Efficient Reachable Sets on Lie Groups Using Lie Algebra Monotonicity and Tangent Intervals

TL;DR

Using interval bounds of the Baker-CampbellHausdorff formula, reachable set estimates are extended to arbitrary time horizons in an efficient Runge-Kutta-Munthe-Kaas integration algorithm.

Abstract

In this paper, we efficiently compute overapproximating reachable sets for control systems evolving on Lie groups, building off results from monotone systems theory and geometric integration theory. We consider intervals in the tangent space, which describe real sets on the Lie group through the exponential map. A local equivalence between the original system and a system evolving on the Lie algebra allows existing interval reachability techniques to apply in the tangent space. Using interval bounds of the Baker-Campbell-Hausdorff formula, these reachable set estimates are extended to arbitrary time horizons in an efficient Runge-Kutta-Munthe-Kaas integration algorithm. The algorithm is demonstrated through consensus on a torus and attitude control on $SO(3)$.

Figures (4)

• Figure 1: A pictoral representation of Theorems \ref{['thm:smalltreach']} and \ref{['thm:embeddedLiealg']}. An interval in the tangent space $T_{\mathring{x}}\mathcal{X}$, pictured in blue, is evolved to time $t$ using monotone systems theory in the Lie algebra. The tangent interval at $t$ is exponentiated to the Lie group, pictured in red, overapproximating the reachable set at $t$.
• Figure 2: A pictoral representation of recentering using a $\textsf{BCH}$ inclusion function \ref{['eq:BCHincl']} for the Lie group $SO(3)$, with Lie algebra $\mathfrak{so}(3)$ and the simplicial cone $K$ from Section \ref{['subsec:exSO3']}. Left ($SO(3)$): The exponentiated tangent intervals $I\exp([-0.1,0.1]^3)$ and $I\exp([0.2,0.4]^3)$ centered around the identity matrix are pictured in blue, using an evenly spaced meshgrid of $7^3=343$ points exponentiated from the Lie algebra. With $\mathring{\Theta} = [0.3\ 0.3\ 0.3]^T$, the exponentiated tangent interval $\exp(\mathring{\Theta}) \exp(\textsf{BCH}_{-\mathring{\Theta}}([0.2,0.4]^3))$ centered around $\exp(\mathring{\Theta})$ is pictured in purple. Visually, the geometry of the tangent interval centered at the identity before the BCH overapproximation is different than the geometry of the tangent interval centered at $\exp(\mathring{\Theta})$. Right ($\mathfrak{so}(3)$): The box $[0.2,0.4]^3$ and the map $\Theta \mapsto \operatorname{bch}_{-\mathring{\Theta}}(\Theta)$ sampled with $7^3=343$ evenly spaced points in this box are pictured in blue. The box $\textsf{BCH}_{-\mathring{\Theta}}([0.2,0.4]^3)\approx[-1.4,1.4]^3$ from the BCH inclusion function is pictured in purple, overapproximating these outputted points. The exponential map links the left and right plots.
• Figure 3: The reachable set $\{\mathring{x}_t\exp([\ul\Theta_t,\overline{\Theta}_t]_K)\}_t$ for the coupled oscillators is visualized as two arcs on a circle, as well as on a torus embedded in $\mathbb{R}^3$. For this system, the uncertainty in the initial condition of the blue oscillator begins larger than the orange oscillator, but the uncertainty is quickly shared between the two. For the abelian torus, the recentering can be done without any loss of information, allowing the reachable set to remain tight to the true reachable set.
• Figure 4: The overapproximated reachable set $\{\mathring{x}_t\exp([\ul\Theta_t,\overline{\Theta}_t]_K)\}_t$ for the orientation of a satellite evolving on $SO(3)$, for the run constantly recentering using $\textsf{BCH}$. The coordinate frame visualizes $\mathring{x}_t$, where red, green, blue represent the $x$, $y$, and $z$-axes respectively. The point clouds represent the reachable set. Each point is generated from an evenly spaced meshgrid of $7^3=343$ points $\Theta_t \in [\ul\Theta_t,\overline{\Theta}_t]_K$ in the Lie algebra which are exponentiated to the element $\mathring{x}_t\exp(\Theta_t)$ in the Lie group.

Theorems & Definitions (12)

• Definition 1: Differential of $\exp$
• Proposition 2: Canonical coordinates
• Definition 3: Tangent interval
• Theorem 4: Monotone Lie algebra
• proof
• Theorem 5: Embedded Lie algebra
• proof
• Proposition 6: Recentering via $\textsf{BCH}$
• proof
• Proposition 7: Abelian Lie groups