This paper derives both open-loop and closed-loop control policies that steer a finite set of differential-drive robots to desired positions in a two-dimensional workspace, when all robots receive the same control inputs but each robot turns at a slightly different rate. In the absence of perturbation, the open-loop policy achieves zero error in finite time. In the presence of perturbation, the closed-loop policy is globally asymptotically stabilizing with state feedback. Both policies were validated with hardware experiments using up to 15 robots. These experimental results suggest that similar policies might be applied to control micro- and nanoscale robotic systems, which are often subject to similar constraints.
AnsariAMurpheyT (2013) Minimal parametric sensitivity trajectories for nonlinear systems. In: Proceedings of the American controls conference (ACC), Washington DC, USA, 17–19 June 2013, pp. 5011–5016.
2.
BadjićJBalzaniVCrediA. (2004) A molecular elevator. Science303(5665): 1845–1849.
3.
BeauchardKCoronJMRouchonP (2010) Time-periodic feedback stabilization for an ensemble of half-spin systems. In: Proceedings of the 8th IFAC symposium on nonlinear control systems, Bologna, Italy, 1–3 September 2010.
BeckerABretlT (2012a) Approximate steering of a plate-ball system under bounded model perturbation using ensemble control. In: Proceedings of the IEEE/RSJ international conference on intelligent robots and systems (IROS), Vilamoura, Portugal, 7–12 October 2012, pp. 5353–5359.
7.
BeckerABretlT (2012b) Approximate steering of a unicycle under bounded model perturbation using ensemble control. IEEE Transactions on Robotics28(3): 580–591.
8.
BeckerAHabibiGWerfelJ. (2013a) Massive uniform manipulation: Controlling large populations of simple robots with a common input signal. In: Proceedings of the IEEE/RSJ international conference on intelligent robots and systems (IROS), Tokyo, 3–8 November 2013, pp. 520–527.
9.
BeckerAMcLurkinJ (2013) Exact range and bearing control of many differential-drive robots with uniform control inputs. In: Proceedings of the IEEE/RSJ international conference on intelligent robots and systems (IROS), Tokyo, 3–8 November 2013, pp. 3338–3343.
10.
BeckerAOnyukselCBretlT (2012) Feedback control of many differential-drive robots with uniform control inputs. In: Proceedings of the IEEE/RSJ international conference on intelligent robots and systems (IROS), Vilamoura, Portugal, 7–12 October 2012, pp. 2256–2262.
11.
BeckerAOuYKimP. (2013b) Feedback control of many magnetized tetrahymena pyriformis cells by exploiting phase inhomogeneity. In: Proceedings of the IEEE/RSJ international conference on intelligent robots and systems (IROS), Tokyo, 3–8 November 2013, pp. 3317–3323.
BorensteinJFengL (1996) Measurement and correction of systematic odometry errors in mobile robots. IEEE Transactions on Robotics and Automation12(6): 869–880.
14.
BrockettRWKhanejaN (1999) On the control of quantum ensembles. In: DjaferisTSchickI (eds) System Theory: Modeling, Analysis and Control. Boston, MA: Kluwer Academic Publishers.
15.
ChandrasekharanK (1968) An Introduction to Analytic Number Theory. Berlin: Springer-Verlag, p. 23.
16.
ChiangPTMielkeJGodoyJ. (2011) Toward a light-driven motorized nanocar: Synthesis and initial imaging of single molecules. ACS Nano6(1): 592–597.
17.
ChosetHLynchKHutchinsonS. (2005) Principles of Robot Motion. Cambridge, MA: The MIT Press.
18.
De LanauzeDFelfoulOTurcotJP. (2013) Three-dimensional remote aggregation and steering of magnetotactic bacteria microrobots for drug delivery applications. The International Journal of Robotics Research33(3): 359–374.
19.
DillerEFloydSPawasheC. (2012) Control of multiple heterogeneous magnetic microrobots in two dimensions on nonspecialized surfaces. IEEE Transactions on Robotics28(1): 172–182.
20.
DillerEGiltinanJSittiM (2013) Independent control of multiple magnetic microrobots in three dimensions. The International Journal of Robotics Research32(5): 614–631.
21.
DonaldBLeveyCMcGrayC. (2006) An untethered, electrostatic, globally controllable MEMS micro-robot. Journal of MEMS15(1): 1–15.
22.
DonaldBLeveyCPaprotnyI (2008) Planar microassembly by parallel actuation of MEMS microrobots. Journal of MEMS17(4): 789–808.
23.
DonaldBRLeveyCGPaprotnyI. (2013) Planning and control for microassembly of structures composed of stress-engineered mems microrobots. The International Journal of Robotics Research32(2): 218–246.
24.
EinsteinA (1956) Investigations on the Theory of the Brownian Movement. New York: Dover Publications.
25.
FinkJMichaelNKumarV (2007) Composition of vector fields for multi-robot manipulation via caging. In: Proceedings of Robotics: science and systems, Atlanta, GA, USA, 27–30 June 2007.
26.
FloydSDillerEPawasheC. (2011) Control methodologies for a heterogeneous group of untethered magnetic micro-robots. International Journal of Robotics Research30(13): 1553–1565.
27.
GastonKJBlackburnTM (1997) How many birds are there?Biodiversity and Conservation6(4): 615–625.
28.
GhoshAFischerP (2009) Controlled propulsion of artificial magnetic nanostructured propellers. Nano Letters9(6): 2243–2245.
29.
GrahamRLKnuthDEPatashnikO (1990) Concrete Mathematics: A Foundation for Computer Science. Reading, MA: Addison-Wesley.
30.
HasegawaTOgawaNOkuH. (2008) A new framework for microrobotic control of motile cells based on high-speed tracking and focusing. In: Proceedings of the IEEE international conference on robotics and automation (ICRA), Pasadena, CA, 19–23 May 2008, pp. 3964–3969.
31.
JingWPaganoNCappelleriD (2013) A tumbling magnetic microrobot with flexible operating modes. In: Proceedings of the IEEE international conference on robotics and automation (ICRA), Karlsruhe, Germany, 6–10 May 2013, pp. 5514–5519.
32.
KhanejaN (2000) Geometric Control in Classical and Quantum Systems. PhD Thesis. Harvard University, Cambridge, MA, USA.
33.
LaSalleJP (1960) Some extensions of Liapunov’s second method. IRE Transactions on Circuit TheoryCT(7): 520–527.
34.
LevineWS (ed) (1996) Discrete-time linear time-varying systems. In: The Control Handbook. Boca Raton, FL: CRC Press, pp.459–463.
35.
LiJS (2011) Ensemble control of finite-dimensional time-varying linear systems. IEEE Transactions on Automatic Control56(2): 345–357.
36.
LiJSKhanejaN (2009) Ensemble control of Bloch equations. IEEE Transactions on Automation and Control54(3): 528–536.
37.
LucibelloPOrioloG (2001) Robust stabilization via iterative state steering with an application to chained-form systems. Automatica37: 71–79.
38.
LyapunovAM (1992) The General Problem of the Stability of Motion. London: Tayor & Francis.
39.
MajewskiBSHavasG (1994) The complexity of greatest common divisor computations. In: AdlemanLHuangMD (eds) Algorithmic Number Theory (Lecture Notes in Computer Science, Vol. 877). Berlin; Heidelberg, Germany: Springer-Verlag, pp.184–193.
40.
McLurkinJLynchARixnerS. (2010) A low-cost multi-robot system for research, teaching, and outreach. In: Distributed Autonomous Robotic Systems. Berlin; Heidelberg: Springer, pp. 597–609.
41.
MinettAFràysseJGangG. (2002) Nanotube actuators for nanomechanics. Current Applied Physics2(1): 61–64
42.
OlsonE (2011) AprilTag: A robust and flexible visual fiducial system. In: Proceedings of the IEEE international conference on robotics and automation (ICRA), Shanghai, 9–13 May 2011, pp. 3400–3407.
43.
OnyukselC (2012) Feedback control of many differential-drive robots with uniform control inputs. Master’s Thesis. University of Illinois at Urbana-Champaign, USA. Available at: http://hdl.handle.net/2142/34414.
44.
OuYKimDHKimP. (2013) Motion control of magnetized tetrahymena pyriformis cells by magnetic field with model predictive control. International Journal of Robotics Research32(1): 129–139.
45.
PenroseR (1955) A generalized inverse for matrices. Proceedings of the Cambridge Philosophical Society51: 406–413.
46.
PeyerKEZhangLNelsonBJ (2013) Bio-inspired magnetic swimming microrobots for biomedical applications. Nanoscale5(4): 1259–1272.
47.
PlooijMde VriesMWolfslagW. (2013) Optimization of feedforward controllers to minimize sensitivity to model inaccuracies. IEEE International Journal of Robotics and Systems, pp. 3382–3389.
48.
ProakisJGManolakisDG (1996) Digital Signal Processing: Principles, Algorithms, and Applications. Englewood Cliffs, NJ: Prentice Hall.
49.
SchürleSPeyerKEKratochvilBE. (2012) Holonomic 5-DOF magnetic control of 1D nanostructures. In: Proceedings of the IEEE international conference on robotics and automation (ICRA), St Paul, MN, USA, 14–18 May 2012, pp. 1081–1086.
50.
ShannonCE (1949) Communication in the presence of noise. Proceedings of the Institute of Radio Engineers. Reprinted as classic paper in: Proceedings of the IEEE37(1): 10–21.
51.
ShinYKKimYKimJ (2013) Automated microfluidic system for orientation control of mouse embryos. In: Proceedings of the IEEE/RSJ international conference on intelligent robots and systems (IROS), Tokyo, 3–8 November 2013, pp. 496–501.
52.
ShiraiYOsgoodAJZhaoY. (2005) Directional control in thermally driven single-molecule nanocars. Nano Letters5(11): 2330–2334.
53.
SteagerEBSakarMKimDH. (2011) Electrokinetic and optical control of bacterial microrobots. Journal of Micromechanics and Microengineering21(3): 035001.
54.
ThrunSBurgardWFoxD (2005) Probabilistic Robotics (Intelligent Robotics and Autonomous Agents). Cambridge, MA: The MIT Press.
55.
TottoriSZhangLQiuF. (2012) Magnetic helical micromachines: Fabrication, controlled swimming, and cargo transport. Advanced Materials24(6): 811–816.
56.
TungHWFrutigerDRPanèS. (2012) Polymer-based wireless resonant magnetic microrobots. In: Proceedings of the IEEE international conference on robotics and automation (ICRA), St Paul, MN, USA, 14–18 May 2012, pp. 715–720.
57.
VartholomeosPAkhavan-SharifMDupontPE (2012) Motion planning for multiple millimeter-scale magnetic capsules in a fluid environment. In: Proceedings of the IEEE international conference on robotics and automation (ICRA), St Paul, MN, USA, 14–18 May 2012, pp. 1927–1932.
58.
VivesGKangJKellyKF. (2009) Molecular machinery: Synthesis of a “nanodragster”. Organic Letters11(24): 5602–5605.
59.
ZhangLAbbottJJDongLX. (2009a) Artificial bacterial flagella: Fabrication and magnetic control. Applied Physics Letters94(6): 064107.
60.
ZhangLAbbottJJDongLX. (2009b) Micromanipulation using artificial bacterial flagella. In: Proceedings of the IEEE/RSJ international conference on intelligent robots and systems (IROS), St. Louis, MO, USA, 11–15 October 2009, pp. 1401–1406.
61.
ZhangLAbbottJJDongLX. (2009c) Characterizing the swimming properties of artificial bacterial flagella. Nano Letters9(10): 3663–3667.
62.
ZhouYChirikjianG (2003) Probabilistic models of dead-reckoning error in nonholonomic mobile robots. In: Proceedings of the IEEE international conference on robotics and automation (ICRA), Taipei, Taiwan, 14–19 September 2003, Vol. 2, pp. 1594–1599.
