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624
RealTime Motion Planning For Agile Autonomous Vehicles
 AIAA JOURNAL OF GUIDANCE, CONTROL, AND DYNAMICS
, 2000
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The EulerPoincaré equations and semidirect products with applications to continuum theories
 ADV. MATH
, 1998
"... We study Euler–Poincaré systems (i.e., the Lagrangian analogue of LiePoisson Hamiltonian systems) defined on semidirect product Lie algebras. We first give a derivation of the Euler–Poincaré equations for a parameter dependent Lagrangian by using a variational principle of Lagrange d’Alembert type. ..."
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Cited by 149 (66 self)
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We study Euler–Poincaré systems (i.e., the Lagrangian analogue of LiePoisson Hamiltonian systems) defined on semidirect product Lie algebras. We first give a derivation of the Euler–Poincaré equations for a parameter dependent Lagrangian by using a variational principle of Lagrange d’Alembert type. Then we derive an abstract KelvinNoether theorem for these equations. We also explore their relation with the theory of LiePoisson Hamiltonian systems defined on the dual of a semidirect product Lie algebra. The Legendre transformation in such cases is often not invertible; thus, it does not produce a corresponding Euler–Poincaré system on that Lie algebra. We avoid this potential difficulty by developing the theory of Euler–Poincaré systems entirely within the Lagrangian framework. We apply the general theory to a number of known examples, including the heavy top, ideal compressible fluids and MHD. We also use this framework to derive higher dimensional CamassaHolm equations, which have many potentially interesting analytical properties. These
Liegroup methods
 ACTA NUMERICA
, 2000
"... Many differential equations of practical interest evolve on Lie groups or on manifolds acted upon by Lie groups. The retention of Liegroup structure under discretization is often vital in the recovery of qualitatively correct geometry and dynamics and in the minimization of numerical error. Having ..."
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Cited by 110 (19 self)
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Many differential equations of practical interest evolve on Lie groups or on manifolds acted upon by Lie groups. The retention of Liegroup structure under discretization is often vital in the recovery of qualitatively correct geometry and dynamics and in the minimization of numerical error. Having introduced requisite elements of differential geometry, this paper surveys the novel theory of numerical integrators that respect Liegroup structure, highlighting theory, algorithmic issues and a number of applications.
Robust Hybrid Control for Autonomous Vehicle Motion Planning
, 2000
"... The operation of an autonomous vehicle in an unknown, dynamic environment is a very complex problem, especially when the vehicle is required to use its full maneuvering capabilities, and to react in real time to changes in the operational environment. A possible approach to reduce the computationa ..."
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Cited by 101 (9 self)
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The operation of an autonomous vehicle in an unknown, dynamic environment is a very complex problem, especially when the vehicle is required to use its full maneuvering capabilities, and to react in real time to changes in the operational environment. A possible approach to reduce the computational complexity of the motion planning problem for a nonlinear, high dimensional system, is based on a quantization of the system dynamics, leading to a control architecture based on a hybrid automaton, the states of which represent feasible trajectory primitives for the vehicle. This paper focuses on the feasibility of this approach: the structure of a Robust Hybrid Automaton is defined and its properties are analyzed. Algorithms are presented for timeoptimal motion planning in a free workspace, and in the presence of fixed or moving obstacles. A case study involving a small autonomous helicopter is presented: a nonlinear control law for maneuver execution is provided, and a robust hyb...
EulerPoincaré models of ideal fluids with nonlinear dispersion
, 1998
"... Based on recent advances in the theory of EulerPoincare (EP) equations with advected parameters and using the methods of Hamilton's principle asymptotics and averaged Lagrangians, we propose a new class of models for ideal incompressible fluids in three dimensions, including stratification an ..."
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Cited by 66 (19 self)
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Based on recent advances in the theory of EulerPoincare (EP) equations with advected parameters and using the methods of Hamilton's principle asymptotics and averaged Lagrangians, we propose a new class of models for ideal incompressible fluids in three dimensions, including stratification and rotation for GFD applications. In these models, the amplitude of the rapid fluctuations introduces a length scale, #, below which wave activity is filtered by both linear and nonlinear dispersion. This filtering enhances the stability and regularity of the new fluid models without compromising either their large scale behavior, or their conservation laws. These models also describe geodesic motion on the volumepreserving di#eomorphism group for a metric containing the H 1 norm of the fluid velocity. PACS Numbers: 03.40.Gc, 47.10.+g, 03.40.t, 03.40.z 1 EulerPoincare models of ideal fluids 2 Linear dispersion is well known to have profound e#ects on wave mean flow interaction in fluids [...
Mechanical Integrators Derived from a Discrete Variational Principle
"... Many numerical integrators for mechanical system simulation are created by using discrete algorithms to approximate the continuous equations of motion. In this paper, we present a procedure to construct timestepping algorithms that approximate the flow of continuous ODE's for mechanical system ..."
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Cited by 66 (11 self)
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Many numerical integrators for mechanical system simulation are created by using discrete algorithms to approximate the continuous equations of motion. In this paper, we present a procedure to construct timestepping algorithms that approximate the flow of continuous ODE's for mechanical systems by discretizing Hamilton's principle rather than the equations of motion. The discrete equations share similarities to the continuous equations by preserving invariants, including the symplectic form and the momentum map. We girst present a formulation of discrete mechanics along with a discrete variational principle. We then show that the resulting equations of motion preserve the symplectic form and that this formulation of mechanics leads to conservation laws from a discrete version of Noether's theorem. We then use the discrete mechanics formulation to develop a procedure for constructing mechanical integrators for continuous Lagrangian systems. We apply the construction procedure to the rigid body and the double spherical pendulum to demonstrate numerical properties of the integrators.
Kinematic controllability for decoupled trajectory planning in underactuated mechanical systems
 IEEE Transactions on Robotics and Automation
, 2001
"... Abstract — We introduce the notion of kinematic controllability for secondorder underactuated mechanical systems. For systems satisfying this property, the problem of planning fast collisionfree trajectories between zero velocity states can be decoupled into the computationally simpler problems of ..."
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Cited by 65 (16 self)
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Abstract — We introduce the notion of kinematic controllability for secondorder underactuated mechanical systems. For systems satisfying this property, the problem of planning fast collisionfree trajectories between zero velocity states can be decoupled into the computationally simpler problems of path planning for a kinematic system followed by timeoptimal time scaling. While this approach is well known for fully actuated systems, until now there has been no way to apply it to underactuated dynamic systems. The results in this paper form the basis for efficient collisionfree trajectory planning for a class of underactuated mechanical systems including manipulators and vehicles in space and underwater environments.
The Geometric Mechanics of Undulatory Robotic Locomotion
 INTERNATIONAL JOURNAL OF ROBOTICS RESEARCH
, 1996
"... This paper uses geometric methods to study basic problems in the mechanics and control of locomotion. We consider in detail the case of "undulatory locomotion," in which net motion is generated by coupling internal shape changes with external nonholonomic constraints. Such locomotion pro ..."
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Cited by 61 (15 self)
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This paper uses geometric methods to study basic problems in the mechanics and control of locomotion. We consider in detail the case of "undulatory locomotion," in which net motion is generated by coupling internal shape changes with external nonholonomic constraints. Such locomotion problems have a natural geometric interpretation as a connection on a principal fiber bundle. The properties of connections lead to simplified results for studying both dynamics and issues of controllability for locomotion systems. We demonstrate the utility of this approach using a novel "Snakeboard" and a multisegmented serpentine robot which is modeled after Hirose's Active Cord Mechanism.
High order RungeKutta methods on manifolds
 APPL. NUMER. MATH
, 1999
"... This paper presents a family of RungeKutta type integration schemes of arbitrarily high order for differential equations evolving on manifolds. We prove that any classical RungeKutta method can be turned into an invariant method of the same order on a general homogeneous manifold, and present a fa ..."
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Cited by 56 (11 self)
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This paper presents a family of RungeKutta type integration schemes of arbitrarily high order for differential equations evolving on manifolds. We prove that any classical RungeKutta method can be turned into an invariant method of the same order on a general homogeneous manifold, and present a family of algorithms that are relatively simple to implement.
On the Construction of Lyapunov Functions using the Sum of Squares Decomposition
, 2002
"... A relaxation of Lyapunov's direct method has been proposed recently that allows for an algorithmic construction of Lyapunov functions to prove stability of equilibria in nonlinear systems, but the search is restricted to systems with polynomial vector fields. In this paper, the above technique ..."
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Cited by 55 (9 self)
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A relaxation of Lyapunov's direct method has been proposed recently that allows for an algorithmic construction of Lyapunov functions to prove stability of equilibria in nonlinear systems, but the search is restricted to systems with polynomial vector fields. In this paper, the above technique is extended to include systems with equality, inequality, and integral constraints. This allows certain nonpolynomial nonlinearities in the vector field to be handled exactly and the constructed Lyapunov functions to contain nonpolynomial terms. It also allows robustness analysis to be performed. Some examples are given to illustrate how this is done.