Tactical Game-theoretic Decision-making with Homotopy Class Constraints

TL;DR

Simulation-based testing in a Model Predictive Control framework demonstrates the capability of the framework in achieving globally optimal solutions while yielding a 78% average decrease in the computational time with respect to an implementation without the homotopic constraints.

Abstract

We propose a tactical homotopy-aware decision-making framework for game-theoretic motion planning in urban environments. We model urban driving as a generalized Nash equilibrium problem and employ a mixed-integer approach to tame the combinatorial aspect of motion planning. More specifically, by utilizing homotopy classes, we partition the high-dimensional solution space into finite, well-defined subregions. Each subregion (homotopy) corresponds to a high-level tactical decision, such as the passing order between pairs of players. The proposed formulation allows to find global optimal Nash equilibria in a computationally tractable manner by solving a mixed-integer quadratic program. Each homotopy decision is represented by a binary variable that activates different sets of linear collision avoidance constraints. This extra homotopic constraint allows to find solutions in a more efficient way (on a roundabout scenario on average 5-times faster). We experimentally validate the proposed approach on scenarios taken from the rounD dataset. Simulation-based testing in receding horizon fashion demonstrates the capability of the framework in achieving globally optimal solutions while yielding a 78% average decrease in the computational time with respect to an implementation without the homotopic constraints.

Figures (11)

• Figure 1: An example of a roundabout scenario with four players. The non-convex problem is represented in Frenet coordinate frame (lower half) which allow to easily incorporate homotopy class constraints to determine the optimal passing sequences. The conflict regions are used to construct collision avoidance areas and the binary variables $h^{1,2},h^{1,3}$, and $h^{2,4}$ encode the order in which vehicles enter their conflict regions, representing different homotopy classes. The lower section showcases two different tactical options and their corresponding trajectories. In the $s^1-s^3$ progress plane, fixing an homotopy class corresponds to activate different sets of linear constraints.
• Figure 2: (a) Shows a player $\nu$ traversing a path $r^\nu(s)$ defined by vectors $T^\nu(s)$ and $N^\nu(s)$. (b) Shows the path envelope $E^\nu$ and its boundary $\partial_B E^\nu$. (c) Shows an approximation of the envelope and boundary as a result of the discretization procedure.
• Figure 3: (a) Shows the conflict region for two players navigating an intersection, (b) A close-up of the conflict regions highlighting relevant points for its definition
• Figure 4: (a) Defines the area where an $(s^\nu(k), s^\mu(k))$ pair is a collision for the players $\nu$ and $\mu$ in the $s^\nu, s^\mu$ plane. (b) Shows the corresponding limits of the collisions area of the homotopy diagram in the global world frame.
• Figure 5: (a), (b) Show trajectories that belong to homotopy class 1 and 2 respectively, highlighting the active constraints in each case and the passing sequence.

Theorems & Definitions (8)

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