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Research Papers

# Design, Analysis, and Characterization of a Two-Legged Miniature Robot With Piezoelectric-Driven Four-Bar LinkageOPEN ACCESS

[+] Author and Article Information
Audelia G. Dharmawan

Engineering Product Development,
Singapore University of Technology and Design,
Singapore 487372
e-mail: audelia@sutd.edu.sg

Hassan H. Hariri

Engineering Product Development,
Singapore University of Technology and Design,
Singapore 487372
e-mail: hassan_hariri@sutd.edu.sg

Gim Song Soh

Engineering Product Development,
Singapore University of Technology and Design,
Singapore 487372
e-mail: sohgimsong@sutd.edu.sg

Shaohui Foong

Engineering Product Development,
Singapore University of Technology and Design,
Singapore 487372
e-mail: foongshaohui@sutd.edu.sg

Kristin L. Wood

Engineering Product Development,
Singapore University of Technology and Design,
Singapore 487372
e-mail: kristinwood@sutd.edu.sg

1Corresponding author.

Manuscript received September 14, 2017; final manuscript received December 4, 2017; published online February 1, 2018. Assoc. Editor: Andrew P. Murray.

J. Mechanisms Robotics 10(2), 021003 (Feb 01, 2018) (8 pages) Paper No: JMR-17-1295; doi: 10.1115/1.4038970 History: Received September 14, 2017; Revised December 04, 2017

## Abstract

This paper presents the design and development of a new type of piezoelectric-driven robot, which consists of a piezoelectric unimorph actuator integrated as part of the structure of a four-bar linkage to generate locomotion. The unimorph actuator replaces the input link of the four-bar linkage, and motion is generated at the coupler link due to the actuator deflection. A dimensional synthesis approach is proposed for the design of four-bar linkage that amplifies the small displacement of the piezoelectric actuator at the coupler link. The robot consists of two such piezo-driven four-bar linkages, and its gait cycle is described. The robot's speed is derived through kinematic modeling and experimentally verified using a fabricated prototype. The robot prototype's performance in terms of its payload capability and nominal operating power is also characterized experimentally. These results will be important for developing a motion planning control strategy for a autonomous robot locomotion, which will be part of future work.

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## Introduction

Miniature mobile robots offer many advantages due to their small size. They are able to enter and move in tight spaces that are unreachable by big robots. In addition, due to their light weight, they can be easily transported and deployed in hard-to-reach environments. This has vast applications in surveillance, exploration, or search and rescue operations. Piezoelectric actuators, due to their small size, are a potential enabling technology for driving miniature robots. Such actuators have been studied by a variety of researchers for use in robot locomotion, and a variety of designs have been proposed.

For instance, Sahai et al. [1] uses two clamped-free piezoelectric bimorph actuators in conjunction with a hexapod structure to move a robot with alternating tripod gait. Ho and Lee [2] used two pieces of lightweight piezoceramic composite curved actuator [3] to drive four legs of a robot with bounding gait locomotion. Baisch [4] used six piezoelectric actuators developed by Wood et al. [5] for a quadrupedal structure where each contralateral pair of legs were driven by three piezoelectric actuators arranged to generate lift and swing through a linkage mechanism. Avirovik et al. [6] used eight piezoelectric bimorphs to drive a millipede-inspired miniature robot. Hariri et al. [7] used two thin piezoelectric ceramic patches to generate a traveling wave on a robot with beam structure. Rios et al. [8] utilized piezoelectric bimorph benders to drive a six-legged robot, where each leg consists of two bimorph benders mounted side-by-side joined at the tip by a flexure and an end-effector to generate walking motion.

The design of piezoelectric-driven walking robots, in general, consists of a structural body with either active or passive legs. The former uses legs made of piezoelectric materials to actively drive the robot, but it easily faces issues of improper gait motion if the legs are not properly assembled [9]. However, they have the advantage of modeling simplicity over passive legs design due to the direct interaction between the piezo legs and the ground. The latter type uses rigid or semirigid legs attached to a piezoelectric actuator directly or through a mechanism to create locomotion. Several robots have been designed based on this approach. Some are over-actuated where the number of piezoelectric actuators required to control the motion exceeds the degrees-of-freedom [10,11], while others are under-actuated [12,13].

A significant challenge that must be addressed in developing piezoelectric-driven robots is the limited displacement of the piezoelectric actuators [14]. Piezoelectric cantilevers usually have submillimeter displacements and thus require amplification through linkage mechanism in order to achieve larger stroke [15]. The piezoelectric miniature robots developed by Wood and coworkers [16] and Goldfarb et al. [17] both used five-bar linkage to amplify the leg motion. Sitti [18] used multiple four-bar linkages to enable stroke amplification of the wings of a piezo-driven micromechanical flying insect. It is important to note that while most of the above robotic systems show satisfactory performance, the geometric parameters of the linkages are chosen arbitrarily or through some optimization process to obtain an optimal stroke amplification. There still does not exist a technique for a task-oriented design of piezo-driven mechanism that achieves a required stroke amplification or gait trajectory. Presenting such a technique is the objective of this paper.

In this paper, we describe our design methodology and show how it can be applied to the development of a two-legged piezoelectric mobile robot for bidirectional motion. This is our first step toward the development of novel piezo-driven robotic systems that can potentially recreate biological motion with simplified gait controls. To design the robot, we offer an approach to synthesize four-bar linkages as a constrained robotic system that achieves a required stroke amplification given the limited displacement of the input link. To predict the robot motion, the kinematic analysis is performed to estimate the speed of the robot based on an input voltage and frequency. A prototype is fabricated based on the design, and the experimental results for the speed versus applied voltages and frequencies are given and compared with our kinematic model to verify its validity. Finally, the robot prototype's performance in terms of its payload capability and nominal operating power is characterized.

## Dimensional Synthesis of Piezo-Driven Four-Bar Linkage With Required Stroke Amplification

Figure 1 shows the three-dimensional (3D) model of our proposed robot design. It consists of two cantilever piezoelectric unimorph actuators, a central body, and two pairs of four-bar linkage legs at opposite ends of the robot's body. Each of these actuators drives the robot in one particular direction. A cantilever piezoelectric unimorph actuator consists of a layer of piezoelectric material and a layer of elastic material bonded together where one end is fixed and the other end is free. When an electric field is applied to the piezoelectric material in the direction of the poling axis, the piezoelectric material will expand. On the other hand, when an electric field is applied to the piezoelectric material in the opposite direction of the poling axis, the piezoelectric material will contract. The expansion and contraction of the piezoelectric layer result in bending deflection due to the stress difference between the two layers. Hence, if we replace the input of a four-bar linkage with a unimorph actuator, articulated motion can be created at the coupler link, as depicted in Fig. 2, moving the robot.

To design the robot's leg, we consider how a unimorph actuator can be mechanically constrained by an RR chain to form a one degree-of-freedom four-bar linkage as shown in Fig. 3. This requires the designer to select the various link dimensions of the pseudo-rigid-body model of the unimorph actuator for task identification and geometric synthesis. To identify the task for dimensional synthesis, the vibration motion of the actuator is analyzed to establish its joint limits $θ1,max$ and $θ1,min$, and they are subsequently used in the specification of five task positions that incorporate the required stroke amplification factor. From the five task positions, an RR chain is designed to yield a piezo-driven four-bar linkage.

###### Joint Limit Identification.

To identify the actuator joint limits, its equation of motion is derived based on Euler–Bernoulli hypothesis that the stress in the x-direction is uniaxial, the deformation is small, and the cross section remains perpendicular to the neutral axis after deformation. It is also assumed that the electric field is uniformly distributed in the z-direction, i.e., $E=Ez(x,t)$. Then, the displacement field becomesDisplay Formula

(1)$u={ux(x,y,z,t)≈−(z−zn)∂xw(x,t)uy(x,y,z,t)=0uz(x,y,z,t)≈w(x,t)$

where the geometric parameters for the system are given in Fig. 4, and w(x, t) is the transverse displacement of the actuator about its neutral axis zn, whose location can be obtained usingDisplay Formula

(2)$zn=12cmtm2+cptp2+2cptptmcmtm+cptp$

where $cp=(1/sxxE)$, cm is the Young's modulus of the elastic layer, and $sxxE$ is the elastic compliance of the piezoelectric layer. See Ref. [19].

It was found in Ref. [20] that in static operation, the equation of motion becomes equivalent to the static beam equationDisplay Formula

(3)$(EI)eq∂2w(x)∂x2=−qepEz$

where

$(EI)eq=(Ipcp+Imcm)Im=b∫0tm(z−zn)2dz=b13[(tm−zn)3+zn3]Ip=b∫tmtp+tm(z−zn)2dz=b13[(tp+tm−zn)3−(tm−zn)3]q=b∫tmtp+tm(z−zn)dz=b2[(tp+tm−zn)2−(tm−zn)2]$
$ep=(dzx/sxxE)$, and dzx is the piezoelectric charge constant.

Now, with fixed-free boundary conditions, the transverse displacement of the actuator is given byDisplay Formula

(4)$w(x)=12kx2$

where $k=((−qepEz)/(EI)eq)$.

The piezoelectric material will be actuated at its first resonant frequency to exploit maximum transverse deflection. The first resonant frequency of the actuator can be calculated asDisplay Formula

(5)$f1=(β1l)22πl2(EI)eq(ρA)eq$

where $β1l=1.875, (ρA)eq=b(ρptp+ρmtm)$, and ρp and ρm are the density of the piezoelectric and elastic material, respectively. See Ref. [21].

To determine the joint limits, we leverage on the fact that the piezoelectric deflection is very small. Hence, small angle approximation applies, and the angular displacement of the piezoelectric cantilever, as shown in Fig. 5, can be approximated asDisplay Formula

(6)$ζ≈w(l)l$

where ζ is the maximum angular deformation of the actuator beam. Let $θ1,0$ be the initial angle of the input crank of the RR link when no electric field is applied to the piezoelectric actuator. Then, the range of angular displacement of the input crank is given byDisplay Formula

(7)$θ1,min=θ1,0−ζθ1,max=θ1,0+ζ$

where $θ1,max$ corresponds to the input angle when the piezoelectric cantilever flexes downward, and $θ1,min$ corresponds to the input angle when the piezoelectric cantilever flexes upward.

###### Stroke Amplification Through Task Specification.

To specify the stroke amplification requirements at the leg, we generate the positions reached by the unimorph actuator using the kinematics equation of a planar 2R chain, taking into account its joint limits as calculated above and the required amplification factor at the leg. The kinematic equations of a planar 2R chain equate the 3 × 3 homogeneous transformation $[T]$ between the end-effector and the base frame to the sequence of local coordinate transformations around the joint axes and along the links of the serial chainDisplay Formula

(8)$[T]=[G][Z(θ1)][X(a)][Z(θ2)][H]$

The parameters θi define the movement of each joint, and a defines the length of the link. The transformation $[G]$ defines the position of the base $O$ of the chain relative to the world frame, and $[H]$ locates the task frame M relative to the end-effector frame. See Fig. 3.

The required stroke amplification of the unimorph actuator to the leg can be specified by introducing a scaling factor α, and we use the following relationship to define its joints limits:Display Formula

(9)$θ2,min=θ2,0−αζθ2,max=θ2,0+αζ$

We now choose five joint parameters from this range to obtain five task positions using Eq. (8) for the synthesis of an RR chain to yield a four-bar linkage.

###### Synthesis of an RR Constraint.

The synthesis of an RR link is well known [22], and to design an RR link that reaches the above set of five task positions, we solve the following set of design equations:Display Formula

(10)$(B2−C)·(B2−C)−(B1−C)·(B1−C)=0(B3−C)·(B3−C)−(B1−C)·(B1−C)=0(B4−C)·(B4−C)−(B1−C)·(B1−C)=0(B5−C)·(B5−C)−(B1−C)·(B1−C)=0$
to obtain C and $B1$, where C denotes the fixed pivot and $Bi$ denotes the moving pivot at the ith position. The moving pivot is related to its first position byDisplay Formula
(11)$Bi=[Ti][T1]−1B1, i=2,…,5$

## Robot's Gait Cycle for Locomotion

To drive our synthesized piezoelectric robot, we use a gait cycle as denoted in Figs. 6(a)6(g). One unimorph actuator will be operated to drive the robot to move in one particular direction using a sinusoidal input. This cycle was chosen purely for simplicity so as to establish a proof-of-concept that controllable-independent motion can be achieved. We intend to explore more complex coordinated gait locomotion as part of our future work. In this gait cycle explanation, the left unimorph actuator is the active actuator. This moves the robot to the left. We assume that the leg in contact with the ground does not slip and double-leg flight phase does not occur, i.e., at least one of the robot's legs touches the ground.

The gait cycle starts with the left unimorph actuator flexing maximally upward at time t = 0 and both robot's legs on the ground (Fig. 6(a)). From this configuration, the piezoelectric layer begins to expand and starts to deflect the unimorph actuator downward toward its neutral position at $t=T/4$ (Fig. 6(b)) and to its maxima at $t=T/2$. This pulls the robot's body upward and leftward by pivoting about point L as shown in Fig. 6(c). This is followed by an instantaneous clockwise rotation of $γ′$ about L due to gravity, until the robot's right leg touches the ground again at $L′$ (Fig. 6(d)). To complete the gait cycle, the unimorph actuator starts to deflect upward again, passing through its neutral position at $t=3T/4$ (Fig. 6(e)) and its minima at t = T (Fig. 6(f)); this time supported at $L′$ as shown in Figs. 6(e)6(f). Similarly, immediately after this, the whole robot will experience an anticlockwise rotation of γ about $L′$ due to gravity, until the left leg touches the ground again at L. This completes the gait cycle to its original state as shown in Fig. 6(g). The cycle then repeats, moving the robot to the left.

## Gait Analysis and Mathematical Modeling

To mathematically model the robot's motion based on the gait cycle, the nonactuated leg will be treated as a rigid body with the robot's body. To analytically calculate the robot's displacement after one gait cycle, the gait is analyzed in two halves: the front leg stance half cycle (Figs. 6(a)6(d)) and the back leg stance half cycle (Figs. 6(d)6(g)).

###### Front Leg Stance Gait Analysis.

To determine the robot displacement during its first half of gait, consider a leg kinematic model as shown in Fig. 7. Let the fixed and moving pivots of the input link be O and A, and the fixed and moving pivots of the output link be C and B, respectively. Denote a fixed base frame F with its origin at O and its x-axis directed along OC. In the following analysis, pivot O will be used to define the displacement of the robot after a gait cycle. Its location can be obtained by finding the intersection of the actuator neutral axis zn with the actuator fixed end.

Now, consider a four-bar linkage with link lengths calculated from the robot's geometry such thatDisplay Formula

(12)$a=|A−O|, b=|B−C|, g=|C−O|, h=|B−A|$

Using the analysis approach of McCarthy and Soh [22], the coordinates of point $FL=(X,Y)T$, expressed in base frame F is given byDisplay Formula

(13)${X(θ)Y(θ)1}=[ cos(θ+ϕ)−sin(θ+ϕ)a cos θ sin(θ+ϕ) cos(θ+ϕ)a sin θ001]{xy1}$

where $(x,y)T$ is the coordinates of a point L on the robot's leg frame M, θ is the angle of the input link, and the coupler angle isDisplay Formula

(14)$ϕ(θ)=arctan(b sin ψ−a sin θg+b cos ψ−a cos θ)−θ$

(15)$ψ(θ)=arctan(BA)±arccos(CA2+B2)A(θ)=2ab cos θ−2gbB(θ)=2ab sin θC(θ)=g2+b2+a2−h2−2ag cos θ$

Next, define a contact frame W at L and a body frame V at O, as shown in Fig. 8. Let β be the angle between frame V and F. The coordinate of L in frame V is then given byDisplay Formula

(16)$VL(θ)={Lx(θ)Ly(θ)}=[ cos β−sin β sin β cos β]{XY}$

If the relationship is inverted with contact frame W being the fixed reference frame, the displacement of O can be obtained asDisplay Formula

(17)$WO(θ)=−VL(θ)={−Lx(θ)−Ly(θ)}$

This defines the robot trajectory due to the leg contact at L as it moves from Figs. 6(a)6(c). Hence, the displacement of $Oa$ and $Oc$, which corresponds to the robot displacement at t = 0 and $t=T/2$ can be obtained usingDisplay Formula

(18)$Oa= WO(θmin)$
Display Formula
(19)$Oc= WO(θmax)$

As shown in Fig. 6(d), the robot will return back to double leg stance due to gravity, by a clockwise rotation of $γ′$ pivoted about L. To determine the overall robot displacement, consider a parallel frame $V′$ located on the rear leg piezoelectric actuator. Its origin is at $O′$ and its x-axis parallel to that of V. Define another contact frame $W′$ with its origin at $L′$, as shown in Fig. 8. Since the rear leg design is symmetrical to the front leg, the coordinate of $L′$ as seen in frame $V′$ is an exact mirror image of its front leg with initial angle θ0. Thus, from Eq. (16), this yieldsDisplay Formula

(20)$V′L′={−Lx(θ0)Ly(θ0)}$

To relate this back to the contact frame W, consider the vector sumDisplay Formula

(21)$WL′(θ)= WO(θ)+ VL′= WO(θ)+{d0}+ V′L′={d−Lx(θ0)−Lx(θ)Ly(θ0)−Ly(θ)}$

Since the rotated angle $γ′$ in Fig. 6(c) is very small, we can approximate it from Eq. (21) using trigonometry asDisplay Formula

(22)$tan γ′≈γ′=Ly(θ0)−Ly(θmax)d−Lx(θ0)−Lx(θmax)$

Now, to determine the pivot displacement $Od$ in Fig. 6(d), we rotate the pivot displacement $Oc$ at gait instance Fig. 6(c) about the robot leg contact point L by an angle $γ′$. This yieldsDisplay Formula

(23)$Od=[ cos γ′ sin γ′−sin γ′ cos γ′] Oc$

The overall robot displacement $Sad$ for the front leg stance cycle can then be obtained as the difference between the displacement of O from its contact frame W at gait instance Figs. 6(a) and 6(d), respectively. This givesDisplay Formula

(24)$Sad=Od−Oa$

###### Back Leg Stance Gait Analysis.

To determine the robot displacement during the back leg stance half, note that the contact point of the robot with the ground now shifts to the rear leg at $L′$ as shown in Figs. 6(e)6(f). Hence, for this subsequent gait cycle, we determine the displacement of pivot O with frame $W′$ acting as the fixed reference frame. To express the displacement of pivot $Od$ in frame W$′$, we use Eq. (21) and consider the vector sumDisplay Formula

(25)$W′Od= W′L+Od=−[ cos γ′ sin γ′−sin γ′ cos γ′] WL′(θmax)+Od$

Following similar analysis leading to Eq. (22) as described in Sec. 4.1, we can approximate γ asDisplay Formula

(26)$γ≈Ly(θ0)−Ly(θmin)d−Lx(θ0)−Lx(θmin)$

Since the rear leg is nonactuated and act to support the robot at $L′$ as illustrated in Figs. 6(d)6(f), there is only articulated motion at the front leg and no movement will occur on the robot's body. Hence,Display Formula

(27)$Of=Oe= W′Od$

Similarly, the robot will return back to double leg stance due to gravity, this time by an anticlockwise rotation of γ pivoted at $L′$ as shown in Fig. 6(g). Then, the displacement of pivot O in frame $W′$ at this instance isDisplay Formula

(28)$Og=[ cos γ sin γ−sin γ cos γ] Of$

Now, the overall robot displacement $Sdg$ during the rear leg stance cycle can be obtained as the difference between the displacement of pivot O from its contact frame $W′$ at gait instance Figs. 6(d) and 6(g), respectively. This givesDisplay Formula

(29)$Sdg=Og− W′Od$

###### Combined Gait Analysis.

The total displacement of the robot after one full gait cycle as denoted in Fig. 6 can be calculated asDisplay Formula

(30)$Sgait={SxSy}=Sad+Sdg$

Upon substituting Eqs. (18), (23), (25), and (28) into Eq. (30) and simplifying the result, the average fore-aft speed of the robot given the applied frequency (f) of the piezoelectric actuator can be calculated asDisplay Formula

(31)$Vbot=f*Sx=f*|| Lx(θmin)−cos γ′Lx(θmax)−sin γ′Ly(θmax)+[cos γ′−cos(γ+γ′)]d+[sin γ′−sin(γ+γ′)]Ly(θ0)−[cos γ′−cos(γ+γ′)]Lx(θ0) ||$

γ and $γ′$ are given in Eqs. (26) and (22), respectively, and $Lx(θ)$ and $Ly(θ)$ are given in Eq. (16). In fact, when γ and $γ′$ are very small, the average linear speed of the robot based on this gait cycle can be easily approximated even further byDisplay Formula

(32)$Vbot≈f*||Lx(θmin)−Lx(θmax)||$

## Robot Design and Experimental Results

The piezoceramic material used for the robot is the soft-doped PZT (lead zirconate titanate) NCE55 from Noliac, Inc. (Kvistgaard, Denmark) [23] and the elastic material is aluminum. Table 1 summarizes the properties of the materials used for the piezoelectric unimorph actuator.

###### Robot Design and Fabrication.

We select the link dimensions of the RR robot such that $a12=36.64$ mm. The position of the robot base is chosen as $[G]=(0deg,0,0)$ and the task frame relative to the end effector frame as $[H]=(0 deg,18.48,0)$. Using Eq. (7), the range of angular displacement was found to be $θ1,min=184.928deg$ and $θ1,max=185.024deg$. We choose between these two angles to obtain five input angles θ1. Due to manufacturability reasons, we choose an amplification factor of $α=3.5$ to obtain the range of θ2 using Eq. (9). Now, the task positions for the desired leg motion are obtained using the forward kinematic equations in Eq. (8). They are as listed in Table 2. This yields four solutions and the chosen solution is $C=(2.5,2.83)$ and $B1=(−40.28,−4.50)$. The synthesized geometric parameters of the chosen four-bar linkage are as summarized in Table 3. The front and the rear legs of the robot are symmetrical and follow the same design results.

To verify the validity of the robot's kinematic model, the robot's body and legs are fabricated fully using 3D printer with ABS material. The length of the aluminum is 3 mm longer than the length of the piezoelectric at each side to enable it to be clamped onto the printed robot using standard screws. Figure 9 shows the fabricated robot prototype. The two legs at each end of the robot are connected together at the bottom so that they produce similar gait to ensure straight motion. The final dimension of the robot prototype is 100 × 17.4 × 27.2 mm3, and it weighs 12.17 g.

###### Kinematic Model Verification.

Experiments were conducted to find out the experimental speed of the robot and compare it with the speed calculated using our kinematic model. The robot was let to run in one direction on a flat wooden surface for a distance of 30 cm, and the time taken to complete the task was noted down. The robot's speed was calculated by dividing the distance traveled by the time taken. An image sequence of the piezoelectric robot in motion during one of the trial is shown in Fig. 10.

The first resonant frequency of the actuator is obtained theoretically as 917.8 Hz using Eq. (5), which is a useful guide to experimentally attain the first resonant frequency of the actuator. Experiments were carried out on the robot prototype by varying the frequency of a 100 V input from 600 Hz to 900 Hz at an interval of 50 Hz. Five trials were performed for each frequency, and the results are plotted in Fig. 11. The data points show the mean results, while the shaded areas show that maximum and minimum values of the trials. The first resonant frequency of the actuator was found experimentally to be equal to 800 Hz, and this would be the constant applied frequency used to validate our model.

The experiment and analytical speed calculation were then performed at four different applied voltages: 40 V, 60 V, 80 V, and 100 V. Similarly, the experiments were repeated five times for each applied voltage, and the results are plotted in Fig. 12. The robot is able to achieve an average maximum speed of 3.97 cm/s while consuming 80 mW in doing so. To analytically determine the dynamic displacement, an experimental dynamic factor Q needs to be multiplied to the displacement w(x) produced in static operation. This yields a dynamic displacement ofDisplay Formula

(33)$wd(x)=w(x)*Q$

The scalar Q can be experimentally determined [24,25], and its value will be different for each applied frequency f. At the first resonant frequency, Q was experimentally found to be 0.32 (this value reflects the experimental boundary conditions used for the piezoelectric actuator). It should be noted that this approach is simpler, and in our case, it is more accurate to build the kinematic model as opposed to the direct equation of dynamic displacements. From the plot, it can be seen that the behavior of the prototype corresponds closely to the result as predicted by our kinematic model. The over-estimation of the model at low voltage could be due to the frictional loss in the rotational joints or leg slippage during locomotion, which was not taken into account in the modeling and also may vary slightly with voltage. The material properties of the piezoelectric actuator may also not be perfectly accurate, and they vary with voltage as well. The under-estimation of the model at higher voltage could be due to the presence of double-leg flight phase at higher speed, which was assumed not to occur in the theoretical gait analysis.

## Robot's Characterization

In this section, we characterize the performance of the robot in terms of payload capability, nominal operating mechanical power, and speed on different types of surfaces. The robot is first tested for different payloads, and the results are given in Fig. 13, which shows the speed versus payload at 100 V amplitude on a smooth flat surface. As can be seen from the results, the speed decreases almost linearly with payload. The robot can carry a maximum of 13.6 g at a very low speed of 2.5 mm/s at 100 V.

The speed of the robot is also measured for different surface inclination angles until it is not able to move at 10 deg. At each angle, the dragged force capability is then calculated by the horizontal component of the weight force vector. The speed versus dragged force curve at 100 V applied voltage on a smooth flat surface is given in Fig. 14. From the figure, the maximum nominal operating power is computed, and it is found to be 0.19 mW.

The robot performance is then tested on different surfaces as shown Fig. 15 at 100 V amplitude, and the results are given in Table 4. Based on the tests, the robot's speed is almost always the same on smooth surfaces such as glass, acrylic, steel, smooth, and less smooth wood. However, the robot does not show any movement on a slightly rough surface, such as the rough wood. Based on the characterization, optimization needs to be carried out on the robot in order to improve its performance and capability, which will be part of future work. We are also interested in analyzing the effect of the selection of the amplification factor α on the gait quality, in terms of the robot's velocity, payload capability, force output, etc., in order to obtain the optimal gait.

## Conclusions

The design and theoretical analysis of a new type of piezoelectric legged robot with a four-bar architecture is presented in this paper. Its gait cycle is described, and kinematic modeling is performed to estimate its speed based on an input voltage. A robot prototype is fabricated to test the validity of the derived kinematic model, and it is found that the behavior of the physical prototype corresponds with the model. The derived model will be useful for controlling and planning the motion of the robot based on a control input. The performance of the robot is also characterized to find out the capability of the robot. The end goal of this work is to have a controllable multi-degree-of-freedom miniature robot using piezoelectric actuators. As such, we also have in mind the concept of a multi-degrees-of-freedom quadrupedal robot as shown in Fig. 16, whereby each leg drives the robot toward different direction, as the next iteration of this work.

## Acknowledgements

The authors gratefully acknowledge the support of TL@SUTD-Systems Technology for Autonomous Reconnaissance & Surveillance and SUTD-MIT International Design Center.2

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Hariri, H. H. , Soh, G. S. , Foong, S. H. , Wood, K. L. , and Otto, K. , 2015, “Miniature Piezoelectric Mobile Robot Driven by Standing Wave,” 14th IFToMM World Congress, Taipei, Taiwan, Oct. 25–30, pp. 325–330.
Hariri, H. H. , Prasetya, L. A. , Foong, S. , Soh, G. S. , Otto, K. N. , and Wood, K. L. , 2016, “A Tether-Less Legged Piezoelectric Miniature Robot Using Bounding Gait Locomotion for Bidirectional Motion,” IEEE International Conference on Robotics and Automation (ICRA), Stockholm, Sweden, May 16–21, pp. 4743–4749.
Varma, V. K. , and Dixon, W. E. , 2002, “Design of a Piezoelectric Meso-Scale Mobile Robot: A Compliant Amplification Approach,” IEEE International Conference on Robotics and Automation (ICRA)), Washington, DC, May 11–15, pp. 1137–1142.
Ozcan, O. , Baisch, A. T. , Ithier, D. , and Wood, R. J. , 2014, “Powertrain Selection for a Biologically-Inspired Miniature Quadruped Robot,” IEEE International Conference on Robotics and Automation (ICRA)), Hong Kong, China, May 31–June 7, pp. 2398–2405.
Baisch, A. T. , Sreetharan, P. S. , and Wood, R. J. , 2010, “Biologically-Inspired Locomotion of a 2g Hexapod Robot,” IEEE/RSJ International Conference on Intelligent Robots and Systems (IROS), Taipei, Taiwan, Oct. 18–22, pp. 5360–5365.
Goldfarb, M. , Gogola, M. , Fischer, G. , and Garcia, E. , 2001, “Development of a Piezoelectrically-Actuated Mesoscale Robot Quadruped,” J. Micromechatronics, 1(3), pp. 205–219.
Sitti, M. , 2003, “Piezoelectrically Actuated Four-Bar Mechanism With Two Flexible Links for Micromechanical Flying Insect Thorax,” IEEE/ASME Trans. Mechatronics, 8(1), pp. 26–36.
Ballas, R. G. , 2007, Piezoelectric Multilayer Beam Bending Actuators: Static and Dynamic Behavior and Aspects of Sensor Integration, Springer Science & Business Media, New York.
Hariri, H. , Bernard, Y. , and Razek, A. , 2011, “Analytical and Finite Element Model for Unimorph Piezoelectric Actuator: Actuator Design,” Sixth International Conference on Electroceramics for End-Users, Sestriere, Italy, Feb. 28–Mar. 2, pp. 71–75.
Graff, K. F. , 1975, Wave Motion in Elastic Solids, Courier Corporation, North Chelmsford, MA.
McCarthy, J. M. , and Soh, G. S. , 2010, “Geometric Design of Linkages,” Interdisciplinary Applied Mathematics, 2nd ed., Springer Science & Business Media, New York.
Noliac, 2017, “Noliac Acquired by CTS,” Noliac, Denmark, UK, accessed Jan. 19, 2017,
Nashif, A. D. , Jones, D. I. , and Henderson, J. P. , 1985, Vibration Damping, Wiley, New York.
Yang, Y. , Lemaire-Semail, B. , Giraud, F. , Amberg, M. , Zhang, Y. , and Giraud-Audine, C. , 2015, “Power Analysis for the Design of a Large Area Ultrasonic Tactile Touch Panel,” Eur. Phys. J. Appl. Phys., 72(1), p. 11101.
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## References

Sahai, R. , Avadhanula, S. , Groff, R. , Steltz, E. , Wood, R. , and Fearing, R. S. , 2006, “Towards a 3g Crawling Robot Through the Integration of Microrobot Technologies,” IEEE International Conference on Robotics and Automation (ICRA), Orlando, FL, May 15–19, pp. 296–302.
Ho, T. , and Lee, S. , 2009, “Implementation of a Piezoelectrically Actuated Self-Contained Quadruped Robot,” SPIE Defense, Security, and Sensing, Orlando, FL, Apr. 15–19, Paper No. 73320E .
Yoon, K. J. , Shin, S. , Park, H. C. , and Goo, N. S. , 2002, “Design and Manufacture of a Lightweight Piezo-Composite Curved Actuator,” Smart Mater. Struct., 11(1), p. 163.
Baisch, A. T. , 2013, “Design, Manufacturing, and Locomotion Studies of Ambulatory Micro-Robots,” Ph.D. thesis, Harvard University, Cambridge, MA.
Wood, R. J. , Steltz, E. , and Fearing, R. S. , 2005, “Optimal Energy Density Piezoelectric Bending Actuators,” Sens. Actuators A: Phys., 119(2), pp. 476–488.
Avirovik, D. , Butenhoff, B. , and Priya, S. , 2014, “Millipede-Inspired Locomotion Through Novel U-Shaped Piezoelectric Motors,” Smart Mater. Struct., 23(3), p. 037001.
Hariri, H. , Bernard, Y. , and Razek, A. , 2015, “Dual Piezoelectric Beam Robot: The Effect of Piezoelectric Patches Positions,” J. Intell. Mater. Syst. Struct., 26(18), pp. 2577–2590.
Rios, S. A. , Fleming, A. J. , and Yong, Y. K. , 2017, “Miniature Resonant Ambulatory Robot,” IEEE Rob. Autom. Lett., 2(1), pp. 337–343.
Muscato, G. , 2004, “The Collective Behavior of Piezoelectric Walking Microrobots: Experimental Results,” Intell. Autom. Soft Comput., 10(4), pp. 267–276.
Simu, U. , and Johansson, S. , 2006, “Analysis of Quasi-Static and Dynamic Motion Mechanisms for Piezoelectric Miniature Robots,” Sens. Actuators A: Phys., 132(2), pp. 632–642.
Son, K. J. , Kartik, V. , Wickert, J. A. , and Sitti, M. , 2006, “An Ultrasonic Standing-Wave-Actuated Nano-Positioning Walking Robot: Piezoelectric-Metal Composite Beam Modeling,” J. Vib. Control, 12(12), pp. 1293–1309.
Hariri, H. H. , Soh, G. S. , Foong, S. H. , Wood, K. L. , and Otto, K. , 2015, “Miniature Piezoelectric Mobile Robot Driven by Standing Wave,” 14th IFToMM World Congress, Taipei, Taiwan, Oct. 25–30, pp. 325–330.
Hariri, H. H. , Prasetya, L. A. , Foong, S. , Soh, G. S. , Otto, K. N. , and Wood, K. L. , 2016, “A Tether-Less Legged Piezoelectric Miniature Robot Using Bounding Gait Locomotion for Bidirectional Motion,” IEEE International Conference on Robotics and Automation (ICRA), Stockholm, Sweden, May 16–21, pp. 4743–4749.
Varma, V. K. , and Dixon, W. E. , 2002, “Design of a Piezoelectric Meso-Scale Mobile Robot: A Compliant Amplification Approach,” IEEE International Conference on Robotics and Automation (ICRA)), Washington, DC, May 11–15, pp. 1137–1142.
Ozcan, O. , Baisch, A. T. , Ithier, D. , and Wood, R. J. , 2014, “Powertrain Selection for a Biologically-Inspired Miniature Quadruped Robot,” IEEE International Conference on Robotics and Automation (ICRA)), Hong Kong, China, May 31–June 7, pp. 2398–2405.
Baisch, A. T. , Sreetharan, P. S. , and Wood, R. J. , 2010, “Biologically-Inspired Locomotion of a 2g Hexapod Robot,” IEEE/RSJ International Conference on Intelligent Robots and Systems (IROS), Taipei, Taiwan, Oct. 18–22, pp. 5360–5365.
Goldfarb, M. , Gogola, M. , Fischer, G. , and Garcia, E. , 2001, “Development of a Piezoelectrically-Actuated Mesoscale Robot Quadruped,” J. Micromechatronics, 1(3), pp. 205–219.
Sitti, M. , 2003, “Piezoelectrically Actuated Four-Bar Mechanism With Two Flexible Links for Micromechanical Flying Insect Thorax,” IEEE/ASME Trans. Mechatronics, 8(1), pp. 26–36.
Ballas, R. G. , 2007, Piezoelectric Multilayer Beam Bending Actuators: Static and Dynamic Behavior and Aspects of Sensor Integration, Springer Science & Business Media, New York.
Hariri, H. , Bernard, Y. , and Razek, A. , 2011, “Analytical and Finite Element Model for Unimorph Piezoelectric Actuator: Actuator Design,” Sixth International Conference on Electroceramics for End-Users, Sestriere, Italy, Feb. 28–Mar. 2, pp. 71–75.
Graff, K. F. , 1975, Wave Motion in Elastic Solids, Courier Corporation, North Chelmsford, MA.
McCarthy, J. M. , and Soh, G. S. , 2010, “Geometric Design of Linkages,” Interdisciplinary Applied Mathematics, 2nd ed., Springer Science & Business Media, New York.
Noliac, 2017, “Noliac Acquired by CTS,” Noliac, Denmark, UK, accessed Jan. 19, 2017,
Nashif, A. D. , Jones, D. I. , and Henderson, J. P. , 1985, Vibration Damping, Wiley, New York.
Yang, Y. , Lemaire-Semail, B. , Giraud, F. , Amberg, M. , Zhang, Y. , and Giraud-Audine, C. , 2015, “Power Analysis for the Design of a Large Area Ultrasonic Tactile Touch Panel,” Eur. Phys. J. Appl. Phys., 72(1), p. 11101.

## Figures

Fig. 1

3D model of the proposed piezoeelectric robot

Fig. 2

Locomotion principle of the piezoelectric robot

Fig. 3

The pseudo-rigid-body model of the unimorph actuator OA, constrained by an RR chain CB, to form a four-bar linkage

Fig. 4

Geometric parameters of the piezoelectric unimorph actuator

Fig. 5

Angular displacement approximation of the actuator deflection

Fig. 6

Gait cycle model of the piezoelectric robot

Fig. 7

Kinematic model of the robot's leg

Fig. 8

Frames assignment for the robot's body and legs

Fig. 9

Prototype of the fabricated miniature piezoelectric robot

Fig. 10

Image sequence of the piezoelectric robot in one of the trials

Fig. 11

Plot of the robot's speed versus applied frequency on a flat surface

Fig. 12

Plot of the robot's average speed versus applied voltage on a flat surface

Fig. 13

Plot of the robot's velocity versus payload on a flat surface

Fig. 14

Plot of the robot's velocity versus dragged force

Fig. 15

Robot tested on different surfaces

Fig. 16

Concept of the future multi-degrees-of-freedom robot

## Tables

Table 1 Properties of the actuator materials
Table 2 Five task positions for the end-effector of a RR chain
Table 3 The chosen and synthesized geometric parameters of the robot
Table 4 Velocity of the robot at 100 V amplitude on different surfaces

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