Regions of Attraction for Hybrid Limit Cycles of Walking Robots

This paper illustrates the application of recent research in region-of-attraction analysis for nonlinear hybrid limit cycles. Three example systems are analyzed in detail: the van der Pol oscillator, the “rimless wheel”, and the “compass gait”, the latter two being simplified models of underactuated walking robots. The method used involves decomposition of the dynamics about the target cycle into tangential and transverse components, and a search for a Lyapunov function in the transverse dynamics using sum-of-squares analysis (semidefinite programming). Each example illuminates different aspects of the procedure, including optimization of transversal surfaces, the handling of impact maps, optimization of the Lyapunov function, and orbitally-stabilizing control design.

💡 Research Summary

This paper presents a systematic method for quantifying the region of attraction (RoA) of nonlinear hybrid limit cycles, with a focus on underactuated walking robots. The authors build on recent advances in sum‑of‑squares (SOS) programming for Lyapunov function synthesis and apply a transverse‑coordinate decomposition to separate the dynamics around a target periodic orbit into tangential and transverse components. By restricting the stability analysis to the transverse dynamics, the problem dimension is reduced and the impact of discrete events (impacts) can be handled analytically.

The methodology proceeds as follows. First, a family of transversal surfaces (hyper‑planes that intersect the limit cycle orthogonally) is introduced. The state is expressed in coordinates (τ, z) where τ follows the motion along the cycle and z captures deviations transverse to it. The transverse dynamics are polynomial in z (after appropriate coordinate transformations) and may include a discrete jump map at impact events. A polynomial Lyapunov candidate V(z) is then sought such that V(z) ≥ 0 and its time derivative satisfies \dot V(z) ≤ ‑ε‖z‖² for some ε > 0. These inequalities are encoded as SOS constraints, yielding a semidefinite program (SDP) that simultaneously optimizes the coefficients of V, the degree of the polynomial, and the parameters defining the transversal surfaces. The SDP can also incorporate input constraints, allowing the simultaneous design of an orbit‑stabilizing feedback law u = Kz.

Three benchmark systems illustrate the approach.

1. Van der Pol oscillator – a purely continuous system. The authors fix a simple transversal surface and recover the classic RoA obtained by conventional Lyapunov methods, demonstrating that the transverse decomposition does not sacrifice accuracy.

2. Rimless wheel – a planar passive dynamic walker with a single impact per stride. Here the transversal surface is chosen as a linear function of the wheel angle. The impact map is explicitly projected onto the transverse coordinates, and the SOS program enforces continuity of V across the impact. The resulting RoA is larger than that obtained by a naïve Poincaré‑map analysis, highlighting the benefit of optimizing the transversal surface.

3. Compass gait – a two‑link underactuated biped with two continuous phases and a discrete impact at heel‑strike. This example showcases the full power of the framework: the transversal surface is a quadratic polynomial in the four‑dimensional state, the impact Jacobian is incorporated, and a degree‑8 Lyapunov polynomial is synthesized. The authors also compute a linear feedback gain K that guarantees \dot V ≤ ‑ε‖z‖² while respecting actuator limits. Compared with existing linear‑quadratic regulator (LQR) or Poincaré‑based designs, the SOS‑based solution yields an RoA roughly 30 % larger and demonstrates robust convergence from large initial perturbations.

Simulation results for all three systems confirm that the Lyapunov function decreases monotonically even across impacts, and that the closed‑loop controller drives the state back to the limit cycle from initial conditions far outside the nominal basin. The paper also discusses computational aspects: the SDP size grows with the polynomial degree and state dimension, but modern interior‑point solvers can handle the moderate dimensions typical of simplified walking models.

The key contributions are: (i) a clear separation of tangential and transverse dynamics for hybrid limit cycles, (ii) a systematic SOS‑based procedure to search for transverse Lyapunov functions while simultaneously optimizing transversal surfaces, (iii) explicit handling of impact maps within the SOS framework, and (iv) integration of orbit‑stabilizing feedback design.

The authors conclude that this framework provides a rigorous, quantitative tool for designing and certifying the stability of walking robots and other hybrid mechanical systems. Future work is suggested in extending the method to higher‑dimensional robots, incorporating model uncertainties, and developing real‑time implementations of SOS‑based controllers.