1
Minimizing Energy Consumption and Peak Power
of Series Elastic Actuators: a Convex Optimization
Framework for Elastic Element Design
Edgar Bol´ıvar, Student Member, IEEE, Siavash Rezazadeh, Member, IEEE, and Robert Gregg, Senior Member,
IEEE
Abstract—Compared to rigid actuators, Series Elastic Ac-
tuators (SEAs) offer a potential reduction of motor energy
consumption and peak power, though these benefits are highly
dependent on the design of the torque-elongation profile of the
elastic element. In the case of linear springs, natural dynamics
is a traditional method for this design, but it has two major
limitations: arbitrary load trajectories are difficult or impossible
to analyze and it does not consider actuator constraints. Paramet-
ric optimization is also a popular design method that addresses
these limitations, but solutions are only optimal within the space
of the parameters. To overcome these limitations, we propose a
non-parametric convex optimization program for the design of
the nonlinear elastic element that minimizes energy consumption
and peak power for an arbitrary periodic reference trajectory.
To obtain convexity, we introduce a convex approximation to
the expression of peak power; energy consumption is shown
to be convex without approximation. The combination of peak
power and energy consumption in the cost function leads to a
multiobjective convex optimization framework that comprises the
main contribution of this paper. As a case study, we recover
the elongation-torque profile of a cubic spring, given its natural
oscillation as the reference load. We then design nonlinear SEAs
for an ankle prosthesis that minimize energy consumption and
peak power for different trajectories and extend the range of
achievable tasks when subject to actuator constraints.
Index Terms—Series elastic actuator, convex optimization,
energy minimization, peak power minimization, human-robot
interaction.
I. INTRODUCTION
H
UMAN-ROBOT interaction represents one of the biggest
challenges of this decade for the software and hardware
design of robotic systems. From the hardware perspective,
elasticity represents a mechanical principle that empowers
the current generation of human-friendly robotic systems. In
general, elasticity can be considered in multiple components
of the robot; one popular strategy considers an elastic element
connected in series between traditional rigid actuators and
the load. This concept, pionered by Pratt and Williamson [1],
is known as series elastic actuators (SEAs). In general, the
This work was supported by the National Science Foundation under Award
Number 1830360 and the National Institute of Child Health & Human
Development of the NIH under Award Number DP2HD080349. The content
is solely the responsibility of the authors and does not necessarily represent
the official views of the NIH or the NSF. R. D. Gregg holds a Career
Award at the Scientific Interface from the Burroughs Wellcome Fund. E.
Bol´ıvar, S. Rezazadeh, and R. D. Gregg are with the Departments of
Bioengineering and Mechanical Engineering, The University of Texas at
Dallas, Richardson, TX, 75080, USA. Email: {ebolivar, rgregg}@ieee.org,
siavash.rezazadeh@utdallas.edu.
rigid actuator could be an electric motor or a hydraulic or
pneumatic cylinder; in this work, the analysis considers only
electric motors. Powered prosthetic legs [2], [3], [4], as well
as humanoid [5] and manufacturing robots working in close
contact with human users [6], are important applications of this
technology. In traditional SEAs, the motor is connected to a
high-ratio linear transmission, then an elastic element connects
the transmission’s output to the load [7]. Designs with a low
ratio transmission are less common, but still possible due to
the increasing supply of high torque motors [8], [9].
The architecture of SEA’s offers important benefits to the
actuation of robotic systems. The elastic element in an SEA
decouples the reflected inertia of the rigid actuator and the
inertia of the load [5]. In addition, the spring can store elastic
energy and release it with enormous power. SEAs also work
as a soft load cell, suitable for measuring and controlling force
generation [10]. Robots using SEAs exploit these important
characteristics in order to reduce the energy lost during impacts
[5], improve the safety of the human and robots [11], move
loads with higher velocities [12], reduce energy consumption
of the system [13], [14], [15], and decrease peak motor power
[16], [14], [2], so a smaller/lighter motor can be used. All
these benefits are subject to the design of the SEA’s elastic
element and the motion task.
Traditionally, the design of elastic elements for SEAs to
minimize energy consumption and/or peak power follows
two main approaches: natural dynamics [17] or parameterized
optimization [2]. Natural dynamics associates first principles
with the benefits of elastic elements, which provides intuition
for the designer. However, this formulation is limited by
the kind of trajectories that can be analyzed. For instance,
the desired motion of the load may not correspond to the
natural frequency of a conservative elastic element. Even if
the required motion matches a natural frequency of oscillation,
holding the motor’s initial position may require (depending
on the transmission and the load) a reactionary torque which
dissipates energy by Joule heating (i.e., copper losses). For
this configuration, a more accurate formulation considers the
natural frequency of the double-mass single-spring system
formed by the elastic element, the motor, and load inertia
[18]. Note that the natural frequency of this system may not
have an analytic solution once nonlinear elastic elements are
considered. Natural dynamics also do not explain how to design
the elastic element when actuator constraints are considered.
Maximum deflection of the elastic element, maximum torque,
arXiv:1811.05057v3  [cs.RO]  6 Feb 2019
2
and maximum velocity of the motor are important constraints
imposed by the construction of the device; neglecting them
may lead to an infeasible design.
The second approach, parameterized optimization, describes
the elastic element as a set of parameters, e.g., stiffness constant
for the case of linear springs, and optimizes over the set of
parameters [3]. In contrast to natural dynamics, constraints are
explicitly included in the optimization problem. Parameterized
optimization, however, has limitations. It only guarantees a
minimum value within the scope of the parameter space;
optimization results are typically local and sensitive to initial
conditions, and rely on gradient-descent methods that assume
differentiability of the objective function with respect to the
parameters [19, p. 9]. In addition, parameterization assumes
a specific shape for the elastic element, which may limit
the types of springs that can be considered. For instance, a
nonlinear conservative spring could further reduce the energy
consumption compared to a parameterized quadratic spring. The
same limitations arise when the objective is to simultaneously
minimize peak power and energy; thus the design of elastic
elements that reduce both energy consumption and peak power
for SEAs remains an open question.
Our Contribution
In this work, the elastic element of an SEA is defined as a
function f : R →R, where
τela = f(δ),
(1)
and δ is the elongation and τela the torque of the elastic element.
Our contribution is to specify f(δ) as the solution of a convex
quadratically constrained quadratic program (convex-QCQP),
such that it minimizes energy consumption and peak power
of the electric motor while satisfying actuator constraints.
A preliminary version of this work [18] discussed a similar
formulation to minimize only energy consumption. In this paper,
we extend this formulation to minimize both peak power and
energy consumption as a multiobjective optimization problem.
Convexity of the optimization problem relies on a convex
approximation to the expression of peak power. Convexity
is important because it guarantees that the solution of the
optimization problem is a global optimum, i.e., if the cost is
only energy consumption, the resulting elastic element reduces
more of this energy than any other conservative elastic element.
We constrain f(δ) to be a strictly increasing monotonic
function to ensure that it represents a conservative elastic
element. In contrast to previous design methods, we do not
assume that f(δ) is defined by a set of parameters or that the
reference motion of the load resembles natural dynamics. Our
design approach also considers actuator constraints, allowing
the design of nonlinear SEAs that can perform tasks that would
be infeasible for linear SEAs or rigid actuators. In addition, our
methodology considers arbitrary periodic trajectories, which
can include a combination of periodic tasks. For example, an
SEA designed for the ankle joint of a powered prosthetic leg to
minimize energy consumption and peak power during walking
and running.
The content is organized as follows. A description of
the energy flow, expressions of peak power, and modeling
Fig. 1: Energy flow of an SEA: Dashed lines indicate that the energy
path may or may not exist depending on the construction of the device.
For instance, energy flowing from the drive to the battery requires
drivers capable of regeneration. Energy flowing from the load to the
electric motor requires that the load is high enough to backdrive the
motor-transmission system.
of an SEA are provided in Section II. This introduces the
multiobjective convex optimization problem that supports the
design of the elastic element in Section III. Section IV considers
two different tasks for the application of the optimization
framework: the natural oscillation of a nonlinear spring and the
design of an SEA for the ankle joint of a powered prosthetic leg.
Section V discusses and concludes the proposed methodology
and corresponding results.
II. SYSTEM DESCRIPTION AND MODELING
SEAs are mechatronic devices that transduce electrical
energy into mechanical and vice versa. From the energy per-
spective, they are similar to traditional electric motors; however,
their capability to store and release elastic energy creates an
additional opportunity to reduce the energy consumption and
peak power of their electric motors. This section describes the
energy flow of SEAs as an introduction to our formulation.
In this work, we analyze SEAs powered by a battery and
an electric motor, a typical scenario for portable devices such
as wearable robots [5]. The corresponding energy flow and
main components are illustrated in Fig. 1. In practice, every
component in the system is capable of dissipating energy. For
example, the battery self-discharges, the motor drive produces
Joule heat, friction in the transmission generates heat, and the
elastic elements are not purely elastic (i.e., dissipate energy
through their viscous behavior). We concentrate on the energy
consumed by the motor since it is the largest consumption in
the system. In addition, the energy consumed by other elements
of the SEA may be lumped in the expressions of energy of the
motor. For instance, viscous friction at the transmission can
be considered as additional motor viscous friction.
Energy flow in an electric motor occurs in two principal
modes of operation: actuator and generator mode. As an
actuator, the electric motor receives electrical energy from
the battery/driver and converts it into mechanical energy and
3
Electric Motor
+
Transmission
Load
Fig. 2: Diagram of an SEA. Equations (4)-(5) illustrate the system’s
equations of motion.
heat. Traditional motor drives and mechanical transmissions
are designed so that the electric motor can always operate
in this mode. A more interesting scenario occurs when the
motor works as a generator. For example, when the motor is
decelerating a load, the kinetic energy of the load and elastic
energy of the spring is transferred to the motor’s rotor to store
it as electrical energy in the battery.
However, traditional SEAs may not function in generator
mode. These designs typically have linear transmissions with
high reduction ratios. The reflected inertia of the motor after
the transmission, which is proportional to the reduction ratio
squared, is normally very high compared to the load. For
example, three recent SEA designs reflect output inertia of
360 kg, 270 kg, and 294 kg for the UT-SEA [7], Valkyrie’s
SEA [20], and THOR-SEA [21] respectively, as indicated by
[9]. As a consequence, the system requires a high load to
backdrive. An additional limitation is the motor driver. In order
to regenerate energy, motor drivers should be selected such
that the electrical energy recovered from the motion of the
rotor can flow back to charge the battery [22].
In this work, we assume the SEA has been designed such
that energy can flow from the load to the energy source and
vice versa. In other words, the load is high enough to backdrive
the motor and adequate electronics allow energy to flow to
and from the battery. In this case, the energy it consumes, Em,
is given by
Em =
Z tf
t0
τ 2
m
k2m
|{z}
Winding
Joule
heating
+ τm ˙qm
| {z }
Rotor
mechanical
power
dt,
(2)
where t0 and tf are the initial and final times of the trajectory
respectively, km is the motor constant, τm the torque produced
by the motor, and ˙qm the motor’s angular velocity. Notice
that energy associated with Joule heating can be also written
as i2
mR, since τm = imkt and km = kt/
√
R, where im is
the electric current flowing through the motor, R the motor
terminal resistance, and kt the motor torque constant [23].
In addition to energy consumption, mechanical peak power
of the motor is an important property to analyze because it is
related to the weight and size of the actuator. Its mathematical
expression is written as follows:
Peak power = ∥τm ˙qm∥∞.
(3)
A. Modeling
Figure 2 illustrates the configuration of an SEA. Using the
Newton-Euler method, the corresponding balance of torques
at the motor and load side provides the following equations of
motion
Im¨qm = −bm ˙qm + τm + τela
ηr ,
(4)
τela = g(ql, ˙ql, ¨ql, τext),
(5)
where Im is the inertia of the motor, bm its viscous friction
coefficient, τela the torque produced by the elastic element,
r the transmission ratio, η the efficiency of the transmission,
and g(ql, ˙ql, ¨ql, τext) defines the load dynamics as a function
of the corresponding load position ql, load velocity ˙ql, load
acceleration ¨ql, and the external torque applied to the load τext
[24], [23], [25]. For instance, in the case of an inertial load
with viscous friction and an external torque, the load dynamics
are defined by g(ql, ˙ql, ¨ql, τext) = −Il¨ql −bl ˙ql + τext, where
Il is the inertia of the load, and bl its corresponding viscous
friction coefficient. The variables ql, ˙ql, ¨ql, τext represent the
given load trajectory. The elongation of the elastic elastic
element is defined as δ = ql −qm/r. As seen in (4)-(5), the
elastic element cannot modify the torque required to perform
the motion, τela, but it can modify the position of the motor
such that Im¨qm + bm ˙qm reduces the torque of the motor, τm.
If periodic motion is considered, τm from (4) can be
replaced in the expressions of energy (2) to make the following
simplification
Z tf
t0
τm ˙qmdt =
Z tf
t0

Im¨qm + bm ˙qm −τela
ηr

˙qmdt,
=
Z tf
t0

bm ˙q2
m −τela ˙qm
ηr

dt +

*0
Z
˙qmf
˙qm0
Im ˙qmd ˙qm
,
=
Z tf
t0

bm ˙q2
m −τela
ηr ( ˙qm −r ˙ql + r ˙ql)

dt,
=
Z tf
t0

bm ˙q2
m −τela ˙ql
η

dt +

*0
Z δf
δ0
τela
η dδ
,
=
Z tf
t0

bm ˙q2
m −τela ˙ql
η

dt,
(6)
where ˙qmf = ˙qm0 and δf = δ0 due to periodic motion.
This simplification illustrates two concepts. First, canceling
the kinetic energy of the motor’s rotor for periodic motion
will be key to show convexity of energy consumption; when
considering non-periodic motion, the expression of energy
consumption will not be convex in our optimization framework.
Second, the mechanical energy provided to or absorbed from the
motion of the load (i.e.,
R tf
t0
τela ˙ql
η
dt) is provided or absorbed
by the electric motor regardless of the elastic element, i.e.,
series elasticity can only reduce the dissipated energy of the
motor. This is relevant for tasks that are mainly dissipative
such as level-ground locomotion [22].
B. Dynamics of the System in Discrete Form
Equations (4)-(5) illustrate the dynamics of the system for
continuous time derivatives of qm ∈R. In order to formulate
this in a convex optimization framework, we approximate the
continuous time derivative with a discrete time representation.
4
We discretize time for one period of the reference motion into
n points and approximate time derivatives using the following
matrix operation ˙qm
≈Dqm, where qm, ˙qm ∈Rn is the
discrete-time representation of qm and ˙qm, D ∈Rn×n is
D =


0
1
0
0
· · ·
−1
−1
0
1
0
· · ·
0
...
...
...
1
· · ·
−1
0


1
2∆t,
(7)
and ∆t is the sample rate. Dqm is the discrete time derivative
of qm based on the central difference method. The first and last
rows of D assume that qm represents a periodic trajectory, i.e.,
qm(n+1) = qm(1), where qm(i) is the ith element of the vector
qm. Then the equations of motion, (4)-(5), can be approximated
as
τm = (ImD2 + bmD)qm −τela
1
ηr,
(8)
τela = g(ql, ˙ql, ¨ql, τext),
(9)
δ = ql −qm
1
r ,
(10)
where D2 ∈Rn×n is the matrix that computes the second
order derivative, and τm, τela, ql, ˙ql, ¨ql, τext, δ ∈Rn represent
the discrete-time versions of the motor torque, torque of the
elastic element, load position, load velocity, load acceleration,
external torque applied to the load, and elongation of the spring,
respectively. The function g(·) defines the required values of
torque from the elastic element given a reference trajectory,
but it does not relate them with the elongation of the spring.
The objective in our optimization is to define the elastic
element. This is equivalent to finding the function f(δ) that
relates the given τela and the elongation δ. Since the elongation
is partially defined by the reference load position (10), the
optimization problem can be interpreted as finding the position
of the motor, qm, such that the energy consumption and/or
peak power of the motor are minimized. Once the position of
the motor is established, the deflection of the elastic element
is defined and can be used in conjunction with the given τela
to generate f(δ).
III. ELASTIC ELEMENT DESIGN
In this section, we introduce three optimization problems
that guide the design of the elastic element. Each optimization
problem has a different cost function: energy consumption,
peak power, or a combination of these two. However, the three
optimization problems share the same constraints to guarantee
that a conservative elastic element is obtained and the electric
motor is capable to generate the motion. The outcome of each
optimization problem is the function f(δ) in (1) to define the
elastic element of the SEA. The general idea of the optimization
strategy can be summarized as follows:
minimize
Energy consumed by the motor
and/or peak power of the motor,
subject to
Actuator constraints.
In this work, convexity of the optimization problem is an
important property for the formulation and is exploited when-
ever possible. In addition to polynomial-time solutions [26],
convexity guarantees the existence of a global minimum, and
consistent results can be found regardless of the initial point of
the optimization solver [19]. Regarding the reference motion,
the given task is defined as a reference trajectory qlref(t), its
corresponding time derivatives ˙qlref(t), ¨qlref(t), and the external
torque τext(t). In order to draw conclusions independent
of the controller design, we assume perfect tracking, i.e.,
ql(t) ≡qlref(t), ˙ql(t) ≡˙qlref(t), and ¨ql(t) ≡¨qlref(t).
A. Actuator Constraints
To represent a conservative elastic element, it is sufficient to
constrain f(δ) to be a strictly-increasing monotonic function of
δ, i.e., dτela/dδ > 0. In discrete form, dτela/dδ ≈∆τela/∆δ,
where ∆τela(i) = τela(i) −τela(i−1). Using the definition of
elongation (10), we impose monotonicity using the following
equality and inequality constraints:
A1qm < b1,
A2qm = b2,
(11)
where the non-zero entries of the sparse matrices A1 and A2
are defined as follows:
if
∆τela(i) > 0,



A1(i,i) = 1/r
A1(i,i−1) = −1/r
b1(i) = ∆ql(i)
;
if
∆τela(i) < 0,



A1(i,i) = −1/r
A1(i,i−1) = 1/r
b1(i) = −∆ql(i)
;
if
∆τela(i) = 0,



A2(i,i) = 1/r
A2(i,i−1) = −1/r
b2(i) = ∆ql(i)
;
for i = 2, ..., n. The case of i = 1 is defined similarly replacing
i −1 by n, which exploits periodicity of the trajectory. In
this work, vector inequalities such as A1qm < b1 denote
componentwise inequalities between vectors, i.e., a generalized
inequality with respect to the nonnegative orthant [19].
Our last set of constraints consider the limitations of the
electric motor and elastic element. In our framework, maximum
torque, maximum angular speed of the electric motor, and
maximum elongation of the elastic element can be constrained
within the optimization. Using (8)-(10), we impose these
limitations as the following vector inequality constraint:
h(qm) ≤umax,
(12)
where
h(qm) =


max{(ImD2 + bmD)qm −τela(ηr)−1}
max{Dqm}
max{δ}

,
umax =
τmax, ˙qmax, δmax
T ,
τmax, ˙qmax are the maximum torque and velocity of the electric
motor, and δmax the maximum elongation of the elastic element.
Notice that each element of h(qm) is the composition of the
max function and affine functions of qm [19, p. 67], which
5
represents a convex function of qm [19, p. 79]. By adding
a slack variable, one can rewrite this convex constraint into
an affine inequality constraint [19, pp. 150, 151]. This can be
used to represent all the constraints in this section as affine
equalities and inequalities with respect to qm.
B. Minimizing Energy Consumption
In this section, we use the constraints from Section III-A
along with (2) to formulate the convex optimization problem
that defines the elastic element that globally minimizes energy
consumption of the motor.
1) Cost function: Energy consumed by the motor: Using
the discrete-time formulation of the dynamics (8)-(10) and the
simplification of rotor mechanical power in (6), the energy
required by the motor (2) can be approximated in discrete form
as
Em ≈
n
X
i=1
 
τ 2
m(i)
k2m
+ τm(i) ˙qm(i)
!
∆t,
=
τ T
mτm
k2m
+ bmqT
mDT Dqm −τ T
ela ˙ql
η

∆t.
(13)
By defining τm/km = Fqm + c, where F = (ImD2 +
bmD)/km and c = −τela/(ηkmr), we rewrite (13) as
Em =

∥Fqm + c∥2
2 + bmqT
mDT Dqm −τ T
ela ˙ql/η

∆t,
= qT
mQeqm + Aeqm + ce,
(14)
where
Qe =
F T F + bmDT D


∆t,
(15)
Ae =
2cT F


∆t,
(16)
ce =
cT c −τ T
ela ˙ql/η


∆t.
(17)
2) Optimization problem: Using the definition of energy
in (14) and the constraints in Section III-A, we write the
optimization problem as follows:
minimize
qm
qT
mQeqm + Aeqm + ce,
subject to
A1qm < b1,
A2qm = b2,
h(qm) ≤umax.
(18)
which corresponds to a convex quadratic program (QP) [19].
A similar optimization problem was formulated by the authors
in the preliminary work [18].
3) Convexity of the optimization problem: The Hessian
of the cost function (14), 2Qe, is a positive semi-definite
matrix, therefore the function is convex with respect to qm [19].
Positive semi-definiteness can be shown from the definition
of Qe in (15). The Gramian matrices of F and D, i.e., F T F
and DT D, are positive semi-definite as can be seen from their
singular value decomposition. The sum of these two matrices
is also positive semi-definite. All the constraints can be written
as affine functions of qm, therefore the optimization problem
is convex.
C. Minimizing Peak Power
Design of the elastic element to minimize peak power follows
the same principles as in the case of energy consumption. The
same constraints apply to both cases, but with a different cost
function. The analysis starts with the definition of mechanical
power of the electric motor, pm ∈Rn, as a function of qm:
pm(i) = τm(i) ˙qm(i),
=

(ImD2(i,∗) + bmD(i,∗))qm −τela(i)
1
ηr

D(i,∗)qm,
= qT
m

ImDT
2(i,∗)D(i,∗) + bmDT
(i,∗)D(i,∗)

|
{z
}
Gi
qm
−τela(i)
1
ηrD(i,∗)
|
{z
}
−Hi
qm,
(19)
= qT
mGiqm + Hiqm,
(20)
for i = 1, 2, ..., n, where D(i,∗) refers to the ith row of the
matrix D ∈Rn×n. In other words, every element of the power
vector, pm, is a quadratic expression of the motor position.
In contrast to energy consumption, the following subsection
will show that peak power, ∥pm∥∞, is not a convex function
of qm. To keep the advantages of a convex optimization
problem, we propose a convex approximation to the expression
of peak power. This approximation neglects the torque due
to inertia, and maximum power is considered instead of
peak power, i.e., we do not take the absolute value. Other
approximations of peak power are reported in the literature for
numerical optimization, for instance, pseudo-power has been
defined in [27]. Our simulation results indicate that our convex
approximation resembles the actual expression of peak power,
and that minimizing the convex version minimizes the actual
expression as well.
1) Optimization problem: Using the definitions from the
previous section, we write the optimization problem as
minimize
qm
∥pm∥∞,
subject to
A1qm < b1,
A2qm = b2,
h(qm) ≤umax.
(21)
Peak power, ∥pm∥∞, is not a convex function of qm because,
in general, the infinity norm of a set of quadratic functions
is not convex. In addition, every quadratic function may not
be convex as shown by its Hessian, Gi, in (19). This matrix
is non-definite because the matrix ImDT
2(i,∗)D(i,∗) may have
positive and negative eigenvalues.
2) Simplification to obtain a convex optimization problem:
Peak power can be approximated with maximum power to
obtain a convex version of the cost function. The max function,
max{f1, f2, ..., fn}, is convex when each function f1, f2, ..., fn
is also convex [19]. This justifies the use of the max function
instead of the infinity-norm. In our case, each function fi
corresponds to the quadratic expression (20). These expressions
are convex if and only if Gi is positive semi-definite. From
its definition (19), the matrix Gi is positive semi-definite if
6
inertial torques are neglected. With this in mind, we define
pcvx
m
∈Rn, a convex approximation of actual power pm, as
follows:
pcvx
m(i) := qT
m

bmDT
(i,∗)D(i,∗)

|
{z
}
Gcvx
i
qm −τela(i)
1
ηrD(i,∗)qm,
= qT
mGcvx
i
qm + Hiqm.
With this approximation the convex optimization problem is
written as
minimize
qm
max{pcvx
m },
subject to
A1qm < b1,
A2qm = b2,
h(qm) ≤umax.
(22)
Convexity can be shown since every element in the vector pcvx
m
is a convex-quadratic function of qm, i.e., the matrix Gcvx
i
is
positive semi-definite for all i since it is the Gramian matrix
of D(i,∗). The maximum of a set of convex functions is also
convex [19, p. 80]. All the constraints are affine with respect
to qm, therefore the optimization problem is convex.
3) Regularization to avoid high accelerations: The solution
to the optimization problem (22) may result in a position
trajectory, qm, with very high accelerations. This has a great
impact in the calculation of the actual peak power, especially
for SEAs using a high reduction-ratio transmission. One way
to solve this is to penalize solutions with high accelerations.
This can be achieved by including acceleration of the motor
in the cost function as follows:
minimize
qm
max{pcvx
m } + γ1 ∥D2qm∥∞,
subject to
A1qm < b1,
A2qm = b2,
h(qm) ≤umax,
(23)
where γ1 is a scalar constant that controls the influence of
the peak acceleration relative to the maximum of the convex
expression of power. Section IV elaborates on the selection
of γ1. The optimization problem (23) remains convex. The
term γ1 ∥D2qm∥∞corresponds to the infinity-norm function,
which is convex, composed by the linear expression D2qm.
The composition of a convex function with an affine function
results in a convex function [19].
D. Multiobjective Optimization: Energy Consumption and Peak
Power
This section describes the optimization problem that simulta-
neously minimizes energy consumption and peak power. These
two objectives are not always competing; however, when they
do, a trade-off curve is useful to guide the design process. For
example, a global minimum that reduces energy consumption
can lead to higher peak powers compared to actuators without
elastic elements. This motivates a multiobjective optimization
framework where the designer can choose the appropriate
trade-off based on the design specifications. The proposed
methodology combines the optimization problems introduced
in Sections III-B2 and III-C3 to generate the following
multiobjective program:
minimize
qm
θγ2
qT
mQeqm + Aeqm + ce


+ (1 −θ) (max{pcvx
m } + γ1 ∥D2qm∥∞) ,
subject to
A1qm < b1,
A2qm = b2,
h(qm) ≤umax,
(24)
where γ2 is a the factor that scales the magnitude of energy
consumption relative to peak power, and θ ∈[0, 1] is the factor
that controls the trade-off between energy consumption and
peak power in the solution of the optimization problem. Using
θ = 0 indicates that only peak power will be minimized, and
θ = 1 minimizes only energy consumption. It can be shown that
the solution to the optimization problem in (24) corresponds to
a pareto optimal point [19, p. 178]. The optimization problem
remains convex since it is the positive sum of convex functions.
IV. CASE STUDIES
In this section, we apply the convex optimization framework
in (24) to design the SEA’s elastic element for two different
reference trajectories: natural oscillation of a nonlinear spring,
and motion of the human ankle during level-ground walking
and running. The first case is of interest because it compares the
numerical results of the optimization with the well established
analysis of a mass-spring system. This comparison validates
the optimization results. For example, if the reference position
of the SEA corresponds to the natural oscillation of a nonlinear
spring and the cost function is the energy dissipated by viscous
friction, the optimal elastic element should be the same as
the nonlinear spring used to generate the trajectories. Section
IV-A illustrates this case and compares it with the optimization
results using two different cost functions: winding losses and
total energy consumption.
Section IV-B discusses the design of the SEA’s elastic
element for the case of the ankle joint of a powered prosthetic
leg. For this application, minimizing peak power and energy
consumption using SEAs is challenging when considering
multiple locomotion tasks. For example, design for level ground
walking and running normally leads to different elastic elements
for these tasks [28]. In our case study, we design a single
nonlinear elastic element that minimizes peak power and energy
consumption for both tasks subject to the parameters and
constraints for the specific motor-transmission-load system.
This illustrates that SEAs with nonlinear elastic elements
can be designed for multiple tasks, which is a scenario that
traditionally utilizes variable stiffness actuators (VSA) [29].
Section IV-B2 includes elongation constraints for the elastic
element to demonstrate that SEAs using nonlinear springs can
perform tasks that a rigid actuator or an SEA with linear springs
cannot perform.
For each case study, we used the following configuration. The
trajectory of the load, i.e., ql, ˙ql, ¨ql, and τext in (5), is given and
the optimization problem is numerically solved using CVX, a
package for specifying and solving convex programs [30], [31].
In all simulations, CVX executed the solver Mosek [32] with
7
TABLE I: Simulation parameters: motor ILM85x26 from RoboDrive.
Parameter
ILM85x26
Units
Motor torque constant, kt
0.24
N·m/A
Motor terminal resistance, R
323
mΩ
Rotor inertia, Imr
1.15
kg·cm2
Rotor assembly, Ima
0.131
kg·cm2
Motor inertia, Im = Imr + Ima
1.246
kg·cm2
Gear ratio, r
22
Motor viscous friction, bm
60
µN·m·s/rad
Max. motor torque, τmax
8.3
N·m
Max. motor velocity, ˙qmax
1500
rpm
Nominal power output, @48 V
410
W
Peak power output, @(48 V, τmax)
1259
W
Load
 
Load
 
Motor and 
Transmission
 
(a)
(b)
Fig. 3: (a) Single-mass spring system. The elastic element describes
the nonlinear spring with τe = αq3
l . (b) Double-mass single-spring
system. The equilibrium position of the elastic element is ql = qm/r,
elongation is defined as δ = ql −qm/r. Motor and transmission are
considered to be backdrivable.
precision settings cvx_precision best. For the two case
studies, we used the parameters of a commercial frameless
motor (Model: ILM 85x26, RoboDrive, Seefeld, Germany,
Table I), motivated by the motor selection of the second-
generation of the powered prosthetic leg at the University of
Texas at Dallas [33]. This motor has a high nominal and peak
torque, requiring a lower reduction ratio. This configuration
favors backdrivability of the SEA.
A. Natural Oscillation of a Nonlinear Spring
This case study analyzes the design of the elastic element
to minimize energy consumption, which can be separated into
the following cost functions: energy associated with viscous
friction, winding heat losses, and the total dissipated energy.
In this section, we solve the optimization problem for each of
these cost functions. Figure 3-(a) describes a single mass-spring
system with a nonlinear spring, τela = αq3
l , and corresponding
equation of motion τela = −Il¨ql, where Il = 125 g·m2 is the
inertia of the load and α = 40 N·m/rad3. The main objective
of this case study is to validate the optimization results against
the known natural-oscillation of a cubic spring. To simplify
the analysis, we consider no actuator constraints and a no-loss
mechanical transmission, i.e., η = 1. Section IV-B describes
the design with actuator constraints and inefficiencies in the
transmission. Figure 4 illustrates the motion of the load for an
initial displacement, ql(0) = π/2 rad. This natural vibration
is defined as our reference motion. The SEA, Fig. 3-(b),
can generate this motion with the motor holding its initial
position if the elastic element matches the nonlinear spring in
TABLE II: Energy expenditure. Natural oscillation of cubic spring.
Cost
Joule
Viscous
Total
function
heating [J]
friction [J]
Energy [J]
Viscous friction
20.175
0.000
20.175
Winding losses
0.008
20.511
20.519
Total energy
4.972
4.607
9.579
Fig. 3-(a). However, this approach may not be energetically
efficient. If the load is high enough to backdrive the system,
the motor must apply a reactionary torque to hold its initial
position. This torque requires a current that generates heat
losses at the motor’s winding due to Joule heating. In contrast,
we can solve the optimization problem in (24) to find the
elastic element that minimizes the total energy expenditure
(i.e., winding losses and viscous friction). To evaluate the
proposed methodology, we solved the optimization problem
for each of the following cost functions: energy dissipated by
winding Joule heating, energy dissipated by viscous friction,
and total energy consumption. Each of these cost functions
is formulated from appropriate modifications to the matrices
Qe and Ae in (14). The resulting elastic elements, torques,
and positions of the motor are illustrated in Fig. 5. Table II
summarizes the energy expenditure for each case.
0
100
200
−100
0
100
τext
Time [ms]
τext [N·m]
−1
0
1
ql
ql [rad]
Fig. 4: The reference trajectory of the load is defined by the natural
oscillation of the single mass-spring system in Fig. 3-(a) with
α =40 N·m/rad3, Il =125 g·m2, and ql(0) = π/2 rad.
Minimizing viscous friction leads to the same elastic element
as in Fig. 3-(a), which validates the numerical results with
respect to first principles. The energy required to produce the
motion is then 20.175 J, which is all dissipated in the motor’s
winding. In contrast, minimizing the total energy consumption
results in a cost of 9.579 J, 52% less compared to the previous
case. The elastic element is nonlinear but is not defined by
τela = 40δ3, and the motor no longer remains stationary.
Minimizing only the energy dissipated by the motor’s winding
leads to an elastic element that minimizes as much motor-torque
as possible, as seen in Fig. 5. This SEA spring approximates
the natural dynamics of the double-mass single-spring system
defined by the inertia of the load and the motor.
B. Ankle Joint of a Powered Prosthetic Leg
This section presents the design of the elastic element of an
SEA for the ankle joint of a powered prosthetic leg. In this
case study, we constrain the maximum absolute value of torque
and velocity of the motor to be within the specifications of the
datasheet (Table I). In Section IV-B1, we design the elastic
8
−4
−2
0
2
4
−100
0
100
≈40δ3
δ [rad]
τela [N·m]
Viscous friction
Winding losses
Total energy
0
5 · 10−2 0.1
0.15
0.2
0.25
−5
0
5
Time [ms]
τm [N·m]
0
5 · 10−2 0.1
0.15
0.2
0.25
−50
0
50
Time [ms]
qm [rad]
Fig. 5: Optimization results considering natural oscillation of a
nonlinear spring as the reference motion. The solid line corresponds
to the elastic element that minimizes the energy consumption due to
viscous friction. It matches τela = 40δ3, the nonlinear spring used in
the single mass-spring system. The dotted line describes the elastic
element that minimizes winding losses due to Joule heating. The
dashed line describes the elastic element that minimizes both winding
losses and viscous friction, i.e., total energy. The corresponding energy
expenditure is shown in Table II.
element without constraining its elongation. In Section IV-B2,
we constrain the maximum elongation of the spring to show
that nonlinear springs can extend the range of operation of
SEAs, i.e., allow SEAs to perform tasks that are not possible
with linear springs or rigid actuators. These sections analyze a
system with energy losses at the transmission, i.e., η = 0.8.
1) Unconstrained elongation: One of the advantages of the
proposed methodology is the capability to analyze arbitrary
periodic reference trajectories. Taking advantage of this flexi-
bility, the design of the elastic element considers three different
tasks for the prosthetic leg: level-ground walking or running
as shown in Fig. 6, and a combination of walking and running.
The walking and running trajectory combines four steps of
walking and one of running, which corresponds to the case
where the user runs 20% of the time and walks during the
remaining portion of operation.
The multiobjective optimization involves analysis of the
0
0.5
ql [rad]
Walking
Running
0
20
40
60
80
100
−100
−50
0
Percentage of gait cycle [%]
τext [N·m]
Fig. 6: Motion of the human ankle during level ground walking [34]
and running [35]. The gait cycle begins with heel contact of one foot
and finishes with the subsequent occurrence of the same foot. In the
lower figure, the external torque, τext, is defined for a 75 kg subject.
trade-off curves in Fig. 7. Results are reported relative to the
peak power and dissipated energy of a rigid actuator performing
the same task. The dissipated energy with a rigid actuator is
59.27 J and 33.75 J per gait cycle for walking and running
tasks, respectively, while its peak power reaches 325.29 W
and 1111.12 W for these tasks. Note that the dissipated energy
values are per cycle, and because the walking period is longer
than running (1.14 s walking [34] and 0.66 s for running [35]),
it dissipates more energy, even though the running peak torques
are higher (Fig. 6).
To generate the trade-off curves in Fig. 7, CVX solved
the optimization problem (24) using 30 different values for
θ ∈[0, 1]. The points between θ = 0 and θ = 1 were sampled
from a sigmoid function to have an adequate distribution of
points in the trade-off curves. In the proposed methodology, γ1
and γ2 in (24) control the relative magnitude of the two costs.
γ1 scales the maximum acceleration with respect to the convex
simplification of peak power and γ2 scales energy consumption
relative to peak power. Comparing the relative magnitude of
peak power and energy consumption, we defined γ1 = 0.02
and γ2 = 300.
For the given trajectories and motor configuration, there is a
correlation between reduction of convex power and peak power,
as shown in Fig. 7. In addition, the relationship between energy
consumption and peak power represents a weak trade-off in the
multiobjective optimization [19, p. 182], i.e., a small increase
in the dissipated energy will imply a significant reduction
of peak power. For example, in the walking gait trade-off
curve of Fig. 7, point (a) represents a reduction of 4.3% of
dissipated energy and an increase of 187% of peak power
when compared to a rigid actuator. A significant reduction of
peak power is achieved by consuming a little more of energy,
as seen in point (b) where dissipated energy and peak power
reduce 1.61% and 65.91% respectively. The other two tasks
9
−3.7
−1.1
106.5
−65.9
a: -4.3%, 187%
b,c: -1.6%, -65.91%
Walking
max{pcvx
m }
∥pm∥∞
Linear
−11.5
−5.4
−1.9
−24.2
Running
a: -12.3%, 2.7%
b: -11.9%, -24.4%
c: -6.5%, -23.1%
Peak power relative to rigid case [%]
−4
−1.9
−1.9
−24.2
Walking & running
a: -4.6%, 2.7%
b: -4.2%, -24.4%
c: -2.3%, -22.3%
Dissipated energy relative to rigid case [%]
Fig. 7: Trade-off curves for the tasks of level ground walking, running,
and walking and running. Point (a) in each graph indicates the results
when the cost function is only energy consumption, i.e., θ = 1 in (24).
Point (b) represents the optimal point based on the trade-off analysis.
Point (c) represents the results when the cost function is only maximum
convex power, i.e., θ = 0 in (24). Relative percentage is computed
as 100(xoptim −xrigid)/xrigid, where xoptim and xrigid are the
dissipated energy or peak power from the optimization algorithm
and rigid case respectively. Peak power and its corresponding convex
approximation are denoted by ∥pm∥∞and max{pcvx
m } respectively.
As a reference, we include the energy and peak power savings using
a linear series spring (legend: Linear).
use the same principle to define point (b), the solution of
the multiobjective optimization problem. The selection of this
point depends on the priorities of the designer. For example, if
energy consumption is the main cost to minimize, then point
(a) should be selected. For the case of peak power, point (c)
may represent the best choice of the elastic element. Point
(c) is defined with respect to the convex approximation of
peak power, max{pcvx
m }. Thus, it is possible to find optimal
solutions with lower peak power and θ ̸= 0, e.g., point (b) for
running and walking & running. Figure 8 illustrates the optimal
elastic element for each of the optimal trade-off points (b).
These conservative elastic elements are nonlinear and satisfy
the constraints defined in Section III-A.
2) Constrained elongation: In this section, we design the
series elastic element constraining its maximum elongation.
This exemplifies the case where nonlinear series elasticity
can achieve tasks that linear SEAs or rigid actuators cannot
0
0.2
0.4
0.6
0.8
1
0
50
100
Walking
Running
Walking & running
Elongation elastic element, δ, [rad]
τela [N·m]
Fig. 8: Each elastic element corresponds to the solution described by
the optimal points (b) in Fig. 7, i.e., represents the optimal point based
on our trade-off analysis for each trajectory. Each solution weights
energy consumption and peak power differently as the value of θ is
different in each case.
accomplish. In this case study, the task is defined by the
kinematics and kinetics for the ankle joint of an 85 kg subject
during running [35]. In addition to the constraints in torque
and velocity of the motor (less than 8.3 N·m and 1500 rpm
respectively, Table I), we limit the maximum elongation of
the spring to be less than 0.4 rad. This constraint may be
imposed by the geometry of the mechanism or maximum
elongation of the spring. For the rigid actuator, the task will
require a peak torque of 9 N·m and maximum absolute speed of
1674 rpm, which is outside the motor’s specifications using a
48 V power source. The solution of the linear spring approaches
the characteristics of the rigid actuator as we constrain the
elongation. In contrast, the nonlinear spring in Fig. 9 elongates
less than 0.4 rad, while the motor torque and velocity remain
within specifications. The dissipated energy using the optimal
nonlinear spring is 40.67 J per cycle and the peak power is
1161.96 W.
0
0.1
0.2
0.3
0.4
0
50
100
150
2562 kN·m/rad
303 N·m/rad
Elongation elastic element, δ, [rad]
τela [N·m]
Fig. 9: Optimal nonlinear elastic element subject to elongation
and motor constraints. Local values of stiffness are reported in
the graph. At small elongations the elastic element is almost rigid
(2562 kN·m/rad). A parametric representation of this elastic element
would involve a polynomial of high degree, which may be cumbersome
for existing methods of design.
V. DISCUSSION AND CONCLUSION
The proposed framework formulates the design of elastic
elements for SEAs as a convex optimization problem to
minimize peak power and energy consumption. The formulation
considers arbitrary periodic trajectories, actuator constraints,
and a non-parametric description of the torque-elongation
relationship of the elastic element. The case studies illustrate
10
practical implications of the trade-off analysis between peak
power and energy consumption, the accuracy of the convex
approximation of peak power, and how to increase the range
of operation of SEAs using nonlinear springs.
The trade-off between peak power and energy consumption
depends on the characteristics of the load, motor, and transmis-
sion. For instance, for the results in Section IV, minimizing
peak power produced a reduction in both peak power and energy
consumption for all the cases studied. However, minimizing
energy alone increased peak power. Using a high-speed low-
torque motor, as in [18], had the opposite effect, where
minimizing energy consumption also decreased peak power.
Though results depend on the characteristics of the load, motor,
and transmission, our simulation results show the importance
of considering peak power for design, as seen by the weak
trade-off between peak power and energy consumption in Fig. 7.
Similar results are reported in [36] using linear springs.
Convexity of energy consumption indicates that the optimal
point corresponds to a global optimum, which is exploited to
obtain the maximum amount of energy that can be reduced
using SEAs. This upper bound can also guide the multiobjective
version of the optimization. The designer can trade an increase
in energy consumption for a reduction of peak power using as a
reference the maximum amount of energy that can be reduced.
For example, for the given motor-transmission configuration
during level ground walking, the maximum amount of energy
that can be reduced is about 4.3% with a corresponding increase
of 187% of peak power. Reducing dissipated energy to 1.61%
correlates to a decrease of 65.91% of peak power. Saving 1.61%
of the dissipated energy may seem small, but comparing it
with the maximum savings of 4.3% gives a better perspective.
The convex approximation of peak power is close to the
actual expression of power depending on the motor configura-
tion. For all the cases considered in this article (Fig. 7), the
approximation was accurate enough to provide a significant
reduction of the actual expression of peak power. However,
using a high transmission ratio increases the relevance of
inertial torques in the definition of peak power. In this case
the convex approximation is less likely to yield an accurate
estimation. The designer can evaluate the performance of the
approximation offline to guide the design process. Using the
convex approximation has significant advantages with respect
to the quality of the solution and the time required to solve the
optimization problem. In this case, the proposed convex QCQP
program can be efficiently solved in polynomial-time [26],
which is useful for an actuator that can modify its stiffness
during operation, e.g., a VSA.
The capability to analyze arbitrary periodic trajectories is
useful for the design of spring profiles that minimize energy
consumption and peak power for a variety of tasks, e.g., the
walking and running trajectories in Section IV-B1. In addition,
nonlinear SEAs can accomplish tasks that are impossible for
linear SEAs or rigid actuators, as shown in Section IV-B2. This
property of nonlinear springs and the possibility to analyze any
set of periodic trajectories enables our methodology to design
SEAs that perform a larger set of tasks.
In practice, our methodology presents some challenges. The
most important is the fabrication of elastic elements with arbi-
trary torque-elongation behavior. For instance, manufacturing
the spring prescribed in Fig. 9 requires additional mechanical
design as off-the-shelf springs normally have a linear torque-
elongation relationship. However, manufacturing of nonlinear
springs has been already explored in robotic applications
[37]. Jutte and Kota [38] introduced a generalized nonlinear
spring synthesis methodology for any prescribed nonlinear
load-displacement function. Custom springs can also be de-
signed using topology optimization, especially for hyperelastic
structures [39]. Vanderborght et al. [40] summarized available
techniques to produce nonlinear springs; cam, hypocycloid, and
double-slider mechanisms coupled with linear springs are some
examples of these techniques [41], [42], [43]. It is important
to note that some of these techniques may introduce energy
losses, such as dissipated heat due to friction in the mechanisms,
and result in elastic elements that may not be conservative.
Materials with inherent nonlinear elasticity, such as polymers,
may exhibit viscoelastic behavior with its respective energy
losses [4]. To account for these losses, our cost function can
be extended to include viscous friction losses in the spring,
and the optimization problem remains convex. However, we
did not include these losses explicitly as elastic elements are
very efficient in practice [44].
In summary, the proposed methodology designs conservative
elastic elements that minimize energy consumption and peak
power for arbitrary periodic reference trajectories subject to
actuator constraints. Simultaneous minimization of energy
consumption and peak power is not trivial and motivates
the trade-off analysis described in Section IV. The use of
general conservative elastic elements may increase the range
of operation in which the benefits of SEAs apply. Future work
will focus on the implementation of the optimal elastic element
and the formulation of a motor control algorithm to perform
a task with minimum energy consumption that satisfies the
torque-speed characteristics of the motor.
REFERENCES
[1] Pratt, G., and Williamson, M., 1995. “Series elastic actuators”. In Proc.
1995 IEEE/RSJ Int. Conf. Intell. Robot. Syst. Hum. Robot Interact. Coop.
Robot., Vol. 1, IEEE Comput. Soc. Press, pp. 399–406.
[2] Hollander, K. W., Ilg, R., Sugar, T. G., and Herring, D., 2006. “An
Efficient Robotic Tendon for Gait Assistance”. J. Biomech. Eng., 128(5),
p. 788.
[3] Rouse, E. J., Mooney, L. M., and Herr, H. M., 2014. “Clutchable series-
elastic actuator: Implications for prosthetic knee design”. Int. J. Rob.
Res., 33(13), pp. 1611–1625.
[4] Bolivar, E., Allen, D., Ellson, G., Cossio, J., Voit, W., and Gregg, R.,
2016. “Towards a Series Elastic Actuator with Electrically Modulated
Stiffness for Powered Ankle-Foot Orthoses”. In 2016 IEEE Int. Conf.
Autom. Sci. Eng., IEEE, pp. 1086–1093.
[5] Hurst, J., and Rizzi, A., 2008. “Series compliance for an efficient running
gait”. IEEE Robot. Autom. Mag., 15(3), pp. 42–51.
[6] Bischoff, R., et al., 2010. “The KUKA-DLR Lightweight Robot arm-a
new reference platform for robotics research and manufacturing”. In
Int. Symp. Robot. and German Conf. Robot., C. Loughlin, ed., Vol. 34,
pp. 1–8.
[7] Paine, N., Oh, S., and Sentis, L., 2014. “Design and Control Consid-
erations for High-Performance Series Elastic Actuators”. IEEE/ASME
Trans. Mechatronics, 19(3), pp. 1080–1091.
[8] Hirzinger, G., et al., 2002. “DLR’s torque-controlled light weight robot
III - are we reaching the technological limits now?”. In IEEE Int. Conf.
Robot. Autom., Vol. 2, IEEE, pp. 1710–1716.
[9] Schutz, S., et al., 2016. “RRLAB SEA A highly integrated compliant
actuator with minimised reflected inertia”. In IEEE Int. Conf. Adv. Intell.
Mechatronics, Vol. 2016-Septe, IEEE, pp. 252–257.
11
[10] Robinson, D., Pratt, J., Paluska, D., and Pratt, G., 1999. “Series elastic
actuator development for a biomimetic walking robot”. In IEEE/ASME
Int. Conf. Adv. Intell. Mechatronics, IEEE, pp. 561–568.
[11] Bicchi, A., and Tonietti, G., 2004. “Fast and ”Soft-Arm” Tactics”. IEEE
Robot. Autom. Mag., 11(2), pp. 22–33.
[12] Braun, D. J., et al., 2013. “Robots Driven by Compliant Actuators:
Optimal Control Under Actuation Constraints”. IEEE Trans. Robot.,
29(5), pp. 1085–1101.
[13] Vanderborght, B., et al., 2006. “Exploiting Natural Dynamics to Reduce
Energy Consumption by Controlling the Compliance of Soft Actuators”.
Int. J. Rob. Res., 25(4), pp. 343–358.
[14] Jafari, A., Tsagarakis, N. G., and Caldwell, D. G., 2013. “A novel
intrinsically energy efficient actuator with adjustable stiffness (AwAS)”.
IEEE/ASME Trans. Mechatronics, 18(1), pp. 355–365.
[15] Ding, Y., and Park, H.-w., 2017. “Design and Experimental Implementa-
tion of a Quasi-Direct-Drive Leg for Optimized Jumping”. In IEEE/RSJ
Int. Conf. Intell. Robot. Syst.
[16] Jafari, A., Tsagarakis, N. G., and Caldwell, D. G., 2011. “Exploiting
natural dynamics for energy minimization using an Actuator with
Adjustable Stiffness (AwAS)”. In 2011 IEEE Int. Conf. Robot. Autom.,
IEEE, pp. 4632–4637.
[17] Verstraten, T., et al., 2016. “Series and Parallel Elastic Actuation: Impact
of natural dynamics on power and energy consumption”. Mech. Mach.
Theory, 102, pp. 232–246.
[18] Bol´ıvar, E., Rezazadeh, S., and Gregg, R., 2017. “A General Framework
for Minimizing Energy Consumption of Series Elastic Actuators With
Regeneration”. In Dyn. Syst. Control Conf., ASME, p. V001T36A005.
[19] Boyd, S., and Vandenberghe, L., 2004. Convex Optimization. Cambridge
University Press, Cambridge.
[20] Zhao, Y., Paine, N., Kim, K. S., and Sentis, L., 2015. “Stability and
Performance Limits of Latency-Prone Distributed Feedback Controllers”.
IEEE Trans. Ind. Electron., 62(11), pp. 7151–7162.
[21] Knabe, C., Lee, B., Orekhov, V., and Hong, D., 2014. “Design of a
Compact, Lightweight, Electromechanical Linear Series Elastic Actuator”.
In Vol. 5B 38th Mech. Robot. Conf., ASME.
[22] Seok, S., et al., 2015. “Design Principles for Energy-Efficient Legged Lo-
comotion and Implementation on the MIT Cheetah Robot”. IEEE/ASME
Trans. Mechatronics, 20(3), pp. 1117–1129.
[23] Verstraten, T., Mathijssen, G., Furn´emont, R., Vanderborght, B., and
Lefeber, D., 2015. “Modeling and design of geared DC motors for energy
efficiency: Comparison between theory and experiments”. Mechatronics,
30, pp. 198–213.
[24] Spong, M. W., 1987. “Modeling and Control of Elastic Joint Robots”. J.
Dyn. Syst. Meas. Control, 109(4), p. 310.
[25] Verstraten, T., et al., 2017. “On the Importance of a Motor Model
for the Optimization of SEA-driven Prosthetic Ankles”. In Wearable
Robot. Challenges Trends. Biosyst. Biorobotics, Vol. 16. Springer, Cham,
pp. 403–407.
[26] Nesterov, Y., and Nemirovskii, A., 1994.
Interior-Point Polynomial
Algorithms in Convex Programming. Society for Industrial and Applied
Mathematics.
[27] Hong, Y., Kim, J., and Park, F. C., 2017. “Comparative analysis of
energy-based criteria for dynamics-based robot motion optimization”. In
2017 IEEE Conf. Control Technol. Appl., IEEE, pp. 175–180.
[28] Eslamy, M., Grimmer, M., and Seyfarth, A., 2012.
“Effects of
unidirectional parallel springs on required peak power and energy in
powered prosthetic ankles: Comparison between different active actuation
concepts”. IEEE Int. Conf. Robot. Biomimetics, pp. 2406–2412.
[29] Wolf, S., et al., 2016. “Variable Stiffness Actuators: Review on Design
and Components”. IEEE/ASME Trans. Mechatronics, 21(5), pp. 2418–
2430.
[30] Grant, M., and Boyd, S., 2014. CVX: Matlab Software for Disciplined
Convex Programming, version 2.1.
[31] Grant, M. C., and Boyd, S. P., 2008.
“Graph Implementations for
Nonsmooth Convex Programs”. In Recent Adv. Learn. Control, Vol. 371.
Springer London, London, pp. 95–110.
[32] Aps, M., 2017. MOSEK Optimization Suite.
[33] Elery, T., Rezazadeh, S., Nesler, C., Doan, J., Zhu, H., and Gregg, R.,
2018.
“Design and Benchtop Validation of a Powered Knee-Ankle
Prosthesis with High-Torque, Low-Impedance Actuators”. In IEEE Int.
Conf. Robot. Autom.
[34] Winter, D., 1991. The biomechanics and motor control of human gait:
normal, elderly and pathological, 2nd ed. University of Waterloo Press.
[35] Novacheck, T. F., 1998. “The biomechanics of running”. Gait Posture,
7(1), pp. 77–95.
[36] Grimmer, M., and Seyfarth, A., 2011. “Stiffness adjustment of a Series
Elastic Actuator in an ankle-foot prosthesis for walking and running:
The trade-off between energy and peak power optimization”. In 2011
IEEE Int. Conf. Robot. Autom., IEEE, pp. 1439–1444.
[37] Hawkes, E. W., and Cutkosky, M. R., 2018. “Design of Materials and
Mechanisms for Responsive Robots”. Annu. Rev. Control. Robot. Auton.
Syst., 1(1), pp. 359–384.
[38] Jutte, C. V., and Kota, S., 2008. “Design of Nonlinear Springs for
Prescribed Load-Displacement Functions”.
J. Mech. Des., 130(8),
p. 081403.
[39] Chen, F., Wang, Y., Wang, M. Y., and Zhang, Y., 2017. “Topology
optimization of hyperelastic structures using a level set method”. J.
Comput. Phys., 351, pp. 437–454.
[40] Vanderborght, B., et al., 2013. “Variable impedance actuators: A review”.
Rob. Auton. Syst., 61(12), pp. 1601–1614.
[41] Realmuto, J., Klute, G., and Devasia, S., 2015. “Nonlinear Passive
Cam-Based Springs for Powered Ankle Prostheses”. J. Med. Device.,
9(1), p. 011007.
[42] Thorson, I., and Caldwell, D., 2011. “A Nonlinear Series Elastic Actuator
for Highly Dynamic Motions”. In IEEE/RSJ Int. Conf. Intell. Robot.
Syst., IEEE, pp. 390–394.
[43] Park, J., Kim, H., and Song, J., 2009. “Safe robot arm with safe joint
mechanism using nonlinear spring system for collision safety”. In 2009
IEEE Int. Conf. Robot. Autom., IEEE, pp. 3371–3376.
[44] Hubicki, C., et al., 2016. “ATRIAS: Design and validation of a tether-
free 3D-capable spring-mass bipedal robot”. Int. J. Rob. Res., 35(12),
pp. 1497–1521.
Edgar Bol´ıvar (S16) received the B.S. degree in
Mechatronics Engineering from the Universidad
Nacional de Colombia (2011). Currently, he is a Ph.D.
student in Mechanical Engineering at the University
of Texas at Dallas. He is with the Locomotor Control
Systems Lab (LoCoLab) at the University of Texas
at Dallas. His goal at the LoCoLab is to provide
wearable robots with actuators and control systems
that allow synergy between human and machine.
Prior to joining the LoCoLab, he was undergraduate
research assistant at the University of Wisconsin-
Milwaukee and research engineer at the Universidad Nacional de Colombia.
Siavash Rezazadeh received his B.Sc. and M.Sc.
from Sharif University of Technology, and his PhD
from University of Alberta. Currently, he is with
the Locomotor Control Systems Lab at University
of Texas at Dallas, working on design and control
prosthetic legs. Before joining Locolab, he worked
at Dynamic Robotics Laboratory of Oregon State
University on control of ATRIAS, a bipedal robot,
for DARPA Robotics Challenge (DRC).
Robert D. Gregg IV (S08-M10-SM16) received
the B.S. degree (2006) in electrical engineering and
computer sciences from the University of California,
Berkeley and the M.S. (2007) and Ph.D. (2010)
degrees in electrical and computer engineering from
the University of Illinois at Urbana-Champaign.
He joined the Departments of Bioengineering and
Mechanical Engineering at the University of Texas at
Dallas as an Assistant Professor in June 2013. Prior to
joining UT Dallas, he was a Research Scientist at the
Rehabilitation Institute of Chicago and a Postdoctoral
Fellow at Northwestern University. His research is in the control of bipedal
locomotion with applications to autonomous and wearable robots.