Dynamic Structural and Contact Modeling for a Silicon Hexapod Microrobot

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Dynamic Structural and Contact Modeling for a Jinhong Qu1 Vibration and Acoustics Laboratory, Silicon Hexapod Microrobot Department of Mechanical Engineering, University of Michigan, Ann Arbor, MI 48109 This paper examines the dynamics of a type of silicon-based millimeter-scale hexapod, e-mail: [email protected] focusing on interaction between structural dynamics and ground contact forces. These microrobots, having a 5 mm 2 mm footprint, are formed from silicon with integrated Jongsoo Choi thin-ﬁlm lead–zirconate–titanate (PZT) and high-aspect-ratio parylene-C polymer micro- actuation elements. The in-chip dynamics of the microrobots are measured when actuated

Vibration and Acoustics Laboratory, Downloaded from http://asmedigitalcollection.asme.org/mechanismsrobotics/article-pdf/9/6/061006/6404124/jmr_009_06_061006.pdf by guest on 29 September 2021 Department of Mechanical Engineering, with tethered electrical signal to characterize the resonant behavior of different parts of University of Michigan, the robot and its piezoelectric actuation. Out-of-chip robot motion is then stimulated by Ann Arbor, MI 48109 external vibration after the robot has been detached from its silicon tethers, which e-mail: [email protected] removes access to external power but permits sustained translation over a surface. A dynamic model for robot and ground interaction is presented to explain robot locomotion Kenn R. Oldham in the vibrating field using the in-chip measurements of actuator dynamics and additional dynamic properties obtained from finite element analysis (FEA) and other design infor- Mem. ASME mation. The model accounts for the microscale interaction between the robot and ground, Vibration and Acoustics Laboratory, for multiple resonances of the robot leg, and for rigid robot body motion of the robot Department of Mechanical Engineering, chassis in five degrees-of-freedom. For each mode, the motions in vertical and lateral University of Michigan, direction are coupled. Simulation of this dynamic model with the first three resonant Ann Arbor, MI 48109 modes (one predominantly lateral and two predominantly vertical) of each leg shows a e-mail: [email protected] good match with experimental results for the motion of the robot on a vibrating surface, and allows exploration of influence of small-scale forces such as adhesion on robot locomotion. Further predictions for future autonomous microrobot performance based on the dynamic phenomena observed are discussed. [DOI: 10.1115/1.4037802]

1 Introduction dynamic or resonant gaits during robot locomotion. However, it has been previously observed that the interaction of elastic reso- Microscale walking robots, typically with maximum nant behavior in microstructures and impact dynamics between dimensions on the order of a centimeter or smaller, have been small micro-actuators or microrobotic legs and underlying terrain proposed or developed over recent decades based on a variety of can give rise to highly varying, complex, and sometimes even electromechanical actuation principles, including electrostatic nearly chaotic dynamic behavior [9–12]. The authors have previ- [1], electrothermal [2–4], magnetic [5], shape-memory alloy [6], ously described a multiple-modes model to explain the dynamics and piezoelectric [7] transduction. Thin-film piezoelectric of microrobots with similar actuation principle and materials ceramic actuators [8], as a class of smart materials for micro- selection, but at the centimeter-scale where small-scale forces electromechanical systems, share several advantages with bulk have limited effect on foot–terrain interaction [9]. A preliminary piezoelectric materials. These include large work densities and extension to a micro-scale robot was presented in Ref. [13], but substantial force generation. As thin-film, piezoelectric actuation this did not address robot foot dynamics, body contact, and can also be achieved with modest actuation voltages (typically model-based performance estimation. The modeling process in 5–20 V) and over comparatively large deflections via beam this work is guided by previous models for centimeter-scale bending. Drawbacks of thin-film piezoelectric microactuators robots, but converts the system dynamics to a hybrid dynamic include complicated fabrication requirements, limitations on model that can account for the distribution of modal motion in material compatibility with some processes or other materials, and two directions for each mode and additional microscale ground relative fragility compared to many semiconductor or polymer interaction phenomena that become significant at microscales. materials. Recently, it was shown that some degree of fragility Preliminary modeling based on this approach had shown reasona- and processing complexity could be compensated by integrating ble agreement between identified robot properties and global robot lead–zirconate–titanate (PZT) thin-films with high-aspect ratio motion (body vertical and lateral displacements), but model polymer microstructures and coatings based on parylene-C after refinements and new validation based on experimental measure- PZT deposition [7]. Parylene-C films and microbeams were ments of individual robot foot behavior in this work provide fur- shown to help protect the fragile piezoelectric layer and amplify ther insight into the influence of individual foot–terrain contacts actuator stroke in compliant mechanisms, producing more robust on cumulative robot locomotion. thin-film PZT microrobots than previously demonstrated. There have been several prior studies of fundamental dynamic A further benefit of thin-film piezoelectric actuation for concepts related to small-scale walking or running locomotion. microrobotic applications is that these actuators can achieve high This includes studies originating in locomotion of biological bandwidths with low damping ratios, which raises possibilities for organisms, such as insects [14], as well as tests of robots intended to operate on these principles, typically at the size-scale of several centimeters [14,15]. These works typically develop a lumped- 1Corresponding author. Manuscript received April 12, 2017; final manuscript received August 11, 2017; parameter model for leg dynamics and apply it with relatively published online September 18, 2017. Assoc. Editor: Larry L. Howell. simple ground interaction modeling, alternating between firm

Journal of Mechanisms and Robotics Copyright VC 2017 by ASME DECEMBER 2017, Vol. 9 / 061006-1 contact and motion of the robot feet in air. This reflects basic con- vibratory actuation from a shaker. The robot motion simulated by cepts of legged robot dynamics such as their foot–terrain and the dynamic model is compared with the empirical results to eval- foot–body interaction, which have some different features when uate the model with respect to robot translation due to vibratory examining dynamics of even smaller, silicon-micromachined response of its legs and with respect to small-scale force effects robots. Regarding the former, foot–terrain interaction can be char- on this motion. acterized regardless the number of legs in a robot, but it is signifi- This paper is organized as follows: Section 2 introduces the cantly influenced by the scale of robots. At the microscale, robot design and details of the millimeter-scale hexapod proto- nonlinear air damping and adhesion become significant factors types. Section 3 presents the dynamic model. Section 4 discusses that should be modeled accurately to estimate the robot dynamics. the experimental results for static, in-chip resonant and out-of- The foot–body interaction, meanwhile, in micromachined struc- chip resonant responses, and simplifications for parameter identifi- tures depends significantly on elastic structural resonances. These cation within the model. The robot dynamic model is validated may exhibit coupling between vibration modes in the legs and the with the out-of-chip measurement of vertical motion of its feet body itself, or at least coupling of resonances in multiple legs that and body and of average forward motion on the vibrating surface. may vary based on the number and location of legs with respect to Section 5 discusses validation testing of the dynamic model with robot body. Again at scales of several centimeters, elastic vibra- comparisons between experimental and simulated results for cer- Downloaded from http://asmedigitalcollection.asme.org/mechanismsrobotics/article-pdf/9/6/061006/6404124/jmr_009_06_061006.pdf by guest on 29 September 2021 tion has been used to generate piezoelectric robot locomotion, but tain feasible testing conditions. Section 6 discusses implications the further coupling of these dynamics with foot–ground interac- for potential future autonomous microrobot locomotion based on tion at small-scales has not been studied in detail [10,15–18]. observations from the dynamic model. Section 7 presents the con- Several contributions toward the understanding of millimeter- clusions of this work. scale microrobots are then included in this work. At a high level, while prior works have explored effects of small-scale forces and 2 Robot Design impacts of individual microrobot legs or mesoscale representations, those works had yet to validate modeling of multilegged dynamic An example of the microrobot design used for dynamic testing impact behavior on millimeter-scale, resonance-driven microrobot is shown in Fig. 1, both before and after removal from the silicon locomotion with significant small-scale force contributions. To wafer in which it is fabricated. Details on robot actuator design, effectively capture robot motion, multiple vibration modes are fabrication, and testing have been presented in Refs. [8] and [19]. included in this model at all legs: three modes in the robot structure In brief, the robot consists of a central silicon chassis or body evaluated here, with further extension of modes possible with this (30 lm thick silicon), surrounded by six nominally identical legs. modeling technique. Also, for each vibration mode, the direction Each leg contains two actuation elements based on thin-film pie- of motion is considered, so the motion in lateral and vertical direc- zoelectric actuators. The piezoelectric actuators are formed on a tions are coupled, where prior modeling assumed minimal direc- silicon dioxide base layer (0.5 lm) with two electrode layers of tional coupling between modes. The dynamics of adhesion and rest platinum (0.1 lm and 0.2 lm) on either side of the PZT thin-film between the feet and ground are further distinguished for a better layer (1.0 lm). The hip actuator consists of a PZT unimorph con- understanding of the adhesion influence on microrobot dynamics. strained to act in lateral contraction [7,20], coupled to a high- From this model, the robot forward motion and individual foot aspect-ratio parylene-C microbeam that leverages contraction into motion of future autonomous robots may be better predicted given in-plane rotation. The knee actuator consists of three PZT unim- specific design parameters and prefabrication analysis. orphs acting in pure bending to rotate the robot foot out-of-plane. Model validation during this study is performed using a In-plane and out-of-plane actuators are connected in parallel elec- millimeter-scale microfabricated hexapod robot prototype. trically and thus must actuate simultaneously, while legs are Figure 1 shows examples of these microrobots before and after addressed either individually or in-pairs from electrical intercon- detachment from a silicon chip. To interpret both active and pas- nects on the robot body. Conceptually, robot locomotion would be sive microrobot dynamics, a dynamic model is constructed that based on alternating actuation of legs, say in a tripod gait, using includes: nonlinear foot–terrain interactions, coupled structural variations in natural frequency and response time between lateral resonances of the robot’s actuators and structures, and rigid body and vertical motion at the foot to create elliptical-type foot motion of the robot chassis with multiple degrees-of-freedom. motions. Conceptual foot motion with ground contact, when the The microrobots as initially fabricated in the silicon chip are sus- PZT elements in each leg are actuated in parallel, is demonstrated pended by silicon tethers used to support the microrobots through with Fig. 2. In a prototypical cycle, the robot foot will spend time their fabrication process, which also provide electrical signals for both in contact with ground and moving in air within a single characterization. Several locations on microrobots are measured actuation cycle, or robot “step,” though substantial variation in under different loading conditions to identify robot structural behavior will be seen in the presence of contact dynamics and dynamics. For ground interaction testing, the microrobot is small-scale forces. Tested static amplitudes at the robot foot are detached from its chip, and motion is excited on other surfaces. about 50–100 lm at actuation voltages of 10–20 V. Much larger Since the detaching process breaks the electrical connection to the foot displacements are possible near resonance, as measured microrobot, out-of-chip dynamics are measured under external

Fig. 1 Photo of (left) a silicon die containing tethered hexapod Fig. 2 Side view of nominal robot foot motion when ground is microrobots; and (right) sample hexapod microrobot detached present; the foot remains stationary with respect to the ground from its wafer; the coordinate system for dynamic model is also for a certain period of time when actuated downward and labeled; with the z-direction pointing out of plane moves in air when actuated upward

061006-2 / Vol. 9, DECEMBER 2017 Transactions of the ASME damping ratio for the leg structures is near 0.05, but maximum leg–foot system various vertical and lateral motion ratios by range-of-motion before failure has not been evaluated. choosing the actuation frequency carefully. The ratio of vertical to As fabricated in a silicon chip, serpentine silicon springs or lateral motion at each resonance, as motions of the two actuators tethers support the robot body and provide electrical signals to the are not perfectly orthogonal at their respective resonances, and robot legs. This allows characterization of active robot leg motion, thus, vertical to lateral motion ratio at other frequencies is a criti- but does also influence robot dynamics, as compliance of the robot cal factor in generating robot locomotion from this architecture. body or chassis itself is constrained by the tethers, and a rigid The vertical to lateral motion ratio was determined with COM- body mode of the entire robot oscillating on the tethers is intro- SOL to be 1:15 at the first lateral resonance and 17:1 at the first duced. Sections 3–5 examine the various resonant vibration vertical resonance. These ratios are used to model the dynamics of behaviors that occur across the robot legs and body, and use this microrobots when associated leg resonances are excited by impact information to interpret forward robot locomotion in passive after the robot is detached from the wafer. walking on a vibrating field after silicon tethers have been sev- ered, and the robot is permitted to translate freely across a surface. 3.2 Hybrid Leg Dynamics. Individual robot foot motion in the time domain is distributed into three dynamic or hybrid Downloaded from http://asmedigitalcollection.asme.org/mechanismsrobotics/article-pdf/9/6/061006/6404124/jmr_009_06_061006.pdf by guest on 29 September 2021 3 Dynamic Model modes. The first dynamic mode describes the free, in-air motion A model for microrobot dynamics is generated as a hybrid of a robot leg. This mode shows up in both in-chip and out-of- dynamic system. In this process, microrobot dynamics are first chip motion. No foot–terrain contact is included in this mode, so studied with finite element analysis (FEA) at the individual leg the dominant external force from ground interaction is the air level. Using FEA results and in-chip characterization of the damping force at the microscale, i.e., squeeze-film damping. microrobots, the full dynamic model is built to account for multi- Mode 2 describes the impact phenomena of the robot foot. If the ple resonant modes of the leg–foot system, nonlinear foot–ground robot foot is not capable of leaving the ground because of adhe- interactions, and body motion in five degrees-of-freedom. The sion forces or gravity, modes 3 and 4 are used to describe leg resulting dynamic model includes two parts to completely motion. Again we note that structural dynamics in each mode describe the robot motion. The first part is a leg–foot model with originate in the modal vibratory dynamics of the system; as such, three different dynamic motion modes (“dynamic modes” used when referring to modal dynamics associated with specific here in a hybrid system sense as opposed to “vibration modes” or resonances, we will refer to “vibration modes,” and to “dynamic “resonant modes” associated with the elastic compliance of the modes” when referring to the three modes of motion comprising robot). These dynamic modes are an in-air mode, an impact mode, the hybrid system model. and an adhesion mode, to cover possible robot foot motions. The second part is a body model to describe the robot body/chassis 3.2.1 Mode 1: Leg Motion in Air. The free in-air motion of motion in five degrees-of-freedom, requiring appropriate motion the robot foot is described with a state-space model. The in-air state transformations between relevant reference systems. The motion represents the foot motion when tested in-chip, as well as dynamic model is then used to predict and compare robot in-plane out-of-chip motion after bouncing (mode 2) or release from and out-of-plane motion when actuated by external vibration from ground after adhesion (mode 3). The vertical and lateral motions a shaker. for each mode are coupled with a relation defined by preliminary FEA results. The state-space form in both lateral and vertical directions shares a similar format as follows: 3.1 Preliminary FEA Modeling. A dynamic model of the 2 3 2 32 3 2 3 microrobot’s legs is first constructed in COMSOL finite element A 0 analysis software to understand the frequency response of both X1;jþ1 6 1 7 X1;j B1 4 Ӈ 5 4 .. 54 Ӈ 5 4 Ӈ 5 hip and knee actuators. Both actuators are simulated to predict the ¼ 0 . 0 þ Fext (1) X X B structural resonance. With nominal robot design parameters and n;jþ1 0 An n;j n fabrication results as reported in Ref. [19], the first lateral resonant 2 3 frequency of the hip actuators is simulated to be 498 Hz; the first 00 vertical frequency of knee actuators is 3671 Hz. Figure 3 shows 01 6 7 xi;j Ai ¼ ; Bi ¼ 4 1 1 5; Xi;j ¼ (2) the simulated structural deformation of these first two modes ki bi vi;j of the robot legs. It is worth recalling that both actuators in an mi;imp mi;dis individual leg are connected electrically and thus are actuated ( simultaneously. While it is not possible to stimulate both actuators zi ¼ Cz;ixi vzi ¼ Cz;ivi at their maximum motion amplitude simultaneously with a single- (3) frequency actuation signal, it is possible to actuate the robot yi ¼ Cy;ixi vyi ¼ Cy;ivi

in which ki and bi are the spring constant and damping coefficient for resonance i, normalized by the effective mass of that vibration mode; xi;j and vi;j are the displacement and velocity for the ith vibration mode at jth time step. Vibration mode matrices Ai are constructed independently for vertical and lateral directions based on resonant frequencies with dominant influence in those directions. It should be noted here that the robot motion states in the state space form represents the robot foot motion with respect to the robot body. The displacement and velocity in the lateral (y-) and vertical (z-) directions (respectively, zi, vzi, yi, and vyi) are expressed as proportional to a combined modal displacement and velocity, xi and vi, using motion coupling coefficients Cz;i and Cy;i. The motion coupling coefficients mea- Fig. 3 (Left) the compliant structure of a sample robot leg with sure the amount of motion in lateral and vertical direction con- a high aspect ratio link connecting the hip and knee actuators; also shown are the COMSOL-simulated mode shapes of the tributed by ith mode of the leg: Cz is the ratio of motion in the first lateral mode of the leg (middle), originating in pivot about vertical direction to the overall motion of the ith mode, and Cy the hip actuator, and the first vertical mode of the leg (right), is the ratio of the motion in the lateral direction to the overall originating in bending of knee actuators motion of the ith mode.

Journal of Mechanisms and Robotics DECEMBER 2017, Vol. 9 / 061006-3 Fext is a vector of external forces acting on the leg and foot, of robot feet in the lateral direction during this interaction [9]. divided into those that are concentrated at the foot and those that are However, unlike robot motion with an electrical input, which can distributed over the leg; and mi;imp and mi;dis are the effective mass directly drive lateral actuation, shaker motion is in the vertical for the ith mode with respect to those point and distributed external direction alone. As it excites the individual legs’ motion, coupling forces. Forces able to be approximated as distributed over the leg between forces in the two axes produces lateral motion of the foot and/or foot include gravity on the leg (ml) and foot (mf), nonlinear with little opportunity for slip, so friction is treated as sufficient to air damping (Fdamp), and effective internal piezoelectric force (Fact) maintain fixed contact laterally during ground interaction in the current testing scenario. Therefore, the lateral velocity of a robot 0 Fext ¼½Fact þ Fdamp þ mlg mf g (4) foot after impact is enforced to be zero when impact happens, as a combination of two nonzero lateral modal velocity. Because in which g is a gravitational vector. Robot foot velocity (vzf b and motion in the vertical and lateral directions is coupled, for a post- vyf b) and displacement (zfb and yfb) relative to the robot body aris- impact vertical velocity, the mode velocity is estimated as if only ing from the combined effect of vibration modes included in the the first vertical and lateral modes are influenced by impact, fol- model are expressed as lowing Ref. [12]. Using the motion coupling coefficients (Cz and Downloaded from http://asmedigitalcollection.asme.org/mechanismsrobotics/article-pdf/9/6/061006/6404124/jmr_009_06_061006.pdf by guest on 29 September 2021 8 X X Cy) for two modes (subscript 1 for first vertical mode and 2 for > first lateral mode) and vertical velocity after impact (v ), the <> vzf b ¼ vzi zfb ¼ zi zfb; f Xi Xi overall velocity for the first vertical (vz) and lateral mode (vy) after > (5) impact are :> vyf b ¼ vyi yfb ¼ yi 8 i i > Cz;2vzfb;f <> vz ¼ The foot velocities (vzf g and vyf g) and displacements (zfg and yfg) Cy;1Cz;2 Cz;1Cy;2 > (8) relative to ground take the body motion into account as well. > Cz;1vzfb;f Equations (13)–(16) in the prior work [9] explain the conversion : vy ¼ Cy;1Cz;2 Cz;1Cy;2 between velocity and displacement with respect to ground (referred to as vz , vy , z , and y ) versus motion relative to the fg fg fg fg In which the total velocities are related to the dominant modes in body (v , v , z , and y , as above). zf b yf b fb fb their respective directions, the first mode for vertical velocity and At the current millimeter robot scale, some other nonlinear and/ the second mode for lateral velocity. Displacement is distributed or small-scale forces may affect robot dynamics, even in the into these two modes through a similar calculation. In other absence of ground contact. The most influential of these is found words, instead of a force calculation based on normal or frictional to be nonlinear effects of air flow during motion of the leg near forces, state are directly updated when impact occurs. the ground (Fdamp), which is modeled as an air drag force (Fdb) and an effective squeeze film damping force (Fsb for the robot 3.2.3 Mode 3: Sustained Contact With Adhesion. After each body and Fsf for the robot foot), acting in the vertical direction foot–terrain interaction in mode 2, one more check step is added and nonlinearly dependent on distance from ground [21,22]. The to determine whether the robot foot is able to leave the ground air damping affects both the robot feet and body, calculated with afterward. In some situations, the robot foot will stay in contact different geometrical parameters and velocity profiles. The coeffi- with ground due to large ground upward velocity or large body cients for the robot body and each foot are calculated separately. downward velocity, while at other times it may be restricted by a These forces are expressed as small downward adhesion force, hypothesized to occur due to 8 small electrostatic and intermolecular adhesion forces between the > > bgf vzf g robot foot and the underlying surface. From the previous work, > Fsf ¼ <> zg zfg the adhesion force was found to be time dependent in microscale for similar geometries and materials [23]. Therefore, the adhesion bgbvzb (6) > Fsb ¼ force is linearized as > zg zb > : 2 F c t c (9) Fdb ¼ avzf g ad ¼ ad;t þ ad in which v and v are the vertical velocity with respect to ground in which t is the contact duration; cad,t and cad are coefficients for zf g zb adhesion force. If net upward force on the robot foot is smaller for the robot foot and robot body, respectively, bgf and bgb are the coefficient of squeeze film damping for the robot foot and body, than threshold adhesion force, then the foot velocity (vfg;j) and dis- respectively, calculated from their component geometries, air prop- placement (xfg;j) in both vertical and lateral directions are enforced erties, and nominal distance from ground; z is the ground height; to be same as ground motion. Then, the foot motion (vfb;j and xfb;j) g relative to body is inferred from body motion. To calculate the and zfg and zb are the vertical height of robot foot and robot body. motion states (Xi;j), only the resonant mode with lowest frequency is actuated at mode 3, as identified in Refs. [21] and [22]. 3.2.2 Mode 2: Impact. Mode 2 represents impacts between For practical purposes, in simulation an additional check is foot and ground. Comparatively instantaneous foot–terrain inter- required at the beginning of a time step if at the previous time step action is the fundamental feature of this mode. The impact the foot was in mode 3. This check step is to confirm whether the between foot and ground is approximated by a coefficient of resti- foot could leave ground, computed as tution model 8 vzfb;f vzg;f ¼ðvzfb;o vzg;oÞcr (7) 2 3 > X1;j >6 7 > 4 Ӈ 5 in which the velocity of robot foot before (vzfb;o) and after (vzfb;f ) > K1 B1 … Kn Bn þ Fext Fnet > Fad vzfb;f < 0 <> impact are related to shaker ground velocity before (vzg;o) and X 2 n;j 3 after (vzg;f ) impact according to a coefficient of restitution, cr.In > > X1;j Eq. (7), the robot foot velocity is relative to the fixed ground con- >6 7 > Ӈ dition, which is calculated from robot body motion and foot > K1 B1 … Kn Bn 4 5 þ Fext Fnet < Fad vzfb;f > 0 :> motion with respect to the robot body, from Eq. (5). X Friction effects can influence contact interactions during this n;j interaction, as sufficiently light vertical contact can result in slip (10)

061006-4 / Vol. 9, DECEMBER 2017 Transactions of the ASME in which Ki and Bi are spring constant and damping coefficient of the ith mode, in this case not normalized by effective mass. The robot will remain in mode 3 if either of these two inequalities is fulfilled. Once a time step is determined to remain in mode 3, the robot foot displacement and velocity are forced equal to the ground condition during that time step. 3.2.4 Mode 4: Rest. The last mode for a robot foot is a rest or normal contact mode. In this mode, the robot foot rests on the ground with a positive normal force, because the net force on the foot is downward. The dynamics of this mode are identical to those of mode 3. However, it is useful to distinguish this mode during simulation for the further study on the influence from adhesion force, so that behavior specific to the presence of adhesive forces can be easily assessed. In the dynamic model, if a robot Downloaded from http://asmedigitalcollection.asme.org/mechanismsrobotics/article-pdf/9/6/061006/6404124/jmr_009_06_061006.pdf by guest on 29 September 2021 foot would move below ground for a previous step while in modes 2, 3, or 4, the foot’s dynamics remain in mode 4. Fig. 4 Photo of the experimental setup: (left) LDV measuring in-chip dynamics; (right) robot out-of-chip measurement with 3.3 Robot Body Model. The robot body or chassis is treated shaker as a rigid body having five relevant degrees-of-freedom; lateral translation in the y-direction is neglected. Based on the coordinate system defined in Fig. 1, the robot rotation in x, y, and z directions voltage input to bond pads at the base of the silicon tethers to the piezoelectric actuators. A laser Doppler vibrometer or LDV (Poly- (xbx; xby; and xbz) and translational motion in y and z direc- tec OFV 3001 S Controller & 303 Sensor head), and a stereoscope tions (vby and vbz) are treated as being generated from the total moments in the z-direction and forces in x- and y-direction trans- are used to acquire out-of-plane velocity and in-plane displace- mitted from the robot legs ment measurements, respectively. The test setup for out-of-chip X X X dynamics testing is shown in Fig. 4 (right). Data acquisition is a performed as above, while a shaker (BK Vibration Exciter Type Fzly Fzlx Fylz x_ bx ¼ ; x_ by ¼ ; x_ bz ¼ (11) 4809) is used to excite motion of the detached microrobot in a Ix Iy Iz Teflon tray. The out-of-chip motion is measured with both the X X LDV and camera. The LDV measurement is used to understand Fyl Fzl mg Fdb Fsb the vertical motion of robot, and the video recorded with the cam- v_ ¼ ; v_ ¼ (12) by m bz m era is used to characterize the lateral speed of the robot. It is worth noting that the quality of the LDV measurement is dependent on In Eq. (13), m is the mass of the robot body, and Ix, Iy, and Iz the size and materials at the surface of the robot at various mea- are the moments of inertia in the x-, y-, and z-directions. Fdb and surement points. As the polymer structures forming the foot do Fsb are the linear air drag and squeeze-film air damping terms of not reflect the LDV laser effectively, the motion of parylene-C the robot body from Eq. (6); Fyl and Fzl are the force from each foot cannot be directly measured. Thus, all experimental measure- robot leg to body; xj, yj, and zj are the distances between the con- ments of robot leg behavior are taken at least a short distance nection point of the body and the jth leg to the position of the from the foot itself, for instance at the outer edge of the robot robot center of mass in the x-, y-, and z directions, respectively. knee joint. When estimating foot displacement or relating simu- Because of the finite gap between ground and the robot body, it is lated foot motion to experimental results, the displacement of the also possible that robot body can have contact with ground if the foot relative to the robot body is assumed to scale from measure- downward motion of body is large. This contact between robot ments taken near the knee according to the geometries of mode body and ground is modeled with a coefficient of restitution, as shapes generated by FEA. for foot impact in the vertical direction. The foot–terrain interaction is calculated based on the foot 4.2 In-Chip Testing. First, functionality of microrobot actua- velocity with respect to ground as described under mode 2 of the tors during in-chip testing was assessed visually. Static displace- leg model. The body–foot interaction is calculated from the foot ment of individual robot feet was measured to be on the order of velocity and displacement with respect to the robot body. There- tens of micrometers using a 15 Vpp step input at 1 Hz. This step fore, the transformation between these two velocities is important amplitude is observable with microscope. Therefore, it is conven- during simulation of body dynamics, which was shown in the pre- ient to identify the actuator functionality with a low frequency vious work [9] as mentioned above. input (1 Hz or lower). Then, microrobot in-chip characterization is focused on characterizing its dynamics under different frequen- 4 Testing and System Identification cies and loading conditions. Measurements of robot motion were taken at various locations While initial FEA modeling provides a nominal set of parame- on the robots during swept-sine excitation of the robot legs to ver- ters for model parameters associated with specific resonance ify and refine resonance behavior predicted by the FEA model. It modes (i.e., effective mass and stiffness parameters), completed was known that the silicon tether that connects the microrobots to robots may deviate from the model due to fabrication nonideal- the wafer could also have a large influence on robot dynamics by ities, and a number of parameters associated with damping and introducing a mass-spring rigid body vibration mode to the system external forces must be identified empirically. This section dis- and stiffening the chassis structure; this was evaluated in part by cusses dynamic frequency-domain testing. examining the effects of constraining the vertical motion of the robot chassis with a micromanipulator probe. 4.1 Experimental Setup. Both in-chip and out-of-chip Figure 5 plots the frequency response of out-of-plane velocity dynamics of microrobots are measured, as excited by internal measured by the LDV at the robot chassis, at the robot “hip” (just piezoelectric and external vibration stimuli, respectively. The after the in-plane actuator), and at the robot “knee” (just after the experimental setup for in-chip characterization is shown in Fig. 4 vertical actuator). Although the elastic structure of the robot (left). A LabVIEW frequency sweep program interfaces a power allows resonances to be transmitted throughout its structure, com- supply to two micromanipulator probes, which apply the resulting parison of locations at which various resonances become most

Journal of Mechanisms and Robotics DECEMBER 2017, Vol. 9 / 061006-5 existence of signiﬁcant foot off-axis tilting motion. This had previously been predicted to be small during FEA analysis. While this was not intentionally designed for microrobot locomotion, such motion has potential to be beneﬁcial if it can supplement or be used in-place of originally designed in-plane foot rotation originating in the hip actuator.

4.3 Out-of-Chip Testing. Since the eventual goal of microrobot motion is to realize autonomous, after characterization of in- chip dynamics one robot was detached to evaluate dynamic response during interaction with ground. As extracting the robot at this stage in development removes access to an active actuation signal, motion for out-of-chip testing was excited by a shaker.

Figure 6 (left) shows the absolute motion amplitude of the tray, Downloaded from http://asmedigitalcollection.asme.org/mechanismsrobotics/article-pdf/9/6/061006/6404124/jmr_009_06_061006.pdf by guest on 29 September 2021 which is used as a reference to calculate the relative motion of three locations on the robot (chassis, hip, and knee), as shown in Fig. 6 (right) under small amplitude ground excitation. It should be noted that the vibration velocity from the shaker decreases with frequency amplitude of robot motion is often small, such that only velocity measurements under about 3 kHz could be clearly Fig. 5 Resonance measurement of three different locations on distinguished from background noise. Unfortunately, given a finite the microrobot: body (pink dash line), hip (blue dotted line), availability of amplification settings for the current supply to the and knee (black solid line) indicate mode shapes associated shaker, while a frequency sweep versus input voltage can be read- with the hip near 438 Hz, uniformly generated on the body near ily tested, only a small range of frequencies permit multiple 830 Hz, and most strongly associated with the knee near 3.4 kHz ground amplitudes to be excited at the same frequency, and minimal robot motion could be observed at those settings. significant allows their sources to be verified. Testing results indi- In Fig. 6 (right), the relative velocities of three locations on the cate that the robot chassis rigid body resonance on the tethers robots with respect to ground motion are plotted versus frequency. occurs around 830 Hz; the hip or primary lateral resonance is The largest motion amplitude is observed between 1.5 and around 438 Hz (versus 498 Hz from COMSOL); the knee or pri- 2.5 kHz. This mode is predominantly in the vertical direction as mary vertical resonance is around 3.4 kHz (versus 3671 Hz from observable in the experimental setup. It is taken to be the primary COMSOL). The FEA and measured results matched well for both vertical vibration leg of the mode after release from the wafer, actuators in the frequency domain. The body or chassis resonance being reduced from in-chip testing due to additional elastic chas- at 830 Hz was confirmed as a rigid body mode through compari- sis bending that is constrained by the silicon tethers when they son of velocities at additional locations on the chassis, showing were present in-chip. Additional vertical motion is measured nearly pure translation. These measurements also confirmed that around 800 Hz and 1.2 kHz, believed to be a resonance of body elastic modes of the chassis itself are small below 3 kHz when motion itself since these modes were not measured when a probe constrained by the silicon tethers. Further confirmation was pro- was pressed to the robot body in the chip. vided by adding additional stiffness supporting the robot with a micromanipulator probe, which increased the associated resonant 4.4 Parameter Identification. The overall conclusion of fre- peak from 830 Hz to 1.8 kHz, with negligible effect on the modes quency response testing was that after extraction from the wafer, attributed to the individual legs. the most significant resonant behaviors were present near 438 Hz, One significant difference between FEA modeling and experi- 1.2 kHz, and 1.8 kHz, attributed to vibration modes associated mental testing was observed with regard to vertical motion of the with in-plane rotation at the hip, foot tilting motion, and out-of- knee or out-of-plane actuator. The end points of the three unim- plane leg bending due to compliance through the released body, orph PZT beams comprising the knee actuator were measured and leg, and knee, respectively. To simplify the dynamic model, an found to exhibit unequal velocity amplitudes, which indicates the additional 800 Hz resonance was not considered in simulation,

Fig. 6 Frequency sweep results of a detached hexapod microrobot: (left) the absolute velocity of the robot body; (right) the velocity of robot body (pink dotted line), robot hip (blue dashed line), and robot knee (black solid line) relative to the tray motion

061006-6 / Vol. 9, DECEMBER 2017 Transactions of the ASME Table 1 Table of coefﬁcients for damping and adhesion

Coefficient Value

19 2 bg 1.5 10 Nm s (body) 2.5 1019 Nm2 s (leg) a 4.55 106 Ns2/m2 6 cad 50 10 N 7 cad,t 1 10 N/s

Table 2 Table of simulation parameters in dynamic model Fig. 7 Hexapod microrobot location at (left) time 5 0 s and Parameter Value (right) time 5 1 s when the tray is externally vibrated by the Downloaded from http://asmedigitalcollection.asme.org/mechanismsrobotics/article-pdf/9/6/061006/6404124/jmr_009_06_061006.pdf by guest on 29 September 2021 shaker at 240 Hz Total robot mass 0.33 mg Robot body mass 0.28 mg Normalized spring constant (N/m/kg) Mode 1: 7.57 106 Mode 2: 5.68 107 Mode 3: 1.28 108 Normalized damping coefﬁcient (N s/m/kg) Mode 1: 1.38 103 Mode 2: 1.76 103 Mode 3: 2.97 103 Modal motion ratio, lateral to vertical motion Mode 1: 15 Mode 2: 1/17 Mode 3: 1/15

because it was only found with in-chip testing, and attributed to robot oscillation on its tethers. The resonant frequencies and amplitudes obtained were used to tune stiffness, mass, and input parameters in the state-space model for robot structural dynamics. The squeeze film damping coefficients are calculated for both robot body and foot, as shown in Table 1, from models in Refs. [21] and [22]. The coefficients for time-dependent adhesion force are estimated from the previous work [23].

5 Dynamic Analysis Results Fig. 8 The relationship between shaker frequency and average robot speed; the simulations are shown as the black dashed Using the identiﬁed system model and parameters generated line, and the experimental results are shown as individual red through the experiments described in Secs. 3 and 4, the effective- data points with error bars ness of the proposed dynamic model at describing global robot motion in the presence of foot–ground interaction could be examined. The parameters used in simulation are listed in simulated speeds are shown in Fig. 8. The error bar of the simula- Table 2, including the mass of robot, resonant parameters. Two tion is generated from the uncertainty of robot parameters, and the primary sets of data are available for model assessment: measure- error of measurement is caused by the nonuniformity of robot ments of average lateral speed of the robot for various vertical motion speed. When the actuation frequency is higher than 400 ground excitation frequencies, and detailed time-domain measure- Hz, no lateral motion was observed. The speed trends for simula- ments of vertical robot leg displacement, measured at the knee or tion and experiments are in qualitative agreement, showing a fast hip. decline in speed from 200 Hz, and at around 300 Hz there was an increase in robot speed found both in simulation and experiments. 5.1 Body Lateral Motion. At key frequencies, the detached The experimental setup (i.e., LDV and stereoscope supports) hexapod microrobot was found capable of fast lateral motion exhibits resonance when the shaker is actuated at a frequency within the tray when excited by vertical vibration. Figure 7 shows lower than 150 Hz, preventing lower-frequency characterization two frames of sample motion recorded when the shaker was oper- using the current setup. ating at 240 Hz with an amplitude of 25 lm. Sample robot motion Also shown in Fig. 8, the simulation correctly predicts that no video stills are shown with a time separation of 2 s. The speed of motion will occur at frequency ranges from 500 Hz up to 1.8 kHz, the robot is about 4 mm/s, with a small counter-clockwise turn where experimental tests showed no measurable locomotion. This observed, possibly due to one robot leg showing signs of damage is despite the presence of various vibration modes in that region, during the detachment process. Because of the turn, the speed, but consistent with those vibration modes acting effectively out- instead of velocity, of robot is used for further validation of the of-plane with only small in-plane components. Ability to excite dynamic model. locomotion is also limited at higher frequencies, because maxi- The existence of lateral motion of the robot near 300 Hz mum amplitudes achievable by the shaker become smaller. The suggests that its locomotion during ground excitation is generated robot speed is mildly overestimated through the entire frequency by external vibration coupling to the hip actuation mode below range, which may indicate the existence of other microscale forces its resonant frequency, which is also the mode of lateral foot or more complex adhesion behavior beyond the contents of the actuation. Simulated robot motion at different frequencies was current model. Nonetheless, the model does accurately estimate then compared with experimental measurement. Measured and the overall speed trend.

Journal of Mechanisms and Robotics DECEMBER 2017, Vol. 9 / 061006-7 Downloaded from http://asmedigitalcollection.asme.org/mechanismsrobotics/article-pdf/9/6/061006/6404124/jmr_009_06_061006.pdf by guest on 29 September 2021

Fig. 9 The vertical motion (velocity and displacement) of robot knees measured with the LDV at different legs, with commonly observed motion patterns: (top left and top right) ﬁrm-contact pattern with constant foot–ground contact, (bottom left) partial ﬁrm-contact pattern with extended adhesion mode but eventual detachment from ground, and (bottom right) jumping pattern with long time in-air mode. The red line in the bottom right is a sample tray motion beside the measured location on the body. The shaker is actuated at 2 V (top left), 4 V(top right), 6 V (bottom left), and 8 V (bottom right), respectively.

5.2 Leg Vertical Motion. More detailed comparison of experimental and model results, and insight into effects of small- scale forces, was obtained from LDV measurements of vertical motion of individual robot legs. Different motion patterns were observed as characteristic for tests at varying frequencies and amplitudes. Three representative patterns are shown in Fig. 9. The first example (Fig. 9 (top-left and top-right)) is a firm-contact pattern, in which the robot foot has constant contact with ground for small ground oscillations. Figure 9 (bottom-left) shows a partial firm-contact pattern at increased amplitudes: the robot stays in adhesion mode for some time length of each period, even beyond the point at which body motion would have otherwise pulled the leg free of the ground, but the foot does detach for an approximately uniform amount of time in each cycle. Figure 9 (bottom-right) is a jumping or bounding pattern when the shaker is actuated at higher amplitudes. Leg motion is stimulated to a much greater extent by the external vibration, and individual legs stay in the air for more than one actuation period, with substantially higher amplitudes (200 lm) than the underlying ground amplitude (25 lm). The appearance of different patterns depends on the actuation condition and initial condition of the robot in tray. The firm-contact pattern is measured when robot is only Fig. 10 The vertical motion of the robot knee (dashed red) and moving vertically when the shaker is actuated at 4 V peak-to- tray (solid black), measured with the LDV when the shaker is peak; the jumping pattern is measured at a temporary pause within actuated with 4 V input voltage

061006-8 / Vol. 9, DECEMBER 2017 Transactions of the ASME robot lateral in-tray motions when the shaker is actuated at 8 V additional adhesion behavior as inferred from robot speed com- peak-to-peak. parisons. For 8 V peak-to-peak actuation (bottom-right plot in Figure 10 compares the simulated vertical motion of robot knee Fig. 11), a jumping pattern with minimal adhesion time is with various external vibrating amplitudes, to support the assump- observed in both experiments and simulation. tion of distinct motion modes in the hybrid model. With small external vibration, the robot foot is accurately predicted to stay in contact with ground throughout motion. Larger external vibration 5.3 Small-Scale Forces. The relative importance of various could take off robot leg and show the in-air mode, in which the small-scale forces is also examined through comparison between robot knee moves with a different velocity than the ground. By full and simplified simulations. Sample predictions of foot vertical comparing them together, examples of adhesion mode behavior motion in the presence of various possible small-scale forces are are identified. The abrupt velocity change from tracking ground to shown in Fig. 12. Both adhesion force and squeeze film damping motion in-air mode is representative of a downward adhesive force are small-scale, nonlinear factors that substantially influence force that is broken suddenly when sufficient upward force is the simulated results for robot leg motion and are critical to gener- exerted from the body to the leg being observed. ating comparable foot motion patterns to those seen in experi- Figure 11 shows resulting motion profiles of firm-contact, ments. Without the presence of a finite adhesion force, the partial Downloaded from http://asmedigitalcollection.asme.org/mechanismsrobotics/article-pdf/9/6/061006/6404124/jmr_009_06_061006.pdf by guest on 29 September 2021 partial firm-contact, and jumping patterns appearing in the simula- firm-contact breaks the similarity between each actuation cycle, tion. It should be emphasized that these conditions are only and no fully periodic foot motion is generated. It is important to observed in simulation results when accounting for the three note that “adhesion force” as treated in this work makes no dis- major out-of-chip vibration modes, external air damping identified tinction between various intermolecular forces that might influ- from out-of-chip measurement, squeeze-film damping estimated ence its magnitude, but if even a small amount of adhesive from robot geometry, and adhesion forces estimated from prior attraction is present between the robot foot and its terrain, it is pre- works. Figure 11 (top-left) shows the firm-contact pattern when dicted to have a substantial influence on robot locomotion. At the shaker is actuated with 2 V peak-to-peak, as in experiments. small locomotion amplitudes, it can prevent effective movement, Partial firm-contact starts to appear when the voltage signal and at larger locomotion amplitudes, the adhesion force transfers increase to 4 V peak-to-peak (top-right plot in Fig. 11). This simi- the additional surface motion into the robot’s compliant structure, lar motion pattern is observed in experiments, though it is first which can be beneficial for forward motion. In contrast, squeeze- observed only above 4 V/below 6 V, again implying some film damping reduces all individual leg motions to a substantial

Fig. 11 Robot leg vertical motion simulated with the proposed dynamic model: (top left) ﬁrm- contact pattern, (top right and bottom left) partial-contact and an intermediate, semi-periodic, pattern, and (bottom right) the jumping pattern. Red dashed line in all plots is the tray motion in simulation. The shaker is actuated at 2 V (top left), 4 V (top right), 6 V (bottom left), and 8 V (bottom right), respectively.

Journal of Mechanisms and Robotics DECEMBER 2017, Vol. 9 / 061006-9 Downloaded from http://asmedigitalcollection.asme.org/mechanismsrobotics/article-pdf/9/6/061006/6404124/jmr_009_06_061006.pdf by guest on 29 September 2021

Fig. 12 Robot leg vertical motion simulated with the dynamic model under different hypothetical microscale forcing scenarios: (left) simulated without squeeze film damping force for nei- ther robot leg nor body; (right) simulated without adhesion force between robot foot and ground. Robot leg motion is shown with the black solid line, and ground motion is shown with the red dashed line. degree, so its collective influence is found to reduce robot speed. As a passive walker, this dynamic model can be compared with Therefore, understanding microscale forces are critical for the a smaller and faster robot modeled in Ref. [24]. The model techni- estimation of robot dynamics. ques of Ref. [24] have a decent match for free leg motion between experiments and simulation, but are not able to give any predic- 6 Discussion tion on the contact performance and forward motion. Compared to a single leg model from Ref. [12] which does include contact Based on a dynamic model for structural and contact dynamics, interaction, the leg motion prediction on its own is less accurate, the motion of a piezoelectrically actuated microrobot has been but this work permits estimation of robot forward motion and syn- simulated. Microscale forces, such as adhesion and squeeze film thesis of multiple foot contacts, which was not done previously. damping, are studied to understand their influence on robot loco- As a candidate autonomous walking robot technology, compari- motion. The robot speed is simulated to be in the scale of mm/s, son of the current robot’s range of motion has been done in with predicted speed in agreement with experimental speed in Ref. [19], with a basic projection of velocity, though findings passive locomotion stimulated by vertical ground vibration. Error from this work would suggest a reduction in robot speed and pay- bars are simulated accounting for the resolution of fabrication and load capacity due to squeeze-film damping and adhesion. variation in forward motion over the duration of various simula- Toward the goal of predicting requirements and capabilities tions and are likewise consistent with observed robot locomotion for dynamic locomotion of such an autonomous microrobot in variability on a sample surface. dynamic locomotion under piezoelectric actuation, we also applied the model developed above to a hypothetical robot being driven by an on-board power supply. To simulate autonomous locomotion, ground height is held constant in the dynamic model, and the internal piezoelectric forcing (Fact in the hybrid system model) are taken to be a square wave around the resonance of the actuators, to produce a tripod gait. Figure 13 shows the simulated influence of adding robot payload when driving robot motion through on-board actuation near the first vertical resonance. The current microrobot with structure and actuators is about 0.3 mg. In simulation, when the robot total mass is larger than 1.7 mg, the robot stop moving. In some cases, locomotion is predicted to achieve a higher speed with a payload, as it improves uniformity of foot impact during simulation of a tripod gait. This is seen in Fig. 13 as projected robot speed increases beginning when the robot mass is larger than 1.2 mg, because the robot foot has more contact with ground and fewer instantaneous rebound or bouncing impacts. When the total mass of a robot is larger than 1.7 mg, the actuation force is not large enough to support sustained vertical motion on all steps. Around 1.7 mg, the robot speed estimation has a large variation, because the robot can move quickly with lots of small bouncing impact to gain forward speed, or stick to the ground with almost no motion, heavily dependent on initial conditions and random perturbations to the robot actuation force or geometry. Fig. 13 Simulated relation between robot mass and microro- Figure 14 shows an example of robot speed trends as a function bot motion when actuated with PZT actuators at 1.8 kHz (first of frequency, in this case near the robot vertical resonance vertical resonance in simulation); Robot mass is about 0.33 mg (1.8 kHz), associated with primarily lateral motion. This differs with structure and actuators from ground excitation measurements, where vertical ground

061006-10 / Vol. 9, DECEMBER 2017 Transactions of the ASME emphasize the importance of resonance characterization and oper- ation near one or another of the robot’s structural resonances. Fur- ther details of other contributing factors to microscale surface interactions, such as electrostatic forces and foot and/or ground plastic deformation, may improve the accuracy of dynamic model prediction. Further analysis will also examine other actuator inputs having potential for sustained efﬁcient locomotion. Future tasks for practical microrobot development include the validation of payload capability and integration with on-board battery or wirelessly coupled power supplies.

Acknowledgment

The authors thank the Lurie Nanofabrication Facility staff for Downloaded from http://asmedigitalcollection.asme.org/mechanismsrobotics/article-pdf/9/6/061006/6404124/jmr_009_06_061006.pdf by guest on 29 September 2021 assistance in development of microfabication processes, and Ms. Hermione Li and Mr. Mayur Birla for additional help with experimental testing and image preparation.

Funding Data

National Science Foundation (NSF) (Award Nos. CMMI 1435222 and IIS 1208233). Fig. 14 Simulated microrobot motion when actuated with PZT actuators at different frequency around the first lateral resonance (438 Hz) with same voltage amplitude (20 V) References [1] Hollar, S., Flynn, A., Bellew, C., and Pister, K., 2003, “Solar Powered 10 mg Silicon Robot,” The Sixteenth Annual International Conference on Micro Elec- amplitude easily breaks foot-to-ground contact at various frequen- tro Mechanical Systems (MEMS-03), Kyoto, Japan, Jan. 19–23, pp. 706–711. cies, but high-frequency excitations are both small amplitude [2] Ebefors, T., Mattsson, J. U., K€alvesten, E., and Stemme, G., 1999, “A (limited by the shaker) and excite only out-of-plane oscillation. 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061006-12 / Vol. 9, DECEMBER 2017 Transactions of the ASME