Christophe Louembet

Brief paper: Motion planning for flat systems using positive B-splines: An LMI approach

In this paper, the motion planning problem is studied for nonlinear differentially flat systems using B-spline parameterization of the flat output history. In order to satisfy the constraints continuously in time, the motion planning problem is transformed into a B-spline positivity problem. The latter problem is formulated as a convex semidefinite programming problem by means of a non-negative piecewise polynomial function description based on sum of squares decomposition. The contribution of the paper is thus a one-step design procedure for motion planning that satisfies constraints continuously in time where usual B-spline and collocation techniques need post-analysis. Finally, an example of flexible link manipulator motion is presented to illustrate the overall approach.

A New Mixed Iterative Algorithm to Solve the Fuel-Optimal Linear Impulsive Rendezvous Problem

The optimal fuel impulsive time-fixed rendezvous problem is reviewed. In a linear setting, it may be reformulated as a non-convex polynomial optimization problem for a pre-specified fixed number of velocity increments. Relying on variational results previously published in the literature, an improved mixed iterative algorithm is defined to address the issue of optimization over the number of impulses. Revisiting the primer vector theory, it combines variational tests with sophisticated numerical tools from algebraic geometry to solve polynomial necessary and sufficient conditions of optimality. Numerical examples under circular and elliptic assumptions show that this algorithm is efficient and can be integrated into a rendezvous planning tool.

Designing Continuously Constrained Spacecraft Relative Trajectories for Proximity Operations

This paper presents a new method for designing an optimal plan of impulsive maneuvers for spacecraft proximity operations. The proposed method accounts for the presence of linear continuous constraints on the spacecraft relative trajectory. Impulsive control and continuous constraints are brought together through the parameterization of the spacecraft relative trajectory between two consecutive maneuvers. This parameterization is used in order to develop a finite convex description of all the admissible trajectories. It enables the transformation of the continuous constraints on the spacecraft relative trajectory into a finite number of polynomial nonnegativity constraints. The resulting optimal control problem can be solved using semidefinite programming.

Minimizing the Effects of Navigation Uncertainties on the Spacecraft Rendezvous Precision

The ability to robustly and precisely control the spacecraft relative motion will play an important role in future on-orbit inspection and on-orbit servicing missions. Model predictive control (MPC) is considered to be an effective control strategy for these types of spacecraft operations, which can easily handle mission specificconstraintswhileexplicitlyminimizingthefuelconsumption. The maneuvers plan is obtained by solving a finite horizon open- loop optimal control problem starting from the spacecraft relative state, and the optimal solution consists of a series of control actions out of which only the first one is executed. Some ideas from tube-basedMPCare used in this Note to solve the robust fixed-time spacecraft rendezvous problem for eccentric reference orbits. The purpose is to obtain a sequence of feedback policies that steers the spacecraft from an initial relative state toward an ellipsoidal set centered around a desired final state, in the presence of navigation uncertainties. This must be done while respecting the actuators saturation constraints and while pursuing a double objective: minimize the fuel cost of the mission and minimize the size of the arrival set to guarantee a good rendezvous precision. The control policies are restricted to affine disturbance feedback policies to ensure a convex formulation of the control synthesis problem. The obtained sequence of feedback policies drives the system to the guaranteed arrival set without any need for recurrent optimization.

Robust Rendezvous Planning Under Maneuver Execution Errors

The problem of designing rendezvous guidance maneuver plan robust to thrusting errors is addressed in this paper. The aim of this paper is to develop tractable and robust guidance algorithms. Solving the rendezvous guidance problem via a direct approach leads to uncertain optimization problems while accounting for the Guidance, Navigation and Control (GNC) systems uncertainties and errors. A worst-case ap-proach is considered in order to obtain tractable robust counterparts. The robustness certificates derived from these guidance algorithm provide the means to analyze the effects of the considered errors on the rendezvous mission. Several types of missions tested in a linear environment are used to illustrate the methodology.

Convex inner approximations of nonconvex semialgebraic sets applied to fixed-order controller design

We describe an elementary algorithm to build convex inner approximations of nonconvex sets. Both input and output sets are basic semialgebraic sets given as lists of defining multivariate polynomials. Even though no optimality guarantees can be given (e.g. in terms of volume maximisation for bounded sets), the algorithm is designed to preserve convex boundaries as much as possible, while removing regions with concave boundaries. In particular, the algorithm leaves invariant a given convex set. The algorithm is based on Gloptipoly 3, a public-domain Matlab package solving nonconvex polynomial optimisation problems with the help of convex semidefinite programming (optimisation over linear matrix inequalities, or LMIs). We illustrate how the algorithm can be used to design fixed-order controllers for linear systems, following a polynomial approach.

Constrained periodic spacecraft relative motion using non-negative polynomials

A new method for obtaining constrained periodic relative motion between spacecraft on Keplerian orbits is presented. The periodic relative trajectory is required to evolve autonomously inside a predefined tolerance region. Unlike the classical time-sampling approaches, our method guarantees continuous satisfaction of the constraints over an infinite horizon. This is done by reformulating the tolerance region constraints on the relative trajectory as conditions of non-negativity of some polynomials. The resulting problem is solved using semi-definite programming.

A hybrid control framework for impulsive control of satellite rendezvous

We focus on the problem of satellite rendezvous between two spacecraft in elliptic orbits. Using a linearized model of the relative dynamics, we first propose a periodic similarity transformation based on Floquet-Lyapunov theory, leading to a set of coordinates under which the free motion is linear time-invariant. Then we address the problem of impulsive control of satellite rendezvous as a hybrid dynamical system, and we show that the arising elegant representation enables designing impulsive control laws with different trade-offs between computational complexity and fuel consumption. The adopted hybrid formalism allows us to prove suitable stability properties of the proposed controllers. The results are comparatively illustrated on simulation examples.

Predictive control algorithm for spacecraft rendezvous hovering phases

The paper proposes a predictive control law for orbital rendezvous hovering phases. The proposed algorithm provides an impulsive control that steer the vehicle to a set of the periodic relative orbits enclosed in a hovering zone. The control law is computed by determining a point lying in the intersection between a semi-algebraic set and a hyperplane using an alternating projections algorithm. The present work proposes several improvement compare to previous published works based on a two-impulse strategy: the control consists of one impulse instead of two, the periodic and admissible orbit that the chaser is steered to is not defined a priori, and the budget and saturation conditions are taken into account. The efficiency of the proposed predictive control algorithm is bench tested on a nonlinear simulator.

A two-impulse method for stabilizing the spacecraft relative motion with respect to a periodic trajectory

The article presents an analytical method for computing a two-impulse control law that stabilizes the spacecraft relative motion with respect to an invariant set. The invariant set contains the states belonging to a desired periodic relative trajectory and is described using linear equations. The two impulses are computed analytically based on the prediction of the evolution of the relative trajectory. Closed-loop tests are conducted using the non-linear relative dynamics, for different eccentricities of the reference orbit and for different levels of navigation uncertainty. Encouraging results are obtained using a control strategy that requires very few computational effort.