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A supervised stochastic learning method called the Gaussian Process Regression (GPR) is used to design an autonomous guidance law for low-thrust spacecraft. The problems considered are both of the time- and fuel-optimal regimes and a methodology based on "perturbed back-propagation" approach is presented to generate optimal control along neighboring optimal trajectories which form the extremal bundle constituting the training data-set. The use of this methodology coupled with a GPR approximation of the spacecraft control via prediction of the costate n-tuple or the primer vector respectively for time- and fuel-optimal trajectories at discrete time-steps is demonstrated to be effective in designing an autonomous guidance law using the open-loop bundle of trajectories to-go. The methodology is applied to the Earth-3671 Dionysus time-optimal interplanetary transfer of a low-thrust spacecraft with off-nominal thruster performance and the resulting guidance law is evaluated under different design parameters using case-studies. The results highlight the utility and applicability of the proposed framework with scope for further improvements. [ABSTRACT FROM AUTHOR]

Equipping a spacecraft with multiple solar-powered electric engines (of the same or different types) compounds the task of optimal trajectory design due to presence of both real-valued inputs (power input to each engine in addition to the direction of thrust vector) and discrete variables (number of active engines). Each engine can be switched on/off independently and "optimal" operating power of each engine depends on the available solar power, which depends on the distance from the Sun. Application of the Composite Smooth Control (CSC) framework to a heliocentric fuel-optimal trajectory optimization from the Earth to the comet 67P/Churyumov-Gerasimenko is demonstrated, which presents a new approach to deal with multiple-engine problems. Operation of engine clusters with 4, 6, 10 and even 20 engines of the same type can be optimized. Moreover, engine clusters with different/mixed electric engines are considered with either 2, 3 or 4 different types of engines. Remarkably, the CSC framework allows us 1) to reduce the original multi-point boundary-value problem to a two-point boundary-value problem (TPBVP), and 2) to solve the resulting TPBVPs using a single-shooting solution scheme and with a random initialization of the missing costates. While the approach we present is a continuous neighbor of the discontinuous extremals, we show that the discontinuous necessary conditions are satisfied in the asymptotic limit. We believe this is the first indirect method to accommodate a multi-mode control of this level of complexity with realistic engine performance curves. The results are interesting and promising for dealing with a large family of such challenging multi-mode optimal control problems. • Optimization of low-thrust interplanetary trajectories with multiple engines. • Engine clusters with the same or different/mixed engine types are considered. • Same-type engine cluster with 4, 6, 10 and 20 engines are investigated. • Different-type engine clusters with 2, 3 and 4 engines are investigated. • Fuel-optimal trajectories from Earth to comet 67P/Churyumov-Gerasimenko. [ABSTRACT FROM AUTHOR]

Efficient performance of a number of engineering systems is achieved through different modes of operation - yielding systems described as "hybrid", containing both real-valued and discrete decision variables. Prominent examples of such systems, in space applications, could be spacecraft equipped with 1) a variable- I sp , variable-thrust engine or 2) multiple engines each capable of switching on/off independently. To alleviate the challenges that arise when an indirect optimization method is used, a new framework — Composite Smooth Control (CSC) — is proposed that seeks smoothness over the entire spectrum of distinct control inputs. A salient aftermath of the application of the CSC framework is that the original multi-point boundary-value problem can be treated as a two-point boundary-value problem with smooth, differentiable control inputs; the latter is notably easier to solve, yet can be made to accurately approximate the former hybrid problem. The utility of the CSC framework is demonstrated through a multi-year, multi-revolution heliocentric fuel-optimal trajectory for a spacecraft equipped with a variable- I sp , variable-thrust engine. • Optimization of low-thrust interplanetary trajectories with variable specific impulse, variable thrust engines. • A new framework, called Composite Smooth Control (CSC), is developed to handle various types of state-triggered constraints. • Indirect optimization method is used to formulate fuel-optimal problems. • Complex-based derivative approach is used to construct co-state dynamics numerically. • Multi-revolution, multi-year trajectory optimization is analyzed. [ABSTRACT FROM AUTHOR]

Abstract An adaptive Collision Risk Assessment Tool (CRATER) is introduced in order to analyze conjunctions featuring non Gaussian distributions, long encounter times, and considerable model uncertainty. The algorithm makes use of the Fokker Planck equation to quantify the error in various assumptions, refine approximations of the propagated probability density functions (PDFs), and assess the need for external validation in some scenarios. A Local Gaussian approximation of the non-Gaussian propagated PDFs is introduced and shown to be comparable to Gaussian Mixture methods (GMM) in terms of accuracy, while exhibiting superior performance in terms of speed. The algorithm is made adaptive through the use of a non linearity index, which is used to determine the number of mixture components needed in a GMM or the spacing of local Gaussians when approximating the propagated PDFs. Coppola's method for calculating the probability of collision (PC) between space objects is modified to considering parametric model uncertainty. The PC integral is then incorporated into CRATER. CRATER is then used to compute the PC for 8 different test cases and is compared to Monte Carlo results. Highlights • New algorithm for probability of collision analysis is tested on eight test cases. • Model uncertainty shown to affect collision probability by orders of magnitude. • Alternative to gaussian mixture method shown to be faster with similar accuracy. • Error in assumptions are quantified with respect to Liouville equation. [ABSTRACT FROM AUTHOR]

A modified Chebyshev Picard iteration method is proposed for solving orbit propagation initial/boundary value problems. Cosine sampling techniques, known as Chebyshev-Gauss-Lobatto (CGL) nodes, are used to reduce Runge’s phenomenon that plagues many series approximations. The key benefit of using the CGL data sampling is that the nodal points are distributed nonuniformly, with dense sampling at the beginning and ending times. This problem can be addressed by a nonlinear time transformation and/or by utilizing multiple time segments over an orbit. This paper suggests a method, called a multisegment method, to obtain accurate solutions overall regardless of initial states and albeit eccentricity by dividing the given orbit into two or more segments based on the true anomaly. [ABSTRACT FROM AUTHOR]

The halo orbits around the Earth-Moon L2 libration point provide a great candidate orbit for a lunar communication satellite, where the satellite remains above the horizon on the far side of the Moon being visible from the Earth at all times. Such orbits are generally unstable, and station-keeping strategies are required to control the satellite to remain close to the reference orbit. A recently developed Modified Chebyshev-Picard Iteration method is used to compute corrective maneuvers at discrete time intervals for station-keeping of halo orbit satellite, and several key parameters affecting the mission performance are analyzed through numerical simulations. Compared with previously published results, the presented method provides a computationally efficient station- keeping approach which has a simple control structure that does not require weight turning and, most importantly, does not need state transition matrix or gradient information computation. The performance of the presented approach is shown to be comparable with published methods. [ABSTRACT FROM AUTHOR]

Abstract: In this work we present a methodology for intelligent path planning in an uncertain environment using vision-like sensors, i.e., sensors that allow the sensing of the environment non-locally. Examples would include a mobile robot exploring an unknown terrain or a micro-UAV navigating in a cluttered urban environment. We show that the problem of path planning in an uncertain environment, under certain assumptions, can be posed as the adaptive optimal control of an uncertain Markov decision process, characterized by a known, control-dependent system, and an unknown, control-independent environment. The strategy for path planning then reduces to computing the control policy based on the current estimate of the environment, also known as the “certainty-equivalence” principle in the adaptive control literature. Our methodology allows the inclusion of vision-like sensors into the problem formulation, which, as empirical evidence suggests, accelerates the convergence of the planning algorithms. Further we show that the path planning and estimation problems, as formulated in this paper, possess special structure which can be exploited to significantly reduce the computational burden of the associated algorithms. We apply this methodology to the problem of path planning of a mobile rover in a completely unknown terrain. [Copyright &y& Elsevier]

Indirect optimization methods convert optimal control problems (OCPs) into two- or multi-point boundary-value problems. A highly desirable feature of indirect methods, specifically for space applications, is that high-resolution trajectories can be generated, which satisfy the first-order necessary conditions of optimality. A recently developed Composite Smoothing Control (CSC) framework is utilized to formulate and solve the problem of simultaneous trajectory optimization and propulsion sub-system design of spacecraft. A reasonable parameterized breakdown of the spacecraft mass is adopted, which captures the impact of power produced by the solar arrays and its contribution to the total spacecraft mass. Thus, the implicit trade-offs can be considered in the indirect optimization approach. The function space co-optimization problem of spacecraft power subsystem parameters along with the main trajectory is solved with the objective to maximize the payload delivered. The proposed framework amounts to an invariant embedding that reduces the original, difficult-to-solve, multi-point boundary-value problem into a two-point boundary-value problem with continuous, differentiable control inputs. Utility of the proposed construct is demonstrated through a low-thrust, multi-revolution, multi-year rendezvous maneuver to asteroid Dionysus with a variable-specific-impulse, variable-thrust modeled engine. This is the first time that indirect optimization methods have tackled such a complex co-optimization problem using the CSC framework. • Co-optimization of trajectory and solar electric propulsion system parameters. • Comparison of constant and variable- l s p thruster for low-thrust trajectory design. • Solve multi-point boundary value problem using Composite Smoothing Control (CSC). • Optimization over bounds of the l s p , variable thruster efficiency, and power is achieved. • Linear relation between thruster efficiency and l s p is near-accurate for early studies. [ABSTRACT FROM AUTHOR]

Impulsive trajectories provide time- and ΔV -reachability insights. Two novel homotopy-based methods are proposed for generating optimal many-impulse, multirevolution maneuvers. The first method is based on the continuation over the specific impulse value, which is shown to enhance convergence performance of the resulting two-point boundary-value problems. The second method is based on the formulation of impulsive trajectories using a linear acceleration term. The two methods are used in a hybrid framework. The utility of the proposed methods is demonstrated on four problems: Two interplanetary trajectories 1) from Earth to Mars, 2) from Earth to asteroid Dionysus, 3) a planet-centric transfer maneuver from a geostationary transfer orbit (GTO) to geostationary orbit (GEO) consisting of 21 revolutions, and 4) a 50-revolution transfer from the considered GTO to a circular orbit at the first Lagrange point (L1) of the Earth-moon system. The last two problems leading to 18 and 50 impulses are tackled to give an optimal near-impulsive solution. The impulsive solution with 18 impulses is shown to satisfy Lawden's conditions. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the manifolds of three Near-Rectilinear Halo Orbits (NRHOs) and optimal low-thrust transfer trajectories using a high-fidelity dynamical model. Time- and fuel-optimal low-thrust transfers to (and from) these NRHOs are generated leveraging their 'invariant' manifolds, which serve as long terminal coast arcs. Analyses are performed to identify suitable manifold entry/exit conditions based on inclination and minimum distance from the Earth. The relative merits of the stable/unstable manifolds are studied with regard to time- and fuel-optimality criteria, for a set of representative low-thrust family of transfers. [ABSTRACT FROM AUTHOR]

Abstract: In this paper, a semianalytic partition of unity finite element method (PUFEM) is presented to solve the transient Fokker–Planck equation (FPE) for high-dimensional nonlinear dynamical systems. Meshless spatial discretization of PUFEM with local -refinement (discussed in a previous paper) is employed to develop linear ordinary differential equations for the time varying coefficients of local shape functions. A similarity transformation to modal coordinates is shown to reveal numerous spurious modes in the eigenspace of the discretized FPE operator. Identification and elimination of these modes leads to an analytical solution of ODEs obtained from spatial discretization in terms of the remaining admissible modes, and a significant reduction in the size of the discretized transient problem. Initial equation error resulting from the set of admissible modes is shown to be an upper bound for all time, meaning that the reduced admissible set is sufficient for the FPE approximation for all time. [Copyright &y& Elsevier]

Abstract: The problem of real-time, on-board star pattern identification is considered which must precede any spacecraft attitude estimation algorithm based on measured line of sight directions to stars. Following the use of the search-less -vector method to access feasible candidate stars (detailed in a previous paper) a tiered logical structure is introduced in which inter-star angles for pairs, triples and general elementary star configurations are used to match measured patterns to corresponding patterns in a star catalog. Analytical expressions are developed for the expected frequency of matching an observed star pattern with an incorrect pattern in the star catalog due to measurement error. Such knowledge of erroneous match frequencies can be used to rigorously terminate the star identification process with a virtually certain match, while obviating the need for expensive Monte Carlo simulations. Developments shown are supported with a few simulation results. [Copyright &y& Elsevier]

Abstract: An iterative approach for the solution refinement of the stationary Fokker–Planck equation is presented. The recursive use of a modified norm induced on the solution domain by the most recent estimate of the stationary probability density function, is shown to significantly improve the accuracy of the approximation over the standard -norm based Galerkin error projection. The modified norm is argued to be naturally suited to the problem, and hence preferable over the standard -norm, because the former requires substantially fewer degrees of freedom for the same order of approximation accuracy, making it immediately attractive for the Fokker–Planck equation in higher dimensions. Additionally, it is shown that the modified norm can be utilized to progress through a homotopy of dynamical systems, , in order to determine the domain of the stationary distribution of a nonlinear system of interest, (corresponding to ) by starting with a known dynamical system (corresponding to ) and working upwards. The partition of unity finite element method is used for numerical implementation. The meshless nature of this technique facilitates the application of the modified-norm approach to higher dimensional problems. [Copyright &y& Elsevier]

Star trackers are the most accurate attitude sensors to perform 3-axis attitude estimation of spacecraft. However, even though they are specifically designed to estimate attitude, they can also be used to estimate the local coordinates of the camera on the Earth as well as the direction of East. This requires us to align the camera optical axis with the local direction of the gravity and to have knowledge of the time, that can be provided by an accurate clock. The resulting system is the Compass Star Tracker (CST), that is, a star tracker used as a global surface navigation system. This system would certainly not substitute the Global Positioning System which, in turn, does not suffer of the limitation night-only operation and in clear weather conditions. On the contrary, the CST does not require any satellite information or other support from ground stations to be used in lieu of the traditional Global Positioning System. [ABSTRACT FROM AUTHOR]

Direction-dependent scaling, shaping, and rotation of Gaussian basis functions are introduced for maximal trend sensing with minimal parameter representations for input output approximation. It is shown that shaping and rotation of the radial basis functions helps in reducing the total number of function units required to approximate any given input-output data, while improving accuracy. Several alternate formulations that enforce minimal parameterization of the most general radial basis functions are presented. A novel "directed graph" based algorithm is introduced to facilitate intelligent direction based learning and adaptation of the parameters appearing in the radial basis function network. Further, a parameter estimation algorithm is incorporated to establish starting estimates for the model parameters using multiple windows of the input-output data. The efficacy of direction-dependent shaping and rotation in function approximation is evaluated by modifying the minimal resource allocating network and considering different test examples. The examples are drawn from recent literature to benchmark the new algorithm versus existing methods. [ABSTRACT FROM AUTHOR]

Star identification can be accomplished by several different available algorithms that identify the stars observed by a star tracker. However, efficiency and reliability remain key issues and the availability of new active pixel cameras requires new approaches. Two novel algorithms for recursive mode star identification are presented here. The first approach is derived by the spherical polygon search (SP-search) algorithm, here used to access all the cataloged stars observed by the sensor field-of-view (FOV) and recursively add/remove candidate cataloged stars according the predicted image motion induced by camera attitude dynamics. Star identification is then accomplished by a star pattern matching technique which identifies the observed stars in the reference catalog. The second method uses star neighborhood information and a catalog neighborhood pointer matrix to access the star catalog. In the recursive star identification process, and under the assumption of "slow" attitude dynamics, only the stars in the neighborhood of previously identified stars are considered for star identification in the succeeding frames. Numerical tests are performed to validate the absolute and relative efficiency of the proposed methods. [ABSTRACT FROM AUTHOR]

Focuses on study which utilized the structured model reference adaptive control (SMARC) method for the active suppression of limit cycle oscillations for a typical wing section with a structural nonlinearity. Investigation on nonlinear aeroelastic control; Steps in adaptive feedback linearization; Advantages of SMARC over conventional adaptive methods.

In this work, end-to-end low-thrust transfers from a GTO orbit to a low-altitude lunar orbit by exploiting the manifolds of a chosen Earth–Moon L 1 halo orbit has been studied. The practicality of piece-wise, minimum-time transfers that exploit halo orbit manifolds is demonstrated, which offers more flexibility to meet mission objectives. It is known that the structure of the manifolds varies substantially due to the presence of the Sun and its contribution has to be considered to obtain more realistic trajectories. To incorporate Sun's perturbation, we study (1) manifolds' behavior within a Bi-Circular Problem (BCP) dynamics and (2) Sun's impact on the previously converged trajectories obtained using the standard Circular Restricted Three-Body Problem (CR3BP). Comparisons of the resulting trajectories using the CR3BP and BCP are presented. • Analyzed invariant manifolds in the CR3BP and BCP. • Methodology to analyze patch points on manifold for transfer trajectories. • Demonstrated feasibility of low-thrust transfer using invariant manifolds. • Designed Earth GTO to 1000 km circular lunar polar orbit via patching to manifolds. • Framework to merge periodic orbits and manifolds with indirect optimal control. [ABSTRACT FROM AUTHOR]

Introduces an approach for planning minimum-control optimal trajectories for a two-wheeled mobile robot in the absence of obstacles. Formulation of the trajectory planning in minimum-control, fixed-time and optimal control problems with no terminal cost; Presentation of some of the simulation results of the relaxation method; Solving of the two-point boundary value problem; Discussion of the procedure for constants of motion.

Abstract: In this paper, the stationary Fokker–Planck equation (FPE) is solved for nonlinear dynamical systems using a local numerical technique based on the meshless partition of unity finite element method (PUFEM). The method is applied to stationary FPE for 2-, 3- and 4-D systems and is argued to be an excellent candidate for higher dimensional problems and the transient problem. Local refinement is applied by introducing higher order polynomials in selected subdomains (local -refinement) to keep the problem size small while ensuring high approximation accuracy. Various local approximations are blended using novel pasting functions that provide any desired order of continuity. Results are compared with existing global and local techniques. Local -refinement is touted as an important step towards breaking the curse of dimensionality in numerical solution of FPE. [Copyright &y& Elsevier]

We are developing an autonomous mobile robotic system to emulate six degree of freedom relative spacecraft motion during proximity operations. A mobile omni-directional base robot provides x, y, and yaw planar motion with moderate accuracy through six independently driven motors. With a six degree of freedom micro-positioning Stewart platform on top of the moving base, six degree of freedom spacecraft motion can be emulated with high accuracy. This paper presents our approach to dynamic modeling, control, and simulation for the overall system. Compared with other simulations that introduced significant simplifications, we believe that our rigorous modeling approach is crucial for the high fidelity hardware in-the-loop emulation. [ABSTRACT FROM AUTHOR]

Inspired by the original developments of recursive power series by means of Lagrange invariants for the classical two-body problem, a new analytic continuation algorithm is presented and studied. The method utilizes kinematic transformation scalar variables differentiated to arbitrary order to generate the required power series coefficients. The present formulation is extended to accommodate the spherical harmonics gravity potential model. The scalar variable transformation essentially eliminates any divisions in the analytic continuation and introduces a set of variables that are closed with respect to differentiation, allowing for arbitrary-order time derivatives to be computed recursively. Leibniz product rule is used to produce the needed arbitrary-order expansion variables. With arbitrary-order time derivatives available, Taylor series-based analytic continuation is applied to propagate the position and velocity vectors for the nonlinear two-body problem. This foundational method has been extended to also demonstrate an effective variable step size control for the Taylor series expansion. The analytic power series approach is demonstrated using trajectory calculations for the main problem in satellite orbit mechanics including high-order spherical harmonics gravity perturbation terms. Numerical results are presented to demonstrate the high accuracy and computational efficiency of the produced solutions. It is shown that the present method is highly accurate for all types of studied orbits achieving 12–16 digits of accuracy (the extent of double precision). While this double-precision accuracy exceeds typical engineering accuracy, the results address the precision versus computational cost issue and also implicitly demonstrate the process to optimize efficiency for any desired accuracy. We comment on the shortcomings of existing power series-based general numerical solver to highlight the benefits of the present algorithm, directly tailored for solving astrodynamics problems. Such efficient low-cost algorithms are highly needed in long-term propagation of cataloged RSOs for space situational awareness applications. The present analytic continuation algorithm is very simple to implement and efficiently provides highly accurate results for orbit propagation problems. The methodology is also extendable to consider a wide variety of perturbations, such as third body, atmospheric drag and solar radiation pressure. [ABSTRACT FROM AUTHOR]