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Energy Efficiency in Friction-Based Locomotion Mechanisms for Soft and Hard Robots: Slower can be Faster


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Many recent designs of soft robots and nano robots feature locomotion mechanisms that cleverly exploit slipping and sticking phenomena. These mechanisms 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 featuring stick-slip friction is more efficient than a counterpart that only features slipping. Our analysis also provides a framework to establish how optimal locomotion mechanisms can be selected.
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Nonlinear Dynamics Published Online August 2014
Energy Efficiency in Friction-Based Locomotion
Mechanisms for Soft and Hard Robots: Slower can be
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
Tel.: +510-642-0877
Fax: +510-643-5599
Carmel Majidi
Department of Mechanical Engineering
Carnegie Mellon University
Pittsburg PA 15213
Tel.: +412-268-2492
Fax: +412-268-3348
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].
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-
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 0(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 0(t), the center
of mass of the system can be made to move backwards
or forwards.
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.
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-
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
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)) X2X1
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:
The corresponding angle between the directions of Fs
and a unit vector in the horizontal direction, E1, is
θ= arcsin D
L,0θ < π
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
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=M2gFs·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 0
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-
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 0(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=tg/ ˆ
Land introduce some new
dimensionless parameters
M, m2=M2
L, x2=X2
L, =L
L, d =D
L, ℓ0=L0
g, k =Kˆ
Mg , c =C
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(0) 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)
fx=k(0) cos(θ),and
θ= arcsin
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,
|˙x1|µdn1= 0
|˙x2|µdn2= 0 )(x1,˙x1, x2,˙x2)/∈ B1,2.
The dimensionless total energy eof the system is
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
with the rigid body mode ω0:
ω0= 0, ωn1=sk(m1+m2)
For Modes 2 and 3, the system shoud behave as a sin-
gle mass system whose natural frequencies are, respec-
, ωn3=rk
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
0=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 0(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 0(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(0) 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
ω= 0.2ωn1ω= 2.3ωn1
6 6
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, ¯
= 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.
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, ¯
= 3, d= 0.5, µs= 0.7, µk= 0.5, m1=m2= 1, and
ωn1= 10.
For all the above and subsequent numerical simula-
tions, we set the initial conditions as
x1(t= 0) = 0, x2(t= 0) = ¯
˙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: µsµk. 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 µsµkare not significant when the dynamic
coefficient µkis fixed.
ω= 0.2ωn1
ω= 2.3ωn1
Fig. 7 Influence of µsµkon 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, ¯
= 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:
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:
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 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 0(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
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, ¯
= 3, d= 0.5, k= 50, c= 0.01,
m1=m2,µk= 0.5, µs= 0.7 and ωn1= 10.
250 140
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 ¯
= 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 a0.08, 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.
Fig. 11 Schematic of a particle mmoving under the influence of
a resultant periodic force f(t). The sawtooth profile of fis also
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
47 53 49.750.3
ω= 0.2ωn1ω= 2.3ωn1
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, ¯
= 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 0(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
¯v=v0+Fmax Tf
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 ω3ωn1when
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
Fig. 12 Numerical results that are used to analyze the energy-efficiency of the model in Figure 4for different mass distributions
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,
= 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
opportunity to thank an anonymous reviewer for their construc-
tive criticisms.
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 3
2. Denny, M.: The role of gastropod pedal mucus in lo-
comotion. Nature 285(1), 160–161 (1980). URL
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
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).
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
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 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
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
10. Majidi, C.: Soft robotics: A perspective - current trends and
prospects for the future. Soft Robotics 1(P), 5–11 (2013).
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).
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
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
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
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).
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
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
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
20. Sitti, M.: Miniature devices: Voyage of the micro-
robots. Nature 458(7242), 1121–1122 (2008). URL
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
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
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
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)
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The forward rectilinear motion of a system of two rigid bodies along a horizontal plane is considered. Forces of dry friction act between the bodies and the plane, and the motion is controlled by internal forces of interaction between the bodies. A periodic motion in which the system moves along a straight line is constructed. The optimum parameters of the system and a control law are found corresponding to the maximum mean velocity of motion of the system as a whole.