Actuated dynamic walking in biped robots: Control approach, robot design and experimental validation

Abstract

This paper presents and experimentally demonstrates a control approach for actuated dynamic walking in biped robots. Rather than utilizing trajectory tracking, we proposed a control approach which employs state dependent control torques generated by low-gain spring-damper couples to encourage patterned motion of the robot. In order to verify that the control idea provides natural looking (human-like) dynamic walking, a seven-link biped robot was designed with backdrivable joint actuators, which allows passive leg motion. Following a discussion on robot design and the real-time control implementation, experimental data is presented to validate the proposed methodology.

Full text

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Actuated Dynamic Walking in Biped Robots: Control Approach, Robot
Design and Experimental Validation
David J. Braun Member, IEEE, Jason E. Mitchell and Michael Goldfarb, Member, IEEE
Abstract—This paper presents and experimentally demon-
strates a control approach for actuated dynamic walking in
biped robots. Rather than utilizing trajectory tracking, we
proposed a control approach which employs state dependent
control torques generated by low-gain spring-damper couples
to encourage patterned motion of the robot. In order to verify
that the control idea provides natural looking (human-like)
dynamic walking, a seven-link biped robot was designed with
backdrivable joint actuators, which allows passive leg motion.
Following a discussion on robot design and the real-time control
implementation, experimental data is presented to validate the
proposed methodology.
Index Terms—Biped, dynamic walking.
I. INTRODUCTION
I
are proposed in the literature, see Hurmuzlu et al. [1] for a
review, most of the control methods either utilize the zero
moment point (ZMP) paradigm, or rely on the (passive)
dynamic walking principle.
The ZMP approach introduced by Vukobratovi´ c et al. [2],
[3], [4] and applied or further developed by numerous authors
[5], [6], [7], [8], [9], [10], is a well established method
for biped locomotion synthesis. Application of this method
has been shown to provide effective, robust, and versatile
locomotion for biped robots, [11], [6]. As was pointed out
in the literature however, utilization of trajectory tracking,
relying on a characteristic bent knee stance support, and
preferring a flat foot restriction, while walking, makes the
ZMP-based walking control distinct from the one generally
employed by humans, see [12], [13].
In order to achieve an efficient and natural-looking bipedal
gait, several researchers have investigated a “dynamic walk-
ing principle” that emphasizes the significance of the passive
(natural) dynamics of the robot through walking. On one end
of this spectrum are fully passive dynamic walkers which rely
on specific geometries and precisely tuned inertial robot de-
sign, and can walk on a slight downward slope powered only
by gravity, [14], [15]. The same idea motivated development
of actuator-assisted dynamic walkers (that utilizes a passive
robot design) which have shown to posses human-like and
energy efficient gait [16], [17], [18].
Actuated dynamic walking approaches which do not uti-
lize the ZMP method (neither rely on a passive or a nearly
passive robot design), have also been proposed in literature,
see [19], [20]. In this context, a concept of “hybrid zero
N recent years there has been considerable research effort
devoted to bipedal locomotion. While various approaches
The authors are with the Department of Mechanical Engineering, Van-
derbilt University, Nashville, TN 37235 USA (david.braun@vanderbilt.edu;
jason.mitchell@vanderbilt.edu; michael.goldfarb@vanderbilt.edu).
dynamics” [21] and “virtual control constraints” [22] were
used to develop and experimentally verify a walking control
approach [23].
In order to synthesize natural looking compliant dynamic
walking (without using considerable energy), backdrivable
actuation has been recognized as important. In this light, Pratt
et al. [24] have utilized “virtual model control” and “series-
elastic actuators” to allow practical control realization which
may not dictate, but (depending on the choice of control
parameters) rather exploits the natural dynamics of the biped.
There are two main preconditions which allow natural-
looking and energy-efficient realization of actuated dynamic
walking. The first, related to the control approach, precludes
enforcing a predefined reference trajectory and may also
not favor enforcing state dependent kinematic constraints, or
other attributes of the walking cycle (such as stride length,
stepping frequency of average forward speed) with high gain
control. This condition motivated us to develop a control
framework which utilizes state-dependent control torques
(generated by low-gain spring-damper couples) to provide
motion coordination without prespecifying the response of
the system. The second precondition (not related to control)
depends on joint actuation which should not suppress passive
joint motion (i.e., joints should be highly backdrivable such
as human joints). Utilization of backdrivable joint design
allows the inertial motion of the robot to be exploited through
walking rather then being suppressed by the actuation units.
Accordingly, to validate the control approach, a 7-link biped
was designed with backdrivable joints which meets the
second precondition (defined above) required for realization
of actuated dynamic walking.
In the remainder of this paper, we describe our con-
trol approach, discuss the design of the seven link biped
robot, comment on the real-time control implementation,
and present data from walking experiments to demonstrate
feasibility of the presented method.
II. MODEL OF THE BIPED
In order to facilitate the controller development, we derive
the dynamic model of the biped illustrated in Fig.1. The
configuration of the biped is defined with nine coordinates,
q = [x,y,θ,θ1,θ2,θ3,θ4,θ5,θ6]T, where the first two coor-
dinates characterize the translational motion of the robot in
the inertial frame while the last seven angular coordinates
reference the orientation of the links with respect to the
inertial frame. The biped is considered actuated at each joint
(i.e., right and left hip, knee, and ankle joints), such that, the
dynamics of the robot are affected by six actuator torques,

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u = [u1,u2,u3,u4,u5,u6]T, which are considered positive
in the same (counterclockwise) direction as the joint angles.
In order to support the forthcoming discussion, we will also
define the joint angles (relative angles between the links) as:
ϕ = [ϕ1,ϕ2,ϕ3,ϕ4,ϕ5,ϕ6]T= [θ1− θ,θ2− θ1,θ3− θ2+
π/2,θ4− θ,θ5− θ4,θ6− θ5+ π/2]T.
Fig. 1. Seven-link biped with the absolute coordinates q, and the associated
geometric and inertial parameters of the robot and the actuator units which
generate the control torques u. The Cartesian coordinates (xi,yi), i ∈
{1,2,3,4}, represent the position of the toe and the heel for the left and
right leg.
As follows, we derive the mathematical model by consid-
ering the biped as a constrained mechanical system, [25],
[26], [27]. This model contains the differential equations of
the flight phase motion, and the (algebraic and differential)
relations which define the kinematic (physical) constraints
along the motion. The model provides necessary information
for the closed-loop control design.
1) Unconstrained Dynamics: The equations of motion for
the 9-DoF (unconstrained) “flying” biped, can be written as
M(q)¨ q + h(q, ˙ q) + G(q) = Qu,
where M ∈ ?9×9is a symmetric and positive definite mass
matrix, h ∈ ?9represents the normal and Coriolis inertial
forces, G ∈ ?9represents the gravity terms, while Qu =
Eu ∈ ?9is a generalized actuator force, calculated using
a (constant) matrix E ∈ ?9×6which defines the actuator
arrangement on the robot.
2) Kinematic Constraints: For the biped in Fig.1, neither
foot can penetrate the ground, the knee joints cannot extend
beyond the fully straight position, and both feet are assumed
not to slide when in contact with the ground. Since each toe
and heel are independently characterized by non-penetration
and no-slip with the ground, the flight phase dynamics
(1) can be subject to the following kinematic (physical)
constraints,
⎡
y3
y4
ϕ2
ϕ5
(1)
Φh(q) =
⎢⎢⎢⎢⎢
⎣
y1
y2
⎤
⎥⎥⎥⎥⎥
⎦
= 0,Φn(q, ˙ q) =
⎡
⎣
⎢
˙ x1
˙ x2
˙ x3
˙ x4
⎤
⎥
⎦ = 0,
(2)
where (xi,yi), i ∈ {1,2,3,4} are the toe and heel coor-
dinates while ϕ2and ϕ5are the relative angles at the knee
joint, see Fig.1. Instead of using (2) directly, the forthcoming
control development only requires the information contained
in the constraint matrix which is defined as,
A = [(∂Φh/∂q)T,(∂Φn/∂˙ q)T]T.
(3)
Depending on the configuration of the robot, the constraints
in (2) and (3) are active when they restrict the motion, and
inactive when they do not influence the dynamics. In order to
ensure that A only contains the active constraints, the con-
figuration of the robot is monitored to identify and eliminate
the inactive constraints by zeroing the corresponding row in
(3). The constraint matrix obtained in this way carries the
kinematic information from the configuration of the biped
utilized in the forthcoming control development.
III. CONTROL APPROACH
In this paper, the control methodology recently introduced
for actuated dynamic walking in biped robots, [26], [27],
is utilized. In the following, we briefly recall the main
idea, and discuss the necessary ingredients for real-time
implementation of the closed-loop walking controller.
The control approach considered here can be discussed
in two stages. In the first stage, the robot is equipped with
generalized control forces Qd which are (partially) refer-
enced to the inertial frame to make generation of patterned
movement intuitive. On a real robot however there is no
associated control actuator which can realize the generalized
control forces (referenced to the inertial frame) directly.
Accordingly, in the second stage, Qd is recomputed to
equivalent joint torques, u, which are directly commanded
through the robot joint actuators. As follows, we provide a
systematic description of the outlined control idea on a seven
link robot.
A. Generalized Control Forces
In order to generate patterned movement without any
intent to utilize trajectory tracking, the seven link robot
is provided with seven control elements which are spring-
damper couples with fixed equilibrium points, see Fig1.
Each control element can be characterized with three control
parameters, a stiffness constant, a damping constant, and an
equilibrium angle. These parameters are changed as piece-
wise constant functions through four separate states along
the walk using a configuration-based switching controller.
1) Computing the Generalized Control Forces: For a
given set of control parameters, the desired generalized
control force is computed as
Qd= −Kd(φ − φd) − Bd˙φ,
(4)
where φ = [θ,θ1,ϕ2,ϕ3,θ4,ϕ5,ϕ6]Tis obtained by posi-
tion feedback, ˙φ is known from a corresponding velocity
feedback, while the control parameters concatenated in the
stiffness matrix, Kd, damping matrix Bdand the equilibrium
angles φd = [θd,θd1,0,ϕd3,θd4,0,ϕd6]Tare assigned by

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Fig. 2.The control elements and the control parameters on a 7-link robot.
the configuration-based switching controller as discussed in
the forthcoming subsection.
While (4) has the same form as a usual PD control law,
application of (4) in the present context is entirely different.
Specifically, we utilize piecewise constant (fixed) equilibrium
angles φdalong the motion of the robot instead of tracking
a predefined desired trajectory φd = φd(t). While this
difference may not seem significant, one can recognize that
it precludes high gains to be used by definition. Practically,
high gains, with fixed angular references as depicted in Fig.2,
would destabilize the robot instead of providing a good
tracking performance usually required in trajectory tracking
approaches.
2) Parameter Modulation based on the Robot Configu-
ration: In order to achieve a walking motion, the control
parameters are selected depending on the configuration of the
robot. In this light, we define four separate states for each
leg depending on whether the toe and/or the heel touches
the ground and whether the leg is fully extended at the knee
joint, see Fig.3.
Fig. 3.
states. The particular state-flow, S1 → S2 → S3 → S4 → S1..., together
with the switching events ,which correspond to normal walking, is indicated
with dashed lines.
The configuration-based switching logic with the four separate
Utilizing the control elements which partially reference
the inertial frame supports intuitive parameter tuning. As
follows, we describe a biologically inspired approach for
the parameter selection with the intention to mimic muscle
activation of walking human subjects, [28], [29].
While walking, one of the primary objectives is to keep
the upper body in a upright vertical position. Utilizing the
control element which acts between the body and the inertial
reference frame one can set the stiffness parameter and the
equilibrium angle to provide a near upright position for the
upper body. A similar idea can be used to generate leg
oscillation (with respect to a fixed inertial reference) by using
the control elements attached to the thigh. Specifically, in
swing, low stiffness and low damping elements pull the leg
towards a fixed hip flexion configuration (specified with an
equilibrium angle), while in stance, a higher stiffness and
higher damping is assigned to the same control element,
which encourages the stance leg to move towards a fixed
hip extension angular configuration. The knee stiffness and
damping are also modulated, by means of relatively high
values in stance (to support the body with the help of
the knee stop), and by employing only slight damping to
generate nearly ballistic swing. The ankle is controlled by
mimicking the strategy taken by humans. Accordingly, the
ankle stiffness is used to accumulate elastic energy from the
middle stance and provide a characteristic push-off at late
stance. The main control parameter at the swinging ankle is
an equilibrium angle which should be adjusted to provide
slight dorsiflexion to avoid stumbling and/or scuffing.
B. Actuator Torques
While Qdis straightforward to define, it does not represent
the joint torques, and as such, it cannot be directly used
to coordinate the motion of the robot. Practically, one may
want to find the joint (motor) torques u which, while directly
commanded through the actuators, provide the same motion
the robot would have by applying the desired generalized
control forces Qd. In order to compute these torques, we
propose a closed form analytical relation between Qdand u
as
u = (NR−TE)+NR−TQd,
(5)
where R is the upper triangular Cholesky factorization of
the mass matrix, N = I − (AR−1)+(AR−1) is the null-
space projection operator of the inertially-weighted con-
straint matrix, while E is a constant matrix which specifies
the actuator arrangement on the robot. This control solution
is derived with a primary objective to minimize the “accel-
eration energy” (¨ q − ¨ qd)TM(¨ q − ¨ qd) between the motion
generated by the desired generalized control forces ¨ qd and
the motion controlled by the actuator torques ¨ q, and also
to provide a minimum joint torque solution (minimize the
squared Euclidean norm of the control torques) when the
primary objective does not specify u uniquely, see [26], [27].
Following the section on robot design, we will address the
real-time control implementation where computation of (5)
is further discussed.
IV. ROBOT DESIGN
The 7-link biped robot, shown in Fig.4, is an experimen-
tal prototype which is 1.2m tall and weighs 14.3kg. The

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geometric parameters and mass distribution on the robot are
specified in Table.I. Below, we discuss the robot design,
including a description of the modular joint design, foot
design, the upper body and the sensory-system on the robot.
Fig. 4.
the Vanderbilt University - Center for Intelligent Mechatronics
Experimental prototype of a 7-link dynamic walker developed at
TABLE I
GEOMETRIC AND INERTIAL PARAMETERS OF THE ROBOT WITH TOTAL
MASS OF M = 14.3kg AND HEIGHT OF L = 1.2m.
Structure no. (∗)
−
1
2
3
l∗
[m]
0.390
0.295
0.298
0.183
lc∗
[m]
0.185
0.147
0.14
−
m∗
[kg]
6.12
0.67
0.55
0.36
Jc∗
[kgm2]
0.021
0.0096
0.0069
0.0007
Body
Thigh
Shank
Foot
Foot
a[m]
0.137
b[m]
0.046
h[m]
0.055
cx[m]
0.014
cy[m]
0.035
Actuators
Hip
Knee
Ankle
no. (∗)
1
2
3
n∗
21
12
21
ma∗[kg]
0.84
0.84
0.84
Ja∗[kgm2]
0.0067
0.0022
0.0067
A. Upper Body
The robot has a characteristic upper body which carries a
4.54kg, (10lb), mass located 0.2m from the hip joints, see
Fig.4. The body is provided with a single-axis gyroscope
(Analog Devices, ADXR150 with ±150o/s measurement
range) which directly measures the angular velocity of the
body (in the sagittal plane). To reduce the noise level, the
analog signal provided by the gyroscope is filtered with a
first order (50Hz cutoff frequency) filter. The gyroscope
is the only inertial sensor used in the proposed control
implementation.
B. Joint Design
The seven link robot has an upper body, hip, knee, ankle
and human-like foot. In the current solution, a unified design
principle is utilized for the hip, knee and ankle joints with
slight modifications made for differences in range of motion
and attachment points. Fig.5 depicts the specific design
solution of the knee joint. The robot is actuated with six
150W brushed DC-motors (Maxon RE40) through low gear
ratio planetary reducers (Maxon GP42C), specifically 21 : 1,
12 : 1, 21 : 1 for the hips, knees and ankles respectively.
A low gear ratio drive allows passive motion of the joints
which is a precondition to leverage natural dynamics through
actuated dynamic walking in biped robots. Each joint is
provided with an incremental quadrature encoder (Maxon,
ENC-MR-L-1024CPT) attached to the motor shaft. The
reference position for each of the six encoders is identified
(in a static stance phase during initialization) using two
acceleration sensors located on the upper body and the upper
right leg.
Fig. 5.Knee joint on the robot.
C. Foot Design and the Foot Sensors
The foot of the robot is constructed from ABS plastic, each
of which is instrumented with four force sensing resistors
(Interlink, 402 FSR), specifically, two FSR’s on each toe
and heel. These sensors are located between the body of
the foot and a thin foot-plate made from spring steel, Fig.6.
Whenever the toe and/or the heel touches the ground, the
circular rubber touch-pad (located on the foot-plate) touches
the foot sensors. The corresponding signal is relayed to
generate an on/off switch type output. This output serves
to identify the contact configuration between the foot and
the environment. Near to the toe and the heel (which are
the expected contact areas) the foot-plate is supplemented
with silicon rubber pads with appropriate shock absorption
capability.
Fig. 6.The ankle joint and the foot of the robot.

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D. Comment on Planar Walking
For the purpose of an experimental verification the robot
is attached through its hip to a lever arm which keeps the
biped on a circular path, with a 1.6m radius while walking.
This solution although not an ideal realization of a sagittal
plane walk, was also utilized by [24] and [23] as a convenient
platform for experimentation.
V. REAL-TIME CONTROL IMPLEMENTATION
The proposed closed-loop control approach is developed
on a desktop PC with the real-time interface provided by
MATLAB/Simulink Real Time Workshop. As follows, we
describe implementation of the closed-loop control.
A. Feedback Information
The feedback information for the closed-loop control is
provided with the contact foot sensors (FSR’s), angular joint
sensors (encoders) and attitude body sensor (gyroscope).
The foot sensors are used to identify the active (and
inactive) constraints along the motion, imposed by foot
contact with the ground. If the contact configuration is
known, using the encoder measurements on the joint angles
ϕ and the gyroscope on the upper body, the orientation of the
body θ can calculated (or precisely estimated). Specifically,
whenever the robot is not underactuated the absolute angular
orientation for the upper body θ is calculated using the
six (encoder) measurements ϕ. Otherwise, the integrated
gyroscope signal is used to provide an estimate of the ab-
solute orientation of the upper body. Note that underactuated
configurations in which case the gyroscope signal is required
are expected to be short, which mitigates drift and phase lag
issues with the gyroscope. Once θ is computed, the absolute
angular orientation for all the links can be obtained and the
control torque computation can be started.
B. Computing the Actuator Torques, u
The real-time implementation of (5) is developed utiliz-
ing standard Matlab code with Simulink/Embedded Matlab
Functions. Using the model parameters E, A and M, u
are calculated from (5) in real-time based on two standard
linear algebra routines: the Cholesky factorization, and the
pseudoinversion, [30], [31]. Although pseudoinversion is a
relatively expensive operation, the complete control approach
is shown to be real-time capable with 1000Hz sampling rate
on an Intel Core 2 Quad 2.4Ghz, desktop computer.
VI. DYNAMIC WALKING EXPERIMENT
The control approach described is applied to the robot. The
experimental data from the conducted walking experiment
is depicted on Fig.7. An extracted frame sequence which
represents two subsequent steps is depicted on Fig.8. The
walking is characterized with: average step length of Lstep≈
0.5m, average stepping frequency of fstep≈ 1Hz, average
forward speed of vavg ≈ 0.5m/s (Froude number Fr =
vavg/?gLleg ≈ 0.2), specific mechanical cost of transport
six times higher than human efficiency (also reproduced by
cmt≈ 0.32. Comparatively,this efficiency is (approximately)
the Cornell dynamic walker), while it is five times lower
than the value estimated for the Asimo robot, [16]. During
the walking experiment, the biped utilized a characteristic
ankle push-off, preferred by humans. The experiment also
verified proper coordination of the robot through short (0.1s)
underactuated motion phases (when only the forward heel
was on the ground).
VII. CONCLUSION
The authors have proposed an approach for the control of
biped walking that enables dynamic walking in a fully actu-
ated biped robot. Rather than prescribe kinematic trajectories,
state dependent control torques are utilized that “encourage”
patterned movement through the natural dynamics of the
biped. Validation of the control method is performed on a
seven link biped robot developed at the Vanderbilt University
- Center for Intelligent Mechatronics. Experimental results
are provided to demonstrate dynamic walking generated with
the proposed control methodology.
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