Synergistic Potential Functions from Single Modified Trace Function on SO(3)

TL;DR

The paper addresses globally robust attitude tracking on $SO(3)$ despite topological obstacles by constructing centrally synergistic potential functions from a single modified trace function through angular warping. It introduces a refined switching rule that evaluates a subset of the family using the refined synergistic gap $\\pi_V$, and derives explicit lower bounds via the warping angle function. It provides a complete set of warping directions according to the eigenstructure of $M$ to guarantee central synergy for all $M$ with rank at least 2, reducing computation by restricting the number of functions that must be evaluated per update. Simulations demonstrate robust global attitude tracking with unwinding avoidance and faster convergence under measurement noise and switching faults, validating the practical impact of the refined centrally synergistic control framework.

Abstract

This paper is about the construction of a family of centrally synergistic potential functions from a single modified trace function on SO(3). First, we demonstrate that it is possible to complete the construction through angular warping with multiple directions, particularly effective in the unresolved cases in the literature. Second, it can be shown that for each potential function in the family, there exists a subset of the family such that the synergistic gap is positive at the unwanted critical points. This allows the switching condition to be checked within the selected subsets while implementing synergistic hybrid control. Furthermore, the positive lower bound of synergistic gap is explicitly expressed by selecting a traditional warping angle function. Finally, we apply the proposed synergistic potential functions to obtain robust global attitude tracking.

Figures (8)

• Figure 1: Logic implemented by synergistic feedback control. $\dot{q} = 0$ represents that the switch is not triggered; $q^+$ denotes the new value of $q$ when the switch is triggered; $G(X,q) = \left\{p \in \mathcal{Q}: {\arg\min}_{p \in \mathcal{Q}} V(X,p) \right\}$.
• Figure 2: The block diagram of constructing the synergistic potential functions.
• Figure 3: The block diagram of the closed-loop system for attitude tracking.
• Figure 4: Synergistic potential functions of the item (2) in Theorem \ref{['thm:SelectDirctions']}. The input attitude $X = \mathcal{R}_a(\theta, u)$ where $u = [0.37, 0, 0.93]^\top$ for $\theta \in [0,2\pi]$, $u = [0.25, 0.69, 0.69]^\top$ for $\theta \in [2\pi,4\pi]$, and $u = [0.25, -0.69, 0.69]^\top$ for $\theta \in [4\pi,6\pi]$. The dash y-line is $V(X,q) = 2 \lambda^G$ and the shadow area is $2 \lambda^G \leq V(X,q) \leq 2 \lambda^G - \bar{\delta}_q$.
• Figure 5: Synergistic potential functions of the item (3) in Theorem \ref{['thm:SelectDirctions']}. The input attitude $X = \mathcal{R}_a(\theta, u)$ where $u = [0.37, 0, 0.93]^\top$ for $\theta \in [0,2\pi]$, $u = [0.25, 0.69, 0.69]^\top$ for $\theta \in [2\pi,4\pi]$, and $u = [0.25, -0.69, 0.69]^\top$ for $\theta \in [4\pi,6\pi]$. The dash y-line is $V(X,q) = 2 \lambda^G$ and the shadow area is $2 \lambda^G \leq V(X,q) \leq 2 \lambda^G - \bar{\delta}_q$.