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Geometric PID Controller for Stabilization of Nonholonomic Mechanical Systems on Lie Groups [article]

Rama Seshan, Ravi N Banavar, D. H. S. Maithripala, Arun D. Mahindrakar
2021 arXiv   pre-print
developed for mechanical systems evolving on Lie groups.  ...  The PID controller is an elegant and versatile controller for set point tracking in double integrator systems of which mechanical systems evolving on Euclidean space constitute a large class.  ...  Geometric control methods have been developed for a wide variety of systems. There have been works that generalize PID controller to very specific class of mechanical systems without any constraints.  ... 
arXiv:2111.07061v1 fatcat:yuugtldspncjlp2mz53xnbo56y

Equivariant constrained symplectic integration

R. I. McLachlan, C. Scovel
1995 Journal of nonlinear science  
These results provide an elementary construction of symplectic integrators for Lie-Poisson systems and other Hamiltonian systems with symmetry.  ...  We use recent results on symplectic integration of Hamiltonian systems with constraints to construct symplectic integrators on cotangent bundles of manifolds by embedding the manifold in a linear space  ...  In this case it shows that the constraint algorithm can provide a symplectic integrator for Lie-Poisson systems: Many important groups in mechanics are defined as matrix groups and the constraint functions  ... 
doi:10.1007/bf01212956 fatcat:uef43dxf3zbslde2jsg2zy7bju

Mechanical Systems with Symmetry, Variational Principles, and Integration Algorithms [chapter]

Jerrold E. Marsden, Jeffrey M. Wendlandt
1997 Current and Future Directions in Applied Mathematics  
We recall some general features of how to reduce variational principles in the presence of a symmetry group along with general features of integration algorithms for mechanical systems.  ...  This paper studies variational principles for mechanical systems with symmetry and their applications to integration algorithms.  ...  Acknowledgments We thank Francisco Armero, Oscar Gonzalez, Abhi Jain, Ben Leimkuhler, Andrew Lewis, Robert MacKay, Richard Murray, George Patrick and Shmuel Weissman for useful discussions or comments.  ... 
doi:10.1007/978-1-4612-2012-1_18 fatcat:uc2metaegnaozp2wkf3qe55h5i

Optimization on manifolds: A symplectic approach [article]

Guilherme França, Alessandro Barp, Mark Girolami, Michael I. Jordan
2021 arXiv   pre-print
thereof, and the other for constrained submanifolds that is based on a dissipative generalization of the famous RATTLE integrator.  ...  Moreover, we construct two dissipative generalizations of leapfrog that are straightforward to implement: one for Lie groups and homogeneous spaces, that relies on the tractable geodesic flow or a retraction  ...  "Rate-matching" geometric integrators We now state general results for geometric integrators of nonconservative and constrained Hamiltonian systems that generalize classical results for symplectic integration  ... 
arXiv:2107.11231v1 fatcat:owppycnxmzf4dfrzmii6xzclz4

Geometric discretization of nonholonomic systems with symmetries

Gaurav Sukhatme, Jerrold Marsden, Marin Kobilarov
2009 Discrete and Continuous Dynamical Systems. Series S  
The paper develops discretization schemes for mechanical systems for integration and optimization purposes through a discrete geometric approach.  ...  A family of nonholonomic integrators that are general, yet simple and easy to implement, are then obtained and applied to two examples-the steered robotic car and the snakeboard.  ...  More abstractly, Lie group integrators preserve symmetry and group structure for systems with motion invariants.  ... 
doi:10.3934/dcdss.2010.3.61 fatcat:agaxid5pyfalbmqzjuuzx75nt4

Lie group variational integrators for the full body problem

Taeyoung Lee, Melvin Leok, N. Harris McClamroch
2007 Computer Methods in Applied Mechanics and Engineering  
We use Lagranges method for constrained problems in the calculus of variations to derive the discrete-time necessary conditions.  ...  This paper develops the theory of abelian Routh reduction for discrete mechanical systems and applies it to the variational integration of mechanical systems with abelian symmetry.  ... 
doi:10.1016/j.cma.2007.01.017 fatcat:4gasexilivapdnjg2egh6gbglq

Structure-Preserving Constrained Optimal Trajectory Planning of a Wheeled Inverted Pendulum [article]

Klaus Albert, Karmvir Singh Phogat, Felix Anhalt, Ravi N Banavar, Debasish Chatterjee, Boris Lohmann
2019 arXiv   pre-print
In this article we derive a discrete-time model of the WIP system using discrete mechanics and generate optimal trajectories for the WIP system by solving a discrete-time constrained optimal control problem  ...  ensuing stable motion of the WIP system.  ...  where involved in the development, building, modeling, controller and observer design process of the WIP with their term-and masters thesis's for their valuable help.  ... 
arXiv:1811.12819v2 fatcat:c3etmtvq2jbkdjhevfrfakdm7m

Efficient Computation of Higher-Order Variational Integrators in Robotic Simulation and Trajectory Optimization [article]

Taosha Fan, Jarvis Schultz, Todd Murphey
2019 arXiv   pre-print
This paper addresses the problem of efficiently computing higher-order variational integrators in simulation and trajectory optimization of mechanical systems as those often found in robotic applications  ...  We develop O(n) algorithms to evaluate the discrete Euler-Lagrange (DEL) equations and compute the Newton direction for solving the DEL equations, which results in linear-time variational integrators of  ...  Scalable variational integrators for constrained mechanical systems in generalized coordinates. Claude Lacoursiere.  ... 
arXiv:1904.12756v1 fatcat:zc5v7sg5gfbhbie5c3wbnwnb3m

Structure-preserving discrete-time optimal maneuvers of a wheeled inverted pendulum [article]

Karmvir Singh Phogat, Ravi Banavar, Debasish Chatterjee
2017 arXiv   pre-print
on a discrete-time maximum principle applicable to mechanical systems whose configuration manifold is a Lie group.  ...  In this article we present a successful effort in this direction: We employ geometric mechanics to obtain a discrete-time model of the system, followed by the synthesis of an energy-optimal control based  ...  Let Q be the configuration space of a nonholonomic mechanical system. Suppose G × Q (ḡ, q) → Φḡ(q) ∈ Q is a group action of a Lie group G on the manifold Q.  ... 
arXiv:1710.10932v2 fatcat:tfbfa6vtevgkxf4zjt4bu5ajia

The variational discretization of the constrained higher-order Lagrange-Poincaré equations [article]

Anthony Bloch, Leonardo Colombo, Fernando Jiménez
2018 arXiv   pre-print
In this paper we investigate a variational discretization for the class of mechanical systems in presence of symmetries described by the action of a Lie group which reduces the phase space to a (non-trivial  ...  Optimal control problems for underactuated mechanical systems can be viewed as higher-order constrained variational problems.  ...  local truncation error of the numerical methods).  ... 
arXiv:1801.00577v2 fatcat:knwfi64aq5czlossyt3ncumz3q

High-speed event-based camera tracking

William Chamorro, Juan Andrade-Cetto, Joan Solà
2020 British Machine Vision Conference  
Our method is capable of tracking either camera motion or the motion of an object in front of it, using an error-state Kalman filter formulated in a Lie-theoretic sense.  ...  The method includes a robust mechanism for the matching of events with projected line segments with very fast outlier rejection.  ...  Excellence to IRI (MDM-2016-0656, and by a scholarship from SENESCYT, Republic of Ecuador to William Chamorro.  ... 
dblp:conf/bmvc/ChamorroAS20 fatcat:34aha5zbkzannkqzoevyuyxofy

The variational discretization of the constrained higher-order Lagrange-Poincaré equations

Anthony Bloch, Leonardo Colombo, Fernando Jiménez
2019 Discrete and Continuous Dynamical Systems. Series A  
In this paper we investigate a variational discretization for the class of mechanical systems in presence of symmetries described by the action of a Lie group which reduces the phase space to a (non-trivial  ...  Optimal control problems for underactuated mechanical systems can be viewed as higher-order constrained variational problems.  ...  local truncation error of the numerical methods).  ... 
doi:10.3934/dcds.2019013 fatcat:qjursz5gbrdxbnhbt5zxljm2vu

Symplectic groupoids and discrete constrained Lagrangian mechanics

Ari Stern, David Martín de Diego, Juan Marrero
2014 Discrete and Continuous Dynamical Systems. Series A  
In this article, we generalize the theory of discrete Lagrangian mechanics and variational integrators in two principal directions.  ...  Next, we use this framework -- along with a generalized notion of generating function due to Sniatycki and Tulczyjew -- to develop a theory of discrete constrained Lagrangian mechanics.  ...  We are grateful to Eduardo Martínez for providing helpful feedback at various stages of this research.  ... 
doi:10.3934/dcds.2015.35.367 fatcat:ph3p4j7rz5cxzceow64uuvn3ci

On the constraints violation in forward dynamics of multibody systems

Filipe Marques, António P. Souto, Paulo Flores
2016 Multibody system dynamics  
It is known that the dynamic equations of motion for constrained mechanical multibody systems are frequently formulated using the Newton-Euler's approach, which is augmented with the acceleration constraint  ...  This formulation results in the establishment of a mixed set of partial differential and algebraic equations, which are solved in order to predict the dynamic behavior of general multibody systems.  ...  The authors also would like to acknowledge the considerable contributions of Professor Javier Cuadrado from University of A Coruña, Spain, for sharing with us some thoughts and material for the numerical  ... 
doi:10.1007/s11044-016-9530-y fatcat:yltdxfncfba4dbijmawxceqxge

Lie Group Spectral Variational Integrators

James Hall, Melvin Leok
2015 Foundations of Computational Mathematics  
We demonstrate the construction of one such variational integrator for the rigid body, and discuss how this construction could be generalized to other related Lie group problems.  ...  We present a new class of high-order variational integrators on Lie groups.  ...  Proof of Theorem 3.2. In §3.1.2, we stated Theorem 3.2 but did not provide a proof.  ... 
doi:10.1007/s10208-015-9287-3 fatcat:dk2h2bmbmjdczfanh5b6ysw52q
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