Exponentially stabilizing continuous-time controllers for periodic orbits of hybrid systems: Application to bipedal locomotion with ground height variations
Restricted accessResearch articleFirst published online July, 2016
Exponentially stabilizing continuous-time controllers for periodic orbits of hybrid systems: Application to bipedal locomotion with ground height variations
This paper presents a systematic approach for the design of continuous-time controllers to robustly and exponentially stabilize periodic orbits of hybrid dynamical systems arising from bipedal walking. A parameterized family of continuous-time controllers is assumed so that (1) a periodic orbit is induced for the hybrid system, and (2) the orbit is invariant under the choice of controller parameters. Properties of the Poincaré map and its first- and second-order derivatives are used to translate the problem of exponential stabilization of the periodic orbit into a set of bilinear matrix inequalities (BMIs). A BMI optimization problem is then set up to tune the parameters of the continuous-time controller so that the Jacobian of the Poincaré map has its eigenvalues in the unit circle. It is also shown how robustness against uncertainty in the switching condition of the hybrid system can be incorporated into the design problem. The power of this approach is illustrated by finding robust and stabilizing continuous-time feedback laws for walking gaits of two underactuated 3D bipedal robots.
AizermanMAGantmacherFR (1958) Determination of stability by linear approximation of a periodic solution of a system of differential equations with discontinuous right-hand sides. The Quarterly Journal of Mechanics and Applied Mathematics11(4): 385–398.
2.
Akbari HamedKBussBGrizzleJ (2014) Continuous-time controllers for stabilizing periodic orbits of hybrid systems: Application to an underactuated 3D bipedal robot. In: 2014 IEEE 53rd annual conference on decision and control (CDC), pp. 1507–1513.
3.
Akbari HamedKGrizzleJ (2014) Event-based stabilization of periodic orbits for underactuated 3-D bipedal robots with left-right symmetry. IEEE Transactions on Robotics30(2): 365–381.
4.
Akbari HamedKSadatiNGruverWDumontG (2012) Stabilization of periodic orbits for planar walking with noninstantaneous double-support phase. IEEE Transactions on Systems, Man and Cybernetics, Part A: Systems and Humans42(3): 685–706.
5.
AltendorferRKoditschekDEHolmesP (2004) Stability analysis of legged locomotion models by symmetry-factored return maps. The International Journal of Robotics Research23(10–11): 979–999.
6.
AmesA (2014) Human-inspired control of bipedal walking robots. IEEE Transactions on Automatic Control59(5): 1115–1130.
7.
AmesAGallowayKSreenathKGrizzleJ (2014) Rapidly exponentially stabilizing control Lyapunov functions and hybrid zero dynamics. IEEE Transactions on Automatic Control59(4): 876–891.
8.
AmesAGreggRSpongM (2007) A geometric approach to three-dimensional hipped bipedal robotic walking. In: 46th IEEE conference on decision and control, pp. 5123–5130.
9.
AmesADSinnetRWWendelED (2009) Three-dimensional kneed bipedal walking: A hybrid geometric approach. In: MajumdarRTabuadaP (eds.) Hybrid Systems: Computation and Control (Lecture Notes in Computer Science, vol. 5469). Berlin: Springer, pp. 16–30.
10.
AndersonBBoseNJuryE (1975) Output feedback stabilization and related problems-solution via decision methods. IEEE Transactions on Automatic Control20(1): 53–66.
11.
AnkaraliMMSaranliU (2011) Control of underactuated planar pronking through an embedded spring-mass hopper template. Autonomous Robots30(2): 217–231.
12.
ArnoldV (1996) Geometrical Methods in the Theory of Ordinary Differential Equations, 2nd edition. New York: Springer.
BainovDSimeonovP (1989) Systems With Impulse Effect: Stability, Theory and Applications. Ellis Horwood Ltd.
15.
BuehlerMKoditschekDEKindlmannPJ (1994) Planning and control of robotic juggling and catching tasks. The International Journal of Robotics Research13(12): 101–118.
16.
BurdenSRevzenSSastryS (2015) Model reduction near periodic orbits of hybrid dynamical systems. IEEE Transactions on Automatic Control (in press).
17.
BussBGRamezaniAAkbari HamedKGriffinBAGallowayKSGrizzleJW (2014) Preliminary walking experiments with underactuated 3D bipedal robot Marlo. In: IEEE/RSJ international conference on intelligent robots and systems, pp. 2529–2536.
18.
CarverSGCowanNJGuckenheimerJM (2009) Lateral stability of the spring-mass hopper suggests a two-step control strategy for running. Chaos19(2): 026106.
19.
ChevallereauCAbbaGAoustinYPlestanFWesterveltECanudas-de WitCGrizzleJ (2003) Rabbit: a testbed for advanced control theory. IEEE Control Systems23(5): 57–79.
20.
ChevallereauCGrizzleJShihCL (2009) Asymptotically stable walking of a five-link underactuated 3-D bipedal robot. IEEE Transactions on Robotics25(1): 37–50.
21.
DaiHTedrakeR (2012) Optimizing robust limit cycles for legged locomotion on unknown terrain. In: IEEE 51st annual conference on decision and control, pp. 1207–1213.
22.
DiehlMMombaurKNollD (2009) Stability optimization of hybrid periodic systems via a smooth criterion. IEEE Transactions on Automatic Control54(8): 1875–1880.
23.
FreidovichLMettinUShiriaevASpongM (2009) A passive 2-DOF walker: Hunting for gaits using virtual holonomic constraints. IEEE Transactions on Robotics25(5): 1202–1208.
24.
GoebelRSanfeliceRTeelA (2012) Hybrid Dynamical Systems: Modeling, Stability, and Robustness. Princeton, NJ: Princeton University Press.
25.
Gounaridis-MinaidisCKalouptsidisN (1986) Stability of discrete-time bilinear systems with constant inputs. International Journal of Control43(2): 663–669.
26.
GreggRLenziTHargroveLSensingerJ (2014) Virtual constraint control of a powered prosthetic leg: From simulation to experiments with transfemoral amputees. IEEE Transactions on Robotics30(6): 1455–1471.
27.
GreggRSensingerJ (2014) Towards biomimetic virtual constraint control of a powered prosthetic leg. IEEE Transactions on Control Systems Technology22(1): 246–254.
28.
GreggRSpongM (2008) Reduction-based control with application to three-dimensional bipedal walking robots. In: American control conference, pp. 880–887.
29.
GreggRTiltonACandidoSBretlTSpongM (2012) Control and planning of 3-D dynamic walking with asymptotically stable gait primitives. IEEE Transactions on Robotics28(6): 1415–1423.
30.
GriffinBGrizzleJ (2015) Walking gait optimization for accommodation of unknown terrain height variations. In: American control conference (in press).
31.
GrimesJHurstJ (2012) The design of ATRIAS 1.0 a unique monoped, hopping robot. In: 2012 international conference on climbing and walking robots and the support technologies for mobile machines, pp. 548–554.
GrizzleJAbbaGPlestanF (2001) Asymptotically stable walking for biped robots: Analysis via systems with impulse effects. IEEE Transactions on Automatic Control46(1): 51–64.
34.
GrizzleJW (2006) Remarks on event-based stabilization of periodic orbits in systems with impulse effects. In: Second international symposium on communication, control and signal processing.
35.
GrizzleJWChevallereauCSinnetRWAmesAD (2014) Models, feedback control, and open problems of 3D bipedal robotic walking. Automatica50(8): 1955–1988.
36.
HaddadWChellaboinaV (2008) Nonlinear Dynamical Systems and Control: A Lyapunov-Based Approach. Princeton, NJ: Princeton University Press.
37.
HaddadWChellaboinaVNersesovS (2006) Impulsive and Hybrid Dynamical Systems: Stability, Dissipativity, and Control. Princeton, NJ: Princeton University Press.
38.
HenrionDLofbergJKocvaraMStinglM (2005) Solving polynomial static output feedback problems with PENBMI. In: 44th IEEE conference on decision and control, and European control conference, pp. 7581–7586.
39.
HiskensIPaiM (2000a) Hybrid systems view of power system modelling. In: Proceedings the 2000 IEEE international symposium on circuits and systems, vol. 2, pp. 228–231.
40.
HiskensIPaiM (2000b) Trajectory sensitivity analysis of hybrid systems. IEEE Transactions on Circuits and Systems I: Fundamental Theory and Applications47(2): 204–220.
41.
HobbelenDWisseM (2007) A disturbance rejection measure for limit cycle walkers: The gait sensitivity norm. IEEE Transactions on Robotics23(6): 1213–1224.
42.
HolmJLeeDSpongM (2007) Time-scaling trajectories of passive-dynamic bipedal robots. In: IEEE international conference on robotics and automation, pp. 3603–3608.
43.
HurmuzluYMarghituDB (1994) Rigid body collisions of planar kinematic chains with multiple contact points. The International Journal of Robotics Research13(1): 82–92.
44.
IsidoriA (1995) Nonlinear Control Systems,
3rd edition. New York: Springer.
45.
LackJPowellMAmesA (2014) Planar multi-contact bipedal walking using hybrid zero dynamics. In: IEEE international conference on robotics and automation, pp. 2582–2588.
46.
LofbergJ (2004) YALMIP: a toolbox for modeling and optimization in Matlab. In: 2004 IEEE international symposium on computer aided control systems design, pp. 284–289.
47.
MaggioreMConsoliniL (2013) Virtual holonomic constraints for Euler Lagrange systems. IEEE Transactions on Automatic Control58(4): 1001–1008.
48.
ManchesterIRMettinUIidaFTedrakeR (2011) Stable dynamic walking over uneven terrain. The International Journal of Robotics Research30(3): 265–279.
49.
MartinAESchmiedelerJP (2014) Predicting human walking gaits with a simple planar model. Journal of Biomechanics47(6): 1416–1421.
50.
MorrisBGrizzleJ (2009) Hybrid invariant manifolds in systems with impulse effects with application to periodic locomotion in bipedal robots. IEEE Transactions on Automatic Control54(8): 1751–1764.
51.
ParkerTChuaL (1989) Practical Numerical Algorithms for Chaotic Systems. New York: Springer.
52.
RaibertMH (1986) Legged robots. Communications of the ACM29(6): 499–514.
53.
RamezaniAHurstJAkbai HamedKGrizzleJ (2013) Performance analysis and feedback control of ATRIAS, a three-dimensional bipedal robot. Journal of Dynamic Systems, Measurement, and Control December, ASME136(2).
54.
RemyCD (2011) Optimal Exploitation of Natural Dynamics in Legged Locomotion. PhD Thesis, ETH Zurich.
55.
SaglamCBylK (2013) Switching policies for metastable walking. In: IEEE 52nd annual conference on decision and control, pp. 977–983.
56.
SeipelJHolmesP (2007) A simple model for clock-actuated legged locomotion. Regular and Chaotic Dynamics12(5): 502–520.
57.
SeyfarthAGeyerHHerrH (2003) Swing-leg retraction: a simple control model for stable running. The Journal of Experimental Biology206(15): 2547–2555.
58.
ShihCLGrizzleJWChevallereauC (2012) From stable walking to steering of a 3D bipedal robot with passive point feet. Robotica30: 1119–1130.
59.
ShiriaevAFreidovichLGusevS (2010) Transverse linearization for controlled mechanical systems with several passive degrees of freedom. IEEE Transactions on Automatic Control55(4): 893–906.
60.
ShiriaevASandbergACanudas de WitC (2004) Motion planning and feedback stabilization of periodic orbits for an acrobot. In: 43rd IEEE conference on decision and control, vol. 1, pp. 290–295.
61.
SohC (1990) Schur stability of convex combination of matrices. Linear Algebra and its Applications128: 159–168.
62.
SpongMBulloF (2005) Controlled symmetries and passive walking. IEEE Transactions on Automatic Control50(7): 1025–1031.
63.
SreenathKParkHWPoulakakisIGrizzleJ (2013) Embedding active force control within the compliant hybrid zero dynamics to achieve stable, fast running on MABEL. The International Journal of Robotics Research32(3): 324–345.
64.
SreenathKParkHWPoulakakisIGrizzleJW (2011) Compliant hybrid zero dynamics controller for achieving stable, efficient and fast bipedal walking on MABEL. The International Journal of Robotics Research30(9): 1170–1193.
65.
TedrakeRZhangTSeungH (2004) Stochastic policy gradient reinforcement learning on a simple 3D biped. In: Proceedings 2004 IEEE/RSJ international conference on intelligent robots and systems, vol. 3, pp. 2849–2854.
66.
TokerOOzbayH (1995) On the NP-hardness of solving bilinear matrix inequalities and simultaneous stabilization with static output feedback. In: Proceedings of the 1995 American control conference, vol. 4, pp. 2525–2526.
