Explicit Solution to the Full Nonlinear Problem for Satellite Formation-keeping

peer reviewedThis paper presents simple and exact formation-keeping guidance schemes that use a new method that is rooted in some recent advances in analytical dynamics. As a result of this new approach, explicit control inputs to exactly maintain a given formation configuration are easily determined using continuous thrust propulsion systems. The complete nonlinear problem is addressed, and no linearizations and/or approximations are made. The approach provides a marked improvement over existing results in that the control forces, which cause geometric formation-keeping constraints to be exactly satisfied for arbitrary reference orbits, are found in closed form. For Keplerian reference orbits, a much simpler and explicit expression for the control needed to exactly satisfy formation-keeping constraints than hereto available is obtained. The paper also includes explicit control results when the follower is inserted into orbit with incorrect initial conditions, as usually happens in practice. The Hill reference frame, which is often more intuitive for formation-keeping, is used in the analysis. While this paper takes an example of a projected circular formation, the methodology that is developed can be applied to any desired configuration or orbital requirements. Extensive computational simulations are performed to demonstrate the ease of implementation, and the numerical accuracy provided by the approach developed herein

First Integrals and Solutions of Duffing-Van der Pol Type Equations

A simple transformation is used to obtain the first integrals and the solutions of the Duffing–van der Pol type equation under certain conditions. It is shown that the system can be totally integrable and this total integrability admits new solutions. The new solutions require weaker conditions on the system’s parameters than hereto known

Chattering-free Sliding Mode Control for Propellantless Rendez-vous using Differential Drag

peer reviewedThis paper develops a differential drag-based sliding mode controller for satellite rendez-vous. It is chattering-free and avoids bang-bang type control to adjust the relative motion more efficiently. In spite of uncertain nonlinear perturbations and disturbances, it is shown that the in-plane relative motion between two satellites can be effectively controlled by regulating the drag difference. An adaptive tuning rule is also presented such that the errors are suppressed to lie within a desired error box. The proposed controller is simple and easy to implement in a small satellite, and numerical simulations are carried out to demonstrate its effectiveness in a high fidelity environment

Six-degree-of-freedom Optimal Feedback Control of Pinpoint Landing using Deep Neural Networks

Machine learning regression techniques have shown success at feedback control to perform near-optimal pinpoint landings for low fidelity formulations (e.g. 3 degree-of-freedom). Trajectories from these low-fidelity landing formulations have been used in imitation learning techniques to train deep neural network policies to replicate these optimal landings in closed loop. This study details the development of a near-optimal, neural network feedback controller for a 6 degree-of-freedom pinpoint landing system. To model disturbances, the problem is cast as either a multi-phase optimal control problem or a triple single-phase optimal control problem to generate examples of optimal control through the presence of disturbances. By including these disturbed examples and leveraging imitation learning techniques, the loss of optimality is reduced for pinpoint landing scenario

Stability of Deep Neural Networks for Feedback-Optimal Pinpoint Landings

The ability to certify systems driven by neural networks is crucial for future rollouts of machine learning technologies in aerospace applications. In this study, the neural networks are used to represent a fuel-optimal feedback controller for two different 3-degree-of-freedom pinpoint landing problems. It is shown that the standard sum-ofsquares Lyapunov candidate is too restrictive to assess the stability of systems with fuel-optimal control profiles. Instead, a parametric Lyapunov candidate (i.e. a neural network) can be trained to sufficiently evaluate the closed-loop stability of fuel-optimal control profiles. Then, a stability-constrained imitation learning method is applied, which simultaneously trains a neural network policy and neural network Lyapunov function such that feedback-optimal control is achieved, and Lyapunov stability is verified. Phase-space plots of the Lyapunov derivative show the improvement in stability assessment provided by the neural network Lyapunov function, and Monte Carlo simulations demonstrate the stable, feedback-optimal control provided by the policy

INVERSE PROBLEM FOR LAGRANGIAN DYNAMICS FOR MULTI-DEGREE-OF- FREEDOM SYSTEMS WITH LINEAR DAMPING

ABSTRACT This paper deals with the inverse problem for Lagrangian dynamics for linear multi-degree-of-freedom systems. New results for linearly damped systems are obtained using extensions of results for single-degree-of-freedom systems. First, for a two-degree-of-freedom linear system with linear damping, the conditions for the existence of a Lagrangian are explicitly obtained by solving the Helmholtz conditions. Next, since the Helmholtz conditions are near-impossible to solve for general n-degree-of-freedom systems, a new simple procedure that does not require the use of the Helmholtz conditions and that is easily extended to n-degree-of-freedom linear systems, is developed. The emphasis is on obtaining the Lagrangians for these multi-degree-of-freedom systems in a simple manner, using insights obtained from our understanding of the inverse problem for single-and two-degree-of-freedom systems. Specifically we include systems that commonly arise in linear vibration theory with positive definite mass matrices, and symmetric stiffness and damping matrices. This method yields several new Lagrangians for linear multi-degree-of-freedom systems. Finally, conservation laws for these damped multidegree-of-freedom systems are found using the Lagrangians obtained

Leveraging the Moon and stable Libration point orbits around L4/L5 to observe the Solar corona

There is a significant interest in studying the Solar corona to gather information about the Sun. This investigation provides an efficient approach to observe the So- lar corona by using the Moon as an occulter to suppress the blinding luminosity of the Sun’s surface. Another objective is an analysis and comparison of diffraction patterns created by Lunar occultations (LO) from L4 and from Earth. By exploit- ing the Libration point L4 within the Cislunar region, a spacecraft (s/c) would be within proper position to observe the Solar corona every sidereal month

Gain Adaptation for Continuous Sliding Mode Control

peer reviewedIn this paper a novel adaptive sliding mode controller design is presented for robust control of nonlinear uncertain systems. A continuous control law to compensate for the uncertainties is first developed that is completely free from chattering. Focusing on the relation between the tried gain value and the resultant sliding variable, a new method for estimating the uncertainty bounds is then derived, leading to an adaptive law for gain-tuning by which the error eventually lies within a user-specified region in a finite time. Unlike other existing approaches, the new adaptive rule only requires the magnitude of the control input in the previous time step, which greatly eases the application of the proposed algorithm to real-world systems. An inverted pendulum system is simulated to demonstrate the accuracy and effectiveness of the proposed control strategy