1
3LP: a linear 3D-walking model including torso and swing dynamics
Salman Faraji∗and Auke J. Ijspeert∗
Abstract— In this paper, we present a new model of biped
locomotion which is composed of three linear pendulums (one
per leg and one for the whole upper body) to describe stance,
swing and torso dynamics. In addition to double support, this
model has different actuation possibilities in the swing hip and
stance ankle which could be widely used to produce different
walking gaits. Without the need for numerical time-integration,
closed-form solutions help finding periodic gaits which could
be simply scaled in certain dimensions to modulate the motion
online. Thanks to linearity properties, the proposed model can
provide a computationally fast platform for model predictive
controllers to predict the future and consider meaningful
inequality constraints to ensure feasibility of the motion. Such
property is coming from describing dynamics with joint torques
directly and therefore, reflecting hardware limitations more
precisely, even in the very abstract high level template space.
The proposed model produces human-like torque and ground
reaction force profiles and thus, compared to point-mass models,
it is more promising for precise control of humanoid robots.
Despite being linear and lacking many other features of human
walking like CoM excursion, knee flexion and ground clearance,
we show that the proposed model can predict one of the
main optimality trends in human walking, i.e. nonlinear speed-
frequency relationship. In this paper, we mainly focus on de-
scribing the model and its capabilities, comparing it with human
data and calculating optimal human gait variables. Setting up
control problems and advanced biomechanical analysis still
remain for future works.
I. INTRODUCTION
Humanoid robots are challenging to control mainly due to
their complex structure and floating base. During locomotion
tasks, these systems introduce another complexity compared
to wheeled or flying robots, which is the hybrid nature
of stepping where the continuous model changes in each
phase. Since the beginning, it has always been challenging
to balance these robots considering the fact that they can
only establish unilateral support with the environment. In
addition, creating a sequence of motion, associated timing
and the required control architecture in each phase of motion
are other important topics in controlling humanoid robots.
The main objectives are therefore being human-like, energy
efficient, versatile and of course agile like humans. In this
paper, we are proposing a new template model that describes
main aspects of walking while being computationally very
efficient. Such model can be very useful in modern control
architectures from the computational perspective. It can also
go beyond conventional template models such as Linear
Inverted Pendulum (LIP) by producing more natural motions
in faster walking speeds, resembling human locomotion.
∗Biorobotics Laboratory, Ecole Polytechnique F´ed´erale de Lausanne
(EPFL), Lausanne, Switzerland
The design of controllers should address many concerns
like fast implementation, stability, robustness to unknown
model parameters and the degree of dependency on sensory
data. It has always been appreciated if transition to different
speeds and large disturbance rejection are handled in the
same control framework. Therefore, candidate methods are
normally coming with proper identification of the basin of
attraction regarding system states, actuator limitations and
violation of model assumptions. This identification is not
always straight forward due to its nonlinear nature, though
it has been postulated that two steps are enough to stabilize
in almost all conditions [1]. In this regard, Model Predictive
Control (MPC) is a powerful framework as it can find optimal
policies constrained to certain actuation and state limitations.
It can also predict if there is no feasible solution, in order
to let the algorithm take a different decision.
A. Hierarchical controllers
Recently, hierarchical control approaches are becoming
popular, where a simple template model determines the
overall dynamics in an abstract way and then, a detailed
full-body inverse dynamics controller converts this behavior
to individual actuator inputs [2]–[4]. In dynamical systems,
prediction of future evolution is mainly sensitive to the model
and sensory data precision [5]. In hierarchical approaches
similarly, dynamical matching between the template and the
full model is crucial to ensure precise execution of the
abstract plan.
B. Inverted Pendulums
One of the earliest template models that roughly described
human in single support was Inverted Pendulum (IP) [18]
with fixed leg length. In this model, a single mass rolls
over a contact point, established by a massless leg. IP is
widely used to analyze passive walkers [18] and human
motion [19]. Inspired by this idea, there are various simple
robots built to walk naturally with minimal energy, pumped
either in push-off or swing hip or both [20]. Later this
model was simplified to Linear Inverted Pendulum (LIP)
[21], mainly favoring availability of analytical solutions
instead of numerical integration. With proper modulation of
Zero Moment Point (ZMP) [22], many position controlled
robots like ASIMO [23] can perform walking through inverse
kinematics methods. These algorithms are normally able to
produce slow to moderate walking speeds. However complex
robots using the LIP method usually walk with crouched
knees to keep the Center of Mass (CoM) at constant height.
In addition to increasing energy consumption, it is harmful
for the robot in long term and less human like, though
providing full controllability.
arXiv:1605.03036v1  [cs.RO]  10 May 2016
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[6]
[7]
[8]
[9]
[8]
[10]
[11]
name
SLIP
IP
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LIP
knee
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x
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steering
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x
3D
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x
x
smooth
x
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x
torso
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swing
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x
x
x
x
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linear
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x
[12]
[13]
[14]
[15]
[16]
[17]
proposed
name
BSLIP
FMCH
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3LP
knee
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x
x
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steering
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x
(x)
3D
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x
x
smooth
x
x
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x
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x
torso
rotating
rotating
rotating
rotating
rotating
rotating
fixed
swing
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x
x
x
x
x
linear
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x
Fig. 1: Different key models introduced in literature for walking. In this table, the model is standing on the left leg and the right leg is in swing motion.
Solid arrows show the direction of motion and degrees of freedom while gray arrows show actuations torques or push-off forces. For models without swing
dynamics, we show the swing leg also in gray color to implicitly show that attack angle is a control authority. Note that some of these models are only
in 2D while more advanced models are in 3D. Some models simulate pelvis width, torso dynamics or ground clearance as well. Most of these models
produce compass gait, however some have ankle actuation or arc foot. In the comparison table, we mention important features such as knee flexion (for
ground clearance), steering capabilities, 3D model, smooth profiles, inclusion of torso and swing dynamics and providing linear equations. By smoothness
we mean no collision and push-off impulse, but possibly describing double support phase. Many of these models allow for torso pitch while we keep it
fixed in 3LP for simplicity. Note that 3LP can describe steering like our previous work with LIP [2] only if pelvis width is set to zero. Overall, 3LP offers
many features that are not existing in other template models, although it is still linear like LIP.
C. Multi-link pendulums
Apart from these two models, there are other nonlinear
extensions solved numerically. In [8], [9] the IP model is
extended to have two separate masses for each leg as well
as a single mass at hip. Using similar actuation schemes,
this model produces compass gaits on 2D-constrained simple
robots. In [8], same model is modified to have another De-
gree of Freedom (DoF) in the knee for the swing leg to avoid
foot scuffing. The stance leg however always stays straight.
In [24], this advanced model is augmented with a torso and
later, it is used also by [16] to perform natural walking on
uneven terrain using a library of motion primitives. Another
model with four masses in legs, hip and torso is proposed in
[25] without any DoF in the knees, trying to slightly turn in
3D.
Targeting impact-less walking, a simpler model with two
passive springs in the hips is proposed by Gomes [15].
Addition of these springs is mainly motivated by elastic
properties of human muscles. By exploiting torso motions,
Gomes can find zero energy gaits that means impact-losses
are less important in energetics of walking. Another very
interesting extension of IP-based models is proposed in [17].
In this new model, the pelvis has a width in 3D with a mass
in the center. The swing leg is also having another DoF in
the knee. This new 3D model is allowed to take advantage
3
of a limited transversal wrench in the contact point for better
turning. However finding a periodic gait for such complicated
model is difficult and computationally expensive.
D. Spring-Loaded Pendulums
It is always questionable which template model produces
more realistic motion. This can be inspected from the view-
point of geometry, torques or energy. The aforementioned
categories mainly address energetic and geometric similari-
ties. However ground reaction forces and elastic behaviors
are other favorite aspects addressed by another category
based on Spring Loaded Inverted Pendulum (SLIP). This
model is composed of two massless springs (legs) connected
to a single mass. One can expect better description of energy
exchange in this model over faster walking speeds and
running, mimicking compliant properties of human tendons.
Based on this model, Iida [6] built a hip-actuated robot
walking in 2D with various springs, similar to human mus-
cles. Properties of the passive version without hip actuation
were later widely explored in [26]. Using the concept of
Virtual Pivot Point (VPP) to stabilize the torso, the model
was also extended to have an upper body [13]. In Figure.1,
we have briefly shown key models proposed for walking in
the literature. Note that in some of them, passive springs
are added to the hip actuators for energy storage, similar to
humans. In this paper however, we do not investigate elastic
behaviors and energy storage.
E. Control difficulty
Except LIP, all other models presented before require
numerical integration to obtain time trajectories. Therefore,
the Jacobian around a nominal solution linearizes the model
and provides the framework for Floquet analysis or discrete
controller design [26]. This approach can be used to create an
optimal library of primitives [16], [17], [27]. However online
reaction to disturbances as well as inclusion of other inequal-
ity constraints that are often ignored in calculating a stable
basin of attraction, limit the generality of this framework.
MPC on the other hand is powerful in this regard, however
it requires simple and possibly linear models to provide
online performance, depending on the hardware platform.
LIP model therefore fits best in the MPC framework [2], [28].
MPC, its simpler version, LQR and sometimes Discretized
LQR (DLQR) [29] are used a lot on nonlinear models to
stabilize walking gaits and recover pushes [9], [27]. Either a
library of optimal policies is generated off-line or a discrete
transition model is considered at specific events like CoM
apex or heel-strike.
F. Why 3LP?
In this paper, we propose a more general version of LIP
[21] with three linear pendulums (called 3LP) that captures
torso and swing dynamics in 3D. This model allows predic-
tion of future at any time in closed form which is favorable
by limited computational resources and MPC. Compared
to LIP, the planned motion with 3LP has instantaneous
accelerations that are easier to track by inverse dynamics
block in the hierarchical control. In other words, the motion
of CoM is more natural for the humanoid robot as swing
and torso dynamics are taken into account. Additionally, the
swing trajectory is more natural compared to many other
template models that track an imposed angle of attack with
a stiff controller [9], [20], [27]. It should be noted however
that the CoM height in our model will still be constant similar
to LIP.
The 3LP model supports many different inputs, i.e. hip
and ankle actuation authorities. These input dimensions let us
find various types of gaits with simple algebraic routines, not
optimizations. 3LP as a template model is in fact very useful
for motion planning. Besides describing falling dynamics
like IP [19] and LIP [30], hip torques required to keep the
torso upright are also part of the model like [12]. The most
outstanding feature of 3LP is considering swing dynamics
that allow us to calculate natural cycles for the model.
Considering Figure.1 again, there are few models in the
literature that consider this integral part of walking. In these
models due to nonlinearity, numerical integration is always
needed to search for periodic gaits. In 3LP however we do
not need to impose the timing, nor optimizing hip torque
trajectories.
This paper merely focuses on introducing 3LP and its
capabilities, while setting up control problems remain for
future work. The main motivation behind 3LP is for control
however, as it can provide more natural CoM and swing
motions. It suits MPC control better than LIP in our previous
work [2], because we do not need to define a desired
footstep plan. The plan comes out of 3LP just by modulating
hip torques. Closed form formulations also let the MPC
controller change phase timings in case of extreme push
recovery for example. Although the MPC problem becomes
nonlinear in this case, one can define meaningful hip torque
magnitude and rate limitations instead of putting vague
timing boundaries on the step time which do not precisely
reflect physical facts about the real hardware.
In the next section, we will explain the model details and
assumptions behind as well as compare it to the existing
models. Next, a method to find different periodic gaits is
explained based on symmetry ideas. In the end we show that
for a human-like gait, actuation profiles of 3LP are similar
to those of human. Additionally, we also show quantitatively
that 3LP can explain main optimality criteria of human
walking, i.e. speed-frequency relation, despite being linear.
II. 3LP DYNAMICS
Motivated by the fact that CoM motion is influenced
by swing and torso dynamics, we have added two other
pendulums to the normal linear inverted pendulum model,
connected together with a pelvis of certain width. In this
model, as shown in Figure.2, there are 2-DoF actuators in
each hip and ankle. We demonstrate feet of limited size in
Figure.2 merely to mention availability of ankle actuation.
The upper body is represented by a single mass referred to as
torso and each leg is represented by a single inertia-less mass.
4
X3,M3,F3
X2,M2,F2
x3,τ3,f3
x2,τ2, f2
X1
τ1,f1
m3
m2
m1
z1
z2
z3
y3
y2
w/2
y1
Swing
Stance
Fig. 2: A schematic of 3LP model with all variables and parameters. The
bottom plane shows level ground and all upper planes of fixed height show
where the three masses and the pelvis are constrained to move. The torso is
always upright and the pelvis is along y axis, by model construction. The
swing foot remains inside the bottom plane, i.e. sliding on the ground with
zero force during swing phase. Note that in single support all contact forces
for the swing leg (F2,M2) are zero. The arbitrary actuation inputs are τ2
and M3 shown in red and bold.
By construction (assuming ideal controllers), masses stay in
horizontal planes of constant height and the torso is always
upright without transversal rotation. These assumptions are
used in [17] as well to decouple sagittal and lateral dynamics.
Since the torso is connected to an accelerated frame (i.e.
pelvis), the τ1 torque is always found to keep the torso
upright. Such hip torques in literature are alternatively calcu-
lated by virtual pendulum concept in the models which let the
torso pitch or roll freely during the motion [13]. Inspired by
the fact that rolling contact constraint produces more human
like walking [31], we also allow for transversal wrenches at
stance contact to keep the pelvis orientation fixed. In 3LP
model, we do not consider turning properties as they make
the model nonlinear. It is however practically easy to turn
the robot using inverse dynamics layer, as we showed in [2]
using a simple LIP model. There is no need to add heel-
strike and push-off impulses since our double support phase
smoothly takes care of transition. In SLIP based models [13],
the compliant springs automatically produce a double support
phase and perform weight transition without impulses. In
3LP however, switching to double support is triggered when
both horizontal components of the swing foot velocity are
zero. This assumption is typically used in models with swing
dynamics and double support phase like [15], though unlike
3LP, Gomes removes double support for more simplicity.
In Figure.2, all external/internal forces and torques as
well as positions are shown for each interesting point of
the model, i.e. contacts, hips and pelvis center. These vector
variables in our model are expressed in Cartesian frame. One
can easily write geometric relations as:
X1
=
[X1,x,X1,y,z1]T
X2
=
[X2,x,X2,y,0]T
X3
=
[X3,x,X3,y,0]T
x2
=
X1 +[0, wd
2 ,0]T
x3
=
X1 +[0,−wd
2 ,0]T
y1
=
X1 +[0,0,z3]T
y2
=
x2 + z2
z1
(X2 −x2)
y3
=
x3 + z2
z1
(X3 −x3)
(1)
Where w stands for pelvis width and d = ±1, depending on
left or right support. We can write total force equations for
each mass i=1,2,3:
mi(¨yi +g) = fi +Fi
(2)
and total moment as:
(X1 −y1)× f1 +M1 +τ1 = 0
(X2 −y2)×F2 +(x2 −y2)× f2 +M2 +τ2 = 0
(X3 −y3)×F3 +(x3 −y3)× f3 +M3 +τ3 = 0
(3)
Finally, we can write these equations for our mass-less pelvis
around the center point:
−f1 −f2 −f3 = 0
−τ1 −τ2 −τ3 −wd
2 (f2 −f3) = 0
(4)
which link all other variables together. In these equations,
we consider X1,X2,X3 as independent variables and solve
for others as dependent variables. Actuation possibilities for
3LP are selected as stance foot M3 and swing hip torques
τ2 in sagittal and lateral directions. Note that the variables
F1,M1 stand for disturbances added to our model to simulate
external forces. Now, the differential equations governing
the system behavior could be obtained for different phases.
Features of a full stride, consisting of a double support
followed by a single support are defined in Table.I. In double
support, the weight is transfered from F2 to F3 and in single
support, the leg with variables of subscripts 2 will swing
forward. We are going to find transfer matrices that relate
the states of the system together at any instance of time over
a stride phase.
Full stride =
double support +
single support
duration
Tds
Tss
timing order
1
2
stance leg
subscripts 3
subscripts 3
swing leg
X
subscripts 2
control input
ankle
hip/ankle
controllability
redundant
full
TABLE I: Information about the two consecutive phases that form a full
stride phase.
5
A. Single support
In this phase, the swing foot does not have any external
forces, meaning:
F2 = [0,0,0]T
M2 = [0,0,0]T
(5)
Also stance foot is fixed on the ground, meaning X3 = const.
The position variable X, disturbance vector W and contact
position P are defined as:
X =


X2,x
X2,y
X1,x
X1,y

,W =


F1,x
F1,y
M1,y
M1,x

,P =

X3,x
X3,y

,
(6)
And the inputs to the system are defined as:


τ2,y
τ2,x
M3,y
M3,x

=


Mh,y
Mh,x
Ma,y
Ma,x

+ t
Tss


rMh,y
rMh,x
rMa,y
rMa,x

= U + t
Tss
rU
(7)
Note that we consider two modes of input torques, constant
and linearly increasing with time (ramp profile). Therefore
we have eight inputs to the system. Now for single support
phase, we write linear differential equations and using Maple
software [32], we solve them analytically to find the state
evolution after arbitrary time with respect to the initial state.
¨X = C1X +C2(t)

PT
UT
rUT
W T
dTT
Q(t) = Hss(t)Q(0)
(8)
where C1 is a constant matrix and C2(t) is a linear function
of time. Here the vector Q contains both inputs and states
together:
Q(t) =

X(t)T
˙X(t)T
PT
UT
rUT
W T
dTT
(9)
Note that inputs U and rU are constant during a single
support phase. As expected, Hss(t) could be written as:
Hss(t) = Hss
0 +
4
∑
i=1
Hss
i ewss
i t +Hss
5 t
(10)
where wss
i
and Hss
i
variables are complicated functions of
system model parameters, though very sparse. Once these
individual matrices are calculated off-line, Hss(t) can be
easily calculated online by few arithmetic operations. It is
also worth mentioning that lateral and sagittal motions are
decoupled, meaning that in Hss(t), only elements with both
odd or both even column and row index are nonzero. We
also like to emphasize that the variables P,U,rU,W,d are
assumed to be constant during the whole single support
phase and therefore, the corresponding rows on Hss(t) only
have 1 on the diagonal. We keep everything packed for
simpler manipulation of matrices in future. Note that in ˙X(0),
initial foot velocities are zero, starting to swing from rest
conditions. It should also be noted that the switch between
left and right support only changes the variable d and contact
positions P, not the system transition matrix Hss(t).
B. Double support
In this phase, the two feet are fixed and contact forces
are being transfered from (F2,M2) to (F3,M3). Note that
although there are two constraints added regarding the new
fixed foot, 6 other forces and torques (F2,M2) are none-
zero and should be found. Therefore we need 8 additional
equations to replace (5) and (7). In our double support, we
decide to linearly transfer the weight from one leg to another.
This means the vertical component of the Ground Reaction
Force (GRF) in previous stance leg (which is constant during
single support) will go linearly to zero during double support
time. Such policy will ensure smoother torque profiles which
are easier to track on the real robot. The linear policy
also makes simpler equations in analytic form (compared
to quadratic or other forms).
Note that all contact forces are calculable at any time
including right at the end of single support (as a function of
state variables). If we want to preserve continuity of other
horizontal components of GRF in phase transitions as well,
in the right hand side of the differential equations (8), terms
like tx(t) will appear. Although these forms are still linear,
they make finding analytical solutions difficult. Instead, we
keep the Center of Pressure (CoP) for each foot constant
during double support. Note that in single support, ground
reaction torques in the stance foot are determined by (7) and
the vertical GRF is constant. The CoP position can be then
simply preserved in double support by linearly decreasing
contact reaction moment M2 in stance leg (and increasing
M3 accordingly). Overall, equations being considered addi-
tionally for the CoP constraints are:

M2,y
M2,x

=
(1−t
Tds
)

Ma,y +rMa,y
−Ma,x −rMa,x

(11)

M3,y
M3,x

=
t
Tds

Ma,y
Ma,x

(12)
Linear transfer of weight and transversal contact torques as
well as continuity of hip torque profiles require 4 more
equations. We solve all equations except 4, and replace
variables in these 4 equations. This results in a set of 4
equations E with 8 unknown variables, 6 hip torques and
2 vertical components of GRFs. Inspired by linear weight
transfer idea, we obtain the following other 4 equations in
(13), which transfer torques and forces uniformly during the
double support:
[]V2 =

τT
2 ,F2,z
T
V3 =

τT
3 ,F3,z
T
ˆV2 =

Mh,x,
Mh,y,
0,
0
T
ˆV3 =

−Mh,x −rMh,x,
Mh,y +rMh,y,
0,
0
T
1
1−
t
Tds
∂E
∂V2
(V2 −ˆV2) =
1
t
Tds
∂E
∂V3
(V3 −ˆV3)
(13)
Here we calculate the Jacobian of these equations with
respect to V2 and V3, multiply them by these variables again
and divide each side by proper time to induce linear force
transition. Combining the last equation set in (13) with E (to
have 8 equations for 8 variables) and general equations of
6
the robot mentioned before, similar to single support case,
Maple finds analytical solutions of the form:
Q(t) = Hds(t)Q(0)
(14)
Note that variables X2,x and X2,y (foot positions) are constant
during double support phase and therefore, they are correctly
linked to the right-hand side vector in this equation. Again
as before:
Hds(t) = Hds
0 +
4
∑
i=1
Hds
i ewds
i t +Hds
5 t
(15)
Which ensures fast implementation of this matrix. At this
stage, we would also like to define row selection matrices
that if multiplied from left, they output corresponding rows.
If their transpose is multiplied from right, they output corre-
sponding columns alternatively, if dimensions match. These
selection matrices are listed in Table.II.
Matrix
selected variables
SXP ∈R8×23
all states and P except ˙X2
S ˙X2 ∈R2×23
˙X2
SX2,x ∈R1×23
sagittal component of X2
SU ∈R8×23
all inputs
SMh ∈R2×23
constant hip torques
SMa ∈R2×23
constant contact torques
SrMa ∈R2×23
time-increasing contact torques
Sd ∈R1×23
d
TABLE II: Different selection matrices used hereafter.
C. Full stride
Once we have matrices for both phases, we can link them
together to find the transfer matrix for the full stride phase:
H(t) =

Hds(t)
t ≤Tds
Hss(t −Tds)Hds(Tds)
0 < t −Tds ≤Tss
Note that although we have used parameters Tss and Tds
in calculating transfer matrices, they are defined by other
methods explained later. The variable Tds is crucial for double
support calculations, though Tss only determines the rate
of time-increasing input components in single support. The
duration of a stride phase is therefore defined as Tstride =
Tds + Tss. Apart from the forward transfer matrix H(t), we
can also define another back transfer matrix in a similar way:
Hss(t +∆t) = Gss(t,∆t)Hss(t)
Hds(t +∆t) = Gds(t,∆t)Hds(t)
(16)
One can easily show that the matrices Gss and Gds could be
written as:
Gss(t,∆t) =
(∑4
i=0
1Gss
i ewss
i ∆t + 1Gss
5 ∆t)
+
t(∑4
i=0
2Gss
i ewss
i ∆t + 2Gss
5 ∆t)
Gds(t,∆t) =
(∑4
i=0
1Gds
i ewds
i ∆t + 1Gds
5 ∆t)
+
t(∑4
i=0
2Gds
i ewds
i ∆t + 2Gds
5 ∆t)
(17)
which favor fast implementation. Now having Gss and Gds
matrices, we can define a back transfer matrix as:
G(t) =

Gss(Tss)Gds(t,Tds −t)
t ≤Tds
Gss(t −Tds,Tss +Tds −t)
0 < t −Tds ≤Tss
0
Tds+Tss
Tds
τ 
t
Tds+t 
Hds(t)
Hss(t )
H(Tds+Tss)
Gss(t , t)
Gds(t, t)
G(τ)
H(τ)
 t
Fig. 3: Definition of all transition matrices. Note that in this figure, ∆t
represents arbitrary time duration that could be as small as the time-step
for example, although there is no need to integrate the system. We use
these matrices only for simulating intermittent pushes (discussed later) or
for visualization. Otherwise the matrix H(Tstride) is enough to find the state
transition.
Which always satisfies:
G(τ)H(τ) = H(Tstride)
(18)
This equation indicates that any intermediate state could be
evolved to the end of the phase simply by multiplying the
matrix G(t). For the purpose of visualization or time integra-
tion, one can also use individual matrices Gss and Gds with
sufficiently small ∆t. Figure.3 gives a better understanding
of all transition matrices introduced so far as well as the
composition of individual single and double support phases
to make the full stride phase. Note that for an arbitrary state
vector, it is straightforward to calculate all internal forces
and torques in closed form as linear and quadratic functions
of the state vector respectively.
The matrix H(t) is used to find evolution of the system
state over time, specially at the end of the full stride where
t = Tstride. Although the fact that the two feet are fixed in
double support is already encoded in the matrix, one should
consider another constraint for the system which forces the
foot velocity to zero at the end of the stride. This means
two input dimensions should be dedicated all the time and
actively being regulated to ensure satisfaction of foot velocity
constraint. For this purpose, we dedicate constant hip torque
components for example:
H′(t) = H(t)−H(t)ST
Mh(S ˙X2H(t)ST
Mh)−1S ˙X2H(t)
(19)
This equation in fact selects foot velocity rows, calculates
the hip torques in terms of other variables and then replaces
hip torques in other equations. Now using H′(Tstride), foot
velocity is always zero at the end of the stride, though two
input dimensions are lost as expected. One can do the same
to find G′(t) which is used to find the evolution of current
state (measured at time t) until the end of the phase. In
next sections we will use these new augmented matrices for
simplicity.
7
So far in this section, we have found transition matrices
for the system and defined a full phase, consisting of a
double support followed by a single support. We have also
formulated matrices such that there are very fast for online
calculation. It should be noted that for the purpose of MPC,
only a single off-line computation of the matrix H(Tstride) is
enough. In next section, we are going to find different open-
loop periodic gaits based on the type of actuation desired.
III. FINDING PERIODIC GAITS
All we need in this section is the matrix H′(t) as the
system is linear and fully expressed with this matrix. For
the purpose of this paper unlike [26], we only focus on
symmetric gaits observed in normal human walking. We find
different classes of vectors that produce symmetric gaits. The
concept of symmetry could be encoded in a single matrix
along with the constraint of zero foot velocity at the end
of the stride. Consider foot position X2, pelvis position X1,
pelvis velocity ˙X1 and stance foot position P packed in a
vector:
h
XT
2
XT
1
˙X1
T
PT
iT
(20)
This vector can be of course extracted from the full vector Q
(9) with the selection matrix SXP. After a stride, we define
relative vectors between base, swing foot and stance foot
which are linked to those in the beginning in a certain way.
These vectors could be defined in the following matrix M,
if multiplied by (20):
M =


−1
.
1
.
.
.
.
.
.
−1
.
1
.
.
.
.
.
.
1
.
.
.
−1
.
.
.
.
1
.
.
.
−1
.
.
.
.
1
.
.
.
.
.
.
.
.
1
.
.


Comparing these quantities before and after a symmetric
stride, in sagittal plane, components are equal and in lateral
plane, they are linked with a negative sign. This fact could
be explained in a matrix O:
O = diag([1,−1,1,−1,1,−1])
(21)
Now we should also consider the transition matrix T that
exchanges the swing and support contact points after a stride
as:
T =


.
.
.
.
.
.
1
.
.
.
.
.
.
.
.
1
.
.
1
.
.
.
.
.
.
.
.
1
.
.
.
.
.
.
.
.
1
.
.
.
.
.
.
.
.
1
.
.
1
.
.
.
.
.
.
.
.
1
.
.
.
.
.
.


With these matrices, we can define the matrix R which
enfolds valid periodic gaits in its null-space:
R = −MSXP +OMTSXPH′(Tstride)
(22)
This matrix in fact compares aforementioned vector quan-
tities before and after a stride. Note that apart from state
vectors, the hip and contact torques U and Ur, the distur-
bance vector W and the variable d could also be considered
in calculating the null-space. However, it is meaningless in
practice to consider a periodic gait with constant external
disturbance. It should also be mentioned that for a valid
solution vector, initial swing velocities are zero (impact-less
model assumption) and contact positions P are also set to
zero to avoid redundant null-space dimensions.
Remember that in the previous section, there were two
variables to decide: Tss and Tds. In this section, we find
various types of gaits by selecting different combinations of
actuation and timing. We do so by considering the matrix R
which is in fact a function of timing variables. Any periodic
solution (a vector containing initial states and actuation
inputs) should lie in the null-space of R matrix. Note that
only columns attributed to non-zero values in the solution
vector are selected. We basically exclude columns related
to initial foot velocity, contact position and disturbances to
obtain the reduced matrix R0 ∈R8×15. We then find solution
manifolds by combining different actuation dimensions. In-
deed, possible actuations are swing hip and stance contact
torques (in sagittal and lateral directions) and also their time-
increasing modes. In the rest of this paper, we use human-
like body parameters for numerical simulations, where mass
distributions and geometries are taken from [33]. TableIII
lists these parameters for two adult-size and kid-size models,
used further in this paper.
Model
adult-size
kid-size
unit
Total mass
70
30
kg
Body length
1.7
1.0
m
z1
0.89
0.52
m
z2
0.32
0.19
m
z3
0.36
0.22
m
m1
45.7
19.6
kg
m2=m3
12.15
5.2
kg
w/2
0.1
0.06
m
TABLE III: Parameters of 3LP model for adult-size and kid-size models
used for simulations in this paper.
A. Pseudo-passive gaits manifold
The first choice is to see whether the system has any
pseudo-passive walking pattern or not. By pseudo-passive
we mean a gait in which swing hip and stance contact
torques are zero. The term pseudo is indeed indicating that in
stance hip, the actuators are producing or dissipating power.
It also refers to the fact that in our linear model, the legs
are stretched or shortened by prismatic actuators, as part of
model construction. To find pseudo-passive gaits, we look
at the reduced matrix R1 ∈R8×7 (not necessarily square)
extracted from R0 by excluding also the 8 columns attributed
to hip/ankle torques. Since any valid solution should lie in
the null-space of this matrix, we are interested in inspecting
singular values. We normally calculate singular values of
RT
1 R1, since any vector in the null-space of this new matrix
8
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
1.1
1.2
Tds + Tss
0
50
100
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
1.1
1.2
0
2
4
|eig(R1)|
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
1.1
1.2
0
0.5
1
Fig. 4: Square-root of the singular values of the matrix RT
1 R1 with respect
to the stride time Tstride, plotted for a adult-size model. In these plots, we
have fixed the double support time Tds = 0.3s. It is notable that around
Tstride = 0.86s, the system shows a zero singular value which corresponds to
a null-space containing infinite number of periodic solutions. These solutions
are all without swing hip or stance contact actuation, referred to as pseudo-
passive gaits.
lies in the null-space of R1 as well. The resulting singular
values are shown in Figure.4 over time.
One can clearly see that there is a time Tstride = Trelax =
0.86s where the system shows two zero singular values.
Trelax can be found by a simple root-finding algorithm. It
is obvious that one of these singular values refers to the
sagittal and the other to lateral dynamics. We can simply
calculate corresponding eigenvectors of RT
1 R1 and find the
null-space manifold, composed of v1 and v2. Note that there
is only one lateral solution as the value of d should only be
±1. However the solution in the sagittal plane can be scaled
by any arbitrary positive or negative value to obtain different
modulated speeds. Therefore the manifold of pseudo-passive
compass gaits in this case is only 1-dimensional. From
such analysis, we can also conclude that if any other stride
time is chosen, the robot cannot demonstrate pseudo-passive
forward progression and only steps in place. A demonstration
of normal pseudo-passive compass gaits can be found in
Figure.6.
B. Actuated gaits manifold
In this part we are going to find manifolds of motion which
can benefit from swing hip actuation and CoP modulation
as well. These inputs are of course containing a constant
and a time varying components for both sagittal and lateral
dynamics, as discussed in the previous section. With this
ability, we can pump energy to the swing leg and brake at
the end to produce faster swing motions. We can also apply
contact torques which modulate the CoP and resemble the
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
1.1
1.2
Tds + Tss
0
50
100
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
1.1
1.2
0
2
4
|eig(R0)|
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
1.1
1.2
0
0.02
0.04
0.06
Fig. 5: Square-root of the singular values of the matrix RT
0 R0 with respect to
the stride time Tstride, plotted for a adult-size model. In these plots, we have
fixed the double support time Tds = 0.3s. It is notable that around 0.9s, the
system does not show a zero singular value like before. However there are
7 default zero eigenvalues that can produce gaits for any choice of Tstride.
These gaits are indeed actuated, with many possibilities of swing hip or
stance contact torque profiles.
fact the CoP in human goes forward from the heel to the
toes over a swing phase. We simply perform all calculations
on R0 itself. The resulting singular values using the same
method described earlier are shown in Figure.5 over time.
It is surprising that the system does not have a distinct
zero singular value at Trelax like before. However, it has 7
singular values equal to zero that produce a null-space at any
given stride time. Each of the corresponding singular vectors
vi where 1 ≤i ≤7 have similar dimensions with Q in (9),
though with P, ˙X2 and W to be zero. These are nominal initial
states with contact point at origin, resting swing foot and no
disturbance of course. This null-space is not 7-dimensional
however. The variable d which should always be ±1 reduces
the total dimensions to 6 like before. Now one can simply
decide the timing and active actuators in order to reduce this
high dimensionality and find a unique solution.
Now for any desired speed, we have the possibility to
choose Tds, Tss and a linear combination of resulting 7 null-
space vectors. As demonstrated in pseudo-passive gaits, at
Trelax, a certain combination of these vectors can results
in zero-torque gaits. Now what if we calculate the 7-
dimensional null-space using Trelax? Could we still find a
pseudo-passive linear combination? In fact it is possible,
although no distinct zero singular value is observed in
Figure.5. The reason is that the rank of actuation space at
Trelax is equal to 5 in the 7-dimensional null-space manifold.
This means if we constrain all of them to zero for pseudo-
passive walking, we only loose 5 ranks. The other 2 ranks
are therefore dedicated to the variable d and the desired
speed, like before. So the null-space manifold of actuated
9
gaits already encompasses the one for pseudo-passive gaits
and we do not need to calculate them separately. In the next
subsection, we are going to show a few examples of walking
gaits using the null-space calculated.
C. Numerical examples
Apart from pseudo-passive walking which has a certain
timing Trelax, we are going to show same speed walking
solutions with different timing, once using only hip and
once using hip and ankle torques. From singular value
analysis, we have 7 singular vectors that can produce mo-
tion. We pack them together column-wise in a matrix V =

v1
v2
...
v7

. We also select a more human-like choice
of Tds = 0.1s and Tss = 0.6028s calculated from pseudo-
passive walking. The choice of speed will be vdes = 1m/s.
We setup an optimization problem to find linear coeffi-
cients αi which produce walking gaits of desired speed vdes
with minimal input torques. As mentioned before, this is just
a demonstration and does not have any precise meaning in
terms of energy. The purpose of this paper is not to find a
cost function for energy that produce human-like motions. It
is part of our future work to compare this model with human
gaits and their associated timing. In [8], [14] however, limit
cycles for the real robot are found through dynamic energy
minimization which is interesting and inspiring for future
works. By considering a vector of α =

α1
α2
...
α7
T,
our optimization is formulated as:
minimize
α
|SUVα|2
subject to
SdVα = ±−1
SX2,xVα = −vdes(Tstride)
(23)
Now consider the following scenarios:
• Pseudo-passive walk: which is being calculated as
mentioned before. Note that it can be shown if Trelax
is chosen, the result of our optimization is the same
pseudo-passive gait, as obtained before.
• Long double support: in this case we enforce ankle
torques to zero by adding another constraint to the
optimization:
SM3Vα = 0
(24)
Keeping the same stride time, we double Tds and
decrease Tss accordingly. Note that now the walking
cannot be pseudo-passive anymore, so the optimization
will find minimal hip torques to produce the same speed
and stride length.
• Stage walk: here we constrain ankle torques to zero
like before. Now instead of optimizing torques, we
optimize the lateral velocity in the cost function. The
optimization will give hip torques that produce a motion
with minimal lateral bounce. In this case, the biped
walks on a straight line without lateral bounce.
• CoP modulated: given the length of human foot, the
total weight and the timing of single support, we can
calculate a constantly increasing ankle torque
t
Tss τCoP
acting in sagittal plane to move the CoP to the toes
gradually during single support. In this scenario, we
force other components of ankle torques to zero and
the time-increasing component to τCoP by adding the
following constraint to the optimization:

SMa
SrMa

Vα =

0
0
τCoP
0
T
(25)
The result is a gait with time-increasing ankle torque
profile and proper hip actuation, walking at the same
frequency and speed like before.
• LIP like: in this case, keeping the original timing, we
change the model of the robot. We move most of the
weight of each leg to the torso, and also move the 3
masses closer to the pelvis by decreasing z2 and z3.
The idea is to see a behavior similar to LIP. Again we
disable all ankle torques as well.
It should be noted that these optimizations are very fast,
in the order of microseconds. It is also always possible to
find closed-form solutions as the optimizations are equality-
constrained quadratic optimizations. The accompanied Mul-
timedia Extension demonstrates different features of 3LP
while Multimedia Extension shows movies of five previously
mentioned scenarios. The 3D geometry of resulting gaits
are shown in Figure.7 while a detailed diagram of each
stride is shown in Figure.6. Our flexible model can produce
periodic gaits with different actuation schemes. We consider
piecewise linear profiles for each actuator to produce more
human-like torque and ground reaction profiles, investigated
in the final section.
It can be concluded from Figure.7 that changing different
parameters does not have major effect on the overall geom-
etry of walking. However, since our target is to develop a
template model that is easy for inverse dynamics to track,
we are interested to investigate dynamic properties of these
walking scenarios. For this purpose, we have shown CoM
velocities for all scenarios in Figure.8.
Although CoM trajectories look similar in Figure.7, they
have very different characteristics in terms of velocity varia-
tions. The LIP like model shows a large variation in sagittal
velocity. It is not so obvious how swing and torso dynamics
affect this motion at first glance. Remember that by torso
dynamics, we mean the torque required by the stance hip to
keep the torso always upright. This torque is not necessarily
zero, since the pelvis is an accelerated frame. Therefore,
hip torque can affect CoM motion considerably, specially
since the torso is relatively heavy. Moreover, although the
swing foot has smaller weight compared to other parts of
the body, it can be seen from Figure.7 that swing leg has
a faster motion during single support. Such fast motion
quadratically increases kinetic energy and therefore results in
a considerable work flow. In our model, we have described
these effects in a simplified and linear fashion, but capturing
important couplings between the 3 pendulums.
Taking a closer look at Figure.8 reveals that even maximal
CoP modulation still does not change velocity profile consid-
erably. This basically means the difference between pseudo-
passive walking and LIP is way larger than that between
pseudo-passive and CoP modulated walking. In other words,
10
Pseudo-passive walk
long double support
stage walk
CoP modulation
LIPM like
Fig. 7: Snapshots of different walking scenarios at 1m/s and 1.5step/s. These snapshots are taken at the switching times from double support to single
support or vice versa. Feet trajectories are plotted along with the projection of the CoM trajectory on the ground. In pseudo-passive walking, there is no
actuation. However one can clearly see that the model is able to produce CoM trajectory, lateral bounces and swing dynamics. In long double support case,
the motion is geometrically quite similar. Stage walking is basically producing no lateral bounce and uses proper hip torques to let the model step only on
a single straight line. On the real robot however, one should avoid foot scuffing and this motion is not feasible. CoP modulation also shows quite similar
geometry to pseudo-passive walking, though CoM trajectory (the green line on the ground) starts a bit further from the trailing leg, producing a more
symmetrical gait. The influence of CoP modulation therefore is mainly on sagittal symmetry and less variations in the sagittal speed (Figure.8). Finally,
the motion of LIP like model is also rather similar to pseudo-passive case, but we will see that it has higher CoM speed variations (Figure.8). In this case,
we impose lateral footstep distances to be similar to other scenarios. Otherwise, in LIPM pelvis width is not modeled. All corresponding walking movies
could be found in Multimedia Extension.
CoP authority can at most convert the pseudo-passive gait
to the CoP modulated gait. The available CoP authority is
hardly enough to convert the pseudo-passive gait to LIP
gait and this gap increases mainly in faster walking speeds.
Although here the speed is moderate, we can easily infer that
LIP as a template model can only operate in a very limited
range of walking speeds. Remember that in fact an inverse
dynamics or kinematics approach eventually synthesizes the
template motion with the full model by exploiting all control
authorities of the robot (including CoP). This motivates
therefore not to modulate the CoP in template level and leave
the control authority free for full-body controllers to mimic
the template motion as precisely as possible.
In this section we discussed an easy method to find man-
ifolds of periodic motions without any numerical forward
simulation of the system. Once these manifolds were found,
we also showed how to find individual solutions, based on
the type of actuation and timing desired. We only considered
gaits with minimal hip torques here. However to go further,
we would like to investigate the effect of timing and walking
speed as well. Such investigation reveals interesting energetic
properties of 3LP, discussed in the next section.
IV. COMPARISON WITH HUMAN DATA
Compared to LIP, the 3LP model is much more similar to
human locomotion, because it describes falling dynamics,
swing motion, torso balance, lateral stepping and double
support features all together. In addition to geometric sim-
ilarities, we plot ground reaction forces and joint torques
to compare the underlying dynamics that result in such
geometric similarity. For this purpose, regarding the available
data from human subjects [34], we select similar model
parameters and timing, calculate periodic manifolds and
find solutions with the same speed and CoP modulation
pattern. Such comparison is demonstrated in Figure.9. In the
following, we discuss various similarities observed in this
figure.
A. Sagittal dynamics
From the last column of Figure.9, one can observe a good
match of hip extensor and ankle plantarflexor torques as
well as Anterior-Posterior ground reaction forces. This is
despite the fact that walking speed is relatively fast and the
step length is about 80% of the leg length. Note also that
the constant and time-increasing components for hip/ankle
torques are roughly enough to describe major trends in
human curves. Nonlinear profiles of 3LP are however related
to the stance leg and generally those degrees of freedom
which are not directly controlled by desirable input torques.
The LIP model does not have hip torques and produces larger
A/P GRF, because of different CoM trajectories shown in
Figure.8.
B. Vertical GRF
By model design, the CoM height is constant and we
do not expect two peaks in the model profiles, similar to
[13], [26]. But the general trapezoidal shape is preserved,
thanks to our double support phase. The main consequence
of such profile is walking with crouched knees which looks
less human-like compared to other template models. Note
that LIP model produces the same profile as expected.
C. Transversal rotation torques
The transversal rotation torques are preserving the general
trend of the human data, but not matching very well, spe-
cially in the ankle. One major reason is that arm motions and
pelvic rotations are not considered in the model. The other
important reason is that here we just minimized all hip/ankle
torques to find a unique solution out of the manifold of
11
0
20
40
60
80
100
−0.5
0
0.5
1
1.5
hip abductor
torque [Nm/kg]
0
20
40
60
80
100
−0.2
−0.1
0
0.1
0.2
hip ext rotator
0
20
40
60
80
100
−2
−1
0
1
2
hip extensor
0
20
40
60
80
100
−0.2
−0.1
0
0.1
0.2
ankle evertor
torque [Nm/kg]
0
20
40
60
80
100
−0.4
−0.2
0
0.2
0.4
ankle ext rotator
0
20
40
60
80
100
−0.5
0
0.5
1
1.5
2
ankle plantarflexor
0
20
40
60
80
100
−0.5
0
0.5
1
M/L GRF
force [N/kg]
cycle [%]
0
20
40
60
80
100
−5
0
5
10
vertical GRF
cycle [%]
0
20
40
60
80
100
−2
−1
0
1
2
A/P GRF
cycle [%]
 
 
LIP model
Human data
3LP model
Fig. 9: Comparing dynamic profiles of 3LP and LIP with normalized human data, taken from [34]. This data is for male subjects with average weight
of 77.2kg and height of 1.8m, walking at 1.6m/s and 108steps/min. In these curves, we have demonstrated hip/ankle torques as well as ground reaction
forces. Note that our model does not have any knee and we consider ankle torques to be approximately equal to contact wrenches. Although all profiles of
3LP match the human data quite well, lateral and transversal dynamics have some discrepancies. The LIP model is however unable to describe hip torques
as it does not include swing and torso dynamics. Here we use the same CoP modulation for both LIP and 3LP for better comparison.
all available solutions. This minimization is not necessarily
realistic and human-like, as it causes wider lateral stepping,
larger lateral motion and therefore more ground moment. A
better cost function on energy might produce more human-
like gaits, although it is doubted in [35] that optimal gaits
are defined merely by energy terms. They might be highly
influenced by posture balance, at least in lower speeds. In
faster walking speeds like the human data demonstrated here
[34], humans tend to take laterally closer steps, compared to
our model. Note that the transversal moment is required to
keep the torso upright and straight ahead during the swing
phase, compensating the moment produced by the swing leg.
In LIP however, since there is no pelvis and inertia around
the yaw axis, we do not expect transversal torques.
D. Lateral dynamics
Again due to previously mentioned problem in optimizing
lateral motion, we see that our model keeps general trends,
like double peaks in the Medio-Lateral GRF, but cannot pre-
cisely describe other torque profiles. Humans normally tend
to step as close as possible to minimize lateral motions and
energy [10] (remember stage walking in Figure.7). However
humans swing their foot over an arc shape to avoid scuffing
as well. Such fine motion requires better objective functions
to find proper solutions from the available manifolds. Note
that the sagittal swing motion can influence lateral dynamics
as well [10], [20], possibly through transversal moments.
This might be another reason for the discrepancy observed
between different lateral curves. Our model however com-
pletely decouples lateral and sagittal dynamics. The only
linking parameter is stride time which relates to the zero-
velocity assumption for the swing foot. In the LIP model,
there is no pelvis modeled. However we consider a gait with
the same lateral foot-stepping for better comparison. We can
observe that although LIP does not explain hip torques, M/D
GRF forces are yet similar to 3LP.
E. Energetics
Apart from dynamics profiles, it is always interesting to
investigate energy flow in the model and compare it with
the real human. Remember that one of the main motivations
behind developing 3LP is to match humanoid dynamics as
precise as possible in the template space. Such matching
could be viewed form the power-flow perspective, inspired by
the fact that humans can walk very efficiently. Compared to
LIP, 3LP can additionally describe swing and torso dynamics
which are important aspects of locomotion, specially in faster
speeds. Since the pelvis has accelerations in the sagittal and
lateral directions, the whole upper-body requires a hip torque
to remain upright during locomotion. Our model considers
12
Fig. 6: A detailed demonstration of a full stride phase in pseudo-passive
walking where snapshots are taken every 30ms. Black arrows show the
direction of motion and the swing leg is shown in red. In this figure, lateral
bounces could be seen on right while velocities can be inferred from the
snapshots on left. The swing leg speeds up and slows down during a stride
phase while the torso has minimal speed when the swing foot is at maximum
speed. It can also be observed that the swing foot approximately follows a
straight line while the swing hip bounces laterally.
0.8
0.9
1
1.1
1.2
1.3
sagittal CoM speed [m/s]
-0.15
-0.1
-0.05
0
0.05
0.1
0.15
lateral CoM speed [m/s]
pseudo-passive walking
long double support
stage walk
CoP modulation
LIPM like
Fig. 8: Sagittal vs lateral CoM velocity trajectories for different scenarios
discussed. Note that LIP like model produces large sagittal variations.
Pseudo-passive walking is still showing high variations in both directions.
Larger lateral motions in pseudo-passive walking with respect to LIP might
be explained by higher altitude of CoM. It can be concluded however that
swing and torso dynamics clearly reduce these variations. Long double
support also reduces variations in both directions. By modulating CoP,
although lateral motion is the same as pseudo-passive walking, sagittal
variations are reduced even more and the motion is smoother. Finally, one
can see that stage walking has no lateral motion, however it has similar
sagittal motion to pseudo-passive walking. In general, we can conclude that
increasing double support time has similar effect on CoM speed variations
as CoP modulation. However it does not induce any argument on energy
efficiency.
stance hip torques to fulfill this requirement and of course
describes the influence of these torques on horizontal mo-
tions. We do not model torso movements in 3LP and assume
they are negligible, but they might induce additional hip
torques too. Although the weight of swing leg is relatively
small, its peak velocity is about two times larger than the
torso which has a heavier mass. Therefore, the peak kinetic
energy of the swing leg is quite comparable with the torso.
Such energy comes from both accelerated pelvis where the
swing leg is attached to and the swing hip torques that
can change swing dynamics. In trade off with the work
required for accelerations and decelerations of the torso,
this phenomenon explains optimal speed-frequency relations
observed in metabolic cost of human walking [36].
Although 3LP does not describe many important walk-
ing features like heel-toe motions, knee flexions and CoM
excursions, it is still very surprising that it can capture the
main optimality trend in human walking. To demonstrate this
capability, we consider two main parameters of locomotion,
stepping frequency and forward speed. Inspired by [36],
we calculate the economy of walking for different speed-
frequency combinations. Such Economy is in fact the inverse
of cost of transport, which is the total energy consumed per
unit mass, per unit distance traveled. There are many ways
to calculate such mechanical power in our model. In fact,
3LP does not simulate muscles and exact geometry of a
human and is thus, unable to precisely reconstruct the human
economy surface based on the metabolic cost. However by
integrating the mechanical power on CoM, we are able to
approximate a portion of this energy which still plays an
important role in the overall energy, according to [37]. We
are not able to model other costs like muscle activation,
maintenance and shortening heat rates though, since we do
not have muscles in 3LP.
By doing a systematic search over a grid of different
speed-frequency combinations, we calculate the net positive
work performed on the CoM per unit distance, divided by
the total mass. The calculation of such energy is rather easy
in our model, as it corresponds to the difference between
minimum and maximum kinetic energy. Since the velocity
of the CoM is a linear function of the state variable, it can
be simply related to the known initial state through H(t)
matrix which is a combination of exponential functions of
time. (10,15). The problem reduces to solving two simple
maximization and minimization problems which are fast,
thanks to the simplicity of closed form solutions. Figure.10.A
demonstrates economy contours for the choice of
Tds
Tstride =
10%. It is surprising that the model is showing a peak line
within the desired range of frequency-speeds. Such peak
in fact demonstrates the trade-off between hip torques and
contact switching acceleration/deceleration costs. The former
increases in higher frequencies as energy needs to be pumped
into the swing leg and taken out by braking at the end
of swing [38]. The latter however increases by step-length
which results more variations in the CoM velocity.
We have also repeated the same process for two other
double support time choices of 20% and 30%. The resulting
peak lines are demonstrated on Figure.10.A as well as the
13
optimal trend of human data, taken from [36] (the case of
constrained speed). Here we use the same average weight
(66kg) and height (1.7m) of human subjects in [36] as well as
average weight distribution reported in [33]. Although 3LP
is successful in identifying the trade-off, using the choice
of constant double support time ratio, it fails to match the
human data.
It can be postulated that in slow speeds, humans have
larger double support time ratios compared to higher speeds
(refer to Figure.10.A). Experiments on real humans actually
verify this postulation, suggesting a linear relation between
double support time and the walking speed [39]. Here we
implement similar simple law, making
Tds
Tstride a linear function
of walking velocity v:
Tds
Tstride
= 0.12+(2.5−v)×0.09
(26)
This relation gives a ratio of 27% at v = 0.8m/s and
12% at v = 2.5m/s, close to our three initial conjectures.
Repeating the same search process with this particular choice
of double support time, we obtain Figure.10.C with the
actual economy surface shown in Figure.10.D. It is surprising
that 3LP is able to predict almost precisely the energy-
optimal relation between frequency and speed in human
walking. Although 3LP itself cannot probably explain the
optimal double support time, it already shows that minimal
knowledge of human data is enough. The dependence of (26)
on body weight and geometry is yet to be explored in future
works. If not dependent, the equation (26) and 3LP seem to
be enough to predict optimal frequency-speed relation, given
specific body parameters.
Figure.10 is very promising and important, despite the fact
that:
• No impact or push-off is considered.
• CoM height is constant.
• No foot clearance is modeled.
• Heel-toe motions and knee flexion are not included.
• Upper body is assumed to be fixed.
• The torso has no rotation in any direction.
• Weight loading cost on stance leg is not considered.
• Walking-independent metabolic cost is not modeled.
Therefore, although the overall frequency-speed relation
is correctly predicted, the 3D surface of economy is not
matching the human data shown in Figure.10.B, taken from
[36]. The correct prediction in high-frequency and high-
speed walking conditions is surprising. Linear models are
conventionally expected to be a linearization of the actual
non-linear system, thus performing well in near-stance (small
stride-length) conditions where the geometry of 3LP is more
close to the actual system. However here, although 3LP does
not model heal-toe-knee motions and intrinsically simplifies
them with an extensible prismatic actuator, it appears to
demonstrate the same mechanical energy flow, even with
large stride length. In future works, it is still interesting to ex-
plore other sub-components of walking energy, keeping 3LP
as a core model. Inclusion of other costs might reconstruct
the actual human economy surface more precisely.
1
1.5
2
2.5
Frequency [step/s]
0.5
1
1.5
2
Speed [m/s]
A) 3LP economy, fixed Tds ratio
Human
3LP, Tds=10%
3LP, Tds=10%
3LP, Tds=20%
3LP, Tds=30%
1
1.5
2
2.5
Frequency [step/s]
0.5
1
1.5
2
Speed [m/s]
C) 3LP economy contours, variable Tds ratio
Human
3LP, Variable Tds
3LP, Variable Tds
Speed 
[m/s]  
0
2
1.5
20
1
Economy
Frequency [step/s]
0.5
2.5
2
1.5
1
D) 3LP economy surface, variable Tds ratio
40
Human
3LP, Variable Tds
3LP, Variable Tds
B) Human economy contours
1
1.5
2
2.5
Frequency [step/s]
2
1.5
1
0.5
Speed [m/s]
Fig. 10: Economy of walking, i.e. the inverse of Cost of Transport (CoT)
calculated as positive work on CoM over unit distance traveled, normalized
by body mass. Here we use the body mass of 66kg and height of 1.7m with
normal human-like mass distribution. A) Contours of 3LP walking economy,
calculated for the choice of
Tds
Tstride = 10%. Optimal peaks for this ratio as
well as other choices of 20% and 30% are plotted versus optimal relation in
human. Here 3LP gives an optimal peak, but not matching with the human
data. B) The smoothed 3D surface of human walking economy, taken from
[36]. C) 3LP Economy, using the natural choice of (26), obtained from
human data. Now, 3LP almost precisely shows similar trend with human.
C) The actual 3D surface of 3LP economy, associated with the plot C.
14
Overall, compared to LIP, while being still linear and
slightly more complicated, the proposed model is able to
describe much more features of human walking. It is very
important in hierarchical control architectures that template
models produce as accurate motions as possible to make
tracking more precise. Despite being linear and of course
operating in a more limited region of feasible states, this
model is better than LIP to describe human dynamics and
consequently, more precise for controlling humanoids. The
energy flow of 3LP is also more similar to human than LIP,
making the planned motion more natural for the humanoid
robot which has similar body features.
V. CONCLUSION
Compared to most of other template models listed in
Figure.1, our proposed model considers swing and torso
dynamics in a linear formulation. On the other hand, it is
computationally similar to LIP which is vastly used in the
literature to control real robots over relatively slow walking
speeds [23]. Other nonlinear models are also used in simpler
robots [20], but over a limited range of speeds. Template
models try to describe major dynamics of the robot in an
abstract way. This can be used either for analysis of human
motion or control of a complex robot, probably in a hierarchy
with more complex controllers. In such control paradigm,
it is important to keep computational costs as minimal as
possible, favoring future prediction.
In 3LP model, the pelvis width links linearly with the
lateral motion. This parameter can be used to find lateral
ankle/hip torques required for balancing and to determine
natural lateral foot placement. This is compared to most
of other methods like [2] where the two feet are enforced
to be apart to avoid scuffing. In literature, the timing and
footstep locations are mainly enforced by other desired
trajectories. In our model however since swing dynamics is
included, we introduce zero foot velocity assumption and let
natural periodic gaits come out of equations. This assumption
which relieves the need to calculate impact forces actually
determines the timing and periodicity conditions as well.
The proposed model can predict human walking profiles
quite well, even for relatively fast speeds where the linearity
assumption might be quite limiting for operation on the
real robot. Although IP-based models can demonstrate CoM
excursions quite well, their nonlinear nature makes them
less suitable for highly complex robots that require online
planning. There are more advanced versions of IP-based
models, including torso and swing dynamics. However again,
nonlinear equations cannot be used for per time-step future
prediction over a wide range of speeds, although they might
better describe energetics of human walking. The proposed
model keeps a trade-off between geometric and dynamic
similarities, favoring fast computation properties.
We also showed that 3LP can approximately describe
the exchange of energy despite keeping the CoM height
constant which is a less realistic assumption in fast speeds.
We would like to mention that the vertical excursion of CoM
in human locomotion, even in very fast walking at 2m/s is
still about 5cm [40] which is quite negligible compared to
the leg length of about 1m (pelvis excursion is about 7cm
however). CoM excursion is mainly determined by the stride
length which is not much larger in faster speeds. Humans in
fact increase stepping frequency together with step length
to walk faster, which highly affects swing dynamics and
requires more energy pumped by hip torques. Therefore in
faster speeds, apart from the double-peaked vertical GRF, one
should consider swing dynamics as well due to considerable
exchange of energy. LIP of course fails to describe swing
dynamics, but 3LP successfully captures them.
We have not used it for extensive bio-mechanical analysis
in this paper. Rather, we focus on the fact that such similarity
can be inspiring for generating more precise abstract plans,
used to control humanoid robots. In brief, advantages of the
3LP can be listed as following:
+
Swing dynamics.
+
Torso balancing.
+
Double support.
+
Optimal frequency-speed relation similar to human.
+
Hip/ankle actuation possibilities.
+
Computationally fast.
+
Possibility to consider hip torque limits.
+
Natural lateral motion.
+
Natural periodic gaits.
+
Pseudo-passive compass gait.
While disadvantages are:
-
Flat vertical GRF profiles.
-
Stretching legs.
-
No steering capability yet.
-
No arm motion.
-
No torso pitch/roll DoF.
It should be noted that without pelvis width like LIP,
steering is possible as demonstrated in our previous work [2].
Steering makes 3LP nonlinear, but one can compromise the
lateral motion and let inverse dynamics find proper actuation
patterns to turn around the yaw axis. In future works, we
are going to replace the LIP with the proposed model and
design better controllers to improve the performance on a
full humanoid robot. We would also like to setup MPC
with all aforementioned policies and inspirations to produce
feasible plans for faster walking. 3LP can be extended to
have two more degrees of freedom for the torso to describe
asymmetries observed in human over fast walking speeds. It
can also include actuation inputs proportional to the square
of time to produce more precise torque profiles. Among
many different advantages of 3LP, we favor its capability
to produce more natural motions. In this way, we can expect
our inverse dynamics layer to track the template model more
precisely and therefore, being able to produce more human-
like motions. This paper is accompanied with a multimedia
extension, demonstrating general features of 3LP and the dif-
ferent gaits it can produce. All the codes used in this article
as well as the multimedia extension are available online at
http://biorob.epfl.ch/page-99800-en.html.
15
VI. ACKNOWLEDGMENTS
This work was funded by the WALK-MAN project (Eu-
ropean Community’s 7th Framework Programme: FP7-ICT
611832).
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