Irregular-shaped Asteroid Research Articles

Convex optimization of stochastic path-constrained trajectories near asteroids

The environment around asteroids is of significant uncertainties, which could cause great challenges of exploration missions to asteroids. This paper presents a novel approach to stochastic optimal problem in path-constrained trajectory optimization near asteroids based on convex approach. The path constraints that prevent the spacecraft from colliding with asteroid in different scenarios are composed of keep-out zone constraint and glide-slope constraint, which are formulated in chance-constraint forms due to the uncertain state. To convexify these constraints, the first-order Taylor expansion is applied for linearization and the nonconvexity is tackled with the introduction of additional intermediate variable and virtual buffer. Thereafter, a convex optimization based method is proposed to solve stochastic optimal problem. The effectiveness of the proposed method is validated through simulations of soft landing and hopping transfer scenarios near irregular-shaped asteroid 1996HW1. The results indicate that the proposed method can generate path-constrained trajectories under uncertainties using convex optimization.

A continuation approach for computing periodic orbits around irregular-shaped asteroids. An application to 433 Eros

The orbital dynamics in the gravitational environment of irregular asteroids is an important issue for space missions and a quite complex problem. In this paper we propose a methodological approach for computing periodic orbits around rotating, irregular-shaped asteroids. Our study starts from the families of periodic orbits of a triaxial ellipsoid model that can approximate the gravitational field of an irregular body. The families of periodic orbits of the triaxial ellipsoid have particular structures and symmetries. By using a particular periodic orbit, which belongs in a family of the ellipsoid model, we introduce and apply the method of shape continuation in order to obtain a corresponding periodic orbit in the gravitational field of an irregular body, which in our study is approximated with a sufficient number of mascons. Then, by applying analytic continuation to this orbit, we can compute a family of periodic orbits in the asymmetric gravitational field. This family can be assumed as a perturbed family, with respect to that of the ellipsoid, with the strength of the perturbation being dependent on the irregular shape of the asteroid and the distance of the orbits from its surface. We apply our methodology to the asteroid 433 Eros and present results on particular planar and 3D orbits. Similarities and differences between families of the ellipsoid and the asteroid model are indicated and discussed.

Landing of hopping rovers on Irregularly-shaped small bodies using attitude control

A hopping rover that is driven only by internal or external attitude actuators is an ideal mobility approach for surface exploration of small solar system bodies. Without thrust control and grasping mechanisms, a hopping rover is mechanically simple to design and less prone to mechanical failures, but faces challenges during soft landing. It may rebound from the surface, causing deviations from its original landing site. In this paper, landing of a hopping rover on the surface of an asteroid is investigated, and a strategy using only attitude control to shorten the landing distance is proposed. Based on rigid body impact dynamics, the edge impact configuration is investigated in detail. The factors that affect the impact states of a cube-shaped hopping rover are studied. Then, controlled edge landing is analyzed, in which the post-impact velocity of the hopping rover is changed by controlling its attitude prior to impact. Three guidance schemes are developed, followed by attitude profile generation and finite time stable attitude control. Finally, simulations are performed on an ideal flat surface and uneven terrain. The results show that controlled edge landing can effectively reduce the landing distance and settling time, compared with uncontrolled landing. This study on hopping motion on the surface of an irregular-shaped asteroid with attitude control, can provide a reference for hopper trajectory plan in future asteroid surface explorations.

Orbital maneuver strategy design based on piecewise linear optimization for spacecraft soft landing on irregular asteroids

Recently, asteroid exploration becomes an important branch of human’s deep space activities. In this paper, a piecewise linear optimal orbital maneuver strategy is designed for a spacecraft soft landing on irregular-shaped asteroids. First, the space around an irregular asteroid is converted into several grid units, and the gravitational field of the asteroid is linearly fitted in each unit. Then, the soft-landing orbital maneuver strategy design problem is formulated as a piecewise linear optimal problem, and further transferred into a family of two-point boundary value problems, which can be solved using collocation method. Finally, a corresponding algorithm is developed to obtain the piecewise linear optimal maneuver strategy, which is proved to be able to achieve the soft-landing mission well. Simulation results show that the error of the model linearization is small enough, while the calculation efficiency is remarkably improved, and the robustness of maneuver strategy is also improved.

Use of Tetrahedral Finite Element Method for Computing the Gravitation of Irregular-Shaped Asteroid

Asteroids are one of the most important targets for deep space exploration. In previous asteroid missions, the accurate estimate of gravity has proved to have a strong influence on the design of the approach orbit and navigation strategy. The wield gravitational field of an asteroid is mainly caused by the irregular overall shape and possible heterogenous mass distribution the interior. We propose to use the finite element method to compute the gravity of irregularly shaped asteroids; this method combines the advantages of the conventional mascon method and the polyhedral method. The tetrahedral meshes can be generated following the conventional division technique. Taking asteroid 216 Kleopatra as an example, we calculate the exterior gravitational field using the above mentioned methodology. We then compare the results from the finite element method and the polyhedral method under a degenerated case, i.e., with constant density. Then, four different density distribution assumptions are given, and the gravitational fields are calculated respectively. The comparative study and the density distribution assumptions indicate that the proposed method is suitable for modeling an arbitrary asteroid with nonuniform mass distribution. This method is expected to provide reliable gravity data for the design of guidance, navigation, and control systems in future asteroid missions.

Adaptive double-saturated control for hovering over an asteroid

Hovering over an irregular-shaped asteroid is particularly challenging due to the large gravitational uncertainties and various external disturbances. An adaptive control scheme considering commanded acceleration and its change-rate saturation for hovering is developed in this paper. Taking full advantage of terminal sliding-mode control theory, first, we convert the double-saturated control problem to a new equivalent system by introducing a special bounded function, in which just control input saturation needs to be considered. Then, a continuous finite-time saturated controller is designed for the new system with the assistance of an constructed auxiliary subsystem. Additionally, an adaptive law is devised for the controller to avoid the requirement of the unknown upper bounds of the disturbances, rendering the control scheme especially suitable to asteroid hovering missions. The finite-time stability of the whole closed-loop system is proved via Lyapunov analysis. Numerical simulation studies are carried out, and the results demonstrate the design features and the desired performance.

New image processing algorithm for terminal guidance of multiple kinetic-energy impactors for disrupting hazardous asteroids

This paper describes the preliminary study results of developing a hypervelocity terminal intercept guidance system of a multiple kinetic-energy impactor vehicle (MKIV). The proposed MKIV system is intended to fragment or pulverize an asteroid of smaller than approximately 150 m in diameter that is detected with a mission lead time of shorter than 10 years, without using nuclear explosive devices. This paper focuses on the development of a new image processing algorithm based on Otsu’s method for the coordinated terminal intercept guidance and control of multiple kinetic-energy impactors employing visual and/or infrared sensors. A scaled polyhedron shape model of asteroid (216) Kleopatra is used as a fictional target asteroid. GPU-based simulation results demonstrate the feasibility of impacting a small irregular-shaped asteroid by using the proposed new image processing algorithm and a classical pulsed TPN (true proportional navigation) terminal guidance law.

The motion of surface particles for the asteroid 101955 Bennu

The asteroid 101955 Bennu is the target asteroid of the ongoing asteroid sample return mission OSIRIS-REx from NASA. In the mission, the spacecraft is scheduled to rendezvous with Bennu in August 2018. Investigating the dynamics of surface motion can offer guidance for further hopping landers or even sample return missions regarding Bennu or similarly irregular-shaped asteroids. This paper presents the free motion of sample particles on and above the surface of Bennu. The concepts and equations that govern the motion of a particle on and above the surface of an asteroid are briefly introduced. Trajectories of a number of particles' free motion are numerically calculated by considering four scenarios with different restitution coefficients. Initial positions of those particles are all randomly given on the surface of Bennu and their initial velocities are set as zero. The distribution of final locations of those sample particles is summarized and analyzed. Both the gravitational potential and its effective potential on the surface of Bennu are calculated to give a possible reason for the distributing characteristics.

Global search for periodic orbits in the irregular gravity field of a binary asteroid system

Periodic orbits are important keys to understand the motion of a massless particle in the vicinity of a binary asteroid system. Due to the complex gravity generated by the irregular-shaped asteroids, it is difficult to generate periodic orbits with an analytical method, except with the linearized dynamics in a small region around the libration points. This paper has presented a numerical method to search the three-dimensional periodic orbits in a global space. The grid searching method is used to find nearly periodic orbits, and then the shooting method is used to get the accurate periodic orbits. The method is applied to the binary asteroid (66391) 1999 KW4 to solve the periodic orbits in the vicinity. The periodic orbits are then classified into five categories according to their shapes and locations. The search method can also be used to find periodic orbits in the vicinity of other binary systems.

Relative equilibria and stability of a dumbbell spacecraft about an asteroid

The gravitational orbit-attitude coupling becomes pronounced in spacecraft dynamics about small asteroids, due to the large ratio of the spacecraft's dimension to the orbital radius. In this study, relative equilibria and their stability of the gravitationally coupled orbit-attitude dynamics (full dynamics) of a dumbbell spacecraft about an irregular-shaped asteroid are investigated. The shape of the asteroid is represented by a homogeneous polyhedron, and its non-spherical gravity is calculated accordingly. By using geometric mechanics, the Hamiltonian structure and equations of motion of the full dynamics are derived. Then, the orbit-attitude equilibria about asteroids 4769 Castalia, 433 Eros, and 6489 Golevka are calculated numerically, respectively, and the drift of orbit-attitude equilibria with respect to the dumbbell's varying length is also discussed. By comparing with the equilibria of a point mass, it has been found that a classical point-mass equilibrium can generate three orbit-attitude equilibria located nearby. With the increase of the dumbbell's length, the effects of orbit-attitude coupling become stronger, and the orbit-attitude equilibria diverge further from the classical point-mass equilibria. The spectral stability of the orbit-attitude equilibria is determined numerically and analyzed. It has been found that the gravitational orbit-attitude coupling is likely to make the equilibria unstable in our simulations. The dumbbell is a good approximation of a spacecraft consisting of two modules that are connected by trusses, and its orbit-attitude equilibria have potential applications in asteroid proximity operations, such as body-fixed hovering.

Artificial Equilibrium Points near Irregular-Shaped Asteroids with Continuous Thrust

Artificial equilibrium points near irregular-shaped asteroids are generated by using continuous thrust to enlarge the feasible regions of body-fixed hovering missions. The dynamics of artificial equilibrium points and feasible sets with constraints of the control magnitude and direction are investigated. The linearized motion near artificial equilibrium points is derived with the generalized effective potential for analysis of topological classification, bifurcations, and stability. Collision and annihilation of equilibrium points and Hopf bifurcation are found during numerical continuation. A parametric study is conducted to investigate the influence of the control magnitude on the sets of artificial equilibrium points. The linearly stable regions are obtained by a grid searching. Moreover, a method based on the polyhedral model of asteroids is proposed to solve the feasible artificial equilibrium points with constraints of the control direction and observation. The observable area at feasible artificial equilibrium points is also obtained by the proposed method. In numerical simulations, 433 Eros is selected as a representative irregular-shaped asteroid to validate the effectiveness of the methods.

Soft landing on an irregular shape asteroid using Multiple-Horizon Multiple-Model Predictive Control

This study has introduced a predictive framework including a heuristic guidance law named Predictive Path Planning and Multiple-Horizon Multiple-Model Predictive Control as the control scheme for soft landing on an irregular-shaped asteroid. The dynamical model of spacecraft trajectory around an asteroid is introduced. The reference-landing trajectory is generated using Predictive Path Planning. Not only does the presented guidance law satisfy the collision avoidance constraint, but also guarantees the landing accuracy and vertical landing condition. Multiple-Horizon Multiple-Model Predictive Control is employed to make the spacecraft track the designed reference trajectory. The proposed control approach, which is a Model Predictive Control scheme, utilizes several prediction models instead of one. In this manner, it heritages the advantages of optimality and tackling external disturbances and model uncertainties from classical Model Predictive Control and at the same time has the advantage of lower computational burden than Model Predictive Control. Finally, numerical simulations are carried out to demonstrate the feasibility and effectiveness of the proposed control approach in achieving the desired conditions in presence of uncertainties and disturbances.

Multiple-hopping trajectories near a rotating asteroid

We present a study of the transfer orbits connecting landing points of irregular-shaped asteroids. The landing points do not touch the surface of the asteroids and are chosen several meters above the surface. The ant colony optimization technique is used to calculate the multiple-hopping trajectories near an arbitrary irregular asteroid. This new method has three steps which are as follows: (1) the search of the maximal clique of candidate target landing points; (2) leg optimization connecting all landing point pairs; and (3) the hopping sequence optimization. In particular this method is applied to asteroids 433 Eros and 216 Kleopatra. We impose a critical constraint on the target landing points to allow for extensive exploration of the asteroid: the relative distance between all the arrived target positions should be larger than a minimum allowed value. Ant colony optimization is applied to find the set and sequence of targets, and the differential evolution algorithm is used to solve for the hopping orbits. The minimum-velocity increment tours of hopping trajectories connecting all the landing positions are obtained by ant colony optimization. The results from different size asteroids indicate that the cost of the minimum velocity-increment tour depends on the size of the asteroids.

An efficient algorithm for global periodic orbits generation near irregular-shaped asteroids

Periodic orbits (POs) play an important role in understanding dynamical behaviors around natural celestial bodies. In this study, an efficient algorithm was presented to generate the global POs around irregular-shaped uniformly rotating asteroids. The algorithm was performed in three steps, namely global search, local refinement, and model continuation. First, a mascon model with a low number of particles and optimized mass distribution was constructed to remodel the exterior gravitational potential of the asteroid. Using this model, a multi-start differential evolution enhanced with a deflection strategy with strong global exploration and bypassing abilities was adopted. This algorithm can be regarded as a search engine to find multiple globally optimal regions in which potential POs were located. This was followed by applying a differential correction to locally refine global search solutions and generate the accurate POs in the mascon model in which an analytical Jacobian matrix was derived to improve convergence. Finally, the concept of numerical model continuation was introduced and used to convert the POs from the mascon model into a high-fidelity polyhedron model by sequentially correcting the initial states. The efficiency of the proposed algorithm was substantiated by computing the global POs around an elongated shoe-shaped asteroid 433 Eros. Various global POs with different topological structures in the configuration space were successfully located. Specifically, the proposed algorithm was generic and could be conveniently extended to explore periodic motions in other gravitational systems.

A Nonhomogeneous Immersed-Finite-Element Particle-in-Cell Method for Modeling Dielectric Surface Charging in Plasmas

We present a particle-in-cell (PIC) method using a nonhomogeneous immersed-finite-element (IFE) field solver for modeling dielectric surface charging of complex-shaped objects in plasmas. The IFE solver allows PIC codes using a Cartesian mesh applied to simulations involving arbitrarily shaped objects with a similar accuracy as that using a body-fitting mesh. The object surface is treated as an interface. Surface charging is calculated directly from charge deposition at the interface, and the electrostatic fields on both sides of the interface are resolved self-consistently. The capability of the nonhomogeneous IFE-PIC method is demonstrated by a simulation study of the charging of an irregular-shaped asteroid in the solar wind.

Bifurcation of equilibrium points in the potential field of asteroid 101955 Bennu

The stability and topological structure of equilibrium points in the potential field of the asteroid 101955 Bennu have been investigated with a variable density and rotation period. A dimensionless quantity is introduced for the nondimensionalization of the equations of motion, and this quantity can indicate the effect of both the rotation period and bulk density of the asteroid. Using the polyhedral model of the asteroid Bennu, the number and position of the equilibrium points are calculated and illustrated by a contour plot of the gravitational effective potential field. The topological structure and the stability of the equilibrium points are also investigated using the linearized method. The results show that there are nine equilibrium points in the potential field of the asteroid Bennu, eight in the exterior of the body and one in the interior of the body. Moreover, bifurcation will occur with a variation of the density and rotation period. Different equilibrium points will encounter each other and mix together. Thus, the number of equilibrium points will change. The stability and topological structure of the equilibrium points will also change because of the variation of the density and rotation period of the asteroid. When considering the error of the density of Bennu, the range of the dimensionless quantity covers the critical values that will lead to bifurcation. This means that the stability of the equilibrium points is uncertain, making the dynamical environment of Bennu much more complicated. These bifurcations can help better understand the dynamic environment of an irregular-shaped asteroid.

Two-impulse transfer orbits connecting equilibrium points of irregular-shaped asteroids

The transfer orbits connecting equilibrium points of irregular-shaped asteroids are studied. A new method is proposed to generate the transfer orbits connecting any two equilibrium points near an arbitrary irregular asteroid. This new method has three steps, including the approximate searching, the first transfer orbit correction and the flight time continuation. Specially, this method is applied to the asteroid 433 Eros and all twelve families of transfer orbits connecting every two of its four equilibrium points are founded in its vicinity. The results indicate that the tendency of the total velocity-increment with respect to the flight time is decreasing and the transfer orbits with the lowest total velocity-increment have the longest flight time in each family. Besides, the velocity-increments required for transferring between two equilibrium points can be larger than the one obtained by the difference of the orbital energy. Moreover, four minimum-velocity-increment tours of equilibrium points starting at each equilibrium point are obtained by patching the generated transfer orbits and the minimum-velocity-increment tour cost about 18 m/s.