Deep Reinforcement Learning (RL) has shown remarkable success in robotics with complex and heterogeneous dynamics. However, its vulnerability to unknown disturbances and adversarial attacks remains a significant challenge. In this paper, we propose a robust policy training framework that integrates model-based control principles with adversarial RL training to improve robustness without the need for external black-box adversaries. Our approach introduces a novel Hamilton-Jacobi reachability-guided disturbance for adversarial RL training, where we use interpretable worst-case or near-worst-case disturbances as adversaries against the robust policy. We evaluated its effectiveness across three distinct tasks: a reach-avoid game in both simulation and real-world settings, and a highly dynamic quadrotor stabilization task in simulation. We validate that our learned critic network is consistent with the ground-truth HJ value function, while the policy network shows comparable performance with other learning-based methods.

Detecting human actions is a crucial task for autonomous robots and vehicles, often requiring the integration of various data modalities for improved accuracy. In this study, we introduce a novel approach to Human Action Recognition (HAR) using language supervision named LS-HAR based on skeleton and visual cues. Our method leverages a language model to guide the feature extraction process in the skeleton encoder. Specifically, we employ learnable prompts for the language model conditioned on the skeleton modality to optimize feature representation. Furthermore, we propose a fusion mechanism that combines dual-modality features using a salient fusion module, incorporating attention and transformer mechanisms to address the modalities’ high dimensionality. This fusion process prioritizes informative video frames and body joints, enhancing the recognition accuracy of human actions. Additionally, we introduce a new dataset tailored for real-world robotic applications in construction sites, featuring visual, skeleton, and depth data modalities, named VolvoConstAct. This dataset serves to facilitate the training and evaluation of machine learning models to instruct autonomous construction machines for performing necessary tasks in real-world construction sites. To evaluate our approach, we conduct experiments on our dataset as well as three widely used public datasets: NTU-RGB+D, NTU-RGB+D 120, and NW-UCLA. Results reveal that our proposed method achieves promising performance across all datasets, demonstrating its robustness and potential for various applications. The code, dataset, and demonstration of real-machine experiments are available at: https://mmahdavian.github.io/ls_har/

This paper presents a novel shared autonomy and baseline policy adapting framework for human-robot interactions in high-level context-aware robotic tasks. With a unique methodology that leverages hierarchies in decision-making as well as variational analysis of human policy, we propose a mathematical model of shared autonomy policy. The framework aims at interpretable high-level decision-making for efficient robot operation with human in the loop. We modeled the decision-making process using hierarchical Markov decision processes (MDPs) in an algorithm we calledpolicy adapting, where the autonomous system policy is adapted, and hence, shaped by incorporating design variables contextual to the robot, human, task, and pre-training. By integrating deep reinforcement learning within a multi-agent hierarchical context, we present an end-to-end algorithm to train a baseline policy designed for shared autonomy. We showcase the effectiveness of our framework, and particularly the interplay between different design elements and human's skill level, in a pilot study with a human user in a simulated sequence of high-level pick-and-place tasks. The proposed framework advances the state-of-the-art in shared autonomy for robotic tasks, but can also be applied to other domains of autonomous operation.

The maritime industry's continuous commitment to sustainability has led to a dedicated exploration of methods to reduce vessel fuel consumption. This paper undertakes this challenge through a machine learning approach, leveraging a real-world dataset spanning two years of a passenger vessel in west coast Canada. Our focus centers on the creation of a time series forecasting model given the dynamic and static states, actions, and disturbances. This model is designed to predict dynamic states based on the actions provided, subsequently serving as an evaluative tool to assess the proficiency of the vessel's operation under the captain's guidance. Additionally, it lays the foundation for future optimization algorithms, providing valuable feedback on decision-making processes. To facilitate future studies, our code is available at https://github.com/pagand/model_optimze_vessel/tree/AAAI.

In end-to-end autonomous driving, current sensor fusion and navigational control techniques used by imitation learning algorithms are insufficient in challenging scenarios involving multiple dynamic agents and result in poor driving capabilities. To tackle this issue, we introduce DMFuser, a transformer-based algorithm that employs knowledge distillation between multi-task student and single-task teachers and combines attention and convolutions to fuse multiple RGB-D camera representations to produce vehicular navigational commands (throttle, steering and brake). Our model incorporates two modules. The first module, perception, encodes data from RGB-D cameras for tasks like semantic segmentation, semantic depth cloud (SDC) mapping, and traffic light state recognition. To enhance feature extraction and fusion from both RGB and depth sources, we harness local and global capabilities of convolution and transformer modules. We employ an attention-CNN fusion structure to effectively learn and fuse RGB and SDC map features. Subsequently, the control module decodes these features along with supplementary data, containing environment’s static and dynamic information, to predict waypoints and vehicular control actions. We evaluate the model and conduct a comparative analysis, in various scenarios, weather conditions, and traffic situations, spanning from normal to adversarial in the CARLA simulator. We achieve better or comparable results in term of driving score (DS) and other metrics with respect to our baselines. Also, our ablation studies demonstrate the effectiveness of our contributions to improve the driving skills. Our code is available at the following github page: https://github.com/pagand/e2etransfuser

Long-term human trajectory prediction is a challenging yet critical task in robotics and autonomous systems. Prior work that studied how to predict accurate short-term human trajectories with only unimodal features often failed in long-term prediction. Reinforcement learning provides a good solution for learning human long-term behaviors but can suffer from challenges in data efficiency and optimization. In this work, we propose a long-term human trajectory forecasting framework that leverages a guided diffusion model to generate diverse long-term human behaviors in a high-level latent action space, obtained via a hierarchical action quantization scheme using a VQ-VAE to discretize continuous trajectories and the available context. The latent actions are predicted by our guided diffusion model, which uses physics-inspired guidance at test time to constrain generated multimodal action distributions. Specifically, we use reachability analysis during the reverse denoising process to guide the diffusion steps toward physically feasible latent actions. We evaluate our framework on two publicly available human trajectory forecasting datasets: SFU-Store-Nav and JRDB, and extensive experimental results show that our framework achieves superior performance in long-term human trajectory forecasting.

In this paper, we greatly expand the capability of robots to perform the follow-ahead task and variations of this task through development of a neural network model to predict future human motion from an observed human motion history. We propose a non-autoregressive transformer architecture to leverage its parallel nature for easier training and fast, accurate predictions at test time. The proposed architecture divides human motion prediction into two parts: 1) the human trajectory, which is the 3D positions of the hip joint over time, and 2) the human pose which is the 3D positions of all other joints over time with respect to a fixed hip joint. We propose to make the two predictions simultaneously, as the shared representation can improve the model performance. Therefore, the model consists of two sets of encoders and decoders. First, a multi-head attention module applied to encoder outputs improves human trajectory. Second, another multi-head self-attention module applied to encoder outputs concatenated with decoder outputs facilitates the learning of temporal dependencies. Our model is well-suited for robotic applications in terms of test accuracy and speed, and compares favorably with respect to state-of-the-art methods. We demonstrate the real-world applicability of our work via the Robot Follow-Ahead task, a challenging yet practical case study for our proposed model. The human motion predicted by our model enables the robot follow-ahead in scenarios that require taking detailed human motion into account such as sit-to-stand, stand-to-sit. It also enables simple control policies to trivially generalize to many different variations of human following, such as follow-beside. Our code and data are available at the following Github page: https://github.com/mmahdavian/STPOTR

Human body motion prediction is a fundamental part of many human-robot applications. Despite the recent progress in the area, most studies predict human body motion relative to a fixed joint and only limit their model to predict one possible future motion, or both. However, due to the complex nature of human motion, a single prediction cannot adequately reflect the many possible movements one can make. Also, for any robotics application, prediction of the full human body motion including the absolute 3D trajectory - not just a 3D body pose relative to the hip joint - is needed. In this paper, we try to address these two shortcomings by proposing a transformer-based generative model for forecasting multiple diverse human motions. Our model generates$N$future possible body motions given the human motion history. This is achieved by first predicting the pose of the body relative to the hip joint as was done in prior work. Then, our proposed Hip Prediction Module predicts the trajectory of the hip position relative to a global reference frame for each predicted pose frame, an aspect of human body motion neglected by previous work. To obtain a set of diverse predicted motions, we introduce a similarity loss that penalizes the pairwise sample distance. Our system not only outperforms the state-of-the-art in human motion prediction, but also is able to predict a diverse set of future human body motions, including the hip trajectory.

Although the Bayesian paradigm offers a formal framework for estimating the entire probability distribution over uncertain parameters, its online implementation can be challenging due to high computational costs. We suggest the Adaptive Recursive Markov Chain Monte Carlo (ARMCMC) method, which eliminates the shortcomings of conventional online techniques while computing the entire probability density function of model parameters. The limitations to Gaussian noise, the application to only linear in the parameters (LIP) systems, and the persistent excitation (PE) needs are some of these drawbacks. In ARMCMC, a temporal forgetting factor (TFF)-based variable jump distribution is proposed. The forgetting factor can be presented adaptively using the TFF in many dynamical systems as an alternative to a constant hyperparameter. By offering a trade-off between exploitation and exploration, the specific jump distribution has been optimised towards hybrid/multi-modal systems that permit inferences among modes. These trade-off are adjusted based on parameter evolution rate. We demonstrate that ARMCMC requires fewer samples than conventional MCMC methods to achieve the same precision and reliability. We demonstrate our approach using parameter estimation in a soft bending actuator and the Hunt-Crossley dynamic model, two challenging hybrid/multi-modal benchmarks. Additionally, we compare our method with recursive least squares and the particle filter, and show that our technique has significantly more accurate point estimates as well as a decrease in tracking error of the value of interest.

Nowadays, transportation networks face the challenge of sub-optimal control policies that can have adverse effects on human health, the environment, and contribute to traffic congestion. Increased levels of air pollution and extended commute times caused by traffic bottlenecks make intersection traffic signal controllers a crucial component of modern transportation infrastructure. Despite several adaptive traffic signal controllers in literature, limited research has been conducted on their comparative performance. Furthermore, despite carbon dioxide (CO2) emissions' significance as a global issue, the literature has paid limited attention to this area. In this report, we propose EcoLight, a reward shaping scheme for reinforcement learning algorithms that not only reduces CO2 emissions but also achieves competitive results in metrics such as travel time. We compare the performance of tabular Q-Learning, DQN, SARSA, and A2C algorithms using metrics such as travel time, CO2 emissions, waiting time, and stopped time. Our evaluation considers multiple scenarios that encompass a range of road users (trucks, buses, cars) with varying pollution levels.

We propose a novel methodology for robotic follow-ahead applications that address the critical challenge of obstacle and occlusion avoidance. Our approach effectively navigates the robot while ensuring avoidance of collisions and occlusions caused by surrounding objects. To achieve this, we developed a high-level decision-making algorithm that generates short-term navigational goals for the mobile robot. Monte Carlo Tree Search is integrated with a Deep Reinforcement Learning method to enhance the performance of the decision-making process and generate more reliable navigational goals. Through extensive experimentation and analysis, we demonstrate the effectiveness and superiority of our proposed approach in comparison to the existing follow-ahead human-following robotic methods. Our code is available at https://github.com/saharLeisiazar/follow-ahead-ros.

Cooperative multi-agent problems often require coordination between agents, which can be achieved through a centralized policy that considers the global state. Multi-agent policy gradient (MAPG) methods are commonly used to learn such policies, but they are often limited to problems with low-level action spaces. In complex problems with large state and action spaces, it is advantageous to extend MAPG methods to use higher-level actions, also known as options, to improve the policy search efficiency. However, multi-robot option executions are often asynchronous, that is, agents may select and complete their options at different time steps. This makes it difficult for MAPG methods to derive a centralized policy and evaluate its gradient, as centralized policy always select new options at the same time. In this work, we propose a novel, conditional reasoning approach to address this problem and demonstrate its effectiveness on representative option-based multi-agent cooperative tasks through empirical validation. Find code and videos at: https://sites.google.com/view/mahrlsupp/

Collaborative autonomous multi-agent systems covering a specified area have many potential applications. Traditional approaches for such problems involve designing model-based control policies; however, state-of-the-art classical control policy still exhibits a large degree of sub-optimality. We present a combined reinforcement learning (RL) and control approach for the multi-agent coverage problem involving agents with second-order dynamics, with the RL component being based on the Multi-Agent Proximal Policy Optimization Algorithm (MAPPO). Our proposed network architecture includes the incorporation of LSTM and self-attention, which allows the trained policy to adapt to a variable number of agents. Our trained policy significantly outperforms the state-of-the-art classical control policy. We demonstrate our proposed method in a variety of simulated experiments.

Although human navigational intent inference has been studied in the literature, none have adequately considered both the dynamics that describe human motion and internal human parameters that may affect human navigational behaviour. In this paper, we propose a general probabilistic framework to infer the probability distribution over future navigational states of a human. Our framework incorporates an extended Dubins car dynamics to model human movement, which captures differences in human navigational behaviour depending on their position, heading, and movement speed. We assume a noisily rational model of human behaviour that incorporates a) human navigational intent that may change over time, b) how optimal a person's actions are given the navigational intent, and c) how far ahead in time a person considers when choosing navigational actions. These parameters are recursively and continuously updated in a Bayesian fashion. To make the Bayesian update and inference tractable, we exploit properties of the time-to-reach value function from optimal control and the extended Dubins car dynamics to construct a utility function on which the human policy is based, and employ particle representations of probability distributions where necessary. We demonstrate the effectiveness of our method by comparing our results with a recent approach using synthetic data and validate it on real world data.

Robots and artificial agents that interact with humans should be able to do so without bias and inequity, but facial perception systems have notoriously been found to work more poorly for certain groups of people than others. In our work, we aim to build a system that can perceive humans in a more transparent and inclusive manner. Specifically, we focus on dynamic expressions on the human face, which are difficult to collect for a broad set of people due to privacy concerns and the fact that faces are inherently identifiable. Furthermore, datasets collected from the Internet are not necessarily representative of the general population. We address this problem by offering a Sim2Real approach in which we use a suite of 3D simulated human models that enables us to create an auditable synthetic dataset covering 1) underrepresented facial expressions, outside of the six basic emotions, such as confusion; 2) ethnic or gender minority groups; and 3) a wide range of viewing angles that a robot may encounter a human in the real world. By augmenting a small dynamic emotional expression dataset containing 123 samples with a synthetic dataset containing 4536 samples, we achieved an improvement in accuracy of 15% on our own dataset and 11 % on an external benchmark dataset, compared to the performance of the same model architecture without synthetic training data. We also show that this additional step improves accuracy specifically for racial minorities when the architecture's feature extraction weights are trained from scratch.

Co-speech gestures are a principal component in conveying messages and enhancing interaction experiences between humans and critical ingredients in human-agent interaction, including virtual agents and robots. Existing machine learning approaches have yielded only marginal success in learning speech-to-motion at the frame level. Current methods generate repetitive gesture sequences that lack appropriateness with respect to the speech context. To tackle this challenge, we take inspiration from successes in natural language processing on context and long-term dependencies, and propose a new framework that views text-to-gesture as machine translation, where gestures are words in another (non-verbal) language. We propose a vector-quantized variational autoencoder structure as well as training techniques to learn a rigorous representation of gesture sequences. We then translate input text into a discrete sequence of associated gesture chunks in the learned gesture space. Ultimately, we use translated gesture tokens from the input text as an input to the autoencoder's decoder to produce gesture sequences. Subjective and objective evaluations confirm the success of our approach in terms of appropriateness, human-likeness, and diversity. We also introduce new objective metrics using the quantized gesture representation.

Safety is an important topic in autonomous driving since any collision may cause serious injury to people and damage to property. Hamilton-Jacobi (HJ) Reachability is a formal method that verifies safety in multi-agent interaction and provides a safety controller for collision avoidance. However, due to the worst-case assumption on the car’s future behaviours, reachability might result in too much conservatism such that the normal operation of the vehicle is badly hindered. In this paper, we leverage the power of trajectory prediction and propose a prediction-based reachability framework to compute safety controllers. Instead of always assuming the worst case, we cluster the car’s behaviors into multiple driving modes, e.g. left turn or right turn. Under each mode, a reachability-based safety controller is designed based on a less conservative action set. For online implementation, we first utilize the trajectory prediction and our proposed mode classifier to predict the possible modes, and then deploy the corresponding safety controller. Through simulations in a T-intersection and an 8-way roundabout, we demonstrate that our prediction-based reachability method largely avoids collision between two interacting cars and reduces the conservatism that the safety controller brings to the car’s original operation.

This paper investigates a hybrid solution which combines deep reinforcement learning (RL) and classical trajectory planning for the "following in front" application. Here, an autonomous robot aims to stay ahead of a person as the person freely walks around. Following in front is a challenging problem as the user’s intended trajectory is unknown and needs to be estimated, explicitly or implicitly, by the robot. In addition, the robot needs to find a feasible way to safely navigate ahead of human trajectory. Our deep RL module makes decisions at a high level by implicitly estimates the human trajectory and produces short-term navigational goals to guide the robot. These goals are used by a trajectory planner, which is responsible for low-level execution, to smoothly navigate the robot to the short-term goals, and eventually in front of the user. We employ curriculum learning in the deep RL module to efficiently achieve a high return. Our system outperforms the state-of-the-art in following ahead and is more reliable compared to end-to-end alternatives in both the simulation and real world experiments. In contrast to a pure deep RL approach, we demonstrate zero-shot transfer of the trained policy from simulation to the real world.

Hamilton-Jacobi (HJ) reachability analysis is a powerful technique used to verify the safety of autonomous systems. HJ reachability is ideal for analysing nonlinear systems with disturbances and flexible set representations. A drawback to this approach is that it suffers from the curse of dimensionality, which prevents real-time deployment on safety-critical systems. In this paper, we show that a customized hardware design on an Field Programmable Gate Array (FPGA) could accelerate 4D grid-based HJ reachability analysis up to 14 times compared to an optimized implementation and 103 times compared to state-of-the-art MATLAB toolboxes on a 16-thread CPU. Because of this, we are able to achieve guaranteed real- time collision avoidance in dynamic environments that abruptly change with a 4D car model by re-solving the HJ partial differential equation (PDE) at a frequency of 4Hz on an FPGA. Our design can overcome the complex data access pattern while taking advantage of the parallel nature of the computations for solving the HJ PDE. The low latency of our computation is consistent, which is crucial for safety-critical systems. The methodology presented here is without loss of generality: it can potentially be applied to different systems dynamics, and more- over, leveraged for higher dimensional systems. We validate our approach in real world collision avoidance experiments with a robot car in a changing environment. We also provide the code of our hardware design and an AWS AFI image.

Understanding human navigational intent is essential for robots to be able to interact with and navigate around humans safely and naturally. Current methods typically perform inference through only one mode of perception such as human motion trajectory, and a single theoretical framework such as a learning-based or classical approach. In contrast, this paper studies prediction of human navigational intent using multimodal perception within a hybrid framework. Our framework consists of two modules: a) a learning-based prediction module to predict a human’s future goal position, and b) a classical control theory-inspired reconstruction module to reconstruct a possible future trajectory or a set of possible future positions using the predicted future goal position. For the prediction module, we propose an end-to-end LSTM-CNN hybrid neural network for predicting a human’s future position in the real world, given human motion, human body pose and head orientation. This visual information from an egocentric perspective is used to make predictions of a human’s future position in world space, essential for robotic navigation algorithms and planning. In the reconstruction module, we present two control theoretic methods to reconstruct possible future trajectories of human: trajectory generation for differentially flat system and reachability analysis. We evaluate the performance of our framework on a newly collected dataset called SFU-Store-Nav. Experimental results reveal that our method outperforms various baselines especially when a relatively small amount of data is available.

Model-free reinforcement learning (RL) is a powerful approach for learning control policies directly from high-dimensional state and observation. However, it tends to be data-inefficient, which is especially costly in robotic learning tasks. On the other hand, optimal control does not require data if the system model is known, but cannot scale to models with high-dimensional states and observations. To exploit benefits of both model-free RL and optimal control, we propose time-to-reach-based (TTR-based) reward shaping, an optimal control-inspired technique to alleviate data inefficiency while retaining advantages of model-free RL. This is achieved by summarizing key system model information using a TTR function to greatly speed up the RL process, as shown in our simulation results. The TTR function is defined as the minimum time required to move from any state to the goal under assumed system dynamics constraints. Since the TTR function is computationally intractable for systems with high-dimensional states, we compute it for approximate, lower-dimensional system models that still captures key dynamic behaviors. Our approach can be flexibly and easily incorporated into any model-free RL algorithm without altering the original algorithm structure, and is compatible with any other techniques that may facilitate the RL process. We evaluate our approach on two representative robotic learning tasks and three well-known model-free RL algorithms, and show significant improvements in data efficiency and performance.

Model-free Reinforcement Learning (RL) offers an attractive approach to learn control policies for high dimensional systems, but its relatively poor sample complexity often necessitates training in simulated environments. Even in simulation, goal-directed tasks whose natural reward function is sparse remain intractable for state-of-the-art model-free algorithms for continuous control. The bottleneck in these tasks is the prohibitive amount of exploration required to obtain a learning signal from the initial state of the system. In this work, we leverage physical priors in the form of an approximate system dynamics model to design a curriculum for a model-free policy optimization algorithm. Our Backward Reachability Curriculum (BaRC) begins policy training from states that require a small number of actions to accomplish the task, and expands the initial state distribution backwards in a dynamically-consistent manner once the policy optimization algorithm demonstrates sufficient performance. BaRC is general, in that it can accelerate training of any model-free RL algorithm on a broad class of goal-directed continuous control MDPs. Its curriculum strategy is physically intuitive, easy-to-tune, and allows incorporating physical priors to accelerate training without hindering the performance, flexibility, and applicability of the model-free RL algorithm. We evaluate our approach on two representative dynamic robotic learning problems and find substantial performance improvement relative to previous curriculum generation techniques and naive exploration strategies.

Hamilton-Jacobi (HJ) reachability provides a flexible framework for the verification of safety in robotic systems: it accounts for nonlinear system dynamics and provides safety-preserving controllers. However, computational scalability limits its direct application to systems of less than five continuous state dimensions. To alleviate this computational burden, system decomposition methods have been proposed; however, safety guarantees are lost in situations involving “leaking corners which arise when there are conflicting controls between subsystems. In this paper, a coupled HJ formulation is presented, which addresses leaking corners and guarantees safety, while incorporating dimensionality reduction. We demonstrate our method in two examples, one of which is a vehicle obstacle avoidance problem with a 5D car model, whose HJ computation was previously considered to be intractable.

Reach-avoid problems involve driving a system to a set of desirable configurations while keeping it away from undesirable ones. Providing mathematical guarantees for such scenarios is challenging but have numerous potential practical applications. Due to the challenges, analysis of reach-avoid problems involves making trade-offs between generality of system dynamics, generality of problem setups, optimality of solutions, and computational complexity. In this paper, we combine sum-of-squares optimization and dynamic programming to address the reach-avoid problem, and provide a conservative solution that maintains reaching and avoidance guarantees. Our method is applicable to polynomial system dynamics and to general problem setups, and is more computationally scalable than previous related methods. Through a numerical example involving two single integrators, we validate our proposed theory and compare our method to Hamilton-Jacobi reachability. Having validated our theory, we demonstrate the computational scalability of our method by computing the reach-avoid set of a system involving two kinematic cars.

Hamilton-Jacobi (HJ) reachability is a method that provides rigorous analyses of the safety properties of dynamical systems. These guarantees can be provided by the computation of a backward reachable set (BRS), which represents the set of states from which the system may be driven into violating safety properties despite the system's best effort to remain safe. Unfortunately, the complexity of the BRS computation scales exponentially with the number of state dimensions. Although numerous approximation techniques are able to tractably provide conservative estimates of the BRS, they often require restrictive assumptions about system dynamics without providing an exact solution. In this paper we propose a general method for decomposing dynamical systems. Even when the resulting subsystems are coupled, relatively high-dimensional BRSs that were previously intractable or expensive to compute can now be quickly and exactly computed in lower-dimensional subspaces. As a result, the curse of dimensionality is alleviated to a large degree without sacrificing optimality. We demonstrate our theoretical results through a 3D Dubins Car model and a 6D Acrobatic Quadrotor model.

We consider a reach-avoid differential game, in which one of the players aims to steer the system into a target set without violating a set of state constraints, while the other player tries to prevent the first from succeeding; the system dynamics, target set, and state constraints may all be time-varying. The analysis of this problem plays an important role in collision avoidance, motion planning and aircraft control, among other applications. Previous methods for computing the guaranteed winning initial conditions and strategies for each player have either required augmenting the state vector to include time, or have been limited to problems with either no state constraints or entirely static targets, constraints and dynamics. To incorporate time-varying dynamics, targets and constraints without the need for state augmentation, we propose a modified Hamilton-Jacobi-Isaacs equation in the form of a double-obstacle variational inequality, and prove that the zero sublevel set of its viscosity solution characterizes the capture basin for the target under the state constraints. Through this formulation, our method can compute the capture basin and winning strategies for time-varying games at virtually no additional computational cost relative to the time-invariant case. We provide an implementation of this method based on well-known numerical schemes and show its convergence through a simple example; we include a second example in which our method substantially outperforms the state augmentation approach.