Abstract: When designing autonomous systems, we need to consider multiple trade-offs at various abstraction levels, and choices of single (hardware and software) components need to be studied jointly. For instance, the design of future mobility solutions (e.g., autonomous vehicles) 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 policies. Co-designing autonomous systems is a complex task for at least two reasons. First, the co-design of interconnected systems (e.g., networks of cyber-physical systems) involves the simultaneous choice of components arising from heterogeneous fields, while satisfying systemic constraints and accounting for multiple objectives. Second, components are connected via interactions between different stakeholders. I will present a framework to co-design such systems, leveraging a monotone theory of co-design. The framework will be instantiated in applications in mobility and autonomy. Through various case studies, I will show how the proposed approaches allow one to efficiently answer heterogeneous questions, unifying different modeling techniques and promoting interdisciplinarity, modularity, and compositionality. I will then discuss open challenges for compositional systems design optimization.

Abstract: We introduce an innovative approach for analyzing strategic interactions in transportation networks featuring MoD services. This study focuses on achieving company-traveler equilibria, whereby a single company optimizes pricing and routing decisions to maximize profitability while considering travelers’ mode choices, modeled via a multinomial logit model (MNL). Although profit maximization problems have been extensively studied in the field of revenue management across various domains, their application to transportation networks poses unique challenges, such as the influence of network topology and additional constraints (e.g., flow conservation, rebalancing, etc.). To address the inherent non-linear relationship arising from endogenous travel demand, we shift our domain space from price to market share. Subsequently, we derive prices using a direct one-to-one correspondence within the MNL. This work is the first effort in leveraging such novel techniques in the context of transportation network analysis. Remarkably, the proposed reformulation results in an equivalent problem exhibiting convexity, offering computational efficiency and interpretability. By solving the KKT conditions, we characterize user equilibrium with the generalized route cost, which incorporates the operating cost by rebalancing and travelers’ disutility caused by congestion. Our approach is empirically validated through a numerical analysis conducted on the widely recognized Sioux Falls network. The results underscore the effectiveness and practical applicability of our method in analyzing transportation networks featuring MoD services, and opens the stage for important future investigations.

Abstract: 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.

Abstract: The flow of information through a complex system can be readily understood with category theory. However, negative information (e.g., what is not possible) does not have an immediately evident categorical representation. The formalization of nategories using unconventional composition addresses this issue, and lets imposed limitations on categories be considered. However, traditional nategories abandon core categorical constructs and rely on extensive mathematical development. This creates a divide between the consideration of positive and negative information composition. In this work, we show that negative information can be considered in a natural categorical manner. This is aided by functor string diagrams, a novel flexible diagrammatic approach that can intuitively show the operation of hom-functors and natural transformations in expressions. This insight reveals how to consider the composition of negative information with foundational categorical constructs without relying on enrichment. We present diagrammatic means to consider not only nategories, but preorders more broadly. This paper introduces diagrammatic methods for the consideration of triangle inequalities and co-designs \displaystyle \mathbfDP/Feas_Bool, showing how important cases of negative information composition can be categorically and diagrammatically approached. In particular, we develop systematic tools to rigorously consider imposed limitations on systems, advancing our mathematical understanding, and present intuitive diagrams which motivate widespread adoption and usage for various applications.

Abstract: Traditional round-trip car rental systems mandate users to return vehicles to their point of origin, limiting the system’s adaptability to meet diverse mobility demands. This constraint often leads to fleet under-utilization and incurs high parking costs for idle vehicles. To address this inefficiency, we propose a N-user matching algorithm which is designed to facilitate one-way trips within the round-trip rental framework. Our algorithm addresses the joint problem of optimal pricing and user matching through a Two-Stage Integer Linear Programming (ILP)-based formulation. In the first stage, optimal rental prices are determined by setting a risk factor that governs the likelihood of matching a set of N-user. The second stage involves maximizing expected profit through a novel ILP-based user-matching formulation. Testing our algorithm on real-world scenarios demonstrates an approximate 35% increase in demand fulfillment. Additionally, we assess the model’s robustness under uncertainty by varying factors such as the risk factor (probability of user’s ride acceptance at the offered price), cost factor (rental cost-to-fare ratio), and maximum chain length.

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Abstract: This paper introduces an automatic control method designed to enhance the operation of electric vehicles, besides the speed tracking objectives, by including reliability and lifetime requirements. The research considers an automotive power converter which supplies electric power to a permanent magnet synchronous motor (PMSM). The primary control objective is to mitigate the thermal stress on the power electronic Insulate Gate Bipolar Transistors (IGBTs), while simultaneously ensuring effective speed tracking performance. To achieve these goals, we propose an extended \mathcalH_∞design framework, which includes reliability models. The method is tested in two distinct scenarios: reliability-aware, and reliability-free cases. Furthermore, the paper conducts a lifetime analysis of the IGBTs, leveraging the Rainflow algorithm and temperature data.

Abstract: We present a study focusing on the joint optimization of the sizing of hardware components as well as strategic decisions for a race car in a Formula 1 setting. Our research leverages a monotone theory of co-design, which allows for hardware and software considerations to achieve optimal, synergistic performance improvements. We aim to identify the Pareto optimal curves that illustrate the optimal balance between conflicting objectives, such as speed, energy allocation, and component choice, within the tight constraints imposed by the regulations. The results of the study demonstrate the versatility of our framework by showing optimal component sizing on two structurally different track layouts on a single lap. Moreover, by increasing the amount of laps under consideration, we show the ability of our tool to consider strategic energy allocation decisions.

Abstract: The design of future mobility solutions (autonomous vehicles, micromobility solutions, etc.) 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 future mobility systems in terms of different objectives. This paper presents a framework to address such co-design problems. In particular, we leverage the recently developed 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 such as shared bikes and e- scooters, and public transit, in terms of fleets sizing, vehicle char- acteristics, 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. 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 policy. 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 inform policy-making in the future.

Abstract: 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 → 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.

Abstract: 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 sociotechnical 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 an "larger" game among stakeholders of the mobility ecosystem (e.g., municipality, Mobility Service Providers (MSPs), and citizens), effectively giving rise to tools to inform strategic interventions and policy-making in the mobility ecosystem.

Abstract: Battery Electric Vehicles (BEVs) offer a sustainable alternative to Internal Combustion Engine Vehicles (ICEVs). This paper addresses some of the challenges faced by the automotive industry and the scientific community in defining the technology for the next generation of automotive power converters. The focus is on achieving reduced CO2 emissions, improved energy efficiency, and reliability, while minimizing costs to enable large-scale adoption of BEVs and Hybrid Electric Vehicles (HEVs). The paper leverages an automotive converter equipped with the recently developed Adjustable Hybrid Switch (AHS) based electric gear and proposes a reliability-based control algorithm for operating the converter E-Gear (EG) of BEVs. By integrating reliability control principles, the proposed algorithm minimizes system damage over time and enhances the converter’s lifetime. The case studies, based on standardised driving cycles, demonstrate the benefits of the presented approach in terms of energy losses and lifetime expectations. Overall, this work contributes a novel approach to drivetrain control in BEVs, highlighting the potential of the proposed control strategy to improve energy efficiency and reliability. The research findings provide valuable insights for the development of next-generation automotive power converters.

Abstract: 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.

Abstract: Visual navigation for insect-scale robots is very challenging because in such a small scale, the size, weight, and power (SWaP) constraints do not appear to permit visual navigation techniques such as SLAM (Simultaneous Localization and Mapping) because they are likely to be too power-hungry. We propose to use a biology-inspired approach, which we term the bilinear optic flow approximation, that is more computationally efficient. We build on previous work that has shown that the bilinear approximation can be used for visual servoing. Here, we show that a bilinear approximator can be learned that is able to stabilize the heading of a robot while performing continuous forward motion in a corridor-shaped environment. This is a necessary capability for confined-space navigation that insect-sized robots are likely to perform. In this work, we describe the underlining methodology of the method and built a 2D visual simulation environment and omnidirectional camera model to validate our results.

Abstract: 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.

Abstract: 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.

Abstract: Challenged by urbanization and increasing travel needs, existing transportation systems call for 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, both from the operational and the planning perspective. 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.

Abstract: 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.

Abstract: 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.

Abstract: 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.

Abstract: 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.

Abstract: 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.

Abstract: 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.

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Abstract: The contemporary era struggles with the intricate challenge of designing “complex systems”. These systems are characterized by intricate webs of interactions that interlace their components, giving rise to multifaceted complexities, springing from at least two sources. First, the co-design of complex systems (e.g., a large network of cyber-physical systems) demands the simultaneous selection of heterogeneous components (e.g., hardware vs. software parts), while satisfying system constraints and accounting for multiple objectives. Second, different components are interconnected through interactions, and their design cannot be decoupled (e.g., within a mobility system). Navigating this complexity necessitates innovative approaches, and this thesis responds to this imperative by focusing on the theory of co-design. Our exploration extends from the design of individual platforms, such as autonomous vehicles, to the orchestration of entire mobility systems built upon such platforms. In particular, we delve into the theoretical foundations of a monotone theory of co-design, establishing a robust mathematical framework, leveraging category theory to elucidate key concepts, including compositionality and functorial solution schemes in co-design. Notably, this thesis offers not only an understanding of the theoretical underpinnings, but also practical guidance for applying them to a diverse array of real-world problems, revolving around the domain of embodied intelligence. The presented toolbox empowers efficient computation of optimal design solutions tailored to specific tasks and, in its novelty, paves the way for several possibilities for future research.

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