Using distributed magnetometry in navigation of heavy launchers and space vehicles

Recently, a new technique has emerged to address the general problem of reconstructing the inertial velocity of a rigid body moving in a magnetically disturbed region. The contribution of this paper is to apply the developed method, in a prospective spirit, to a case of space navigation in view of estimating the performance improvement that could be obtained using state-of-the-art magnetometer technology onboard heavy launchers and other space vehicles. The main underlying idea of the approach is to estimate the inertial velocity by readings of the magnetic field at spatially distributed (known) locations on the rigid body. Mathematically, through a chain-rule differentiation involving variables commonly appearing in classic inertial navigation, an estimate of this velocity can be obtained. In this paper, we show the potential of this method in the field of navigation of heavy launchers passing through particular regions of the Earth magnetosphere as considered, e.g., for upcoming Galileo missions. Numerical results based on the specifications of candidate embedded magnetic sensors stress the relevance of the approach

Asymptotic expansion of the optimal control under logarithmic penalty: worked example and open problems

We discuss the problem of expansion of optimal control, state and costate when a logarithmic penalty is applied to constraints. We show that, in a simple case, that the variation of (a regular) junction point, and of the optimal control, state and costate is of order \eps\log \eps, where \eps is the penalty parameter

Computation of order conditions for symplectic partitioned Runge-Kutta schemes with application to optimal control

We discuss the derivation of order conditions for the discretization of (unconstrained) optimal control problems, when the scheme for the state equation is of Runge-Kutta type. This problem appears to be essentially the one of checking order conditions for symplectic partitioned Runge-Kutta schemes. We show that the the computations using bi-coloured trees are naturally expressed in this case in terms of oriented free tree. This gives a way to compute them by an appropriate computer program. Our software is able to compute conditions up to order 7 (we display them up to order 6). The results are in accordance with those of Hager (where they were computed for order up to 4) as well as those of Murua where the number of conditions up to order 7 is stated

Asymptotic expansion of the optimal control under logarithmic penalty: worked example and open problems

We discuss the problem of expansion of optimal control, state and costate when a logarithmic penalty is applied to constraints. We show that, in a simple case, that the variation of (a regular) junction point, and of the optimal control, state and costate is of order \eps\log \eps, where \eps is the penalty parameter

An Interior-Point Approach to Trajectory Optimization

This paper presents an interior-point approach for solving optimal control problems. Combining a flexible refinement scheme with dedicated linear algebra solvers, we obtain an efficient algorithm. Numerical results are displayed for various problems, among them several variants of atmospheric reentry

Using distributed magnetometry in navigation of heavy launchers and space vehicles

Recently, a new technique (magneto-inertial navigation, MINAV) has emerged to address the general problem of reconstructing the inertial velocity of a rigid body moving in a magnetically disturbed region. The contribution of this paper is to apply the developed method, in a prospective spirit, to a case of space navigation in view of estimating the performance improvement that could be obtained using state-of-theart magnetometer technology onboard heavy launchers and other space vehicles. The main underlying idea of the approach is to estimate the inertial velocity by readings of the magnetic field at spatially distributed (known) locations on the rigid body. Mathematically, through a chainrule differentiation involving variables commonly appearing in classic inertial navigation, an estimate of this velocity can be obtained. This paper presents the potential of this method in the field of navigation of heavy launchers passing through particular regions of the Earth magnetosphere as considered, e. g., for upcoming Galileo missions. Numerical results based on the specifications of candidate embedded magnetic sensors stress the relevance of the approach. The presented methodology is patent pending and has been partially funded by CNES