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Robust Stabilizing Continuous-Time Controllers for Periodic Orbits of Hybrid Systems : Application to Bipedal Robots
| Content Provider | Semantic Scholar |
|---|---|
| Author | Hamed, Kaveh Akbari Buss, Brian G. Grizzle, Jessy W. |
| Copyright Year | 2014 |
| Abstract | This paper presents a systematic approach for the design of continuous-time controllers to robustly and expo nentially stabilize periodic orbits of hybrid dynamical systems. A parameterized family of continuous-time controllers is asumed 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 Poincaŕ e map and its firstand second-order derivatives are used to translate the problemof exponential stabilization of the periodic orbit into a set of Bilinear Matrix Inequalities (BMIs). A BMI optimization problem is t hen set up to tune the parameters of the continuous-time control ler so that the Jacobian of the Poincaŕe map has its eigenvalues in the unit circle. It is also shown how robustness against uncerta inty in the switching condition of the hybrid system can be incorpor ated into the design problem. The power of this approach is illust rated by finding robust and stabilizing continuous-time feedbacklaws for walking gaits of two underactuated 3D bipedal robots. I. I NTRODUCTION This paper addresses the problem of designing continuoustime controllers to robustly and exponentially stabilize p riodic orbits of hybrid dynamical systems. Hybrid systems exhibit characteristics of both continuous-time and discr etetime dynamical systems and are used to model a large range of processes [1][4] including power systems [5] and mechanic al systems subject to impacts [6][22]. Our motivation is to design robust stabilizing continuous-time controllers fo r 3D bipedal robots with high degrees of underactuation, but the results we present apply to non-hybrid as well as hybrid systems [23][25]. The most basic tool to investigate the stability of hybrid periodic orbits is the method of Poincaré sections [23][2 5], [3], [6]. In this approach, the evolution of the system on the Poincaré section, a hypersurface transversal to the perio dic orbit, is described by a discrete-time system referred to as the Poincaré return map. In general, there is no closed-for m expression for the Poincaré map, and this complicates the design of continuous-time controllers. Hence, stabilizat ion of periodic orbits for hybrid systems is often achieved with mu ltilevel feedback control architectures, in which continuous time feedback laws are employed at the lower levels of the control scheme to create the periodic orbit. As the lower-le vel B. G. Buss was supported by NSF Graduate Student Research Fel lowship under Grant No. DGE 1256260. K. Akbari Hamed and J. W. Grizzle were supported by NSF Grants ECCS-1343720 and ECCS-1231171. K. Akbari Hamed, B. G. Buss, and J. W. Grizzle are with the Depa rtment of Electrical Engineering and Computer Science, Universit y Michigan, Ann Arbor, MI, USA, {kavehah,bgbuss,grizzle}@umich.edu controllers may not ensure exponential stability of the orb it, a set of adjustable parameters is introduced to the continuou stime controllers. These parameters are then updated by high erlevel event-based controllers when state trajectories cro s the Poincaré section [26], [19], [13], [27], [28]. The event-b ased controllers are designed to render the Jacobian of the Poinc aré map around the fixed point a Hurwitz matrix. One drawback of achieving stability via event-based controllers is the potentially large delay between the occurre nce of a disturbance and the event-based control effort. Altern ative approaches attempt to achieve stability at the first level. Reference [19] made use of a nonlinear optimization problem to minimize the spectral radius of the Jacobian of the Poinca ré map for simultaneous design of periodic orbits and continuo ustime controllers. Diehlet al. [29] introduced a smoothed version of the spectral radius and a nonlinear optimization problem to generate maximally stable periodic orbits. This approach was employed to design parameters and optimal control inputs of a fully actuated bipedal robot with 2 degrees of freedom (DOF). Both methods require recomputation of the Jacobian matrix at each iteration of the optimization. For mechanical systems with many degrees of freedom and underactuation (such as the 3D bipedal robot ATRIAS [27], which has13 DOF and6 actuators), the cost of numerically computing the Poincaré map and its Jacobian make these methods impractical. The contribution of this paper is to present a method based on sensitivity analysis and bilinear matrix inequalities ( BMIs) to design continuous-time controllers that provide robust exponential stability of a given periodic orbit without relyi ng on event-based controllers. The approach assumes that a famil y of parameterized continuous-time controllers has been desig n d so that (1) the periodic orbit is an integral curve of the clos edloop system and (2) the orbit is invariant under the choice of parameters in the controllers. By investigating the proper ties of the Poincaré map and its firstand second-order derivative s, a sensitivity analysis is presented. On the basis of the sensi tivity analysis, the problems of robust and exponential stability are translated into a set of BMIs. A BMI optimization problem is then set up to tune the parameters of the continuoustime controllers. Finally, this approach is illustrated to design continuous-time controllers for two underactuated 3D bipe dal robots with8 and13 DOF, respectively. Some of the results in this paper (namely, those illustratin g exponential stabilization of periodic orbits for the 8 DOF bipedal robot) were already presented without mathematica l CONFIDENTIAL. Limited circulation. For review only Preprint submitted to IEEE Transactions on Robotics. Received: July 16, 2014 15:37:01 PST |
| File Format | PDF HTM / HTML |
| Alternate Webpage(s) | http://web.eecs.umich.edu/~grizzle/papers/BMI2014_AkbariHamedBussGrizzle.pdf |
| Alternate Webpage(s) | http://www-personal.umich.edu/~bgbuss/files/TRO2014_AkbariHamedBussGrizzle.pdf |
| Language | English |
| Access Restriction | Open |
| Content Type | Text |
| Resource Type | Article |
