Banavara N. Shashikanth

The Hamiltonian structure of a two-dimensional rigid circular cylinder interacting dynamically with N point vortices

This paper studies the dynamical fluid plus rigid-body system consisting of a two-dimensional rigid cylinder of general cross-sectional shape interacting with N point vortices. We derive the equations of motion for this system and show that, in particular, if the vortex strengths sum to zero and the rigid-body has a circular shape, the equations are Hamiltonian with respect to a Poisson bracket structure that is the sum of the rigid body Lie–Poisson bracket on Se(2)*, the dual of the Lie algebra of the Euclidean group on the plane, and the canonical Poisson bracket for the dynamics of N point vortices in an unbounded plane. We then use this Hamiltonian structure to study the linear and nonlinear stability of the moving Foppl equilibrium solutions using the energy-Casimir method.

Optimal approach to and alignment with a rotating rigid body for capture

This paper addresses a feed-forward optimal control problem for one rigid body to approach to and align with another arbitrarily rotating rigid body, with an application to the satellite rendezvous problem. In particular, we focus on the satellite rendezvous strategy of finding an optimal trajectory, and the required thrust force profiles, which will guide the chasing spacecraft to approach the tumbling satellite such that the two vehicles will eventually have no relative rotation and thus a subsequent docking or capture operation can be safely performed with a normal docking or capture mechanism. Our approach is to model the system using rigid-body dynamics and apply Pontryagin’s Maximum Principle for the optimal control. A planar problem is presented as a case study, in which together with the Maximum Principle, the Lie algebras associated with the system are used to examine the existence of singular extremals for the time-optimal control problem. Also, optimal trajectories and the corresponding set of control force/torque profiles are numerically generated for the time/fuel-consumption optimal control problem.

Symmetric pairs of point vortices interacting with a neutrally buoyant two-dimensional circular cylinder

The dynamic interaction of N symmetric pairs of point vortices with a neutrally buoyant two-dimensional rigid circular cylinder in the inviscid Hamiltonian model of Shashikanth et al. [Phys. Fluids 14, 1214 (2002)] and Shashikanth [Reg. Chaotic Dyn. 10, 1 (2005)] is examined. The model may be thought of as a section of an inviscid axisymmetric model of a neutrally buoyant sphere interacting with N coaxial circular vortex rings and has possible applications to problems such as fish swimming. The Hamiltonian structure of this half-space model is first presented. The cases N=1 and N=2 are then examined in detail. Equilibria and bifurcations are studied, and for both these cases an important bifurcation parameter involving the total linear “momentum” of the system, the strength of the vortex pairs, and the radius of the cylinder emerges. For N=1, it is shown that there exist the moving Foppl equilibrium and the moving normal line equilibrium (in which the vortices in the pair are located on the top and bottom...

Optimal Control for Spacecraft to Rendezvous with a Tumbling Satellite in a Close Range

One of the most challenging tasks for satellite on-orbit servicing is to rendezvous and capture a non-cooperative satellite such as a tumbling satellite. This paper presents an optimal control strategy for a servicing spacecraft to rendezvous (in close range) with a tumbling satellite. The strategy is to find an optimal trajectory which will guide the servicing spacecraft to approach the tumbling satellite such that the two vehicles will eventually have no relative rotation. Therefore, a subsequent docking or capture operation can be safely performed. Pontryagins maximum principle is applied in generation of the optimal approaching trajectory and the corresponding set of control force/torque profiles. A planar satellite chasing problem is presented as a case study, in which together with the maximum principle, the Lie algebras associated with the system are used to examine the existence of singular extremals for optimal control. Optimal trajectories for minimum fuel consumption are numerically simulated

Dynamics and control of the system of a 2-D rigid circular cylinder and point vortices

The Hamiltonian system of a 2-D rigid circular cylinder dynamically interacting with N point vortices in its vicinity (Shashikanth et al., 2002) is an idealized example of coupled solid-fluid systems interacting in the presence of vorticity and has applications to problems in engineering, such as locomotion of autonomous underwater vehicles, and in nature, such as swimming of fish. The first half of the paper presents simulation results and analysis of the system with N = 1 and N = 2, where the latter case is more relevant to the swimming of fish. The second half of this paper is devoted to theoretical analysis of the time-optimal control of the system. The control input is a bounded external force acting through the cylinder center-of-mass. The maximum principle and its later extensions, in particular those due to Sontag and Sussmann, are applied to this system for the case of the cylinder interacting with a single point vortex to arrive at conclusions regarding characteristics of the time-optimal controller

Dissipative N-point-vortex Models in the Plane

A method is presented for constructing point vortex models in the plane that dissipate the Hamiltonian function at any prescribed rate and yet conserve the level sets of the invariants of the Hamiltonian model arising from the SE (2) symmetries. The method is purely geometric in that it uses the level sets of the Hamiltonian and the invariants to construct the dissipative field and is based on elementary classical geometry in ℝ3. Extension to higher-dimensional spaces, such as the point vortex phase space, is done using exterior algebra. The method is in fact general enough to apply to any smooth finite-dimensional system with conserved quantities, and, for certain special cases, the dissipative vector field constructed can be associated with an appropriately defined double Nambu–Poisson bracket. The most interesting feature of this method is that it allows for an infinite sequence of such dissipative vector fields to be constructed by repeated application of a symmetric linear operator (matrix) at each point of the intersection of the level sets.

Discussion on: “Symmetry Reduction and Control of the Dynamics of a 2-D Rigid Circular Cylinder and a Point Vortex: Vortex Capture and Scattering”

Symmetry reduction and control of the Hamiltonian system of a 2D rigid circular cylinder dynamically interacting with a point vortex external to it is presented. This dynamic model is an idealized example in an inviscid framework of fully coupled solid–fluid systems interacting in the presence of vorticity and has potential applications to problems in engineering and in nature involving the interaction of coherent vortices with bodies moving (primarily) under their influence. The dynamics of the system generically gives rise to two types of vortex orbits relative to the moving cylinder: bound and scattering orbits. The control input of a bounded external force acting through the center of mass of the cylinder is then added. Exploiting the S 1 -symmetry in the system, symplectic reduction is employed to formulate an S 1 -invariant control system, that preserves the momentum map, on the 2D symplectic reduced space. On this reduced space, both nonoptimal and optimal controllers, the latter using Pontryagins maximum principle, are investigated with the control objective of changing the vortex orbit from a bound to a scattering type and vice versa.

Erratum: “The Hamiltonian structure of a two-dimensional rigid circular cylinder interacting dynamically with N point vortices” [Phys. Fluids 14, 1214 (2002)]

where ^,& is the standard Euclidean inner product on t plane. There are two small errors in the statement of the Pro sition on pages 1216–1217. The set B in the definition of the phase space on page 1217 should be B, i.e.,N copies ofB. In the last mathematical term in the statement on page 1 ~just before the Proof begins !, there is a plus sign instead o a minus sign in front of the summation of terms denoting point vortex bracket.