Masters Thesis Written Report:
Mechanical Engineering Conference and Bennett Competition
April 15, 2005, Carnegie Mellon University, Pittsburgh, PA
A PIEZOELECTRIC UNIMORPH ACTUATOR BASED PRECISION POSITIONING MINIATURE
WALKING ROBOT
Kwon Joong Son
Masters Candidate
NanoRobotics Laboratory
Department of Mechanical Engineering
Carnegie Mellon University, Pittsburgh, PA 15213
Email: iammrson@cmu.edu
Venkataraman Kartik
Ph.D. Candidate
Dynamic Systems Laboratory
Department of Mechanical Engineering
Carnegie Mellon University, Pittsburgh, PA 15213
Email: vkartik@andrew.cmu.edu
Jonathan A. Wickert
Professor
Dynamic Systems Laboratory
Department of Mechanical Engineering
Carnegie Mellon University, Pittsburgh, PA 15213
Email: wickert@andrew.cmu.edu
Metin Sitti
Assistant Professor
NanoRobotics Laboratory
Department of Mechanical Engineering
Carnegie Mellon University, Pittsburgh, PA 15213
Email: sitti@cmu.edu
ABSTRACT
This paper proposes a precision positioning miniature
walking robot using piezoelectric unimorph actuators. The
theoretical working equations of a uniform piezoelectric
unimorph beam are derived based on the linear piezoelectric
relations, Hamilton’s principle, and Euler-Bernoulli beam
equation.
The modal superposition method is used to
determine the response of the forced transverse vibration of the
beam under the effect of electrical signal inputs to the patterned
piezoelectric element. Two standing waves corresponding to
the third and fourth bending vibration modes are utilized to
achieve the bi-directional walking mechanism for a miniature
positioning robot. Design strategies and the fabrication
method for the proposed walking robot are introduced.
Preliminary performance tests of the robot prototype are carried
successfully, and the robot can move forward with the speed of
5.86 cm/sec velocity and backward with 3.37 cm/sec.
INTRODUCTION
Recent advances in micro/nanorobotic technology have
necessitated the development of novel actuation mechanisms
and highly precise measurement systems with submicron and
nanometer resolution. Miniature walking robots with highprecision positioning capability have applications as
transportation/positioning platforms for nano-tools such as
probes for Atomic Force Microscopy, or for manipulation
mechanisms [1, 2]. The concept of a mobile micro robot
which can serve as a platform for interchangeable tools would
be extremely useful in a number of applications. A significant
advantage of such a suitably small-sized tool carrier robot
would be the possibility of being able to employ multiple such
robots as a team to simultaneously work on a material sample
[3].
Many designs and mechanisms for precision positioning
walking robots have been developed [4 - 12]. In this research,
inspired by conventional ultrasonic motors (USM) [13, 14] and
their excellent performance in applications for positioning
stages, a prototype of the mobile frame was constructed that
uses standing waves induced by the bending vibration of a
piezoelectric (PZT in this research) unimorph actuator to propel
the robot by means of suitably attached legs. The vibration
analysis of the piezo-metal structure is, thus, necessary to
predict the dynamic behavior of the robot, and to evolve an
effective design. The coupled nature of piezo-electricity also
needs to be considered.
We derive an analytical working equation for the piezo-metal
composite actuator and employ it as a theoretical basis to
design the robot. Predictions of optimal configurations and
suitable operating conditions for the robot are obtained.
PRINCIPLE OF OPERATION
The proposed design uses high frequency standing waves
generated through the piezoelectric effect. Figure 1 shows a
schematic diagram of a standing wave in a beam with two fixed
ends. Three projections (legs) are attached to the beam at the
points A, B, and C. These points are located toward the right
of the anti-nodes of the respective waves. If the projections
are assumed to be sufficiently stiff, they will always be
perpendicular to the shape of the standing wave at the points of
attachment. When a standing wave is induced in the beam as
shown, the tips of the projections undergo linear diagonal
motions. For the configuration shown in the figure, the effect
of the standing wave is to produce motion that moves the beam
toward the left. On the down-stroke, the friction between the
leg tip and the surface propels the robot forward. However,
there is no such force on the up-stroke, thus ensuring that the
robot does not return to the position from where it started.
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Copyright © 2005 by ASME
Equation of Motion and Boundary Conditions
For a system such as the one depicted in Fig. 3, Hamilton’s
principle [16] yields
t
t2
1
Fig.1. Locomotion principle of a walking robot using
a standing wave induced in a beam with fixed ends
The shortcoming of this design is the fact that only onedirectional motion is possible using a given standing wave
pattern. A modified design has been proposed to achieve bidirectional motion using two different bending modes [15].
Here, we select the third and fourth bending modes of the
beams transverse vibration (henceforth, denoted as B3 and B4,
respectively). Bi-directional motion can be achieved if the
legs are located toward the right of the anti-node of one mode
and simultaneously towards left of the anti-node of another
mode. Two possible combinations of such regions where legs
can be attached are depicted in Fig. 2 as regions A and B.
Thus, exciting one of the modes would lead to motion in one
direction, while exciting the other mode would result in motion
in the opposite direction. The desired direction of motion can
be achieved by modulating the operating frequency to the
appropriate resonance frequency of the beam.
Fig.2. Two permissible combinations of regions for
attachment of three legs to achieve bi-directional
motion
(L Wd W p )dt 0
(1)
where u(x,t), δL, δWd, and δWp are respectively the vertical
deflection field of the beam, the variation of the Lagrangian,
the virtual work due to the external disturbance fd(x,t), and the
virtual work due to the applied voltage Vi on the patterned
electrodes of the piezoelectric layer. In this research, fd(x,t) is
assumed to be negligible.
The variation of u(x,t) at each known instant is zero.
Thus,
(2)
u ( x, t1 ) u ( x, t 2 ) 0
Fig.4. (a) Corss-section of the piezo-metal
composite beam structure (b) equivalent crosssection of the beam
The location of the neutral axis of the piezo-metal
composite beam is obtained using the equivalent cross-section
method. From Fig. 4(b), the position of the neutral axis,
denoted by its distance form the bottom surface y , satisfies the
relation,
tp
t
(3)
y ( Abt s bt p ) s ( Abt s ) (t s )bt p
2
2
where ts, tp and b are the height of the PZT layer, the height of
the metal layer and the width of the beam, respectively. From
the above equation, y can be alternatively expressed as
MODEL OF A PIEZO-METAL COMPOSITE BEAM
ACTUATOR
The vibration analysis of a composite layer consisting of a
piezoelectric material and a metal plate is essential for a proper
design of the USM based walking robot. Figure 3 depicts
such a composite plate in schematic form. The plate is
supported at opposite ends by hinges and torsional springs.
1 AB 2 2 B 1
(4)
tp
2
AB 1
The position y is chosen so that the maximum
transduction from electrical energy to mechanical energy can
occur [17]. The maximum energy transduction condition is
achieved when the following condition is satisfied.
tp
Es
(5)
or AB 2 1
ts
Ep
Fig.3. Schematic diagram of a piezo-metal composite
structure supported with torsional springs at the
ends
where Es and Ep are the Young’s moduli of the steel and PZT
ceramic. The two dimensionless numbers in the above
equation are defined as A=Es/Ep and B= ts / tp. Geometrically,
if B is chosen such that AB 2 1 , the neutral axis would
become coincident with the interface of the PZT and metal
layers, i.e.,
y ts
(6)
If materials with specific mechanical properties are selected, it
follows from equation (3) that y depends only on the
thickness of each layer.
Therefore, the thickness ratio
y
2
Copyright © 2005 by ASME
between the PZT and the metal substrate layers can be
determined by equation (6).
The total kinetic energy of the composite beam is
L1
( p t p s t s )bu,t 2 dt
T Ts T p
0 2
(7)
L
1
2
T p t p b( BR 1) u,t dt
0
2
where ρp and ρs are the densities of the PZT ceramic and the
metal substrate respectively, and R=ρp/ρs..
Fig.5. (a) Linear strain profile, and (b) piecewise linear
stress profile in the cross-section of a beam
The total strain energy in the beam can be expressed if the
strain and stress distributions are known through the whole
domain. The Euler-Bernoulli beam theory is adopted in this
derivation procedure. The beam theory makes the basic
assumption that a normal strain, at a certain position, is linearly
proportional to the distance from the neutral axis (Fig. 5).
Taking the total strain energy in the integral form
U strain dV , the total strain energy for the composite beam
can be written in the following form.
L tp 1
0 1
E s yu,xx yu,xx dy bdx
U
E p yu, xx yu, xx dy
t s 2
0 0 2
L
1
(8)
U E p t p 3b( AB 3 1) u, xx 2 dx
0
6
Comparing equations (7) and (8) with the expressions for the
total kinetic energy and the total strain energy for a single-layer
L1
uniform
beam,
i.e.
and
T
Au,t 2 dx
0 2
L1
U
EIu, xx 2 dx , we find the equivalent constants (ρA)eq and
0 2
(EI)eq for the given piezo-metal composite beam.
(9)
( A) eq p t p b( BR 1)
1
(10)
E p t p 3b( AB 3 1)
3
The above quantities are associated with the equivalent mass
and the equivalent stiffness for the transverse vibration of a
beam, and they are useful in describing the beam dynamics
succinctly. For example, the Lagrangian of the system can be
expressed as
L
L
1
1
(11)
L T U ( A) eq u,t 2 dx ( EI ) eq u, xx 2 dx
0
0
2
2
The corresponding variation of the Lagrangian is
Fig.6. (a) Linear strain profile (b) stress profile
generated by the applied voltage Va on the PZT layer
The work due to a voltage applied on the patterned
electrodes on the PZT layer (Fig. 3) results from the energy
transduction which occurs in the piezoelectric material. The
piezoelectric constant d31, which is defined as the ratio of the
resultant strain in the x-direction to the electric field intensity in
the y-direction, enables the quantitative estimation of the work
done in this case. Figure 6 shows the stress profile generated
in the beam due to the applied voltage Va. The stress in the
PZT layer σp is derived using Hooke’s law and the linear
relation between the applied electric field intensity and the
induced strain profile which is assumed to be unidirectional.
N separated electrodes on the upper piezo layer are considered
here. If a sequential voltage Vk (k=1, 2, ··· N) is applied to
each electrode (Fig. 3), Va can be expressed as the following
summation
Va Vk H ( x xk ) H ( x xk 1 )
N
k 1
(14)
where N is the number of electrodes and H(x) is the Heaviside
step function. The work done by the applied voltage,
therefore, is
L
1
(15)
W p p dV E p t p bd 31 Va u, xx dx
0
2
Hence, the virtual work due to the applied voltage Va is
L
1
(16)
W p E p t p bd 31 Va (u, xx )dx
0
2
( EI ) eq
L ( A) eq u,t (u,t )dx ( EI ) eq u, xx (u, xx )dx
L
L
0
0
(12)
After successively integrating twice by parts, equation (12) can
be rewritten with respect to the variation of u(x,t) as,
L ( A) eq u,tt ( EI ) eq u, xxxx udx
L
0
(13)
Fig.7. External disturbance function fd(x,t)
Substituting equations (13) and (16) into equation (1), the
equation of motion is obtained by applying the variational
principle:
2V
1
(17)
( A) eq u,tt ( EI ) eq u, xxxx E p t p bd 31 2a
2
x
In addition to the above field equation of motion, boundary
conditions are required to fully characterize the dynamic
behavior of the system. Besides the vanishing displacement
conditions at x=0 and L, the other two required boundary
conditions can be obtained by balancing a bending moment at
each end. The four boundary conditions for the given
configuration are summarized below:
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u 0, ( EI ) eq u, xx kT u, x 0 at x 0
(18)
u 0, ( EI ) eq u, xx kT u, x 0 at x L
(19)
Natural Frequencies and Mode Sahpes
From equation (17), we may write the homogenous
equation as
(20)
( A) eq u,tt ( EI ) eq u, xxxx 0
Considering a variable-separable form of the solution with a
harmonic time variation,
u ( x , t ) ( x ) e i t
(21)
Equation (20) may be re-written as,
'''' 4 0
(22)
where, 4
( A) eq
( EI ) eq
2 .
The general solution of equation (22) has the following
form:
( x) a1 cos(x) a2 sin(x) a3 cosh(x) a4 sinh(x) (23)
Applying boundary conditions (18) and (19), we get the system
of simultaneous equations in a1, a2, a3 and a4:
(24)
a1 a3 0
( EI ) eq 2 a1 kT a 2 ( EI ) eq 2 a3 kT a 4 0
(25)
a1 cos(L) a2 sin(L) a3 cosh(L) a 4 sinh(L) 0 (26)
( EI ) cos(L) k sin(L)a
( EI ) sin(L) k cos(L)a
( EI ) cosh(L) k sinh(L)a
( EI ) sinh(L) k cosh(L)a
2 sinh (cosh cos )
Cn
(sinh sin )
a2n
a3n C n
(31)
(32)
2 sin (cosh cos )
Cn
(33)
(sinh sin )
Hence, the eigen-function, or mode shape, corresponding
to the solution of the characteristic equation can be written as
n ( x) C n cos( n x) Dn sin( n x)
L
L
(34)
n
n
cosh(
x) En sinh(
x)
L
L
where
2 sinh (cosh cos )
(35)
Dn
(sinh sin )
2 sin (cosh cos )
En
(36)
(sinh sin )
The normalization condition is,
1
2
(37)
Cn L
2
x
dx
(
)
n
a4n
0
where n ( x)
( x)
Cn
.
2
eq
1
T
2
eq
T
2
(27)
2
eq
T
3
T
4
2
eq
0
Equations (24)-(27) can be represented as the matrix equation.
[Q]{a} 0
(28)
where, {a}=[a1 a2 a3 a4]T.
The characteristic equation which guarantees a nontrivial set of
the vector {a}, i.e. det[Q]=0 can be expressed as below with
kT
and
respect to two dimensionless parameters
( EI ) eq / L
L :
2 2 sin sinh 2 sin cosh
fn
t p n2
E p ( AB 3 1)
u ( x, t )
n ( x) n (t )
(38)
n 1
For the modal analysis to obtain the forced response, the
generalized force in the η coordinate system is expressed by
using the virtual work principle:
W p Qn n
(39)
n1
From equation (17), the generalized forces are
2 cos sinh cos cosh 0
Let βn and Cn be the positive nth solution of the characteristic
equation and the corresponding coefficient a1 respectively.
From the computed values of βn , the natural frequencies can be
derived as follows.
t 2 E ( AB 3 1)
2 ( EI )
n n2 ( A)eq p 2n 3p ( BR 1)
eq
p
L
L
or
2
Forced Vibration Response
The steady-state response of the piezo-metal composite
beam can be represented by the generalized coordinate ηn in the
eigen-function space.
3
(29)
(30)
2L2 3 p ( BR 1)
In addition, the coefficients a2, a3, and a4 associated with
the index n can be represented with respect to Cn as follows.
Qn (t )
N
1
E p t p bd 31 Vk (t ) n ' ( xk 1 ) n ' ( xk )
2
k 1
(40)
Therefore, the transformed equation of motion in the
generalized coordinate η can be written in the following form:
(41)
n n 2 n Qn (t )
Consider a sinusoidal electrical input with identical amplitude,
but with different polarity, for each electrode of the form
~
Vk (t ) Vk sin(t ). Defining n (t ) ~n sin(t ) and assuming
zero initial conditions u(x,0)=u,t=0 (or equivalently
n (0) n (0) 0 ), the amplitude of ηn can be obtained from
equation (41) as,
N ~
V ' ( xk 1 ) n ' ( xk )
1
~
k 1 k n
(42)
n (t ) E p t p bd 31
2
n 2 2
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Finally, we obtain the steady-state response of the beam under a
sinusoidal electrical input as,
u ( x, t ) n ( x) n (t )
n 1
N ~
1
Vk n ' ( xk 1 ) n ' ( xk ) ( x) sin(t )
E p t p bd 31 k 1
n
2
n 2 2
n 1
(43)
Fig.8. Analytical frequency response of the
composite beam under a uniform sinusoidal input
with an amplitude of 10 Volts on the PZT layer
DESIGN AND FABRICATION OF THE ROBOT
Design
If the material properties and geometric features of the
PZT and metal layers are well defined, equation (43) enables us
to analytically predict the dynamics of the given system (Fig.
3). The design process, aided by this theoretical analysis, is
described below.
TABLE 1
PROPERTIES FOR THE PZT-5A AND STAINLESS STEEL LAYERS
Stainless
PZT-5A
Steel
(Type 304)
Young’s modulus E (GPa)
66
200
7800
7800
Density ρ (Kg/m3)
-12
Piezoelectric Constant d31 (m/V)
-190 × 10
Length × Width × Thickness (mm3)
30 × 6 ×
30 × 5 ×
0.508
0.300
In this research, industry type piezo-layer (PZT-5A, T110A4E-602 ceramic single sheet, Piezo Systems, Inc.) with
300nm thick vacuum sputtered nickel electrodes on both sides,
and Type 304 stainless with a high quality factor were chosen
as the components of the actuating parts. Table 1 shows their
material specifications and dimensions. From equation (5),
the desirable thickness of the steel layer is found to be 0.3048
mm for the selected PZT-5A. However, we used 0.3 mmthick stainless steel sheet because of its commercial
availability.
An important parameter required in design process is the
torsional spring constant kT. This parameter has a significant
effect on the beam’s dynamics, i.e. its natural frequencies and
mode shapes, because the solutions of equation (29) strongly
The value of kT
was determined
depend on kT.
experimentally by adjusting two resonance frequencies for the
fourth mode vibration, one from the experiment and the other
from equation (30). The measured resonance frequencies
were 14.76 kHz for the third mode and 24.80 kHz for the fourth
mode. These values were obtained by sweeping the input
frequency from 10 to 30 kHz and simultaneously monitoring
the amplitude of one specific position (x=4.83 mm) where one
of the leg will be placed. The reason for using the fourth
mode used instead of the third mode for the frequency
measurement is that the fourth mode shows more distinct
resonance behavior in the neighborhood of the corresponding
natural frequency than the third mode, where the peak has a
wider base, as shown in Fig. 8.
The above frequency responses were computed from
equation (43) using the same operating conditions as with the
experiment in which a sinusoidal electric signal with 10V
amplitude was uniformly applied to the entire electrode on top
of the PZT layer. Thus, kT was set to be 6.623×10-3 Nm so
that the analytical fourth natural frequency equal the
experimental result, i.e. 24.80 kHz. Theoretical resonance
frequencies using this value of kT were 14.37 kHz for the third
mode and 24.80 kHz for the fourth mode.
(a)
(b)
Fig.9. Patterned electrode and operation schemes (a)
for exciting the B3 vibration mode and (b) for exciting
the B4 vibration mode
For the proposed application, it is desired that the same
piezo-metal plate be used for producing both forward and
reverse locomotion. To achieve this, the electrodes are
polarized such that with one polarization scheme, one bending
mode is induced, while the second scheme yields a different
bending mode. Five insulation gaps for the six electrodes of
different sizes are etched on to the piezo layer to realize the
alternative excitation schemes. Figure 9 shows the relative
polarities of the electrodes that are required to excite the B3 and
B4 modes, respectively. To excite the B3 mode, electrodes 3
and 4 need to be opposite in polarity to the other four
electrodes. On the other hand, in order to obtain the B4 mode,
electrodes 2, 3, and 6 need to be opposite in polarity to the
other three electrodes. The application of a sinusoidallyvarying voltage gives rise to the two forcing functions.
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Copyright © 2005 by ASME
Fig.10. Analytical B3 and B4 mode shapes and the
corresponding positions of the three legs, when
kT=6.623×10-3 Nm
The positions of the insulation gaps are the same as those
of the five nodal points for the B3 and B4 mode shapes. The
locations of the insulation gaps, computed with the expansion
of the first 10 eigenfunctions, are found to be 8.03 mm, 10.72
mm, 15.40 mm, 20.40 mm, and 22.80 mm (Fig. 10).
contaminants which might act as defects in the bonded region.
The width of the piezo-element is made 1 mm longer than that
of the metal substrate in order to provide a space to apply the
ground voltage to the bottom side of the PZT. The ratio of
layer thicknesses, calculated from equation (5), results in the
neutral axis being located at the interface between the two
layers. Because maximum shear stresses are induced in that
region, strong bonding between the surfaces is essential to the
fabrication of the piezo-metal composite actuator.
To
accomplish this, epoxy adhesive (Super Instant Epoxy 90,
Pacer Technology, Inc.) was applied at each interface and then
cured for 90 sec at room temperature. Two such piezo-metal
composite beams are placed opposite to each other to form a
pair. For each pair of beams, there is a 180◦ phase difference
between the voltage inputs to the individual beams such that, at
any given instant, a total of three feet (one from one beam, and
two from the other) are always in contact with the ground, thus
ensuring stability. This locomotive principle is bio-mimetic of
six-legged insect walking [7]. A signal distributing circuit
was mounted on top of the main frame, and six ball-shaped legs
(Soda-Lime ball with 1.5875 mm diameter) were bonded onto
the appropriate positions of the beams. In order to minimize
any interruption of the robot’s motion, thin and light Teflon
coated silver wires with 0.14 mm diameter (A-M Systems, Inc.)
were connected to the circuit board to feed electric signals to
the robot. The complete structure after wiring is shown in Fig.
12. The overall size and weight of the prototype are 53 x 45 x
19 mm3 and 23.25g respectively.
(a)
(b)
Fig.11. Predicted frequency responses of the three
leg positions (a) under B3 electrical input and (b)
under B4 electrical input configuration
To realize bi-directional locomotion, three legs are placed
at the intersection points of two modes shapes (Fig. 10). Each
leg has a different tip trajectory because of the different
displacement amplitude, as shown in Fig. 11.
This
inconsistency of tip trajectories will be discussed later.
Fabrication
Two piezoelectric layers with the dimensions given in
Table 1 were sliced manually from a large single sheet.
Insulation gaps were etched by means of a laser cutting system
(New Wave Research, Inc.). The thin stainless steel plates
were machined by wire EDM (electrical discharge machining).
Each surface was cleaned with acetone removing organic
Fig.12. Photograph of the fabricated prototype of the
miniature walking robot
EXPERIMENT
Experiments were carried out on a 6" (100) silicon wafer.
Two out-of-phase 10V sinusoidal signals obtained from two
synchronized function generators were applied to the patterned
electrodes according to the operational scheme stated above.
To excite two different bending modes for the bi-directional
locomotion, input frequencies were specified as 14.2 kHz for
B3 mode and 24.7 kHz for B4 mode. The displacement for
each direction was measured by means of a laser measurement
system (LS-3034, Keyence, Inc.). Figure 13 shows the
experimental measurements of the robot’s motion and the
corresponding results from linear curve fitting. The slopes of
the fitted curves, i.e. the velocities, are determined to be 5.86
cm/sec and 3.37 cm/sec for forward motion and reverse motion,
respectively, under the specified input conditions. The results
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Copyright © 2005 by ASME
reveal that the single step displacements for motion in each
direction are 4.13 μm/step for forward motion (B3 mode
excitation) and 1.36 μm/step for reverse motion (B4 mode
excitation).
robot equipped with two pairs of piezo-metal composite
actuators.
ACKNOWLEDGMENTS
The authors thank Professor Adnan Akay and Zhaoshun
Xu for their valuable support and assistance with the modal
experiments, and Thomas Cimprich and Mustafa Emre
Karagozler for their help with the performance tests.
REFERENCES
[1]
[2]
[3]
Fig.13. Measured displacement of the robot in the two
directions and the associated linear fitting curves
We will continue this research in order to build a walking
robot with sub-micron step resolution.
The primary
advantages of our design arise from the fact that equations (30)
and (43) indicate the possibility of achieving this by reducing
the speed of the robot or increasing the operating frequencies.
The speed can be slowed down by reducing the vibration
amplitude of the piezoelectric unimorph actuator, or by feeding
input voltages with lower amplitudes. Because there is some
threshold voltage, and, hence, a threshold amplitude of the
vibration, below which the robot cannot move due to the static
friction between the leg and surface [18], the amplitude
modulation strategy is limited to some degree. In addition, a
further miniaturized robot is also expected to achieve finer
motion since the natural frequencies of the actuators can be
made higher.
CONCLUSION
This paper has presented the analytical working equation
for a piezo-metal composite beam actuator with generalized
torsional springs at the ends. The theoretical results which can
predict kinematics of the standing wave patterns generated in
the beam structure, especially the B3 and B4 modes of
vibration for bi-directional locomotion, were directly used in
the design of the walking robot prototype. The torsional
spring constant was determined experimentally and the
positions of the legs and insulation gaps on the piezoelectric
layer were determined based on the predicted mode shapes.
The fabrication process for the robot has also been introduced.
The feasibility of the proposed design was shown by the
demonstration of the prototype on a smooth silicon wafer
surface. Even if each leg has slightly different dynamic
characteristic (Fig. 11) which might cause oblique walking,
cyclic and out-of-phase operation of two parallel actuators turns
out to compensate probable oblique walking at each period in
the long run if the surface condition is perfectly uniform.
Future work will target the collection of experimental data to
enable a more thorough understanding of the system dynamics,
the development of on-board driving circuitry, and the
construction of a three degree-of-freedom (X-Y-) walking
[4]
[5]
[6]
[7]
[8]
[9]
[10]
[11]
[12]
[13]
[14]
[15]
[16]
[17]
[18]
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