Robust Optimal Design of Energy Efﬁcient Series
Elastic Actuators: Application to a Powered
Prosthetic Ankle
Edgar Bol´ıvar, Siavash Rezazadeh, Tyler Summers, and Robert D. Gregg
Abstract—Design of robotic systems that safely and efﬁciently
operate in uncertain operational conditions, such as rehabilitation
and physical assistance robots, remains an important challenge in
the ﬁeld. Current methods for the design of energy efﬁcient series
elastic actuators use an optimization formulation that typically
assumes known operational conditions. This approach could lead
to actuators that cannot perform in uncertain environments
because elongation, speed, or torque requirements may be beyond
actuator speciﬁcations when the operation deviates from its
nominal conditions. Addressing this gap, we propose a convex
optimization formulation to design the stiffness of series elastic
actuators to minimize energy consumption and satisfy actuator
constraints despite uncertainty due to manufacturing of the
spring, unmodeled dynamics, efﬁciency of the transmission, and
the kinematics and kinetics of the load. In our formulation,
we express energy consumption as a scalar convex-quadratic
function of compliance. In the unconstrained case, this quadratic
equation provides an analytical solution to the optimal value of
stiffness that minimizes energy consumption for arbitrary peri-
odic reference trajectories. As actuator constraints, we consider
peak motor torque, peak motor velocity, limitations due to the
speed-torque relationship of DC motors, and peak elongation of
the spring. As a simulation case study, we apply our formulation
to the robust design of a series elastic actuator for a powered
prosthetic ankle. Our simulation results indicate that a small
trade-off between energy efﬁciency and robustness is justiﬁed to
design actuators that can operate with uncertainty.
I. INTRODUCTION
Series elastic actuators (SEAs) [1] have the potential to be
a safe, energy efﬁcient, and easy to control actuation scheme
for human-robot interaction in rehabilitation and physical
assistance robots. SEAs are well suited for force control and
impedance control in human-robot interaction, as the elastic
element behaves as a soft load cell [2], [3]. SEAs encourage
safety in robots by potentially reducing the mass of the
actuator [4] and its associated kinetic energy during impacts.
Additionally, SEAs enable elastic collisions for greater safety
where impacts might occur [5]. Elastic collisions can also
improve energy efﬁciency in applications subject to periodic
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, T. Summers is with the Department of Mechanical
Engineering, The University of Texas at Dallas, Richardson, TX, 75080, USA.
Email:{ebolivar, rgregg}@ieee.org
impacts such as bipedal locomotion [6], [7]. In addition, com-
pared to rigid actuators, SEAs can reduce energy dissipated
by the actuator for periodic tasks [8], [9].
The SEA torque and speed bandwidths, reachable virtual
and mechanical impedance, stored elastic energy, tolerance to
impact loads, peak power output, and energy efﬁciency [3],
[10] depend on the selection of the SEA’s spring stiffness. The
requirements of the application ultimately determine which of
these criteria should be used for the stiffness design. In this
paper, we focus on reducing energy consumption of the SEA,
which has the potential to reduce mass and increase battery
life of robots for rehabilitation and physical assistance.
Existing methods to design the stiffness that minimizes
energy consumption of SEAs, such as natural dynamics [11]
and optimization formulations [9], [12], assume nominal ref-
erence kinematic and kinetic trajectories of the load. These
nominal trajectories easily change during operation in human-
robot interaction. For example, in the design of an SEA for
a powered prosthetic leg, the reference kinematic and kinetic
trajectories change as the subject walks with different speeds
or wears different accessories, such as a backpack. When the
load conditions deviate from their nominal values, the energy
consumption and peak power of SEAs may not be optimal
[13]. Additionally, the speed and torque requirements for the
motor may be outside the motor’s speciﬁcations, i.e., the task
becomes infeasible. For instances when changing stiffness of
the SEA makes the task feasible again, a possible solution
is to replace SEAs with variable stiffness actuators (VSAs)
[8]. However, VSAs require additional mechanisms to operate,
increasing the mechanical complexity and potentially the mass
of the actuator. Thus, we are interested to know if a ﬁxed-
stiffness SEA could satisfy the actuator constraints despite
uncertainty, and what trade-off a robust design would have
with energy consumption.
Acknowledging uncertainty in the reference trajectories
leads to more realistic and robust designs. For example, in
[13], the optimal design of series stiffness considered deviation
from the nominal trajectories for the application of powered
prosthetic legs. The SEA was optimized over both walking and
running reference trajectories, but the design did not consider
the wide range of other possible tasks and did not analyze the
feasibility of the actuator. Brown and Ulsoy [14] considered
task uncertainty by deﬁning the reference trajectory as a
sample from a probability distribution of reference trajectories.
Their stochastic approach to designing linear parallel elastic
arXiv:1812.04771v2 [cs.RO] 6 Feb 2019
elements provided energy savings and constraint satisfaction
for 90% of the reference trajectories, but worst-case scenarios
would violate the strict safety requirements of a co-robot.
More arbitrary reference motions resulted in stiffer optimal
solutions, converging to a rigid actuator for totally arbitrary
motion [14]. However, increasing stiffness may not be the
solution when actuator constraints must be satisﬁed despite
uncertainty, as demonstrated in Section V. Thus, a robust
formulation is required to guarantee feasibility of the actuator
even in the worst case conditions that could manifest during
operation.
Our Contribution
We present a convex optimization formulation to design
the stiffness of an SEA to minimize energy consumption
and satisfy actuator constraints despite uncertainty due to
manufacturing of the spring, unmodeled dynamics, efﬁciency
of the transmission, and the kinematics and kinetics of the
load. As actuator constraints we consider: 1) peak motor
torque, 2) peak motor velocity, 3) limitations due to the speed-
torque relationship of DC motors, and 4) maximum elongation
of the spring. We use existing robust optimization techniques
to design robust-feasible stiffness values, i.e., satisfy the con-
straints despite uncertainty [15].
In general, any optimization problem beneﬁts from a ro-
bust feasible solution. However, only a few robust feasible
optimization problems, especially convex, can be computa-
tionally tractable [15]. Part of our contribution is to write
energy consumption of an SEA’s electric motor as a convex-
quadratic function of compliance, the inverse of stiffness. This
convex-quadratic formulation is not only useful for a robust
feasible design, but it also provides an analytical solution for
the optimal stiffness in the unconstrained case. Our convex-
quadratic formulation allows computation of an optimal value
of stiffness within polynomial time [16], which is beneﬁcial
for systems that can modify their mechanical stiffness during
operation, such as VSAs.
Our formulation applies to any application with periodic
motion. We apply our methods to the actuator design of
powered prosthetic legs. Battery life and device mass inﬂuence
the performance of these devices. Reducing prosthesis mass
is paramount [17], especially because mass attached to distal
parts of the human body increases the metabolic energy
consumed by the user. For example, in [18], a 2 kg load
placed on each foot increased the rate of oxygen uptake 30%,
which is an indirect measure of metabolic energy consumption.
SEAs have the potential to reduce mass in two ways: 1)
extending battery life by reducing the energy dissipated by
the actuator will allow the use of smaller batteries, and 2)
reducing the speed-torque requirements for the motor will
allow the use of smaller, lighter motors. As discussed in [9],
an elastic element connected in series is passive and cannot
reduce the energy required by the load, but it has the potential
to reduce the energy dissipated for a given task. Reduction
of dissipated energy is important, especially in tasks that are
mainly dissipative such as level-ground walking. For example,
the ankle of a 75 kg human provides about 17 J per stride
during normal walking, but a rigidly actuated prosthetic ankle
consumes 33 J [9]. The same motor connected in series with
an elastic element requires about 25 J per stride, i.e., a 50%
reduction in the energy dissipated [9]. Thus, SEAs represent a
viable actuator alternative for the design of powered prosthetic
legs, with some examples reported in [12], [19], [20].
The explanation of our formulation starts in Section II with
an introduction to the modeling of SEAs with an emphasis
on their energy consumption. Section III describes energy
consumption as a convex-quadratic function of compliance.
This quadratic expression is the cost function of our formu-
lation. Section IV describes our robust design methodology
along with the actuator constraints. This section illustrates
how to reformulate the optimization problem in order to ﬁnd
a solution that satisﬁes the constraints despite uncertainty.
Section V applies our methodology to the robust feasible
design of SEAs for a powered prosthetic ankle. Section VI
includes the discussion and conclusion of our work.
II. MODELING OF SERIES ELASTIC ACTUATORS
In this work, we assume that the SEA uses an electric DC
motor, and energy can ﬂow from the load to the energy source
and vice versa. In other words, suitable electronics allow
energy to ﬂow to and from the battery. The corresponding
energy ﬂow and main components of an SEA are illustrated
in Fig. 1. We also assume that the energy consumption of
the SEA refers to the energy consumption of the motor and
energy lost at the transmission. The energy dissipated by
viscous friction at the transmission can be lumped with the
viscous friction of the motor’s shaft. Energy losses in the
power electronics and battery exist in practice, but we do not
include them in our formulation. This is because they are either
proportional to the energy losses of the motor (and hence can
be lumped with winding losses) or are independent of the
system’s dynamics and hence are not affected by the spring
stiffness. In addition, most of the losses occur at the motor’s
winding and transmission. For example, in the MIT Cheetah
robot [21], 76% of the total energy consumption is attributed
to heat loss from the motor; the remaining energy is dissipated
by friction losses and impacts [21]. Under these assumptions,
the energy consumption of the SEA is equivalent to the energy
consumption of the motor including losses at the transmission,
Em, which is given by [22]
Em =
Z tf
t0
τ 2
m
k2m
|{z}
Winding
Joule
heating
+ τm ˙qm
| {z }
Rotor
mechanical
power

dt,
(1)
where t0 and tf are the initial and ﬁnal times of the reference
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 also be
written as i2
mR, since τm = imkt and km = kt/
√
R, where
im is the electric current ﬂowing through the motor, R the
Battery
Drive
Motor &
Transmission
Spring
Load
Winding Joule Heating , Viscous Friction
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 load to the battery requires that the load is high
enough to backdrive the motor-transmission system.
motor terminal resistance, and kt the motor torque constant
[23].
Using the Newton-Euler method, the corresponding balance
of torques at the motor and load side provides the following
equations of motion [23], [24]:
Im¨qm = −bm ˙qm + τm + τl
ηr + τu,
(2)
τl = g(ql, ˙ql, ¨ql, τext),
(3)
where Im is the inertia of the motor; bm its viscous friction
coefﬁcient; r the transmission ratio; η the efﬁciency of the
transmission; τu is the unmodeled dynamics torque that lumps
unmodeled effects at the motor and load side, e.g., cogging
torque and friction; ql,
˙ql,
¨ql, τl represent the position,
velocity, acceleration and torque of the load respectively; and
g(ql, ˙ql, ¨ql, τext) deﬁnes the torque of the load based on the
load dynamics and external torque, τext. 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. Because of the
connection in series, the torque of the spring, τs, is equal to
the torque of the load, τs = τl. For a linear spring, the torque
of the spring is proportional to its elongation, τs = kδ, where
elongation is deﬁned as
δ = ql −qm
r .
(4)
As seen in (2)-(3), the elastic element cannot modify the torque
required to perform the motion, τs, but it can modify the
position of the motor such that Im¨qm + bm ˙qm reduces the
torque of the motor, τm.
III. ENERGY CONSUMPTION AS A CONVEX-QUADRATIC
FUNCTION OF COMPLIANCE
In the case of a linear spring, elongation and torque are
related by τs = k(ql −qm/r), where k is the stiffness
constant. Using this relationship, the position of the motor and
corresponding time derivatives can be expressed as a function
of the given load position and the load torque τl as follows:
qm = (ql −τl/k)r, ˙qm = ( ˙ql −˙τl/k)r, and ¨qm = (¨ql −¨τl/k)r.
Replacing these expressions into (2) and deﬁning compliance
as the inverse of stiffness, α := 1/k, the expression of motor
torque can be written as an afﬁne function of compliance as
follows:
τm = γ1α + γ2,
(5)
where
γ1 = −(Im¨τlr + bm ˙τlr) ,
(6)
γ2 = Im¨qlr + bm ˙qlr −τs
ηr −τu,
(7)
are known constants that depend on the reference trajectory.
Using the deﬁnition of τm in (5), assuming periodic motion,
and neglecting the uncertain torque, τu = 0, we write the ex-
pression of energy consumption of the motor as the following
convex-quadratic function of compliance:
Em =
Z tf
t0
τ 2
m
k2m
+ τm ˙qm

dt,
=
Z tf
t0
τ 2
m
k2m
+ bm ˙q2
m −τs ˙qm
ηr

dt +
Z tf
t0
Im ˙qmd ˙qm,
= aα2 + bα + c,
(8)
where
a =
Z tf
t0
γ2
1
k2m
+ bmr2 ˙τ 2
s

dt,
b =
Z tf
t0
2γ1γ2
k2m
−2bmr2 ˙ql ˙τs

dt,
c =
Z tf
t0
γ2
2
k2m
+ bm ˙q2
l r2 −˙qlτs
η

dt.
Properties of the Convex-Quadratic Function of Compliance:
1) d2Em/dα2 = 2a ≥0, which follows from the deﬁnition
of a. Therefore (8) is a convex function of compliance
[25, p. 71].
2) Parameter c is the energy consumed by a rigid actuator
performing the same task without an elastic element, i.e.,
lim
k→∞Em = c.
3) The optimal value of compliance that minimizes energy
consumption for any periodic trajectory is α = −b/(2a),
neglecting actuator constraints. This optimal value can
be computed in polynomial time. Note that the integrals
in the deﬁnition of a and b can be approximated with
discrete-time summations.
4) The sign of b determines if the reference trajectories and
motor conﬁguration will beneﬁt from series elasticity
in order to reduce energy consumption. The quadratic
cost function (8) leads to two possible scenarios for the
effect of compliance α on motor energy (Fig. 2). In the
ﬁrst case, dEm/dα is negative at α = 0, thus series
elasticity improves actuator efﬁciency over some range
of compliance. In the second case, this slope is positive
at α = 0, so energy increases with compliance, i.e., there
is no energetic beneﬁt to linear series elasticity for the
0
−b/a
b
c
E.S.
α
Em
0
b > 0
α
Em
Fig. 2. Left: Energy consumption as a function of compliance, α, where the
energy savings (E.S.) region 0 ≤α ≤−b/a provides Em below the rigid
level c. Right: Case of motor and load that would not beneﬁt energetically
from series elasticity.
given task. Thus, the necessary condition for an SEA to
be energetically beneﬁcial is b < 0 in (8), i.e.,
Z tf
t0

2γ1γ2
k2m
−2bmr2 ˙ql ˙τs

dt < 0.
(9)
IV. ROBUST STIFFNESS DESIGN
This section presents our convex optimization formulation
for the robust feasible design of the SEA’s linear spring. The
convex-quadratic function in (8) is the cost function of our
optimization problem. The constraints in our formulation are:
1) peak motor torque, 2) peak motor velocity, 3) limitations
due to speed-torque relationship of DC motors, and 4) peak
elongation of the spring. Below we discuss the deﬁnition,
convexity, and uncertainty of the constraints.
A. Actuator Constraints
1) Elongation Constraint: Limited elongation of the elastic
element is typical in SEA applications. An elastic element
reaching its maximum elongation could be dangerous for co-
robots. When the spring bottoms out, the elastic collisions
with the environment become inelastic which may be harmful
for the user and the robot itself. We express the elongation
constraint as ∥τsα∥∞
≤δs, where δs is the maximum
elongation of the spring. This results in the constraint

m(τl/m)α

∞≤δs,

m(τl/m)
−m(τl/m)
T
α ≤1δs,
d1α ≤e1,
(10)
where
d1 =
τl/m
−τl/m

, e1 = 1δs
m,
and τl/m is a normalized expression of the load torque per
unit of m, i.e., τl = mτl/m.
2) Torque Constraint: In our formulation, we express the
limitations in peak torque of the motor as ∥τm∥∞≤τmax,
where τmax is the maximum peak value of torque. Recall that
the torque of the motor can be written as an afﬁne function of
compliance (5), τm = γ1α + γ2. Thus, constraining the peak
torque is equivalent to the following afﬁne constraint:
∥γ1α + γ2∥∞≤τmax,
γ1
−γ1

α ≤1τmax +
−γ2
γ2

,
d2α ≤e2,
(11)
where
d2 =
Im¨τl/mr + bm ˙τl/mr
−Im¨τl/mr −bm ˙τl/mr

,
e2 =


τl/m
ηr
−τl/m
ηr

+
−Imr¨ql −bmr ˙ql + 1(τmax + τu)
Imr¨ql + bmr ˙ql + 1(τmax −τu)
1
m.
3) Speed-Torque Relationship Constraint: As an actuator,
a DC motor simultaneously operates as an electric generator
producing a back-emf voltage. This back-emf voltage, which
is proportional to the motor’s speed of rotation, limits the
current that can ﬂow through the motor’s winding, which
is proportional to the torque produced by the motor. As
a consequence, the torque that a DC motor generates is a
function of the rotational speed [26, p. 536]. This phenomenon
is summarized by the equation τm(R/kt) = vin−kt ˙qm, where
vin is the input voltage to the electric motor. Then for a
DC motor to be feasible τm(R/kt) ≤vin −kt ˙qm [22]. The
same inequality applies for positive and negative values of
speed and torque, therefore in total there are four inequalities
to express the torque-velocity relationship constraints. The
following afﬁne constraint represents these inequalities:
τm ≤1vin
kt
R −k2
t
R ˙qm,
γ1α + γ2 ≤1vin
kt
R −k2
t
R ( ˙ql −˙τlα)r,
d3aα ≤e3a,
(12)
where
d3a = Im¨τl/mr + bm ˙τl/mr −k2
t r
R ˙τl/m,
e3a = τl/m
ηr
+

vin
kt
R + τu

−Imr¨ql−
bmr ˙ql −k2
t r
R ˙ql
1
m.
Using positive and negative values of torque and speed we can
deﬁne three similar versions of the inequality (12), which we
will denote using the vectors d3b,c,d and e3b,c,d. Summarizing,
the torque and velocity relationship constraints can be lumped
into the single vector inequality constraint
d3α ≤e3,
(13)
where
d3 = [dT
3a, dT
3b, dT
3c, dT
3d]T , e3 = [eT
3a, eT
3b, eT
3c, eT
3d]T .
4) RMS Torque and Maximum Speed: Long-term operation
of an electric motor can generate excessive heat and can be
harmful for the actuator. Constraining the RMS torque is a
typical method to guarantee that long-term operation is safe for
the device. In our formulation, the square of the RMS torque
can be written as a convex-quadratic function of compliance,
and therefore can be included as a constraint. However, RMS
torque also appears in our cost function (8). Therefore, it is
redundant to include it as a constraint. The constraint (13)
already considers the maximum speed of rotation of the motor,
which is equivalent to τm(R/kt) ≤vin−kt ˙qm when the motor
torque, τm, is zero.
5) Lumping the Constraints: Peak motor torque, peak mo-
tor velocity, speed-torque relationship constraints, and max-
imum elongation of the spring can be represented as the
following vector inequalities:
dα ≤e
(14)
where
d = [dT
1 , dT
2 , dT
3 ]T , e = [eT
1 , eT
2 , eT
3 ]T .
(15)
B. System Uncertainty
Feasibility of the constraints is subject to the selection
of the spring compliance and uncertainty in the deﬁnition
of the constraints. Uncertainty in our formulation means
that the reference kinematics and kinetics of the load, the
manufacturing accuracy of the spring, the efﬁciency of the
transmission, and the unmodeled dynamics are not precisely
known but are restricted to belong to an uncertainty set, U. In
our formulation, U is deﬁned as the Cartesian product
U = Uql × U ˙ql × U¨ql × Um × Uη × Uτu × Ud,
where the uncertainty sets Uql, U ˙ql, U¨ql, Um, Uη, Uτu, and Ud
express all the possible realizations for the load position,
velocity, and acceleration; the multiplicative factor of the
load torque; the efﬁciency of the transmission; the unmod-
eled dynamics; and the manufacturing accuracy of the spring
respectively.
For the position of the load, the uncertainty set is deﬁned
as follows:
Uql = {ql ∈Rn : ¯ql −1εql ≤ql ≤¯ql + 1εql},
where ¯ql ∈Rn and εql ∈R represent the nominal load
trajectory and uncertainty of the load position, respectively.
Inequalities for vectors are element-wise. In other words, the
position of the load, ql, is within ¯ql±1εql. Using the respective
nominal and uncertainty values (˙¯ql, ¨¯ql, ¯η, ¯τu, ε ˙ql, ε¨ql, εη, ετu),
we use the same deﬁnition for U ˙ql, U¨ql, Uη, and Uτu. Uncer-
tainty in the manufacturing of the spring is deﬁned as the factor
(1 ± εd) that multiplies the spring compliance. Because it is a
multiplicative factor, uncertainty in the manufacturing of the
spring is equivalent to uncertainty in the coefﬁcient vector d,
as seen in (14). Therefore the corresponding uncertainty set is
deﬁned by
Ud = {d ∈Rp : d −εd|d| ≤d ≤d + εd|d|},
where p is the number of constraints. Inequalities and absolute
values for vectors are element-wise. This uncertainty in the
manufacturing of the spring implies that its stiffness is in the
set
k ∈{k ∈R : [(1 + εd)α]−1 ≤k ≤[(1 −εd)α]−1}.
Uncertainty in the kinetic reference trajectories is deﬁned
by a nominal value and a uncertain multiplicative factor.
Precisely, the reference torque of the load τl is considered
to be τl
= m(τl/m), where τl/m is a nominal value of τl
per unit of m. Our uncertain multiplicative factor, m, could
be any element within the set
Um = {m ∈R : 0 < ¯m −εm ≤m ≤¯m + εm},
where ¯m ∈R is the nominal value of m and εm ∈R is its
corresponding uncertainty. In other words, τl = ( ¯m±εm)τl/m.
C. The Robust Formulation of the Constraints
A robust feasible design should satisfy the constraints (14)
for all possible realizations of the uncertainty within the
uncertainty set. Note that the uncertainty in the manufacturing
of the spring manifests as uncertainty in the vector d in (14).
Thus, a robust feasible design results in an optimal selection
of α that satisﬁes
dα ≤e, ∀ql, ˙ql, ¨ql, m, η, τu, d ∈U.
(16)
Because α > 0, a robust feasible solution is equivalent to
¯dα ≤e,
(17)
where ¯d and e are the vectors that represent the worst case
representation of the uncertainty. These vectors are deﬁned as
follows:
¯d = d + εd|d|, e = [eT
1 , eT
2 , eT
3 ]T ,
(18)
where
e1 = 1
δs
¯m + εm
,
e2 =

τl/m
−τl/m

1
(¯η ± εη)r + f
1
¯m ± εm
,
f =
−Imr(¨ql + ε¨ql) −bmr( ˙ql + ε ˙ql) + 1(τu + τmax)
Imr(¨ql −ε¨ql) + bmr( ˙ql −ε ˙ql) + 1(τu + τmax)

,
e3 = [eT
3a, eT
3b, eT
3c, eT
3d]T ,
e3a =
τl/m
(¯η ± εη)r + (−Im(¨ql + ε¨ql)r −bm( ˙ql + ε ˙ql)r+
1

vin
kt
R + τu

−k2
t r
R ( ˙ql + ε ˙ql))
1
( ¯m ± εm),
and the values for e3b, e3c , and e3d are deﬁned using positive
and negative values of torque and speed in the deﬁnition
of the torque-speed constraints. The sign of 1/( ¯m ± εm)
Electric motor
Transmission
Spring
SEA
Fig. 3. Schematic SEA for powered prosthetic ankle.
depends on the elements of the vector that it multiplies, as
the multiplication applies element-wise. When the element of
the vector is positive, then the multiplication factor becomes
1/( ¯m + εm); when the element is negative, 1/( ¯m −εm). The
same idea applies to 1/( ¯m ± εη), which describes the worst
possible scenario to satisfy the inequality (17).
D. The Optimization Problem
Combining the deﬁnition of the cost function in (8), and the
constraints in (17), the optimization problem becomes
minimize
α
aα2 + bα + c,
subject to
¯dα ≤e,
(19)
also known as a convex-quadratic program with afﬁne inequal-
ity constraints. This is a convex-optimization problem; the cost
function is convex as shown in Section 1 and the constraints
represent a closed interval which is a convex set described by
the afﬁne inequality (17). The solution of this optimization
problem is a value of compliance that is robust feasible, i.e.,
it satisﬁes the actuator constraints despite uncertainty in the
load.
V. CASE STUDY: SIMULATION OF A POWERED
PROSTHETIC ANKLE
In this section, we apply our methods to the design of
an SEA for a powered prosthetic ankle to minimize energy
consumption while satisfying actuator constraints despite un-
certainty. Figure 3 illustrates a schematic of the prosthesis.
Traditionally, actuator designs for powered prostheses use
average kinetic and kinematic trajectories [12], [17], [19],
[27]. However, load conditions during human locomotion vary
signiﬁcantly even for a single subject [28]. Robust design
is important in this application as human locomotion and
manufacturing methods are inherently uncertain. For instance,
the ankle joint position during human locomotion varies with
a standard deviation of ±5◦[29], and the stiffness of a
manufactured spring has a standard deviation of about ±10%
from the desired stiffness value [20].
In our formulation, we take advantage of the connection
between uncertainty in the kinematics and kinetics of the
load and our deﬁnition of uncertainty sets to obtain a ro-
bust feasible design. The parameters εql, ε ˙ql, ε¨ql deﬁne the
TABLE I
MOTOR PARAMETERS (EC-30 FROM MAXON MOTOR).
Parameter
EC30
Units
Motor torque constant, kt
13.6
mN·m/A
Motor terminal resistance, R
102
mΩ
Motor inertia, Im
33.3
g·cm2
Gear ratio, r
600
Efﬁciency transmission, η
0.8
Motor viscous friction, bm
1.665
µN·m·s/rad
Max. motor torque, τmax
337.5
mN·m
Max. motor velocity, ˙qmax
21065
rpm
Voltage, vin
30
V
TABLE II
UNCERTAINTY BASED ON THE VARIANCE IN [20], [29].
Uncertainty in
Units
Mass, εm
±8.8 kg
Reference position, εql
±5◦
Reference velocity, ε ˙ql
±30 % rms average trajectory
Reference acceleration, ε¨ql
±30 % rms average trajectory
Transmission efﬁciency, εη
±20 %
Unmodeled dynamics, ετu
±13.5 mN·m
Manufacturing of spring, εd
±20 %
uncertainty in the kinematics U{ql, ˙ql,¨ql}. In this simulation
case study, we deﬁne these parameters to be equal to the
reported standard deviation of the joint kinematics in [29].
Our formulation of uncertainty in the kinetics has a practical
meaning in biomechanics. The reference torque of the ankle
joint is traditionally normalized by the mass of the user
[30]. Because our deﬁnition of uncertainty in the kinetics
is multiplicative, it is equivalent to uncertainty in the user’s
mass. As a result, it becomes relevant to rehabilitation and
physical assistance robots where users can vary or a single user
can wear additional accessories, such as backpacks. We select
the uncertainty in the mass, εm, to be equal to the reported
standard deviation of the subjects’ mass in [29]. Figure 4
illustrates the reference trajectories and corresponding bounds
of uncertainty. Uncertainty in the manufacturing of the spring,
εd, is equal to twice the standard deviation of the SEA spring
stiffness of the open-source prosthetic leg at University of
Michigan [20]. The uncertain torque, ετu, is equal to 10 %
of the maximum continuous motor torque. Uncertainty in
the efﬁciency of the transmission is based on our experience
aiming for a realistic simulation case. Table I illustrates the
parameters of the actuator and Table II the parameters of
uncertainty. The parameters of the actuator are inspired by
the design of the ﬁrst-generation powered prosthetic leg at the
University of Texas at Dallas [31], [32]. Using the actuator
parameters and reference trajectories, we used CVX, a package
for specifying and solving convex programs [33], [34], to solve
the optimization problem (19).
−0.4
−0.2
0
0.2
ql [rad]
0
0.2
0.4
0.6
0.8
1
1.2
0
50
100
Time [s]
τl [N·m]
Fig. 4. Position (top) and torque (bottom) of the human ankle during level
ground walking [29]. The solid line indicates the mean values for a 69.1 kg
subject [29]. The shaded region around the nominal trajectory illustrates the
uncertainty in the position εql = ±5◦and the mass of the subject εm =
±8.8 kg. This uncertainty corresponds to the standard deviation reported in
[29].
Results
To contextualize our results, we analyze three possible
actuator designs: (a) a rigid actuator Maxon EC-30 without
series elasticity, (b) an SEA using the same motor with optimal
stiffness that satisﬁes constraints only for the nominal data, and
(c) an SEA with the same motor that satisﬁes actuator con-
straints despite uncertainty using our robust formulation. Using
(2) and (3) we compute the motor speed and torque trajectories
considering the ankle kinematics and kinetics as the load.
Figure 5 illustrates these trajectories in a torque-speed plot.
For the actuator (a), the required speed and torque do not stay
within the speciﬁcations of the motor and therefore the rigid
actuator is infeasible. Including series elasticity, the design (b)
makes the actuator feasible and dissipates 30.8% less energy
compared to (a). This justiﬁes the use of series elasticity, not
only for the reduction of energy consumption, but also to
maintain the requirements within the actuator speciﬁcations.
The optimal stiffness of design (b) is 217.4 N·m/rad. However,
this design becomes infeasible when the reference trajectory
deviates within the uncertainty set, as shown in Fig. 5. Using
our robust formulation, design (c) satisﬁes the constraints
despite uncertainty using a spring stiffness of 243.4 N·m/rad.
Design (c) reduces 30.45% of the dissipated energy compared
to a 30.8% reduction in the case (b), where the reported energy
savings are relative to the rigid case. The small trade-off in
performance using the robust SEA is justiﬁed when feasibility
of the constraints is satisﬁed. Table III summarizes the results.
VI. DISCUSSION AND CONCLUSION
In this paper, we introduced a convex optimization for-
mulation to compute the stiffness of SEAs that minimizes
−2,000 −1,000
0
1,000
2,000
−0.2
0
0.2
Motor velocity, qm [rad/s]
Motor torque, τm [N·m]
(a)
(b)
(c)
Fig. 5. Speed and torque requirements of different actuators for a powered
prosthetic ankle. The region enclosed by the dotted line describes the speeds
and torques that satisfy the speciﬁcations of the motor, i.e., feasible region.
Figure shows three possible actuator designs: (a) rigid actuator Maxon EC-30
without series elasticity, (b) SEA using the same motor with optimal stiffness
that satisﬁes constraints only for the nominal data, and (c) SEA with the
same motor that satisﬁes actuator constraints despite uncertainty using our
formulation. The robust design (c) is the only actuator that satisﬁes the actuator
constraints for all possible values of uncertainty.
TABLE III
OPTIMIZATION RESULTS THAT INDICATE WEAK TRADE-OFF BETWEEN
ROBUSTNESS AND ENERGY SAVINGS. ENERGY SAVINGS ARE RELATIVE TO
DISSIPATED ENERGY OF THE RIGID ACTUATOR 11.7 J.
Design
Optimal Stiffness
Energy Savings
nominal (b)
217.4 N·m/rad
30.8%
robust feasible (c)
243.4 N·m/rad
30.45%
energy consumption and satisﬁes actuator constraints despite
uncertainty due to manufacturing of the spring, unmodeled
dynamics, efﬁciency of the transmission, and the kinematics
and kinetics of the load. The methodology relies on the
following two concepts: a scalar convex-quadratic function
of compliance to express motor energy consumption, and
deﬁning uncertainty sets that represent tractable solutions of
the optimization problem. As shown in our simulation case
study, series elasticity can reduce energy consumption and also
modify the speed and torque of the motor so that it becomes
feasible.
Our simulation case study illustrated the robust feasible
SEA design for a powered prosthetic ankle. Uncertainty from
the recorded biomechanics naturally connected with our deﬁ-
nition of the uncertainty sets. The results illustrate that a small
trade off between robustness and energy consumption justiﬁes
a robust feasible design. It is important to note that the robust
solution satisﬁes actuator constraints despite the uncertainty
described in Table II. Previous research [14] did not consider a
robust feasible solution of the optimization problem, however,
they analyzed the effect of uncertainty in the energetic cost.
Their results indicate that as the required motion of an SEA
becomes more arbitrary, the optimal spring stiffness that
minimizes power consumption approaches inﬁnity, showing
that the best design for a completely arbitrary task is a system
without spring. In general, our results indicated a similar trend:
the more arbitrary or the bigger the uncertainty sets, the stiffer
the optimal design. However, when considering feasibility of
the actuator, inﬁnite spring stiffness may lead to an infeasible
actuator. Thus, a robust feasible optimal solution cannot be
obtained simply by increasing stiffness. Instead, it requires
proper treatment of uncertainty as presented in our convex
optimization method.
The convex-quadratic expression of compliance in (8) is
beneﬁcial beyond our robust formulation. The convexity and
simplicity of the expression allow optimization algorithms to
ﬁnd the optimal value of stiffness in polynomial time [16].
This could be exploited by VSAs to calculate their reference
stiffness values during operation. In the unconstrained case,
the proposed convex-quadratic expression has an analytical
solution, which is useful to study the principles of series
elasticity. For instance, (9) describes the necessary conditions
for periodic trajectories so that series elasticity can reduce
energy consumption. Future work will focus on experimental
applications and extend the presented robust formulation to the
robust design of SEAs that use nonlinear springs. The design
of nonlinear springs for SEAs can be formulated as a discrete-
time convex optimization problem [35], which is desirable for
a robust formulation.
REFERENCES
[1] G. Pratt and M. Williamson, “Series Elastic Actuators,” in Proc. 1995
IEEE/RSJ Int. Conf. Intell. Robot. Syst. Hum. Robot Interact. Coop.
Robot., vol. 1.
IEEE Comput. Soc. Press, 1995, pp. 399–406.
[2] N. Hogan, “Impedance Control: An Approach to Manipulation: Part I-
Theory,” J. Dyn. Syst. Meas. Control, vol. 107, no. 1, p. 1, 1985.
[3] D. P. Losey, A. Erwin, C. G. McDonald, F. Sergi, and M. K. O’Malley,
“A Time-Domain Approach to Control of Series Elastic Actuators:
Adaptive Torque and Passivity-Based Impedance Control,” IEEE/ASME
Trans. Mechatronics, vol. 21, no. 4, pp. 2085–2096, 2016.
[4] K. W. Hollander et al., “An Efﬁcient Robotic Tendon for Gait Assis-
tance,” J. Biomech. Eng., vol. 128, no. 5, p. 788, 2006.
[5] A. Bicchi and G. Tonietti, “Fast and “Soft-Arm” Tactics,” IEEE Robot.
Autom. Mag., vol. 11, no. 2, pp. 22–33, 2004.
[6] J. Hurst and A. Rizzi, “Series compliance for an efﬁcient running gait,”
IEEE Robot. Autom. Mag., vol. 15, no. 3, pp. 42–51, 2008.
[7] S. Rezazadeh, A. Abate, R. L. Hatton, and J. W. Hurst, “Robot leg
design: A constructive framework,” IEEE Access, vol. 6, no. 1, pp.
54 369–54 387, 2018.
[8] G. Grioli et al., “Variable stiffness actuators: The user’s point of view,”
Int. J. Rob. Res., vol. 34, no. 6, pp. 727–743, 2015.
[9] E. Bol´ıvar, S. Rezazadeh, and R. D. Gregg, “A General Framework
for Minimizing Energy Consumption of Series Elastic Actuators With
Regeneration,” in ASME Dynamic Systems and Control Conference,
2017, p. V001T36A005.
[10] N. Paine, S. Oh, and L. Sentis, “Design and Control Considerations
for High-Performance Series Elastic Actuators,” IEEE/ASME Trans.
Mechatronics, vol. 19, no. 3, pp. 1080–1091, 2014.
[11] T. Verstraten et al., “Series and Parallel Elastic Actuation: Impact of
natural dynamics on power and energy consumption,” Mech. Mach.
Theory, vol. 102, pp. 232–246, 2016.
[12] E. J. Rouse, L. M. Mooney, and H. M. Herr, “Clutchable series-elastic
actuator: Implications for prosthetic knee design,” Int. J. Rob. Res.,
vol. 33, no. 13, pp. 1611–1625, 2014.
[13] M. Grimmer and A. Seyfarth, “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 IEEE Int. Conf.
Robot. Autom., 2011, pp. 1439–1444.
[14] W. R. Brown and A. G. Ulsoy, “A Maneuver Based Design of a Passive-
Assist Device for Augmenting Active Joints,” J. Mechanisms Robotics,
vol. 5, no. 3, p. 031003, 2013.
[15] A. Ben-Tal, L. El Ghaoui, and A. Nemirovski, Robust Optimization, ser.
Princeton Series in Applied Mathematics.
Princeton University Press,
October 2009.
[16] Y. Nesterov and A. Nemirovskii, Interior-Point Polynomial Algorithms in
Convex Programming. Society for Industrial and Applied Mathematics,
1994.
[17] T. Lenzi, M. Cempini, L. Hargrove, and T. Kuiken, “Design, develop-
ment, and testing of a lightweight hybrid robotic knee prosthesis,” Int.
J. Rob. Res., vol. in press, p. 027836491878599, 2018.
[18] R. L. Waters and S. Mulroy, “The energy expenditure of normal and
pathologic gait,” Gait Posture, vol. 9, no. 3, pp. 207–231, 1999.
[19] S. K. Au and H. M. Herr, “Powered ankle-foot prosthesis,” IEEE Robot.
Autom. Mag., vol. 15, no. 3, pp. 52–59, 2008.
[20] A. F. Azocar, L. M. Mooney, L. J. Hargrove, and E. J. Rouse, “Design
and Characterization of an Open-Source Robotic Leg Prosthesis,” in
2018 7th IEEE Int. Conf. Biomed. Robot. Biomechatronics, 2018, pp.
111–118.
[21] S. Seok et al., “Design Principles for Energy-Efﬁcient Legged Loco-
motion and Implementation on the MIT Cheetah Robot,” IEEE/ASME
Trans. Mechatronics, vol. 20, no. 3, pp. 1117–1129, 2015.
[22] S. Rezazadeh and J. W. Hurst, “On the optimal selection of motors and
transmissions for electromechanical and robotic systems,” in IEEE/RSJ
Int. Conf. Intell. Robot. Syst., 2014, pp. 4605–4611.
[23] T. Verstraten et al., “Modeling and design of geared DC motors
for energy efﬁciency: Comparison between theory and experiments,”
Mechatronics, vol. 30, pp. 198–213, 2015.
[24] M. W. Spong, “Modeling and control of elastic joint robots,” ASME J.
Dynamic Systems, Measurement, and Control, vol. 109, pp. 310–319,
1987.
[25] S. P. Boyd and L. Vandenberghe, Convex Optimization.
New York,
NY: Cambridge University Press, 2004.
[26] J. E. Carryer, R. M. Ohline, and T. W. Kenny, Introduction to Mecha-
tronic Design.
Prentice Hall, 2011.
[27] T. Elery, S. Rezazadeh, C. Nesler, J. Doan, H. Zhu, and R. D. Gregg,
“Design and Benchtop Validation of a Powered Knee-Ankle Prosthe-
sis with High-Torque, Low-Impedance Actuators,” in IEEE Int. Conf.
Robotics & Automation, 2018.
[28] K. R. Embry, D. J. Villarreal, and R. D. Gregg, “A uniﬁed parameteriza-
tion of human gait across ambulation modes,” in IEEE Eng. Med. Bio.
Conf., 2016, pp. 2179–2183.
[29] D. A. Winter, “Biomechanical Motor Patterns in Normal Walking,” J.
Mot. Behav., vol. 15, no. 4, pp. 302–330, dec 1983.
[30] ——, Biomechanics and motor control of human movement. John Wiley
& Sons, 2009.
[31] D. Quintero, D. J. Villarreal, and R. D. Gregg, “Preliminary experimental
results of a uniﬁed controller for a powered knee-ankle prosthetic leg
over various walking speeds,” in IEEE Int. Conf. Intelligent Robots
Systems, 2016.
[32] D. Quintero, D. J. Villarreal, D. J. Lambert, S. Kapp, and R. D.
Gregg, “Continuous-Phase Control of a Powered Knee-Ankle Prosthesis:
Amputee Experiments Across Speeds and Inclines,” IEEE Trans. Robot.,
vol. 34, no. 3, pp. 686–701, 2018.
[33] M. Grant and S. Boyd, “CVX: Matlab Software for Disciplined Convex
Programming, version 2.1,” 2014.
[34] M. C. Grant and S. P. Boyd, “Graph Implementations for Nonsmooth
Convex Programs,” in Recent Adv. Learn. Control.
London: Springer
London, 2008, vol. 371, pp. 95–110.
[35] E. Bol´ıvar, S. Rezazadeh, and R. D. Gregg, “Minimizing Energy
Consumption and Peak Power of Series Elastic Actuators: A Convex
Optimization Framework for Elastic Element Design,” IEEE/ASME
Trans. Mechatronics, under review.