Validation of multi-body modelling methodology for reconfigurable underwater robots

This paper investigates the problem of employing reconfigurable robots in an underwater setting. The main results presented is the experimental validation of a modelling methodology for a system consisting of N dynamically connected robots with heterogeneous dynamics. Two distinct types of experiments are performed, a series of hydrostatic free-decay tests and a series of open-loop trajectory tests. The results are compared to a simulation based on the modelling methodology. The modelling methodology shows promising results for usage with systems composed of reconfigurable underwater modules. The purpose of the model is to enable design of control strategies for cooperative reconfigurable underwater systems

Foundations of the Geometric Mechanics Udwadia-Kalaba Framework for Rigid Body Constrained Motion Analysis

Presented herein are multiple tools for constrained motion analysis extended to different dynamical frameworks. The Udwadia-Kalaba (UK) formalism for the constrained motion analysis of a point mass is a well-documented and applied methodology. Here, UK formulation is generalized to the dynamics of rigid bodies on nonlinear manifolds in the geometric mechanics framework. This approach simultaneously treats rotational and translational motion in a unified method without encountering singularites or non-uniqueness, issues that would arise were attitude parameterization sets used. The viability of this geometric mechanics UK formalism is demonstrated for the cases of fully and underconstrained systems. The nominal UK formalism requires the complete knowledge of the system dynamics. In the presence of unmodeled dynamics or uncertainties in the system, the stability of the system cannot be assessed using the nominal UK formulation. Therefore, a controller is presented that stabilizes the system under unmodeled dynamics and external perturbations. In addition, the UK formulation has been historically applied to systems with equality constraints. However, it has not been formulated for usage with inequality constraints. Here, the implementation of slack and excess variables to treat this class of constraints is presented for usage within the UK formulation for the point mass constrained motion with inequality constraints. Also contained within is an extension of pre-existing work which models the gravitational force acting on a rigid body from a nonuniform gravitational field that holds for any degree and order of spherical harmonics

Constrained Motion Analysis of Spacecraft Trajectory in Restricted Three Body Problem

Due to the popularity of libration points, many satellites are being maintained on their desired trajectory. Indian space research organization has planned to launch the Aditya-L1 spacecraft to study about the Sun by 2021. James Webb Space Telescope has also been designed to observe deep space at L2 in the Sun-Earth system by 2021. The combined gravity of the Earth and the Sun keep satellite’s orbit locked at libration points. Though satellites enjoys an uninterrupted view of Sun and Earth all the time, they are affected by the solar radiation pressure (SRP) continuously. Due to the instability of collinear libration points, the certain amount of thrust is required to maintain the desired trajectory. This thesis introduces the Udwadia-Kalaba (UK) formulation of constrained dynamics as applied to the restricted three-body problem of the Sun-Earth-Spacecraft. A dynamic model of the restricted three-body system is presented to analyze the unconstrained motion of spacecraft. The results show the instability due to perturbation from the SRP. Then, the Udwadia-Kalaba formulation is applied to derive the equation of motion of spacecraft with additional constraints such that spacecraft maintains the desired trajectory at libration points. The results of exact amount of control acceleration with the directions are provided for spacecraft for the following three cases: i) maintaining position at the L1 considering the Earth’s circular orbit (i.e. CR3BP) ii) maintaining position at the L1 and L4 or L5 considering the Earth’s elliptical orbit (i.e. ER3BP) and iii) maintaining the three-dimension halo orbit around the L1 and L2 in CR3BP. The UK formulation is modified using Baumgarte’s stabilization method to allow to compute the results for the incorrect initial conditions (i.e. initial state deviations). In this case, the results are analyzed for the underdamped, overdamped and critically-damped systems. In addition, the delta-v is compared for the transient response with time-varying linear quadratic regulator (LQR). For fully constrained system, the control accelerations required to maintain the desired trajectory obtained by the UK technique are shown identical to those obtain via feedforward part of the time-varying LQR, as expected

Constrained Motion Analysis and Control of Spacecraft Asteroid Hovering with Formulation Extension in Geometric Mechanics Framework

This thesis studies the constrained motion for a spacecraft hovering over an asteroid, where the Udwadia-Kalaba (UK) formulation is applied for nominal control, and an adaptive controller is developed to account for unknowns in the dynamics. Then, the formulation is extended in the geometric mechanics framework to account for rigid body spacecraft asteroid hovering. Constraints are developed and applied for fully constrained and under-constrained asteroid hovering. The fully constrained solutions provided by the UK fundamental equation are compared to an optimal linear quadratic regulator. An adaptive controller is designed using the UK fundamental equation as a basis in the form of a model reference adaptive controller. The controller is proven to have asymptotic tracking of the reference system designed by the desired constraints on the spacecraft. The convergence of the tracking error dynamics is studied using the Lyapunov’s direct method. It is shown that the controller, with accurate estimation of the unknown parameters, results in the minimum required control response due to its basis on the UK equation. The parameters are successfully estimated using a finite-time estimation method. Furthermore, the extension of the UK formulation into the geometric mechanics framework is developed to account for rigid-body spacecraft, where the formulation also allows orientation constraints to be applied on the spacecraft. Constraints with a basis on the Lie algebra of special Euclidean group SE(3) are developed to fully constrain a spacecraft’s position and orientation for hovering over an asteroid. The geometric mechanics UK formulation successfully gives the required angular and translational accelerations to maintain the desired configuration (pose) of the rigid-body spacecraft. The developments above are discussed for a spacecraft hovering over the asteroid Bennu and the closed-loop response of the system, control inputs, and control efforts are provided and discussed

Udwadia-Kalaba Approach for Three Link Manipulator Dynamics With Motion Constraints

Aiming to dynamic modeling of a three-link manipulator subjected to motion constraints, a novel explicit approach to the dynamical equations based on Udwadia-Kalaba (UK) theory is established. The motion constraints on the three-link manipulator can be regarded as external constraints of the system. However, it is not easy to obtain explicit equations for the dynamic modeling of constrained systems. For a multibody system subjecting to motion constraints, it is common to introduce Lagrange multipliers, but obtaining an explicit dynamical equation using traditional Lagrange multipliers is difficult. In order to obtain such equations more simply, motion constraints are handled using the UK equation. Compared with the Lagrange method, the UK approach can simplify the analysis and solution of a constrained system, without the need to introduce additional auxiliary variables to solve the Lagrange equation. Based on a more real-life nominal system (whose parameters are known) model considering the uncertain environment, this paper develops a nonlinear controller that satisfies the required trajectory. This controller allows the nonlinear nominal system to track the desired trajectory exactly without linearizations or approximations. These continuous controllers compensate extra force to eliminate the errors caused by uncertainties. The controllers are based on a generalization of sliding surfaces. Error bounds on tracking caused by uncertainties are analytically obtained. The numerical results show the simplicity and efficacy of the proposed methodology, and the reliability of the error bounds