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Logo of nihpaAbout Author manuscriptsSubmit a manuscriptNIH Public Access; Author Manuscript; Accepted for publication in peer reviewed journal;
Int J Rob Res. Author manuscript; available in PMC Apr 9, 2013.
Published in final edited form as:
Int J Rob Res. Feb 1, 2013; 32(2): 218–246.
doi:  10.1177/0278364912467486
PMCID: PMC3621805
NIHMSID: NIHMS435733

Planning and Control for Microassembly of Structures Composed of Stress-Engineered MEMS Microrobots

Abstract

We present control strategies that implement planar microassembly using groups of stress-engineered MEMS microrobots (MicroStressBots) controlled through a single global control signal. The global control signal couples the motion of the devices, causing the system to be highly underactuated. In order for the robots to assemble into arbitrary planar shapes despite the high degree of underactuation, it is desirable that each robot be independently maneuverable (independently controllable). To achieve independent control, we fabricated robots that behave (move) differently from one another in response to the same global control signal. We harnessed this differentiation to develop assembly control strategies, where the assembly goal is a desired geometric shape that can be obtained by connecting the chassis of individual robots. We derived and experimentally tested assembly plans that command some of the robots to make progress toward the goal, while other robots are constrained to remain in small circular trajectories (closed-loop orbits) until it is their turn to move into the goal shape.

Our control strategies were tested on systems of fabricated MicroStressBots. The robots are 240–280 μm × 60 μm × 7–20 μm in size and move simultaneously within a single operating environment. We demonstrated the feasibility of our control scheme by accurately assembling five different types of planar microstructures.

1 Introduction

Microrobotics systems have the potential to enable applications in many different areas including medical science (Dario et al., 1992), surveillance (Kahn et al., 1999), and assembly (Popa & Stephanou, 2004). Our goal is to develop a microrobotic system capable of coordinated self-assembly. In this paper we present a multi-microrobotic system composed of several stress-engineered MEMS microrobts (MicroStressBots), together with planning and control strategies for directing these robots to assemble into planar shapes using a single, global, control signal. In (Donald et al., 2006), we described a globally-controllable 240 μm × 60 μm × 10 μm mobile stress-engineered MEMS microrobot (MicroStressBot). This robot, although complex to fabricate, contains only simple electromechanical 2-bit memory and logic. Such simplicity is common in other microrobotic systems as well (e.g., (Yesin et al., 2006; Frutiger et al., 2008).) Control of several such simple devices is desirable, but presents a significant challenge when only a single, global control signal can be used to control all the devices: the resulting system is highly underactuated. In this paper we present algorithms enabling independent control of several MicroStressBots, and directing the robots (also referred to throughout this paper as devices) as components that follow complex paths to assemble into larger, planar structures.

Our control scheme fits within the paradigm of global control, selective response (GCSR), where the individual devices are controlled by exploiting fabricated differences in their response to a broadcast-type control signal (Donald, 2007; Donald et al., 2008b). The GCSR paradigm enables the control of small and simple devices with their individual size approaching the scale-limitations of the fabrication processes. Until microfabrication technology advances to the point where one can embed on-board power and complex control on micro- or nanorobots, GCSR allows for scalable implementation of multi micro- and nanorobotic systems.

In this paper, we show that designing microrobots that move along different trajectories in response to voltage levels in the broadcasted control signal allows us to independently maneuver multiple microrobots to assemble into desired planar structures. Although the robots move in parallel, our algorithms decouple their motion by reducing the parallel motion of n microrobots to parallel motion of two robots, followed by sequential motion of single devices. Each robot either orbits, or pursues a trajectory towards the goal. Changes in the global control signal cause the robots to selectively switch (based on their differentiated physics) between orbiting trajectories and goal trajectories.

We implemented microassembly using GCSR by fabricating groups of MicroStressBots (240–280 μm ×60 μm × 7–20 μm in size) with different steering-arm designs. A scanning-electron micrograph of one such microrobot is shown in Figure 1.a, while four microrobots in the process of assembling a shape are shown in Figure 1.b. Each robot consists of an untethered scratch-drive actuator (USDA) (Donald et al., 2003) that provides forward motion, and a steering-arm actuator (i) that determines whether the robot moves in a straight line or turns. Each device receives both power and a control signal remotely without the use of restrictive wires or tethers, using a capacitive coupling with an underlying engineered substrate. The substrate comprises the robot’s operating environment. The robot is non-holonomic, and although it is not small-time locally controllable, in (Donald et al., 2006) we proved that interleaving of straight-line and turning motions is sufficient to ensure global controllability.

Figure 1
Scanning-electron micrograph of a stress-engineered MEMS microrobot (MicroStressBot) (a), and optical micrograph of four microrobots (b). a: The microrobot consists of an untethered scratch-drive actuator (USDA) that provides forward motion, and a curved ...

The steering-arm can be either raised to cause the robot to move in a straight line, or lowered to cause the robot to turn. We call the position of the steering-arm actuator the hysteresis state of the microrobot (arm raised, hysteresis state = 0; arm lowered, hysteresis state = 1). A system of n MicroStressBots contains 2n possible hysteresis states (all 2n combinations of n steering-arms being raised or lowered). However, in general, not all of the 2n hysteresis states are electromechanically accessible, depending on the physical design of the steering arm actuators. For example, the steering arms may be designed such that we can raise the steering arms for some devices only while simultaneously raising the steering arms of certain other robots. Such couplings can constrain the number of reachable hysteresis states to be significantly less than 2n. In this paper we show that during microassembly, it is sufficient to use only n + 1 hysteresis states to control n robots. While the control algorithm that uses n + 1 hysteresis states states has poor time complexity, it requires the smallest number of independent voltage levels in the control signal. This is significant, since the number of independent voltage levels in the control signal, called the control bandwidth, is the scarcest resource in our microrobotic system.

Details of fabrication, designs, and testing were reported in (Donald et al., 2008b). In this work, we present the control strategies and algorithms that allow us to implement simultaneous control of multiple microrobots to perform assembly of planar structures. These algorithms are crucial to the system, and are not covered in (Donald et al., 2008b). Enabling these control algorithms are new theorems, proven below, that minimize the required control voltage bandwidth; these were essential for controlling multiple untethered microrobots to move and assemble independently.

The major contributions of this paper can be summarized as follows:

  • Control Signal Engineering for Sub-linear Control Voltage Bandwidth: Design of the control primitives i.e. waveforms encoding the hysteresis states of the MicroStressBots to minimize the number of independent control voltage levels. A key contribution of this paper is novel engineering of control primitives that requires only 2n voltage levels to control n microrobots. This is a dramatic improvement over our previously proposed requirements of 2n voltage levels.
  • Heuristic Control Strategies: A comprehensive description of the two-stage heuristic control strategy based on motion planning under uncertainty. The strategy maneuvers the microrobots towards the goal while reducing the accumulating control error. The control heuristic includes compliance and collision avoidance, and, as we show in the presented experimental results, allows us to generate virtually defect-free assemblies.

The remainder of this paper is organized as follows. Sec. 2 discusses related work. In Sec. 3 we describe the stress-engineered MEMS microrobots (MicroStressBots) used in our work. In Sec. 4 we show how a control signal can be engineered to maneuver our microrobots independently despite the coupling of their motion. Sec. 5 describes a heuristic to construct the assembly plan, which allows us to determine the target configurations of the individual microrobots within the assembling shape. In Sec. 6 we describe how we used the control matrix engineered in Sec. 4, to implement a two-stage Error-Detection and Recovery (EDR) control strategy, allowing us to control the robots from their initial configuration to their final configuration within the assembling shape specified by the assembly plan generated in Sec. 5. In Sec. 7 we show the experimental results with fabricated devices to demonstrate the control strategies, and present the results of the assembly of several types of planar structures. Finally, in Sec. 8 we conclude by discussing the potential for application of our control algorithms to a broader class of underactuated microrobotic systems. A preliminary and abbreviated report of the ideas in this paper appeared in the workshop WAFR (Donald et al., 2008e).

2 Related Work

2.1 Microrobotics

Small-scaled robotic devices are envisioned by previous authors to have numerous applications within biomedicine (Dario et al., 1992), surveillance (Kahn et al., 1999), or microassembly (as we show in (Donald et al., 2008b) and this paper). Small, often completely autonomous mobile robotic devices, with the size on the order of centimeters (Byrne et al., 2002; Stefanini et al., 2006; Jianghao et al., 2007; Churaman et al., 2011) are called miniature robots. Miniature robots containing microfabricated components (Kladitis & Bright, 2000; Hollar et al., 2003; Linderman & Bright, 2000; Mohebbi et al., 2001) are called MEMS robots. The components of MEMS robots are often manufactured separately and then assembled to form complete systems. The size of the MEMS robots ranges from few micrometers to several centimeters. In (Donald et al., 2006, 2008b) and this paper, we use the term microrobot to denote mobile untethered MEMS robots with their dimensions strictly confined within a 1 mm3 cube. Several magnetically-actuated microrobots have been presented in (Yesin et al., 2006; Floyd et al., 2008; Frutiger et al., 2008; Pawashe et al., 2008; Frutiger et al., 2010).

2.2 Multirobotic Systems and Control

There are many examples of macroscopic multi-robotic systems that aggregate form or functionality through cooperation or mutual interaction. Self-reconfiguring robots are examples of physically-connected robotic systems that can reconfigure their shape and functionality (Kotay & Rus, 1999; Rus, 1998) to match the task at hand. The self-reconfiguration process has been controlled using distributed algorithms and message-passing (Butler et al., 2002). Multi-robotic cooperation can be also achieved in absence of explicit communication (Donald et al., 1997; Rus et al., 1995; Stilwell & Bay, 1993; Pagello et al., 1999). Specifically, (Donald et al., 1997; Rus et al., 1995) show how implicit communication can be used to coordinate several robots to manipulate larger objects. Coordination of multiple microrobots can be also achieved through the use of local rules (Jadbabaie et al., 2003; Martinoli & Agassounon, 2004; Pallottino et al., 2007).

Most previous work on decentralized multi-robotic control assumes the robots have sufficient onboard hardware resources to receive and process sensory inputs and/or communicate with other devices. However, our microrobots have a much simpler structure, and can only partially demultiplex individual commands transmitted through the global control signal. Bretl (Bretl, 2007, 2012) presented a related theoretical motion planning approach for systems of robots with limited controllability, showing that even simple devices controlled through a global signal can perform useful tasks. Our MicroStressBots are also simple, however, unlike the robots envisioned in (Bretl, 2007, 2012) they are fabricated to behave differently during portions of the control waveform. Control strategies presented later in this paper (Sec. 6) exploit this differentiation to maneuver the robots independently toward the goal. We call this control concept Global Control, Selective Response (GCSR) (Donald, 2007; Donald et al., 2008b). The problem of controlling the evolution of a dynamical system through a single control signal was proposed by Li and Khaneja (Li & Khaneja, 2005; Li, 2006), originally for controlling spin ensembles in NMR spectroscopy and magnetic resonance imaging (MRI). GCSR is related to the control of inhomogeneous ensembles coined by Li (Li & Khaneja, 2006), with the added difference that the devices that are being controlled in our work are specifically designed and fabricated to have sufficiently differentiated behavior to ensure controllability (as opposed to dynamic systems with intrinsic differences in behavior.) Previous work does not show how to design device physics for ensemble control, since in NMR the physics (e.g., nuclear spin) is determined by Nature. We use nanofabrication to explicitly design for different device physics to ensure differentiated responses that enable GCSR. We present scalable ways of designing this differentiation in MicroStressBots in Sec. 4. Recently, our group achieved independent control of multiple microrobots using differences in turning-rates, as opposed to transition voltages of the steering arms, to independently maneuver multiple robots on a planar substrate (Paprotny et al., 2012).

A similar class of control problem emerges in sensorless manipulation using vibrational surfaces (Bohringer et al., 2000; Branicky et al., 1999) or distributed manipulation (Böhringer et al., 1994; Bohringer et al., 1999; Suh et al., 1999), where global force-fields can be designed to geometrically filter parts differentiated by their geometry (similar to (Berretty et al., 1998)), implementing either GCSR or Ensemble Control. It can be argued however whether a global force-field applied through hundreds or even thousands (Böhringer et al., 1994) of actuators can be considered a single control signal.

2.3 Microassembly

Microassembly has been introduced in the literature as an application for cooperating microrobotic systems (Popa & Stephanou, 2004). Microassembly can be performed using pick-and-place macroscale robotic micromanipula-tors (Dechev et al., 2004; Skidmore et al., 2004), distributed manipulation (Böhringer et al., 1994; Bohringer et al., 1999; Suh et al., 1999), or via parallel but less controllable self-assembly (Whitesides & Grzybowski, 2002; Rothemund, 2000; Liang et al., 2004)

Pick-and place assembly using macroscale robotic micromanipulators has been shown to successfully assemble several microstructures (Skidmore et al., 2004). However, the actuator mass is several orders of magnitude larger than its payload. This restricts the speed of this assembly method compared with microactuator-driven microassembly described below. (For example, we have demonstrated virtually instantaneous acceleration and de-acceleration of our USDAs to speeds up to 1.8 mm/sec, due to their very low mass (Donald et al., 2003).) Magnetically-actuated microrobots have recently been shown to be able to manipulate microscale objects (Frutiger et al., 2010; Pawashe et al., 2011).

In distributed manipulation (Böhringer et al., 1994; Bohringer et al., 1999; Suh et al., 1999), arrays of micro-manipulators are embedded into a surface (so called active surface) and can be used independently to translate, orient and assembled parts. Virtual force fields, created by synchronous motion of actuators in an active surface, have been shown to orient and filter parts through sensorless manipulation. Optical manipulation using a pixelized light source (Ohta et al., 2007) is another form of distributed manipulation.

Self-assembly (SA) is an increasingly viable autonomous method for assembling micro- and nanoscale components (Gracias et al., 2000). In some instances, the complexity of the generated shapes is quite astounding (see (Rothemund, 2006)), however, unless one uses DNA with base-pair sequence matching (Seeman, 1998), the shape complexity is often inversely proportional to the yield due to difficulties in designing unique local energy minima for the various docking configurations. An interesting direction in SA is Directed Self-Assembly (DSA), which implies some control over the assembly process. This is done either through the use of optimization and distribution of docking sites (for example (Saeedi et al., 2006)) (also called templated self-assembly), or by actively changing the docking site affinity (Onoe et al., 2004). The latter approach has great potential for generation of complex shapes (Klavins, 2004) with high yield.

In this work, we implement microassembly by maneuvering multiple microrobots to dock together and form larger structures. This type of microassembly was first presented by our group in (Donald et al., 2008b), however related multi-microrobotic assembly using magnetically-actuated robots was recently presented in (Diller et al., 2011). The magnetic assembly (and disassembly) described in (Diller et al., 2011) relies on selective electrostatic clamping using the substrate as a temporary anchor to achieve independent control. A related multi-robotic control mechanism that does not rely on a specialized substrate was recently published in (Floyd et al., 2011). Similar to the mechanism presented in (Donald et al., 2008b) and this work, (Floyd et al., 2011) relies on engineered differences in the response of the microrobots to the same global control signal. The robots in (Floyd et al., 2011) are differentiated by how their entire chassis interact with the global magnetic field. However, large design differences were necessary to ensure significantly different motion to enable independent control. The need for such large design differences limited the number of robots that could be controlled simultaneously. In contrast, the robots presented in (Donald et al., 2008b) and this work decode a sequence of control pulses using the one-bit onboard memory stemming from the steering-arm hysteresis gap. As we show below, this allows us to greatly increase the number of simultaneously-controllable MicroStressBots. Furthermore, the mobility of the microcomponents in our assembly scheme is not restricted by discretization of the substrate, as in the case of (Diller et al., 2011) or distributed manipulation. In contrast with SA, our microassembly scheme relies on intersecting trajectories, rather than component affinity and energy minimization to promote structure aggregation. This permits the control of defect formation through collision avoidance and non-intersecting trajectories. However, while our robots move in parallel, our assembly algorithms use a largely sequential order, and lack the vast parallelism present in biological SA systems.

3 Stress-engineered MEMS Microrobot (MicroStressBot)

All the control and assembly algorithms presented in this paper are implemented using groups of parallel-actuated stress-engineered MEMS micrororobots (MicroStressBots) (Donald et al., 2006). A MicroStressBot has two actuated internal degrees of freedom (DOF); an untethered scratch-drive actuator (USDA) (Donald et al., 2003) that provides forward locomotion, and a steering-arm actuator that determines whether the robot moves in a straight-line or turns. The steering-arm consists of a cantilever beam with a circular pad and a .75–1.2 μm deep dimple. The cantilever beam is curved out-of-plane using a stress-engineering process (Donald et al., 2006), which determines the deflection of the steering arm. The microrobot operates on fields of zirconia-insulated interdigitated electrodes. When a voltage is applied across these electrodes, the electrodes and the conductive microrobot chassis form a capacitive circuit inducing an electric potential on the microrobot body. This voltage (waveform) is varied over time to provide power to the untethered scratch-drive actuator and to control the state of the steering-arm. This waveform is called the control waveform. Figure 2 illustrates one cycle of the control waveform. The waveform is divided into a control cycle, containing j ≥ 1 control pulses (Va, j), that sets the state of the steering-arm actuator, and a power-delivery cycle that provides power to the scratch-drive actuator. USDA. The power delivery cycle consisting of 250 stepping pulses, alternating between a maximum (Vs) and a minimum (Vb) (the subscripts a, s, and b, are abbreviations of arm, step, and bias). In order for the USDA to actuate reliably, VsV flx and VbVrel, where V flx is the minimum voltage at which the backplate of the USDA obtains enough curvature to produce a forward step, while Vrel is the maximum voltage at which that curvature is sufficiently relieved to generate forward motion. The relationship between V flx and Vrel is described in more detail in (Akiyama & Shono, 1993; Linderman & Bright, 2000; Donald et al., 2006).

Figure 2
The control and power delivery waveform for a single MicroStressBot.

Similar to an electrostatic cantilever-beam (Nathanson et al., 1967), the steering-arm of the MicroStressBot has two distinct voltage-levels at which it abruptly changes state. These voltages are called the transition voltages. While the state of the microrobot includes the state of the steering arm and the state of the scratch-drive actuator, for the purpose of this section it suffices to consider only the states of the steering-arm actuators, which we call the robots’ hysteresis states. Consequently, a single actuated MicroStressBot can be in one of only two hysteresis states; the steering-arm can be either raised (0) or lowered (1). When the voltage supplied to the robot reaches the steering-arm’s snap-down transition voltage (Vd), the arm is pulled into contact with the substrate. When the voltage is reduced past the release transition voltage (Vu), the arm is released from the substrate. Vu is less than Vd because the electrostatic attraction is a strongly non-linear function of the gap between the steering arm and the substrate. The transition voltages are a function of the design of the individual steering-arm actuators: for example, a smaller air gap or larger steering pads primarily reduce Vd and Vu, respectively. Microrobots with identical steering-arms are classified as belonging to the same microrobot species. The difference between the snap-down and release voltage of a steering-arm is called the hysteresis gap.

The microrobot moves forward during the power-delivery cycle of the control waveform (Akiyama & Shono, 1993) with an average step size of 10 – 20 nm, such that many stepping pulses are required to produce micron-scale displacement. We have successfully operated our microrobots using stepping frequencies as high as 20 kHz, achieving speeds of up to 200 μm/s, while the USDAs in isolation have been shown to travel at stepping frequencies of up to 100 kHz with speeds up to 1.8 mm/s. Applying the power-delivery cycle while the arm is lowered causes the microrobot to turn. As the robot turns, a portion of the arm remains in flat contact with the substrate, providing a resistive force. This force generates a moment on the scratch-drive actuator, causing it to turn. The robot turns around a fixed radius of curvature, which is defined by the design of the steering-arm actuator and the voltage Vb of the stepping cycle. If the power-delivery cycles are applied while the arm is raised, the robot moves in straight line. Both turning and straight line trajectories are shown in Fig. 3.

Figure 3
Kinematics of a MicroStressBot.

For a single robot, a control waveform defined by a specific voltage triple (Va, Vb, Vs) (see Fig. 2) is called a control primitive. A control primitive with a Va > Vd, and Vb > Vu keeps the steering-arm lowered through the stepping cycle, causing the robot to turn. A control primitive with Va < Vd or Vb < Vu causes the robot to move in straight-line motion. For simultaneous control of n microrobots, the control primitive may contain up to 2n control pulses, and has the form (Va, 1, ···,Va,k, Vb, Vstep), where k [set membership] {1, ···, 2n}. In both cases, the control primitive defines the control and power-delivery waveform that is supplied to the robots through the operating environment.

The trajectory of a microrobot is a concatenation of straight-line motion and turning, and is generated through the execution of a control sequence. A control sequence is a sequence of several control primitives, where each primitive is applied for a specific length of time. For example, a control sequence S consisting of two control primitives, P1 and P2, applied for 10 and 20 seconds respectively, is written as S = (P1(10), P2(20)). We refer to a nominal (error-free) microrobot trajectory as the ideal trajectory of a control sequence executed in the absence of the control error. We summarize the technical terms introduced in this section, and Section 4, in Table 1. The design types (NHG,STRING, ESat, and SESat) may appear unfounded. However, we will show that they have provable properties that are crucial for multirobot assembly and are also possible to implement in hardware within the constraints of the microfabrication process. Hence, these design classes are a melding of theoretical and physical constraints.

Table 1
Summary of technical terms.

The kinematics of our robot is illustrated on Fig. 3. The configuration of the robot is given by the vector q = (x, y, θ)T in configuration space (C-space (LaValle, 2006)). The configuration of the robot is measured at the point Zo in the middle of its bushing. The velocity of the robot is q.=v(sinθ,cosθ,ahr)T, where h [set membership] {−1, 1} and denotes whether the steering arm is on the right or the left side, v is the velocity of the scratch-drive actuator, r is the turning radius and a [set membership] {0, 1} is the state of the steering arm (0 = up, 1 = down). The velocity v can be varied by changing the frequency of the stepping pulses, however for the remainder of this paper we will consider v to be a positive constant (positive because the robot can not back up). We define a distance r* as the turning radius of the most extreme point on the microrobot chassis, i.e., the point farthest away from the center of curvature, Zc, for the robots turning motion. The radius r* allows us to define the necessary separation between the orbiting microrobots (see Appendix Sec. C.4). Our robot is not small-time locally controllable: it can only turn one way ( θ.=ahr) and cannot back up (v > 0).

4 STRING Theory: Control Signal Engineering

The control strategies presented in Sec. 6 depend on a set of control primitives that couple the motion of the microrobots in a specific way (later called STRING control primitives). In this section (Sec. 4), we show how to design such set of control primitives given the snap-down and release voltages of a group of MicroStressBots. First, we provide formulas for generating STRING control primitives. Next, we prove two theoretical results; (1) we show that STRING control primitives can be created for any non-degenerate (distinct snap-down and release voltages) system of microrobots, and (2) we show that in certain cases the control signal can be engineered to achieve sub-linear ( O(n)) complexity of independent voltage levels required to control n micrororobts. For example, this last result permits the control of 100 microrobots with only 20 control voltage levels, compared to 200 voltage levels from our previously proposed algorithm ((Donald et al., 2006)) that used Nested Hysteresis-gaps (NHG).

The snap-down and release voltages are a function of the robots design parameters, such as width of the steering arms, diameter of the steering pads, or the height of the steering arm dimple. In Secs. 4.1 and 4.2 we assume that these parameters are set, and the corresponding snap-down and release voltages are given. We refer the interested reader to Appendix F for a list of design parameters used for the systems of microrobots presented in this paper. In Sec. 4.3 we show how to maximize the number of independently controllable MicroStressBots with our control strategies by designing the robots snap-down and release voltages in a specific way.

4.1 The Control Voltage Bandwidth

We now define the concepts of the control voltage bandwidth and the control voltage bandwidth requirement, which will help us to discuss the scalability of our microassembly scheme. For a given system of n microrobots, let Vd,i and Vu,i denote the snap-down and release voltages of microrobot i. Let VΩ be the breakdown voltage of the operating environment. For the robots to function properly, their snap-down and release voltages must conform to the following constraints:

  1. Vd,i < VΩ : snap-down voltage cannot exceed the break-down voltage of the operating environment.
  2. Vd,i > Vu,i : dictated by the electromechanics of cantilever beams.
  3. Vrel > Vu,i for all i : ensures the microrobot can receive power during all the hysteresis states. Recall that Vrel is the voltage at which the backplate of the USDA actuator relaxes, allowing the USDA to take a step forward during the power delivery cycle. If VrelVu,i, the USDA will not be able to actuate the microrobot during turning.
  4. VflxminiVd,i: Ensures that the USDA flexes sufficiently to produce forward motion during the power i delivery cycle. Recall that V flx is the minimum voltage at which the backplate of the USDA generates enough flexure to produce a forward step.

We define the control voltage bandwidth ξ; of a MicroStressBot system as the number of independent electromechanically-addressable transition voltage levels of the control signal. ξ; depends on four parameters: 1) the break-down voltage of the operating environment, VΩ, 2) the inherent variability of the power coupling between the robot and the underlying substrate, 3) the precision of the fabrication process, and 4) the minimum range of voltages required to reliably power the USDA, VS DA (VS DA = V flxVrel).

The variability in the power coupling causes deviations in the potential induced between the steering arm and the substrate, while inaccuracies in the fabrication process cause deviations in ms. Let δv be the maximum deviation of the transition voltage manifested during the microrobot operation, determined by these two parameters. We define two transition voltages to be significantly independent if they are separated by at least 2δv. Note that, although in general, VS DA = V flxVrel, it is possible for the stepping pulse to overlap with the lowest snap-down voltage, miniZnVd,i where Zn = {1, ···, n}. Consequently, we define VSDA as the additional control voltage gap required by the power delivery cycle to ensure reliable actuation of the USDA, as follows:

VSDA={0,ifVflx-Vrel2δv,Vflx-Vrel-2δv,otherwise.
(1)

The control voltage bandwidth of a microrobot system is then ξ=VΩ-VSDA2δv, assuming V flx and Vrel can vary by at least 2δv (otherwise ξ=Vrel2δv+VΩ-Vflx2δy). It tells us how many independent voltage levels are available for control. How much of ξ; need actually be used for controlling our microrobots is related to the number of accessible hysteresis states. We define the control voltage bandwidth requirement, ξ;n, of a n-microrobot system as the number of significantly-independent transition voltage levels necessary to implement microassembly under the control strategy presented in Sec. 6. Clearly, in order to be able to achieve microassembly, it must hold that ξ;nξ;. In general, a microrobotic system with fewer accessible hysteresis states has a lower control voltage bandwidth requirement. More specifically, the accessibility of the hysteresis states depends on the relation between the hysteresis gaps of the individual robots.

For example, consider a system of two microrobots, D1 and D2, with steering arms that have Nested Hysteresis Gaps (NHG) ((Donald et al., 2006)). Fig. 4(a) shows the relation between the transition voltages for such system. The snap-down and release voltages are shown as circles and rectangles, respectively. Each transition voltage corresponds to an independent voltage level of the control signal (labeled Vα, Vβ, Vγ, and Vδ). Fig. 4(b) shows the programming cycles for the four control primitives that access the four hysteresis states (11), (10), (01) and (00) (we assume Vb = Vrel). More generally, we classify the system of n steering arms, sorted according to ascending Vd,i, as having NHG when (Vd,i + 2δv < Vd,j) and (Vu,i − 2δv > Vu,j), for all i < j. NHG systems can access all 2n hysteresis states. However each device requires two unique control voltage levels, and so the control voltage bandwidth requirement of this system is ξ;n = 2n.

Figure 4
(a) Transition voltages for a system of two microrobots with nested hysteresis gaps (NHG), and (b) programming cycles for the four control primitives that access all system hysteresis states.

4.2 The STRING Control Matrix

We now describe how to generate control primitives to be used by the control strategies presented in Sec. 6 to implement microassembly using fewer control voltage levels than in the case of NHG. The recursive formulas (Eq. (2), p. 11, and Eq. (4), p. 12) generate a set of such control primitives given a set of snap-down and release voltages.

Assume a system of m microrobots. The application of a control primitive will cause either straight-line or turning motion in each of the m robots. A mapping between the control primitives and the motion of the individual microrobots is expressed through a control matrix M of size n × m, where each entry Mi,j contains the hysteresis state of microrobot j during the application of the control primitive i. The control matrix corresponds to the coupling of the microrobot motion as a function of the control signal, providing a layer of abstraction between the electromechanical functionality of the steering-arm with the motion of the individual devices. An example of the control matrix is shown in Eq. (3) on p. 12. The control strategies presented in Sec. 6 require the control matrix to be structured such that the robots progressively start turning as the control primitives with higher index i are applied.

NHG is sufficient but not necessary, to control multiple devices during assembly. Consider a two-robot system where the hysteresis gaps of the robots are not nested, as shown in Fig. 5(a). In this particular system, only three hysteresis states are electromechanically accessible. The programming cycles that access all three hysteresis states ((00), (10) and (11)) are shown in Fig. 5(b). Control state (01) can not be accessed, because pulling down the steering-arm of D2 also pulls down the steering-arm of D1, and the steering-arm of D1 can not be released without also releasing the arm of D2.

Figure 5
(a) Transition voltages for a system of two microrobots with non-nesting hysteresis gaps, and (b) programming cycles for the control primitives that access three of the hysteresis states for the system.

It is convenient for us to define a lexicographic sorting of the robots, using two keys. In general, an n-microrobotic system, primarily sorted according to ascending values of Vd, and secondarily sorted according to ascending values of Vu, has non-nested hysteresis gaps if (Vd,iVd,j) and (Vu,iVu,j), for all i < j. However, in the case when Vd,jVd,i < 2δv and Vu,jVu,i < 2δv, the behavior of robots i and j is indistinguishable, and such two devices cannot be controlled independently. We call such two robots a degenerate pair. Let a STRIctly Non-nested hysteresis Gaps (STRING) system be a non-nested hysteresis gap system with no degenerate pairs of devices.

Lemma 4.1

An n-robot STRING system has exactly n +1 accessible hysteresis states.

The proof for Lemma 4.1 is provided in Appendix A. We now construct the control primitives and corresponding control matrix that can access the n + 1 hysteresis states of a n-robot STRING system. The ordering of the robots is determined by the transition voltages of the steering arms, i.e., the robots must be primarily sorted according to increasing order of Vu,i and secondarily sorted according to increasing order of Vd,i. We construct the control primitive P j such that it snaps down the arms of devices Di for ij, and releases the arms of devices Di for all i > j. P j is defined by a control cycle containing two control pulses, P j = (Va,1, Va,2), assume Vb = Vrel and Vs = V flxVd,1. Consider the STRING system shown on Fig. 6, where Vα, ···, Vε represent significantly independent control voltage levels. We define P j as:

Pj={(V0,V0),ifj=0;(Vd,j,Vu,j+),ifjZn-1;(Vd,n,Vd,n),ifj=n,
(2)

where Vu,j+ = Vu,j + 2δv. In practice, Vu,j+ is the next significantly independent release voltage above Vu,j. Also, note that in order for Vd,j to cause reliable snap down, it must be δv above the designed (nominal) Vd,j level. Correspondingly, Vu,j must be δv below the designed (nominal) Vu,j level to ensure reliable steering arm release.

Figure 6
Construction of a STRING control matrix.

The first control pulse (Va,1) snaps down the steering arms of all the devices Di, i [set membership] Z j, as well as any devices Dk, k > j with Vd,k = Vd,j. The second control pulse (Va,2) releases all the devices Dk, k > j that were snapped down by the first control pulse, because in the case when Vd,k = Vd,j, it must hold that Vu,j < Vu,j+Vu,k. An example control cycle is also shown in Fig. 6.

The n +1 control primitives generated by P j form a (n + 1) × n control matrix M. An example of such control matrix for four devices is:

M=(00001000110011101111).
(3)

We refer to M as the STRING control matrix, the n +1 control primitives contained in M as the STRING control primitives, and the n +1 hysteresis states accessible using these control primitives as the STRING hysteresis states. Note that because adding a new control state to a STRING system requires the addition of another independent voltage level (per Lemma 4.1), the control bandwidth requirement for a STRING system is ξ;n = n + 1.

4.3 Beyond STRING: Sublinear Reduction of ξ;n through SESat

The reduction of the control bandwidth requirements from ξ;n = 2n (NHG) to ξ;n = n + 1 (STRING) enabled our implementation of microassembly using a group of four microrobots. We can however further reduce ξ;n, assuming we can freely design the snap-down and release voltages around the constraints presented in Sec. 4.1.

Lemma 4.2

Any stress-engineered n-microrobotic system with no degenerate pairs of robots can be sorted such that all n +1 STRING hysteresis states are accessible.

For any n microrobots sorted primarily according to increasing release voltage Vu,i and secondarily sorted according to increasing snap-down voltage Vd,i, the n + 1 STRING control primitives can be generated using the following recursive formula:

Pj={(V0,V0),ifj=0;(Pj-1,Vd,j,Vu,j+),ifjZn.
(4)

The complete proof of Lemma 4.2 is provided in Appendix B.

Theorem 4.3

A system of n STRING microrobots contains the minimum number (n + 1) of electromechanically accessible hysteresis states of any stress-engineered microrobot system without degeneracy.

Proof

Per Lemma 4.1; an n-microrobot STRING system has exactly n + 1 accessible hysteresis states, and by Lemma 4.2, any n stress-engineered microrobotic system without degenerate pairs of robots contains at least n +1 hysteresis states.

Theorem 4.4

An algorithm that can plan the motion (i.e. finds the control sequence S) for a STRING system can be applied to plan the motion for any non-degenerate system of stress-engineered microrobots.

Proof

A consequence of Lemma 4.2; a STRING control matrix can be constructed for any n-robot stress-engineered microrobotic system.

Theorem 4.4 allows us to further reduce the control bandwidth requirements (ξ;n) for a microrobotic implementing microassembly using the control strategies described in Sec. 6. The control voltage bandwidth requirement for a microrobot system with k independent snap-down voltage levels and [ell] independent release voltage levels is ξ;n = k + [ell]. In an electromechanically saturated (ESat) system (ESat contains all permutations of Vd,j and Vu,j, see Appendix Sec. B for a complete definition), the number of non-degenerate microrobots, is n = k[ell]. It follows that n is maximized when [ell] = k = ξ;n/2, and n=ξn2/4. We call such system symmetric electromechanically saturated, or SESat. As a consequence, the control bandwidth requirement for an ESat system under our control strategy is ξn2n, but it is merely ξn=2n for an SESat system. As a consequence of Theorem 4.4 we can control SESat and ESat systems using a STRING control matrix and control algorithms that we will discuss in detail in Sec. 6, greatly reducing the control voltage bandwidth requirement for microassembly (under control strategy from Sec. 6). It follows that an SESat system maximizes the number of simultaneously controllable microrobots.

Table 2 compares the control voltage bandwidth requirements (under the control strategy from Sec. 6) and the number of control pulses in the control cycle of the STRING control primitives for the three classes of micro-robotic systems: a) NHG, b) STRING, c) SESat. Note that although the NHG system can access all the possible hysteresis states using O(2n) control pulses, only a single control pulses per control primitive is required to access all the STRING hysteresis states. Although the SESat system has the smallest control voltage bandwidth requirements under our control strategy, the control cycle of each of its control primitives requires O(2n) control pulses. In contrast, each control cycle in every control primitive for a STRING system requires only two control pulses. Clearly, which type of system is preferable is dictated by the number of robots, n, and the available control voltage bandwidth. Table 2 also shows the number of robots that can be independently controlled using NHG, STRING, and SESat, respectively, under the assumption that the transition voltages require a minimum separation 2δv = 10 V, and assuming VΩVS DA = 200 V, which is consistent with our experimental setup. Note that under these conditions SESat in principle enables the control of 100 robots, 10 times more microrobots than can be simultaneously controlled using NHG.

Table 2
Comparison of the control voltage bandwith requirements, ξ;n, and the number of control pulses in the STRING control primitives of n-robot NHG, STRING and SESat systems.

Under the control strategies presented in Section 6, the time and physical space (area of assembly) complexity of both NHG, STRING, and SESat are similar, because they are all based on sequential ordering (two robots in parallel, followed by one robot at a time) and consequently sequential assembly of a goal shape An external file that holds a picture, illustration, etc.
Object name is nihms435733ig1.jpg. This sequential motion is promoted by the structure of the STRING control matrix, but is also important to enable collision avoidance and error correction during during the assembly process. However, the ability of an NHG system to access all of its 2n hysteresis states provides a potential advantage compared to both STRING and SESat. Recall that for both STRING and SESat, the index i of the microrobot Di, and correspondingly the order in which the robots are maneuvered towards the goal shape, is unique and predetermined by the structure of the STRING control matrix. Because an NHG system can access all of its hysteresis states, one can generate a STRING control matrix with arbitrary bijection to the set of all n microrobots. This allows the robots to assemble the target shape in arbitrary order, enabling any robot to occupy any configuration within the goal-shape An external file that holds a picture, illustration, etc.
Object name is nihms435733ig1.jpg. This may be of great advantage if the robots have the same shape (chassis) but are non-homogenous, such as carrying different payloads.

Furthermore, the merely sub-linear control voltage bandwidth requirement of a SESat system requires that the microrobot USDAs can be fabricated such that VS DA/2 + Vrel = VΩ/2, or that the steering-arm actuators can be fabricated such that the number of significantly independent snap-down and release voltages is equal.

5 The Assembly Plan

An assembly plan (de Mello & Sanderson, 1990) contains an ordering of the assembly process, specifying precedence constraints between the assembling parts (certain parts must be assembled before others). In this section, we describe a simple method to generate an assembly plan for our microrobots given a desired target shape by enumerating the accessible (i.e., stable) shapes. Note, that the order of the assembly is set by the STRING control matrix. Consequently, the target configuration for the robots will contain enough information to represent the assembly plan.

For simplicity of exposition, we assume that all the robots are identical, and do not consider collisions between the out-of-plane curved steering arms. The input to the algorithm is the target shape, An external file that holds a picture, illustration, etc.
Object name is nihms435733ig1.jpg, represented as a list of robot configurations, An external file that holds a picture, illustration, etc.
Object name is nihms435733ig1.jpg = (g1, g2, ···, gP), where gi represents the configuration of ith robot within the overall goal-shape An external file that holds a picture, illustration, etc.
Object name is nihms435733ig1.jpg. The output of the algorithm is the goal configuration g = (g1, g2, ···, gn), which contains the individual goal configurations of the n robots composing An external file that holds a picture, illustration, etc.
Object name is nihms435733ig1.jpg, sorted according to the order of motion imposed by the structure of the STRING control matrix.

The goal-shape is progressively assembled using compliant-stable structures. A compliant-stable structure is a structure where the forces and moments generated by all the microrobots and friction sum to zero. Consequently, since MEMS systems such as ours are quasi-static, compliant-stable structures do not change their configuration while USDAs are powered, allowing them to be used as building-blocks to form larger assemblies.

We calculate the assembly sequence for a target shape An external file that holds a picture, illustration, etc.
Object name is nihms435733ig1.jpg by enumerating all the intermediate stable structures leading up to An external file that holds a picture, illustration, etc.
Object name is nihms435733ig1.jpg. Our stability model dictates there exists only one feasible structure for n = 2, An external file that holds a picture, illustration, etc.
Object name is nihms435733ig3.jpg, called the initial seed shape. Consequently, stable structures composed of three robots (n = 3) are generated by adding a single microrobot to each of the restricted docking locations (Fig. 7(b) and (c)) of An external file that holds a picture, illustration, etc.
Object name is nihms435733ig3.jpg, testing for stability, and discarding the unstable structures. This procedure is repeated generating stable structures containing progressively more robots, until An external file that holds a picture, illustration, etc.
Object name is nihms435733ig1.jpg is generated. The resulting tree-structure, as shown on Fig. 7.c for n = 4, has An external file that holds a picture, illustration, etc.
Object name is nihms435733ig3.jpg at the root, while each branch represents the successive addition of a single robot at a permitted docking location. An external file that holds a picture, illustration, etc.
Object name is nihms435733ig1.jpg is located in one of the leaves of the generated tree-structure. A path from An external file that holds a picture, illustration, etc.
Object name is nihms435733ig1.jpg to An external file that holds a picture, illustration, etc.
Object name is nihms435733ig3.jpg (the root of the tree) generates an assembly plan for An external file that holds a picture, illustration, etc.
Object name is nihms435733ig1.jpg (For example see An external file that holds a picture, illustration, etc.
Object name is nihms435733ig2.jpg in Fig. 7.c).

Figure 7
(a,b) docking locations for the MicroStressBot. Restricted docking locations simplify the calculation of the stable shapes by ensuring that all torques sum to zero. Robots can dock either aligned along the long edge of a single device (a), or aligned ...

Note that the set of goal configurations (planar shapes) which can be reached (assembled) by our system is also restricted by the kinematics of our robots (the robots only move forward) and the reliance on force closure to maintain the position of the assembling shape. To relax both the force closure and unidirectional motion restrictions would require a redesign of our microrobots, an important direction for future work. (For instance, to relax the force closure requirement, our robots would require the ability to stop without pushing against other objects.)

6 Control Strategies for Microassembly

In this section (Sec. 6) we describe the control strategies that allow us to implement microassembly using groups of MicroStressBots. In our implementation, a control strategy consists of a control algorithm used to maneuver the robot to its goal, and recognizable goal and failure regions of C-space (Donald, 1987). (Recognizable region are regions where the robots’ entry can be reliably measured.) All our control strategies were implemented using an iterative replanting control algorithm, RePlan, described in Sec. 6.1.

Two important features of our control strategies enable the control of our microrobots to assemble into planar shapes. First, we use self-aligning compliance (a form of pairwise self-assembly) to facilitate the generation of the initial seed shape An external file that holds a picture, illustration, etc.
Object name is nihms435733ig3.jpg, simplifying simultaneous control of two robots to assemble An external file that holds a picture, illustration, etc.
Object name is nihms435733ig3.jpg. Second, we use the structure of the STRING control matrix M to reduce the parallel motion of n robots to parallel motion of two robots, followed by sequential motion for single robots (while awaiting their turn, the robots are confined to circular trajectories called closed-loop orbits). Sequential motion towards the goal permits the correction of control error in the trajectories of only one or two robots at a time, which simplifies our solution to the accumulating control error.

We use planning under uncertainty (Lozano-Perez et al. (1984)), Error Detection and Recovery (EDR) (Donald (1987)), and compliance to define intermediate goal regions Ai, extended final goal regions Gi, and to construct global and local control strategies (σG and σL) that guarantee our robots to enter Ai and Gi (A brief review of the EDR terminology is presented in Appendix D). The resulting two-step EDR control strategy, combined with compliance, allows our robots to achieve high docking accuracy, as demonstrated by the experiments described in Sec. 7.2. In particular, Figs. 11, ,14,14, and and1616 highlight how the concepts presented in the sections below were used in our experiments. In these figures, the theoretical model is illustrated as an annotated line drawing on the left, while on the right we show optical micrographs (of actual microrobots) that were captured during the corresponding physical experiments performed to validate the theory.

Figure 11
Self-aligning compliance of An external file that holds a picture, illustration, etc.
Object name is nihms435733ig3.jpg: (a) At initial contact, the robots are misaligned by an offset ws and a relative rotation α. Under subsequent application of straight-line motion control primitives (this occurs automatically when actuating robot ...
Figure 14
Compliant interaction of a single robot docking with the assembling shape: (a) The mechanism of compliance between a docking robot and the assembling shape. The robot Di approaches the assembling shape along a local trajectory defined by σL (i). ...
Figure 16
Local control strategy σL: (a) The trajectory of the local control strategy projected onto the x-y plane of the C-space. A nominal (error-free) interpolated turning trajectory S L (ρa) is generated by interleaving straight line motion ...

To model the inherent uncertainty in the pose and control of the robots described using the EDR framework, we define Ri to be the starting region for robot i, typically a ball around the nominal initial configuration ri. Correspondingly, let Gi be the region of goal configurations for robot i, typically an open set around the nominal goal region gi. Our objective is to maneuver the robots from their start region R = R1 × R2 × ··· × Rn, to their goal region G = G1 × G2 × ··· × Gn.

The process of assembling the target goal shape An external file that holds a picture, illustration, etc.
Object name is nihms435733ig1.jpg described using the EDR framework is shown in Fig. 8. The n STRING microrobots are labeled Di, where i [set membership] {1,…, n} according to the convention described in Sec. 4. Initially, two robots (Dn and Dn−1) are simultaneously maneuvered to form the initial seed shape An external file that holds a picture, illustration, etc.
Object name is nihms435733ig3.jpg through self-aligning compliance. Following the assembly of An external file that holds a picture, illustration, etc.
Object name is nihms435733ig3.jpg, single robots (Di where i [set membership] {n − 2,…, 1}) are maneuvered sequentially, progressively assembling the target shape An external file that holds a picture, illustration, etc.
Object name is nihms435733ig1.jpg. A summary of the symbols used throughout this section is presented in Table 3.

Figure 8
Flow chart illustrating the process of assembling the target goal shape An external file that holds a picture, illustration, etc.
Object name is nihms435733ig1.jpg using EDR framework. The shape is assembled by first generating the initial seed shape, and then by iteratively implementing a two-step EDR control strategy to sequentially maneuver ...
Table 3
Summary of symbols used in this section.

6.1 Implementing Control Strategies: Iterative Re-planning Control Algorithm (RePlan)

All our control strategies were implemented using an iterative re-planning control algorithm (RePlan), which is illustrated by Algorithm 1. RePlan consists of planning and execution stages that are repeated iteratively until the robots reach a goal or a failure region. In the planning stage, RePlan constructs a nominal trajectory, described through control sequence S, from a current (q [set membership] Q) to a goal (g [set membership] G) configuration using some planning function An external file that holds a picture, illustration, etc.
Object name is nihms435733ig4.jpg(q,g), which depends on the control strategy that is being implemented. Following the planning stage, the sequence S is partially executed by actuating the robots for a duration of tx seconds (in our experiments tx varied between 1 and 20 seconds). After the execution, the new position of the robots (q) is registered (using an optical microscope and a digital camera), a new sequence S is generated, and the cycle is repeated. This loop continues until the robots enter the goal region G or the failure region H, where the algorithm exits.

Algorithm 1
RePLan:Iterative Re-planning Control Algorithm

6.2 Generating the Initial Seed Shape An external file that holds a picture, illustration, etc.
Object name is nihms435733ig3.jpg

The initial seed shape An external file that holds a picture, illustration, etc.
Object name is nihms435733ig3.jpg provides the initial structure to which the consecutive robots dock to progressively assemble the target shape An external file that holds a picture, illustration, etc.
Object name is nihms435733ig1.jpg, and provides a nucleus for further assembly. The seed shape An external file that holds a picture, illustration, etc.
Object name is nihms435733ig3.jpg is generated using force closure, and by necessity requires the simultaneous control of two microrobots. Using the order imposed by the STRING control matrix, robots Dn and Dn−1 generate An external file that holds a picture, illustration, etc.
Object name is nihms435733ig3.jpg, while Di, i [set membership] {1,…, n − 2} are confined to following closed-loop orbits. We further restrict An external file that holds a picture, illustration, etc.
Object name is nihms435733ig3.jpg to be assembled from robots with steering-arms on opposing sides, such that their steering arms remain confined to one side of the seed shape An external file that holds a picture, illustration, etc.
Object name is nihms435733ig3.jpg, minimizing interference during subsequent assembly process.

The structure of the STRING control matrix dictates that the motion of Dn and Dn−1 is coupled; Dn can only turn if Dn−1 is also turning. Although it is possible to generate an EDR control strategy for An external file that holds a picture, illustration, etc.
Object name is nihms435733ig3.jpg that takes into account this coupling, we leave this derivation for future work. Instead, we used a simplified control heuristic that directs Dn and Dn−1 towards a common intersecting point of their trajectories, and rely on self-aligning compliance to align the robots to form An external file that holds a picture, illustration, etc.
Object name is nihms435733ig3.jpg. This simplified control heuristic, which implements RePlan as a subroutine, is shown on the flow-chart shown on Fig. 9. It consists of three steps: In step one, the robots were oriented (Fig. 10.a), such that they can collide using a single Dubins trajectory in the the center region of the field. If the robots were already aligned properly at the start of the assembly, this step was omitted. In step two, the robots were maneuvered along a modified Dubins trajectory (Dubins (1957)) to make initial contact (Fig. 10.b). The initial contact may include large misalignment of the two robots, consequently in step 3, self-aligning compliance (step 3) was used to align the robots Dn and Dn−1 to generate an accurate seed shape An external file that holds a picture, illustration, etc.
Object name is nihms435733ig3.jpg.

Figure 9
A flow chart outlining the control heuristic that was used to assembles the initial seed shape An external file that holds a picture, illustration, etc.
Object name is nihms435733ig3.jpg using robots Dn and Dn−1.
Figure 10
Assembly of the seed shape An external file that holds a picture, illustration, etc.
Object name is nihms435733ig3.jpg: (a) Orienting robots Dn and Dn−1 to allow them to collide by following a single Dubins trajectory (i). (b) Maneuvering of Dn and Dn−1 towards a common point using a modified Dubins trajectory ((ii), extra ...

Self-aligning compliance is a form of pairwise self-assembly. In our experiments, self aligning compliance was assumed successful if the maximum allowed lateral misalignment ws was less than 90 μm (2/3 of the bushing width), while the relative rotational misalignment α was less than 60° (see Fig. 11.a). Fig. 11.b shows a series of optical micrographs taken from the initial contact (i), and up to completed alignment (iv) of the robots forming An external file that holds a picture, illustration, etc.
Object name is nihms435733ig3.jpg. Self-aligning compliance requires that the robots Dn and Dn−1 are actuated with their steering arms lifted (i.e. straight-line motion). The structure of the STRING matrix however ensures that the next robot (Dn−2) is controlled using primitives Pn−2 and Pn−3, which both cause straight-line motion of Dn and Dn−1. Consequently, the robots Dn and Dn−1 are self-aligned as consecutive devices are maneuvered to dock with the seed shape An external file that holds a picture, illustration, etc.
Object name is nihms435733ig3.jpg.

6.3 Two-step EDR Control Strategy for Sequential Assembly of An external file that holds a picture, illustration, etc.
Object name is nihms435733ig1.jpg

We now construct an EDR control strategy to assemble the goal shape An external file that holds a picture, illustration, etc.
Object name is nihms435733ig1.jpg by sequentially adding single robots to the seed shape An external file that holds a picture, illustration, etc.
Object name is nihms435733ig3.jpg. To simplify the construction of the control strategy, the robots will be maneuvered to goal Gi via an intermediate region Ai. A control strategy σG, called the global control strategy is used to maneuver the robots from Ri to Ai. The global control strategy lacks sufficient accuracy to complete the docking operation, but in the absence of obstacles or collisions, it can maneuver the robots anywhere within their operating environment. From region Ai, the robots are maneuvered by control strategy σL, also referred to as the local control strategy, towards the goal region Gi. This strategy permits further reduction of the control error, however unlike σG, σL requires the robots to start in a specific region (Ai) in order to guarantee that they reach Gi. Region Ai contains configurations that are provably reachable from Ri using σG, and from which the robots are guaranteed to enter the goal using σL. The complete σG-σL control strategy is called a two-step EDR control strategy, and can be represented by the following reachability diagram: RiσGAiσLGi. Fig. 12 shows the projection of the C-space onto the x-y plane with annotated regions Ri, Ai, Gi, the trajectories representing both the global and the local control strategies, the configuration of the microrobot Di at the start of σG (a), at the end of σG and start of σL (b), and at the end of σG.

Figure 12
Theory of the two-step EDR control strategy: Regions Ri, Ai, B, CB, and Gi, as well as the trajectories for σG and σL illustrated on a x-y plane projection of the C-space. The region Ai is restricted by the error bounds (i) and CB. The ...

In EDR, the failure region H signals the failure of assembly. Because our control strategies were implemented using manual registration of microrobot configurations, entry to the H region was visually recognized when the robots were stuck in a configuration that was not one of the intermediate goal shapes. In all our experiments we observed entry to H only once, resulting from an intermediate goal shape destabilizing and not the accuracy of the control strategies.

The iterative implementation of the two-step EDR control strategy on robots Di, i [set membership] {n−2,…, 1} for assembling the shapes An external file that holds a picture, illustration, etc.
Object name is nihms435733ig5.jpg, l [set membership] {2,…, k} is shown by the flow chart in Fig. 13. The individual sub-blocks of the flow-chart are described in Secs. 6.3.1 – 6.3.4 below. Because the pose of each intermediate assembly An external file that holds a picture, illustration, etc.
Object name is nihms435733ig5.jpg depends on the pose of An external file that holds a picture, illustration, etc.
Object name is nihms435733ig3.jpg), the regions Gi, Gi, Ai, and Ai are derived at the start of the motion of the individual robots Di, i [set membership] {n−2,…, 1} towards their goal.

Figure 13
Flow chart illustrating the implementation of the two-step EDR control strategy for progressive assembly of the goal shape An external file that holds a picture, illustration, etc.
Object name is nihms435733ig1.jpg using robots Di, i [set membership] {n − 2,…, 1}.

6.3.1 Expanding the Goal-regions Gi to Gi through Compliance

The goal region Gi is derived geometrically based on the configuration of the robots Dn, …, Di+1 that make up the thus far assembled shape. We use compliance between the docking robots and the assembling shape to expand the goal region Gi to an expanded goal region Gi from which the robot Di is guaranteed to enter Gi via compliant interaction with the assembling shape. The phenomenon of compliance is illustrated in Fig. 14. When a single robot docks with an assembling shape, the docking robot will successfully align the front of its scratch-drive actuator with the edge of the shape as long as its angle of approach θ [set membership] [θ0α, θ0 + α], where θ0 is the nominal docking angle, and α is the angle at which the corner will slip. In our experiments, α was found to be approximately 45°. The tolerance for angular misalignment is smaller than in the case of the self-aligning compliance (Section 6.2) because only the docking robot is free to rotate and align. This alignment enlarges the goal configuration Gi to an expanded goal configuration Gi that include all the valid (θ [set membership] [θ0α, θ0 + α]) approach angles. Because the center of rotation as the robot aligns passes through the robot’s corner, the expanded goal configuration Gi is a torus in C-space.

6.3.2 Deriving the Intermediate Goal-regions A and A*

Region A marks the change from the global control strategy σG to a local control strategy σL, and is chosen such that it lies outside the proximity space (CB) of the assembling shape, which is the region of C-space where the robots may collide with the assembling shape B under control strategy σG. CB can be defined geometrically by expanding the boundary of B in C-space by a distance d = zr* where z ≈ 1.5 in our experiments. A robot inside CB may not be able to avoid colliding with the assembling shape under σG, however a robot outside CB will be always able to avoid colliding with B under σG. Let CF be the region outside the proximity space of B, CB, CF = CCB. Simply put, a robot Di may abort and re-attempt the docking while it is under σG and outside of CB, however once robot Di is inside the CB and progressing to Gi under σL, Di is generally committed to attempt to enter Gi or enter H (fail).

We define Ai to be the intersection of CF, the strong preimage of Gi under σL, and the forward projections of Ri under σG (from Ai); formally Ai=PAi,σL(Gi)CFFσG(Ri). In addition, for Ai to be guaranteed reachable from Ri, it must hold that Ri is the subset of the preimage of Ai under σG (from Ri); formally Ri [subset or is implied by] PRi,σG (Ai). Ai must contain the forward projection of the global control strategy, σG, from Ri, FσG (Ri). Because σG is implemented using a re-planning control algorithm (RePlan), FσG (Ri) is a cylinder around a target configuration ai, Ca,i = Bra,i (ai) × [θa,iha,i, θa,i + ha,i] [subset or is implied by] IR2 × S 1, where Bra,i (ai) is a ball of radius ra,i around ai. The bounds ra,i and ha,i on FσG (Ri) are described in detail in Appendix E. In order to ensure that FσG (Ri) is completely contained by the preimage of Gi,PAi,σL(Gi), while remaining outside CB, we define the region Ai which contains all the configurations ai that are at least ra,i away from the boundaries of PAi,σL(Gi) and CB in IR2; ha,i in S 1. Note that if Ai= we can not guarantee that the robot will reach Gi. The regions Ai, Ai, CF, and FσG (Ri) projected onto the x-y plane of the C-space are illustrated on Fig. 15.

Figure 15
Regions Ai, Ai, CF, and FσG (Ri) defining the two-stage EDR control strategy to maneuver the individual robots to their expanded goal regions Gi.

During our experiment the robots were maneuvered towards a configuration aiAi manually selected to fit within PAi,σL(Gi) while sufficiently far away from the assembling shape to allow the robots to complete a full turn without risking a collision with B, i.e. allow docking to be aborted if exurban amount of control error would cause our robot to risk entering H. However, the necessity to abort docking never occurred during in our experiments.

6.3.3 The Global Control Strategy σG

The global control strategy σG is implemented using RePlan algorithm towards aiAi until robot Di entered Ai or H. In principle any global path-planning algorithms can be used as An external file that holds a picture, illustration, etc.
Object name is nihms435733ig4.jpg to generate control sequence S from Qi to Ai; we chose a Dubins trajectory (Dubins (1957)) that was manually adjusted to avoid collisions with other robots and the assembling shape. A more general trajectory planning scheme which includes a collision avoidance heuristic is presented in Appendix C.

6.3.4 The Local Control Strategy σL

The local control strategy σL allows for more precise control of the robot Di towards the expanded goal configuration Gi, however it requires Di to start in a specific region of C-space in the vicinity of the assembling shape (region A). Strategy σL uses iterative re-planning of an interpolated turning trajectory (Donald et al., 2006). Interpolated turning interleaves straight-line and curved trajectory segments to effectively vary the turning radius ri of the microrobot between ∞ (straight-line motion) and, ri (the maximum turning radius of the microrobot). The trajectory of σL follows a single arch along ri that intersects the expanded goal region Gi. As the trajectory of the robot is perturbed by the control error during microrobot motion, ri is adjusted such that it again passes through Gi. This in turn induces a change in the docking approach angle θ, however this accumulating error is later removed through compliance.

Recall the current configuration of Di as qi = (xq,i, yq,i, θq,i) and a goal configuration gi = (xg,i, yg,i, θg,i). The radius ri of a local trajectory that allows Di to reach a goal location pi=(xg,i,yg,i),piGi from qi is

ri=Δx2+Δy22(Δxcosθq,i-Δysinθq,i),
(5)

where Δx = xg,ixq,i and Δy = yg,iyq,i. Let the control sequence S L represent an interpolated trajectory of Di, as S L = (Pi,a, Pi−1,b, ···, Pi,a, Pi−1,b), where a and b denotes the length of time the control primitives Pi and Pi−1 are applied, respectively. Let ρa and ρb be the fraction of the time primitives Pi and Pi−1 are applied in the control sequence SL; ρa=aa+b and ρb=ba+b. The radius of curvature ri for the trajectory defined by S L can now be fully described by either ρa or ρb, since ρa + ρb = 1, defined as:

ri=r(1+ρaρb).
(6)

Eqs. (5) and (6), implemented in the RePlan algorithm as An external file that holds a picture, illustration, etc.
Object name is nihms435733ig4.jpg, can now be used to construct a control sequence S = S L between qi [set membership] Qi and piGi. We use the shorthand notation S L(ρa) to denote the control sequence S L defined by ρa. S L(1) denotes a straight line trajectory while S L(0) denotes turning at r′ = r.

In our experiments, the replanning interval tx in RePlan was reduced to approximately 1 second as robot Di was approaching the assembling shape. As mentioned in Sec. 6.3.1, it was simpler to track the configuration of the corner that will first make contact with the assembling shape, qi. As the robot approached the assembling shape during our experiments, qi was tracked in lieu of qi, adjusting S L to ensure that qi entered Gi. Fig. 16.b shows optical micrographs overlayed with planned local trajectories during our experiments as a robot approaches the assembling shape ((i) – (iii)). Although the control primitive pair Pi and Pi−1 can be repeated many times in S L, we discovered that repeating the pair three times yielded satisfactory results.

7 Experimental Results

The control strategies described in previous sections have been tested experimentally on groups of fabricated MicroStressBots. This section (Sec. 7) uses experimental data that has been previously reported in (Donald et al., 2008b), but describes how this data validates the algorithms above, and gives the explicit construction of the control matrices necessary to replicate the results. The control strategies in Secs. 4–6 were not described in (Donald et al., 2008b) and form the heart of this paper. However, this section (Sec. 7) is necessary to present experimental validation of the algorithms. All data and images where obtained using real robots; there are no simulation results in this paper.

7.1 STRING Microrobots

We fabricated 15 MicroStressBots classified into five microrobot species. The microrobot species are differentiated by the designs of their steering arm actuators. Two of the species (1a and 1b) form a degenerate pair, i.e. although their designs are different, the snap-down and release voltages of their steering arms can not be differentiated using the available control primitives. While this degeneracy was initially unintentional, it is interesting to observe that two robots with vastly different steering-arm designs can exhibit such a similar behavior. Because of this degeneracy, microrobot species 1a and 1b were never simultaneously used in any of our microassembly experiment. This limited the number of microrobots that could be independently controlled at any given time to four. Fig. 17 shows scanning-electron micrographs of all the five microrobot species, including the two degenerate species 1a and 1b.

Figure 17
Scanning-electron micrographs of the five microrobot species used to implement microassembly. Yellow color is used to highlight the areas of the steering-arms covered by the layer of chromium. Reprinted with permission from (Donald et al., 2008b).

The robots were fabricated such that the transition voltages of their steering arms, (Vd,i and Vu,i), were reproducibly confined to the ranges shown in Fig. 18(a). Snap-down voltages (Vd,i) are marked by circles, while the rectangles denote the release voltages (Vu,i). The ranges marked with two vertical dots signifying that the respective bound is not fixed or measured. Groups of robots from species 1a or 1b,2,3,4 form a four-robot STRING system and are independently controllable. The exact parameters of the steering-arms defining all five species are described in (Donald et al., 2008b), for reference these parameters are (with permission) reproduced in the Appendix F. The waveforms (control pulse and three power delivery pulses only) for the five control primitives P0, ···, P4 used to control the microrobots are show in Fig. 18(b). Average Va, Vb and Vs voltage levels across all species are shown. The actual voltage-levels used to control the individual groups of microrobots could vary by up to ± 10 V.

Figure 18
(a) Transition voltage ranges and (b) corresponding control primitives used to control the five microrobot species. Based on data from (Donald et al., 2008b).

Matrix Mr (Eq. (7)) contains experimentally recorded average and standard deviation of the turning radii r (in μm), across three independent stress-engineering runs for all five species, and represent the experimentally-measured behavior of the individual microrobots during the application of the control primitives. Notice that by replacing the cells representing turning behavior (r < 550 μm) with 1, and the cells representing straight-line motion (rμm) with 0 (thresholding), we obtain the STRING control matrix (Eq. (3)). The only difference between M and Mr is that the columns 1 and 2 in the thresholded Mr are linearly dependent, corresponding to degeneracy between species 1a and 1b. Consequently, a STRING control matrix will contain columns 1,3,4,5 or 2,3,4,5 corresponding to the four independently-controllable microrobot species.

Mr=(1699±348>20001570±7201457±742>2000211±43273±31650±6061549±3731720±484157±1227±8172±7>20001406±595160±4195±42195±23322±1121990±17158±7202±32210±32316±92167±14)
(7)

The reproducibility of motion during the application of the five control primitives (P0, ···, P4) is shown in Fig. 19. Each track represents a single, independent experiment. In this figure, a total of 140 tracks are shown, with 28 tracks for each of the 5 control primitives. For each of the device species, two tracks are shown for each of the robots fabricated through independent stress-engineering runs. The species 1a, 1b and 4 are left-handed, hence they turn counter-clockwise. The species 2 and 3 are right-handed, and turn clockwise. Species 3 and 4 show a slight tendency to turn in the opposite direction of their steering-arms when the arms are elevated, however in all cases the radii of curvature are larger than 550 μm.

Figure 19
Reproducibility of motion during the application of the five control primitives P0, P1, P2, P3, P4. Reprinted with permission from (Donald et al., 2008b).

7.2 Microassembly

We applied the control algorithms described in Sec. 6 to groups of four STRING microbots (1a or 1b,2,3,4), thereby assembling a total of 14 planar structures. The assembled structures belong to five types of target shapes, labeled An external file that holds a picture, illustration, etc.
Object name is nihms435733ig3.jpgAn external file that holds a picture, illustration, etc.
Object name is nihms435733ig6.jpg. Optical micrographs of microstructures for each type of target shape is shown on Fig. 20.

Figure 20
Optical micrographs of five types of target shapes assembled using our microrobots. Reprinted with permission from (Donald et al., 2008b).

The robots were operated on a 2 mm2 environment, and their position was recorded using a digital video-camera attached to an optical microscope (6.7 × objective lens). The position of the devices was measured with a precision of ± 2.1 μm. The humidity was kept below 4% RH using a continuous stream of dry nitrogen. The waveforms for the control primitives were produced using an Agilent 33120A arbitrary waveform generator, and amplified with a Trek PZD700-1 high-voltage power amplifier with a gain of 200. The duration of the individual primitives was manually controlled during the execution of the control sequence S. We considered the assembly a success as long as the assembled shape is a rigid-body transformation of the target goal shape, i.e. the rotation and position of the assembled shape on the operating environment was not important.

Table 4 shows the average match (portion of the target structure covered by the assembled shape) for the five assembled shapes, An external file that holds a picture, illustration, etc.
Object name is nihms435733ig3.jpgAn external file that holds a picture, illustration, etc.
Object name is nihms435733ig6.jpg. The assembly experiments were conducted starting from two different classes of initial configurations: An external file that holds a picture, illustration, etc.
Object name is nihms435733ig7.jpg – robots are arranged along the corners of a rectangle with sides 1 by 0.9 mm, all devices oriented along the y-axis (see Fig. 21(a) for a representative example), and An external file that holds a picture, illustration, etc.
Object name is nihms435733ig8.jpg – robots are arranged in a line with average separation of 360 μm, and with variable orientation. The initial position of the microrobots was set using batch-transfer structures called transfer-frames (Donald et al., 2008b) and microprobes. We used common geometric shapes (a line and a rectangle) to demonstrate the ability to achieve successful assembly from arbitrary different initial configurations.

Figure 21
Composite optical micrograph of experimental assembly-data using four robot Di, i [set membership] {1,…, 4} from species 1b,2,3 and 4, respectively. a: Initial configuration ( An external file that holds a picture, illustration, etc.
Object name is nihms435733ig7.jpg) of the microrobots. b: Docking of robot D3 and D4 to form An external file that holds a picture, illustration, etc.
Object name is nihms435733ig3.jpg, while robot ...
Table 4
Precision of Microassembly.

The results in Table 4 do not include completely failed assemblies. We recorded a 11% failure rate during the consecutive assembly of nine structures over the course of three assembly experiments. This reflects that the assembly of one of the nine structures failed due to the loss of stability of an intermediate structure, which we attribute to an initial unfortunate misalignment between the microrobots forming the intermediate assembly. Fig. 21 illustrates a representative assembly experiment. In this experiment, the target shape An external file that holds a picture, illustration, etc.
Object name is nihms435733ig6.jpg is generated via the assembly of An external file that holds a picture, illustration, etc.
Object name is nihms435733ig3.jpg and An external file that holds a picture, illustration, etc.
Object name is nihms435733ig11.jpg. The experiment terminated when all four microrobots were successfully incorporated in the assembled structure. A movie of this assembly experiment is freely available online at (Donald et al., 2008c,d,a).

Table 5 shows the average misalignment after docking (position only), representing the remaining control error. For the assembly of An external file that holds a picture, illustration, etc.
Object name is nihms435733ig3.jpg, we report misalignment before and after self-aligning compliance, underscoring its contribution to the reduction of the control error. In order to further test the self-aligning compliance, large (> 50 μm) misalignment was purposefully introduced to robots Dn and Dn−1 in two of the five assembly experiments. Because compliance during docking of single robots corrects only error in θ, and does not affect the position misalignment, only the misalignment after compliance was recorded during the assembly of structures An external file that holds a picture, illustration, etc.
Object name is nihms435733ig9.jpgAn external file that holds a picture, illustration, etc.
Object name is nihms435733ig6.jpg.

Table 5
Docking Accuracy.

8 Conclusions

In this work, we addressed the planning and control challenges for achieving independent microrobot control of stress-engineered MEMS microrobots (MicroStressBots) assembling into planar structures. We presented control strategies that build on robotic planning under uncertainty, EDR, and compliance, allowing us to plan for and execute the microassemby of several types of planar shapes. The experimental data, reprinted from (Donald et al., 2008b), shows that our control scheme is feasible. To our knowledge, the microassemblies described in (Donald et al., 2008b) were the first implementation of a planar multi-microrobotic system capable of coordinated self-assembly using a single, global, control signal. The present paper describes the control strategies used for these microassemblies, which were not described in (Donald et al., 2008b).

We have shown that our control scheme minimizes the control voltage bandwidth requirements of an n-microrobotic system. The sub-linear ( O(n)) control voltage bandwith requirement is a large improvement over 2n in our previously-proposed approach (Donald et al., 2006). Reducing the control voltage bandwidth requirements below the previous 2n bound was the enabling technology that allowed us to experimentally demonstrate simultaneous control of four devices.

We were able to efficiently implement error correction while controlling our robots towards the assembling shape using the STRING control matrix to reduce parallel motion of n robots to parallel motion of only two robots, followed by sequential motion of single devices. Furthermore, we used planning under uncertainty and EDR to construct control strategies to maneuver the robots to regions from which they could reliably enter the goal configuration, using compliance to correct for accumulating control error. These strategies have allowed us to demonstrate precise assembly of planar microstructures, achieving control accuracy on the order of the minimum feature size of the microfabrication process. The implementation and experiments we reported above use some heuristic steps for motion planning and EDR. However, rigorous and combinatorial versions of most of these algorithms have been reported in literature (e.g., (Donald, 1990b,a, 1993; Donald et al., 1993)). It would be interesting future work to replace the heuristic steps of our system with these provable algorithms to obtain an implementation that was end-to-end provable.

The assembly scheme presented in this work has two main limitations. First, the inability of our robots to move backwards or stop in place restricts the set of goal configurations (shapes) that can be reached (assembled) by our robots. This limitation is reflected in both our planning and control algorithms throughout this paper; our assembly scheme can only form shapes which are maintained through force closure and can be assembled by sequential addition of single microrobots to an initial seed shape formed by a pair of robots. It is easy to design a robot to stop on collision. It may be interesting future work to revisit this restriction and to investigate how our microrobots could be redesigned to stop in free space, for example a 2-armed robot could potentially stop in place by lowering both of its arms at the same time. Second, the lack of a control mechanism to correct the trajectories of orbiting robots necessitates sufficient separation to account for the drift of their closed-loop orbits due to accumulating control error. This impacts scalability, since the orbiting robots will require large separation to avoid collisions as the number of robots increases (resulting in an increasing size of their orbits). However, our data suggest that the control error during turning is much smaller than the control error during straight-line motion, limiting their drift while following closed-loop orbits.

Although not completely general, the methodology presented in this paper tackles the control of stress-engineered multi-microrobotic systems by selectively addressing the behavior of individual devices through a global common control signal. We believe the concept of GCSR will be important for controlling future multi-microrobotic systems. The ability for a subset of the robots to follow a closed-loop orbit, i.e. to remain within a limited spatial region, while other devices progress toward the goal, is an example of how fabricated differences in device physics can be exploited to complete a cooperative task (such as microassembly) in a multi-microrobotic systems. GCSR allows for control of devices with limited computational resources, which will become important as we further reduce the size of the individual robots comprising multi-microrobotic systems. GCSR has interesting biological parallels, and may be the paradigm of choice for controlling groups of future micro- and nano-robots.

Acknowledgments

This work was supported by a grant to B.R.D. from the Duke Institute of Brain Sciences, grant numbers GM-65982, GM-78031, and NS-79929 to B.R.D. from NIH, and 2000-DT-CX-K001 to B.R.D., from the Office for Domestic Preparedness, Department of Homeland Security, USA. The electron micrographs were taken at the Dartmouth Ripple Electron Microscopy Laboratory, with the help of C. P. Daghlian. We further thank D. Balkcom, C. Bailey-Kellogg, A. Lebeck, J. Reif, and K. Böhringer for their advice and suggestions.

Appendices

A Proof of Lemma 4.1

Lemma 4.1

An n-robot STRING system has exactly n +1 accessible hysteresis states

Proof

(By induction.)

Base case: a STRING system with n = 1, has two accessible hysteresis states, (0 - arm up) and (1 - arm down).

Inductive step: adding a single device, (changing the size of the system from n to n +1) extends the number of accessible control state by exactly one, provided that both the n and n +1 microrobotic systems remain STRING.

Let n micrororobots, labeled D1, ···, Dn, be a STRING system sorted according to Vu,i and Vd,i. Without loss of generality, Vd,nVd,n+1 and Vu,nVu,n+1 (If this is not the case, we can simply relabel the voltages and generate an equivalent system sorted as described above). Fig. A.1 shows the ranges for the transition voltages of Dn+1, such that the new, n + 1 robotic system retains STRING. Let Vα, ···, Vδ be significantly independent transition voltage levels, ordered such that Vδ < Vγ < Vu < Vβ < Vα < VΩ. Let Vd,n = Vα and Vu,n = Vγ. It follows that the snap-down voltage Vd,n+1 can have a value V1 in the range [Vα,VΩ], or voltage V2 = Vα. Similarly, the release voltage, Vu,n+1, can have the value V3 in the range [Vrel − 2δv, Vγ], or voltage V4 = Vγ (Vu,n+1 can not exceed Vrel − 2δv without risking that Vrel might release the steering arm during the power delivery cycle). Consequently, for the (n + 1) robot system to remain STRING, one of the following combinations of the snap-down and release voltages for Dn+1 must hold: (V1, V3), (V1, V4) and (V2, V3). We examine each case separately:

(V1, V3): Because the snap-down voltage of Dn+1 is greater than the snap-down voltage of D1 ··· Dn, Vd,n+1 > Vd,i, i [set membership] Zn where Zn = {1, ···, n}, we can only snap-down the arm of Dn+1 after we snap-down the arms of all other devices. Since the release voltage of Dn+1 is greater then the release voltage of D1, ···, Dn, Vu,n+1 > Vu,i, i [set membership] Zn, we can only release the arm of D1, ···, Dn after we have released the arm of Dn+1. Consequently, we can only change the state of Dn+1 when D1, ···, Dn are in state 1. During all other states of the system, the state of Dn+1 must remain 0. Consequently, the number of accessible hysteresis states increases by exactly one.

(V1, V4): This case is identical to (V1, V3), except that the arm of Dn is released at the same time as the arm of Dn+1. As long as Vd,n+1 > Vd,i, we can snap down the arm of Dn+1 only after all other devices D1, ···, Dn are in state 1. As a consequence, the number of accessible hysteresis states increases by one.

(V2, V3): The snap-down voltage of Dn+1 is equal to the snap-down voltage of Dn, Vd,n+1 = Vd,n. In this case, the arm of Dn+1 is snapped down at the same time as the arm of Dn. Because the release voltage of Dn+1 is greater than the release voltage of D1, ···, Dn, Vu,n+1 > Vu,i, where i [set membership] Zn, we can only release the arm of Dn (or any other devices) after we release the arm of Dn+1. As in the (V1, V3) case, the state of Dn+1 must be 0 except when D1, ···, Dn are all snapped down, then Dn+1 can be in either 0 or 1 by varying the release voltage. Consequently, the number of accessible hysteresis states increases by one.

We have now shown that adding a device to STRING system, such that the resulting system remains an STRING system, increases the number of accessible hysteresis states by exactly one. Combined with the base case (n = 1, two hysteresis states), it follows by induction that every n-robot STRING system has exactly n + 1 accessible hysteresis states.

B Proof of Lemma 4.2

Lemma 4.2

Any stress-engineered n-microrobotic system with no degenerate pairs of robots can be sorted such that all n +1 STRING hysteresis states are accessible.

Figure A.1

An external file that holds a picture, illustration, etc.
Object name is nihms435733f22.jpg

Illustration of the proof of Lemma 4.1.

Figure B.1

An external file that holds a picture, illustration, etc.
Object name is nihms435733f23.jpg

Example of an ESat system with k = 3 and l = 2.

Proof

By construction. Consider a microrobot system with k independent snap-down voltages, and [ell] independent release voltages. Assuming no degenerate pairs of devices, it follows that nk[ell]. In the case when n = k[ell], the n steering arms encode all the possible k[ell] combinations of snap-down and release voltages. We call such system for electromechanically saturated (ESat). We can enumerate the hysteresis gaps for an ESat system given both k and [ell]. Consider an ESat system, sorted primarily according to increasing release voltage Vu,i and secondarily sorted according to increasing snap-down voltage Vd,i. Fig. B.1(b) shows such a system when k = 3 and l = 2. Note that sorting ensures a monotonic increase of Vu,i with increasing index i. For such an order, there exists a recursive formula, shown in Eq. (B.1), that generates all n + 1 STRING control primitives. The control cycle for each control primitive defined by Eq. (B.1) contains a sequence of up to 2n control pulses (in contrast with 2 control pulses in Eq. (2)). Again, we assume Vb = Vrel and Vs = V flxVd,1. We construct the control primitive P j,

Pj={(V0,V0)ifj=0;(Pj-1,Vd,j,Vu,j+),ifjZn.
(B.1)

P j generates n + 1 control primitives that form a STRING control matrix, by causing devices Di (ij) to be in state 1, while robots Di (i > j) are in state 0. Consider the base case P0, where all D j, (j [set membership] Zn) is in state 0. We make the inductive argument that after application of the recursive part of P j, P j−1, all D1, ···, D j−1 robots are in control state 1. It is clear that Vd,j,Vu,j+, (j [set membership] Zn), will snap down D j. The Vu,iVd,i sorting implies that, for a device Dk, k > j, only two cases are possible with respect to its transition voltages: (a) Vd,j < Vd,k (e.g., j = 2 and k = 3 in Fig. B.1), or (b) Vu,j < Vu,j+Vu,k (e.g., j = 3 and k = 5 in Fig. B.1). It is clear that in case (a), Vd,j sets D j to state 1, while Dk, k > j is in state 0. The sorting ensures that any previously applied control primitive Pi, i < j with Va,1Vd,k (which also inadvertently snaps down the arm of Dk) must have have been followed by a control pulse Va,2Vu,k − 2δv (which would release the arm of Dk). In case (b), Vu,j+ releases any devices Dk, k > j.

Note that because the devices are sorted according to Vu,i and Vd,i, Eq. (B.1) also holds for any microrobotic system, even one that is not ESat.

C Motion Planning for Microassembly

This section is included for completeness, and describes how the structure of the STRING control matrix can be used to plan the motion of our robots (in the absence of the control error) from an initial an initial configuration r = (r1, r2, ···, rn), to a goal configuration g = (g1, g2, ···, gn). The output of the motion planning is a control sequence S which describes the trajectory of the robots from r to g using control primitives in M. The structure of M is used to reduce the parallel motion of n robots to parallel motion of two robots, followed by sequential motion for single devices. The robots that are not progressing toward the goal are confined to follow closed-loop orbits.

The motion planning presented in this section can plan the assembly of an arbitrary shape generated by assembly planning from Sec. 5 from a set of STRING robots with the center of curvature of their turning trajectories are separated by at least 3r* (A minimum requirement to enable collision avoidance, see Sec. C.4.) A simplified version of this motion planning was used in experimental validation of the control strategies presented in Sec. 7. The complete motion planning scheme is presented below for completeness.

The assembly planning is very similar to the implementation of our control scheme presented in Sec. 6. The initial stable shape An external file that holds a picture, illustration, etc.
Object name is nihms435733ig3.jpg is assembled using two robots, while the remaining n − 2 robots orbit without making progress towards the goal (Sec. C.1). Consecutive robots are then sequentially added to the assembling structure; the robots that have already docked with the assembling shape are kept in place by force closure, while the robots that are awaiting their turn to dock orbit without making progress towards the goal (Sec. C.2). The assembly of a structure composed of n robots takes place in n − 1 steps.

C.1 Step 1: Assembly of An external file that holds a picture, illustration, etc.
Object name is nihms435733ig3.jpg

An external file that holds a picture, illustration, etc.
Object name is nihms435733ig3.jpg is always assembled through the simultaneous motion of the two robots with the highest indexes, i.e. Dn and Dn−1 (the two robots that turn during the smallest number of control primitives), using primitives Pn, Pn−1 and Pn−2 only. This allows the remaining robots D1, ···, Dn−2 to follow closed-loop orbits without making progress towards the goal. The assembly of An external file that holds a picture, illustration, etc.
Object name is nihms435733ig3.jpg is divided into two stages, as shown on Figs. C.1(a) and C.1(b), respectively (for clarity, the orbits of robots Dn−2 ··· D1 are shown in Fig. C.1.c but are not depicted in Figs. C.1.a and C.1.b).

During stage 1 (Fig. C.1(a)) microrobot Dn is maneuvered from an initial configuration, rn, to an intermediate goal configuration an, using only control primitives Pn and Pn−1 (trajectory T1). The reason for maneuvering robot Dn to an rather that directly to its goal gn is that Dn will only be able to move in a straight line during stage 2. Thus during stage 1, Dn must be maneuvered to a configuration from which it can enter the goal, gn, using solely straight-line motion in stage 2 (trajectory T2). As Dn is maneuvered to the intermediate configuration an, robot Dn−1 orbits without making any progress towards the goal An external file that holds a picture, illustration, etc.
Object name is nihms435733ig3.jpg (because control primitives Pn and Pn−1 invoke only turning motion in Dn−1.) In order to calculate the length of the trajectory of robot Dn−1 in stage 2 (trajectory T3), and hence the intermediate configuration an to which we must drive robot Dn, we must know where is Dn−1 at the beginning of stage 2. To achieve this, we ensure that robot Dn−1 always orbits back to initial configuration rn1 at the end of stage 1 by adjusting the length of T1 and correspondingly the lengths of the orbit of Dn−1 using the method described in Sec. C.3. This allows us to use the initial configuration rn1 as the starting configuration for planning the trajectory of robot Dn−1 at the beginning of stage 2.

Figure C.1

An external file that holds a picture, illustration, etc.
Object name is nihms435733f24.jpg

Step 1 of microassembly; the assembly of the initial stable shape, An external file that holds a picture, illustration, etc.
Object name is nihms435733ig3.jpg using microrobots Dn and Dn−1. a: Stage 1: Dn is maneuvered to an while Dn−1 orbits. b: Stage 2: Dn−1 is maneuvered to gn1 while Dn moves in straight line to gn. c: Progressive docking of single microrobots to the assembling stable shape. D j orbits while Di is maneuvered to dock with the stable shape.

In stage 2, microrobot Dn−1 is maneuvered from rn1 to its target configuration, gn1, using only primitives Pn−1 and Pn−2 (see Fig. C.1(b)). Both these primitives are sufficient to maneuver robot Dn−1 to an arbitrary configuration, but cause only straight-line motion in Dn. However, as described above, we ensured that the intermediate configuration an is chosen such that Dn moves into its target configuration gn during its straight-line motion along T2.

Whenever the configuration of the robot swept along the trajectories T1, T2, and T3 would intersect the 2r* region swept by the orbiting robots, a collision might occur. The collision heuristic described in Sec. C.4 must be used, to ensure that the robot stays outside the orbiting region. If the length of T3 changed, T1 must be adjusted again per. Sec. C.3.

C.2 Steps 2: ···, n −1. Subsequent addition of single robots

After the assembly of An external file that holds a picture, illustration, etc.
Object name is nihms435733ig3.jpg, single robots are sequentially maneuvered towards their goal configurations while the remaining robots are either docked or orbiting along limit cycles (See Fig. C.1.c.). The structure of the control matrix M allows robot Di to be maneuvered to its target configuration gi using control primitives Pi and Pi−1, while robots D j, j < i, orbit in place. Control primitives Pi and Pi−1 normally would also cause straight-line motion in robots D j, j > i, but, since our robots are assembled in decreasing order of i, these robots are already docked and immobilized as part of an intermediate stable structure shape.

As in step 1, if the trajectory if robot Di intersects the 2r* region swept by the orbiting robots, the collision heuristic described in Sec. C.4 must be used to ensure that the robot Di stays outside the orbiting regions.

C.3 Adjusting Trajectory T1 such that Robot Dn−1 returns to rn−1

Here we modify control sequence S 1, corresponding to the trajectory T1 (Fig. C.1.a), so that the corresponding trajectory could be easily extended without changing the destination of robot Dn, and then adjusting S 1 such that robot Dn−1 orbits back to rn1. This is done in three steps, shown in Fig. C.2 a, b and c.

In step 1 (Fig. C.2.a), we insert the control primitive Pn,t2π to the middle of a straight segment in S 1, where t2π corresponds to the time it takes robot Dn (and Dn−1) to turn 2π, and Pn,t2πis control primitive Pn applied for t2π time. Naturally, the structure of the control matrix ensures that robot Dn−1 follows a closed-loop orbit during the entire duration of S 1.

Figure C.2

An external file that holds a picture, illustration, etc.
Object name is nihms435733f25.jpg

Three steps of the geometric planning of the control sequence S 1 such that robot Dn reaches the intermediate configuration an at the same time as the robot Dn−1 returns to its initial configuration rn1.

In step two (Fig. C.2.b) we calculate the remaining distance, s′, along the circular closed-loop orbit of Dn−1 between qn1 and its initial configuration rn1. For now we assume that the turning rates of robots Dn and Dn−1 are identical. It follows that the length of the trajectory for Dn must also be increased by s′ in order for Dn−1 to reach rn1 during execution of S 1.

Thus, in step 3 (Fig. C.2.c) we use s′ to adjust S 1 such that robot Dn−1 returns to rn1 as robot Dn reaches an. This is simply done by replacing Pn,t2π with the sequence {Pn,tπ, Pn−1,ts, Pn,tπ}, where tπ is the time it takes robot Dn to turn π, and ts is the time it takes Dn to move distance s′. S 1 can follow the torus around the orbits or assembling shapes required by the collision avoidance heuristic (see Sec. C.4).

C.4 Collision Avoidance

Our collision avoidance heuristic is based on the conservative space requirement; the orbiting devices need π(r*)2 space to orbit, surrounded by a free-space annulus of 2r* thickness (See Fig. C.3). Given an initial trajectory S′, we test for collisions between the space covered by the orbiting robot (i) and the sweep of the area of the robot during it desired trajectory (ii). If these areas intersect, we conservatively declare a collision.

Generating the adjusted trajectory is done as follows; we first convert the initial microrobot trajectory to the trajectory of its center of rotation (iii). The trajectory to maneuver the robot around the space swept by the orbiting device (i) is calculated in the following manner: We plan the motion of the robot along its old trajectory until the robots center of rotation is 2r* away from the center of rotation of the orbiting robot (configuration 1). We then plan turning motion until the steering arm of the robot is pointing towards the center of rotation for the orbiting robot (configuration 2). We then plan the motion of the robot to configuration 3, where the center of rotation for the robot again intersects the initially generated trajectory of its center of rotation (iii). To reach 3 while remaining within the free-space torus, we plan a two-primitive local trajectory (See Sec. 6.3.4) with radius r′ = 2r* + r. Finally, we plan a turning trajectory until the robot faces in its original orientation (configuration 4). The robot can now continue along its initially planned trajectory. A similar approach can be used to avoid the assembling shape.

Figure C.3

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Object name is nihms435733f26.jpg

Geometric collision avoidance heuristic.

D Brief Review of EDR Terminology

We use preimages and forward projection (Lozano-Perez et al. (1984)) to define regions which the robots are guaranteed to enter during the implementation of a control strategy. This terminology is illustrated using the peg-in-a-hole example reproduced in Fig. D.1. Given starting region X and control strategy θ, the (strong) preimage of region Y, PX,θ(Y), is the region of configuration space (C-space) from which the robot is guaranteed To Whom It May Concern: recognizably enter the region Y. A weak preimage of Y, PX,θ(Y), is the region of C-space from which the robot might recognizably enter Y, given fortuitous sensing and control events, when starting in X and applying control strategy θ. The forward projection, Fθ(X) of X under θ is the region of C-space which the robot might reach after the execution of the control strategy θ when starting in region X.

E Error Bounds

Error bounds are primarily used to bound the size of forward-projection regions for the global control strategies, FσG (Ri), defining the regions Ca,i and Ca,i. We derive simple error bounds from the kinematic model shown in Sec. 3 by substituting ω=vahr and adding error components ve and ωe to the microrobot turning rate (ω) and linear velocity (v):

q.=(x.y.θ.)=(sinθcosθ0)(v+ve)+(001)(ω+ωe).
(E.1)

Let v = v +ve, and [omega with circumflex] = ω + ωe. The displacement with error, Δqe(t), can be written as:

Δqe(t)=(v^ω^(cos(θ-π2)+cos(θ-tω^+π2))v^ω^(sin(θ-π2)+sin(θ-tω^+π2))tω^).
(E.2)

Eq. E.2 represents the error bound for a single control primitive. Let δ(t)=SΔqe(t)-Δq(t)dt be the error integrated over the control sequence S, where δ(t) = (δx(t), δy(t), δθ(t))T, and δxy(t)=δx2(t)+δy2(t) (error in IR2).

Figure D.1

An external file that holds a picture, illustration, etc.
Object name is nihms435733f27.jpg

Reproduced from Lozano-Perez et al. (1984), the peg-in-a-hole example illustrates the terminology of preimages and forward projections within a R2 × An external file that holds a picture, illustration, etc.
Object name is nihms435733ig12.jpg C-space. Regions X and Y represent the start and goal regions for the task of inserting a peg (i) into a hole (ii). The angle bracket represents the uncertainty in the commanded velocity (control strategy) θ. Boundaries of the strong and week preimages, as well as forward projection of X are illustrated.

E.1 Bounding FσG (Ri)

The size of the forward-projection of Ri with the global control strategies σG, FσG (Ri), is bounded by the maximum error δ(t) that can accumulate during the execution of the global control strategy. We start by deriving Ca,i as the bound for FσG (Ri) for robot Di, where i [set membership] Zn−2, using the global control strategy σG.

Let tθ= θ/[omega with circumflex] be the time it takes the robot to rotate by angle θ while in control state 1. The forward-projection of Ri using σG for the control of a single robots from Ri to Ai, FσG (Ri) for i [set membership] Zn−1, is equal to δ (t2π). The reason for this is that our microrobots can only turn one way, and correcting 3 error may require a trajectory up to 2πr in length (the robot may have to complete a full circle). Consequently, global control strategies σG will not be able to reduce the control error to below δ (t2π), thus for robots Di with i [set membership] Zn−1, FσG (Ri) is bounded by cylinder Ca,i with ra,i = δxy (t2π) and ha,i = δθ (t2π). Ca,i is centered around the target configuration ai, as mentioned in Sec. 6.3.3.

F Design Specifications for Five STRING Microrobots

This section, provided for reference, is reproduced with permission from (Donald et al., 2008b). The steering-arms in all the five species are fabricated out of 1.5 μm thick polysilicon layer. Table F.1 summarizes the design parameters for the steering-arms defining each of the species. An annotated design of the steering-arm is shown on Fig. F.1 to provide a reference for the parameters in Table F.1. The layer of evaporated chromium is 76 nm thick, except for species 4, where the nominal thickness of chromium ranges from 76 to 92 nm to compensate for design-specific local effects of galvanic attack. The steering-arm designs were determined based on closed-form equations (Donald et al., 2006), finite-element models, and empirical data, such that their transition voltages are reproducibly confined to the voltage ranges shown in Fig. 18(b).

Figure F.1

An external file that holds a picture, illustration, etc.
Object name is nihms435733f28.jpg

Reprinted with permission from (Donald et al., 2008b). Parameters of the steering-arms specified in Table F.1 classifying the five microrobot species.

Table F.1

Reprinted with permission from (Donald et al., 2008b). Design parameters for the steering-arms for the five microrobot species.

SpeciesArm-orientationD [μm]w [μm]L [μm]Lc [μm]Lo [μm]Dimple height [μm]
1aleft side201012033.59.50.75
1bleft side4088059.59.51.2
2right side201012033.51.50.75
3right side36810794.51.50.75
4left side30813091.51.50.75

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