In this article, we study the problem of multiple stochastic agents interacting in a dynamic game scenario with continuous state and action spaces. We define a new notion of stochastic Nash equilibrium for boundedly rational agents, which we call the entropic cost equilibrium (ECE). We show that ECE is a natural extension to multiple agents of maximum entropy optimality for a single agent. We solve both the “forward” and “inverse” problems for the multi-agent ECE game. For the forward problem, we provide a Riccati algorithm to compute closed-form ECE feedback policies for the agents, which are exact in the linear-quadratic-gaussian case. We give an iterative variant to find locally ECE feedback policies for the nonlinear case. For the inverse problem, we present an algorithm to infer the cost functions of the multiple interacting agents given noisy, boundedly rational input and state trajectory examples from agents acting in an ECE. The effectiveness of our algorithms is demonstrated in a simulated multi-agent collision avoidance scenario, and with data from the INTERACTION traffic dataset. In both cases, we show that, by taking into account the agents' game theoretic interactions using our algorithm, a more accurate model of agents' costs can be learned, compared with standard inverse optimal control methods.

We present a novel approach for risk-aware planning with human agents in multi-agent traffic scenarios. Our approach takes into account the wide range of human driver behaviors on the road, from aggressive maneuvers like speeding and overtaking, to conservative traits like driving slowly and conforming to the right-most lane. In our approach, we learn a mapping from a data-driven human driver behavior model called the CMetric to a driver's entropic risk preference. We then use the derived risk preference within a game-theoretic risk-sensitive planner to model risk-aware interactions among human drivers and an autonomous vehicle in various traffic scenarios. We demonstrate our method in a merging scenario, where our results show that the final trajectories obtained from the risk-aware planner generate desirable emergent behaviors. Particularly, our planner recognizes aggressive human drivers and yields to them while maintaining a greater distance from them. In a user study, participants were able to distinguish between aggressive and conservative simulated drivers based on trajectories generated from our risk-sensitive planner. We also observe that aggressive human driving results in more frequent lane-changing in the planner. Finally, we compare the performance of our modified risk-aware planner with existing methods and show that modeling human driver behavior leads to safer navigation.

In this article, we propose a nonlinear receding horizon game-theoretic planner for autonomous cars in competitive scenarios with other cars. The online planner is specifically formulated for a multiple-car autonomous racing game, in which each car tries to advance along a given track as far as possible with respect to the other cars. The algorithm extends previous work on game-theoretic planning for single-integrator agents to be suitable for autonomous cars in the following ways: 1) by representing the trajectory as a piecewise polynomial; 2) incorporating bicycle kinematics into the trajectory; and 3) enforcing constraints on path curvature and acceleration. The game-theoretic planner iteratively plans a trajectory for the ego vehicle and then the other vehicles in sequence until convergence. Crucially, the trajectory optimization includes a sensitivity term that allows the ego vehicle to reason about how much the other vehicles will yield to the ego vehicle to avoid collisions. The resulting trajectories for the ego vehicle exhibit rich game strategies such as blocking, faking, and opportunistic overtaking. The game-theoretic planner is shown to significantly outperform a baseline planner using model-predictive control, which does not take interaction into account. The performance is validated in high-fidelity numerical simulations with three cars, in experiments with two small-scale autonomous cars, and in experiments with a full-scale autonomous car racing against a simulated vehicle (video is available at https://youtube.com/playlist?list=PLmIcLEh8KMje4rYBqRANDuKvqFvj7LCRp).

In this paper, we consider a two-player racing game, where an autonomous ego vehicle has to be controlled to race against an opponent vehicle, which is either autonomous or human-driven. The approach to control the ego vehicle is based on a Sensitivity-ENhanced NAsh equilibrium seeking (SENNA) method, which uses an iterated best response algorithm in order to optimize for a trajectory in a two-car racing game. This method exploits the interactions between the ego and the opponent vehicle that take place through a collision avoidance constraint. This game-theoretic control method hinges on the ego vehicle having an accurate model and correct knowledge of the state of the opponent vehicle. However, when an accurate model for the opponent vehicle is not available, or the estimation of its state is corrupted by noise, the performance of the approach might be compromised. For this reason, we augment the SENNA algorithm by enforcing Permissive RObust SafeTy (PROST) conditions using control barrier functions. The objective is to successfully overtake or to remain in the front of the opponent vehicle, even when the information about the latter is not fully available. The successful synergy between SENNA and PROST-antithetical to the notable rivalry between the two namesake Formula 1 drivers-is demonstrated through extensive simulated experiments.

Modeling the stochastic behavior of interacting agents is key for safe motion planning. In this paper, we study the interaction of risk-aware agents in a game-theoretical framework. Under the entropic risk measure, we derive an iterative algorithm for approximating the intractable feedback Nash equilibria of a risk-sensitive dynamic game. We use an iteratively linearized approximation of the system dynamics and a quadratic approximation of the cost function in solving a backward recursion for finding feedback Nash equilibria. In this respect, the algorithm shares a similar structure with DDP and iLQR methods. We conduct experiments in a set of challenging scenarios such as roundabouts. Compared to ignoring the game interaction or the risk sensitivity, we show that our risk-sensitive game-theoretic framework leads to more timeefficient, intuitive, and safe behaviors when facing underlying risks and uncertainty.

This paper introduces the Buffered Input Cell as a reciprocal collision avoidance method for multiple vehicles with high-order linear dynamics, extending recently proposed methods based on the Buffered Voronoi Cell [1] and generalized Voronoi diagrams [2]. We prove that if each vehicle's control input remains in its Buffered Input Cell at each time step, collisions will be avoided indefinitely. The method is fast, reactive, and only requires that each vehicle measures the relative position of neighboring vehicles. We incorporate this collision avoidance method as one layer of a complete lane change control stack for autonomous cars in a freeway driving scenario. The lane change control stack comprises a decision-making layer, a trajectory planning layer, a trajectory following feedback controller, and the Buffered Input Cell for collision avoidance. We show in simulations that collisions are avoided with multiple vehicles simultaneously changing lanes on a freeway. We also show in simulations that autonomous cars using the BIC method effectively avoid collisions with an aggressive human-driven car.

Kernel method has been applied to many multivariate statistical analysis techniques. In this paper, we investigated the regression properties of Kernel Partial Least Squares (KPLS) and compared it to the standard technique. Basic mathematical algorithms and application of KPLS were shown. We further established regression model based on KPLS and demonstrated the model by a numerical case.

In this paper, an approximation-based adaptive tracking control approach is proposed for a class of multiinput multioutput nonlinear systems. Based on the method of neural network, a novel adaptive controller is designed via backstepping design process. Furthermore, by introducing Nussbaum function, the issue of unknown control directions is handled. In the backstepping design process, the dynamic surface control technique is employed to avoid differentiating certain nonlinear functions repeatedly. Moreover, in order to reduce the number of adaptation laws, we do not use the neural networks to directly approximate the unknown nonlinear functions but the desired control signals. Finally, we provide two examples to illustrate the effectiveness of the proposed approach.