Gioele Zardini, Nicolas Lanzetti, Marco Pavone, Emilio Frazzoli
Challenged by urbanization and increasing travel needs, existing transportation systems need new mobility paradigms. In this article, we present the emerging concept of autonomous mobility-on-demand, whereby centrally orchestrated fleets of autonomous vehicles provide mobility service to customers. We provide a comprehensive review of methods and tools to model and solve problems related to autonomous mobility-on-demand systems. Specifically, we first identify problem settings for their analysis and control, from both operational and planning perspectives. We then review modeling aspects, including transportation networks, transportation demand, congestion, operational constraints, and interactions with existing infrastructure. Thereafter, we provide a systematic analysis of existing solution methods and performance metrics, highlighting trends and trade-offs. Finally, we present various directions for further research.
Gioele Zardini, David I. Spivak, Andrea Censi, Emilio Frazzoli
A compositional sheaf-theoretic framework for the modeling of complex event-based systems is presented. We show that event-based systems are machines, with inputs and outputs, and that they can be composed with machines of different types, all within a unified, sheaf-theoretic formalism. We take robotic systems as an exemplar of complex systems and rigorously describe actuators, sensors, and algorithms using this framework.
Gioele Zardini, Dejan Milojevic, Andrea Censi, Emilio Frazzoli
We consider the problem of co-designing embodied intelligence as a whole in a structured way, from hardware components such as propulsion systems and sensors to software modules such as control and perception pipelines. We propose a principled approach to formulate and solve complex embodied intelligence co-design problems, leveraging a monotone co-design theory. The methods we propose are intuitive and integrate heterogeneous engineering disciplines, allowing analytical and simulation-based modeling techniques and enabling interdisciplinarity. We illustrate through a case study how, given a set of desired behaviors, our framework is able to compute Pareto efficient solutions for the entire hardware and software stack of a self-driving vehicle.
Gioele Zardini, Nicolas Lanzetti, Mauro Salazar, Andrea Censi, Emilio Frazzoli, Marco Pavone
The design of Autonomous Vehicles (AVs) and the design of AVs-enabled mobility systems are closely coupled. Indeed, knowledge about the intended service of AVs would impact their design and deployment process, whilst insights about their technological development could significantly affect transportation management decisions. This calls for tools to study such a coupling and co-design AVs and AVs-enabled mobility systems in terms of different objectives. In this paper, we instantiate a framework to address such co-design problems. In particular, we leverage the recently developed theory of co-design to frame and solve the problem of designing and deploying an intermodal Autonomous Mobility-on-Demand system, whereby AVs service travel demands jointly with public transit, in terms of fleet sizing, vehicle autonomy, and public transit service frequency. Our framework is modular and compositional, allowing to describe the design problem as the interconnection of its individual components and to tackle it from a system-level perspective. Moreover, it only requires very general monotonicity assumptions and it naturally handles multiple objectives, delivering the rational solutions on the Pareto front and thus enabling policy makers to select a solution through political criteria. To showcase our methodology, we present a real-world case study for Washington D.C., USA. Our work suggests that it is possible to create user-friendly optimization tools to systematically assess the costs and benefits of interventions, and that such analytical techniques might gain a momentous role in policy-making in the future.
Gioele Zardini, Zelio Suter, Andrea Censi, Emilio Frazzoli
When designing autonomous systems, we need to consider multiple trade-offs at various abstraction levels, and the choices of single (hardware and software) components need to be studied jointly. In this work we consider the problem of designing the control algorithm as well as the platform on which it is executed. In particular, we focus on vehicle control systems, and formalize state-of-the-art control schemes as monotone feasibility relations. We then show how, leveraging a monotone theory of co-design, we can study the embedding of control synthesis problems into the task-driven co-design problem of a robotic platform. The properties of the proposed approach are illustrated by considering urban driving scenarios. We show how, given a particular task, we can efficiently compute Pareto optimal design solutions.
Gioele Zardini, Nicolas Lanzetti, Giuseppe Belgioioso, Christian Hartnik, Saverio Bolognani, Florian Dörfler, Emilio Frazzoli
The evolution of existing transportation systems,mainly driven by urbanization and increased availability of mobility options, such as private, profit-maximizing ride-hailing companies, calls for tools to reason about their design and regulation. To study this complex socio-technical problem, one needs to account for the strategic interactions of the heterogeneous stakeholders involved in the mobility ecosystem and analyze how they influence the system. In this paper, we focus on the interactions between citizens who compete for the limited resources of a mobility system to complete their desired trip. Specifically, we present a game-theoretic framework for multi-modal mobility systems, where citizens, characterized by heterogeneous preferences, have access to various mobility options and seek individually-optimal decisions. We study the arising game and prove the existence of an equilibrium, which can be efficiently computed via a convex optimization problem. Through both an analytical and a numerical case study for the classic scenario of Sioux Falls, USA, we illustrate the capabilities of our model and perform sensitivity analyses. Importantly, we show how to embed our framework into a "larger" game among stakeholders of the mobility ecosystem (e.g., municipality, Mobility Service Providers, and citizens), effectively giving rise to tools to inform strategic interventions and policy-making in the mobility ecosystem.
Andrea Censi, Emilio Frazzoli, Jonathan Lorand, Gioele Zardini
In many engineering applications it is useful to reason about "negative information". For example, in planning problems, providing an optimal solution is the same as giving a feasible solution (the "positive" information) together with a proof of the fact that there cannot be feasible solutions better than the one given (the "negative" information). We model negative information by introducing the concept of "norphisms", as opposed to the positive information of morphisms. A "nategory" is a category that has "nom"-sets in addition to hom-sets, and specifies the interaction between norphisms and morphisms. In particular, we have composition rules of the form morphism + norphism $\to$ norphism. Norphisms do not compose by themselves; rather, they use morphisms as catalysts. After providing several applied examples, we connect nategories to enriched category theory. Specifically, we prove that categories enriched in de Paiva's dialectica categories GC, in the case C = Set and equipped with a modified monoidal product, define nategories which satisfy additional regularity properties. This formalizes negative information categorically in a way that makes negative and positive morphisms equal citizens.
Gioele Zardini, Nicolas Lanzetti, Mauro Salazar, Andrea Censi, Emilio Frazzoli, Marco Pavone
The design of autonomous vehicles (AVs) and the design of AV-enabled mobility systems are closely coupled. Indeed, knowledge about the intended service of AVs would impact their design and deployment process, whilst insights about their technological development could significantly affect transportation management decisions. This calls for tools to study such a coupling and co-design AVs and AV-enabled mobility systems in terms of different objectives. In this paper, we instantiate a framework to address such co-design problems. In particular, we leverage the recently developed theory of co-design to frame and solve the problem of designing and deploying an intermodal Autonomous Mobility-on-Demand system, whereby AVs service travel demands jointly with public transit, in terms of fleet sizing, vehicle autonomy, and public transit service frequency. Our framework is modular and compositional, allowing one to describe the design problem as the interconnection of its individual components and to tackle it from a system-level perspective. To showcase our methodology, we present a real-world case study for Washington D.C., USA. Our work suggests that it is possible to create user-friendly optimization tools to systematically assess costs and benefits of interventions, and that such analytical techniques might gain a momentous role in policy-making in the future.
Gioele Zardini, Andrea Censi, Emilio Frazzoli
Designing cyber-physical systems is a complex task which requires insights at multiple abstraction levels. The choices of single components are deeply interconnected and need to be jointly studied. In this work, we consider the problem of co-designing the control algorithm as well as the platform around it. In particular, we leverage a monotone theory of co-design to formalize variations of the LQG control problem as monotone feasibility relations. We then show how this enables the embedding of control co-design problems in the higher level co-design problem of a robotic platform. We illustrate the properties of our formalization by analyzing the co-design of an autonomous drone performing search-and-rescue tasks and show how, given a set of desired robot behaviors, we can compute Pareto efficient design solutions.
Gioele Zardini, Nicolas Lanzetti, Laura Guerrini, Emilio Frazzoli, Florian Dörfler
Increasing urbanization and exacerbation of sustainability goals threaten the operational efficiency of current transportation systems and confront cities with complex choices with huge impact on future generations. At the same time, the rise of private, profit-maximizing Mobility Service Providers leveraging public resources, such as ride-hailing companies, entangles current regulation schemes. This calls for tools to study such complex socio-technical problems. In this paper, we provide a game-theoretic framework to study interactions between stakeholders of the mobility ecosystem, modeling regulatory aspects such as taxes and public transport prices, as well as operational matters for Mobility Service Providers such as pricing strategy, fleet sizing, and vehicle design. Our framework is modular and can readily accommodate different types of Mobility Service Providers, actions of municipalities, and low-level models of customers choices in the mobility system. Through both an analytical and a numerical case study for the city of Berlin, Germany, we showcase the ability of our framework to compute equilibria of the problem, to study fundamental tradeoffs, and to inform stakeholders and policy makers on the effects of interventions. Among others, we show tradeoffs between customers satisfaction, environmental impact, and public revenue, as well as the impact of strategic decisions on these metrics.
Alessandro Zanardi, Gioele Zardini, Sirish Srinivasan, Saverio Bolognani, Andrea Censi, Florian Dörfler, Emilio Frazzoli
Modern applications require robots to comply with multiple, often conflicting rules and to interact with the other agents. We present Posetal Games as a class of games in which each player expresses a preference over the outcomes via a partially ordered set of metrics. This allows one to combine hierarchical priorities of each player with the interactive nature of the environment. By contextualizing standard game theoretical notions, we provide two sufficient conditions on the preference of the players to prove existence of pure Nash Equilibria in finite action sets. Moreover, we define formal operations on the preference structures and link them to a refinement of the game solutions, showing how the set of equilibria can be systematically shrunk. The presented results are showcased in a driving game where autonomous vehicles select from a finite set of trajectories. The results demonstrate the interpretability of results in terms of minimum-rank-violation for each player.
Gioele Zardini, Nicolas Lanzetti, Andrea Censi, Emilio Frazzoli, Marco Pavone
The design of future mobility solutions and the design of the mobility systems they enable are closely coupled. Indeed, knowledge about the intended service of novel mobility solutions would impact their design and deployment process, whilst insights about their technological development could significantly affect transportation management policies. This requires tools to study such a coupling and co-design mobility systems in terms of different objectives. We present a framework to address such co-design problems, leveraging a mathematical theory of co-design to frame and solve the problem of designing and deploying an intermodal mobility system, whereby autonomous vehicles service travel demands jointly with micromobility solutions and public transit, in terms of fleets sizing, vehicle characteristics, and public transit service frequency. Our framework is modular and compositional, allowing one to describe the design as the interconnection of simple components and to tackle it from a systemic perspective. Moreover, it requires general monotonicity assumptions and naturally handles multiple objectives, delivering rational, actionable solutions for policy makers. We showcase our methodology in a case study of Washington D.C., USA. Our work suggests the possibility to create user-friendly optimization tools to systematically assess costs and benefits of interventions, and to inform policy-making in the future.
Xinling Li, Xiaotong Guo, Qingyi Wang, Gioele Zardini, Jinhua Zhao
Autonomous Mobility-on-Demand (AMoD) services offer an opportunity for improving passenger service while reducing pollution and energy consumption through effective vehicle coordination. A primary challenge in the autonomous fleets coordination is to tackle the inherent issue of supply-demand imbalance. A key strategy in resolving this is vehicle rebalancing, strategically directing idle vehicles to areas with anticipated future demand. Traditional research focuses on deterministic optimization using specific demand forecasts, but the unpredictable nature of demand calls for methods that can manage this uncertainty. This paper introduces the Deep Uncertainty Robust Optimization (DURO), a framework specifically designed for vehicle rebalancing in AMoD systems amidst uncertain demand based on neural networks for robust optimization. DURO forecasts demand uncertainty intervals using a deep neural network, which are then integrated into a robust optimization model. We assess DURO against various established models, including deterministic optimization with refined demand forecasts and Distributionally Robust Optimization (DRO). Based on real-world data from New York City (NYC), our findings show that DURO surpasses traditional deterministic models in accuracy and is on par with DRO, but with superior computational efficiency. The DURO framework is a promising approach for vehicle rebalancing in AMoD systems that is proven to be effective in managing demand uncertainty, competitive in performance, and more computationally efficient than other optimization models.
Jiarui Li, Alessandro Zanardi, Gioele Zardini
We present a novel algorithm for large-scale Multi-Agent Path Finding (MAPF) that enables fast, scalable planning in dynamic environments such as automated warehouses. Our approach introduces finite-horizon hierarchical factorization, a framework that plans one step at a time in a receding-horizon fashion. Robots first compute individual plans in parallel, and then dynamically group based on spatio-temporal conflicts and reachability. The framework accounts for conflict resolution, and for immediate execution and concurrent planning, significantly reducing response time compared to offline algorithms. Experimental results on benchmark maps demonstrate that our method achieves up to 60% reduction in time-to-first-action while consistently delivering high-quality solutions, outperforming state-of-the-art offline baselines across a range of problem sizes and planning horizons.
Runyu Zhang, Na Li, Asuman Ozdaglar, Jeff Shamma, Gioele Zardini
Risk sensitivity has become a central theme in reinforcement learning (RL), where convex risk measures and robust formulations provide principled ways to model preferences beyond expected return. Recent extensions to multi-agent RL (MARL) have largely emphasized the risk-averse setting, prioritizing robustness to uncertainty. In cooperative MARL, however, such conservatism often leads to suboptimal equilibria, and a parallel line of work has shown that optimism can promote cooperation. Existing optimistic methods, though effective in practice, are typically heuristic and lack theoretical grounding. Building on the dual representation for convex risk measures, we propose a principled framework that interprets risk-seeking objectives as optimism. We introduce optimistic value functions, which formalize optimism as divergence-penalized risk-seeking evaluations. Building on this foundation, we derive a policy-gradient theorem for optimistic value functions, including explicit formulas for the entropic risk/KL-penalty setting, and develop decentralized optimistic actor-critic algorithms that implement these updates. Empirical results on cooperative benchmarks demonstrate that risk-seeking optimism consistently improves coordination over both risk-neutral baselines and heuristic optimistic methods. Our framework thus unifies risk-sensitive learning and optimism, offering a theoretically grounded and practically effective approach to cooperation in MARL.
Jiarui Li, Alessandro Zanardi, Federico Pecora, Runyu Zhang, Gioele Zardini
Multi-Agent Path Finding is a fundamental problem in robotics and AI, yet most existing formulations treat planning and execution separately and address variants of the problem in an ad hoc manner. This paper presents a system-level framework for MAPF that integrates planning and execution, generalizes across variants, and explicitly models uncertainties. At its core is the MAPF system, a formal model that casts MAPF as a control design problem encompassing classical and uncertainty-aware formulations. To solve it, we introduce Finite-Horizon Closed-Loop Factorization (FICO), a factorization-based algorithm inspired by receding-horizon control that exploits compositional structure for efficient closed-loop operation. FICO enables real-time responses -- commencing execution within milliseconds -- while scaling to thousands of agents and adapting seamlessly to execution-time uncertainties. Extensive case studies demonstrate that it reduces computation time by up to two orders of magnitude compared with open-loop baselines, while delivering significantly higher throughput under stochastic delays and agent arrivals. These results establish a principled foundation for analyzing and advancing MAPF through system-level modeling, factorization, and closed-loop design.
Hans Riess, Yujun Huang, Matthew Klawonn, Gioele Zardini, Matthew Hale
Monotone co-design enables compositional engineering design by modeling components through feasibility relations between required resources and provided functionalities. However, its standard boolean formulation cannot natively represent quantitative criteria such as cost, confidence, or implementation choice. In practice, these quantities are often introduced through ad hoc scalarization or by augmenting the resource space, which obscures system structure and increases computational burden. We address this limitation by developing a quantale-enriched theory of co-design. We model resources and functionalities as quantale-enriched categories and design problems as quantale-enriched profunctors, thereby lifting co-design from boolean feasibility to general quantitative evaluation. We show that the fundamental operations of series, parallel, and feedback composition remain valid over arbitrary commutative quantales. We further introduce heterogeneous composition through change-of-base maps between quantales, enabling different subsystems to be evaluated in different local semantics and then composed in a common framework. The resulting theory unifies feasibility-, cost-, confidence-, and implementation-aware co-design within one compositional formalism. Numerical examples on a target-tracking system and a UAV delivery problem demonstrate the framework and highlight how native quantitative enrichment can avoid the architectural and computational drawbacks of boolean-only formulations.
Emma Chollet, Bryce Clarke, Michael Johnson, Maurine Songa, Vincent Wang, Gioele Zardini
Lenses are an important tool in applied category theory. While individual lenses have been widely used in applications, many of the mathematical properties of the corresponding categories of lenses have remained unknown. In this paper, we study the category of small categories and asymmetric delta lenses, and prove that it has several good exactness properties. These properties include the existence of certain limits and colimits, as well as so-called imported limits, such as imported products and imported pullbacks, which have arisen previously in applications. The category is also shown to be extensive, and it has an image factorisation system.
Vincent Abbott, Gioele Zardini
The study of abstraction and composition - the focus of category theory - naturally leads to sophisticated diagrams which can encode complex algebraic semantics. Consequently, these diagrams facilitate a clearer visual comprehension of diverse theoretical and applied systems. Complex algebraic structures - otherwise represented by a forest of symbols - can be encoded into diagrams with intuitive graphical rules. The prevailing paradigm for diagrammatic category theory are monoidal string diagrams whose specification reflects the axioms of monoidal categories. However, such diagrams struggle in accurately portraying crucial categorical constructs such as functors or natural transformations, obscuring central concepts such as the Yoneda lemma or the simultaneous consideration of hom-functors and products. In this work, we introduce functor string diagrams, a systematic approach for the development of categorical diagrams which allows functors, natural transformations, and products to be clearly represented. We validate their practicality in multiple dimensions. We show their usefulness for theoretical manipulations by proving the Yoneda lemma, show that they encompass monoidal string diagrams and hence their helpful properties, and end by showing their exceptional applied utility by leveraging them to underpin neural circuit diagrams, a method which, at last, allows deep learning architectures to be comprehensively and rigorously expressed.
Davide Celestini, Amirhossein Afsharrad, Daniele Gammelli, Tommaso Guffanti, Gioele Zardini, Sanjay Lall, Elisa Capello, Simone D'Amico, Marco Pavone
Effective trajectory generation is essential for reliable on-board spacecraft autonomy. Among other approaches, learning-based warm-starting represents an appealing paradigm for solving the trajectory generation problem, effectively combining the benefits of optimization- and data-driven methods. Current approaches for learning-based trajectory generation often focus on fixed, single-scenario environments, where key scene characteristics, such as obstacle positions or final-time requirements, remain constant across problem instances. However, practical trajectory generation requires the scenario to be frequently reconfigured, making the single-scenario approach a potentially impractical solution. To address this challenge, we present a novel trajectory generation framework that generalizes across diverse problem configurations, by leveraging high-capacity transformer neural networks capable of learning from multimodal data sources. Specifically, our approach integrates transformer-based neural network models into the trajectory optimization process, encoding both scene-level information (e.g., obstacle locations, initial and goal states) and trajectory-level constraints (e.g., time bounds, fuel consumption targets) via multimodal representations. The transformer network then generates near-optimal initial guesses for non-convex optimization problems, significantly enhancing convergence speed and performance. The framework is validated through extensive simulations and real-world experiments on a free-flyer platform, achieving up to 30% cost improvement and 80% reduction in infeasible cases with respect to traditional approaches, and demonstrating robust generalization across diverse scenario variations.