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Geometric stabilization of a quadrotor uav with a payload connected by flexible cable
 in Proceedings of the American Control Conference
"... Abstract — This paper deals with dynamics and control of a quadrotor UAV with a payload that is connected via a flexible cable, which is modeled as a system of seriallyconnected links. It is shown that a coordinatefree form of equations of motion can be derived for an arbitrary number of links acc ..."
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Cited by 7 (4 self)
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Abstract — This paper deals with dynamics and control of a quadrotor UAV with a payload that is connected via a flexible cable, which is modeled as a system of seriallyconnected links. It is shown that a coordinatefree form of equations of motion can be derived for an arbitrary number of links according to Lagrangian mechanics on a manifold. A geometric nonlinear control system is also presented to asymptotically stabilize the position of the quadrotor while aligning the links to the vertical direction. These results will be particularly useful for aggressive load transportation that involves deformation of the cable. The desirable properties are illustrated by a numerical example and a preliminary experimental result. I.
Computational geometric optimal control of rigid bodies
 Communications in Information and Systems, special issue dedicated to R. W. Brockett
, 2008
"... Abstract. This paper formulates optimal control problems for rigid bodies in a geometric manner and it presents computational procedures based on this geometric formulation for numerically solving these optimal control problems. The dynamics of each rigid body is viewed as evolving on a configuratio ..."
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Cited by 6 (4 self)
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Abstract. This paper formulates optimal control problems for rigid bodies in a geometric manner and it presents computational procedures based on this geometric formulation for numerically solving these optimal control problems. The dynamics of each rigid body is viewed as evolving on a configuration manifold that is a Lie group. Discretetime dynamics of each rigid body are developed that evolve on the configuration manifold according to a discrete version of Hamilton’s principle so that the computations preserve geometric features of the dynamics and guarantee evolution on the configuration manifold; these discretetime dynamics are referred to as Lie group variational integrators. Rigid body optimal control problems are formulated as discretetime optimization problems for discrete Lagrangian/Hamiltonian dynamics, to which standard numerical optimization algorithms can be applied. This general approach is illustrated by presenting results for several different optimal control problems for a single rigid body and for multiple interacting rigid bodies. The computational advantages of the approach, that arise from correctly modeling the geometry, are discussed.
Global symplectic uncertainty propagation on SO(3)
, 2008
"... This paper introduces a global uncertainty propagation scheme for the attitude dynamics of a rigid body, through a combination of numerical parametric uncertainty techniques, noncommutative harmonic analysis, and geometric numerical integration. This method is distinguished from prior approaches, a ..."
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Cited by 5 (2 self)
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This paper introduces a global uncertainty propagation scheme for the attitude dynamics of a rigid body, through a combination of numerical parametric uncertainty techniques, noncommutative harmonic analysis, and geometric numerical integration. This method is distinguished from prior approaches, as it allows one to consider probability densities that are global, and are not supported on only a single coordinate chart on the manifold. It propagates a global probability density through the full attitude dynamics, instead of replacing angular velocity dynamics with a gyro bias model. The use of Lie group variational integrators, that are symplectic and remain on the Lie group, as the underlying numerical propagator ensures that the advected probability densities respect the geometric properties of uncertainty propagation in Hamiltonian systems, which arise as consequence of the Gromov nonsqueezing theorem from symplectic geometry. We also describe how the global uncertainty propagation scheme can be applied to the problem of global attitude estimation.
Computational Dynamics of a 3D Elastic String Pendulum Attached to a Rigid Body and an Inertially Fixed Reel Mechanism
, 2009
"... A high fidelity model is developed for an elastic string pendulum, one end of which is attached to a rigid body while the other end is attached to an inertially fixed reel mechanism which allows the unstretched length of the string to be dynamically varied. The string is assumed to have distribute ..."
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Cited by 5 (4 self)
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A high fidelity model is developed for an elastic string pendulum, one end of which is attached to a rigid body while the other end is attached to an inertially fixed reel mechanism which allows the unstretched length of the string to be dynamically varied. The string is assumed to have distributed mass and elasticity that permits axial deformations. The rigid body is attached to the string at an arbitrary point, and the resulting string pendulum system exhibits nontrivial coupling between the elastic wave propagation in the string and the rigid body dynamics. Variational methods are used to develop coupled ordinary and partial differential equations of motion. Computational methods, referred to as Lie group variational integrators, are then developed, based on a finite element approximation and the use of variational methods in a discretetime setting to obtain discretetime equations of motion. This approach preserves the geometry of the configurations, and leads to accurate and efficient algorithms that have guaranteed accuracy properties that make them suitable for many dynamic simulations, especially over long simulation times. Numerical results are presented for typical examples involving a constant length string, string deployment, and string retrieval. These demonstrate the complicated dynamics that arise in a string pendulum from the interaction of the rigid
Dynamics of a 3D Elastic String Pendulum
, 2009
"... This paper presents an analytical model and a geometric numerical integrator for a rigid body connected to an elastic string, acting under a gravitational potential. Since the point where the string is attached to the rigid body is displaced from the center of mass of the rigid body, there exist no ..."
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Cited by 4 (3 self)
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This paper presents an analytical model and a geometric numerical integrator for a rigid body connected to an elastic string, acting under a gravitational potential. Since the point where the string is attached to the rigid body is displaced from the center of mass of the rigid body, there exist nonlinear coupling effects between the string deformation and the rigid body dynamics. A geometric numerical integrator, referred to as a Lie group variational integrator, is developed to numerically preserve the Hamiltonian structure of the presented model and its Lie group configuration manifold. These properties are illustrated by a numerical simulation.
Geometric Numerical Integration for Complex Dynamics of Tethered Spacecraft
"... Abstract — This paper presents an analytical model and a geometric numerical integrator for a tethered spacecraft model that is composed of two rigid bodies connected by an elastic tether. This model includes important dynamic characteristics of tethered spacecraft in orbit, namely the nonlinear cou ..."
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Abstract — This paper presents an analytical model and a geometric numerical integrator for a tethered spacecraft model that is composed of two rigid bodies connected by an elastic tether. This model includes important dynamic characteristics of tethered spacecraft in orbit, namely the nonlinear coupling between deformations of tether, rotational dynamics of rigid bodies, a reeling mechanism, and orbital dynamics. A geometric numerical integrator, referred to as a Lie group variational integrator, is developed to numerically preserve the Hamiltonian structure of the presented model and its Lie group configuration manifold. The structurepreserving properties are particularly useful for studying complex dynamics of a tethered spacecraft over a long period of time. These properties are illustrated by numerical simulations. I.
Dynamics of connected rigid bodies in a perfect fluid
 in Proceedings of the American Control Conference, 2009
"... Abstract — This paper presents an analytical model and a geometric numerical integrator for a system of rigid bodies connected by ball joints, immersed in an irrotational and incompressible fluid. The rigid bodies can translate and rotate in threedimensional space, and each joint has three rotation ..."
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Abstract — This paper presents an analytical model and a geometric numerical integrator for a system of rigid bodies connected by ball joints, immersed in an irrotational and incompressible fluid. The rigid bodies can translate and rotate in threedimensional space, and each joint has three rotational degrees of freedom. This model characterizes the qualitative behavior of threedimensional fish locomotion. A geometric numerical integrator, refereed to as a Lie group variational integrator, preserves Hamiltonian structures of the presented model and its Lie group configuration manifold. These properties are illustrated by a numerical simulation for a system of three connected rigid bodies. I.
FrB15.1 Computational Geometric Optimal Control of Connected Rigid Bodies in a Perfect Fluid
, 2010
"... Abstract — This paper formulates an optimal control problem for a system of rigid bodies that are neutrally buoyant, connected by ball joints, and immersed in an irrotational and incompressible fluid. The rigid bodies can translate and rotate in threedimensional space, and each joint has three rota ..."
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Cited by 2 (1 self)
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Abstract — This paper formulates an optimal control problem for a system of rigid bodies that are neutrally buoyant, connected by ball joints, and immersed in an irrotational and incompressible fluid. The rigid bodies can translate and rotate in threedimensional space, and each joint has three rotational degrees of freedom. We assume that internal control moments are applied at each joint. We present a computational procedure for numerically solving this optimal control problem, based on a geometric numerical integrator referred to as a Lie group variational integrator. This computational approach preserves the Hamiltonian structure of the controlled system and the Lie group configuration manifold of the connected rigid bodies, thereby finding complex optimal maneuvers of connected rigid bodies accurately and efficiently. This is illustrated by numerical computations. I.
HighFidelity Numerical Simulation of Complex Dynamics of Tethered Spacecraft
"... This paper presents an analytical model and a geometric numerical integrator for a tethered spacecraft model that is composed of two rigid bodies connected by an elastic tether. This highfidelity model includes important dynamic characteristics of tethered spacecraft in orbit, namely the nonlinear ..."
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This paper presents an analytical model and a geometric numerical integrator for a tethered spacecraft model that is composed of two rigid bodies connected by an elastic tether. This highfidelity model includes important dynamic characteristics of tethered spacecraft in orbit, namely the nonlinear coupling between tether deformations, rigid body rotational dynamics, a reeling mechanism, and orbital dynamics. A geometric numerical integrator, referred to as a Lie group variational integrator, is developed to numerically preserve the Hamiltonian structure of the presented model and its Lie group configuration manifold. This approach preserves the geometry of the configurations, and leads to accurate and efficient algorithms that have guaranteed accuracy properties that make them suitable for many dynamic simulations, especially over long simulation times. These analytical and computational models provide a method for validating simplifying assumptions that are typically employed in the literature, and numerical simulations are presented that demonstrate the important qualitative differences in the tethered spacecraft dynamics when highfidelity model is employed, as compared to models with additional simplifying assumptions.
Geometric Control of Multiple Quadrotor UAVs Transporting a CableSuspended Rigid Body
"... Abstract — This paper is focused on tracking control for a rigid body payload, that is connected to an arbitrary number of quadrotor unmanned aerial vehicles via rigid links. An intrinsic form of the equations of motion is derived on the nonlinear configuration manifold, and a geometric controller i ..."
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Cited by 1 (0 self)
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Abstract — This paper is focused on tracking control for a rigid body payload, that is connected to an arbitrary number of quadrotor unmanned aerial vehicles via rigid links. An intrinsic form of the equations of motion is derived on the nonlinear configuration manifold, and a geometric controller is constructed such that the payload asymptotically follows a given desired trajectory for its position and attitude. The unique feature is that the coupled dynamics between the rigid body payload, links, and quadrotors are explicitly incorporated into control system design and stability analysis. These are developed in a coordinatefree fashion to avoid singularities and complexities that are associated with local parameterizations. The desirable features of the proposed control system are illustrated by a numerical example. I.