Angles-Only Navigation to a Noncooperative Satellite Using Relative Orbital Elements

Abstract

This work addresses the design and implementation of a prototype relative navigation tool that uses camera-based measurements collected by a servicer spacecraft to perform far-range rendezvous with a noncooperative client in low Earth orbit. The development serves the needs of future on-orbit servicing missions planned by the German Aerospace Center. The focus of the paper is on the design of the navigation algorithms and the assessment of the expected performance and robustness under real-world operational scenarios. The tool validation is accomplished through a high-fidelity simulation environment based on the Multi Satellite Simulator in combination with the experience gained from actual flight data from the GPS and camera systems onboard the Prototype Research Instruments and Space Mission Technology Advancement mission. Nomenclaturê A, A = matrices of the partials of the nonlinear dynamics equations for the complete linear time-varying system and its relative dynamics subpart a = semimajor axis of the servicer satellitê B, B = control input matrices for the complete linear timevarying system and its subpart b = biases vector of the relative orbit estimatê C, C = measurement sensitivity matrices for the complete linear time-varying system and its subpart e = eccentricity of the servicer satellite H = matrix of accumulated partials of measures with respect to the initial state i = inclination of the servicer satellite K = condition number of a matrix m i = ith merit function n = mean angular motion of the servicer satellite P = covariance matrix of the estimation state R y x = rotation matrix from frame x to y r = relative position vector between the servicer and client t = time u = mean argument of latitude of the servicer satellite u c = line-of-sight unit vector in camera frame v = relative velocity vector between the servicer and client W = covariance matrix of the measurements W o = observability Gramian of the linear time-varying systemÂ,Ĉ x = filter state vector y = linearly modeled measurements via sensitivity matrix z = modeled observations: couple of azimuth and elevation Δ•= finite variation of a quantity δα = nondimensional relative orbital elements set δe = nondimensional relative eccentricity vector δi = nondimensional relative inclination vector δλ = nondimensional relative mean longitude δ• = relative quantity ϵ = uncorrelated measurement errors ζ = true observations η = azimuth angle of line-of-sight unit vector from servicer to client θ = ascending node of the relative orbit κ = one pixel camera resolution in radians Λ = information matrix μ = Earth constant ξ = single measure to refer either to azimuth or to elevation angles σ = standard deviation of state and/or measurement errorŝ Φ, Φ, Φ b = state transition matrices for the complete system and its subparts ϕ = mean argument of latitude of the relative perigee ψ = elevation angle of line-of-sight unit vector from servicer to client Ω = right ascension of the ascending node of the servicer satellite ω = argument of perigee of the servicer s...

Simulation Toolkit for Onboard Optimal Maneuver Planning in Active Debris Removal Using Angles-Only Navigation

“…4 maneuvers per hopping loop. The covariance triggered optimal maneuver profile consists of 68 impulsive maneuvers, the maneuver number of each hopping is [8,8,4,4,4,4,5,7,8,8,8] respectively. Covariance triggered maneuvers cost 0.7340m/s more fuel, but the trajectory robustness/dispersions ( Figure 5 and Figure 6) and final relative control accuracy is highly improved.…”

Trajectory Guidance Using Periodic Relative Orbital Motion

“…The phase difference is computed from the colatitude of the sinilaterating node Ξ arctan−2 cos ϒ; sin ϒ (22) and relative amplitude from the colatitude of the sinilaterating node and the slant η − 2 tan σ 1 3 cos 2 Ξ p − 1 3 cos 2 ϒ p tan σ…”

“…It is possible to express the amplitude ratio η and the phase difference Ξ in terms of σ and ϒ [Eqs. (22) and (23)]. This means that all the other elements may be computed from them (with the semimajor axis also needing a factor of κ).…”

“…The nature of the trajectories and frequency of maneuvers will differ between these classes, but the techniques developed here are applicable to both. Gaias et al [22] describe camera-based systems for far-range rendezvous with a client spacecraft and use a guidance system based on the eccentricity and inclination vector differences. The methods described here could potentially improve that work.…”

Optimal Rendezvous Guidance with Enhanced Bearings-Only Observability

“…The equations describing this motion for circular orbits are the well-known Hill equations, whose homogeneous solution is known as the Clohessy-Wiltshire equations [15]. Equivalent equations also exist for elliptical orbits [16] as well as other coordinate systems, such as relative orbital elements [17], in which case, the work in this Note also applies. Moreover, the particular solution for a variety of different types of maneuvers can be found in closed form [15].…”