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Identification of optimal feedback control rules from micro-quadrotor and insect flight trajectories

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Abstract

This paper presents “optimal identification,” a framework for using experimental data to identify the optimality conditions associated with the feedback control law implemented in the measurements. The technique compares closed loop trajectory measurements against a reduced order model of the open loop dynamics, and uses linear matrix inequalities to solve an inverse optimal control problem as a convex optimization that estimates the controller optimality conditions. In this study, the optimal identification technique is applied to two examples, that of a millimeter-scale micro-quadrotor with an engineered controller on board, and the example of a population of freely flying Drosophila hydei maneuvering about forward flight. The micro-quadrotor results show that the performance indices used to design an optimal flight control law for a micro-quadrotor may be recovered from the closed loop simulated flight trajectories, and the Drosophila results indicate that the combined effect of the insect longitudinal flight control sensing and feedback acts principally to regulate pitch rate.

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References

  • Aïoun F, Heniche A, Bourlès H (1994) Maximally-diagonal solution to the inverse LQR problem. In: Proceedings of the 33rd conference on decision and control, pp 1445–1446

  • Beatus T, Guckenheimer JM, Cohen I (2015) Controlling roll perturbations in fruit flies. J R Soc Interface 12(105):20150075

  • Boyd S, Ghaoui LE, Feron E, Balakrishnan V (1997) Linear matrix inequalities in system and control theory (studies in applied and numerical mathematics). Society for Industrial and Applied Mathematics, Philadelphia

    Google Scholar 

  • Camhi JM (1970) Yaw-correcting postural changes in locusts. J Exp Biol 52(3):519–531

    Google Scholar 

  • Cheng B, Fry SN, Huang Q, Deng X (2010) Aerodynamic damping during rapid flight maneuvers in the fruit fly drosophila. J Exp Biol 213:602–612

    Article  PubMed  CAS  Google Scholar 

  • Conroy J, Gremillion G, Ranganathan B, Humbert JS (2009) Implementation of wide-field integration of optic flow for autonomous quadrotor navigation. J Exp Biol 27:189–198

    Google Scholar 

  • David CT (1978) The relationship between body angle and flight speed in free-flying drosophila. Physiol Entomol 3(3):191–195

    Article  Google Scholar 

  • Dickinson MH (1990) Comparison of encoding properties of campaniform sensilla on the fly wing. J Exp Biol 151(1):245–261

  • Dickinson MH, Lighton JR (1995) Muscle efficiency and elastic storage in the flight motor of drosophila. Science 268(5207):87

    Article  PubMed  CAS  Google Scholar 

  • Dickinson MH, Muijres FT (2016) The aerodynamics and control of free flight maneuvers in drosophila. Philos Trans R Soc B 371(20150388)

  • Dickinson MH, Tu MS (1997) The function of dipteran flight muscle. Comp Biochem Physiol A Physiol 116(3):223–238

    Article  Google Scholar 

  • Dickinson MH, Lehmann FO, Sane SP (1954) Wing rotation and the aerodynamic basis of insect flight. Science 284:1999

    Google Scholar 

  • Eberle A, Dickerson B, Reinhall PG, Daniel T (2015) A new twist on gyroscopic sensing: body rotations lead to torsion in flapping, flexing insect wings. J R Soc Interface 12(104):20141088

    Article  PubMed  PubMed Central  CAS  Google Scholar 

  • Faruque IA, Humbert JS (2010a) Dipteran insect flight dynamics .Part 2: lateral–directional motion about hover. J Theor Biol 265(3):306–313

  • Faruque IA, Humbert JS (2010b) Dipteran insect flight dynamics. Part 1: longitudinal motion about hover. J Theor Biol 264(2):538–552

    Article  PubMed  Google Scholar 

  • Fiala J, Kocvara M, Stingl M (2013) PENLAB: a MATLAB solver for nonlinear semidefinite optimization. arXiv preprint arXiv:1311.5240

  • Fry SN, Sayaman R, Dickinson MH (2003) The aerodynamics of free-flight maneuvers in Drosophila. Science 300(5618):495–498

  • Fujii T, Narazaki M (1984) A complete optimality condition in the inverse problem of optimal control. SIAM J Control Optim 22(2):327–341

    Article  Google Scholar 

  • Gao N, Aono H, Liu H (2011) Perturbation analysis of 6DoF flight dynamics and passive dynamic stability of hovering fruit fly Drosophila Melanogaster. J Theor Biol 270:98–111

    Article  PubMed  Google Scholar 

  • Götz KG (1968) Flight control in drosophila by visual perception of motion. Kybernetik 4(6):199–208

    Article  PubMed  Google Scholar 

  • Götz KG (1987) Course-control, metabolism and wing interference during ultralong tethered flight in drosophila melanogaster. J Exp Biol 128(1):35–46

    Google Scholar 

  • Graetzel CF, Nelson BJ, Fry SN (2010) Frequency response of lift control in Drosophila. J R Soc Interface 7(52):1603–1616

    Article  PubMed  PubMed Central  Google Scholar 

  • Hedrick TL (2008) Software techniques for two- and three-dimensional kinematic measurements of biological and biomimetic systems. Bioinspir Biomim 3(3):034001

    Article  PubMed  Google Scholar 

  • Hedrick TL, Robinson AK (2010) Within-wingbeat damping: dynamics of continuous free-flight yaw turns in Manduca sexta. Biol Lett 6:422–425

    Article  PubMed  PubMed Central  Google Scholar 

  • Hedrick TL, Cheng B, Deng X (2009) Wingbeat time and the scaling of passive rotational damping in flapping flight. Science 324:252–255

    Article  PubMed  CAS  Google Scholar 

  • Heisenberg M, Wolf R (2013) Vision in Drosophila: genetics of microbehavior, vol 12. Springer, Berlin

    Google Scholar 

  • Kalman RE (1964) When is a linear control system optimal. J Basic Eng 86D:51–60

    Article  Google Scholar 

  • Kennedy J (1983) Zigzagging and casting as a programmed response to wind-borne odour: a review. Physiol Entomol 8(2):109–120

    Article  Google Scholar 

  • Krapp HG (2009) Ocelli. Curr Biol 19(11):R435–R437

    Article  PubMed  CAS  Google Scholar 

  • MacFarlane K, Bush B, Faruque I, Humbert JS, Baeder J (2011) Quasi-steady and computational aerodynamics applied to hovering Drosophila dynamics. In: AIAA 29th applied aerodynamics conference, June 2011. Honolulu, Hawaii, No. 2011-3793

  • MacFarlane KM, Faruque IA, Humbert JS (2015) Power regulation of kinematic control inputs for forward flying Drosophila. Acta Mech Sin 30(6):809–818

    Article  Google Scholar 

  • Klein V, Morelli E (2002) Aircraft system identification: theory and practice. In: AIAA

  • Muijres FT, Elzinga MJ, Melis JM, Dickinson MH (2014) Flies evade looming targets by executing rapid visually directed banked turns. Science 344:172

    Article  PubMed  CAS  Google Scholar 

  • Muijres FT, Elzinga MJ, Iwasaki NA, Dickinson MH (2015) Body saccades of drosophila consist of stereotyped banked turns. J Exp Biol 218(6):864–875

    Article  PubMed  Google Scholar 

  • Pringle JWS (1948) The gyroscopic mechanism of the halteres of diptera. Philos Trans R Soc Lond B Biol Sci 233(602):347–384

    Article  Google Scholar 

  • Ristroph L, Bergou AJ, Ristroph G, Coumes K, Berman GJ, Guckenheimer J, Wang ZJ, Cohen I (2010) Discovering the flight autostabilizer of fruit flies by inducing aerial stumbles. Proc Nat Acad Sci 107(11):4820–4824

    Article  PubMed  PubMed Central  Google Scholar 

  • Ristroph L, Ristroph G, Morozova S, Bergou AJ, Chang S, Guckenheimer J, Wang ZJ, Cohen I (2013) Active and passive stabilization of body pitch in insect flight. J R Soc Interface 10(85):20130237

    Article  PubMed  PubMed Central  Google Scholar 

  • Sane SP (2003) The aerodynamics of insect flight. J Exp Biol 206:4191–4208

    Article  PubMed  Google Scholar 

  • Sane SP, Dieudonné A, Willis MA, Daniel T (2007) Antennal mechanosensors mediate flight control in moths. Science 315(5813):863–866

    Article  PubMed  CAS  Google Scholar 

  • Schneider D (1964) Insect antennae. Annu Rev Entomol 9(1):103–122

    Article  Google Scholar 

  • Sherman A, Dickinson MH (2003) A comparison of visual and haltere-mediated equilibrium reflexes in the fruit fly Drosophila melanogaster. J Exp Biol 206:295–302

    Article  PubMed  Google Scholar 

  • Sherman A, Dickinson MH (2004) Summation of visual and mechanosensory feedback in Drosophila flight control. J Exp Biol 207:133–142

    Article  PubMed  Google Scholar 

  • Sturm JF (1999) Using SeDuMi 1.02, a MATLAB toolbox for optimization over symmetric cones. Optim Methods Softw 11(1–4):625–653

    Article  Google Scholar 

  • Straw AD, Lee S, Dickinson MH (2010) Visual control of altitude in flying drosophila. Curr Biol 20(17):1550–1556

    Article  PubMed  CAS  Google Scholar 

  • Sun M, Wang J, Xiong Y (2007) Dynamic flight stability of hovering insects. Acta Mech Sin 23:231–246

    Article  Google Scholar 

  • Taylor GK, Thomas ALR (2003) Dynamic flight stability in the desert locust Schistocerca gregaria. J Exp Biol 206:2803–2829

    Article  PubMed  Google Scholar 

  • Thompson RA, Wehling MF, Evers J (2008) Evaluation of the haltere as a biologically inspired inertial rate measurement sensor. In: Guidance, navigation, and control conference

  • van Breugel F, Dickinson MH (2012) The visual control of landing and obstacle avoidance in the fruit fly drosophila melanogaster. J Exp Biol 215(11):1783–1798

    Article  PubMed  Google Scholar 

  • van Breugel F, Dickinson MH (2014) Plume-tracking behavior of flying drosophila emerges from a set of distinct sensory-motor reflexes. Curr Biol 24(3):274–286

    Article  PubMed  CAS  Google Scholar 

  • Vela PA, Burdick JW (2003) Control of biomimetic locomotion via averaging theory. In: Proceedings of the IEEE international conference on robotics and automation, 2003, ICRA’03, vol 1, pp 1482–1489

  • Wehner R (1981) Spatial vision in arthropods. In: Autrum H (ed) Handbook of sensory physiology, vol 7/6C. Springer, Berlin, pp 287–616

  • Weis-Fogh T (1949) An aerodynamic sense organ stimulating and regulating flight in locusts. Nature 164(4177):873. https://doi.org/10.1038/164873a0

  • Wu JH, Mao S (2012) Floquet stability analysis of the longitudinal dynamics of two hovering model insects. J R Soc Interface 9(74):2033–2046

    Article  PubMed  PubMed Central  Google Scholar 

  • Ye Y, Todd MJ, Mizuno S (1994) An O (nL)-iteration homogeneous and self-dual linear programming algorithm. Math Oper Res 19(1):53–67

    Article  Google Scholar 

Download references

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Authors and Affiliations

  1. School of Mechanical and Aerospace Engineering, Oklahoma State University, Stillwater, OK, USA

    Imraan A. Faruque

  2. Department of Aerospace Engineering, University of Maryland, College Park, MD, USA

    Kenneth M. Macfarlane & Andrew Kehlenbeck

  3. WU Animal Sciences, Wageningen UR, Wageningen, The Netherlands

    Florian T. Muijres

  4. Department of Mechanical Engineering, University of Colorado, Boulder, CO, USA

    J. Sean Humbert

Authors
  1. Imraan A. Faruque
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  2. Florian T. Muijres
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  3. Kenneth M. Macfarlane
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  4. Andrew Kehlenbeck
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  5. J. Sean Humbert
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Corresponding author

Correspondence to Imraan A. Faruque.

Additional information

This work sponsored in part by the US Air Force Office of Scientific Research (FA95501410068) and by the US Army Research Laboratory (W911NF-08-2-0004). Special thanks to Prof. Michael Dickinson’s laboratory at Caltech for providing the Drosophila data for analysis.

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Faruque, I.A., Muijres, F.T., Macfarlane, K.M. et al. Identification of optimal feedback control rules from micro-quadrotor and insect flight trajectories. Biol Cybern 112, 165–179 (2018). https://doi.org/10.1007/s00422-017-0742-x

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  • DOI: https://doi.org/10.1007/s00422-017-0742-x

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