR18232,233 , Morph234 and PINO235,236 are examples of small-scale humanoid robots manufactured in Japan.
g
To make the passive dynamic robots walk on level ground, the authors have substituted the gravitational power with simple actuation at the hip and/or the ankles.102
h
There are other prototypes built by Delft Lab (please see Delft Biorobotics Lab: http:// www. dbl.tudelft.nl/).
i
There are other biped prototypes manufactured in Germany such as the small-scale humanoid robots TONI,237 Kidsize238,239 etc.
j
AM12 is a humanoid biped robot manufactured in Republic of Korea based on ZMP criterion.240
k
Other examples of Spanish humanoid biped robot are SILO2241–243 and REEM-B.244

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 A review on walking pattern generators of biped robots 919

Several algorithms have been used to generate stable reference trajectory for the biped mechanism
satisfying the ZMP constraint (see Section 5). This implies that the solution of Eqs. (12) and (13) to find
a relationship between the COM of the biped and the ZMP trajectory requires a lot of computational
effort. Therefore, these algorithms can be applied offline.60,61 Alternatively, most researchers have
used simple models to generate the desired biped walking patterns, such as the linear inverted
pendulum for online implementation (see Section 5.1.3). Biped robot stability approaches based on
ZMP still lack efficiency, robustness, easy handling and natural motion.59,62 For further reading, we
refer to refs. [18, 63–72].

4.2. Periodicity-based gait

For more than 40 years, researchers have tried to understand the locomotion of biped mechanism
using the inverted pendulum model due to similarity between the two mechanical systems both in
static and periodic stability.26,73–77 This demands the investigation of the phase-plane portraits of
generalized coordinates and velocities to see the periodic (cyclic/orbital) stability of the inverted
pendulum.
In 1990, McGeer78 designed a two-dimensional passive walker with two straight legs (without
knees) and semicircular feet. In addition, the knee-jointed legs were designed to imitate the human-
like motion on shallow slops without using any actuators.79 The stability of this simplified prototype
adopted the Poincare map to investigate periodic stability.
The periodic stability exploits the under-actuation resulted from the passive joint of the foot–ground
contact. It is noted that the Hamiltonian dynamical systems cannot have asymptotic stability whereas
the conservative non-holonomic systems can have asymptotic steady stability in some variables.80
Thus, the biped robots based on periodic stability can perform cyclic sequences of steps that are stable
as a whole, but not locally at every instance of time.59 The Poincare map is used to show the stability
of this type of walking as described below.
The transition of the current biped state v k to the successive state after one walking step v k+1 can
be described by the stride function S as follows:

v k+1 = S(v k ). (18)

If the motion is perfectly cyclic, then the state v is a fixed point corresponding to the limit cycle,

v f = S(v f ). (19)

Orbital stability is investigated by using a perturbation around the initial fixed point,
 
S v f + v ≈ v f + Pv, (20)

where P = ∂∂vS denotes the linear return matrix which governs the orbital stability of the biped robot.
If the eigenvalues of P are within the unit circle of the complex plane, the biped system is stable,
otherwise it is unstable.59
In 1996, Honda R & D Co. announced the first self-contained humanoid biped robot called P2.81
It was 1.82-m tall with 210-kg weight and depends on ZMP criterion to generate its desired stable
trajectory. Consequently, a lot of energy can be consumed with unnatural motion due to bent knees.
This motivates the community of the passive dynamic walking to re-examine and extend the work of
McGeer;82–93 see Table II for more details.
The periodicity-based biped robots can be characterized by the following points94 :
r They can have human-like efficiency and actuation requirements.
r Their motions are mostly symmetrical and are extremely difficult to turn, go back, sit etc.
r Compliance plays an important role in regulating walking behavior.
r The velocity of the robot can be controlled by actuating hip and/or ankle joints.

Remark. Although it is well known that the periodic-based gait can be faster than the gait based
on the ZMP criterion, ASIMO95 and Toyota humanoid96,97 (ZMP-based humanoids) can run with
a speed of 9 and 7 km/h respectively, while Johnnie humanoid98,99 can only walk with a speed of
3.34 km/h. In effect, literature can prove contrary among passive walking researchers related to the

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 920 A review on walking pattern generators of biped robots

Fig. 4. Classification of the approaches used for generating walking patterns of biped robot.

speed of passive walking against ZMP-based gait. Chevallereau et al.100 said: “. . . truly dynamic
motions, such as balancing, running, or fast walking, are clearly excluded with this (ZMP) approach,”
whereas Hosoda et al.94 said: “The velocity of the robot can be controlled by actuating hip and/or
ankle joints, but the range of the velocity is relatively limited since the behavior of the robot is strongly
governed by its passive dynamic.”
Remark. The walking speeds of the human, ASIMO and Cornell’s Biped are about 1.3 m/s (optimal
speed for human101 ), 0.75 m/s and 0.44 m/s respectively. The energy efficiency of the biped
locomotion is measured by the specific cost of transport, i.e. (energy used)/(weight) × (distance
traveled). Thus, the energy efficiency of the human, ASIMO and Cornell’s Biped are (0.2, 3.2 and
0.2) respectively.102,103
Methods for generating periodicity-based gait are as follows: central pattern generators (CPGs)-
based gait (Section 5.2.1), using self-excited mechanism,62 optimization techniques104–106 or
exploiting natural dynamics49 (Section 5.3).

5. Walking pattern generators

One of the important issues of the biped locomotion is the generation of the desired paths that ensure
stability while avoiding collision with obstacles.56 Due to the similarity between the biped robot and
the human locomotion, some important aspects should be considered in order to generate natural
biped locomotion, which are as follows107 :
r Learning (training), which needs a certain level of intelligence.
r A high level of adaptability to cope with uneven terrains and external disturbances.
r In specific circumstances, optimal motion to reduce energy consumption during walking.

It is hopeful to make the biped robot compete with the human system. Consequently, it needs at
least an online adaptation system to deal with uncertain environments.108,109 In contrast with this,
there are many researchers who have attempted to design biped models using predefined reference
trajectories which are insufficient for stability and adaptability purposes (Section 5.1.1 is an example
of offline walking patterns). Online walking patterns require knowledge of the path changes online
while guaranteeing stability and motion control of biped mechanism.109 Examples of methods used
to compensate ZMP online include preview control,110 model predictive control,109,111,112 and AI-
based gait.56,113 In general, the model-based methods can give useful explanations about the behavior
of human walking. In contrast, AI can give robust results without explanation.10,114 The first two
methods are described in Section 5.1.4, while the latter method is discussed briefly in Section 5.2.2.
Surely, there are other approaches used in the literature; however, the mentioned approaches serve as
the typical online schemes so far.
There are numerous approaches of generating biped walking patterns. We have established an
overview of walking pattern generators depending on refs. [7, 10, 49, 115]. These approaches can be
classified as illustrated in Fig. 4.

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 A review on walking pattern generators of biped robots 921

5.1. Model-based gait

This method uses analytical schemes for the generation of reference trajectories of joints for biped
mechanism. Then a robust control system is performed to track the desired trajectories.7 The
limitations of the model-based technique are characterized by the need of the full knowledge of
the parameters of the dynamic biped model. These approaches are possibly accompanied with high
burden of computations and large disturbances and noises resulted from measurement devices.
5.1.1. Human motion-capture data. One of the methods to develop walking pattern generators is to
use the human motion-capture data that allows to extract recorded motion data from the examination
of human gait, and provide this data for the target biped robot or the handicapped.116–119 This method
is implemented by attaching markers at the limbs and joints of a human subject to capture human
movement. Once recorded, these data are processed numerically to get the coordinates of the limbs
and joints from which the velocity and acceleration can be obtained. Hemami and Farnsworth119
have applied this technique on a planar 5-link biped robot to analyze the posture and gait stability of
their model. They have concluded that this method is useful in the designing of prosthetic devices for
handicapped and in tuning the model to mimic human movement.
During walking, the human center of gravity oscillates vertically and horizontally. The vertical
oscillation adopts a cycloid path with an amplitude of about 75 mm, while the horizontal oscillation
is a sinusoidal path with an amplitude of about 30 mm.10 Therefore, some researchers have exploited
this feature to provide directly the hip and the feet of the robot model with sinusoidal or cycloid
functions. Although this technique does not include human motion-capture data, it is integrated in
this section for two reasons. First, the sinusoidal trajectories provided for the biped robot are inspired
from the human gait. Second, the two described techniques have the same limitations that will be
illustrated at the end of this section. Juang and Lin120 used a cycloid profile developed by Kurematsu
et al.121 in generating the trajectories for the hip and the ankle joints during the swing leg, while
Nicholls50 used sinusoidal functions to generate paths for the feet and the trunk. However, there exist
kinematic and dynamic inconsistencies, such as length, mass etc., between the human subject, whose
motion is captured, and the proposed model of the biped robot. In fact, the human subject has a high
level of training, adaptability and optimal motion, which are lost in biped robots.7
5.1.2. Interpolation-based gait. This method is widely used in generating reference trajectories for
the end-effectors of industrial robotic arms. It describes the trajectory of the robot manipulator by a
function such as a polynomial satisfying certain continuity conditions and guaranteeing smooth
motion for the manipulator.122 The interpolation can be developed in computer animation for
describing the motion of a sequence of images by interpolating the intermediate key-frames. Brotman
and Netravali proposed that this method can produce unnatural motion due to the absence of the
dynamics of the problem.123 Alternatively, the authors used the optimal control theory for this
purpose.
We should notice that the interpolation of motion for biped robots is associated with either the
optimization problem or stability criteria such as ZMP or both of them. Shih124 realized the walking
of 3D biped robot in different environments. The reference trajectories of the biped robot body and
its feet have been generated by a third-order piecewise cubic polynomial. Huang et al. proposed a
method to generate walking patterns for the hip and the feet of biped mechanism.17,18 The authors
generated the foot trajectory by using a third-order spline function considering the constraints of foot
trajectories. By adjusting the parameters of the foot trajectory, different patterns of foot motion can
be realized. The hip trajectory has been produced by a third-order spline function such that it can
realize the desired ZMP trajectory. A similar work was implemented by Mu and Wu.125
5.1.3. Center of gravity (COG)-based gait. Due to the complex nonlinear dynamics of biped robot,
it is difficult to obtain a closed form solution. Therefore, many researchers have focused their
attention on simplifying the complex dynamics of biped robots in order to use simple algorithms for
generating walking patterns and control. As aforementioned, the inverted pendulum model is used as
an approximate model for biped mechanism due to the similarity between two mechanical systems in
terms of static and periodic stability. However, this approximation can be connected with the periodic
stability of the inverted pendulum. Kajita and Toni126,127 proposed the linear inverted pendulum
mode (LIPM) exploiting the previous related works5,26,73–77,128 . The LIPM includes finding a simple
mathematical relationship between the trajectory of COG of the biped robot and the ZMP to ensure

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 922 A review on walking pattern generators of biped robots

stable biped motion. It is assumed that all masses of the biped model are concentrated in its COG
and there is a reaction force between the ground and the foot without ankle torque.61,129 Depending
on the reference trajectories of COG and the feet, all the motions of the joints can be deduced. Thus,
the relationship between COG and ZMP during the SSP can be computed as7,130–132

H
xZMP = xCOG − ẍCOG , (for sagittal plane) (21)
g

where xCOG is the position of the COG, and H is the height of COG, which is assumed fixed.
Consequently, the COG trajectory motion during SSP can be calculated solving differential Eq. (21),

xCOG = C1 exp(wt) + C2 exp(−wt), (22)

where C1 and C2 are constants that can be obtained from the boundary conditions, and

w= g (23)
H.

In a similar manner, the COG-based model can be used successfully for the modeling of biped
mechanism during DSP. For details, we refer to refs. [130–132].
The LIPM concept could be accepted for quadrupeds due to their slender legs. However, one-third
of the mass of human subjects is contained in the legs. Therefore, it is questionable to use LIPM
in bipeds.8 Therefore, the preview control described in the next section is used as a servo control
technique for compensating the ZMP error produced by the COG-based model. Consequently, Park
and Kim61 suggested to use the gravity-compensated inverted pendulum mode (GCIPM) with two
proposed masses to reduce the errors accompanied by LIPM. One mass denotes the dynamics
of the swing leg and the other represents the stance foot and the trunk. Other modifications to
LIPM have been proposed to avoid the deviation of the desired trajectories of ZMP. These modified
approaches include the Two Masses Inverted Pendulum Mode (TMIPM),133 the Multiple Masses
Inverted Pendulum Mode (MMIPM)134 and the Virtual Height Pendulum Mode (VHIPM).135 For a
comprehensive comparison of the mentioned methods, we refer to ref. [135].
5.1.4. Optimization-based gait. Studies have proved that human beings attempt naturally to minimize
their energy consumption during walking according to the environment.108 Consequently, many
researchers have applied miscellaneous optimization schemes on biped robot to get feasible optimal
design. One should mention that the optimization could be applied to the construction of the biped
robot, its control system or its adaptation to the motion terrain etc.2 For an elaborated discussion
of optimization approaches used in robotics, we refer to refs. [2, 136, 137]. This section discusses
briefly the optimal trajectory of the biped robot using different optimal control methods. In general,
the optimization technique includes the selection of a suitable cost (objective) function that satisfies
some definite constraints such as the step length, the foot clearance, the hip height, or the speed of
walking. Then a set of parameters can be found to achieve the optimal path.108
In general, the optimal control problem can be classified as: dynamic programming, indirect
methods and direct methods as shown in Fig. 5. Although the dynamic programming is less sensitive
to the initial guess of design parameters, it suffers from the curse of dimensionality.138 The indirect
approach represented by Pontryagin’s Maximum Principle (PMP) demands necessary conditions for
optimality that can result in highly nonlinear Ordinary Differential Equations (ODEs) and the two-
boundary value problem.139,140 In addition, the indirect approach is extremely sensitive to the initial
guess of the co-state equations. In effect, the indirect approach can be intricated for complex dynamic
systems such as biped robot.10,141 Despite this difficulty, Rostami and Bessonnet142 and Bessonnet
et al.143 investigated the optimal motion of biped robot during SSP and during the complete gait cycle
respectively using PMP, assuming the boundary conditions of the biped robot are known.
A more flexible method for the optimal control problem is the direct method, by transcribing the
infinite dimension problem into the finite-dimensional nonlinear programming (static or parameter
optimization). This can be implemented by the discretization of controls or states or both, depending
on the selected discretization approach. The solution uses one of the nonlinear programming
algorithms such as sequential quadratic programming (SQP), interior points, genetic algorithm

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 A review on walking pattern generators of biped robots 923

Fig. 5. Classification of optimal control methods.

(GA) etc. Although being easy and robust, this method can only give suboptimal/approximate
solution.139,140,144,145 The formulation of the original optimal control problem can be described
as follows:
Determine: u.
tf
Minimize : J = c0 (x, t) + ∫ L (x (t) , u (t) , t) dt. (24)
t0

Subject to:

ẋ = f (x (t) , u (t) , t), (25)

a1 (x (t0 ) , u (t0 ) , t0 ) ≤ 0
a2 (x (t0 ) , u (t0 ) , t0 ) = 0, (26)
   
b1 (x tf , u tf , tf ) ≤ 0
     
b2 x tf , u tf , tf = 0, (27)
c1 (x (t) , u (t) , t) ≤ 0
c2 (x (t) , u (t) , t) = 0, (28)
ul ≤ u(t) ≤ uu
x l ≤ x(t) ≤ x u , (29)

where u ∈ Rn is the input control vector, c0 and L are scalar functions of the indicated arguments, J
is the scalar performance index, x ∈ Rn is the state vector, t, t0 and tf are the time, initial and final
time respectively, a1 and a2 are the initial constraints, b1 and b2 are the final constraints, c1 and c2 are
the path constrains and Eq. (29) refers to the bound constraints of the input control and the states.
The formulation of the discretized optimal control problem can be described as a nonlinear
programming as follows:
Determine: Y , which may be control variables or states or both.

N−1
Minimize : J = c0 (x(tN )) + lk (x (tk ) , u (tk ) , tk ) t, (30)
k=0

where N is the number of time intervals and t = (tf − t0 )/N. Equation (30) can be solved by any
numerical integration approach such as the trapezoidal or composite Simpson’s rule etc.

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 924 A review on walking pattern generators of biped robots

Subject to:

Z (Y ) = 0, (31)
C l ≤ C(Y ) ≤ C u , (32)
Y l ≤ Y ≤ Y u. (33)

Due to the complexity of the computation of these approaches, these may not be appropriate for
on-line implementations. Moreover, the trajectories obtained may not be the true optimal path. This
scenario may occur either because of the imperfect model, or the solution of the optimization problem
may get stuck at the local minima.56,108 Therefore, a group of researchers prefer to use intelligent
methods for generating the desired paths. This will be discussed in the next section.
There are two interesting approaches associated with the optimal control problem that can give a
feasible solution for online walking adaptation and rejection of disturbances. These two approaches
are the preview control and the model predictive control. The objective of the preview control is to
improve the performance of transient responses of the dynamic system when the future reference
signals or the disturbances are available.7 Kajita et al.110 designed a new method for generating
online walking patterns using the preview control approach of ZMP. The authors approximated the
biped model with a cart-table model for a simple treatment with ZMP. Then they used the preview
control to compensate the ZMP errors produced by the differences between the proposed model of the
biped robot and the actual multi-body model. Alternatively, the model predictive control (MPC) is an
optimal control-based approach that can deal with constrained dynamic systems for realizing online
walking or any affecting disturbance.109,111,112 It is a notable approach for walking adaptation. Using
MPC, the biped mechanism can be controlled without using predetermined reference trajectories.

5.2. Biological mechanism-based gait

5.2.1. Central pattern generators (CPGs). Human and animal locomotion can generate stable
rhythmic patterns of movements. They have versatile capabilities to change their speed or even
their patterns of movements.146 The central pattern generators in the spinal cord, which are neural
networks (NNs), can produce these locomotion rhythms without any sensory signals.146–148 Therefore,
researchers have exploited this biological mechanism with feedback controllers to obtain stable natural
gaits for legged robots. There are two commonly used models adopted by researchers for modeling
CPGs, which are the Matsuoka neural oscillator139 and the Van der Pol oscillator.60,147,149–151 We
will not discuss here the details of the mathematical models of CPGs; instead, we will mention some
merits and disadvantages of this method. The CPGs-based gait does not need the full knowledge of
dynamic biped model. In addition, a legged robot that adopts CPGs for generating walking patterns is
governed by the periodicity-based gait (see Section 4.2). Furthermore, CPGs exhibit natural human-
like motion with less energy consumption. However, there are some limitations included in the
actuator capability,149 and in the highly unstructured environments.113 For further reading, we refer
to Yang et al.152 and Ijspeert.153
5.2.2. Artificial intelligence-based gait. Artificial intelligence approaches include neural networks,
fuzzy logic (FL) and genetic algorithms (GA). In these approaches, the designer does not need a
precise dynamic model to solve target problem. Although AI-based approaches can work in specific
circumstances, they cannot create reliable walking patterns114 because there might be no explanations
of how onegets these interesting results.
Nonetheless, the difficulties encountered in the generation of biped walking patterns using model-
based methods can justify the use of AI-based approaches by many researchers. These difficulties
could include the resulting disturbances and noises of measurement devices and the possible
computational complexity of gait planning. Vundavilli and Pratihar56 made an interesting comparison
study between analytical and AI methods for the gait generation of a planar 7-link biped robot. First,
they solved the problem by using analytical methods, including inverse kinematics and static balance
to generate the reference trajectories of the biped joints during walking in different environments. Then
the same problem was solved by using NN- and FL-based approaches respectively. The databases of
NNs and FL have been optimized using GA off-line. The authors have proposed that the AI-based
approaches are accessible to be used online. For a detailed literature review of this interesting topic,
we refer to Katic and Vukobratovic.154

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 A review on walking pattern generators of biped robots 925

Table III. Comparison of the methods of the walking pattern generators for biped walking.152 .

The approach Advantages Disadvantages

Model-based gait 1. It provides explanations 1. It needs full knowledge of the dynamic

about the behavior of human model of biped robot.
walking. 2. The online walking algorithms require
2. It can adopt ZMP and large computations.
periodicity as stability 3. It rarely employs the natural dynamics of
criteria. Consequently, it can the biped system.
guarantee the dynamic 4. Information of the motion terrain should
stability of biped mechanism. be known.
Biological 1. It does not need precise 1. The limitations of CPG are widely
mechanism-based gait modeling of biped investigated in Yang et al.152
mechanism. 2. Significant results are obtained without
2. Online walking algorithms explanations.
could be applied easily
3. It is robust to disturbances.
4. It can guarantee the stability
of biped mechanism.
Natural dynamics-based 1. It employs the natural 1. No unified strategies could be adopted to
gait dynamics of the biped achieve the desired results.
system. 2. All strategies depend on the experiences
2. Less energy consumption of the designer.
could be produced.
3. It can produce natural
motion.

5.3. Natural dynamics-based gait

This type of gait does not need predefined reference trajectories for biped mechanism and it is
designed according to the natural dynamics of the biped. This strategy is used extensively to generate
the natural human-like passive dynamic walkers.155–157
The intuitive control achieves the walking patterns’ generation and the control of active biped
robots using the physics of the system and the virtual elements such as springs or dampers.158–162
Pieter van Zutven et al.49 proposed a strategy citing intuitive control for designing periodicity-based
gait. The authors used four controllers for this purpose. The first one called “controller symmetries”
enforces the biped robot to imitate the passive dynamic walker compensating the actual gravity force
of the biped system. The second controller prevents feet scuffing, while the third one adds some
compliance for the ankle joint. The last controller regulates the size of walking steps.
The advantage of the intuitive control base gait is that it is easy to apply and is a powerful method
so that the walking of the biped robot is characterized with natural appearance having less energy
consumption. However, there are no unified strategies that a designer can follow to achieve desired
results. Most of these strategies depend on designer’s experience.7,162

6. Conclusions
In this paper, we attempted to collect and introduce a systematic discussion on the modeling,
stability and walking patterns of biped robots. The subject is complicated to deal with, because it is
interdisciplinary. An important remark one should notice is that the designer should decide in advance
the stability criterion that he/she may use for generating walking patterns of biped mechanism. After
that he/she should select the suitable approach described in Fig. 4 for planning the motion of biped
locomotion. Finally, the controller structure should be selected. Table II illustrates with a sequential
history the most distinctive biped/humanoid robots manufactured in different countries with their
characteristics. Table III shows the advantages and disadvantages of the methods for the walking
pattern generators of biped robots.

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 926 A review on walking pattern generators of biped robots

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