1
A Compliant Hybrid Zero Dynamics Controller
for Stable, Efficient and Fast Bipedal Walking
on MABEL
Koushil Sreenath, Hae-Won Park, Ioannis Poulakakis, J. W. Grizzle
Abstract—The planar bipedal testbed MABEL contains
springs in its drivetrain for the purpose of enhancing
both energy efficiency and agility of dynamic locomotion.
While the potential energetic benefits of springs are well
documented in the literature, feedback control designs that
effectively realize this potential are lacking. In this paper,
we extend and apply the methods of virtual constraints
and hybrid zero dynamics, originally developed for rigid
robots with a single degree of underactuation, to MABEL,
a bipedal walker with a novel compliant transmission
and multiple degrees of underactuation. A time-invariant
feedback controller is designed such that the closed-loop
system respects the natural compliance of the open-loop
system and realizes exponentially stable walking gaits. Five
experiments are presented that highlight different aspects
of MABEL and the feedback design method, ranging from
basic elements such as stable walking and robustness under
perturbations, to energy efficiency and a walking speed of
1.5 m/s (3.4 mph). The experiments also compare two feed-
back implementations of the virtual constraints, one based
on PD control as in (Westervelt et al., 2004), and a second
that implements a full hybrid zero dynamics controller. On
MABEL, the full hybrid zero dynamics controller yields
a much more faithful realization of the desired virtual
constraints and was instrumental in achieving more rapid
walking.
Index Terms—Bipedal robots, Hybrid Systems, Zero
Dynamics, Compliance.
I. INTRODUCTION
MABEL is a novel bipedal testbed at the University
of Michigan. The robot is planar, with a torso, two
legs with revolute knees, and four actuators. Two of
its actuators are in series with large springs for the
purpose of enhancing both energy efficiency and agility
of locomotion. The actuators are housed in the torso
and the legs are light, placing the center of mass of the
robot significantly above the hips, as shown in Figure
1. A more detailed description of the robot has been
presented in (Hurst et al., 2007; Hurst and Rizzi, 2008;
Koushil Sreenath and J. W. Grizzle are with the Control Systems
Laboratory, Electrical Engineering and Computer Science Depart-
ment, University of Michigan, Ann Arbor, MI 48109-2122, USA.
{koushils, grizzle}@umich.edu
Hae-Won. Park is with the Mechanical Engineering Depart-
ment, University of Michigan, Ann Arbor, MI 48109-2125, USA.
parkhw@umich.edu
Ioannis Poulakakis is with the Mechanical and Aerospace Engineer-
ing Department, Princeton University, Princeton, NJ 08544-5263, USA.
poulakas@princeton.edu
This work is supported by NSF grants ECS-909300, ECS-0600869.
(a)
−q
Tor
q
LA
q
LS
Virtual
Compliant Leg
−θ
s
(b)
Fig. 1. (a) MABEL, an experimental testbed for bipedal locomotion.
The robot is planar, with a boom providing stabilization in the
frontal plane. The robot weighs 58 kg and is 1 m at the hip. The
robot’s drivetrain contains springs for enhanced power efficiency. (b)
The virtual compliant leg created by the drivetrain through a set
of differentials. The coordinate system used for the linkage is also
indicated. Angles are positive in the counter clockwise direction.
Hurst, 2008), and the identification of its dynamic model
is reported in (Park et al., 2010).
Bipedal robots that are simultaneously robustly sta-
ble, efficient, and fast are extremely rare. The desire
to achieve these traits is driving the introduction of
innovative mechanism designs and feedback control
methods. MABEL was designed to be both a robust
walker and a fast runner. It pushes the state of the art in
bipedal mechanism design and provides an opportunity
for effective control design methodology to maximize the
robot’s efficiency, speed and stability. This paper reports
analytical and experimental results for walking on flat
ground, a very important preliminary stage for running
on flat ground and for walking on uneven ground. In
particular, a Compliant Hybrid Zero Dynamics controller
(HZD) is designed and the HZD controller is experimen-
tally implemented to achieve stable, efficient, and fast
walking.
A. Background
The stability of a biped can be enhanced by introduc-
ing large feet, in which case relatively simple controller
designs can be used. Honda’s ASIMO (Sakagami et al.,
2
2002), Sony’s QRIO (Geppert, 2004), and the HRP
series (Kaneko et al., 2002) have large feet and use
zero moment point-based controllers (Vukobratovi
´
c and
Borovac, 2004) to achieve stable walking. The walking
gaits are flat-footed and the achieved energy efficiency
is low.
Enhanced agility has been demonstrated on hopper-
style robots (i.e., springy, prismatic leg) employing
intuitive controllers, as demonstrated in (Hodgins and
Raibert, 1990; Brown and Zeglin, 1998). These robots
are highly underactuated, though for the most part, their
control systems did not have to deal with stabilization
of significant torso dynamics; indeed, if a torso was
present, its center of mass was coincident with the hip
joint (Poulakakis and Grizzle, 2009b).
The bipedal robot RABBIT was planar, had revo-
lute knees, and a non-trivial torso (Chevallereau et al.,
2003). It was deliberately designed to have point feet
in order to inspire new analytical control approaches to
stabilizing periodic motion in underactuated mechanical
systems, and hence move beyond flat-footed walking
gaits. Research on RABBIT gave rise to the methods
of virtual constraints and hybrid zero dynamics (Grizzle
et al., 2001; Westervelt et al., 2002, 2003; Morris and
Grizzle, 2005; Westervelt et al., 2007), which provide
a systematic method of designing asymptotically sta-
ble walking controllers. A related approach based on
designing a linear feedback controller that stabilizes
the time-varying transverse linearization of a hybrid
system along a periodic orbit has been developed in
(Manchester et al., 2009; Shiriaev et al., 2005, 2010;
Song and
ˇ
Zefran, 2006). Other types of controllers to
achieve stable walking are based on machine learning
and neuronal control, as presented in (Russ Tedrake and
Seung, 2005) and (Manoonpong et al., 2007; Sabourin
et al., 2006), respectively.
The efficiency of bipedal robots is being enhanced by
using minimal actuation, incorporating compliance, or a
combination of the two. Motivated by passive dynamic
walkers which exhibit stable gaits on small downward
slopes, and where gravity compensates for energy losses
at leg impacts, researchers have devised efficient means
of walking on flat ground by injecting minimal amounts
of energy at key points in the gait (Collins et al.,
2005; Kuo, 2002). Another means of enhancing energy
efficiency is by introducing compliant elements. The
energetic benefits of springs in legged locomotion are
well documented (Alexander, 1990). Springs can be used
to store and release energy that otherwise would be lost
as actuators do negative work, and springs can be used
to isolate actuators from shocks arising from leg im-
pacts with the ground. Although these benefits are more
pronounced in running, compliance can also be used
beneficially in walking (Geyer et al., 2006; Iida et al.,
2007, 2008). Enhanced energy efficiency was shown us-
ing pneumatic artificial muscles in (Vanderborght et al.,
2008a,b; Takum et al., 2008), using springs in series with
motors in (Pratt and Pratt, 1998; Schaub et al., 2009),
and using springs in parallel with motors in (Yang et al.,
2008). A combination of both methods, minimalistic
actuation and compliant elements, is employed in the
Cornell Biped (Collins and Ruina, 2005), and the T.U.
Delft bipeds TUlip and Flame (Hobbelen et al., 2008)
in order to improve efficiency. The drawbacks of these
highly efficient walkers are that they cannot lift their legs
over obstacles, readily change speeds, or run.
The speed of a biped can be enhanced by careful
mechanism and control design as suggested in (Koech-
ling and Raibert, 1993), and demonstrated in robots such
as RunBot (Manoonpong et al., 2007).
MABEL achieves stability, efficiency and speed
through a combination of the novel design of its driv-
etrain and the analytical methods being developed to
control it. The robot’s drivetrain uses a set of differentials
to create a virtual prismatic leg between the hip and
the toe such that one actuator controls the angle of
the virtual leg with respect to the torso, and another
actuator controls its length. Moreover, the drivetrain
also introduces a compliant element, a unilateral spring
present in the transmission, that acts along the virtual
leg in series with the actuator controlling the leg length.
A controller that properly utilizes this natural compliant
dynamics will lead to an efficient gait. Further, with the
above mechanical design, it is possible to place all of the
actuators in the torso, thereby making the legs relatively
light and enabling rapid leg motion for fast gaits. More
details on the design philosophy are available in (Grizzle
et al., 2009; Hurst, 2008).
MIT’s Spring Flamingo achieved stable, efficient
and fast walking by employing series elastic actuators
(SEAs) and a virtual model controller (Pratt and Pratt,
1998; Pratt, 2000; Pratt et al., 2001). The virtual model
controller creates virtual components, such as springs,
dashpots, etc., through carefully computed joint torques.
This enables intuitive tuning of parameters of the con-
troller, though no formal stability results exist. The
spring in MABEL may seem similar to that in the SEA,
however the resemblance is only superficial. The SEA
is designed for force control and cannot store significant
amounts of energy. MABEL’s springs provide a revolute
instantiation of a spring-loaded prismatic (pogo-stick)
leg. They can easily absorb 150 J of energy (the equiva-
lent of dropping the robot from a height of 25 cm.) The
spring in the SEA is several orders of magnitude smaller
in size, and is used primarily for filtering and sensing of
external forces, rather than energy storage.
The presence of compliance in MABEL’s transmission
has led to new control challenges that cannot be met
with the initial theory developed for RABBIT. On the
mathematical side, compliance increases the degree of
underactuation, which in turn makes it more difficult to
meet the invariance condition required for a hybrid zero
dynamics to exist. This technical difficulty was overcome
in (Morris and Grizzle, 2009) with a technique called a
“deadbeat hybrid extension”.
A second challenge arising from compliance is how
3
to use it effectively. A first attempt in (Morris and
Grizzle, 2006) at designing a controller for a biped
with springs took advantage of the compliance along a
steady state walking gait, but “fought it” during tran-
sients; the compliance was effectively canceled in the
HZD (for details, see (Poulakakis and Grizzle, 2009b,
p. 1790)). The problem of ensuring that the feedback
action preserves the compliant nature of the system even
during transients was studied in (Poulakakis and Grizzle,
2009b,a; Poulakakis, 2008) for the task of hopping in
a monopod, where the HZD itself was designed to be
compliant.
B. Contributions
The key results of the paper are summarized next.
Firstly, a HZD-based controller is designed for walking
such that the natural compliant dynamics is preserved
in the closed-loop system (robot plus controller). This
ensures that the designed walking gait uses the compli-
ance to do negative work at impact, instead of it being
done by the actuators, thereby improving the energy
efficiency of walking. Stability analysis using the method
of Poincar
´
e is then carried out to check stability of the
closed-loop system. Prior to experimentally testing the
controller, simulations with various model perturbations
are performed to establish robustness of the designed
controller. The controller is then experimentally vali-
dated on MABEL.
Secondly, walking gaits are designed to optimize the
energetic cost of mechanical transport (Collins et al.,
2005; Collins and Ruina, 2005). This results in a gait that
is more than twice as efficient on the testbed than a gait
that we had designed by hand and reported in (Grizzle
et al., 2009). The resulting cost of mechanical transport is
approximately three times more efficient than RABBIT,
and 12 times better than Honda’s ASIMO, even though
MABEL does not have feet. This puts MABEL’s energy
efficiency within a factor of two of T.U. Delft’s Denise
and a factor of three of the Cornell Biped, none of which
can step over obstacles or run; it is also within a factor
of two of the MIT Spring Flamingo which can easily
step over obstacles but cannot run, and within a factor
of three of humans, who can do all of the above.
Thirdly, in preparation for future running experiments,
we turn our attention to fast walking, where each single
support phase may be on the order of 300 to 350 ms.
Very precise control is needed for accurately implement-
ing the virtual constraints of an HZD controller with
these gait times. All experimental implementations of
the virtual constraints reported to date have relied on
local PD controllers (Westervelt et al., 2004). The zero
dynamics controllers provide great tracking accuracy in
theory, but are often criticized for being overly dependent
on high model accuracy, and for being too complex to
implement in real-time. Here we demonstrate, for the
first time, an experimental implementation of a compli-
ant HZD controller. The tracking accuracy attained is far
better than the simple PD controllers used earlier.
Finally, we attack the problem of achieving fast walk-
ing. With a zero dynamics controller, we experimentally
attain a top sustained walking speed of 1.5 m/s (3.4
mph.)
The remainder of the paper is organized as fol-
lows. Section II describes the general features of MA-
BEL’s morphology, and presents the mathematical hy-
brid model used for walking. Section III provides the
systematic procedure based on virtual constraints that
is used to design a suite of walking gaits. Section IV
presents the design of two controllers to realize the gaits
and studies the stability of the fixed points under the
action of the proposed controllers. Section V describes
the experiments performed to demonstrate the validity
of the designed controllers. Section VI discusses various
aspects of the robot and the feedback controllers re-
vealed by the experiments. Finally, Section VII provides
concluding remarks and briefly discusses future research
plans.
II. MABEL TESTBED
This section presents details about the morphology
of MABEL, and develops the appropriate mathematical
models for the study of walking.
A. Description of MABEL
MABEL is a planar bipedal robot comprised of five
links assembled to form a torso and two legs with knees;
see Figure 1. The robot weighs 58 kg, is 1 m at the hip,
and mounted on a boom of radius 2.25 m. The legs are
terminated in point feet. All actuators are located in the
torso, so that the legs are kept as light as possible; this
is to facilitate rapid leg swinging for running. Unlike
most bipedal robots, the actuated degrees of freedom of
each leg do not correspond to the knee and hip angles.
Instead, for each leg, a collection of cable-differentials
is used to connect two motors to the hip and knee joints
in such a way that one motor controls the angle of the
virtual leg consisting of the line connecting the hip to
the toe, and the second motor is connected in series with
a spring in order to control the length or shape of the
virtual leg; see Figure 2. The reader is referred to (Park
et al., 2010; Grizzle et al., 2009; Hurst, 2008) for more
details on the transmission.
The springs in MABEL serve to isolate the reflected
rotor inertia of the leg-shape motors from the impact
forces at leg touchdown and to store energy in the
compression phase of a running gait, when the support
leg must decelerate the downward motion of the robot’s
center of mass; the energy stored in the spring can
then be used to redirect the center of mass upwards
for the subsequent flight phase, when both legs are
off the ground. These properties (shock isolation and
energy storage) enhance the energy efficiency of running
and reduce the overall actuator power requirements.
This is also true for walking as we will demonstrate
experimentally. MABEL has a unilateral spring which
4
Bspring
q
mLS
leg-shape
motor
torso
leg-angle
motor
q
Thigh
q
Shin
q
LS
−
=
q
Thigh
q
Shin
2
=
q
LA
q
Thigh
q
Shin
2
+
Bspring
spring
q
Bsp
Fig. 2. MABEL’s powertrain (same for each leg), all housed in the
torso. Two motors and a spring are connected to the traditional hip
and knee joints via three differentials. On the robot, the differentials
are realized via cables and pulleys (Hurst, 2008) and not via gears.
They are connected such that the actuated variables are leg angle and
leg shape, see Figure 1, and so that the spring is in series with the leg
shape motor. The base of the spring is grounded to the torso and the
other end is connected to the B
spring
differential via a cable, which
makes the spring unilateral. When the spring reaches its rest length,
the pulley hits a hard stop, formed by a very stiff damper. When this
happens, the leg shape motor is, for all intents and purposes, rigidly
connected to leg shape through a gear ratio.
compresses but does not extend beyond its rest length.
This ensures that springs are present when they are useful
for shock attenuation and energy storage, and absent
when they would be a hindrance for lifting the legs from
the ground.
B. Mathematical Model
A hybrid model appropriate for a walking gait, com-
prised of a continuous single support phase and an
instantaneous double support phase, is developed next.
The impact model at double support is based on (Hur-
muzlu and Marghitu, 1994). The single support model
is a pinned, planar, 5-link kinematic chain with revo-
lute joints and rigid links. Because the compliance is
unilateral, it will be more convenient to model it as an
external force when computing the Lagrangian, instead
of including it as part of the potential energy.
1) MABEL’s Unconstrained Dynamics: The config-
uration space Q
e
of the unconstrained dynamics of
MABEL is a simply-connected subset of S
7
× R
2
: five
DOF are associated with the links in the robot’s body,
two DOF are associated with the springs in series with
the two leg-shape motors, and two DOF are associated
with the horizontal and vertical position of the robot
in the sagittal plane. A set of coordinates suitable for
parametrization of the robot’s linkage and transmission
is q
e
:= ( q
LA
st
; q
mLS
st
; q
Bsp
st
; q
LA
sw
; q
mLS
sw
; q
Bsp
sw
;
q
Tor
; p
h
hip
; p
v
hip
), the subscripts st and sw refer to the
stance and swing legs respectively. As in Figure 1 and
Figure 2, q
Tor
is the torso angle, and q
LA
st
, q
mLS
st
,
and q
Bsp
st
are the leg angle, leg-shape motor position,
and B
spring
position, respectively for the stance leg.
The swing leg variables, q
LA
sw
, q
mLS
sw
and q
Bsp
sw
are
defined similarly. For each leg, q
LS
is determined from
q
mLS
and q
Bsp
by
q
LS
= 0.0318q
mLS
+ 0.193q
Bsp
. (1)
This reflects the fact that the cable differentials place
the spring in series with the motor, with the pulleys
introducing a gear ratio. The coordinates p
h
hip
, p
v
hip
are
the horizontal and vertical positions of the hip in the
sagittal plane. The hip position is chosen as an indepen-
dent coordinate instead of the center of mass because
it was observed that this choice significantly reduces
the number of terms in the symbolic expressions for the
dynamics.
The equations of motion are obtained using the
method of Lagrange. The Lagrangian for the uncon-
strained system, L
e
: T Q
e
→ R is defined by
L
e
= K
e
− V
e
, (2)
where, K
e
: T Q
e
→ R and V
e
: Q
e
→ R are the
total kinetic and potential energies of the mechanism,
respectively. The total kinetic energy is obtained by
summing the kinetic energy of the linkage, K
link
e
, the
kinetic energy of the stance and swing leg transmissions,
K
trans
st
e
, K
trans
sw
e
, and the kinetic energy of the boom,
K
boom
e
,
K
e
(q
e
, ˙q
e
) = K
link
e
(q
e
, ˙q
e
) + K
trans
st
e
(q
e
, ˙q
e
) +
K
trans
sw
e
(q
e
, ˙q
e
) + K
boom
e
(q
e
, ˙q
e
) .
(3)
The linkage model is standard. Physically, the boom
constrains the robot to move on the surface of a sphere,
and a full 3D model would be required to accurately
model the robot and boom system. However, we assume
the motion to be planar and, as in (Westervelt, 2003,
p. 94), only consider the effects due to mass and inertia
of the boom. This will introduce some discrepancies be-
tween simulation and experimental results. The symbolic
expressions for the transmission model are available
online at (Grizzle, 2010b).
Similar notation is used for the potential energy,
V
e
(q
e
) = V
link
e
(q
e
) + V
trans
st
e
(q
e
) +
V
trans
sw
e
(q
e
) + V
boom
e
.
(4)
Due to its unilateral nature, the spring is not included in
the potential energy of the transmission; only the mass of
the motors and pulleys is included. The unilateral spring
is considered as an external input to the system.
With the above considerations, the unconstrained
robot dynamics can be determined through Lagrange’s
equations
d
dt
∂L
e
∂ ˙q
e
−
∂L
e
∂q
e
= Γ
e
, (5)
5
where, Γ
e
is the vector of generalized forces acting on
the robot and can be written as,
Γ
e
= B
e
u + E
ext
(q
e
) F
ext
+
B
fric
τ
fric
(q
e
, ˙q
e
) + B
sp
τ
sp
(q
e
, ˙q
e
) ,
(6)
where the matrices B
e
, E
ext
, B
fric
, and B
sp
are derived
from the principle of virtual work and define how the
actuator torques u, the external forces F
ext
at the leg,
the joint friction forces τ
fric
, and the spring torques τ
sp
enter the model, respectively.
Applying Lagrange’s equations (5), with the kinetic
and potential energies defined by (3) and (4), respec-
tively, results in the second-order dynamical model
D
e
(q
e
) ¨q
e
+ C
e
(q
e
, ˙q
e
) ˙q
e
+ G
e
(q
e
) = Γ
e
(7)
for the unconstrained dynamics of MABEL. Here D
e
is
the inertia matrix, the matrix C
e
contains Coriolis and
centrifugal terms, and G
e
is the gravity vector.
2) Dynamics of Stance: For modeling the stance
phase, the stance toe is assumed to act as a passive
pivot joint (no slip, no rebound and no actuation).
Hence, the Cartesian position of the hip,
p
h
hip
, p
v
hip
,
is defined by the coordinates of the stance leg and
torso. The springs in the transmission are appropri-
ately chosen to support the entire weight of the robot,
and hence are stiff. Consequently, it is assumed that
the spring on the swing leg does not deflect, that is,
q
Bsp
sw
≡ 0. It follows from (1) that q
mLS
sw
and
q
LS
sw
are related by a gear ratio; q
mLS
sw
is taken as
the independent variable. With these assumptions, the
generalized configuration variables in stance are taken
as q
s
:=
q
LA
st
; q
mLS
st
; q
Bsp
st
; q
LA
sw
; q
mLS
sw
; q
Tor
.
The stance dynamics is obtained by applying the
above holonomic constraints to the model of Section
II-B1. The stance configuration space is therefore a
co-dimension three submanifold of Q
e
, i.e., Q
s
:=
q
e
∈ Q
e
| q
Bsp
sw
≡ 0, p
h
toe
st
≡ 0, p
v
toe
st
≡ 0
. For later
use, we denote by
q
e
= Υ
s
(q
s
) (8)
the value of q
e
when q
s
∈ Q
s
, and by
q
s
= Π
s
(q
e
) (9)
the value of q
e
projected onto Q
s
⊂ Q
e
, such that, Π
s
◦
Υ
s
= id.
The resulting Lagrangian L
s
: T Q
s
→ R can be
expressed as
L
s
:= L
e
(q
e
, ˙q
e
) |
{
q
Bsp
sw
≡0,p
h
toe
st
≡0,p
v
toe
st
≡0
}
, (10)
and the dynamics of stance are obtained through La-
grange’s equations, expressed in standard form as
D
s
(q
s
) ¨q
s
+ C
s
(q
s
, ˙q
s
) ˙q
s
+ G
s
(q
s
) = Γ
s
, (11)
where, Γ
s
:= B
s
u + B
fric
τ
fric
(q
s
, ˙q
s
) + B
sp
τ
sp
(q
s
, ˙q
s
)
is the vector of generalized forces acting on the robot.
The state-space form of the stance dynamics, with the
state vector x
s
:= (q
s
; ˙q
s
) ∈ T Q
s
, can be expressed as,
˙x
s
:=
˙q
s
¨q
s
=
˙q
s
−D
−1
s
H
s
+
0
D
−1
s
B
s
u
=: f
s
(x
s
) + g
s
(x
s
)u,
(12)
where, f
s
, g
s
are the drift and input vector fields for the
stance dynamics, and H
s
:= C
s
(q
s
, ˙q
s
) ˙q
s
+ G
s
(q
s
) −
B
fric
τ
fric
(q
s
, ˙q
s
) − B
sp
τ
sp
(q
s
, ˙q
s
).
3) Stance to Stance Transition Map: An impact oc-
curs when the swing leg touches the ground, modeled
here as an inelastic contact between two rigid bodies.
In addition to modeling the impact of the leg with
the ground and the associated discontinuity in the gen-
eralized velocities of the robot as in (Hurmuzlu and
Marghitu, 1994), the transition map accounts for the
assumption that the spring on the swing leg is at its
rest length, and for the relabeling of robot’s coordi-
nates so that only one stance model is necessary. In
particular, the transition map consists of three subphases
executed in the following order: (a) standard rigid impact
model (Hurmuzlu and Marghitu, 1994); (b) adjustment
of spring rest length in the new swing leg; and (c)
coordinate relabeling.
Before entering into the details, the spring is dis-
cussed. To meet our modeling assumption of Section
II-B2, the post-transition spring position on the new
swing leg has to be non-deflected. This requirement
makes the pre and post-transition position coordinates
not identical. Physically, the spring being non-deflected
is a well-founded assumption because as soon as weight
of the robot comes off the former stance leg, the spring
rapidly relaxes and the pulley q
Bsp
comes to rest on the
hard stop. This causes a change in torque on the leg-
shape motor, and either the motor shaft or the leg shape
needs to reposition to maintain a balance of torques in
the leg shape differentials. Because the leg shape has a
high reflected inertia at the motor, it is the motor that
repositions. Further, since q
LS
is a linear combination
of q
mLS
and q
Bsp
per (1), we can assume the spring
and motor position change appropriately such that the
linkage positions q
+
LS
, q
−
LS
are still identical. Thus, the
pre and post-transition linkage coordinates still remain
identical.
The robot physically transitions from one stance phase
to the next when the swing toe contacts the ground. It
is assumed that there is no rebound or slip at impact,
and that the old stance leg lifts off from the ground
without interaction. The external forces are represented
by impulses, and since the actuators cannot generate
impulses, they are ignored during impact. Mathemati-
cally, the transition then occurs when the solution of
(12) intersects the co-dimension one switching manifold
S
s→s
:=
x
s
∈ T Q
s
| p
v
toe
sw
= 0
. (13)
The stance to stance transition map, ∆
s→s
: S
s→s
→
