2D Euler elastica in constrained environments - Stavole, Martina; Sato Martin de Almagro, Rodrigo; Capobianco, Giuseppe; Brüls, Olivier; Leyendecker, Sigrid (conference, HFSS International Conference on Highly Flexible Slender Structures (THREAD annual meeting), 2023)

An augmented Lagrangian formulation of the planar elastica in constrained environments - Stavole, Martina; Sato Martin de Almagro, Rodrigo; Capobianco, Giuseppe; Brüls, Olivier; Leyendecker, Sigrid (conference, 7th IMSD International Conference on Multibody System Dynamics, 2024)

A new Lagrangian approach to control affine systems with a quadratic Lagrange term - Leyendecker, Sigrid; Maslovskaya, Sofya; Ober-Blöbaum, Sina; Sato Martin de Almagro, Rodrigo; Szemenyei, Flóra (Journal of Computational Dynamics, 2024)
In this work, we consider optimal control problems for mechanical systems with fixed initial and free final state and a quadratic Lagrange term. Specifically, the dynamics is described by a second order ODE containing an affine control term. Classically, Pontryagin’s maximum principle gives necessary optimality conditions for the optimal control problem. For smooth problems, alternatively, a variational approach based on an augmented objective can be followed. Here, we propose a new Lagrangian approach leading to equivalent necessary optimality conditions in the form of Euler-Lagrange equations. Thus, the differential geometric structure (similar to classical Lagrangian dynamics) can be exploited in the framework of optimal control problems. In particular, the formulation enables the symplectic discretisation of the optimal control problem via variational integrators in a straightforward way.
CRIS-Team Scopus Importer:2024-05-03
Keywords: Hamiltonian system; Lagrangian system; Optimal control problem; Pontryagin’s maximum principle; Variations

A new Lagrangian approach to regular control-affine mechanical systems - Sato Martin de Almagro, Rodrigo; Szemenyei, Flóra; Ober-Blöbaum, Sina; Leyendecker, Sigrid (conference, Workshop on Geometric Control and Mechanics, 2024)

Augmented Lagrangian contact formulation of the 2D Euler elastica - Stavole, Martina; Sato Martin de Almagro, Rodrigo; Brüls, Olivier; Leyendecker, Sigrid (conference, ICIAM 2023 – 10th International Congress on Industrial and Applied Mathematics, 2023)

Convergence of Lobatto-type Runge–Kutta methods for partitioned differential-algebraic systems of index 2 - Sato Martin de Almagro, Rodrigo (Bit Numerical Mathematics, 2021)
In this paper a numerical scheme for partitioned systems of index 2 DAEs, such as those arising from nonholonomic mechanical problems, is proposed and its order for a certain class of Runge–Kutta methods we call of Lobatto-type is proven.
CRIS-Team Scopus Importer:2021-05-21
Keywords: Geometric integration; Index 2 DAEs; Nonholonomic mechanics; Partitioned Runge–Kutta

Discrete adjoint method for variational integration of constrained ODEs and its application to optimal control of geometrically exact beam dynamics - Schubert, Matthias; Sato Martin de Almagro, Rodrigo; Nachbagauer, Karin; Ober-Blöbaum, Sina; Leyendecker, Sigrid (Multibody System Dynamics, 2023)

Discrete Mechanics for Forced and Constrained Systems - Sato Martin de Almagro, Rodrigo (Friedrich-Alexander-Universität Erlangen-Nürnberg, 2019)
Geometric mechanics is a branch of mathematics that studies classical mechanics of particles and fields from the point of view of geometry and its relation to symmetry. One of its most interesting developments was bringing together numerical analysis and geometry by relating what is known as discrete mechanics with numerical integration. This is called geometric integration. In the last 30 years this latter field has exploded with research from the purely theoretical to the strictly applied. Variational integrators are a type of geometric integrators arising naturally from the discretization process of variational principles in mechanics. They display some of the most salient features of the theory, such as symplecticity, preservation of momenta and quasi-preservation of energy. These methods also apply very naturally to optimal control problems, also based on variational principles. Unfortunately, not all mechanical systems of interest admit a variational formulation. Such is the case of forced and nonholonomic mechanical systems. In this thesis we study both of these types of systems and obtain several new results. By geometrizing a new technique of duplication of variables and applying it, we were able to definitely prove the order of integrators for forced systems by using only variational techniques. Furthermore, we could also extend these results to the reduced setting in Lie groups, leading us to a very interesting geometric structure, Poisson groupoids. In addition, we developed new methods to geometrically integrate nonholonomic systems to arbitrary order preserving their constraints exactly. These methods can be seen as nonholonomic extensions of variational methods, and we were able to prove their order, although not through variational means. These methods have a nice geometric interpretation and thanks to their closeness to variational methods, they can be easily generalized to other geometric settings, such as Lie group integration. Finally, we were able to apply these new methods to optimal control problem
Keywords: mathematics; mathematical physics; mechanics; geometry; numerical analysis; variational integrators; nonholonomic mechanics; control

Endoscopes: an experimental characterisation of their constitutive properties - Stavole, Martina; Dörlich, Vanessa; Linn, Joachim; Sato Martin de Almagro, Rodrigo; Leyendecker, Sigrid (conference, ITWM Fraunhofer, 2022)

Frequency-dependent damping as forcing on multisymplectic integrators - Sato Martin de Almagro, Rodrigo; Leyendecker, Sigrid (conference, ICIAM 2023 – 10th International Congress on Industrial and Applied Mathematics, 2023)

Galerkin variational integration of the geometrically exact beam via unit dual quaternion interpolation - Leitz, Thomas; Sato Martin de Almagro, Rodrigo; Leyendecker, Sigrid (conference, GAMM Annual Meeting, 2021)

Galerkin variational integration of the geometrically exact beam via unit dual quaternion interpolation - Sato Martin de Almagro, Rodrigo; Leitz, Thomas; Leyendecker, Sigrid (2021)

Geometric Optimal Trajectory Tracking of Nonholonomic Mechanical Systems - Colombo, Leonardo Jesús; Martín de Diego, David; Nayak, Aradhana; Sato Martin de Almagro, Rodrigo (SIAM Journal on Control and Optimization, 2020)
We study the tracking of a trajectory for a nonholonomic system by recasting the problem as a constrained optimal control problem. The cost function is chosen to minimize the error in positions and velocities between the trajectory of a nonholonomic system and the desired reference trajectory, both evolving on the distribution which defines the nonholonomic constraints. The problem is studied from a geometric framework. Optimality conditions are determined by the Pontryagin Maximum Principle and also from a variational point of view, which allows the construction of geometric integrators. Examples and numerical simulations are shown to validate the result
Keywords: optimal control; trajectory planning; nonholonomic mechanics; variational integrators.

High-order integrators for Lagrangian systems on homogeneous spaces via nonholonomic mechanics - Sato Martin de Almagro, Rodrigo (Journal of Computational and Applied Mathematics, 2023)
In this paper, high-order numerical integrators on homogeneous spaces will be presented as an application of nonholonomic partitioned Runge–Kutta Munthe-Kaas (RKMK) methods on Lie groups. A homogeneous space M is a manifold where a group G acts transitively. Such a space can be understood as a quotient M≅G/H, where a closed Lie subgroup H, is the isotropy group of each point of M. The Lie algebra of G may be decomposed into g=m⊕h, where h is the subalgebra that generates H and m is a subspace spanning the tangent spaces of M. Thus, variational problems on M can be treated as nonholonomically constrained problems on G, by requiring variations to remain on m. Nonholonomic partitioned RKMK integrators are derived as a modification of those obtained by a discrete variational principle on Lie groups, and can be interpreted as obeying a discrete Chetaev principle. These integrators tend to preserve several properties of their purely variational counterparts.
CRIS-Team Scopus Importer:2022-11-04
Keywords: High-order integrators; Homogeneous spaces; Lie groups; Nonholonomic mechanics; Runge–Kutta Munthe-Kaas

High-order integrators on homogeneous spaces via nonholonomic mechanics - Sato Martin de Almagro, Rodrigo (NUMDIFF-16, 2021)

Homogenised stiffness coefficients of unloaded endoscope shafts - Stavole, Martina; Sato Martin de Almagro, Rodrigo; Dörlich, Vanessa; Leyendecker, Sigrid (conference, MaGIC 2023, 2023)

Homogenization of the constitutive properties of composite beam cross-sections - Stavole, Martina; Leyendecker, Sigrid; Sato Martin de Almagro, Rodrigo (conference, MaGIC 2022: Manifolds and Geometric Integration Colloquia, 2022)

Homogenization of the constitutive properties of composite beam cross-sections - Stavole, Martina; Sato Martin de Almagro, Rodrigo; Lohk, Margus; Leyendecker, Sigrid (conference, ECCOMAS - 8th European Congress on Computational Methods in Applied Sciences and Engineering, 2022)

Homogenization of the constitutive properties of composite beam cross-sections - Stavole, Martina; Sato Martin de Almagro, Rodrigo; Lohk, Margus; Leyendecker, Sigrid (Computational Solid Mechanics, 2022)

Homogenized stiffness coefficients of unloaded endoscope shafts - Stavole, Martina; Sato Martin de Almagro, Rodrigo; Dörlich, Vanessa; Leyendecker, Sigrid (conference, ECCOMAS Thematic Conference on Multibody Dynamics, 2023)

Homogenized stiffness coefficients of unloaded endoscope shafts - Stavole, Martina; Sato Martin de Almagro, Rodrigo; Dörlich, Vanessa; Leyendecker, Sigrid (conference, HFSS International Conference on Highly Flexible Slender Structures (THREAD annual meeting), 2023)

Introduction to Numerics (INUMS) - Capobianco, Giuseppe; Huang, Dengpeng; Sato Martin de Almagro, Rodrigo; Leyendecker, Sigrid (invited lecture, FRASCAL Mini Lecture, 2022)

Multisymplectic Galerkin Lie group variational integrators for geometrically exact beam dynamics based on unit dual quaternion interpolation — no shear locking - Leitz, Thomas; Sato Martin de Almagro, Rodrigo; Leyendecker, Sigrid (Computer Methods in Applied Mechanics and Engineering, 2021)
We present a Galerkin multisymplectic Lie group variational integrator. It is suitable for dynamical systems defined on a two dimensional space–time and the integrator allows arbitrary convergence orders independently for both dimensions. As an example we use geometrically exact beam dynamics where a slender structure is modelled as a centre line with a cross section at every point. The Lie group in question is the special Euclidean group in three-dimensional space, SE(3)SE(3)" role="presentation" style="display: inline-block; line-height: normal; font-size: 16.200000762939453px; word-spacing: normal; word-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border-width: 0px; position: relative;">, which we parametrize using unit dual quaternions. This allows a very simple and efficient interpolation method to be used, which additionally prevents shear locking present in more naive discretizations of geometrically exact beams.<br

Neural networks for the approximation of Euler’s elastica - Celledoni, Elena; Çokaj, Ergys; Leone, Andrea; Leyendecker, Sigrid; Murari, Davide; Owren, Brynjulf; Sato Martin de Almagro, Rodrigo; Stavole, Martina (Computer Methods in Applied Mechanics and Engineering, 2025)
Euler’s elastica is a classical model of flexible slender structures relevant in many industrial applications. Static equilibrium equations can be derived via a variational principle. The accurate approximation of solutions to this problem can be challenging due to nonlinearity and constraints. We here present two neural network-based approaches for simulating Euler’s elastica. Starting from a data set of solutions of the discretised static equilibria, we train the neural networks to produce solutions for unseen boundary conditions. We present a discrete approach learning discrete solutions from the discrete data. We then consider a continuous approach using the same training data set but learning continuous solutions to the problem. We present numerical evidence that the proposed neural networks can effectively approximate configurations of the planar Euler’s elastica for a range of different boundary conditions.<br

Optimal Trajectory Tracking of Nonholonomic Mechanical Systems: a geometric approach - Nayak, Aradhana; Sato Martin de Almagro, Rodrigo; Colombo, Leonardo Jesús; Martín de Diego, David (2019 American Control Conference (ACC), 2019)
We study the tracking of a trajectory for a nonholonomic system by recasting the problem as an optimal control problem. The cost function is chosen to minimize the error in positions and velocities between the trajectory of a nonholonomic system and the desired reference trajectory evolving on the distribution which defines the nonholonomic constraints. We propose a geometric framework since it describes the class of nonlinear systems under study in a coordinate-free framework. Necessary conditions for the existence of extrema are determined by the Pontryagin Minimum Principle. A nonholonomic fully actuated particle is used as a benchmark example to show how the proposed method is applie
Keywords: optimal control; geometry; mathematics; trajectory tracking; nonholonomic mechanics

Parallel iterative methods for variational integration applied to navigation problems - Ferraro, Sebastian J.; Martin De Diego, David; Sato Martin de Almagro, Rodrigo (IFAC PAPERSONLINE, 2021)
CRIS-Team WoS Importer:2021-12-03

Parallel iterative methods for variational integration applied to navigation problems - Ferraro, Sebastian J.; Martin de Diego, David; Sato Martin de Almagro, Rodrigo (invited lecture, Geometry, Mechanics and Control Seminar, 2021)

Parallel iterative methods for variational integration and related problems - Ferraro, Sebastian J.; Martín de Diego, David; Sato Martin de Almagro, Rodrigo (invited lecture, Foundations of Computational Mathematics (FOCM), 2023)

Variational integrators and frequency-dependent damping - Sato Martin de Almagro, Rodrigo; Leyendecker, Sigrid (poster, Foundations of Computational Mathematics (FOCM), 2023)

Variational modelling and simulation of complex beams and their optimisation. - Stavole, Martina; Sato Martin de Almagro, Rodrigo; Leyendecker, Sigrid (conference, THREAD Annual Meeting, 2022)

Variational order for forced Lagrangian systems - Martín de Diego, David; Sato Martín de Almagro, Rodrigo Takuro (Nonlinearity, 2018)
We are able to derive the equations of motion for forced mechanical systems in a purely variational setting, both in the context of Lagrangian or Hamiltonian mechanics, by duplicating the variables of the system as introduced by C. Galley. We show that this construction is useful to design high-order integrators for forced Lagrangian systems and, more importantly, we give a characterization of the order of a method applied to a forced system using the corresponding variational order of the duplicated one.<br

Variational order for forced Lagrangian systems II. Euler-Poincare equations with forcing - Martin De Diego, D.; Sato Martin de Almagro, Rodrigo (Nonlinearity, 2020)
In this paper we provide a variational derivation of the Euler-Poincare equations for systems subjected to external forces using an adaptation of the techniques introduced by Galley and others Martin de Diego and Martin de Almagro (2018Nonlinearity313814-3846), Galley (2013Phys. Rev. Lett.110174301), Galleyet al(2014 (arXiv:). Moreover, we study in detail the underlying geometry which is related to the notion of Poisson groupoid. Finally, we apply the previous construction to the formal derivation of the variational error for numerical integrators of forced Euler-Poincare equations and the application of this theory to the derivation of geometric integrators for forced systems.
CRIS-Team WoS Importer:2020-07-03