Reachability-Guaranteed Optimal Control for the Interception of Dynamic Targets under Uncertainty

TL;DR

The paper tackles robust interception of dynamic targets under bounded uncertainty by developing reachability guarantees on SE(3). It introduces the RG-OCP, a forward-reachability–driven, set-valued optimal control framework that uses convex enclosures on SO(3) and time-enclosures to guarantee feasibility and reachability to a target set. Key contributions include strongly convex neighborhoods on SO(3), reconstruction of uncertainty sets from samples, a time-covering theory for continuous dynamics, and a finite-dimensional relaxation that enables practical computation. The authors validate the approach on a spacecraft interception scenario with a tumbling target, demonstrating guaranteed reachability and efficient computation, while acknowledging current online-solver limitations and outlining future improvements.

Abstract

Intercepting dynamic objects in uncertain environments involves a significant unresolved challenge in modern robotic systems. Current control approaches rely solely on estimated information, and results lack guarantees of robustness and feasibility. In this work, we introduce a novel method to tackle the interception of targets whose motion is affected by known and bounded uncertainty. Our approach introduces new techniques of reachability analysis for rigid bodies, leveraged to guarantee feasibility of interception under uncertain conditions. We then propose a Reachability-Guaranteed Optimal Control Problem, ensuring robustness and guaranteed reachability to a target set of configurations. We demonstrate the methodology in the case study of an interception maneuver of a tumbling target in space.

Figures (6)

• Figure 1: Visualization of reachable sets on $SO(3)$ for different realizations of simulated motion on NASA Astrobee robot smith2016astrobee. All possible orientations are enclosed in a set, projected here on the unit-sphere for the robot's body-frame.
• Figure 2: For each $t \in [t_i, t_{i+1}]$, $\bm{F}^t(\boldsymbol{\phi})$ is bounded in time by $L_t$. Between states $\bm{x}_i$ and $\bm{x}_{i+1}$, the locus of the intersection of the two balls defines the region bounding all possible trajectories.
• Figure 3: Sketch of time coverages of the sets $\Xi_i$ () and $\Xi_{i+1}$ () by union of bounding ellipsoids $\mathcal{T}$ () to obtain the boundary of the $\partial \mathcal{R}_i$ ()
• Figure 4: Equation (\ref{['eq:ControlRadiusSet']}) states that the set of admissible inputs $\mathcal{U}(t)$ is conservatively contained by a ball of radius $R_{\delta}$ around the optimal input trajectory $\bm{u}^*(t)$.
• Figure 5: Reachable sets on $SO(3)$ projected onto the unit sphere and lifted to $\mathbb{R}^3$ to construct a convex polygon (). The last set on the lower left is the target of the guaranteed approach.

Theorems & Definitions (10)

• Definition 3.1: Strong Convexity do1992riemannian
• Proposition 3.1: Convex Neighborhoods
• Definition 3.2: GRP
• Lemma 4.1: Convexity Radius in $SO(3)$ hartley2010rotation
• Definition 4.1: Convex Hull on $SO(3)$hartley2010rotation
• Definition 5.1: RTC
• Theorem 5.1: Reachable time-covering contains Reachable sets
• Definition 6.1: RG-OCP
• Lemma 6.1: Convex combination leads to set containment
• Definition 6.2: Finite-dimensional