Local Convergence of the Sequential Quadratic Method for Differential Games

Tanikawa, Akio; Mukai, H.; Xu, Min

doi:10.5687/sss.2012.11

Cited by 3 publications

(4 citation statements)

References 6 publications

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“…While structurally similar to ILQR, our approach solves a LQ game at each iteration instead of a LQR problem. This core idea is related to the sequential linear-quadratic method of [38,39], which is restricted to the two-player zero-sum context. In this paper, we show that LQ approximations can be applied to N -player, general-sum games, provide a theoretical characterization of convergence properties, and, significantly, demonstrate that our approach is faster than existing approaches and is easily real-time for moderate to large-scale problems.…”

Section: Iterative Linear-quadratic (Lq) Methodsmentioning

confidence: 99%

“…Moreover, receding horizon invocations warmstarted every 100 ms can be solved in < 50 (and often < 20) ms. All computation times are reported for single-threaded operation on a 2017 MacBook Pro with a 2.8 GHz Intel Core i7 CPU. For reference, the iterative best response scheme of [44] reports solving a receding horizon two-player zero-sum racing game at 2 Hz, and the method of [39] reportedly takes several minutes to converge for a different two-player zero-sum example. The dynamics and costs in both cases differ from those in Section V (or are not clearly reported); nonetheless, the runtime of our approach compares favorably.…”

Section: Computational Complexity and Runtimementioning

confidence: 99%

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Efficient Iterative Linear-Quadratic Approximations for Nonlinear Multi-Player General-Sum Differential Games

Fridovich-Keil¹,

Ratner²,

Peters³

et al. 2019

Preprint

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Differential games offer a powerful theoretical framework for formulating safety and robustness problems in optimal control. Unfortunately, numerical solution techniques for general nonlinear dynamical systems scale poorly with state dimension and are rarely used in applications requiring realtime computation. For single-agent optimal control problems, however, local methods based on efficiently solving iterated approximations with linear dynamics and quadratic costs are becoming increasingly popular. We take inspiration from one such method, the iterative linear quadratic regulator (ILQR), and observe that efficient algorithms also exist to solve multiplayer linear-quadratic games. Whereas ILQR converges to a local solution of the optimal control problem, if our method converges it returns a local Nash equilibrium of the differential game. We benchmark our method in a three-player generalsum simulated example, in which it takes < 0.75 s to identify a solution and < 50 ms to solve warm-started subproblems in a receding horizon. We also demonstrate our approach in hardware, operating in real-time and following a 10 s receding horizon.

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Section: Iterative Linear-quadratic (Lq) Methodsmentioning

confidence: 99%

Section: Computational Complexity and Runtimementioning

confidence: 99%

Efficient Iterative Linear-Quadratic Approximations for Nonlinear Multi-Player General-Sum Differential Games

Fridovich-Keil¹,

Ratner²,

Peters³

et al. 2019

Preprint

View full text Add to dashboard Cite

show abstract

“…However, iterative bestresponse methods may require a large number of iterations, and; thus, they may be computationally prohibitive. Sequential linear-quadratic methods were applied to two-player zero-sum differential games in [14], [15]. Methods similar to differential dynamic programming were developed for gametheoretic planning.…”

Section: Related Work a Game-theoretic Planningmentioning

confidence: 99%

Efficient Constrained Multi-Agent Interactive Planning using Constrained Dynamic Potential Games

Bhatt¹,

Ayberk²,

Mehr³

2022

Preprint

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Although dynamic games provide a rich paradigm for modeling agents' interactions, solving these games for real-world applications is often challenging. Many real-wold interactive settings involve general nonlinear state and input constraints which couple agents' decisions with one another. In this work, we develop an efficient and fast planner for interactive planning in constrained setups using a constrained game-theoretical framework. Our key insight is to leverage the special structure of agents' objective and constraint functions that are common in multi-agent interactions for fast and reliable planning. More precisely, we identify the structure of agents' cost functions under which the resulting dynamic game is an instance of a constrained potential dynamic game. Constrained potential dynamic games are a class of games for which instead of solving a set of coupled constrained optimal control problems, a Nash equilibrium can be found by solving a single constrained optimal control problem. This simplifies constrained interactive trajectory planning significantly. We compare the performance of our method in a navigation setup involving four planar agents and show that our method is on average 20 times faster than the state-of-the-art. We further provide experimental validation of our proposed method in a navigation setup involving one quadrotor and two humans.

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“…To find equilibria of general differential games, sequential linear-quadratic methods were proposed for two-player zerosum differential games [18,19]. To enable scalable interactive trajectory planning for a broad class of differential games, recently, a local iterative algorithm was proposed in [20] where the analytic solution to the Linear Quadratic games [4] was exploited for approximating the equilibria of general-sum differential games.…”

Section: B Approximate Solutions To Differential Gamesmentioning

confidence: 99%

Potential iLQR: A Potential-Minimizing Controller for Planning Multi-Agent Interactive Trajectories

Kavuncu¹,

Ayberk²,

Mehr³

2021

Preprint

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Many robotic applications involve interactions between multiple agents where an agent's decisions affect the behavior of other agents. Such behaviors can be captured by the equilibria of differential games which provide an expressive framework for modeling the agents' mutual influence. However, finding the equilibria of differential games is in general challenging as it involves solving a set of coupled optimal control problems. In this work, we propose to leverage the special structure of multi-agent interactions to generate interactive trajectories by simply solving a single optimal control problem, namely, the optimal control problem associated with minimizing the potential function of the differential game. Our key insight is that for a certain class of multi-agent interactions, the underlying differential game is indeed a potential differential game for which equilibria can be found by solving a single optimal control problem. We introduce such an optimal control problem and build on single-agent trajectory optimization methods to develop a computationally tractable and scalable algorithm for planning multi-agent interactive trajectories. We will demonstrate the performance of our algorithm in simulation and show that our algorithm outperforms the state-of-the-art game solvers. To further show the real-time capabilities of our algorithm, we will demonstrate the application of our proposed algorithm in a set of experiments involving interactive trajectories for two quadcopters.

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Local Convergence of the Sequential Quadratic Method for Differential Games

Cited by 3 publications

References 6 publications

Efficient Iterative Linear-Quadratic Approximations for Nonlinear Multi-Player General-Sum Differential Games

Efficient Iterative Linear-Quadratic Approximations for Nonlinear Multi-Player General-Sum Differential Games

Efficient Constrained Multi-Agent Interactive Planning using Constrained Dynamic Potential Games

Potential iLQR: A Potential-Minimizing Controller for Planning Multi-Agent Interactive Trajectories

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