Capture Point: A Step toward Humanoid Push
Recovery
Jerry Pratt, John Carff, Sergey Drakunov
Florida Institute for Human and Machine Cognition
Pensacola, Florida 32502
Email: jpratt@ihmc.us
Ambarish Goswami
Honda Research Institute
Mountain View, California 94041
Email: agoswami@honda-ri.com
Abstract— It is known that for a large magnitude push a
human or a humanoid robot must take a step to avoid a fall.
Despite some scattered results, a principled approach towards
“When and where to take a step” has not yet emerged.
Towards this goal, we present methods for computing Capture
Points and the Capture Region, the region on the ground where
a humanoid must step to in order to come to a complete stop.
The intersection between the Capture Region and the Base of
Support determines which strategy the robot should adopt to
successfully stop in a given situation.
Computing the Capture Region for a humanoid, in general, is
very difficult. However, with simple models of walking, compu-
tation of the Capture Region is simplified. We extend the well-
known Linear Inverted Pendulum Model to include a flywheel
body and show how to compute exact solutions of the Capture
Region for this model. Adding rotational inertia enables the
humanoid to control its centroidal angular momentum, much
like the way human beings do, significantly enlarging the Capture
Region.
We present simulations of a simple planar biped that can
recover balance after a push by stepping to the Capture Region
and using internal angular momentum. Ongoing work involves
applying the solution from the simple model as an approximate
solution to more complex simulations of bipedal walking, includ-
ing a 3D biped with distributed mass.
I. INTRODUCTION
Push recovery is important for humanoid robots operating
in human environments. No matter how well we attempt to
shield these robots, it is inevitable that they will occasionally
bump into objects and other people, and will be tripped up
by debris, rocks, and objects that go undetected. Therefore,
their ultimate utility will depend on good algorithms for push
recovery.
Like most aspects of bipedal walking, push recovery is
difficult because bipedal walking dynamics are high dimen-
sional, non-linear, and hybrid. Moreover, a humanoid robot
is underactuated and makes friction-limited unilateral contacts
with the ground. Despite these theoretical difficulties, animals
and humans are very adept at push recovery. As the push force
progressively grows larger, strategies that they use include
moving the Center of Pressure within the foot, accelerating in-
ternal angular momentum through lunging and “windmilling”
of appendages, and taking a step. Although the humanoid
literature contains several analysis and control techniques for
each strategy, there is yet to emerge a principled approach
towards “when and where to step” under a force disturbance.
This paper seeks to contribute in this direction using the
concept of Capture Points.
A Capture Point is a point on the ground where the robot can
step to in order to bring itself to a complete stop. A Capture
Region is the collection of all Capture Points. Fast and accurate
computation of Capture Points may be difficult and closed
form solutions might not exist for a general humanoid robot.
In this paper we examine simple models of walking that can
be used to explain push recovery strategies, and to develop
algorithms for implementing these strategies on humanoid
robots. In particular, we study an extension to the well-known
Linear Inverted Pendulum Model (LIP) [3], [4]. In this model
the biped is approximated as a “hip” point mass which is
maintained at a constant height, supported by a variable length
leg link. For the Linear Inverted Pendulum Model there is a
unique Capture Point corresponding to each state, for which
we can obtain a closed-form expression.
Human movements such as a forward lunge and rapid
arm rotations make use of angular momentum to maintain
balance. However, the Linear Inverted Pendulum Model lacks
rotational inertia and cannot capture this behavior. Therefore,
we replace the point mass by a flywheel to explicitly model
angular momentum about the body Center of Mass (CoM),
resulting in the “Linear Inverted Pendulum Plus Flywheel
Model” shown in Figure 1. By virtue of the ability to accelerate
the Center of Mass by changing the angular momentum stored
in the flywheel, the unique Capture Point extends to a set of
contiguous points, the Capture Region.
During bipedal walking, trajectories in space do not need to
be precise. In fact the only absolute constraints are usually to
get from one point to another without falling down. Therefore,
it is not critical that the feet be placed absolutely precisely.
In addition, having relatively large feet and internal inertia
provide additional control opportunities to make up for im-
precise foot placement. Therefore, solutions for push recovery
from the simple models we present can be successfully applied
as approximations for controlling more complex bipeds. In
ongoing work we are using the computations from the Linear
Inverted Pendulum Plus Flywheel Model to control a three
dimensional biped with distributed mass and feet, and without
the constraint of a constant height Center of Mass.
Accepted for podium presentation in Humanoids 2006
Genoa, Italy, December 2-4.
Fig. 1. Abstract model of a biped in the single support phase with
a flywheel body and massless legs. The swing leg is not shown. The
two actuators of the biped are located at the flywheel center (also the
CoM of the biped) and the leg.
II. BACKGRO UND
Human movements such as a forward lunge and rapid arm
rotations make use of angular momentum to maintain balance.
However, many models of bipedal walking, such as Kajita and
Tanie’s Linear Inverted Pendulum Model [3], treat the entire
humanoid body as a point mass and do not incorporate the
centroidal angular momentum, despite its significant influence
in gait and balance [1], [5], [17].
There have been recent extensions to the Linear Inverted
Pendulum Model that incorporate momentum. In particular,
the Angular Momentum Pendulum Model (AMPM) [6], [7]
is the closest to the model we present as the flywheel in our
model can be seen as a physical embodiment of an angular
momentum generator of the AMPM model.
To explicitly model centroidal angular momentum we
choose a flywheel with centroidal moment of inertia and rota-
tional angle limits. The flywheel (also called a reaction wheel)
is a standard device used to control satellite orientation[15],
[16]. Pratt [13] discussed the addition of a flywheel in simple
biped models, but did not fully explore the models. Kuo
and colleagues have demonstrated that humans use angular
momentum to capture balance after disturbed through a “hip
strategy” [8] and have pointed out the benefits of angular
momentum for lateral stability in walking.
Although applicable to systems with actual physical fly-
wheel devices, our approach to modeling the centroidal mo-
ment of inertia with a flywheel is intended for modeling bipeds
with relatively limited rotation angles and velocities. For an
example of using a traditional flywheel for the control of
balance, refer to the work by Mayer, Farkas, and Asada [9].
Other related work includes the “resolved momentum con-
trol” scheme [5], in which a number of humanoid tasks can
be controlled through the specification of linear and angular
momenta and Hofmann’s work [2] in which push recovery
from significant disturbances is achieved using numerical
optimization techniques.
III. CAPTURE POINTS AND PUSH RECOVERY STRATEGY
Using the concept of Capture Points and the Capture Region
we can determine when and where to take a step to recover
from a push:
• When to take a step: If a Capture Point is situated
within the convex hull of the foot support area (the
Base of Support), the robot is able to recover from the
push without having to take a step, see Figure 2, top.
Otherwise, it must take a step, see Figure 2, middle.
• Where to take a step: In order to stop in one step the
robot must step such that its Base of Support regains an
intersection with the Capture Region.
• Failure: The humanoid will fail to recover from a push in
one step if the Capture Region in its entirety lies outside
the kinematic workspace of the swing foot. In this case
the robot must take at least two steps in order to stop, if
it can stop at all. This is shown in Figure 2, bottom.
Fig. 2. When the Capture Region intersects the Base of Support, a
humanoid can modulate its Center of Pressure to balance and does
not need to take a step (top). When the Capture Region and Base of
Support are disjoint, the humanoid must take a step to come to a stop
(middle). If the Capture Region is outside the kinematic workspace
of the swing foot, the humanoid cannot stop in one step (bottom
figure).
The Capture Region makes it clear when and where a
humanoid must step to in order to stop and therefore may lead
to a more principled approach to humanoid push recovery.
In the next Section we provide a more formal definition of
Capture Points and Capture Regions.
IV. CAPTURE POINTS AND CAPTURE REGIONS
We define a Capture State, a Safe Feasible Trajectory, and
a Capture Point as follows:
Definition 1 (Capture State): State in which the kinetic en-
ergy of the biped is zero and can remain zero with suitable
joint torques. Note that the Center of Mass must lie above
the Center of Pressure in a Capture State. The vertical upright
“home position”[1] is an example of a Capture State.
Definition 2 (Safe Feasible Trajectory): Trajectory through
state space that is consistent with the robot’s dynamics, is
achievable by the robot’s actuators, and does not contain any
states in which the robot has fallen.
Definition 3 (Capture Point): For a biped in state x, a Cap-
ture Point, P , is a point on the ground such that if the biped
covers P (makes its Base of Support include P ), either with
its stance foot or by stepping to P in a single step, and then
maintains its Center of Pressure to lie on P , then there exists
a Safe Feasible Trajectory leading to a Capture State.
The location of a Capture Point is dependent on the trajec-
tory through state-space before and after swinging the leg and
thus is not a unique point. Therefore, there exists a Capture
Region such that if the Center of Pressure is placed inside this
region, then the biped can stop for some state space trajectory.
Definition 4 (Capture Region): The set of all Capture
Points.
For more information on Capture Points, including a dis-
cussion on their usefulness in defining stability margins for
bipedal walking, please refer to [14]. While it is difficult to
compute Capture Points for a general humanoid, we can easily
compute them in closed form for some simplified models
of walking. In the next Section we compute exact closed-
form solutions of the Capture Region for the Linear Inverted
Pendulum Plus Flywheel Model.
V. COMPUTATION OF CAPTURE REGION FOR THE LINEAR
INVERTED PENDULUM PLUS FLYWHEEL MODEL
A. Equations of Motion of Planar Biped with Flywheel
We begin with a biped system abstracted as a planar inverted
pendulum with an inertial flywheel centered at the Center of
Mass as shown in Figure 1. The legs of the biped are massless
and extensible. The biped has two actuators located at the
flywheel and the leg. The equations of motion during the single
support phase are
m¨x = f
k
sin θ
a
−
τ
h
l
cos θ
a
(1)
m¨z = −mg + f
k
cos θ
a
+
τ
h
l
sin θ
a
(2)
J
¨
θ
b
= τ
h
(3)
where m and J are the mass and rotational inertia of the
flywheel, g is the gravitational acceleration constant, x and
z are the CoM coordinates, l is the distance from the point
foot to the CoM, θ
a
and θ
b
are respectively, the leg and the
flywheel angles with respect to vertical, τ
h
is the motor torque
on the flywheel, and f
k
is the linear actuation force on the leg.
B. Linear Inverted Pendulum Plus Flywheel Model
The Linear Inverted Pendulum Plus Flywheel Model can be
derived as a special case of the above model by setting ˙z = 0
and z = z
0
. From Equation 2 we can solve for f
k
as
f
k
=
mg
cos θ
a
−
1
l
sin θ
a
cos θ
a
τ
h
(4)
Replacing cos θ
a
=
z
l
and sin θ
a
=
x
l
, we get
f
k
=
mg
z
0
l −
1
l
x
z
0
τ
h
(5)
Substituting f
k
into Equation 1, we get the equations of motion
for the Linear Inverted Pendulum Plus Flywheel Model,
¨x =
g
z
0
x −
1
mz
0
τ
h
¨
θ
b
=
1
J
τ
h
(6)
Note that these equations of motion are linear, given that z
is constant. This linearity is what makes the Linear Inverted
Pendulum Model and the flywheel extension valuable as an
analysis and design tool.
Before solving for the Capture Region for the Linear
Inverted Pendulum Plus Flywheel Model, we first compute
the Capture Point when the flywheel is not available.
C. Capture Point for Linear Inverted Pendulum Model
By setting τ
h
= 0 we get the equation of motion for the
Linear Inverted Pendulum Model:
¨x =
g
z
0
x (7)
We can derive a conserved quantity called the “Linear
Inverted Pendulum Orbital Energy” [3] by noting that this
equation represents a mass-spring system with unit mass and
a negative-rate spring with a stiffness of −
g
z
0
:
E
LIP
=
1
2
˙x
2
−
g
2z
0
x
2
(8)
If the Center of Mass is moving toward the foot and E
LIP
>
0, then there is enough energy for the CoM to go over the
foot and continue on its way. If E
LIP
< 0, then the CoM
will stop and reverse directions before getting over the foot.
If E
LIP
= 0, then the CoM will come to rest over the foot.
The equilibrium state E
LIP
= 0 defines the two eigenvectors
of the system,
˙x = ±x
r
g
z
0
(9)
Equation 9 represents a saddle point with one stable and one
unstable eigenvector. x and ˙x have opposite signs (the Center
of Mass is moving toward the Center of Pressure) for the stable
eigenvector and the same signs (the CoM is moving away from
the CoP) for the unstable eigenvector.
The Orbital Energy of the Linear Inverted Pendulum re-
mains constant until the swing leg is placed and the feet change
roles. Assuming that the exchange happens instantaneously,
without energy loss, we can solve for foot placement based
on either desired Orbital Energy, or desired speed at a given
value of x [3], [4]. For computing the Capture Point, we are
interested in the foot placement required to obtain an Orbital
Energy of zero and corresponding to the stable eigenvector
from Equation 9,
x
capture
= ˙x
r
z
0
g
(10)
For a given state the Linear Inverted Pendulum Model has a
single Capture Point corresponding to the footstep that would
put the state of the robot onto the stable eigenvector. When the
flywheel is made available, this point will grow to a Capture
Region. If the state is on one side of the stable eigenvector in
phase space, then a clockwise acceleration of the flywheel will
capture balance. If on the other side, then a counterclockwise
acceleration will be required.
We first show two methods for computing an upper bound
on the Capture Region assuming the flywheel can make either
an instantaneous velocity change or an instantaneous position
change. We then derive a more realistic Capture Region based
on a torque-limited and angle-limited flywheel.
D. Instantaneous Flywheel Velocity Change
Suppose we produce an impulsive torque on the flywheel,
that causes a step change in the rotational velocity of the
flywheel of ∆
˙
θ
b
. Then we get a step change in the forward
velocity of ∆ ˙x = −
J
mz
0
∆
˙
θ
b
. In this case, the instantaneous
Capture Region will be
r
z
0
g
( ˙x−
J
mz
0
∆
˙
θ
b
max
) < x
capture
<
r
z
0
g
( ˙x−
J
mz
0
∆
˙
θ
b
min
)
(11)
E. Instantaneous Flywheel Position Change
Suppose we could produce a step change in the rotational
position of the flywheel of ∆θ
b
. This would cause a step
change in the position of the CoM of ∆x = −
J
mz
0
∆θ
b
. In
this case, the instantaneous Capture Region will be
r
z
0
g
˙x −
J
mz
0
∆θ
b
max
< x
capture
<
r
z
0
g
˙x −
J
mz
0
∆θ
b
min
(12)
F. Torque and Angle Limited Linear Inverted Pendulum Plus
Flywheel Capture Region
Assuming step changes in either flywheel angular velocity
or position results in easy to compute and potentially useful
upper bounds on the Capture Region. However, such step
changes are not physically possible. Here we assume the
flywheel is torque limited and has limits on its minimum and
maximum rotation angles.
Torque limits are realistic since most motors are torque
limited and they can achieve the maximum torque nearly
instantaneously when compared to physical time constants. By
using step torque profiles, we can compute the Capture Region
for the Linear Inverted Pendulum Plus Flywheel model fairly
easily since the dynamics are linear and unit steps have simple
Laplace Transforms.
Suppose the robot is moving at ˙x
0
and the flywheel is
spinning at
˙
θ
0
and has an angle of θ
0
with respect to vertical.
We wish to find a flywheel torque profile and a stepping
location that will bring the robot to rest over its foot with
no forward velocity or flywheel angular velocity.
The torque profile that will provide the most influence on
velocity is the one which accelerates the flywheel as hard as
possible in one direction and then decelerates it, bringing it to
a stop at the maximum flywheel angle,
τ(t) = τ
max
u(t) −2τ
max
u(t −T
R1
) + τ
max
u(t −T
R2
) (13)
where τ
max
is the maximum torque that the joint can apply,
u(t−T ) is the unit step function starting at T , T
R1
is the time
at which the flywheel stops accelerating and starts decelerating
and T
R2
is the time at which the flywheel comes to a stop.
Given the torque profile in Equation 13, the flywheel angular
velocity and position will be
˙
θ(t) =
˙
θ
0
+
τ
max
J
u
1
(t) − 2u
1
(t − T
R1
) + u
1
(t − T
R2
)
(14)
θ(t) = θ
0
+
˙
θ
0
t + (15)
τ
max
J
1
2
u
2
(t) − u
2
(t − T
R1
) +
1
2
u
2
(t − T
R2
)
At time T
R2
we want
˙
θ(T
R2
) =
˙
θ
f
and θ(T
R2
) = θ
max
. To
find the extents of the Capture Region,
˙
θ
f
= 0. However, we
keep
˙
θ
f
in the equation since it may be desirable to have a
final velocity which helps return the flywheel to the starting
position. Solving for T
R1
in Equation 16 we get
T
R1
=
1
2
T
R2
+
J
2τ
max
(
˙
θ
f
−
˙
θ
0
) (16)
Substituting T
R1
into Equation 14 and rearranging, we get
a quadratic equation in T
R2
,
h
τ
max
4J
i
T
2
R2
+
1
2
(
˙
θ
f
+
˙
θ
0
)
T
R2
+
θ
0
− θ
max
−
J
4τ
max
(
˙
θ
f
−
˙
θ
0
)
2
= 0 (17)
which can be solved for T
R2
. Note that if
˙
θ
f
=
˙
θ
0
= 0, then
T
R1
=
1
2
T
R2
and
T
R2
=
r
4J
τ
max
(θ
max
− θ
0
) (18)
We can now determine the position and velocity trajectories
of the mass by integrating the equations of motion for the
Linear Inverted Pendulum Plus Flywheel model,
¨x =
g
z
0
x −
1
mz
0
τ
h
(19)
Written in form of Laplace transforms, we have
X(s)
τ(s)
= −
1
mz
0
1
s
2
− w
2
(20)
where w =
q
g
z
0
. The zero input response of this system is
x(t)
ZIR
= x
0
cosh(wt) +
1
w
˙x
0
sinh(wt)
˙x(t)
ZIR
= wx
0
sinh(wt) + ˙x
0
cosh(wt) (21)
The zero state response, given the input in Equation 13 is
x(t)
ZSR
= −
τ
max
mz
0
w
2
[(cosh(wt) −1)u(t)
− 2(cosh(w(t − T
R1
)) − 1)u(t − T
R1
)
+ (cosh(w(t − T
R2
)) − 1)u(t − T
R2
)]
˙x(t)
ZSR
= −
τ
max
mz
0
w
[sinh(w t)u (t)
−2 sinh(w(t − T
R1
))u(t − T
R1
)
+ sinh(w(t − T
R2
))u(t − T
R2
)](22)
Combining the zero input response and the zero state response
at time T
R2
we have
x(T
R2
) = P
1
+ P
2
x
0
˙x(T
R2
) = P
3
+ P
4
x
0
(23)
where
P
1
=
1
w
˙x
0
sinh(wT
R2
)
−
τ
max
mz
0
w
2
[cosh(wT
R2
) − 2 cosh(w(T
R2
− T
R1
)) + 1]
P
2
= cosh(wT
R2
)
P
3
= ˙x
0
cosh(wT
R2
)
−
τ
max
mz
0
w
[sinh(w T
R2
) − 2 sinh(w(T
R2
− T
R1
))]
P
4
= w sinh(wT
R2
) (24)
To solve for a Capture Point, we need the state to lie on the
stable eigenvector of the Linear Inverted Pendulum system
after the flywheel stops at T
R2
,
˙x(T
R2
) = −wx(T
R2
) (25)
Using Equations 23 and 25 and the fact that cosh(y) +
sinh(y) = e
y
, we can solve for x
0
,
x
0
= −
1
w
˙x
0
+
τ
max
mg
(e
wT
R2
− 2e
w(T
R2
−T
R1
)
+ 1)
e
wT
R2
(26)
The Capture Point is then −x
0
. To find the other boundary of
the Capture Region, the above can be repeated with the torque
limit of τ
min
and the angle limit of θ
min
. To find a Capture
Point without the use of angular momentum, one can repeat
the above, except set T
R1
= 0. T
R2
will be long enough to
stop any spin that the flywheel may currently have and x
0
can
be solved as before.
G. Ground Reaction Forces in Linear Inverted Pendulum Plus
Flywheel Model
The ground reaction forces in the Linear Inverted Pendulum
Plus Flywheel model can be computed considering a free body
diagram and examining ¨x and ¨z:
f
z
= mg, f
x
=
mg
z
0
x −
τ
h
z
0
(27)
To prevent slipping, the ground reaction force vector must stay
within the friction cone. Given a coefficient of friction, α ,
−α <
f
x
f
z
=
x
z
0
−
τ
h
mgz
0
< α (28)
If τ
h
= 0 then we get
f
x
f
z
=
x
z
0
, which means that the angle
of the virtual leg from the Center of Mass to the Center of
Pressure must be inside the friction cone. For nonzero τ
h
, the
ground reaction force vector is rotated to produce this torque
about the CoM. For typical coefficients of friction, the above
equation gives us a limit on τ
h
, or equivalently on
¨
θ
b
=
τ
h
J
.
Note that a step change in either
˙
θ
b
or θ
b
, would require an
impulsive torque, which would cause the ground reaction force
to be horizontal, causing slipping on any non-attached surface.
H. Dimensional Analysis
We can perform a dimensional analysis [10] of the state
variables and parameters of the Linear Inverted Pendulum plus
Flywheel Model to reduce the number of variables involved.
Let us define dimensionless position, velocity, time, inertia,
torque, and angles as
x
0
≡
x
z
0
(29)
˙x
0
≡
1
√
gz
0
˙x (30)
t
0
≡ t
r
g
z
0
(31)
J
0
≡
J
mz
2
0
(32)
τ
0
≡
τ
mgz
0
(33)
θ
0
≡ J
0
θ (34)
Note that the dimensionless inertia can also be written as a
ratio of the radius of gyration of the pendulum and the Center
of Mass height, J
0
≡
J
mz
2
0
=
R
2
gyr
z
2
0
. For a point mass, J
0
= 0,
and for a flywheel with all of its mass on the rim, which just
touches the ground, J
0
= 1.0.
With these dimensionless quantities, the equations of mo-
tion for the Linear Inverted Pendulum Plus Flywheel Model
(Equation 6) become
¨
x
0
= x
0
− τ
0
(35)
¨
θ
0
= τ
0
(36)
where time derivatives are with respect to non-dimensional
time:
¨
x
0
≡
d
2
x
0
dt
02
and
¨
θ
0
≡
d
2
θ
0
dt
02
. With this formulation,
the only remaining parameters used in our derivation of
Capture Regions are τ
0
max
, θ
0
max
, τ
0
min
, and θ
0
min
. Therefore,
two Linear Inverted Flywheel Plus Pendulum systems are
dynamically similar if they have the same values of those four
quantities. In order to reformulate the equations for computing
the Capture Region, one can replace all the variables with their
dimensionless versions and set m, g, J and z
0
all to one. For
example, the ground reaction force limits become
−α < x
0
− τ
0
< α (37)
