Hybrid Feedback for Affine Nonlinear Systems with Application to Global Obstacle Avoidance (Extended Version)

Abstract

This paper explores the design of hybrid feedback for a class of affine nonlinear systems with topological constraints that prevent global asymptotic stability. A new hybrid control strategy is introduced, which differs conceptually from the commonly used synergistic hybrid approaches. The key idea involves the construction of a generalized synergistic Lyapunov function whose switching variable can either remain constant or dynamically change between jumps. Based on this new hybrid mechanism, a generalized synergistic hybrid feedback control scheme, endowed with global asymptotic stability guarantees, is proposed. This hybrid control scheme is then improved through a smoothing mechanism that eliminates discontinuities in the feedback term. Moreover, the smooth hybrid feedback is further extended to a larger class of systems through the integrator backstepping approach. The proposed hybrid feedback schemes are applied to solve the global obstacle avoidance problem using a new concept of synergistic navigation functions. Finally, numerical simulation results are presented to illustrate the performance of the proposed hybrid controllers.

Figures (5)

• Figure 1: The architecture of the proposed hybrid feedback strategy.
• Figure 2: Example of the smooth navigation function $V_{nav}$ defined in \ref{['eqn:def_navigation']} with $r_s = 1$ (gray circle) and $\varrho = 16$, and a circular obstacle centered at $p_o = (4,0)$ (black $+$) with radius $r_o=1$. The destination is chosen at $p_d=(0,0)$ (black $\star$), and the undesired (saddle) critical point is located at $p^*=(5.6865,0)$ (red $\ast$). The blue arrows and the light blue lines represent the direction of the gradients and the contour of the navigation function, respectively.
• Figure 3: Geometric representation of the transformation function $\mathcal{T}$.
• Figure 4: (a) Trajectories of the robot using three different controllers with initial position $p(0)=(12,0)$; (b) Time evolution of the distance to the destination and the switching variable $\theta$; (c) Trajectories of the robot using three different controllers from different initial positions.
• Figure 5: Geometric representation of the jump of the switching variable $\theta$ at the critical point $(p,\theta)\in C_{\mathcal{V}_{nav}}$.

Theorems & Definitions (26)

• Definition 1: Bouligand's tangent conebouligand1932introduction
• Definition 2
• Lemma 1
• proof
• Theorem 1
• proof
• Remark 1
• Remark 2
• Remark 3
• Proposition 1