PRE-PRINT ICRA 2015 1

A Comparison of Series and Parallel Elasticity in a Monoped HopperYevgeniy Yesilevskiy, Weitao Xi and C. David Remy, Member, IEEE

. Abstract— In this paper we use optimal control on a gearedelectric DC motor to compare the energetic efficiency of asimulation of conceptual monoped hoppers with either parallelelastic actuation (PEA) or series elastic actuation (SEA). Theenergy is measured using three cost functions: positive actuatorwork, electrical losses, and positive electrical work. For PEA,the presence of the motor inertia in the collision losses leads toincreased collision losses at large transmission ratios, which leadto energetically costly compensatory strategies where the SEAis at its most efficient. At small transmission ratios, the motorforce increases for both cases, leading to increased thermallosses. In agreement with those theoretical predictions, our workshows that for positive actuator work and positive electricalwork the optimal parameter choice for SEA is significantlymore energetically efficient than the optimal choice for PEA.For electrical losses, a suitable choice of the transmission ratiocan lead to negligible cost values for both actuator concepts.

I. INTRODUCTION

Walking, running, and other modes of legged locomotionare almost exclusively based on movements that are periodicin nature and accompanied by repeated fluctuations of poten-tial and kinetic energy. Ideally, the associated energetic fluc-tuations are contained within the system itself. Such systemscan avoid unnecessary negative actuator work by choosingintelligent motion trajectories, and by passively storing andrecovering excessive energy in compliant elements. Humanin-place hopping, for example, can accommodate 70% of thefluctuations in kinetic and potential energy through energystorage in the Achilles tendon [1].

There is broad agreement that elastic elements are simi-larly useful in legged robotic systems [2] as they are in nature[3]. Springs are used in robots for two main purposes: at thehip to tune the leg swing dynamics to a desired frequency[4], and in the legs to mimic the behavior of a Spring LoadedInverted Pendulum model (SLIP) which stores large amountsof energy during the stance phase of a running gait [5]. Theefficacy of both approaches has been studied extensivelyusing conceptual passive dynamic models [6], [7], [8]. Ofcourse, real robots do require actuation to compensate forfriction, negative actuator work, and collision losses, as wellas to enable stabilizing feedback control. This actuation canbe included in parallel or in series to the elastic element.So far, no consensus has been reached on which is themore energetically efficient of these two alternatives. Roboticimplementations exist with both parallel elastic actuation(PEA) [9] and series elastic actuation (SEA) [10], [11], [12](Fig. 1).

The authors are with the Robotics and Motion Laboratory (RAMlab),Department of Mechanical Engineering, University of Michigan, AnnArbor, MI (yevyes@umich.edu, xiweitao@gmail.com,cdremy@umich.edu)

(a) ( )(b)

image courtesy of J.G.D. Karssen

Fig. 1. Both parallel and series elastic actuators have been implementedin legged robotics, but there is still no consensus about which is the moreenergetically efficient of the two. ScarlETH (a) makes use of series elasticactuators in the knee and hip joint [2]. Phides (b) uses a parallel elasticactuator in the knee [9]. Phides also contains a series elastic element anda clutch to prevent damage during collisions. Our work looks to gain adeeper understanding of the benefits of each through a detailed analysis ona conceptual monopod hopper.

The main argument in favor of series elastic actuation isthe fact that in SEA, joint motion and actuator motion arefully decoupled. This decoupling implies that the actuatorinertia does not add to the joint inertia and that gears andother components are fully protected from impacts. In PEA,on the other hand, the actuators have to produce only apart of the overall joint torques, as the torques created bythe parallel springs act additively. This might greatly reducethe thermal losses associated with force creation; particularlyfor electrically driven systems. On the downside, the springtorques might counteract the actuator torques when movinga joint actively; for example, for leg retraction during swing.

At a closer look, the debate becomes more complicated:clutches, small serial elastic elements, and other additionalmechanical components, can help overcome some of theaforementioned shortcomings [13]. The stiffness of the em-ployed compliant elements will influence the overall ener-getics and must be chosen differently for SEA and PEA.Similarly, many effects, such as inertia and thermal losses,depend on the selected transmission ratio, and choosing abetter suited gear box can thus help alleviate their impact.Most importantly, a suitable adaptation of input and motiontrajectories allows for the exploitation of the properties ofeach actuation concept and could thus have a huge impacton the overall power consumption. The actuator inertia in anSEA, for example, can be used to temporarily store kineticenergy [14], and the need for leg retraction with PEA can bereduced by a smart choice of motion trajectories. Due to their

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differing strengths and weaknesses, a SEA driven robot willhence look and move differently than a PEA driven robot. Inother words, rather than asking if one particular robot withSEA is better than the same robot with PEA, we must askwhether the most energetically economical robot with SEAis better than the most energetically economical robot withPEA – given that both are moving in an optimal fashion.

To answer that question, we applied optimal control andparameter optimization to compare SEA and PEA in concep-tual models of a monoped hopper, which can be regarded asthe archetype of running robots. By choosing energeticallyoptimal trajectories, and studying an optimal choice of springstiffnesses and gear ratios, we show theoretical upper boundson the energetic economy of each of the concepts. Theconceptual models that we employ allow a broad general-ization of the obtained results, while being detailed enoughto produce realistic values of power consumption. Our workshows that a SEA hopper with optimal ideal transmission andstiffness values has comparable or better energetic efficiencythan the optimal PEA robot when regarding positive actuatorwork, electrical losses, and positive electrical work. Theseresults are in agreement with our theoretical predictions.

II. METHODS

A. Model

As the basis of our study, we use the simple model of avertical hopper [15]. It consists of a main body with massm1 and a foot segment with mass m2. The motion of thehopper is restricted to a pure vertical movement. Its stateis described by the position and velocity of the main body(given by y and y) and the length of the leg (l and l). Tocreate realistic motions, it is assumed that l is constrainedto be 0.5lo < l < 1.15lo (where lo is the uncompressed leglength). The motion of the foot is coupled to the main bodyby a spring with stiffness k and damping ratio ζ (Fig. 2).Having damping in the springs and a foot with mass meansthat the system is not energetically conservative. As a result,positive net work must be performed by the actuators overthe course of a stride. This work is created by an electric DCmotor that produces a torque of Fmot and has a reflectedinertia of jmot. This actuator is either connected in serieswith the spring (SEA) or in parallel to the spring (PEA).

In the SEA case, the spring is rigidly attached to the footon one end, and attached to a servo-controlled DC motor onthe other end. An additional pair of states (u and u) describesthe motion of this actuator. To create realistic behaviors, theactuator motion is constrained within −0.15lo < u < 0.15lo.For the SEA case, the generalized coordinate vector has threecomponents: qSEA = (y, l, u)

T . Since the actuator is inseries with the spring, the joint force, F , is equal to theforce produced by the spring:

F = k (lo + u− l) + b(u− l

)(1)

b is the damping coefficient of the spring, which is computedfrom the damping ratio ζ. The mass matrix for this systemis given by

MSEA =

m1 +m2 −m2 0−m2 m2 00 0 jmot

. (2)

The dynamics are driven by the generalized torques τSEA =(0, F, Fmot − F )T .

For PEA, the spring is rigidly attached to both the footand the main body. The actuator is mounted in parallel tothis spring, such that the total joint force, F , is the sum ofthe spring force and the actuator force:

F = k (lo − l) + bl + Fmot (3)

Since actuator motion and leg motion are coupled, thevector of generalized coordinates qPEA = (y, l)

T is onlytwo dimensional. The reflected actuator inertia jmot actsalong the coordinate of l, which leads to the following massmatrix for the PEA:

MPEA =

[m1 +m2 −m2

−m2 m2 + jmot

](4)

The dynamics are driven by the generalized torques τPEA =(0, F )

T .In both variations, all states and parameters are normalized

with respect to total mass mo, uncompressed leg length lo,and gravity g (Table I).

During flight, the motion of the hopper is governed bythe equations of motion Mq = h + τ . The differentiableforce vector h includes only the gravitational terms and isgiven by hSEA = (−(m1 + m2)g,m2g, 0) and hPEA =(−(m1 +m2)g,m2g), respectively.

For both models, the main body velocity and the legretraction rate must be equal during ground contact (y = l).That leads to the modified EOMs of y = l = F

m1−g for SEA

and y = l = Fm1+jmot

− m1

m1+jmotg for PEA. This constraint

implies that at the moment of touch-down, a collision bringsthe foot velocity vfoot = y + l to zero. This collision isassociated with energetic losses Ecol., which differ betweenSEA and PEA:

ESEA,col. =1

2m2v

2foot (5)

EPEA,col. =1

2

(m2 +

jmotm1

m1 + jmot

)v2foot.

For PEA, the actuator inertia contributes to the collision.Since we are interested in the energetic efficiency of the

two actuation concepts, we included a detailed motor modelin our simulation. We modeled the actuators as geared DC-motors [16]. To simplify the analysis, we neglected theelectrical dynamics due to the motor inductance as well asfriction in the gears. For a given supply voltage Uo, themaximal input power of a DC motor is given by Pmax =(Uo)

2

R , and its no-load-speed by umax = Uo

nkT. In these

equations n is the transmission ratio, kT the motor constant,and R the armature resistance. Rather than explicitly definingUo, n, kT , and R, we used Pmax and umax to parametrize themotor. Therefore umax is a single parameter that representsthe gear box. With this, the electrical losses can be expressedas a function of the actuator force Fmot:

Ploss = F 2mot

u2max

Pmax(6)

This equation implies that thermal losses are proportionalto the square of the no-load-speed umax (and inverselyproportional to n2). The no-load-speed is an important freeparameter in the optimization since its value has a directimpact on the thermal losses as well as on the reflected motorinertia jmot. This inertia can be computed as

jmot =junsc.

u2max

(7)

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RAMlab

u l

m1

y

m2

k, ζ

RAMlab

lk, ζ

m2

m1

y

g

SEA PEA

Fmot

Fmot

Fig. 2. Using optimal control, this study investigated whether a leggedrobot is more efficient with series elastic actuation (SEA, left) or parallelelastic actuation (PEA, right). For different values of transmission ratiosand spring stiffnesses, we identified optimal periodic motion trajectoriesin a numerical optimization framework. These nominal motions representupper bounds on the energetic efficiency of each actuator variation and thusallow a fair comparison between the two concepts.

TABLE IMODEL PARAMETERS EXPRESSED WITH RESPECT TO TOTAL MASS mo ,

LEG LENGTH lo , AND GRAVITY g.

m1 = 0.95 mo m2 = 0.05mo ζ = 0.2Pmax = 24mog

√glo junsc. = 0.6moglo

It is based on an unscaled inertia junsc. that is given byjunsc. = jrotor

U2o

k2T

[14]. This reduces all important motorequations to the two parameters Pmax and junsc., which werebased on an existing robotic hopper [2] (Table I).

B. Cost Functions

To measure the energetic efficiency of each actuationvariation, we used three cost functions: positive actuatorwork, electrical losses, and positive electrical work. All costfunctions are expressed as an integral over a single hop, fromtime t = 0 until t = T .

1) Actuator Work: The positive actuator work refers to thepositive mechanical work that is performed by the actuator.For the series elastic actuator, it is defined as:

cact =

∫ T

0

max (Fmot · u, 0) dt (8)

For the parallel elastic actuator, the actuator velocity isequal to the joint velocity. Therefore the positive actuatorwork is defined as:

cact =

∫ T

0

max(Fmot · l, 0

)dt (9)

2) Electrical Losses: The electrical losses cost functionexpresses the thermal losses in the motor. For both theparallel and series cases it is given by:

closs =u2max

Pmax

∫ T

0

F 2motdt (10)

3) Electrical Work: Electrical work is the integral of thepositive electrical power used by the actuator. It combinesmechanical work and electrical losses. For the series case,the electrical work is equal to:

cel =

∫ T

0

max

(Fmotu+ F 2

motu2max

Pmax, 0

)dt (11)

and for the parallel case, it is equal to:

cel =

∫ T

0

max

(Fmot l + F 2

motu2max

Pmax, 0

)dt (12)

C. Optimal Control

For these models and cost functions, energy efficientmotions were generated via optimal control. To this end,we analyzed motions from apex transit to apex transit, andenforced periodic boundary conditions to simulate a contin-uous hopping motion. Hopping height was fixed to 1.3 lo.The motor force Fmot (t) was parametrized by a piecewiselinear representation, and both l and u were constrained tobe within their admissible range of motion.

These assumptions and constraints were implemented inthe multiple shooting optimization package MUSCOD [17]to obtain optimal force inputs and periodic states. A range ofoptimizations was conducted for each actuator type and costfunction, while varying the spring stiffness in the range k =10-100mog/lo and actuator no-load-speed in the range umax

= 0.05-5.5√log . Within this parameter-grid, successful

optimizations were used as initial guesses for neighboringgrid points, and optimizations were initiated multiple times.This process ensured that each parameter combination wastested multiple times with slightly different initial conditionsand that local minima were avoided.

III. THEORETICAL PREDICTIONS

From an energetic point of view, the advantage of SEA isthat the actuator inertia does not contribute to the collisionlosses, which reduces the need for active actuator work. Thedownside is that the full joint force F must be supportedby the actuator, which drives up the thermal losses in themotor. Avoiding this is usually stated as the main benefit ofPEA, in which the motor force acts only in addition to thespring force. Yet, both collision losses and thermal losses,depend on the chosen transmission. A larger transmissionratio decreases thermal losses, while driving up the reflectedactuator inertia.

A. Positive Actuator Work

Minimizing the positive actuator work is equivalent tominimizing the sum of the collision losses, the negativeactuator work, and the damping losses [14]. A variety ofstrategies can be used in an optimal motion to minimizethese losses (Fig 3). These strategies require different motionprofiles for SEA and PEA, and do not work equally wellfor both actuator concepts. This difference is caused by theway the two actuator concepts create net positive work. ForSEA, the actuator force is dominated by the force needed tobalance the spring force F . To perform net work, an activeactuator motion u is performed against this force. In contrast,in PEA the actuator motion u is directly coupled to the legmotion and cannot be chosen freely. Work is consequentlyperformed by a modulation of the motor force Fmot. Sincethis has only a small influence on the compression velocityof the leg, it limits strategies that seek to reduce damping

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losses in the spring. In a SEA, for example, the actuatorremains at rest during the first half of stance and thenretracts during the second half. This retraction reduces theexpansion velocity of the spring and thus minimizes thedamping losses, which grow with the square of the spring

extension Pdamp = b(l − u

)2

. Since for a PEA the twomotions are coupled, this strategy cannot be applied and theoverall damping losses are larger.

time � ol g

Acu

tato

r M

otio

nH

op

per

Motio

n

0 0.5 1 1.5 2 2.5

-0.1

0

0.1

0

0.5

1

1.5

y

l

u

Touch Down

I

II

IIIIVV

Upper Leg Length Limit

Lower Leg Length Limit

Positive Actuator Limit

Desired Apex Height

Negative Actuator Limit

STANCEFLIGHT

Fig. 3. Shown are main body position (y, top), leg length (l, top), andactuator motion (u, bottom) of a 1D hopper. Positive mechanical work wasminimized while constraining apex-transit to 1.3 m. The resulting motionextends the leg as much as possible to reduce effective hopping height(I), minimizes damping by moving the actuator at the end of stance (II),reduces negative work by keeping the actuator at rest during the first halfof stance (III), maximizes vertical momentum by maximally extending theleg at apex transit (IV), and reduces collision losses by retracting shortlybefore touch-down (V).

The actuator losses can be expressed as:

cact,SEA = CSEA + ESEA,col (13)cact,PEA = CPEA + EPEA,col

where CSEA and CPEA are constants representing the sumof the negative actuator work and damping losses for eachcase, and ESEA,col and EPEA,col are the collision losses asthey are defined in eqn. (5).

PEA suffers from higher collision losses, as the reflectedactuator inertia contributes to the collision. From eq. (5), wecan directly quantify these additional losses as:

EPEA,col. − ESEA,col. =v2foot2

(junsc.m1

m1u2max + junsc.

). (14)

These losses scale directly with the no-load-speed umax.The only way to compensate for this is by reducing theimpact velocity of the foot. However, this will increase thenegative actuator work and the retraction will pre-load thespring, which will lead to faster spring compression rate, and

therefore increased damping losses. We therefore expect thatthe additional collision losses are directly reflected in thenecessary positive actuator work.

B. Electrical Losses

Electrical losses are merely a function of motor forceand only depend indirectly on the performed work. Tounderstand the implications of this cost function, let usassume an energetically conservative motion, with a masslessfoot (m2 = 0) and no damping in the springs (b = 0).

To sustain such a loss-less motion, a SEA simply has tohold the motor at a fixed position (u ≡ 0) while in a PEA themotor does not create any force (Fmot ≡ 0). In the seriesactuation case, the optimal no-load-speed is thus umax =0, which precludes any thermal loss as they are given byeq. (10). In the parallel elastic case, on the other hand, theoptimal no-load-speed is umax = ∞, which prevents anycollision losses within the actuator (as then the motor inertiais zero, as given by eq. (7)). This shows clearly, that differentactuation concepts require different transmission ratios.

In a more realistic consideration, actuators must performsome net positive work to compensate for energy losses. Thismeans that the SEA has to create motion against the springforce F and that the PEA must apply an active force as theleg is extending. Both requirements lead to a trade-off withrespect to the transmission ratio.

For a SEA that trade-off stems from the motor forceneeded to accelerate the reflected inertia jmot of the motor.Since this inertia is inversely proportional to u2max, it pro-hibits a no-load-speed that is too close to 0. The electricallosses can be expanded to

closs =T

Pmax

((m1gumax)

2 +

(junsc

umax

)2

σ2 (uact)

)(15)

in which the first term accounts for the force required tocompensate for gravity over the course of a hop and thesecond term accounts for the force required to create netpositive work (see [15], Chapter 4 for a detailed derivation).σ2 (uact) is the variance of the active actuator acceleration,which is a consequence of the need for positive net workand an active actuator motion. From this, we can identify anoptimal transmission ratio as:

umax =

√junsc.σ (uact)

m1g(16)

For PEA, the trade-off arises because an infinitly largeno-load-speed would lead to infinite thermal losses if therequired motor force is not equal to 0. While umax directlyscales closs (eq. (10)), decreasing it introduces other un-wanted effects. As shown before, collision losses depend onthe reflected actuator inertia according to eq. (5) and increasefor smaller umax. In addition, a substantial amount of kineticenergy is stored in the reflected rotor inertia at the instance oflift-off (when the leg is extending with maximal velocity).This stored kinetic energy scales with jmot = junsc.

u2max

, andcan lead to negative actuator work that is required to slowdown and reverse the extension of the leg. All these effectsdepend on the chosen motion profile and allow for a largerange of compensatory strategies. For example, the negative

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actuator work needed to slow down the leg extension duringflight can be conducted in part via the leg spring and could bereused for leg retraction before touch-down. These strategies,in turn, create secondary effects, such as an increase of thedamping losses. This makes a more detailed analysis of theoptimal no-load-speed for PEA very difficult. Still, we wouldexpect that an optimal no-load-speed can be found that tradesoff these requirements.

C. Electrical Work

Minimizing electrical work represents a trade-off betweenminimizing actuator work and electrical losses. We thusexpect that the result for this cost function combines the char-acteristics of both previously discussed cases. In particular,we hypothesize, that electrical work has a clear optimum forthe no-load-speed.

IV. NUMERICAL RESULTS

Using optimal control, we minimized each cost functionfor both SEA and PEA. While spring stiffness was not con-sidered as a factor in the analytical considerations presentedin the previous section, we included it in the numericalanalysis to investigate potential differences between SEA andPEA.

As expected, SEA clearly outperformed PEA when con-sidering positive actuator work as a cost function (Fig 4).The ability to reduce damping and collision losses favorsthe series elastic actuation concept. The losses showed aclear dependence on the chosen no-load-speed and increasedfor slower actuators with a higher reflected inertia. To com-pensate for this inertia, the optimizer discovered strategiesthat reduced the amount of energy lost in the collision. Forexample, by retracting the leg in order to decrease the relativevelocity between the foot and the ground. Still, the generaltrend of the resulting cost-values matched the predictionsbased on collision losses, as the compensatory motion led toincreased negative actuator work and damping losses.

For electrical losses, we discovered a clear minimum forboth PEA and SEA as a function of the no-load-speed(Fig. 5). The global minimum PEA value of 0.0059moglo,occuring at a stiffness of 80mog/lo and a maximal no-load-speed of 0.35

√log is slightly lower than the minimum SEA

value of 0.0136moglo, occuring at a stiffness of 100mog/loand a maximal no-load-speed of 0.35

√log. This is a surpris-

ing result. An often hypothesized advantage of using PEA isthat the spring force and motor force act additively, whichcan greatly reduce the required motor force. And since theelectrical losses depend explicitly on the square of the motorforce F 2

mot, one would expect that a cost function basedon electrical losses would strongly favor PEA. In practice,however, the thermal losses of both concepts can be madenearly 0 with the correct choice of umax.

Electrical work as a cost function essentially combinesthe results of positive actuator power and electrical losses.Optimal transmission values can be clearly indentified forboth PEA and SEA, and the optimal parameter choices arevery similar to those found when optimizing electrical losses.

Stiffness

Positive Actuator Work

No-Load Velocity

Posi

tive A

ctua

tor W

ork

PEA

SEA

I

II

Fig. 4. Optimal values of the positive actuator work of a continuoushopping motion are shown for varying parameter choices of leg springstiffness and actuator no-load-speed. In this comparison, SEA always hasa lower value for positive actuator work than PEA, independent of no-load-speed and stiffness. The increase in the cost value for PEA at smallvalues of the no-load-speed is indicative of the increased collision losses thatresult from a higher reflected inertia. The shown predictions (PEA: I, SEA:II) are based on eqns. (13), with fitted values for CSEA = 0.10moglo,CPEA =0.13moglo, and v2foot = 0.53 log.

Stiffness

Electrical Losses

No-Load Velocity

Elec

trica

l Los

ses

PEA

SEA

I

Fig. 5. Optimal values of electrical thermal losses are shown for varyingparameter choices of leg spring stiffness and actuator no-load-speed. Thetheoretical prediction of the SEA cost (I) from eq. (15) was fitted to thedata in the figure (T = 3.5, σ2 (uact) = 0.005). Overall, the optimalvalue for PEA is only slightly better than for SEA, despite the fact thatthe cost function is based on the integral of squared torques. To operateoptimally, the two actuator concepts require particular transmission ratios(here represented by the actuator no-load-speed) and the cost can increasedramatically in the case of a sub-optimal transmission choice.

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Stiffness

Electrical Work

No-Load Velocity

Elec

trica

l Wor

k

PEA

SEA

Fig. 6. Optimal values of the electrical actuator work are shown for varyingparameter choices of leg spring stiffness and actuator no-load-speed. Similarto electrical losses, we can identify clear minima with respect to the no-load-speed. Overall, the minimal value for SEA is significantly smaller thanfor PEA. This reflects that electrical work combines positive actuator workand electrical losses.

Since this cost function considers actuator work, SEA issignificantly more energetically efficient than PEA (Fig 6);provided the right transmission is chosen. This result is anindication that the fundamental differences in collision anddamping losses play a much larger role in differentiatingPEA and SEA than the square of the motor force.

Apart from very low stiffness values, the spring stiffnessdid not have a significant influence on any of the cost func-tions studied. SEA and PEA were always equally sensitive tothe selected spring stiffness, and no dependency between thechosen actuator concept and an optimal stiffness value werefound. This observation largely confirmed previous results[15].

V. CONCLUSION

For a one-dimensional monoped hopper that is driven bya geared DC motor, our results show that with the correctchoice of the transmission parameter, a hopper with SEA ismore energetically efficient than one with PEA when lookingat positive actuator work and electrical work. Electricallosses can be made to be nearly 0 for both actuators. This islargely attributed to the fact, that for a hopper with PEA, themotor inertia contributes to energetic losses due to the groundcontact collisions. Our optimizations showed that even for aforce-squared cost function such as electrical losses, PEA isnot able to overcome the increased collision and dampinglosses to be significantly more efficient than SEA.

The results, as they are presented in this paper meritsome further discussion. To be able to create theoreticalpredictions of the optimization outcomes (and thus ensurethat the obtained solutions are not merely local minima orotherwise flawed results), we had to considerably simplifythe model. Our analysis did, for example, not include theeffects of gear-friction, or the need for active leg retraction asit is required in a multi-legged gait. We therefore believe thatthe value of this paper is not in finding a definite answer tothe question of which actuation concept is better. This answer

will likely depend on a particular robotic implementation.Rather, we believe that the theoretical predictions and resultshelp understand the fundamental processes that are involvedin the energetics of robotic running and can thus guide adetailed system analysis if needed.

With a thorough understanding of the effect of stiffnessand maximal no-load-speed on the monoped hopper, our nextgoal is to extend this work to more complex systems. Byincluding morphological parameters directly in the optimiza-tion, we will be able to expand our method for comparingSEA and PEA to systems such as a planar monoped or biped.

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