Daniele Mortari

A Survey on Star Identification Algorithms

Abstract: The author surveys algorithms used in star identification, commonly used in startrackers to determine the attitude of a spacecraft. Star trackers are a staple of attitude deter-mination systems for most types of satellites. The paper covers: (a) lost-in-space algorithms (when no a priori attitude information is available), (b) recursive algorithms (when some apriori attitude information is available), and (c) non-dimensional algorithms (when the startracker calibration is not well-known). The performance of selected algorithms and support-ing algorithms are compared.Keywords: Star Identification; Attitude Estimation; Star Tracker Algorithms.1. IntroductionThe requirement for attitude (orientation) information of a spacecraft has been the mother of inventionof many devices and algorithms, notably the process of autonomously identifying stars (Star-ID). Thoughthere is much history of devices used to identify stars and compute an attitude that do not use a starcamera, this paper primarily analyzes algorithms that use a star camera with an imaging array and analgorithm to match observed (body) directions of stars with catalog (inertial) directions of stars withoutrequiring reorienting the camera or the spacecraft. These algorithms fall into two basic categories,

Norm-Constrained Kalman Filtering

The problem of estimating the state vector of a dynamical system from vector measurements when it is known that the state vector satisfies norm equality constraints is considered. The case of a linear dynamical system with linear measurements subject to a norm equality constraint is discussed with a review of existing solutions. The norm constraint introduces a nonlinearity in the system for which a new estimator structure is derived by minimizing a constrained cost function. It is shown that the constrained estimate is equivalent to the brute-force normalization of the unconstrained estimate. The obtained solution is extended to nonlinear measurement models and applied to the spacecraft attitude filtering problem.

Euler-q Algorithm for Attitude Determination from Vector Observations

A new cost function for optimal attitude dee nition and the Euler- q algorithm based on this cost function are presented. Theoptimality criterion isderived from theEuler axis rotational property and allowsa fast and reliable computation of the optimal eigenaxis. The mathematical procedure leads to the eigenanalysisof a 3 £ 3 symmetric matrix whose eigenvector, associated with the smallest eigenvalue, is the optimal Euler axis. This eigenvector is evaluated by a simple cross vector, and the singularity is avoided using the method of sequential rotations. The rotational error is then analyzed and dee ned, and an accuracy comparison test is performed between a previously accepted criterion of optimal attitude and the proposed one. Results show that the earlier dee nition of optimality is slightly more precise than Euler- q, which, in turn, demonstrates a clear gain in computational speed.

Flower constellation set theory. Part I: Compatibility and phasing

Flower Constellations are special satellite constellations whose satellites follow the same 3-dimensional space track with respect to assigned rotating reference frame. This paper presents the theoretical foundation of compatibility and phasing of the Flower Constellations. Compatibility is the synchronization property of a Flower Constellation with respect to a rotating reference frame while phasing dictates the satellite distribution property. Compatibility and phasing, which are ruled by a set of five independent integer parameters, constitute the two main properties of the Flower Constellations. In particular, the dual-compatible Flower Constellations theory, which allows a simultaneous synchronization of the Flower Constellation dynamics with two independent rotating reference frames, is introduced. Meaningful examples and potential applications are briefly discussed.

N-Impulse Orbit Transfer Using Genetic Algorithms

T HE orbit transfer problems using impulsive thrusters have attracted researchers for a long time [1]. One of the objectives in these problems is to find the optimal fuel orbit transfer between two orbits, generally inclined eccentric orbits. The optimal two-impulse orbit transfer problem poses multiple local optima, and classical optimization methods find only local optimum solution. McCue [2] solved the problem of optimal two-impulse orbit transfer using a combination between numerical search and steepest descent optimization procedures. Jezewaski and Rozendall [3] developed an iterative method to calculate local minima solutions for the nimpulse fixed time rendezvous problems. Genetic algorithms (GAs) have been used in the literature to search for the global optimal orbit maneuver. Reichert [4] addressed the optimum two-impulse orbit transfer problem for coplanar orbits only. The accuracy obtained using this formulation is not good unless a narrow range, around the optimal value, for each design variable is known in advance [4]. Given narrow ranges for the design variables, the solution obtained using this formulation does not guarantee that the satellite will be inserted exactly into final orbit, but rather there is a small error unless the GA finds exactly the global optimal solution. Kim and Spencer [5] introduced a different formulation to the two-impulse orbit transfer problem by using six design variables for coplanar orbits. This formulation also does not guarantee the satellite is placed exactly in the final orbit. In this note, a new formulation to the problem is introduced. This formulation is general for noncoplanar elliptical orbits. It can also implement any number of thrust impulses. For the case of twoimpulse maneuver, this formulation requires only three design variables for any noncoplanar orbit transfer. The solution obtained by this formulation is guaranteed to insert the satellite in the final orbit exactly, even if the GAs did not converge to the global optimal solution. This formulation requires solving Lambert’s problem to find the parameters of the transfer orbit for a given set of the three design variables. The next section describes the orbit maneuver algorithm. The two-impulse transfer is considered a special case and is presented separately. Validation to this formulation is performed by solving several case studies to which the optimal solution is known.

Constrained Multiple-Revolution Lambert's Problem

A fixed-time, multiple-revolution Lambert’s problem is solved under given constraints. For Nmax revolutions, there exist 2Nmax 1 mathematical solutions to Lambert’s problem. Unfortunately, not all of these solutions are feasible. Practical solutions require that the perigee radius be greater than a minimum value (to avoid Earthimpacting trajectories) and the apogee radius be lower than a maximum value (to avoid expensive changes in eccentricity). In particular, short-path and long-path solutions require different considerations to discriminate betweentheunfeasiblesolutions.Asolutionprocedureforthesemimajor-axisrangeisproposedthattakesthese two constraints into account. Based on the semimajor-axis range, the solutions with a feasible number of revolutions can beeasilyselected.Thecaseofzerorevolutionsisalsodiscussed,asthetrajectorymaybefeasibleevenifnotcomplying withthebounds.Numericalexamplesshowthatthenumberoffeasiblesolutionsgreatlydecreaseswhenconsidering the constraints.

Optimal Linear Attitude Estimator

An optimal linear attitude estimator is presented for the case of a single-point real-time estimation of spacecraft attitude using the minimum-element attitude parameterization: Rodrigues (or Gibbs) vector g. The optimality criterion, which does not coincide with Wahbas constrained criterion, is rigorously quadratic and unconstrained. The singularity, which occurs when the principal angle is π, can easily be avoided by one rotation. The attitude accuracy tests show that the proposed method provides a precision comparable with those fully complying with the Wahba optimality definition. Finally, computational speed tests demonstrate that the proposed method belongs to the class of the fastest optimal attitude estimation algorithms.

Flower constellation set theory part II: Secondary paths and equivalency

In our previous research, the Flower Constellation set theory was introduced but specific details left out. In this work, the particular phasing theory that we have adopted is discussed in full. As a consequence of this choice of parametrization, a new class of orbit theory has emerged: secondary paths (SPs). The theory of SPs is developed and proved in this work. Examples of SPs are presented and discussed. Furthermore, we discuss the equivalency of Flower Constellations and resolve how certain disparate choices of integer parameters can generate identical satellite distributions.

Orbit Design for Ground Surveillance Using Genetic Algorithms

T HE orbits of remote sensing systems usually dictate the ground resolution, area coverage, and frequency of coverage parameters. Lower altitude orbits enable a spacecraft to provide higher resolution measurements, but orbital perturbations are nonnegligible due to atmospheric drag [1]. Electric propulsion systems can be used to compensate for atmospheric drag [2]. However, low altitude orbits lack wide coverage. Strategies to overcome the lack of coverage, by maneuvering the spacecraft using electric propulsion, were proposed in the literature [3]. In this case, the thrusters are not only used to compensate the perturbations, but are also used to continuously maneuver the spacecraft to achieve given coverage requirements over a given set of sites in a given time frame. One example is to visit a set of 20 sites within a time frame tf of 50 days. However, this solution complicates the satellite system. The motivation then is to find a lower cost solution for this kind of mission. This note proposes a new solution in which the spacecraft is placed in a natural orbit such that it visits all the sites within the time frame without maneuvering. An advantage is the short time for visiting all sites. Performing continuous thrust transfers to maneuver the satellite from one site to another often takes more time. The disadvantage is that some sites may not be visited accurately. A tradeoff between the accuracy and the revisit time for all the sites is needed when planning a mission. To find this natural orbit, we use a genetic algorithm (GA) to perform a directed search among all possible orbits. Implementing a genetic algorithm does not guarantee an optimal orbit, however, it has been shown that in subsequent iterations, better solutions will be sampled at exponentially increasing rates [4]. This issue will be discussed in the section titled Genetic Algorithms. GAs have been adopted in the literature to solve several orbital mechanics problems. We chose to solve the ground surveillance problem using genetic algorithms because this problem is characterized by many local minima. Conventional optimization methods (e.g., gradient methods) are not suitable for this kind of problem. GAs use random choice as a tool to guide a highly exploitative search in the design space [5]. Enumerated methods scan the whole domain andfind the optimal solution. Enumeratedmethods can provide good solutions to the problem.However, the efficiency of these algorithms is very low compared with genetic algorithms [5]. TheGA solution is used as a starting point to find a local minimum solution through traditional optimization methodologies. The final solution will be a local minimum in the neighborhood of the GA solution. Two types of constraints are considered. Thefirstmission searches for maximum resolution for each site for a given imaging sensor. The second mission tries to maximize the observation time.

Nondimensional star identification for uncalibrated star cameras

Star identification is the most critical and important process for attitude estimation, given data from any star sensor. The main purpose of the Star Identification (Star-ID) process is to identify the observed/measured stars with the corresponding cataloged stars. The precision of the observed star directions highly depend on the calibrated accuracy of the star camera parameters, mainly the focal length f, and the optical axis offsets (x0, y0). When these parameters are not accurate or when the camera is not well calibrated, the proposed Nondimensional Star-ID method becomes very suitable, because it does not require accurate knowledge of these parameters. The Nondimensional Star-ID method represents a unique tool to identify the stars of uncalibrated or inaccurate parameters cameras. The basic idea derives the identification process from the observed focal plane angles which are, to first order, independent from both the focal length and the optical axis offsets. The adoption of the k-vector range search technique, makes this method very fast. Moreover, it is easy to implement, accurate, and the probability of failing Star-ID is less than 0.1% for typical star tracker design parameters.