obca

Collision Avoidance for Point-Mass Models

Collision-Free Trajectory Generation

$\begin{aligned} \operatorname{dist}(\mathbb{E}(x), \mathbb{O}) & >\mathrm{d}_{\min } \\ & \Longleftrightarrow \exists \lambda \succeq_{\mathcal{K}^*} 0:(A p-b)^{\top} \lambda>\mathrm{d}_{\min },\left\|A^{\top} \lambda\right\|_* \leq 1 \end{aligned}$

K either the standard cone or the second-order cone, which allows us to model polyhedral and ellipsoidal obstacles.
not able to distinguish between "severe" and "less severe" colliding trajectories

Minimum-Penetration Trajectory Generation

$\begin{aligned} \operatorname{pen}(\mathbb{E}(x), \mathbb{O}) & <\mathrm{p}_{\max } \\ & \Longleftrightarrow \exists \lambda \succeq_{\mathcal{K}^*} 0:(b-A p)^{\top} \lambda<\mathrm{p}_{\max },\left\|A^{\top} \lambda\right\|_*=1 . \end{aligned}$

combine:

$\begin{aligned} & \operatorname{sd}(\mathbb{E}(x), \mathbb{O})>\mathrm{d} \\ & \Longleftrightarrow \exists \lambda \succeq \mathcal{K}^* 0:(A p-b)^{\top} \lambda>\mathrm{d},\left\|A^{\top} \lambda\right\|_*=1 . \\ & \end{aligned}$

obca

Collision Avoidance for Point-Mass Models

ref