Jet Transport Technique

The Jet Transport (JT) technique, also known as Differential Algebra (DA), is an automatic differentiation method that allows to obtain high-order Taylor expansions of nonlinear functions. This methodology has its origin in an attempt to solve analytical problems using an algebraic approach. In mathematics, the Jet is an operation that for a given differentiable function produces a Taylor polynomial at each point of its domain. The basic idea behind the JT method is to propagate jets along the vectorfield (map) of a continuous (discrete) dynamical system.

This chapter is devoted to introduce the Jet Transport (JT) technique, and considers both its algorithmic procedures and their implementation in a computer environment. In essence, JT technique is composed of polynomial expansion techniques (in Taylor or Chebyshev basis) and a series of associated algebraic operations defined in the algebra of polynomials. This polynomial approximation technique employs Taylor or Chebyshev polynomials to fit basic nonlinear functions up to an arbitrary order, such as exponentials, powers, logarithms, and trigonometric functions. The specific algebra trivially defines not only polynomial addition, subtraction, multiplication and division operations, but also some useful algorithms, such as polynomial composition, numerical integration, and domain splitting techniques. In what follows, all these operations are introduced in order to build a tailored JT framework for carrying out the polynomial-based semi-analytical method in an automatic, accurate and efficient way.

This chapter studies the application of Jet Transport (JT) to the propagation of geostationary trajectory uncertainties under four dominant perturbations: Solar radiation pressure, Earth’s non-spherical gravity potential, and luni-solar gravitational attractions. A basic scheme is considered and implemented in several coordinate representations and with two kinds of expansions: Taylor and Chebyshev. Taking into account the size of the uncertainty neighborhood, as well as the order and time steps of the integration schemes, a large amount of combinations of the above factors are simulated. A series of look-up tables, with recommendations on the best options to address the propagation, are given.

The particle filter is one of the most powerful methods for nonlinear state estimation of spacecraft on account of its accuracy and stability. However, its heavy computational burden limits its application in real-time estimations. In this chapter, we propose an improvement for the particle filter based on the Jet Transport (JT) method, and apply it to the real-time attitude estimation of tumbling space objects. The main innovation of the Jet Transport particle filter (JTPF) is to use the Jet Transport technique in the particle evolution process, rather than using the numerical integration as the classical particle filter does, so as to reduce the computational burden of the algorithm. Furthermore, the proposed JTPF uses the multiplicative error quaternion to avoid further errors in the normalizing process and the regularization technique to avoid the particle degeneracy. The JTPF is tested in three scenarios with different state and observation dimensions. Monte Carlo simulations demonstrate that the JTPF has a similar accuracy as the classical particle filter, and costs only a \(7 \,\mathrm {\%} \sim 13 \,\mathrm {\%}\) of CPU time of the latter. Moreover, some empirical rules are summarized about the optimal Jet Transport expansion order and particle number of the JTPF.

Based on the Jet Transport technique, this chapter proposes a novel nonlinear Kalman filter for simultaneously estimating the spacecraft state vector and uncertain parameters, either physically related with the spacecraft or with the measurement procedure. Two different coordinate representations, including Cartesian and hybrid geostationary orbital elements, are exploited in the new nonlinear estimators. The performance and sensitivity analyses of the proposed Jet Transport-based nonlinear estimators are assessed by numerical simulations and compared with the classical extended Kalman filter.

Based on the Jet Transport techniques, this chapter proposes a set of efficient nonlinear Kalman filters, implemented at different expansion orders, to achieve autonomous satellite orbit determination via the usage of sparse inter-satellite relative measurement information. Three fault-tolerant algorithms are further put forward to suppress measurement faults.

Variants and improvements of the extended Kalman filter (EKF) always encounter a dilemma between the estimation accuracy delivered and the computational burden associated.

This chapter is devoted to the application of JT to the development of a low-thrust station keeping strategies at libration point orbits. Two families of low-thrust station-keeping control procedures are derived based on the dynamical properties of the phase space near libration point orbits. The first family is obtained as the limit of impulsive maneuvers, and the second one by means of dynamically reshaping Floquet modes. In particular, JT technique allows to control laws to be explicitly written as polynomials in terms of the deviation between the state of the spacecraft and the one of a nominal point on the libration point orbit.