Collision and re-entry analysis under aleatory and epistemic uncertainty

This paper presents an approach to the design of optimal collision avoidance and re-entry maneuvers considering different types of uncertainty in initial conditions and model parameters. The uncertainty is propagated through the dynamics, with a non-intrusive approach, based on multivariate Tchebycheff series, to form a polynomial representation of the final states. The collision probability, in the cases of precise and imprecise probability measures, is computed considering the intersection between the uncertainty region of the end states of the spacecraft and a reference sphere. The re-entry probability, instead, is computed considering the intersection between the uncertainty region of the end states of the spacecraft and the atmosphere

An intrusive approach to uncertainty propagation in orbital mechanics based on Tchebycheff polynomial algebra

The paper presents an intrusive approach to propagate uncertainty in orbital mechanics. The approach is based on an expansion of the uncertain quantities in Tchebicheff series and a propagation through the dynamics using a generalised polynomial algebra. Tchebicheff series expansions offer a fast uniform convergence with relaxed continuity and smothness requirements. The paper details the proposed approach and illustrates its applicability through a set of test cases considering both parameter and model uncertainties. This novel intrusive technique is then comapred against its non-intrusive counterpart in terms of approximation accuracy and computational cost

Collision avoidance as a robust reachability problem under model uncertainty

The paper presents an approach to the design of an optimal collision avoidance maneuver under model uncertainty. The dynamical model is assumed to be only partially known and the missing components are modeled with a polynomial expansion whose coefficients are recovered from sparse observations. The resulting optimal control problem is then translated into a robust reachability problem in which a controlled object has to avoid the region of possible collisions, in a given time, with a given target. The paper will present a solution for a circular orbit in the case in which the reachable set is given by the level set of an artificial potential function

SMART-UQ : uncertainty quantification toolbox for generalised intrusive and non intrusive polynomial algebra

The paper is presenting a newly developed modular toolbox named Strathclyde Mechanical and Aerospace Research Toolbox for Uncertainty Quantification (SMART-UQ) that implements a collection of intrusive and non intrusive techniques for polynomial approximation and propagation of uncertainties. Non intrusive methods build the polynomial approximation of the uncertain states through sampling of the uncertain parameters space and interpolation. Intrusive methods redefine operators in the states model and perform the states evaluation according to the newly defined operators. The main advantage of non intrusive methods is their range of applicability since the model is treated as a black box hence no regularity is required. On the other hand, they suffer from the curse of dimensionality when the number of required sample points increases. Intrusive techniques are able to overcome this limitation since they have lower computational cost than their corresponding non intrusive counterpart. Nevertheless, intrusive methods are harder to implement and cannot treat the model as a black box. Moreover intrusive methods are able to propagate nonlinear regions of uncertainties while non intrusive methods rely on hypercubes sampling. The most widely known intrusive method for uncertainty propagation in orbital dynamics is Taylor Differential Algebra. The same idea has been generalized to Tchebycheff and Newton polynomial basis because of their fast uniform convergence with relaxed continuity and smoothness requirements. However the SMART-UQ toolbox has been designed in a flexible way to allow further extension of the intrusive and non-intrusive methods to other basis. The Generalized Intrusive Polynomial Expansion (GIPE) approach, implemented in the toolbox and presented here in the paper, expands the uncertain quantities in a polynomial series in the chosen basis and propagates them through the dynamics using a multivariate polynomial algebra. Hence the operations that usually are performed in the space of real numbers are now performed in the algebra of polynomials therefore a polynomial representation of the uncertain states is available at each integration step. To improve the computational complexity of the method, arithmetic operations are performed in the monomial basis. Therefore a transformation between the chosen basis and the monomial basis is performed after the expansion of the elementary functions. Non intrusive methods have been implemented for a set of sampling techniques (Halton, Sobol, Latin Hypercube) for interpolation in the complete polynomial basis as well as on sparse grid for a reduced set of basis. In the paper the different intrusive and non intrusive techniques integrated in SMART-UQ will be presented together with the architectural design of the toolbox. Test cases on propagation of uncertainties in space dynamics with the corresponding intrusive and non intrusive approaches will be discussed in terms of computational cost and accuracy

Comparison of non-intrusive approaches to uncertainty propagation in orbital mechanics

The paper presents four different non-intrusive approaches to the propagation of uncertainty in orbital dynamics with particular application to space debris orbit analysis. Intrusive approaches are generally understood as those methods that require a modification of the original problem by introducing a new algebra or by directly embedding high-order polynomial expansions of the uncertain quantities in the governing equations. Non-intrusive approaches are instead based on a polynomial representations built on sparse samples of the system response to the uncertain quantities. The paper will present a standard Polynomial Chaos Expansion, an Uncertain Quantification-High Dimensional Model Representation, a Generalised Kriging model and an expansion with Tchebycheff polynomials on sparse grids. The work will assess the computational cost and the suitability of these methods to propagate different type of orbits

Optimal control of a space-borne laser system for a 100 m asteroid deflection under uncertainties

The paper demonstrates the technical feasibility to deflect a 100 m diameter asteroid using a moderate size spacecraft carrying a 1-20 kW solar-powered class laser. To this purpose, a recent model of the laser ablation mechanism based on the characteristics of both the laser systems and the asteroid has been used to calculate the exerted thrust in terms of direction and magnitude. This paper shows a preliminary deflection uncertainty analysis for two different control logic and assuming different laser mechanism capabilities. In particular, an optimal thrust control direction and fixed laser pointing strategies were considered with two laser optics settings: the first maintaining the focus length fixed and the second able to exactly focus on the surface. Preliminary results show that in general the fixed laser pointing strategy at low power is less able to impart high deflection. Nonetheless, when the power increases, the optimal thrust method produces undesired torques, which reduces the laser momentum coupling as side effects. However, the overall efficiency is higher in the optimal thrust case. Since the collision risk between an impacting asteroid and the Earth depends on the probability distribution of the input uncertainty parameters, it is necessary to study how the overall deflection will be affected. Both aleatory and epistemic uncertainties are taken into account to evaluate the probability of success of the proposed deflection methods

Impact probability under aleatory and epistemic uncertainties

We present an approach to estimate an upper bound for the impact probability of a potentially hazardous asteroid when part of the force model depends on unknown parameters whose statistical distribution needs to be assumed. As case study, we consider Apophis' risk assessment for the 2036 and 2068 keyholes based on information available as of 2013. Within the framework of epistemic uncertainties, under the independence and non-correlation assumption, we assign parametric families of distributions to the physical properties of Apophis that define the Yarkovsky perturbation and in turn the future orbital evolution of the asteroid. We find IP ≤ 5 × 10 - 5 for the 2036 keyhole and IP ≤ 1.6 × 10 - 5 for the 2068 keyhole. These upper bounds are largely conservative choices due to the rather wide range of statistical distributions that we explored

Optimized low-thrust mission to the Atira asteroids

Atira asteroids are recently-discovered celestial bodies characterized by orbits lying completely inside the Earth’s. The study of these objects is difficult due to the limitations of ground-based observations: objects can only be detected when the Sun is not in the field of view of the telescope. However, many asteroids are expected to exist in the inner region of the Solar System, many of which could pose a significant threat to our planet. In this paper, a mission to improve knowledge of the known Atira asteroids in terms of ephemerides and composition and to observe inner-Earth asteroids is presented. The mission is realized using electric propulsion, which in recent years has proven to be a viable option for interplanetary flight. The trajectory is optimized in such a way as to visit the maximum possible number of asteroids of the Atira group with the minimum propellant consumption; the mission ends with a transfer to an orbit with perigee equal to Venus’s orbit radius, to maximize the observations of asteroids in the inner part of the Solar System