Keywords: Nonlinear control, Geometric mechanics, Lagrangian Systems
Abstract: This paper introduces a new notion of differential flatness for mechanical control systems, that we call mechanical flatness, and we study its relations with the mechanical structure of systems and with several variations of the notion of flatness. For the simplest nontrivial case of 3 DOF and 2-inputs, we give a complete classification result in terms of normal forms, examining the interplay between mechanical flatness, config-flatness, and static mechanical feedback linearization.

Keywords: Nonlinear control, Lagrangian Systems
Abstract: We study the exact linearization of configuration flat Lagrangian control systems with p degrees of freedom and p-1 inputs by quasi-static feedback of classical states. First, we present a detailed analysis of the structure of the parameterization of the system variables by the flat output. Based on that, we systematically construct a linearizing quasi-static feedback law of the classical state such that the closed-loop system shows the behavior of decoupled integrator chains. Our approach shows that the construction of a generalized Brunovský state can be completely circumvented. Furthermore, we present a method for determining the lengths of the integrator chains achieved by quasi-static feedback laws that allow for rest-to-rest transitions.

Keywords: Geometric mechanics, Gradient systems, Networked Systems
Abstract: In this work, the notions of formation and rigidity (first developed for formation control in Euclidean spaces) are recast for a mechanical system evolving on a principal fiber bundle. A control law is derived that drives the system to a given desired formation. The inter-agent distance measurements and congruence transformations for systems evolving in a Euclidean space, in the bundle framework, correspond to the projection map onto the base space and Lie group actions on the total space, respectively. Application of the Morse-Bott theory leads to an elegant control law for formation control of second order mechanical systems evolving on principal bundles.

Keywords: Irreversible thermodynamics, Lagrangian Systems, Geometric mechanics
Abstract: The diakoptical method developed by Kron(1963) has been known as a powerful tool in modeling and analyzing large-scale physical systems. The principal idea lies in the process of tearing and interconnection for a physical system, namely, one can reticulate the system into several subsystems, each of which is called a primitive system when it once torn apart from the original system. The original system can be regarded as an interconnected system via the interconnection structure, which is known to be modeled as a Dirac structure. This paper illustrates the idea of a Lagrangian variational modular approach by extending the case in mechanics done by Jacobs and Yoshimura(2014) to the case of thermodynamic systems, by following Gay-Balmaz and Yoshimura (2023), together with examples of non-simple thermodynamic systems with internal heat transfer and mechanical interactions.

Keywords: Dirac structures, Port-Hamiltonian systems, Geometric mechanics
Abstract: We study the geometric structure of port-Hamiltonian systems. Starting with the intuitive understanding that port-Hamiltonian systems are ``in between'' certain closed Hamiltonian systems, the geometric structure of port-Hamiltonian systems must be ``in between'' the geometric structures of the latter systems. These are Courant algebroids; and hence the geometric structures should be related by Courant algebroid morphisms. Using this idea, we propose a definition of an intrinsic geometric structure and show that it is unique, if it exists.

Keywords: Port-Hamiltonian systems, Symplectic geometry
Abstract: We extend the port-Hamiltonian framework de ned with respect to a Lagrangian submanifold and a Dirac structure by augmenting the Lagrangian submanifold with the space of external variables. The new pair of conjugated variables is called energy port. We show that in the most general case, the extension describes constrained Hamiltonian systems whose Hamiltonian function depends on inputs.

Keywords: Lagrangian Systems, Dirac structures, Boundary control
Abstract: We present how the interconnection of distributed and discrete systems can be realized within the setting of variational principles and Dirac structures on the Lagrangian side. For the distributed system we focus on the case when the configuration space is an infinite dimensional vector space and the system is subject to boundary energy flow. A key property of our approach is that it systematically extends the canonical variational and symplectic structures of finite and infinite dimensional Lagrangian and Hamiltonian mechanics.

Keywords: Numerical Methods, Distributed parameter systems, Boundary control
Abstract: A structure-preserving Finite Element Method (FEM) for the transport equation in one- and two-dimensional domains is presented. This Distributed Parameter System (DPS) has non-collocated boundary control and observation, and reveals a scattering-energy preserving structure. We show that the discretized model preserves the aforementioned structure from the original infinite-dimensional system. Moreover, we analyse the case of moving meshes for the one-dimensional case. The moving mesh requires less states than the fixed one to produce solutions with a comparable accuracy, and it can also reduce the overshoot and oscillations of Gibbs phenomenon produced when using the FEM. Numerical simulations are provided for the case of a one-dimensional transport equation with fixed and moving meshes.

Keywords: Port-Hamiltonian systems, Distributed parameter systems
Abstract: We analyze infinite-dimensional Hamiltonian systems corresponding to partial differential equations on one-dimensional spatial domains formulated with formally skew- adjoint Hamiltonian operators and nonlinear Hamiltonian density. In various applications, the Hamiltonian density can depend on spatial derivatives of the state such that these systems can not straightforwardly be formulated as boundary port-Hamiltonian system using a Stokes- Dirac structure. In this work, we show that any Hamiltonian system of the above class can be reformulated as a Hamiltonian system on the jet space, in which the Hamiltonian density only depends on the extended state variable itself and not on its derivatives. Consequently, well-known geometric formulations with Stokes-Dirac structures are applicable. Additionally, we provide a similar result for dissipative systems. We illustrate the developed theory by means of the the Boussinesq equation, the dynamics of an elastic rod and the Allen-Cahn equation.

Keywords: Port-Hamiltonian systems, Distributed parameter systems
Abstract: It is well known that linear and non-linear dissipative port-Hamiltonian systems in finite dimensions admit an energy balance, relating the energy increase in the system with the supplied energy and the dissipated energy. The integrand in the dissipation term is then a function of the state variable. In this note, we answer the question of when this is possible for linear port-Hamiltonian systems in infinite dimensions.

Keywords: Port-Hamiltonian systems, Modeling, Distributed parameter systems
Abstract: This paper is concerned with the port-Hamiltonian modeling of a Timoshenko beam subject geometric nonlinearities through von Kármán strains, material nonlinearity considering hyperelasticity with the assumption of neo-Hookean or Mooney-Rivlin material, in addition to the incompressible deformation constraint that corresponds to the preservation of volume. The model is suitable for representing the behavior of rubber like beams within the range of moderate deformations and rotations. Numerical simulations are carried out to illustrate the accuracy of the proposed model.

Keywords: Robotics, Geometric mechanics
Abstract: We consider problems pertaining to the navigation and localization of mobile mechanical systems such as robots or crafts. We focus on the recently introduced theory of two-frame systems, within the field of invariant Kalman filtering for state estimation of dynamical systems. We show that we can build a two-frame group (TFG) structure not only using rotations as its basic building block, as has been done so far, but also rotations and scalings. This then makes the TFG a generalization of the group of similarilty transformations Sim(d). This allows for new examples to fit into the theory of two-frame systems. Namely, we consider the localization of a car equipped with odometry based on wheels having unknown radius, and inertial navigation where the accelerometers are affected by an unknown scale factor.

Keywords: Nonlinear control
Abstract: In this paper we consider aspects of geometric observability for hypergraphs, extending our earlier work from the uniform to the nonuniform case. Hypergraphs, a generalization of graphs, allow hyperedges to connect multiple nodes and unambiguously represent multi-way relationships which are ubiquitous in many real-world networks including those that arise in biology. We consider polynomial dynamical systems with linear outputs defined according to hypergraph structure, and we propose methods to evaluate local, weak observability.

Keywords: Geometric mechanics, Lagrangian Systems, Numerical Methods
Abstract: Geometric mechanics analyzes mechanical systems in the framework of variational mechanics, while accounting for the geometry of the configuration space. From the late 1970s, developments in this area produced several schemes for geometric control of mechanical systems in continuous time and discrete time. In the mid to late 2000s, geometric mechanics was first applied to state estimation of mechanical systems, particularly systems evolving on Lie groups as configuration manifolds, like rigid body systems. Much of the existing work on geometric mechanics-based estimation has been in continuous time, using deterministic, semi-stochastic and stochastic approaches. While the body of existing literature on discrete-time estimation schemes on Lie groups is not as extensive, the literature on this topic is contemporaneous with continuous-time schemes. This work describes some recent and ongoing research on geometric mechanics-based estimation schemes in continuous and discrete time from the mid-2010s, which were developed using the Lagrange-d’Alembert principle applied to rigid body systems. This approach gives (deterministic) observer designs with strong stability and robustness properties. This work concludes with potential extensions of this approach to mechanical systems with principal fiber bundles as configuration manifolds.

Keywords: Nonlinear control, Lagrangian Systems, Geometric mechanics
Abstract: The trajectory tracking problem is a fundamental control task in the study of mechanical systems. Hamiltonian systems are posed on the cotangent bundle of configuration space of a mechanical system, however, symmetries for the full cotangent bundle are not commonly used in geometric control theory. In this paper, we propose a group structure on the cotangent bundle of a Lie group and leverage this to define momentum and configuration errors for trajectory tracking, drawing on recent work on equivariant observer design. We show that this error definition leads to error dynamics that are themselves “Euler-Poincare like” and use these to derive simple, almost global trajectory tracking control for fully-actuated Euler- Poincare systems on a Lie group state space.