ALGAMES: A Fast Solver for Constrained Dynamic Games (1910.09713v2)

Abstract: Dynamic games are an effective paradigm for dealing with the control of multiple interacting actors. This paper introduces ALGAMES (Augmented Lagrangian GAME-theoretic Solver), a solver that handles trajectory optimization problems with multiple actors and general nonlinear state and input constraints. Its novelty resides in satisfying the first order optimality conditions with a quasi-Newton root-finding algorithm and rigorously enforcing constraints using an augmented Lagrangian formulation. We evaluate our solver in the context of autonomous driving on scenarios with a strong level of interactions between the vehicles. We assess the robustness of the solver using Monte Carlo simulations. It is able to reliably solve complex problems like ramp merging with three vehicles three times faster than a state-of-the-art DDP-based approach. A model predictive control (MPC) implementation of the algorithm demonstrates real-time performance on complex autonomous driving scenarios with an update frequency higher than 60 Hz.

• The paper introduces ALGAMES, a fast Augmented Lagrangian solver for constrained multi-player dynamic games, designed for real-time trajectory optimization in robotics like autonomous driving.
• ALGAMES uses a combination of a quasi-Newton root-finding algorithm and an augmented Lagrangian formulation to find generalized Nash equilibrium strategies while handling nonlinear state and input constraints.
• Numerical results show that ALGAMES is significantly faster than existing methods (e.g., 3x faster than iLQGames), demonstrating robust, real-time performance for complex autonomous driving scenarios.

An Evaluation of ALGAMES: A Solver for Dynamic Games in Robotics

The paper "ALGAMES: A Fast Solver for Constrained Dynamic Games" presents a novel approach to solving constrained multi-player dynamic games, specifically tailored towards applications in robotics such as autonomous driving. This research introduces ALGAMES, an Augmented Lagrangian GAME-theoretic Solver, which is designed to provide real-time trajectory optimization for multiple interacting actors in dynamic environments. The algorithm is structured to identify generalized Nash equilibrium strategies while handling general nonlinear state and input constraints. An essential feature of ALGAMES is its execution speed, demonstrated to be significantly faster than existing methods like iLQGames across various scenarios.

Technical Overview and Methodology

The core innovation of ALGAMES lies in the integration of a quasi-Newton root-finding algorithm with an augmented Lagrangian formulation to rigorously enforce constraints while satisfying first-order optimality conditions. The paper outlines the formulation of the problem using this solver, where multiple agents are modeled as players in a dynamic Nash game. The solver iterates towards a Nash equilibrium point where all agents simultaneously predict and plan their trajectories in response to their environment.

Using a well-defined kinodynamic model integrating road boundary constraints and inter-agent collision avoidance, ALGAMES calculates strategies for agents that are efficient and dynamically feasible. The strategy output is updated in real-time, thereby enabling continuously adaptive feedback via an MPC implementation. This addresses the frozen robot problem and ensures the trajectory is updated at frequencies exceeding 60 Hz to manage rapidly changing scenarios effectively.

Numerical Results and Comparative Analysis

Through comprehensive experiments, ALGAMES demonstrates a robust capacity for rapidly computing solutions. The solver exhibits real-time performance even in complex autonomous driving scenarios, outperforming the state-of-the-art iLQGames solver by factors up to threefold. In scenarios involving ramp merging and intersections, ALGAMES reliably solves for trajectories without intensive tuning requirements due to its adaptive penalty mechanism.

A Monte Carlo analysis further validates the solver by analyzing 1000 samples with randomly perturbed initial states, showcasing ALGAMES’ robust convergence to satisfying Nash equilibria. Its speed and reliability are evidenced by shorter computation times and fewer iterations compared to existing methods. ALGAMES effectively manages constraint compliance despite scenarios with highly nonlinear and non-convex interactions.

Discussion and Implications

The constraints at the intersection of dynamics, robotics, and game theory that ALGAMES navigates make it a valuable tool for trajectory planning in environments replete with interactive agents. Several practical implications arise from this capability, such as enhanced performance in autonomous driving, mobile robotics, and competitive robot interactions (e.g., drone racing). Its efficient solver design offers potential improvements over traditional optimization approaches, providing faster and more reliable planning in dynamic, multi-agent contexts.

Additionally, the MPC framework allows ALGAMES to address discrepancies between assumed and actual agent objective functions. The system adapts its calculations in real-time—a critical feature when dealing with uncooperative or unpredictable agents.

The theoretical implications are significant, emphasizing Nash equilibrium solutions in robotics settings, compared to other equilibrium concepts such as Stackelberg due to their non-hierarchical nature. Future work could explore feedback Nash equilibria, and improvement in the solver's initialization routines to further decrease the occasional failure rate.

Conclusion and Future Directions

ALGAMES stands out as a robust, rapid-response solver for dynamic games involving multiple autonomous agents. Its design balances computational efficiency with flexibility in real-world applications, evidenced by superior performance metrics in various autonomous driving contexts. Looking forward, this algorithmic advancement opens pathways for refining tools capable of managing complex, real-time interactions in robotic systems. Future developments could integrate adaptive learning within the solver, further refining the estimation and response capabilities of ALGAMES for diverse applications in robotics and beyond.