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Nonlinear Dynamics Published Online August 2014

http://dx.doi.org/10.1007/s11071-014-1627-3

Energy Eﬃciency in Friction-Based Locomotion

Mechanisms for Soft and Hard Robots: Slower can be

Faster

Xuance Zhou ·Carmel Majidi ·Oliver M. O’Reilly

Received: 18 March 2014 / Accepted: 29 July 2014/ Published Online: 19 August 2014

Abstract Many recent designs of soft robots and nano

robots feature locomotion mechanisms that cleverly ex-

ploit slipping and sticking phenomena. These mecha-

nisms have many features in common with peristaltic

locomotion found in the animal world. The purpose of

the present paper is to examine the energy eﬃciency

of a locomotion mechanism that exploits friction. With

the help of a model that captures most of the salient

features of locomotion, we show how locomotion featur-

ing stick-slip friction is more eﬃcient than a counterpart

that only features slipping. Our analysis also provides a

framework to establish how optimal locomotion mech-

anisms can be selected.

Keywords Hybrid dynamical systems ·Piecewise-

smooth dynamical systems ·Stick-slip friction ·

Anchoring ·Peristaltic locomotion ·Worm-like

motion ·Robotics

Xuance Zhou ·Oliver M. O’Reilly

Department of Mechanical Engineering

University of California at Berkeley

Berkeley CA 94720-1740

USA

Tel.: +510-642-0877

Fax: +510-643-5599

E-mail: oreilly@berkeley.edu, xuance.zhou@berkeley.edu

Carmel Majidi

Department of Mechanical Engineering

Carnegie Mellon University

Pittsburg PA 15213

USA

Tel.: +412-268-2492

Fax: +412-268-3348

E-mail: cmajidi@andrew.cmu.edu

1 Introduction

Recent advancements in the ﬁeld of robotics include

the development of soft robots [10] and micro-robots

[4,5,20,23]. For some soft robot designs, such as the re-

cent pneumatic quadruped in [19], locomotion can be

achieved by coordinated sticking and slipping of the

limbs. A similar mechanism can be found in certain

micro-robots, such as the ETH-Z¨urich Magmite [8,14,

15], UT-Arlington ARRIpede [12,13], Dartmouth scratch-

drive MEMS robot [3], and magentic micro-robot from

Carnegie-Mellon University [17]. At the macroscale, stick-

slip locomotion is also featured in the Capsubot from

Tokyo’s Denki University [9] and the Friction Board

System [21]. It is also of interest to note that locomo-

tion mechanisms featuring sticking and slipping of limbs

can also be related to the limbless crawling (peristaltic

locomotion [7,11]) observed in a wide variety of species

and bio-inspired robots [16,18] where anchoring (stick-

ing) is realized either by bristles or mucus [2].

E1

E2

FP1

FP2

g

O

P1P2

θ1θ2

AB

Fig. 2 Schematic of a two-degree-of-freedom model for the toy

horse shown in Figure 1. The ﬁgure also indicates the forces FP1

and FP2exerted on the respective links by the pneumatic actu-

ator.

2

Fig. 1 Locomotion of a toy horse. (a) Photo of the stationary toy. (b) Photo of the primary components of the toy horse. (c) A

series of time-lapsed images of the toy horse in motion. The coin in (a) and (b) is a U.S. quarter.

The wealth of designs and implementations in the

aforementioned works make it diﬃcult to gain a per-

spective on the overall energy eﬃciency of a locomotion

scheme controlled by friction. A commercially available

toy, shown in Figure 1, enables us to see the features of

robot locomotion that suﬃce to examine this eﬃciency.

The toy horse has three main components: two limbs

joined by a pair of hinge joints and an air bellows. As

documented in Figure 1(c), by pumping on the bellows,

the toy locomotes forward. One model for this toy is to

represent it as the two-link mechanism shown in Figure

2. The forces due to the bellows (actuator) are modeled

by the forces FP1and FP2acting on the links. As these

forces are varied, the normal forces on the end masses

mAand mBchange. The resulting change in the normal

forces can induce changes to the friction forces acting

on the end masses. If the system is properly designed

and the forces FP1and FP2are properly coordinated,

then, as illustrated in Figure 3, a net forward motion

of the center of mass of the system can be achieved.

We consider the locomotion shown in Figure 3to be

an example of stick-slip locomotion (SSL). However, as

we subsequently discovered, it is also possible to achieve

locomotion with both limbs in a perpetual state of slid-

ing. We deﬁne this locomotion as sliding locomotion

(SL). It is of interest to us to examine how both of these

types of locomotion schemes can be actuated, which one

of them is more energy eﬃcient, and which one of them

enables faster locomotion. To perform such an analy-

sis, we found it essential to simplify the model shown

in Figure 2to the two mass system shown in Figure

4. In this representation, the actuator is modeled as a

spring with a variable initial length ℓ0(t). The spring is

also inclined so that changing the initial length induces

changes to the normal forces on the masses. By chang-

ing the angle of inclination for a given ℓ0(t), the center

of mass of the system can be made to move backwards

or forwards.

g

(a)

(b)

(c)

(d)

actuator

III

A A

A

A

A

B

B

B

C

∆

∆

Fig. 3 Illustration of the locomotion for the model system shown

in Figure 2. In (a) the system is at rest, then in (b) the actuator

extends and the link labeled II moves forward while link Istays

ﬁxed. In (c), the point Astays ﬁxed, the actuator retracts and

link Islips. The net motion ∆of the system after one cycle is

shown in (d). In these ﬁgures the point Cis the center of mass

of the system.

3

While the majority of works in the application area

of interest have addressed hardware design and fabri-

cation, there is an ever increasing number of papers

devoted to a systematic analysis of relevant theoret-

ical models (see, e.g., [1,6,9,22,24]. Such analysis is

challenging because the dynamics are governed by non-

smooth hybrid dynamical systems and recourse to nu-

merical methods is necessary. The present paper ex-

pands such eﬀorts by examining the energetics and per-

formance of devices that feature friction-induced loco-

motion.

The paper is organized as follows: In the next sec-

tion, Section 2, a two degree-of-freedom model for the

locomotion system is described. The model is excited

internally by changing the unstretched spring length

L0(t). The interaction of the resulting normal and fric-

tion forces then leads to locomotion. In Section 3, this

locomotion is classiﬁed into two types and the inﬂu-

ence of some system parameters on the locomotion is

discussed. We then turn to examining the energy eﬃ-

ciency of the SSL and SL mechanisms. Our numerical

analysis features a range of simulations with varying

system parameters. As in [1,24], we show how uneven

friction force distribution can lead to locomotion of the

center of mass. Our analyses conclude with a discussion

of the eﬀects of mass distribution in Section 5. The pa-

per concludes with a set of design recommendations for

balancing time taken by the model to travel a given

distance subject to a given energy dissipation.

2 A Simple Model

While the exploitation of friction to generate locomo-

tion is well known (see, e.g., [4]), analyzing simple mod-

els to examine the features of the implementation of this

locomotion mechanism are rare. The simplest model we

found that could explain the salient features and some

of the challenges of SL and SSL was a two degree-of-

freedom mass-spring system shown in Figure 4(a). As

can be seen from the ﬁgure, the masses are connected

by a spring and are both free to move on a horizontal

surface.

The spring element in the model features an un-

stretched length L0that is to be controlled. This feature

of the model mimics the bellows in the toy horse and

is similar to the active spring used in the recent work

[22] to explain some features of peristaltic locomotion.

We assume that the spring element is linear and exerts

a force −Fson m1and Fson m2:

Fs=−K(L − L0(t)) X2−X1

LE1+D

LE2,(1)

where X1is the displacement of the mass M1and X2is

the displacement of the mass M2. The extended length

Lof the spring features a vertical oﬀset D:

L=q(X2−X1)2+D2.(2)

The corresponding angle between the directions of Fs

and a unit vector in the horizontal direction, E1, is

θ= arcsin D

L,0≤θ < π

2.(3)

We choose the origin so that the position of the centers

of mass for M1and M2are 0 in the vertical E2direction.

Finally, the position vector of the center of mass Cis

XE1=M1X1+M2X2

M1+M2

E1.(4)

The motion Cis crucial to characterizing the eﬃciency

of the locomotion scheme

To mimic the eﬀect of varying normal force that

would be present in a more realistic model of the actual

system, we have tilted the spring at an angle θto the

horizontal. In this way, varying L0induces a change in

the normal forces N1,2on the individual masses:

N1=M1g+Fs·E2, N2=M2g−Fs·E2.(5)

If there is suﬃcient variation, then it can enable a tran-

sition to and from static and dynamic friction. If µs>

µd, then this transition can produces a motion of the

mass particles. For instance in SSL a recurring pattern

of transitions where M1is stuck and M2moves, fol-

lowed by M2being stuck and M1moving towards M2

can occur. We also observe from Figure 4(b) that if ℓ0

is properly controlled, then locomotion of the center of

mass Cof the system is possible.

It is convenient to deﬁne the normalized vertical oﬀ-

set d=D/ ˆ

L, where ˆ

Lis a suitable length scale. Possi-

ble choices of ˆ

Linclude Dand L0(t= 0) 6= 0. As can

be seen from the results shown in Figure 4(b), when

d > 0 (<0), then the center of mass of the model moves

forward (backward) and is stationary when d= 0. In

compiling the results shown in this ﬁgure, we choose

L0(t) = Asin(πt) + ¯

L. It is natural to ask what is the

optimal L0(t) needed to achieve locomotion for a given

average speed of the center of mass C? A related ques-

tion is what is the optimal L0(t) to have the system

perform a prescribed task with minimal power expen-

diture?

4

E1

E2

C

x

(a)(b)

τ

D

dinc.

g

N1

Ff1

N2

Ff2

M1M2

0.8

−0.8

030

Fig. 4 A simple two degree-of-freedom model used to analyze SL and SSL. (a) Schematic of the model showing the normal and

friction forces. (b) The displacement xof the center of mass Cof the system for various values of dand ℓ0(t) = 0.05 sin(πt) + 3.

Referring to (1), the parameters for this model are K= 50,m1=m2= 1,µs= 0.7,µk= 0.5, and d=D/ ˆ

Lis assigned the values

of −0.5,−0.25,0.0,0.25, and 0.50.

2.1 Equations of motion

Before we derive the equations of motion, we deﬁne a

dimensionless time τ=t√g/ ˆ

Land introduce some new

dimensionless parameters

m1=M1

M, m2=M2

M,

x1=X1

ˆ

L, x2=X2

ˆ

L, ℓ =L

ˆ

L,

a=A

ˆ

L, d =D

ˆ

L, ℓ0=L0

ˆ

L,¯

ℓ=¯

L0

ˆ

L,

ω=Ωˆ

L

g, k =Kˆ

L

Mg , c =C

Msˆ

L

g,

n1=|N1|

Mg , n2=|N2|

Mg ,(6)

where we choose M= (M1+M2)/2. In the following

equations, the ˙

() indicates a diﬀerentiation with respect

to τ.

The equations of motion for the simple model form

a hybrid system with state-dependent switching. Here,

we treat the friction as Coulomb friction. For a given

mass Mi, the pair of conditions required for static fric-

tion are

Condition 1. ˙xi(t) = 0.

Condition 2. |k(ℓ−ℓ0) cos(θ)| ≤ µsni(t).

Here, µsis the coeﬃcient of static friction. Based on

the pair of conditions, the switching sets can be deﬁned

as follows:

B1={(x1,˙x1, x2,˙x2)|˙x1= 0 and |fx| ≤ µsn1},

B2={(x1,˙x1, x2,˙x2)|˙x2= 0 and |fx| ≤ µsn2}.(7)

Here,

fx=k(ℓ−ℓ0) cos(θ),and

θ= arcsin

d

q(x2−x1)2+d2

.(8)

Using the switching sets, the equations of motion of the

system can be expressed as follows:

m1¨x1= 0,(x1,˙x1, x2,˙x2)∈ B1,

m2¨x2= 0,(x1,˙x1, x2,˙x2)∈ B2,

m1¨x1+c˙x1−fx+˙x1

|˙x1|µdn1= 0

m2¨x2+c˙x2+fx+˙x2

|˙x2|µdn2= 0 )(x1,˙x1, x2,˙x2)/∈ B1,2.

(9)

The dimensionless total energy eof the system is

e=1

2m1˙x2

1+1

2m2˙x2

2+1

2kq(x2−x1)2+d2−ℓ02

.

(10)

2.2 Analytical modes and natural frequencies

We expect four modes of behavior for the two degree-

of-freedom system:

Mode 0. m1- stick, m2- stick;

Mode 1. m1- slip, m2- slip;

Mode 2. m1- slip, m2- stick;

Mode 3. m1- stick, m2- slip.

It is convenient to deﬁne four natural frequencies that

pertain to the case where d= 0 (i.e., the spring is hor-

izontal) and the system dynamics are assumed to be

linear. In this case, for Mode 1, we expect the system

response to contain the natural frequency ωn1along

5

with the rigid body mode ω0:

ω0= 0, ωn1=sk(m1+m2)

m1m2

.(11)

For Modes 2 and 3, the system shoud behave as a sin-

gle mass system whose natural frequencies are, respec-

tively,

ωn2=rk

m1

, ωn3=rk

m2

.(12)

For the majority of the subsequent analyses, we set

m1=m2. Thus, ωn2=ωn3.

2.3 Internal excitation

In many of the applications of interest, the motion is

controlled by varying a physical parameter of the sys-

tem. For example in the toy horse shown in Figure 1,

the bellows serves as a spring of time-varying length,

in the soft robot in [19], pneumatic cylinders are used

to change an intrinsic curvature, and in peristaltic lo-

comotion a traveling wave is used to induce changes to

the structure’s contact geometry [22]. To model these

eﬀects as simply as possible, we assume that the two

degree-of-freedom system is excited by a time depen-

dent varying intrinsic length

ℓ0=asin(ωt) + ¯

ℓ. (13)

In applications, this type of excitation could be realized

either by a pneumatic cylinder, elastic deformation, or

electromagnetic ﬁelds. Admittedly, other choices of the

function ℓ0(t) are possible and so our numerical inves-

tigation is not exhaustive. Indeed, within the context

of the current system, it would be useful to develop a

framework by which the optimal ℓ0(t) that would min-

imize energy consumption while still ensuring that cer-

tain performance metrics are satisﬁed.

3 Two Types of Locomotion: SL and SSL

In simulations of the simple model presented in Section

2, two types of motion are anticipated: either stick-

slip locomotion (SSL) or slip locomotion (SL). These

two representative locomotion behaviors, which are dis-

cussed extensively in the sequel, are shown in Figure 5.

What distinguishes SL from SSL is that for the latter

one or more of the masses stick for discrete intervals of

time during the motion. That is, for SSL, ∃i∈ {1,2}:

˙xi(t) = 0 and |k(ℓ−ℓ0) cos(θ)| ≤ µsni(t)∀t∈[T1, T2].

In order for this pair of conditions to be satisﬁed, we

found that we need the excitation frequency ωto be

(a)(b)

ττ

ω= 0.2ωn1ω= 2.3ωn1

x1

x1

x2

x2

xx

6 6

00

5555 4545

Fig. 5 Illustration of (a) SSL and (b) SL in the two-degree-

of-freedom model governed by Eqn. (9). In (a) the excitation

frequency is ω= 0.2ωn1and in (b) the excitation frequency is

ω= 2.3ωn1. For both examples, the remaining parameters are

a= 0.05, k= 50, ¯

ℓ= 3, d= 0.5, c= 0.01, m1=m2= 1,

µk= 0.5, µs= 0.7 and ωn1= 10.

lower than the lowest ωn1,2,3and the amplitude of ex-

citation ashould also be small. These results from our

numerical simulations are shown in Figure 6. In these

numerical simulations, we categorize the motion as SL

when no sticking behavior happens during a time in-

terval ∆τ = 25 in the steady-state motion, otherwise,

as SSL. Of all the parameters governing whether the

locomotion was SSL or SL, ωwas the most prepotent.

As can be seen from Figure 6, if ωis suﬃciently large

then the system’s locomotion is SL. The discrete peaks

in the transition curve seen in this ﬁgure also proved

to be very sensitive to tolerances in our numerical inte-

gration schemes.

a

¯

ℓ

0

0

1

0.8

ω

ωn1

SSL

SL

Fig. 6 A graph illustrating the region of SSL with the dimen-

sionless amplitude aand dimensionless frequency ωas the varying

parameters. The other parameter that are kept ﬁxed are k= 50,

c= 0.01, ¯

ℓ= 3, d= 0.5, µs= 0.7, µk= 0.5, m1=m2= 1, and

ωn1= 10.

6

For all the above and subsequent numerical simula-

tions, we set the initial conditions as

x1(t= 0) = 0, x2(t= 0) = ¯

ℓ,

˙x1(t= 0) = ˙x2(t= 0) = 0.(14)

The properties we characterize are based on the be-

havior of the system after the initial transients have

subsided. We assume that a time period of at least 80

periods of the lowest natural frequency is suﬃcient of

these transients to have decayed.

The second parameter which should play a key role

in the occurrence of SSL is the static friction coeﬃcient

µs. To explore the eﬀects of this parameter, we exam-

ined the relationship between the time taken τ5for the

center of mass Cto travel a distance of 5 dimensionless

units and the diﬀerence between the static friction co-

eﬃcient and the dynamic coeﬃcient: µs−µk. In other

words, we are interested in the eﬀects of static friction

on the average speed of locomotion. Referring to Fig-

ure 7, we found that a larger static friction will help

accelerate the system in a certain range. However, in

general, the eﬀects of varying the static friction coef-

ﬁcient µs≥µkare not signiﬁcant when the dynamic

coeﬃcient µkis ﬁxed.

τ5

ω= 0.2ωn1

ω= 2.3ωn1

0

150

0.5

300

µs−µk

Fig. 7 Inﬂuence of µs−µkon the dimensionless time τ5taken

for the center of mass Cto travel a distance of 5 dimensionless

units with ω= 0.2ωn1,a= 0.05, k= 50, c= 0.01, ¯

ℓ= 3, d= 0.5,

m1=m2,µk= 0.5, ωn1= 10 and µsvarying from 0.5 to 1.0.

4 Energetic Considerations

An optimal locomotion scheme could be considered as

one where a ﬁxed distance is travelled in the shortest

time while minimizing energy consumption. To exam-

ine optimality, we ﬁrst need to deﬁne the energy con-

sumption. For the system at hand, the energy consump-

tion can be inspected in two equivalent manners. First,

the energy consumed in actuating the spring is used

to counterbalance the energy dissipated by friction and

the linear damper. The energy dissipated edhas the

dimensionless representation:

ed=Zτ2

τ1µkn1|˙x1|+µkn2|˙x2|+c˙x2

1+c˙x2

2dτ. (15)

The second measure is to consider the work wdone by

the spring force. The work is balanced with the change

in the total energy eof the system and the energy ed

dissipated by the system:

w=ed+e(τ=τ2)−e(τ=τ1).(16)

In the following numerical analysis, edwill be used as

a measure of the energy consumption. The advantage

of choosing edover wis that ednot only indicates the

amount of energy consumed in order to make the sys-

tem move, but also shows the amount of energy con-

verted to heat. Heat dissipation is often a non-trivial

issue for MEMS devices which can be susceptible to

thermal failure.

To explore eﬃciency, we considered the time τztaken

for the center of mass Cto travel a distance zand the

corresponding energy ez. We computed these metrics

for a range of excitation frequencies ωand have com-

piled a representative selection of the results in Figure

8. The results shown in Figure 8challenge our percep-

tion that it is always economical to excite a system at

resonance. Clearly, one attains the minimum time to

travel a given distance when ωis close to the frequency

ωn1and there are also local minima near ωn2=ωn3.

However, the maximum energy dissipated also occurs

when ωis close to ωn1.

In the region ω/ωn1<0.3 where SSL is the observed

locomotion mechanism, several local minima in travel

time τ5occur with minimal changes in e5. However,

there are several disadvantages for those minima in SSL

region. First, these critical points are very sensitive to

changes in ωand, second, the average speed doesn’t

compare to that when ωis close to ωn1. In general, the

results in Figure 8indicate that energy eﬃciency can

never be achieved without lowering the average speed

of the center of mass C.

Another key factor in excitation is the amplitude a

of the spring’s intrinsic length ℓ0(t). In order to draw

some conclusions on the inﬂuence of a, ﬁve excitation

frequencies were selected featuring two low frequencies

(one with SSL and one featuring SL), one frequency

near resonance and two high frequencies. On the whole,

the trend in Figure 9agrees with the results shown

in Figure 8that a higher average speed can only be

achieved with a higher concomitant energy dissipation.1

One feature of particular interest in Figure 9is that

1While the energy e5dissipated for ω= 0.95ωn1does decrease

after a certain amplitude ais reached, this region in parameter

7

(a)(b)

SSLSSL

SLSL

τ5

ω

ωn1

ω

ωn10

0

00

140

e5

3

3

1000

Fig. 8 Numerical results to analyze the eﬃciency of the model in Figure 4as a function of the excitation frequency ω. (a) Nondimen-

sionalized time τ5taken to travel a distance of 5 dimensionless units. (b) The corresponding non-dimensionalized energy consumption

e5during the motion. The parameters for the model used to produce these results were a= 0.05, ¯

ℓ= 3, d= 0.5, k= 50, c= 0.01,

m1=m2,µk= 0.5, µs= 0.7 and ωn1= 10.

0.050.05

0

0

33

250 140

aa

τ5e5

(a)(b)

ω1

ω1

ω2ω2

ω3

ω3

ω4

ω4

ω5

ω5

Fig. 9 Numerical results to analyze the eﬃciency of the model in Figure 4for a varying exciting frequency with three representative

frequencies . (a) Nondimensionalized time τ5travel a distance of 5 dimensionless units. (b) The corresponding dimensionless energy

consumption e5. The parameters for this model are ¯

ℓ= 3, d= 0.5, k= 50, c= 0.01, m1=m2= 1, µk= 0.5, µs= 0.7 and ω1= 2,

ω2= 6, ω3= 9.5, ω4= 16 and ω5= 23 respectively.

when a≤0.08, the system substantially traveled the

ﬁxed distance in the same amount of time with same

amount of energy dissipated for low frequency ω=

0.60ωn1as with a high frequency ω= 1.60ωn1. How-

ever, when a > 0.08, the system excited with a low

frequency ω= 0.60ωn1can travel the ﬁxed distance in

less time and with a smaller energy dissipation than the

system excited with a frequency ω= 1.60ωn1. In other

words, when all the other conditions are equal, the exci-

tation with a lower frequency, namely, a longer period,

appears to allow the system to take more advantage of

the resultant force on the system in the E1direction

than one with a high excitation frequency.

To illustrate the aforementioned comment about the

resultant force in the horizontal direction, we sum the

external forces which are composed of friction and damp-

ing forces in E1direction for the system:

Fex ·E1= (Ff1+Fd1+Ff2+Fd2)·E1.(17)

space is not feasible because when the two mass are too close to

each other there is a possibility that the normal force on one of

them will vanish and that mass would then loose contact with

the ground.

0

−Fmax

Fmax

4Tf

t

f

f(t)

x

m

Fig. 11 Schematic of a particle mmoving under the inﬂuence of

a resultant periodic force f(t). The sawtooth proﬁle of fis also

shown.

We next consider the system and subject it to a periodic

external force. In the ﬁrst case, the excitation frequency

is suﬃciently small that SSL occurs and in the second

case the frequency is suﬃciently high so that SL occurs.

For cases exhibiting SSL and SL, the resultant force

Fex ·E1as a function of time are shown in Figure 10.

The sets of results shown in Figure 10(a),(b) exhibit a

similar average speed for the center of mass C. However,

the amplitude of force for the SL case in Figure 10(b) is

almost twice that for the SSL case in Figure 10(a). Since

all the parameters except the excitation frequency ωare

identical, this eﬀect must be attributed to ω. According

8

0

0

−1.5−1.5

1.51.5

47 53 49.750.3

(a)(b)

ω= 0.2ωn1ω= 2.3ωn1

2Fex·E1

(M1+M2)g

2Fex·E1

(M1+M2)g

˙x1

˙x1

˙x

˙x

˙x2

˙x2

ττ

ττ

Fig. 10 Illustration of the force and velocity proﬁles for the two typical types of motion exhibited by the solutions to Eqn. (9). In

(a), the excitation frequency is ω= 0.2ωn1and in (b) the excitation frequency is ω= 2.3ωn1. For both examples, the remaining

parameters are a= 0.05, k= 50, ¯

ℓ= 3, d= 0.5, c= 0.01, m1=m2= 1, µk= 0.5, µs= 0.7, and ωn1= 10.

to Figure 10, the period of the resultant force Fex ·E1

is half the period of ℓ0(t).

To develop a sense of the role that the period of

the resultant external force Fex ·E1in Figure 10 plays

on the motion of the system, we consider a similar sce-

nario of a particle munder the inﬂuence of the saw-

tooth periodic force in the xdirection (cf. Figure 11).

The sawtooth proﬁle is an approximation of the proﬁle

of Fex ·E1that is visible in Figure 10(b) and the parti-

cle mcan be considered as the system composed of m1

and m2. Based on the above assumption, if the particle

of mass mhas an initial velocity v0, then the corre-

sponding average speed ¯vin one period of the forcing

is

¯v=v0+Fmax Tf

6m=v0+Fmaxπ

3mωf

,(18)

where ωf=2π

Tf. Even though the situation in the two

degree-of-freedom model with friction is far more com-

plicated (because the quantities corresponding to Fand

v0are functions of ωf) we can still use (18) to obtain

some qualitative insights. For instance, assuming that

Fmax and v0have the same sign, then Eqn. (18) shows

that the lower the frequency ωf, the higher the value we

can expect for the average speed ¯v. This simple model

also shows why we should not expect SL to yield faster

locomotion than SSL.

5 The Eﬀects of Mass Distribution

The motion of the system is achieved in part by varying

the normal forces at the contact points with the ground.

These forces are also proportional to the masses m1

and m2, respectively. Consequently, it is of interest to

examine how the mass distribution m1

m2can eﬀect the

locomotion of the system. In this section, we examine

how the time to travel τ5and the energy dissipated e5

are related to the mass parameter m1

m2+m1for a set of

ﬁve representative excitation frequencies.

In Figure 12(b), for high frequency ω=ω5= 23,

we ﬁnd three local minima for τ5near ˆm=m1/(m1+

m2) = 0.05, 0.95, and 0.5. When ˆm= 0.05 or 0.95, then

the natural frequency ωn1=ω. Like the case shown

in Figure 8, the least time needed to achieve a given

distance is near ωn1and this is produced with maximum

energy consumption. We obtain another local minimum

in τ5when ˆm= 0.5 (i.e., m1=m2) and this is produced

with (a local) minimal energy consumption.

The results shown in Figure 12 provide another way

to accelerate our system when changing the excitation

frequency is not possible. For ω=ω4= 16, the trend

follows what happens with ω=ω5= 23 except the

mass ratio where ω=ωn1changes. The third case we

consider is ω=ω3= 9.5. Here, as ω3≈ωn1when

m1=m2, we ﬁnd that the three valleys reduced to a

single wide ﬂat valley. This is a very appealing design

region for applications.

If we continue to decrease the excitation frequency

to ω=ω2= 6, then the ﬁrst natural frequency ωn1can

never be reached regardless of the mass distribution2.

However, as can be seen from Figure 12(b), we still ﬁnd

three local minima of average velocity at ˆm= 0.27,

0.73, and 0.5.

As can be seen in Figure 12(a), with the two mass

distributions ˆm= 0.27 and 0.73, the exciting frequen-

cies are quite close to the approximated frequency cor-

2According to Eqn. (11) and Figure 12(a), the minimum ωn1

is 10 with m1=m2

9

0

0

0

0

0

0

800800

111

m1

m1+m2

m1

m1+m2

m1

m1+m2

τ5e5

30

(a)(b)(c)

ω1

ω1

ω1

ω2ω2

ω2

ω3

ω3

ω3

ω4ω4

ω4

ω5

ω5

ω5

ωn2ωn1

ωn3

Fig. 12 Numerical results that are used to analyze the energy-eﬃciency of the model in Figure 4for diﬀerent mass distributions

m1

m1+m2. (a) Natural frequencies ωn1,2,3as functions of the mass ratio m1

m1+m2. (b) Dimensionless time τ5to travel a distance of 5

dimensionless units. (c) The corresponding dimensionless energy consumption e5. The parameters for this model are k= 50, a= 0.05,

¯

ℓ= 3, d= 0.5, µs= 0.7, µk= 0.5 and ω1= 2, ω2= 6, ω3= 9.5, ω4= 16, and ω5= 23 respectively.

responding to the mode of single mass oscillation ωn2=

5.85 and ωn3= 5.85, respectively. For the other min-

imum at ˆm= 0.5, the excitation frequency ω=ω2is

the closest to ωn1. Of particular interest to us is that its

corresponding energy consumption indicated by Figure

12(c) is also a minimum.

The behavior when ω=ω1= 2 follows what oc-

curred with ω=ω2= 6 except that it does not exhibit

the two valleys for the travel time τ5near the frequency

corresponding to a single mass oscillation. By examin-

ing numerical simulations for the case ω=ω1, we found

that SSL was dominant during the entire motion. With

one of the masses stuck, we have less energy dissipated.

However, the time to reach the ﬁxed distance 5 is longer

in general compared to the other cases ω2,3,4,5,6and is

not signiﬁcantly improved at the minimum m1=m2.

Finally, when ω=ω1= 2, the system can only be

set into motion in a narrow range of mass distributions

near ˆm= 0.5.

6 Conclusions

Based on the numerical simulations and analysis of the

simple model, the following conclusions on locomotion

can be drawn:

1. SSL typically occurs only for frequencies smaller

than ωn1,2,3.

2. SSL is energy eﬃcient, however, it is not always the

fastest form of locomotion.

3. During SSL, the time to travel a given distance is

not very sensitive to the diﬀerence in the coeﬃcients

of static and dynamic friction.

4. To achieve the same average velocity of the center

of mass, especially when the excitation amplitude a

is large, low frequency is better than high frequency

in term of energy eﬃciency.

These observations have potential inﬂuence on how friction-

controlled robots are operated and designed. The de-

sign and operation of these devices include the actu-

ator technology, materials, and geometric dimensions

required to achieve the actuation frequency, amplitude,

kinetic friction, and mass distribution necessary for en-

ergetically eﬃcient locomotion at a prescribed veloc-

ity. Such insights have particularly important implica-

tions in the design and operation of soft robots. In con-

trast to their rigid counterparts, soft robots elastically

conform to a surface and typically engage in friction-

controlled locomotion. Even for designs that cannot be

represented by the models examined here, our analysis

nonetheless identiﬁes the important factors (e.g., ac-

tuation frequency, amplitude) and general advantages

of SSL over SL for accomplishing forward motion with

minimal frictional energy dissipation.

Acknowledgements Support from a Defense Advanced Research

Projects (DARPA) 2012 Young Faculty Award to Carmel Majidi

is gratefully acknowledged. Xuance Zhou is grateful for the sup-

port of a Anselmo Macchi Fellowship for Engineering Graduate

Students and a J.K. Zee Fellowship. The authors also take this

10

opportunity to thank an anonymous reviewer for their construc-

tive criticisms.

References

1. Chernous’ko, F.L.: The optimum rectilinear motion

of a two-mass system. Journal of Applied Math-

ematics and Mechanics 66(1), 1–7 (2002). URL

http://dx.doi.org/10.1016/S0021-8928(02)00002- 3

2. Denny, M.: The role of gastropod pedal mucus in lo-

comotion. Nature 285(1), 160–161 (1980). URL

http://dx.doi.org/10.1038/285160a0

3. Donald, B., Levey, C., McGray, C., Rus, D., Sin-

clair, M.: Power delivery and locomotion of unteth-

ered microactuators. Journal of Microelectrome-

chanical Systems 12(6), 947–959 (2003). URL

http://dx.doi.org/10.1109/JMEMS.2003.821468

4. Driesen, W.: Concept, modeling and experimental character-

ization of the modulated friction inertial drive (MFID) loco-

motion principle: Application to mobile microrobots. Ph.D.

thesis, ´

Ecole Polytechnique F´ed´erale de Lausanne (2008).

URL http://infoscience.epfl.ch/record/121454

5. Driesen, W., Rida, A., Breguet, J.M., Clavel, R.: Friction

based locomotion module for mobile Mems robots. In: In-

telligent Robots and Systems, 2007. IROS 2007. IEEE/RSJ

International Conference on, pp. 3815–3820 (2007). URL

http://dx.doi.org/10.1109/IROS.2007.4399321

6. Edeler, C., Meyer, I., Fatikow, S.: Modeling

of stick-slip micro-drives. Journal of Micro-

Nano Mechatronics 6(3–4), 65–87 (2011). URL

http://dx.doi.org/10.1007/s12213-011- 0034-9

7. Elder, H.Y.: Peristaltic mechanisms. In: H.Y. Elder, E.R.

Trueman (eds.) Aspects of Animal Movement, vol. 5, pp. 71–

92. Society for Experimental Biology: Seminar Series, Cam-

bridge University Press, Cambridge, UK (1985)

8. Frutiger, D., Kratochvil, B., Nelson, B.: MagMites - Micro-

robots for wireless microhandling in dry and wet environ-

ments. In: Robotics and Automation (ICRA), 2010 IEEE

International Conference on, pp. 1112–1113 (2010). URL

http://dx.doi.org/10.1109/ROBOT.2010.5509678

9. Li, H., Furuta, K., Chernousko, F.: Motion generation of the

Capsubot using internal force and static friction. In: Decision

and Control, 2006 45th IEEE Conference on, pp. 6575–6580

(2006). URL http://dx.doi.org/10.1109/CDC.2006.377472

10. Majidi, C.: Soft robotics: A perspective - current trends and

prospects for the future. Soft Robotics 1(P), 5–11 (2013).

URL http://dx.doi.org/10.1089/soro.2013.000

11. McNeil Alexander, R.: Principles of Animal Locomotion.

Princeton University Press, Princeton, New Jersey (2003)

12. Murthy, R., Das, A., Popa, D.O.: ARRIpede: A stick-slip mi-

cro crawler/conveyor robot constructed via 2.5D MEMS as-

sembly. In: Intelligent Robots and Systems, 2008. IROS 2008.

IEEE/RSJ International Conference on, pp. 34–40 (2008).

URL http://dx.doi.org/10.1109/IROS.2008.4651181

13. Murthy, R., Das, A., Popa, D.O., Stephanou, H.E.:

ARRIpede: An assembled die-scale microcrawler.

Advanced Robotics 25(8), 965–990 (2011). URL

http://dx.doi.org/10.1163/016918611X568602

14. Nagy, Z., Frutiger, D., Leine, R., Glocker, C., Nelson, B.:

Modeling and analysis of wireless resonant magnetic microac-

tuators. In: Robotics and Automation (ICRA), 2010 IEEE

International Conference on, pp. 1598–1603 (2010). URL

http://dx.doi.org/10.1109/ROBOT.2010.5509260

15. Nagy, Z., Leine, R., Frutiger, D., Glocker, C., Nel-

son, B.: Modeling the motion of microrobots on sur-

faces using nonsmooth multibody dynamics. IEEE Trans-

actions on Robotics 28(5), 1058–1068 (2012). URL

http://dx.doi.org/10.1109/TRO.2012.2199010

16. Nakazato, Y., Sonobe, Y., Toyama, S.: Development of an

in-pipe micro mobile robot using peristalsis motion. Journal

of Mechanical Science and Technology 24(1), 51–54 (2010).

URL http://dx.doi.org/10.1007/s12206-009-1174-x

17. Pawashe, C., Floyd, S., Sitti, M.: Modeling and ex-

perimental characterization of an untethered mag-

netic micro-robot. The International Journal of

Robotics Research 28(8), 1077–1094 (2009). URL

http://dx.doi.org/10.1177/0278364909341413

18. Seok, S., Onal, C.D., Cho, K.J., Wood, R.J., Rus, D.,

Kim, S.: Meshworm: A peristaltic soft robot with an-

tagonistic nickel titanium coil actuators. Mechatronics,

IEEE/ASME Transactions on 18(5), 1485–1497 (2013). URL

http://dx.doi.org/10.1109/TMECH.2012.2204070

19. Shepherd, R.F., Ilievski, F., Choi, W., Morin, S.A., Stokes,

A.A., Mazzeo, A.D., Chen, X., Wang, M., Whitesides, G.M.:

Multigait soft robots. Proceedings of the National Academy

of Sciences, U.S.A. 108(51), 20,400–20,403 (2011). URL

http://dx.doi.org/10.1073/pnas.1116564108

20. Sitti, M.: Miniature devices: Voyage of the micro-

robots. Nature 458(7242), 1121–1122 (2008). URL

http://dx.doi.org/10.1038/4581121a

21. Suzuki, Y., Li, H., Furuta, K.: Locomotion generation of fric-

tion board with an inclined slider. In: Decision and Control,

2007 46th IEEE Conference on, pp. 1937–1943 (2007). URL

http://dx.doi.org/10.1109/CDC.2007.4434269

22. Tanaka, Y., Ito, K., Nakagaki, T., Kobayashi, R.: Mechanics

of peristaltic locomotion and role of anchoring. Journal of

the Royal Society Interface 9(67), 222—233 (2012). URL

http://dx.doi.org/10.1098/?rsif.2011.0339

23. Wood, R.: The ﬁrst takeoﬀ of a biologically in-

spired at-scale robotic insect. Robotics, IEEE

Transactions on 24(2), 341–347 (2008). URL

http://dx.doi.org/10.1109/TRO.2008.916997

24. Zimmermann, K., Zeidis, I.: Worm-like locomotion as a prob-

lem of nonlinear dynamics. Journal of Theoretical and Ap-

plied Mechanics 45(1), 179–187 (2007)