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Journal of Geometric Mechanics

2011 , Volume 3 , Issue 2

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Lyapunov constraints and global asymptotic stabilization
Sergio Grillo , Jerrold E. Marsden and  Sujit Nair
2011, 3(2): 145-196 doi: 10.3934/jgm.2011.3.145 +[Abstract](50) +[PDF](2034.6KB)
Abstract:
In this paper, we develop a method for stabilizing underactuated mechanical systems by imposing kinematic constraints (more precisely Lyapunov constraints). If these constraints can be implemented by actuators, i.e., if there exists a related constraint force exerted by the actuators, then the existence of a Lyapunov function for the system under consideration is guaranteed. We establish necessary and sufficient conditions for the existence and uniqueness of constraint forces. These conditions give rise to a system of PDEs whose solution is the required Lyapunov function. To illustrate our results, we solve these PDEs for certain underactuated mechanical systems of interest such as the inertia wheel-pendulum, the inverted pendulum on a cart system and the ball and beam system.
Embedded geodesic problems and optimal control for matrix Lie groups
Anthony M. Bloch , Peter E. Crouch , Nikolaj Nordkvist and  Amit K. Sanyal
2011, 3(2): 197-223 doi: 10.3934/jgm.2011.3.197 +[Abstract](44) +[PDF](490.9KB)
Abstract:
This paper is devoted to a detailed analysis of the geodesic problem on matrix Lie groups, with left invariant metric, by examining representations of embeddings of geodesic flows in suitable vector spaces. We show how these representations generate extremals for optimal control problems. In particular we discuss in detail the symmetric representation of the so-called $n$-dimensional rigid body equation and its relation to the more classical Euler description. We detail invariant manifolds of these flows on which we are able to define a strict notion of equivalence between representations, and identify naturally induced symplectic structures.
Sub-Riemannian and sub-Lorentzian geometry on $SU(1,1)$ and on its universal cover
Erlend Grong and  Alexander Vasil’ev
2011, 3(2): 225-260 doi: 10.3934/jgm.2011.3.225 +[Abstract](35) +[PDF](741.6KB)
Abstract:
We study sub-Riemannian and sub-Lorentzian geometry on the Lie group $SU(1,1)$ and on its universal cover SU~(1,1). In the sub-Riemannian case we find the distance function and completely describe sub-Riemannian geodesics on both $SU(1,1)$ and SU~(1,1), connecting two fixed points. In particular, we prove that there is a strong connection between the conjugate loci and the number of geodesics. In the sub-Lorentzian case, we describe the geodesics connecting two points on SU~(1,1), and compare them with Lorentzian ones. It turns out that the reachable sets for Lorentzian and sub-Lorentzian normal geodesics intersect but are not included one to the other. A description of the timelike future is obtained and compared in the Lorentzian and sub-Lorentzain cases.
Euler-Poincaré reduction for systems with configuration space isotropy
Jeffrey K. Lawson , Tanya Schmah and  Cristina Stoica
2011, 3(2): 261-275 doi: 10.3934/jgm.2011.3.261 +[Abstract](28) +[PDF](411.1KB)
Abstract:
This paper concerns Lagrangian systems with symmetries, near points with configuration space isotropy. Using twisted parametrisations corresponding to phase space slices based at zero points of tangent fibres, we deduce reduced equations of motion, which are a hybrid of the Euler-Poincaré and Euler-Lagrange equations. Further, we state a corresponding variational principle.

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