Roberto Armellin

Nonlinear Mapping of Uncertainties in Celestial Mechanics

The problem of nonlinear uncertainty propagation represents a crucial issue in celestial mechanics. In this paper, a method for nonlinear propagation of uncertainty based on differential algebra is presented. Working in the differential algebra framework enables a general approach to nonlinear uncertainty propagation that can provide highly accurate estimate with low computational cost. The nonlinear mapping of the statistics is shown here, adopting the two-body problem as the working framework, including coordinate system transformations. The general feature of the proposed method is also demonstrated by presenting long-term integrations in complex dynamic systems, such as the n-body problem or the simplified general perturbation model.

Aerogravity Assist Maneuvers: Coupled Trajectory and Vehicle Shape Optimization

The aero-gravity assist maneuver is proposed as a tool to improve the efficiency of the gravity assist as, due to the interaction with the planetary atmosphere, the angular deviation of the velocity vector can be definitely increased. Even though the drag reduces the spacecraft velocity, the overall Δv gain could be remarkable whenever a high lift-to-drag vehicle flies. A previous study addressed the 3D dynamic modeling and optimization of the maneuver including heliocentric plane change, heating rate, and structural load analysis. A multidisciplinary study of aero-gravity assist is proposed, focusing on coupled trajectory and vehicle shape optimization. A planar aero-gravity assist of Mars has been selected as a test case with the aim of maximizing the vehicle heliocentric velocity and limiting the heating rate experienced during the atmospheric path. A multiobjective approach has been adopted, and a particle swarm optimization algorithm has been chosen to detect the set of Pareto optimal solutions. The study includes a further refinement of the trajectory for three significant shapes belonging to the Pareto curve. The associated optimal control problem has been solved by selecting a direct method approach. The dynamics has been transcribed into a set of nonlinear constraints and the arising non linear programming problem has been solved through a sequential quadratic programming solver.

Multidisciplinary optimization of aerocapture maneuvers

A multidisciplinary-multiobjective optimization of aerocapture maneuvers is presented. The proposed approach allows a detailed analysis of the coupling among vehicles shape, trajectory control, and thermal protection system design. A set of simplified models are developed to address this analysis and a multiobjective particle swarm optimizer is adopted to obtain the set of Pareto optimal solutions. In order to deal with an unconstrained multiobjective optimization, a two-point boundary value problem is formulated to implicitly satisfy the constraints on the atmospheric exit conditions. The trajectories of themost promising solutions are further optimized in a more refined dynamical system by solving an optimal control problem using a directmultiple shooting transcription method. Furthermore, a more complete vehicle control is considered. All the simulations presented consider an aerocapture at Mars with a polar orbit of 200 km of altitude as target orbit.

Rigorous global optimization of impulsive planet-to-planet transfers in the patched-conics approximation

The rigorous solution of a generic impulsive planet-to-planet transfer by means of a Taylor-model-based global optimizer is presented. Although a planet-to-planet transfer represents the simplest case of interplanetary transfer, its formulation and solution is a challenging task when the rigorous global optimum is sought. A customized ephemeris function is derived from JPL DE405 to allow the Taylor-model evaluation of planets’ positions and velocities. Furthermore, the validated solution of Lamberts problem is addressed for the rigorous computation of transfer fuel consumption. The optimization problem, which consists in finding the optimal launch and transfer time to minimize the required fuel mass, is complex due to the abundance of local minima and relatively high search-space dimension. Its rigorous solution by means of the Taylor-model-based global optimizer COSY-GO is presented considering Earth–Mars and Earth–Venus transfers as test cases.

Pso Algorithm For Planetary Atmosphere Entry Vehicles Multidisciplinary Guidance Design

The paper presents a possible approach to define - within the same global optimization the guidance history and the configuration of each different flight regime an atmospheric Entry-Descent-Landing (EDL) vehicle has to deal with. Precision landing constraints as well as inertial and thermal loads containment are considered. The optimization is focused on detecting a set of possible preliminary solutions to be further investigated by a local finer optimizer. A partially revisited Particle Swarm Optimization technique has been here successfully applied to deal with both a multiobjective and distributed optimization architecture specifically thought to cope with the complexity of the proposed problem. The proposed approach turned out to be powerful in identifying, quite rapidly, a set of feasible good solutions both from the guidance and the configuration point of view for each of the aerodynamics phases the probe goes through.

Multi-step optimization strategy for fuel-optimal orbital transfer of low-thrust spacecraft

An effective method for the design of fuel-optimal transfers in two- and three-body dynamics is presented. The optimal control problem is formulated using calculus of variation and primer vector theory. This leads to a multi-point boundary value problem (MPBVP), characterized by complex inner constraints and a discontinuous thrust profile. The first issue is addressed by embedding the MPBVP in a parametric optimization problem, thus allowing a simplification of the set of transversality constraints. The second problem is solved by representing the discontinuous control function by a smooth function depending on a continuation parameter. The resulting trajectory optimization method can deal with different intermediate conditions, and no a priori knowledge of the control structure is required. Test cases in both the two- and three-body dynamics show the capability of the method in solving complex trajectory design problems.

Global optimization of interplanetary transfers with deep space maneuvers using differential algebra

In this chapter, differential algebra is used to globally optimize multi-gravity assist interplanetary trajectories with deep space maneuvers. A search space pruning procedure is adopted, and the trajectory design is decomposed into a sequence of sub-problems. As far as differential algebra is used, the objective function and the constraints are represented by Taylor series of the design variables over boxes in which the search space is divided. Thanks to the polynomial representation of the function and the constraints, a coarse grid can be used, and an efficient design space pruning is performed. The manipulation of the polynomials eases the subsequent local optimization process, so avoiding the use of stochastic optimizers. These aspects, along with the efficient management of the list of boxes, make differential algebra a powerful tool to design multi-gravity assist transfers including deep-space maneuvers.

Probabilistic Optical and Radar Initial Orbit Determination

Future space surveillance requires dealing with uncertainties directly in the initial orbit determination phase. We propose an approach based on Taylor differential algebra to both solve the initial orbit determination (IOD) problem and to map uncertainties from the observables space into the orbital elements space. This is achieved by approximating in Taylor series the general formula for probability density function (pdf) mapping through nonlinear transformations. In this way the mapping is obtained in an elegant and general fashion. The proposed approach is applied to both angles-only and two position vectors IOD for objects in LEO and GEO.

Efficient evaluation of vapour-liquid equilibria from multi-parameter thermodynamic models using differential algebra

An efficient method is proposed to evaluate the Vapour-Liquid Equilibrium (VLE) curve for complex multi-parameter technical and reference thermodynamic equations of state, in connection with Computational Fluid Dynamics (CFD) simulations of compressible flows of real gases. Differential algebra techniques are used to obtain an approximation of the VLE curve from the reference equation of state of carbon dioxide. Seven fourth-order Taylor polynomials are required to approximate the VLE curve for a reduced pressure between 0.7 and 1, with an error on density below 0.04%, except near the critical point where the error is around 0.1%. The proposed approach is proved to be a suitable alternative to standard Look-Up Table (LUT) techniques, with comparable accuracy and computational burden. Moreover, the explicit use of the model analytical expression in the determination of the polynomial expansions allows to reduce the number of expansion poles and it will possibly simplify the approximation of different fluids, including mixtures.

A differential algebra based importance sampling method for impact probability computation on Earth resonant returns of Near Earth Objects

A differential algebra based importance sampling method for uncertainty propagation and impact probability computation on the first resonant returns of Near Earth Objects is presented in this paper. Starting from the results of an orbit determination process, we use a diferential algebra based automatic domain pruning to estimate resonances and automatically propagate in time the regions of the initial uncertainty set that include the resonant return of interest. The result is a list of polynomial state vectors, each mapping specific regions of the uncertainty set from the observation epoch to the resonant return. Then, we employ a Monte Carlo importance sampling technique on the generated subsets for impact probability computation. We assess the performance of the proposed approach on the case of asteroid (99942) Apophis. A sensitivity analysis on the main parameters of the technique is carried out, providing guidelines for their selection. We finally compare the results of the proposed method to standard and advanced orbital sampling techniques.