Symmetry in legged locomotion: a new method for designing stable periodic gaits

Abstract

International audienceThis paper introduces a method for achieving stable periodic walking for legged robots. This method is based on producing a type of odd-even symmetry in the system. A hybrid system with such symmetries is called a Symmetric Hybrid System (SHS). We discuss the properties of an SHS and, in particular, will show that an SHS can have an infinite number of synchronized periodic orbits. We describe how controllers can be obtained to make a legged robot an SHS. Then the stability of the synchronized periodic orbits of this SHS is studied, where the notion of self-synchronization is introduced. We show that such self-synchronized periodic orbits are neutrally stable in kinetic energy. As the final step in the process of achieving asymptotically stable periodic walking, we show how by introducing asymmetries (such as energy loss at impact) in the system , the synchronized periodic orbits of this SHS can be turned into asymptotically stable periodic orbits. Many numerical examples, including an 8-DOF 3D biped with 2 degrees of underactuation, are studied to demonstrate the effectiveness of the method

Input-to-State Stability of Periodic Orbits of Systems With Impulse Effects via Poincaré Analysis

“…where v 0 is the first element of the sequencev. However, to establish the estimate (10) in Definition 3 we need x 0 to appear in the RHS instead of x(t 0 ). To do this we will use the fact that, by assumption A.3, the map ∆ is continuously differentiable and thus locally Lipschitz.…”

“…This approach has been successful in generating asymptotically stable periodic gaits for bipedal robots through a variety of methods, including hybrid zero dynamics [6], [7], geometric control [8], virtual holonomic constraints [9], to name a few. Recent extensions of these methods resulted in generating continuums of limit-cycle S. Veer gaits for bipedal walkers [10], [11], and switching among them [12]- [14], to enlarge the behavioral repertoire of these robots in order to accomplish tasks that require adaptability to typical human-centric environments [15], and human (or robot) collaborators [16]. Practical use of these robots demands robustness to external disturbances, which has led many researchers-including the authors of the present paper-to analyze [17]- [19] and design [14], [20], [21] controllers that enhance the robustness of limit-cycle walking gaits.…”

On the Existence and Uniqueness of Poincaré Maps for Systems With Impulse Effects

“…This map executes the trajectory of the system from a point on the guard to its next The Poincaré map for systems with impulse effects have been introduced in [19] (see also [36]). It was mainly employed in the search of periodic gaits (limit cycles) of bipedal robots, together with the use of several methods such as geometric abelian Routh reduction, hybrid zero dynamics and virtual constraints, hybrid Hamiltonian systems, symmetries, etc [5], [10], [11], [14], [25], [26], [20], [21], [22], [23], [32], [33], [34]. These methods permit one to gain extra information regarding the behavior of the system, which provide advantages for the construction of the Poincaré map.…”

Symmetries and periodic orbits in simple hybrid Routhian systems

“…Note that in Example 4 we can get, at most a characterization of neutral stability for the periodic solution γ. It would be interesting to consider perturbed simple hybrid Routhian systems, similarly to the framework given in [43], while the perturbation preserves the symmetry, in order that we can turn the neutrally stable periodic orbit into a stable limit cycle. This will be explored in a future work by considering an adaptation of the averaging method and approximate for hybrid systems given in [17] and [50], respectively.…”

“…Since these works, the study of orbital stability for hybrid systems has been the more explored analysis in this field. The method of Poincaré map is frequently used in the legged locomotion community to study orbital stability of walking gaits [21], [25], [42], [43], [52], [53]. In most of the studies analyzed in the literature employing such an approach, one assumes the existence of a periodic solution.…”