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Bounded stability of motions around minor bodies

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Prime contractor
Organisational Unit
Activity Type
Implementation progress
78%
05 March 2020

Duration: 36 months

Publications
ESA’s Hera mission aims to visit binary asteroid Didymos in late 2026, investigating its physical characteristics and the result of NASA’s impact by the DART spacecraft in more detail. Two CubeSats onboard Hera plan to perform a ballistic landing on the secondary of the system, called Dimorphos. For these types of landings the translational state during descent is not controlled, reducing the spacecraft’s complexity but also increasing its sensitivity to deployment maneuver errors and dynamic uncertainties. This paper introduces a novel methodology to analyze the effect of these uncertainties on the dynamics of the lander and design a trajectory that is robust against them. This methodology consists of propagating the uncertain state of the lander using the nonintrusive Chebyshev interpolation (NCI) technique, which approximates the uncertain dynamics using a polynomial expansion. The results are then analyzed using the pseudo-diffusion indicator. This indicator is derived from the coefficients of the polynomial expansion, which quantifies the rate of growth of the set of possible states of the spacecraft over time. The indicator is used here to constrain the impact velocity and angle to values that allow for successful settling on the surface. This information is then used to optimize the landing trajectory by applying the NCI technique inside the transcription of the problem. The resulting trajectory increases the robustness of the trajectory compared to a conventional method, improving landing success by 20% and significantly reducing the landing footprint.
The dynamics of a CubeSat orbiting a binary asteroid system can be quite complex due to the irregular gravitational fields of the two bodies and the significant effect of the Solar radiation pressure. Furthermore, the large distance to Earth and the small size of the bodies result in difficulties in the observational process and brings about significant uncertainties in the modelling of the dynamics. In this work, the robust stability and mission performance of orbits in a binary asteroid system under uncertainty are studied with three novel dynamics indicators. The uncertainty in the dynamics is propagated with a polynomial algebra over the space of Taylor polynomials and the indicators are derived from the coefficients of the propagated polynomials. These indicators are found to show accurately the regions of robust stability and bounded motion around Didymos. In addition to the indicators, the observability of Dimorphos for different initial conditions is also studied. The analysis shows a correspondence between the level of robust stability and the size of the observability bounds, albeit the geometry of certain orbits limits the potential maximum observability.
In this work, two novel dynamics indicators are introduced and used to characterise the uncertain dynamics around a binary asteroid. These indicators are derived from the propa- gated expansion of the states in polynomial series of the uncertainty in initial conditions and dynamical model parameters. Thus, each indicator encapsulates in a single scalar the effect of the uncertainty in multiple model parameters. The first indicator directly calculates the second statistical moment of the propagated uncertainty set. This indicator gives a measure of the rate of divergence of an ensemble of trajectories in phase space. The second indicator estimates the approximation error of the polynomial expansion. Hence, it captures the non- linearity in the distribution of the propagated states that is induced by the uncertainty. The two indicators are then used to create a map in phase space, which relates initial conditions to the sensitivity of the state over time to multiple realisation of the uncertain parameters. The case of the a spacecraft orbiting the binary asteroid system Didymos is considered in this paper. The uncertainty maps proposed in this paper are shown to reveal the characteristics of the motion around Didymos under uncertainty in the masses of both bodies.
Objective

Explorations of minor bodies have attracted lots of attentions, in terms of their scientific and resource values, and also human safety concerns. One challenge for the mission design is to identify the stable motion of spacecraft (s/c) in the highly perturbed and uncertain dynamical environment. Many research focused on identifying the stable region under deterministic perturbations and using the Lyapunov stability criterion that is purely mathematical and cannot be well determined given uncertainties of either the initial state or model parameters. Therefore, considering both perturbations and uncertainties, this research presents an approach to statistically address the bounded stability region, where the s/c can stay within specific bounds for the specific time duration with the aim to meet the practical mission requirements. The uncertainties are divided into two categories: 1) with known/assumed distribution, e.g. the initial state, model parameters; 2) with known/assumed bounds including process noise, maneuverer errors. To balance accuracy and efficiency for facilitating the onboard autonomy, the semi-analytical methods that approximate the nonlinear dynamics with truncated power series algebra based on either Taylor expansion or Chebyshev interpolation, are explored to propagate uncertainties to obtain the state flow, from which the bounds are available for stability evaluation. Further, by comparing the bounded stability region with the deterministic Lyapunov stable region, the stability margin can be characterized. Moreover, the evolution and sensitivities of this bounded stability region to the variations of the perturbations and the uncertainty set (including magnitude and distribution) are investigated, from which the maximum uncertainties that the system can allow for bounded stability can be identified. Therefore, these analyses enhance the robustness of the mission design. A software tool will be developed for systematical analysis.

Contract number
4000130259
Programme
Discovery
OSIP Idea Id
I-2019-01720
Related OSIP Campaign
Novel research co-sponsorship ideas
Main application area
Exploration
Budget
64848€
Bounded stability of motions around minor bodies