Rendezvous Missions to Systems of Small Solar System Bodies Using the Suboptimal SDRE Control Approach

Abstract

In this work, we analyze the suitability of the State-Dependent Riccati Equation (SDRE) suboptimal nonlinear control formulation for the implementation of body-fixed hovering of a spacecraft in the highly nonlinear environment engendered by the faint force fields around single- and multi-body Near-Earth Objects (NEOs), a class of Small Solar System Bodies with high relevance either in scientific, economic, or planetary defense-related aspects. Our results, addressing the hovering of a spacecraft around relative equilibrium points on the effective potential of the Near-Earth Asteroid (16) Psyche and of the much smaller main body (called Alpha) of the triple NEA system (153591) 2001SN263, show that the known effectiveness offered by the flexibility engendered by state-dependent factorization of nonlinear models is also effective when applied in these faint and highly nonlinear force fields. In fact, this work is a qualitative evaluation of the suitability of using SDRE in the highly disturbed environment around Small Solar System Bodies, which has never been undertaken before. We intend to prove that this method is adequate. For real missions, it is necessary to make deeper studies. In particular, our results show the flexibility granted by the SDRE approach in the trade off between maneuvering time against fuel consumption, a central aspect in such space missions. For instance, our simulations showed control effort and time of convergence for two controlled trajectories around (16) Psyche ranging from a half-time convergence with ∼20 times lower cost. Analogously, for the much smaller bodies in the (153591) 2001SN263 triple system, we got two trajectories in which one of them may converge ∼10 times faster but with up to ∼100 times higher cost.

1. Introduction

Besides the Sun, the major planets, and their moons, our Solar System comprises numerous smaller bodies (asteroids, comets, and dust particles) collectively known as Small Solar System Bodies (SSSB), also known as the Minor Planets (MP). These small bodies are remnants from the birth of the Solar System and offer valuable insights into planetary formation and the origins of life [1].

An MP is called a Near-Earth Object (NEO) if its orbit has perihelium less than 1.3 astronomical units (au) and aphelium greater then 0.983 au. According to [2], there is an estimated steady-state population of ${10}^4$ objects with a diameter of the order of 100 m inside the orbit of Neptune (30 au from the Sun) and [3] reports that the Main Asteroid Belt (MAB), between the orbits of Mars and Jupiter, might contain up to 2 Mi bodies with more than 1 km diameter. Lunar cratering data suggests that the NEO population, from dust-sized objects to bodies tens of kilometers in size, has been in steady state for the last 3 G.y. [4] and, according to [5], around $1.2$ Mi MPs were discovered by September 2022 (∼10,000 of them only in 2022), from which 853 NEAs and 152 PHAs were greater than 1 km in diameter.

Rendezvous missions to NEOs may then grant access to pristine (primitive, unweathered, unaltered) samples directly on the target body, meaning that samples were not affected by interactions with Earth’s ambient surroundings, as happens to samples obtained from meteoritic material, thus allowing the collection of accurate information on NEO formation and composition [6,7,8]. This has paved the way for planetary Sample Return Missions (SRM) [9,10,11,12,13,14,15], and made rendezvous missions targeting NEOs a multipurpose endeavor, ranging from the scientific goals of understanding the formation and evolution of our Solar System [16,17,18,19] to planetary defense [20,21,22,23,24,25,26,27,28] and the potential economic value of space resource assessment and exploration of important mining sources of raw materials (metals and volatiles, including water), potentially useful resources for infrastructure constructions, propellant production, and life support for either cis-lunar or deep-space exploration missions [6,8,13,23,29,30,31,32], as is the case of the large MAB asteroid (16) Psyche, the M-class (metallic) target of the NASA Psyche spacecraft [19,33], launched on 13 October 2023 and aiming to approach this asteroid by the beginning of 2029.

Besides the science and technology demands, an additional remarkable benefit favoring rendezvous missions targeting NEOs comes from the fact that, for around 20% of these bodies that periodically approach Earth’s orbit, the velocity impulse $\Delta v\sim$ $6$ km/s required to transfer and insert a spacecraft into an orbit around them is lower than that required for orbiting our Moon [34], and lower than the $\Delta v\sim$ $8$ km/s required to reach around 4000 MBA bodies [35].

Nonetheless, uncertainties arising from a lack of a priori knowledge about these bodies make it challenging to design rendezvous missions aimed at reconnaissance and exploration of NEOs, particularly regarding touch-and-go, descent, or sampling operations, as the spacecraft’s motion in these faint force fields may be easily disturbed by the bodies’ shape/mass asymmetries and by external forces, directly affecting the dynamics and control of orbits around them [36,37], which may lead to collisions with the body or escape from its neighborhood [38]. To easily implement orbit control in these highly nonlinear environments, we will use the suboptimal State-Dependent Riccati Equation (SDRE) control technique, which is based on representing an input affine (linear in control) nonlinear model to resemble a linear structure by adopting state-dependent coefficient (SDC) matrices, in a process called pseudolinearization or extended linearization [39]. This pseudolinear design allows the flexibility of LQR formulation to be directly translated to control nonlinear systems.

Objectives and Paper Outline

The highly nonlinear and highly disturbed environment around NEOs poses stringent difficulties regarding the control of spacecraft around them, an essential task concerning in situ exploration missions targeting these bodies. On the other hand, the state-dependent (SD) factorization used in the SDRE formulation preserves all the nonlinear structure of the involved dynamics while enabling the adaptation of an easily implementable Linear Quadratic approach, either via regulating (LQR) or tracking (LQT) algorithms.

Our objective is to evaluate the suitability of using the SDRE suboptimal control formulation in hovering a spacecraft in the highly nonlinear environment around NEAS, particularly around the relative equilibrium points on the neighborhood of the Near-Earth Asteroid (NEA) (16) Psyche and of the main body (Alpha) of the triple NEA system (153591) 2001SN263, target of the ASTER mission [40], the first proposed Brazilian mission to deep space, which has as one of its main goals to analyze the physical and dynamic structures of the system to understand its origin and evolution. We will also use SDRE to track some stable or long-lasting reference trajectories within the triple NEA system, as such orbits may be used for close observation of the bodies and data acquisition [41].

Within this scope, the intricacies engendered by the highly disturbed environment around NEOs in general will be depicted in Section 2. In Section 3, we present the models for a restricted planar-equatorial motion around symmetric bodies, in which we show the influence of the small bodies’ physical properties on the Zero-Velocity Curves of their effective potential and also a review of the features and power of the suboptimal SDRE control formulation. In Section 4, we present the results of using the SDRE control approach in hovering a spacecraft around the single-body NEA (16) Psyche and the main body (named Alpha) of the triple NEA system (153591) 2001SN263. We used the SDRE control initially for a body-fixed hovering, aiming to keep the spacecraft near the natural equilibrium points around the uniformly rotating single bodies (16) Psyche and (153591)-Alpha, and then used inertial hovering for tracking long-lasting pre-assigned trajectories within the whole triplet (153591) 2001SN263.

2. NEOs Environment

Once NEOs are, in general, relatively small bodies endowed with faint gravitational fields, natural (uncontrolled) orbital trajectories around them may evolve in quite complex fashion [38]. Specifically for the smaller asteroids, there may exist cases of overlapping spheres of influence within the NEO’s gravity field and external disturbance forces, which may lead to unstable or even chaotic orbits for a spacecraft around them [42]. A comprehensive assessment of the attracting body’s major physical parameters, such as mass, shape, density, gravity field, and spin state, is then an essential issue in the construction of accurate models for dynamics and control of orbits around them. This causes the phase of proximity operations to play a central role in rendezvous missions targeting NEOs, as during this phase, valuable information about the physical properties of these bodies can be gathered, allowing mission plans to be properly updated [10,43,44,45,46,47].

Greater concern is needed when considering rendezvous missions to multi-body NEOs. Most of the known binary systems have small mass ratios, but only small enough to allow, at most, the existence of weakly stable orbits that are easily perturbed by external disturbances [42,48]. While stable orbits, or at least long-lasting ones, generally lie outside the secondary’s orbit, the orbits internal to the secondary body are mostly unstable, so their maintenance may require higher fuel consumption. According to [45], better stability can be achieved with retrograde orbits relative to the binary system’s orbital plane. Much more challenging is to orbit a Triple Asteroid System. The extra disturbance added by a very close third body, albeit a small one, adds instability to the environment and makes it much more difficult to flight the spacecraft within the system, mainly when considering fuel limitation constraints.

For instance, this may put stringent requirements in the case of a reconnaissance mission to these multi-body systems. The spacecraft may have to navigate within the whole system for an adequate mapping to collect precise data about the body characteristics, all under all the previously mentioned force-entanglement that generates a highly disturbed environment [36,41].

2.1. Close-Proximity Dynamics

In the complex dynamics of a spacecraft orbiting an asymmetric and irregularly rotating NEO, orbital linear and angular momenta may undergo significant changes in a relatively short period. For instance, assuming a slightly aspheric NEO, the force field engendered by mass asymmetry may cause oscillations in the orbit’s inclination and eccentricity [49], much like those effects experienced by in-orbit servicing geosynchronous Earth satellites [50].

An effective approach is to place the spacecraft around existing stable relative equilibrium or libration points of a Sun–NEO 2-Body-Problem model [51,52,53] or, in the case of a less asymmetric, more massive and regularly spinning central body, to use existing equilibrium points on the Zero-Velocity Surfaces, mimicking a dead-zone control strategy [33,51,54,55,56,57,58]. Thus, any search for equilibria, or their respective periodic orbits, will be fundamentally guided by the NEO’s mass and spin state as these properties are inversely proportional to the distance of the equilibrium points to the attracting body and also affect the stability of these equilibrium points [38].

Alternatively, unstable natural orbits around NEOs may turn into a benefit for controlling fuel expenditure and, consequently, the lifetime of a rendezvous mission. According to [42,59], a spacecraft whose orbit wanders outside the sphere of influence of the attracting NEO ($r_{orb}>R_{{\mathrm{H}}_{\mathrm{ILL}}}$) may fall under the destabilizing influence of external disturbing forces. On the other hand, circular orbits that start with radii smaller than one-half of this reference distance are generally stable against escape from the body [59,60]. Approaching orbits starting outside this sphere of influence, provided they are far enough from the destabilizing external forces [48], may be under weak, unstable dynamics, a condition that can be used to the mission’s benefit as these weak, disturbing accelerations can be actively counteracted with small fuel expenditure while keeping a spacecraft motion relative to the NEO.

When it comes to rendezvous missions targeting multibody-systems, even considering that the spacecraft may have to navigate within the system’s bodies under the effect of previously mentioned force-entanglement that generates a highly disturbed environment, which [38] inputted as the most strongly perturbed astrodynamic environment found in the Solar System, a still effective approach would be to place a spacecraft in an equilibrium point of the system if there exists any. Particularly for binary systems, which generally present small mass ratios, the synchronous orbits relative to a synodic reference frame are good candidates [42]. However, the position may not be fully satisfying regarding the distance for detailed observation of the bodies’ surfaces. For triple systems, a third-body-disturbing version might be used in the case of one or two moonlets presenting small mass ratios, as is the case with (87) Sylvia [60], the first identified triple system, comprised of an asymmetric central body and two tinny moonlets with $\sim {10}^{-5}$ of the main body’s mass and traveling prograde orbits quasi-circular and quasi-coplanar relative to the main body’s equator. This has put on stage the search for naturally stable or at least weakly unstable orbits around NEO systems where to place a spacecraft within the system so as to observe the bodies and proceed to close approaches [43,45,46,61,62,63].

Using the mascon approach [64] to determine Sylvia’s mass properties, ref. [60] found that all the equilibrium points are unstable but there are sufficiently long-lasting prograde orbits after 500 km from main body, approximately half-way to the inner moonlet orbit (∼700 km radius), with variation up to 6 km in semi-axis and up to $0.05$ in eccentricity for a period of 100 days, which may be enough for an initial approach-and-acquaintance phase. For a general three-body problem, ref. [65] found that, for three arbitrary masses and arbitrary non-singular angular momentum, there exists a Lyapunov stable relative equilibrium and the nearby motions are all-time bounded. This notwithstanding, the same concerns about adequate observability of the bodies from such equilibria positions remain valid. In a similar approach, ref. [66] simulated the irregular shape of each object in the triple system (153591) 2001SN263 using uniform-density polyhedra to investigate the system’s gravitational potential [33,67,68]. Their results show that the main system’s body, Alpha, has a peculiar number of 13 equilibrium points, 12 of them external and located very close to the body’ surface. In the cases of Beta and Gamma, they found four external equilibrium points engendered by the irregular-shaped bodies, with just one stable region around Beta, whose dimensions were determined by statistical analysis of orbits in the vicinity of the only equilibrium point whose eigenvalues have no positive real part, as can be seen in Table A1 (online annex of [66]).

Nonetheless, eventual equilibrium points engendered by the whole system in a Sun–Asteroid reference frame may be too far away, which may hinder adequate near-inertial hovering on such environments as a means to data acquisition for mission updates. In either case, the natural stability of a pre-assigned orbit or equilibrium is a central element in this quest for meeting the mission’s requirements with minimal fuel consumption [10,58,69]. By assuming that a spacecraft orbit is within the NEO’s Hill’s sphere of influence, i.e., has an orbital radius within the NEO’s Hill radius $r_{orb}<R_{{\mathrm{H}}_{\mathrm{ILL}}}\approx r_{\mathrm{NEO}}\sqrt[3]{{\displaystyle \frac{m_{\mathrm{NEO}}}{3M_{\mathrm{SUN}}}}}$($r_{\mathrm{NEO}}$ is the mean radius of the NEO’s heliocentric orbit; $M_{(.)}$ and $m_{(.)}$ are the respective masses), it is possible to assume that the NEO’s gravity will be the dominant force acting on the spacecraft.

2.2. Asteroid Hovering

According to [69], hovering can be broadly defined as using active (continuous) control to nullify the total acceleration on a spacecraft in order to keep it at a desired position, generally a stable stationary equilibrium or a periodic orbit in a particular reference frame. In the so-called near-inertial hovering, the spacecraft is regulated at relative equilibria in the Sun–NEO frame, either natural or thrust-induced, irrespective of the NEO’s spin state. As in this case, the points of equilibrium in the Sun–NEO frame are relatively far from the NEO. This approach may be useful in the initial data-acquisition phase for better characterization of the generally poorly known environment. In the body-fixed hovering approach, the spacecraft is parked at a generally closer fixed location relative to a spinning NEO, implying the spacecraft is synchronously accompanying the NEO’s rotation in inertial space. This closer parking is an advantageous option for end-mission approximation tasks such as detailed mappings of the NEO’s surface [58], and also an efficient strategy for creating artificial equilibria via continuous thrust in a soft-landing descent maneuver [69]. Whatever the approach, accurate models for the NEO’s spin state and gravity will be required to allow, as much as possible, to place the spacecraft on equilibrium points or related periodic orbits such that orbit keeping around the NEO may be accomplished with as low as possible fuel consumption.

Despite playing a very important role in the realm of NEO exploration, the hovering approach may not always be suitable, depending strongly on the targeted body’s characteristics. For example, in the case of spacecraft NEAR-Shoemaker orbiting at 50 km distance from the massive NEA (433) EROS (first to be discovered, 30 km diameter, as dense as the Earth) [70], hovering station-keeping for the first 9 months of the mission would have required a total $\Delta V\sim$ 4 km/s, while the actual fuel usage on a classical orbit correction approach was on the order of a few tens of m/s, highlighting the fact that hovering may not be a reasonable approach for massive bodies [42,70]. On the other hand, the SRM of the large area-to-mass spacecraft Hayabusa (former Muses-C) targeting the low-mass asteroid (25143) Itokawa (see

Table 1. Asteroid properties [73,74].

NEO	Discovery	Mass ($\times {10}^{15}$ kg)	Dimension (km)	Rotation Period (h)	Density (g/cm3)
(4) Vesta	1807	259,000	569 × 555 × 453	5.3	3.4
(16) Psyche	1852	24,000	280 × 230 × 190	4.2	4.0
(253) Mathilde	1885	103.3	66 × 48 × 46	417.7	1.3
(433) Eros	1898	6.7	33 × 13 × 13	5.3	2.7
(153591) 2001SN263 (main body “Alpha”)	2001	0.00917	2.8 × 2.7 × 2.9	3.4	1.1

), a mission so highly susceptible to solar radiation disturbances that it could not even be previously assured that an orbital mission would be possible, a successful near-inertial hovering strategy was implemented for orbit maintenance, and body-fixed hovering was used for several descent maneuvers to the asteroid’s surface [69,71,72].

3. Mathematical Modeling

3.1. Equations of Motion

A synodic (co-rotating) reference frame fixed at the center of mass of a regularly rotating symmetric NEO will be adopted for the analysis of the motion of a nearby spacecraft. Assuming that the NEO has constant density and is rotating uniformly about the fixed axis $\widehat{\mathbf{z}}$, which corresponds to the body’s maximum moment of inertia, the equations of motion for the NEO spacecraft 2-Body Problem are written as [75]

$$\begin{array}{cc}\hfill \ddot{x}-2\omega \dot{y}& ={\omega }^{2}x-{\displaystyle \frac{\partial U}{\partial x}}+A_x\hfill \\ \hfill \ddot{y}+2\omega \dot{x}& ={\omega }^{2}y-{\displaystyle \frac{\partial U}{\partial y}}+A_y\hfill \\ \hfill \ddot{z}& =-{\displaystyle \frac{\partial U}{\partial z}}+A_z,\hfill \end{array}$$

(1)

in which $\omega =\parallel \mathit{\omega }\parallel$ is the spin rate of the asteroid, U is a plane-symmetric potential field ($U(x,y,z)=U(x,y,-z)$) and $A_{(·)}$ are the components of an acceleration vector comprising control and possible disturbances.

In general, a quantity associated with the NEO’s dimension, such as its Brillouin radius $r_B$ (the radius of the sphere circumscribing the asteroid’s surface), is taken as the reference distance [10]. When using such a normalization, the Stokes coefficients [76] remain constants, irrespective of the NEO’s physical characteristics, such as size, mass distribution, and rotation state. On the other hand, when adopting as reference distance the point-mass-equivalent synchronous radius (1:1 spin-orbit resonance) for the rotating body

$$R_{\mathrm{s}}=\sqrt[3]{{\displaystyle \frac{Gm}{{\omega }^{2^{\phantom{t}}}}}},$$

(2)

(m is the asteroid’s mass and G the gravitational constant), the Stokes coefficients become a function of these NEO’s physical properties, which allows the study of how the dynamics are affected by these NEO’s characteristics, for instance through the effects of the harmonic coefficients $C_{20}$ and $C_{22}$ usually adopted in studying slightly oblate symmetric bodies [66,77,78,79].

3.1.1. Equilibria and Natural Orbital Dynamics for Symmetric Rotating NEOs

The nature of equilibrium points in the vicinity of regularly rotating NEOs depends on the relative strength of the body’s gravitational and centripetal forces and, as shown in [80], the amount of these equilibrium points in the gravity field of irregular bodies, comprising asteroids and comets, is not fixed, although always existing in at least one of them. Additionally, even if these equilibria exist, they may all be unstable [57,66,81]. Thus, even if relying on scarce pre-mission knowledge about the NEO’s physical parameters, the stability analysis and the search for periodic orbits is a central issue for orbital dynamics in close-approach missions around NEOs.

By defining the Effective Potential [75]

$$V\left(\mathbf{r}\right)=U\left(\mathbf{r}\right)-{\displaystyle \frac{1}{2}}{\parallel \mathit{\omega }\times \mathbf{r}\parallel }^2,$$

(3)

and assuming the frequent case of a small body uniformly rotating around its maximum moment of inertia, Equation (1) transforms to

$$\ddot{\mathbf{r}}+2\mathit{\omega }\times \dot{\mathbf{r}}=-\nabla V\left(\mathbf{r}\right),$$

(4)

which makes the Jacobian integral (mechanical energy) time-invariant, now named Jacobi constant J (Equation (5)),

$$H(\mathbf{r},\dot{\mathbf{r}})={\displaystyle \frac{1}{2}}\left[(\dot{\mathbf{r}}·\dot{\mathbf{r}})-{\parallel \mathit{\omega }\times \mathbf{r}\parallel }^2\right]+U\left(\mathbf{r}\right)={\displaystyle \frac{1}{2}}(\dot{\mathbf{r}}·\dot{\mathbf{r}})+V\left(\mathbf{r}\right)=J.$$

(5)

The contour surfaces defined by the solutions of

$$H(\mathbf{r},\mathbf{0})=V\left(\mathbf{r}\right)=J,$$

(6)

are the so-called Zero-Velocity Surfaces (ZVS), which delimit in space the allowed regions for the spacecraft motion [82]

$$A=\{\mathbf{r}:\dot{\mathbf{r}}·\dot{\mathbf{r}}\ge 0\},$$

(7)

and thus provide a means for classification of the orbital motion into families of orbits that may be identified in correspondence to given values of the Jacobi constant, J, in the time-invariant system defined by Equations (3)–(5).

3.1.2. Zero-Velocity Curves and Equilibrium Points for Some Aspherical Bodies

Changes in body characteristics, like density, rotation speed, geometry, etc., may affect the number, location, and stability of the equilibrium points [83,84], thus affecting the structure/shape of the ZVSs (6), which are intrinsically related to the stability of the critical points of the effective potential (3). The stability analysis for these points will permit the identification of families of orbits, delimited by the ZVS, that will last stable in the vicinity of the NEO or lead the spacecraft to impact on the central body or to escape from its vicinity. The Jacobi integral (energy) may be used to map families of ZVSs for the system in the body-fixed position space $Oxyz$.

In the present work, the near-equilibrium motion will be analyzed assuming a planar-equatorial orbit ($z=0$) around symmetric oblate bodies, thus reducing the analysis to the Zero-Velocity Curve (ZVC) corresponding to $z=0$ at the given ZVS. Several studies have computed four equilibrium points around uniformly rotating small bodies, such as the asteroids (1620) Geographos, (4769) Castalia and (216) Kleopatra [80,85]. Ref. [84] studied the influence of the rotation speed on the behavior of the equilibrium points in the gravitational field of the asteroid (216) Kleopatra. They observed that longer rotation periods engender topological changes and cause some points to collide and annihilate each other until only one point remains inside the object. Ref. [33] presents the characteristics of the geophysical environment of (16) Psyche and studies how density changes affect the distribution of equilibrium points. The linear stability and topological classification of the equilibrium points in the gravity field of more than 20 small celestial bodies were investigated in [80], according to the classification established in [85].

The NEO’s Hill radius $R_H$ will be assumed as the superior limit for the spacecraft’s orbit radius so as to assure that the NEO’s gravitational influence prevails in the dynamics, allowing the disregarding of external perturbations such as those arising from solar radiation pressure or third-body (moonlets) gravitational forces [86]. On the other hand, in order to guarantee the validity of the Harmonic Expansion of the gravity potential in the vicinity of an aspheric NEO surface, its Brillouin radius, $r_B$, will be used as an inferior limit to the spacecraft’s orbit radius. So, it will be assumed the spacecraft’s orbit radius will obey $r_B\le r\le R_H$.

Table 1 presents physical characteristics for some typical asteroids and

Table 2. Stability Parameters for Vesta.

Equilibrium Points	Eigenvalues		Eigenvectors		J
$(\mathit{x},\mathit{y})$ (m)	${\mathit{\lambda }}_{\mathbf{1}}$	${\mathit{\lambda }}_{\mathbf{2}}$	${\mathit{v}}_{\mathbf{1}}$	${\mathit{v}}_{\mathbf{2}}$
(±554,560.98, 0)	$-3.33\times {10}^{-7}$	$1.16\times {10}^{-9}$	$(-1,0)$	$(0,-1)$	$-\mathrm{49,253.2}$
(0, ±551,605.90)	$-3.29\times {10}^{-7}$	$-1.19\times {10}^{-9}$	$(0,1)$	$(-1,0)$	$-\mathrm{49,072.7}$

,

Table 3. Stability Parameters for Psyche.

Equilibrium Points	Eigenvalues		Eigenvectors		J
$(\mathit{x},\mathit{y})$ (m)	${\mathit{\lambda }}_{\mathbf{1}}$	${\mathit{\lambda }}_{\mathbf{2}}$	${\mathit{v}}_{\mathbf{1}}$	${\mathit{v}}_{\mathbf{2}}$
(±221,891.42, 0)	$-5.50\times {10}^{-7}$	$1.21\times {10}^{-8}$	$(-1,0)$	$(0,-1)$	$-\mathrm{12,222.9}$
(0, ± 213,712.06)	$-5.13\times {10}^{-7}$	$-1.46\times {10}^{-8}$	$(0,1)$	$(-1,0)$	$-\mathrm{11,906.8}$

,

Table 4. Stability Parameters for Mathilde.

Equilibrium Points	Eigenvalues		Eigenvectors		J
$(\mathit{x},\mathit{y})$ (m)	${\mathit{\lambda }}_{\mathbf{1}}$	${\mathit{\lambda }}_{\mathbf{2}}$	${\mathit{v}}_{\mathbf{1}}$	${\mathit{v}}_{\mathbf{2}}$
(±727,583, 0)	$-5.24\times {10}^{-11}$	$1.01\times {10}^{-14}$	$(-1,0)$	$(0,-1)$	$-13.860$
(0, ±727,372)	$-5.24\times {10}^{-11}$	$-1.02\times {10}^{-14}$	$(0,1)$	$(-1,0)$	$-13.857$

,

Table 5. Stability Parameters for Eros.

Equilibrium Points	Eigenvalues		Eigenvectors		J
$(\mathit{x},\mathit{y})$ (m)	${\mathit{\lambda }}_{\mathbf{1}}$	${\mathit{\lambda }}_{\mathbf{2}}$	${\mathit{v}}_{\mathbf{1}}$	${\mathit{v}}_{\mathbf{2}}$
(±18,879.64, 0)	$-3.86\times {10}^{-7}$	$3.03\times {10}^{-8}$	$(-1,0)$	$(0,-1)$	$-50.7882$
(0, ±14,975.88)	$-2.29\times {10}^{-7}$	$-9.67\times {10}^{-8}$	$(0,1)$	$(-1,0)$	$-43.6871$
(0, ±9015.31)	$-1.22\times {10}^{-6}$	$8.94\times {10}^{-7}$	$(-1,0)$	$(0,-1)$	$-46.2481$

and

Table 6. Stability Parameters for $2001SN263\_$Alpha.

Equilibrium Points	Eigenvalues		Eigenvectors		J
$(\mathit{x},\mathit{y})$ (m)	${\mathit{\lambda }}_{\mathbf{1}}$	${\mathit{\lambda }}_{\mathbf{2}}$	${\mathit{v}}_{\mathbf{1}}$	${\mathit{v}}_{\mathbf{2}}$
(±1480.39, 0)	$-7.78\times {10}^{-7}$	$9.77\times {10}^{-9}$	$(-1,0)$	$(0,-1)$	$-0.85341$
(0, ±1450.75)	$-7.46\times {10}^{-7}$	$-1.08\times {10}^{-8}$	$(0,1)$	$(-1,0)$	$-0.84237$

show some of the planar ($z=0$) equilibrium points, along with their respective eigenvalues, eigenvectors and Jacobi constants. The shown “Dimension” corresponds to the $x\times y\times z$ full extensions (“diameters”) of the bodies.

Some ZVCs for these bodies are shown in Figure 1, Figure 2, Figure 3, Figure 4 and Figure 5, depicting the saddles (${\lambda }_1<0<{\lambda }_2$) and centers (${\lambda }_1,{\lambda }_2<0$) corresponding to the, respectively, (U)nstable points, crossings of the dashed separatrices, and (S)table equilibrium points. The black (dotted) curves at the center of the figures, except for Icarus and Alpha, represent the equatorial contour of the portrayed body.

From the cases shown here, one can see some intricacies endowed by mass and spin rate on the stability of motion around these small bodies. For instance, (253) Mathilde is ∼250 times less massive than (16) Psyche and ∼2500 times less massive than (4) Vesta but is rotating ∼100 times faster than the other two bodies. One can see that, while the distances of the equilibrium points from the larger bodies’ surfaces correspond to their respective radii, the equilibria of (253) Mathilde are located at a distance corresponding ∼30 times the body’s radius, not prone to pre-close-approach data acquisition.

Then comes the higher asymmetry (∼3-to-1; see Table 1) of the elongate Eros, engendering a more complex diagram of ZVCs and equilibria, as shown in Figure 4.

In between comes the tiny body 2001SN263_“Alpha” (Table 6 and Figure 5 ). Having mass ∼${10}^8\times$ smaller than (4) Vesta’s but rotating at almost the same speed presents equilibria a few tens of meters above the body’s surface.

Further strong evidence of these intricacies can be seen in [66], who used a 3D asymmetric polyhedral model for “Alpha”, the main body in the triplet 2001SN263, and obtained 12 equilibrium points just above the surface of the body, all unstable (either Saddle-Center-Center or Sink-Source-Center), as opposed to the 4 (2 stable and 2 unstable) depicted in Figure 5, obtained from a symmetric, planar top-spin model for the body and based on spherical harmonics.

3.2. Suboptimal Control in Nonlinear Dynamics

Control laws are intrinsically connected to model constructions for physical systems, essentially relying on up-to-date knowledge about the system under study. However, accurate models for real physical systems may be difficult to realize, or even impossible, as the models will be impregnated by uncertainties, mostly coming from unmodeled dynamics, input disturbances, hardware precision and so on, which will engender deviations when comparing the predicted/modeled to real system behaviors. As stated by [87], “No mathematical system can precisely model a real physical system; there is always uncertainty”.

An effective way to correct these deviations is by means of feedback laws (assuming the system is observable), in which the states or outputs depicting actual system behavior are compared to those predicted by the model and, in the case of any discrepancy beyond a pre-assigned limit, correction actions are executed to bring the system back to the prescribed behavior, either via regulation (station-keeping around an equilibrium) or tracking (following a reference trajectory), in due time and considering energy availability. To implement such approaches, consider a dynamical system with a state–space nonlinear model

$$\begin{array}{cc}\hfill \dot{\mathbf{x}}\left(t\right)=& \mathbf{f}(t,\mathbf{x}\left(t\right),\mathbf{u}\left(t\right)),\mathbf{t}\ge t_0,\hfill \\ \hfill \mathbf{x}\left(t_0\right)=& {\mathbf{x}}_0,\hfill \end{array}$$

(8)

in which $\mathbf{x}\in \Omega \subset {\mathbb{R}}^n$ are the allowed or reachable states, $\mathbf{u}\in \mathcal{U}\subset {\mathbb{R}}^m$ are unconstrained admissible or executable controls, and the dynamics $\mathbf{f}:\mathbb{R}\times \Omega \times \mathcal{U}\to {\mathbb{R}}^n$ is assumed sufficiently differentiable. An unconstrained optimization problem for nonlinear regulation (station-keeping) may then be stated based on the determination of a stabilizing feedback control law $\mathbf{u}$ that “regulates the system states $\mathbf{x}$ around a prescribed equilibrium ${\mathbf{x}}_e$” while minimizing a mission-designed cost function, or performance index (PI), usually conditioning the control effort $\mathbf{u}\in \mathcal{U}$, the states $\mathbf{x}\in \Omega$ and the execution time interval $I=[t_0,t_f]\subset \mathbb{R}$, which may be translated to [88,89]

$$PI={\mathbf{\Phi }}_f\left({\mathbf{x}}_f\right)+\underset{t_0}{\overset{t_f}{\int }}\mathrm{\Psi }\left(t,\mathbf{x}\left(t\right),\mathbf{u}\left(t\right)\right)dt,$$

(9)

for some sufficiently differentiable function $\mathrm{\Psi }:\mathbb{R}\times {\mathbb{R}}^n\times {\mathbb{R}}^m\to {\mathbb{R}}^{+}=\{z\in \mathbb{R}:z>0\}$ and a symmetric positive semi-definite weighting matrix ${\mathbf{\Phi }}_f:{\mathbb{R}}^n\to {\mathbb{R}}_o^{+}=\{z\in \mathbb{R}:z\ge 0\}$ which accounts for under-performance costs/penalties due to eventual errors in achieving ${\mathbf{x}}_f$ and may also translate eventual bounds imposed to the final states.

A usual model in many aerospace applications considers converting Equation (8) to a control-affine (linear regarding $\mathbf{u}$) nonlinear dynamic model

$$\begin{array}{cc}\hfill \dot{\mathbf{x}}\left(t\right)=& \mathbf{f}\left(\mathbf{x}\left(t\right)\right)+\mathbf{B}\left(\mathbf{x}\left(t\right)\right)\mathbf{u}\left(t\right),\mathbf{B}:{\mathbb{R}}^n\to {\mathbb{R}}^m,t\ge t_0\hfill \\ \hfill \mathbf{x}\left(t_0\right)=& {\mathbf{x}}_0,\hfill \end{array}$$

(10)

and the adoption of the weighted infinite-time horizon performance index [90,91,92,93]

$$P{I}_{\infty }=\underset{t_0}{\overset{\infty }{\int }}\left({\mathbf{x}}^T\mathbf{Q}\mathbf{x}+{\mathbf{u}}^T\mathbf{R}\mathbf{u}\right)dt,$$

(11)

in which $\mathbf{Q}\in {\mathbb{M}}_{n\times n},\mathbf{R}\in {\mathbb{M}}_{m\times m}$ are weighting symmetric positive-semi-definite and positive-definite matrices, respectively, which makes this cost function quadratic regarding the states $\mathbf{x}$ (assuming equilibrium at the origin ${\mathbf{x}}_e=\mathbf{0}$) and control effort, thus penalizing both the deviations in any of the state variables and control effort.

This unconstrained optimizing design aims to produce the unconstrained optimal control $\tilde{\mathbf{u}}\left(\mathbf{x}\right)$ such that, under the dynamics established in Equation (10), the system is optimally guided along an optimal reference trajectory $\tilde{\mathbf{x}}$ or optimally kept around an equilibrium ${\mathbf{x}}_e=\mathbf{0}$, while minimizing the cost established in Equation (11), i.e.,

$$P{I}_{\infty }(\tilde{\mathbf{x}},\tilde{\mathbf{u}})\le P{I}_{\infty }(\mathbf{x},\mathbf{u}),\forall (\mathbf{x},\mathbf{u})\in \Omega \times \mathcal{U}.$$

(12)

Core results on the resolution of such optimal nonlinear control problems, consisting of minimizing (11) subject to (10), come from the cumbersome task of, given a Bellman/Value function [94]

$$V\left(\mathbf{x}\right)=\underset{\mathbf{u}\in \mathcal{U}}{inf}P{I}_{\infty }(\mathbf{x},\mathbf{u}),$$

(13)

resolving the related Hamilton–Jacobi–Bellman (HJB) partial differential equation

$$-{\displaystyle \frac{\partial V}{\partial t}}\left(\mathbf{x}\right)=\underset{\mathbf{u}}{inf}\left[{\dot{\mathbf{x}}}^T{\displaystyle \frac{\partial V}{\partial \mathbf{x}}}\left(\mathbf{x}\right)+{\mathbf{x}}^T\mathbf{Q}\left(\mathbf{x}\right)\mathbf{x}+{\mathbf{u}}^T\mathbf{R}\left(\mathbf{x}\right)\mathbf{u}\right],$$

(14)

that produces the optimal control

$$\tilde{\mathbf{u}}\left(\tilde{\mathbf{x}}\right)=-{\mathbf{R}}^{-1}\left(\tilde{\mathbf{x}}\right){\mathbf{B}}^T\left(\tilde{\mathbf{x}}\right){\displaystyle \frac{\partial V}{\partial \mathbf{x}}}\left(\tilde{\mathbf{x}}\right),$$

(15)

a task whose feasibility in an analytical fashion depends on the proper construction of an adequate Value Function satisfying Equations (13) and (14), and on the assumption of some stringent smoothness conditions on $\mathbf{f},\mathbf{g},\mathbf{Q},\mathbf{R}$, which may impart some limitation on the usability of this approach [95,96].

For systems whose dynamics are linear and time-invariant (LTI),

$$\begin{array}{cc}\hfill \dot{\mathbf{x}}=& \mathbf{A}\mathbf{x}+\mathbf{B}\mathbf{u},\mathbf{x}\left(t_0\right)={\mathbf{x}}_0\hfill \\ \hfill \mathbf{y}=& \mathbf{C}\mathbf{x}\hfill \end{array}$$

(16)

with constant matrices $\mathbf{A}$, $\mathbf{B}$ and $\mathbf{C}$, the HJB equation (14) reduces to the Algebraic Riccati Equation (ARE) [97]

$$\mathbf{P}\mathbf{A}+{\mathbf{A}}^T\mathbf{P}-\mathbf{P}\mathbf{B}{\mathbf{R}}^{-1}{\mathbf{B}}^T\mathbf{P}+\mathbf{Q}=\mathbf{0},$$

(17)

whose positive-definite solution P engenders the minimization of (11) under (16) via the optimal feedback control

$$\mathbf{u}\left(t\right)=-{\mathbf{R}}^{-1}{\mathbf{B}}^T\mathbf{P}\mathbf{x}\left(t\right)=\mathbf{K}\mathbf{x}\left(t\right),t\ge t_0,$$

(18)

provided the pairs $\{\mathbf{A},\mathbf{B}\}$ and $\{\mathbf{A},\mathbf{C}\}$ are, respectively, controllable and observable, i.e., the controllability and observability matrices [94]

$$\begin{array}{cc}\hfill {\mathbf{M}}_c=& \left[\begin{array}{c}\mathbf{B}\mathbf{A}\mathbf{B}{\mathbf{A}}^2\mathbf{B}...{\mathbf{A}}^{n-1}\mathbf{B}\end{array}\right]\hfill \\ \\ \hfill {\mathbf{M}}_o=& {\left[\begin{array}{c}\mathbf{C}\mathbf{C}\mathbf{A}\mathbf{C}{\mathbf{A}}^2...\mathbf{C}{\mathbf{A}}^{n-1}\end{array}\right]}^T\hfill \end{array}$$

(19)

are such that $rank\left({\mathbf{M}}_c\right)=n$ and $rank\left({\mathbf{M}}_o\right)=n$. Further assumption of non-negativity and symmetry for the matrix $\mathbf{Q}$ will allow the decomposition $\mathbf{Q}={\mathbf{C}}^T\mathbf{C}$, for non-singular $\mathbf{C}$, will guarantee that all states will be led to the stable equilibrium of (16) [97,98].

3.2.1. State-Dependent (SD) Formulation for Suboptimal Nonlinear Control

The nonlinear techniques are very sensitive to the dynamics, leading to a more complex overall control design when compared to linear (ized) ones. The adoption of LQR (or LQT) by means of linearization relies on the use of rather simplified models for the dynamics, which may not be a great limitation when dealing with small deviations from the given references. Nonetheless, as the model errors tend to be instantaneously compensated for by control actions, fuel consumption will increase in the case of large deviations, which makes linear(ized) models not prone for applications, for example, in a rendezvous mission to a NEO, as usual uncertainties concerning the target body’s environment or from reference orbits will induce greater errors on the dynamics. Additionally, if the system is highly nonlinear, the linear(ized) optimal controller does not provide adequate performance as the process of linearization renders rather local validity in some models, and the performance may degrade rapidly as the operation departs from the reference point or reference trajectory, as opposed to global stability/optimality of the autonomous linear models [90,99,100,101].

On the other hand, LTI models are much easier to handle and allow linear optimal control theory to produce easily computable and globally valid solutions for a large class of problems. It looks extremely attractive and promising to extend to nonlinear problems this linear simplicity/effectivity in determining the laws of optimal control, for instance, by means of model linearization around a stationary point (Linear Quadratic Regulation-LQR) or a reference trajectory (Linear Quadratic Tracking-LQT) [87,89,91,93,102,103,104,105].

There begins the search for trade-offs between achieving as easy and inexpensive implementability as occurs in linear/linearized models while keeping as much as possible the real (nonlinear) characteristic of the plants, allowing greater robustness when compared to Taylor or Jacobi linearization, albeit sacrificing some performance (error reduction), as compared to the full nonlinear models. Ref. [106] gave a seminal step in the direction of such an approach by suggesting to approximate the nonlinear optimal control problem by a linear time-varying one, i.e., at fixed instants, the nonlinear system is assumed as linear, enjoying all the mentioned benefits of the linear structure. Much better performance is then achieved from a linear-like construct at the computational cost of repeatedly resolving momentaneously frozen (time-invariant) linear state-dependent problems as the system model evolves in time with the states [107], somehow mimicking a state-related gain-scheduling. The approach relies then on recurrently resolving an Algebraic Riccati Equation (ARE) at the momentaneously fixed points in the state–space, so it was named the Frozen Riccati Equation (FRE) or State-Dependent Riccati Equation (SDRE) technique [100,103,108]. Some alternative approaches, along with their costs and benefits, are presented in [101,103].

The standard SDRE technique is based on representing an input affine nonlinear model in a way that resembles a linear overall structure by adopting state-dependent coefficient (SDC) matrices in a process called pseudo-linearization or extended linearization. Thus, the SDRE design allows the flexibility of LQR/LQT formulation to be directly translated to the control of nonlinear systems. Basically, in the SDRE approach, once the pseudo-linearization of the dynamics in (10) is established via SDC matrices, the resolution of the infinite-horizon feedback regulation/tracking with PI given by (11) is reduced to solving an ARE for the equivalent autonomous LQR/LQT problem at given instants (frozen points) along the trajectory. For the finite-horizon control (9), the solution is time-dependent and leads to the resolution of a Differential Riccati Equation (DRE), instead of an algebraic one, for establishing the control law [101].

3.2.2. SDC Factorization

The objective of an SDC factorization is to find continuous matrix-valued functions $\mathbf{A}:\Omega \subset {\mathbb{R}}^n\to {\mathbb{R}}^{n\times n}$ allowing the rewriting of the nonlinear dynamics in (10) using a (pseudolinear) state-dependent (SD) form,

$$\begin{array}{cc}\hfill \dot{\mathbf{x}}=& \mathbf{A}\left(\mathbf{x}\right)\mathbf{x}+\mathbf{B}\left(\mathbf{x}\right)\mathbf{u},\mathbf{B}\left(\mathbf{x}\right)\ne \mathbf{0},\forall \mathbf{x}\in \Omega \hfill \\ \hfill \mathbf{x}\left(t_0\right)=& {\mathbf{x}}_0,\hfill \\ \hfill \mathbf{y}=& \mathbf{C}\left(\mathbf{x}\right)\mathbf{x},\hfill \end{array}$$

(20)

that preserves all the nonlinear structure of (10) while allowing the use of the easily implementable structure of LQR in LTI systems for the minimization of (11). Concerning the existence of an SDC representation (20) for (10), we refer to [39,105,109,110]:

Proposition 1.

Given $\mathbf{f}:\mathbf{\Omega }\subset {\mathbb{R}}^n\to {\mathbb{R}}^n$, differentiable $({\mathcal{C}}^k,k\ge 1)$ in Ω and with an equilibrium $\mathbf{f}\left(\mathbf{0}\right)=\mathbf{0}$, it is always possible to find $\mathbf{A}:\mathbf{\Omega }\subset {\mathbb{R}}^n\to {\mathbb{R}}^{n\times n}$ such that $\mathbf{f}\left(\mathbf{x}\right)=\mathbf{A}\left(\mathbf{x}\right)\mathbf{x}$.

Proof. 

See [39,105,110].  □

It is interesting to note that, for any of the SDC representations for $\mathbf{f}$, one will have (assuming equilibrium at the origin $\mathbf{x}=\mathbf{0}$)

$${\mathbf{A}}_i\left(\mathbf{x}\right)\mathbf{x}=\mathbf{f}\left(\mathbf{x}\right)\ne \nabla \mathbf{f}\left(\mathbf{0}\right)\mathbf{x},\forall \mathbf{x}\ne \mathbf{0},i=1,2,3,$$

properly demonstrating that an SDC factorization (20) will not be a linearization of (10), thus preserving the full nonlinear structure of the model.

3.2.3. SDRE Infinite-Horizon Regulator

Consider a continuous-time autonomous infinite-horizon nonlinear regulator problem, with full-state observable dynamics and affine(linear) unconstrained inputs, given by (20) now with a state-dependent performance index

$$P{I}_{sd}=\underset{t_0}{\overset{\infty }{\int }}\left({\mathbf{x}}^T\mathbf{Q}\left(\mathbf{x}\right)\mathbf{x}+{\mathbf{u}}^T\mathbf{R}\left(\mathbf{x}\right)\mathbf{u}\right)dt$$

(21)

with a quadratic-like structure in $\mathbf{x}\in \mathbf{\Omega }\subset {\mathbb{R}}^n$ (the reachable/accessible region), and quadratic in $\mathbf{u}\in \mathcal{U}\subset {\mathbb{R}}^m$ (the admissible controls), now admitting also the weighting functions $\mathbf{Q}:\Omega \to {\mathbb{R}}^{n\times n}$ and $\mathbf{R}:\Omega \to {\mathbb{R}}^{m\times m}$ to be ${\mathcal{C}}^k(k\ge 0)$ state-dependent functions. Still enjoying the heritage of the linear models, as the weighting matrices $\mathbf{Q}$ and $\mathbf{R}$ may also be state-dependent, the nonlinear performance index (21) may have now only a quadratic-like structure regarding the states (${\mathbf{x}}^T\mathbf{Q}\left(\mathbf{x}\right)\mathbf{x}$), but still presents a quadratic structure regarding control (${\mathbf{u}}^T\mathbf{R}\left(\mathbf{x}\right)\mathbf{u}$). This state-dependence permits extra flexibility in the control design by allowing extra degrees of freedom in the trade-off between control effort and state errors via tuning $\mathbf{R}\left(\mathbf{x}\right(\tau \left)\right)$ and $\mathbf{Q}\left(\mathbf{x}\right(\tau \left)\right)$ at the frozen instants $\tau \ge t_0$, in the search for better compromise between cost, performance, and robustness [39,91,96,97,108,111].

Analogously to the LTI case, the SDRE approach is based on defining the triplet $\left\{\mathbf{A}\right(\mathbf{x}),\mathbf{B}(\mathbf{x}),\mathbf{C}(\mathbf{x}\left)\right\}$, as well as the weights $\left\{\mathbf{Q}\right(\mathbf{x}),\mathbf{R}(\mathbf{x}\left)\right\}$, so as to guarantee the symmetric positive-definite solution $\mathbf{P}\left(\mathbf{x}\right)$ for the state-dependent version of the ARE (17) [39]. Nonetheless, while $\mathbf{Q}$ and $\mathbf{R}$ are on-demand design parameters, the triplet $\{\mathbf{A},\mathbf{B},\mathbf{C}\}$ must be chosen so as to attain the necessary canonical structure [94,105,108]. Results concerning local stability are established in

Theorem 1.

Given the system (10), in which $\mathbf{f}$ satisfy Proposition 1, assume that a SDC parameterization (20) is chosen such that $\mathbf{A},\mathbf{B},\mathbf{C}$ are continuous and the pairs $\{\mathbf{A},\mathbf{B}\}$, and $\{\mathbf{A},\mathbf{C}\}$ are, respectively, point-wise stabilizable and point-wise detectable, in the linear sense, in some nonempty neighborhood ${\mathcal{X}}_1$ of the origin. Then, the SDRE nonlinear regulator (11) and (20) produces a closed-loop solution for the suboptimal control (15), which makes the system locally asymptotically stable.

Proof. 

see [108].  □

This means the SDRE approach for nonlinear feedback control produces a closed-loop system matrix [112]

$${\mathbf{A}}_{CL}\left(\mathbf{x}\right)=\mathbf{A}\left(\mathbf{x}\right)-\mathbf{B}\left(\mathbf{x}\right)\mathbf{K}\left(\mathbf{x}\right)$$

(22)

which is point-wise Hurwitz (eigenvalues with negative real part), $\forall \mathbf{x}\in {\mathcal{X}}_1$. However, as shown by [112], differently from what happens in the LTI models, this Hurwitz property does not guarantee global asymptotic stability for a nonlinear system (when it is possible to regulate any $\mathbf{x}\in \Omega$ to the origin).

Additionally, Ref. [91] proved that, under the hypotheses of Theorem 1, the SDRE solution is globally asymptotically stable if the closed-loop coefficient matrix (22) for the problem (20) and (21) is symmetric for all $\mathbf{x}\in \Omega$. However, there is no rule of thumb for obtaining such a particular stabilizing matrix. However, even if finding such optimal $\tilde{\mathbf{A}}\left(\mathbf{x}\right)$ is usually not straightforward, it is acknowledged within the practitioner community that such eventual mild off-optimality condition is of minor consequence when faced with the benefits afforded by the SDRE method such as design flexibility, computational simplicity/easy implementability, and its satisfactory simulation/experimental results [39,98,109,113].

3.2.4. Reference Trajectory Tracking Using SDRE Regulation

The regulator problem described in the previous section is naturally adherent to the idea of some system’s state or output being kept at a fixed distance from a given reference, as happens on an oil/gas production platform that keeps its position on the sea surface relative to a seabed oil well, or to a spacecraft hovering over the surface of a planet, or even keeping its position around an equilibrium point close to an asteroid [114,115,116]. Another category of problems concerns the accompaniment or tracking of a reference path, as happens in formation flying for airplanes or artificial satellites, mobile robots, synchronized chaotic systems, or even a space probe following a prescribed stable reference trajectory within the strongly perturbed environment around systems of Small Solar System Bodies [37,61,99,117,118,119,120,121].

The purpose of a tracking-control procedure is to guarantee that the plant’s output will follow a prescribed reference trajectory $\{\mathbf{r}\left(t\right)\in \mathcal{X}\subset \Omega ,t\ge t_0\}$, within assigned tolerances relative to output or state errors and control effort, analogous to the ones established in the regulation problem (20) and (21). However, there are some inherent difficulties related to numerical implementability due to the necessity of simultaneous resolution of the SDRE and feedforward differential equations [122] toward the eventual unboundedness of the cost function and toward the additional requirement that the reference trajectory be generated by (asymptotically) stable dynamics [98,123].

Measures simplifying these tasks can be adopted in order to circumvent some of the implementation difficulties. An effective and easier-to-implement solution is to use the additional formation of the tracking problem as a “regulation with time-varying reference”, which permits the conversion of an SDRE-based LQT problem to a pseudo-LQR one by regulating the output error in SDC form [98,100,118,124,125,126]. For this approach, assume the output of the plant with dynamics (20) must follow a reference-model trajectory driven by stable dynamics

$$\dot{\mathbf{z}}={\mathbf{A}}_r\left(\mathbf{z}\right)\mathbf{z},\mathbf{z}\left(t_0\right)={\mathbf{z}}_0;\mathbf{r}={\mathbf{C}}_r\left(\mathbf{z}\right)\mathbf{z}.$$

(23)

Assuming that both dynamics (20) and (23) are fully observable, one can use the error state vector dynamics $\mathbf{e}=\mathbf{y}-\mathbf{r}$, leading to a new PI

$$\begin{array}{cc}\hfill P{I}_{lqt}=& \underset{t_0}{\overset{\infty }{\int }}\left({\mathbf{e}}^T\mathbf{Q}\left(\mathbf{x}\right)\mathbf{e}+{\mathbf{u}}^T\mathbf{R}\left(\mathbf{x}\right)\mathbf{u}\right)dt,\hfill \end{array}$$

(24)

resulting in the equivalent optimal solution for the error-regulation problem stated by (20), (23), and (24)

$$\tilde{\mathbf{u}}\left(\mathbf{x}\right)=-{\mathbf{R}}^{-1}\left(\mathbf{x}\right){\mathbf{B}}^T\left(\mathbf{x}\right)\left(\mathbf{P}\left(\mathbf{x}\right)\mathbf{x}+\mathbf{v}\left(\mathbf{x}\right)\right),$$

(25)

in which $\mathbf{P}\left(\mathbf{x}\right)\mathbf{x}$ is the feedback portion of the control $\tilde{\mathbf{u}}$, in response to the actual system dynamics, and $\mathbf{v}\left(\mathbf{x}\right)$ is the feedforward portion, endowed by the (a priori) known reference dynamics. $\mathbf{P}$ and $\mathbf{v}$ are obtained as back-propagation solutions (starting on given terminal ($t_f$) conditions and propagated backward to $t_0$) of the SD differential equations (dependence on the states omitted) [98,125]

$$\begin{array}{cc}\hfill \dot{\mathbf{P}}=& -{\mathbf{C}}^T\mathbf{Q}\mathbf{C}+\mathbf{P}\mathbf{B}{\mathbf{R}}^{-1}{\mathbf{B}}^T\mathbf{P}-\mathbf{P}\mathbf{A}-{\mathbf{A}}^T\mathbf{P}\hfill \\ \hfill \dot{\mathbf{v}}=& -{\left(\mathbf{A}-\mathbf{B}{\mathbf{R}}^{-1}{\mathbf{B}}^T\mathbf{P}\right)}^T\mathbf{v}+{\mathbf{C}}^T\mathbf{Q}\mathbf{r},\hfill \end{array}$$

(26)

both with terminal constraints $\underset{t\to \infty }{lim}\mathbf{P}\left(t\right)=\mathbf{0}$ and $\underset{t\to \infty }{lim}\mathbf{v}\left(t\right)=\mathbf{o}$.

It is still possible to avoid this backward along-the-trajectory point-wise resolution of the differential equations above by adopting some simplifications depending on the dynamics of the plant under control. For instance, assuming that $t_f\gg t_0$, which may well be the case when a spacecraft is in the approaching the phase of orbiting an asteroid, the steady-state conditions $\dot{\mathbf{P}}\left(t\right)\approxeq \mathbf{0}$ and $\dot{\mathbf{v}}\left(t\right)\approxeq \mathbf{o}$ can be adopted in (26), which makes these differential equations change much simpler than algebraic ones, resembling an infinite-horizon problem, so that the unknowns $\mathbf{P}$ and $\mathbf{v}$ in (25) may now be obtained by resolving the much simpler algebraic equations [127],

$$\begin{array}{c}\mathbf{P}\mathbf{A}+{\mathbf{A}}^T\mathbf{P}+{\mathbf{C}}^T\mathbf{Q}\mathbf{C}-\mathbf{P}\mathbf{B}{\mathbf{R}}^{-1}{\mathbf{B}}^T\mathbf{P}=\mathbf{0}\\ \\ \mathbf{v}={\left({\left(\mathbf{A}-\mathbf{B}{\mathbf{R}}^{-1}{\mathbf{B}}^T\mathbf{P}\right)}^T\right)}^{-1}{\mathbf{C}}^T\mathbf{Q}\mathbf{r}.\end{array}$$

(27)

3.2.5. Synchronization with Prescribed Performance

Another useful adaptation is to convert an SDRE-tracking problem into a regulation one. At this time, the essential modification is implemented in the system dynamics instead of the cost function. It is based on synchronizing the dynamics established in (20) with a desired reference plant behavior dictated by another model, for instance, (23), by augmenting the state vector as ${\mathbf{X}}^T=\left[{\mathbf{x}}^T{\mathbf{z}}^T\right]$ and returning to a problem equivalent to (20) and (21). This approach is also suitable to rendezvous or formation flying maneuvers in which a chaser/follower body must track a target/leader one [99,118,120,121,128], or for regulation/synchronization of chaotic dynamical systems [100,124,126,129]. Nonetheless, one must remember that the implied assumption on (asymptotic) stability of the reference trajectory (23) might not always be guaranteed [126]. To overcome such mismatching, Ref. [130] proposed an arrangement in the standard LQR cost (21) that will assure the state $\mathbf{X}$ will converge to equilibrium as a result of setting the eigenvalues of the closed-loop matrix ${\mathbf{A}}_{\mathrm{CL}}\left(\mathbf{X}\right)$ (22) within desired/prescribed values, guaranteeing ${\mathbf{A}}_{\mathrm{CL}}$ will be point-wise Hurwitz, which is what [130] called “an optimization process with a prescribed degree of stability (pds)”, also known as discounted cost function.

The rationale behind this technique is to redefine the LQR cost (21) as [100]

$$P{I}_{pds}=\underset{t_0}{\overset{\infty }{\int }}{e}^{2\alpha t}\left({\mathbf{X}}^T\mathbf{Q}\left(\mathbf{x}\right)\mathbf{X}+{\mathbf{u}}^T\mathbf{R}\left(\mathbf{x}\right)\mathbf{u}\right)dt,$$

(28)

for some $\alpha <0$, so as to assure that, under the hypotheses of controllability of the dynamics (20), the original variable $\mathbf{x}$ of the SD (point-wise) linear model must exponentially decay faster than $e^{\alpha t}$ to assure the finiteness of $P{I}_{pds}$.

This approach is then developed by rewriting the suboptimal problem (20) and (21) augmented with (23), considering (28), using the augmented-modified state vector $\mathcal{X}=\mathbf{X}{e}^{\alpha t}$ and modified control vector $\mathcal{U}=\mathbf{u}{e}^{\alpha t}$, which brings back the equivalent easily workable standard LQR problem

$$\begin{array}{ll}\dot{\mathcal{X}}=& \left(\mathcal{A}\left(\mathbf{X}\right)+\alpha \mathbf{I}\right)\mathcal{X}+\mathcal{B}\left(\mathbf{x}\right)\mathcal{U},\mathcal{X}\left(t_0\right)={\mathcal{X}}_0\hfill \\ \hfill \mathbf{Y}=& \mathcal{C}\left(\mathbf{X}\right)\mathbf{X},\hfill \\ \hfill \end{array}$$

(29a)

$$\begin{array}{cc}\hfill P{I}_{pds}=& \underset{t_0}{\overset{\infty }{\int }}\left({\mathcal{X}}^T\mathbf{Q}\left(\mathbf{x}\right)\mathcal{X}+{\mathcal{U}}^T\mathbf{R}\left(\mathbf{x}\right)\mathcal{U}\right)dt\hfill \end{array}$$

(29b)

for the new system model matrices

$$\begin{array}{cccc}\hfill \mathcal{A}\left(\mathbf{X}\right)=& \left[\begin{array}{cc}\mathbf{A}\left(\mathbf{x}\right)& \mathbf{0}\\ \mathbf{0}& {\mathbf{A}}_r\left(\mathbf{z}\right)\end{array}\right],\hfill & \hfill \mathcal{B}\left(\mathbf{x}\right)=& \left[\begin{array}{cc}\mathbf{B}\left(\mathbf{x}\right)& \mathbf{0}\\ \mathbf{0}& \mathbf{0}\end{array}\right],\hfill \\ \\ \hfill {\mathbf{Y}}^T=& \left[{\mathbf{y}}^T{\mathbf{z}}^T\right],\hfill & \hfill \mathcal{C}\left(\mathbf{X}\right)=& \left[\begin{array}{cc}\mathbf{C}\left(\mathbf{x}\right)& \mathbf{0}\\ \mathbf{0}& {\mathbf{C}}_r\left(\mathbf{z}\right)\end{array}\right].\hfill \end{array}$$

The resulting optimal control law is then

$$\tilde{\mathcal{U}}\left(\mathcal{X}\right)=-{\mathbf{R}}^{-1}\left(\mathbf{x}\right){\mathcal{B}}^T\left(\mathbf{x}\right)\mathbf{P}\left(\mathbf{X}\right)\mathcal{X},$$

(30)

in which $\mathbf{P}$ is the symmetric and positive-definite solution of the modified ARE

$$\mathbf{P}(\mathcal{A}+\alpha \mathbf{I})+\mathbf{P}{(\mathcal{A}+\alpha \mathbf{I})}^T+{\mathcal{C}}^T\mathbf{Q}\mathcal{C}-\mathbf{P}\mathcal{B}{\mathbf{R}}^{-1}{\mathcal{B}}^T\mathbf{P}=\mathbf{0}.$$

(31)

In such approach the eigenvalues of the new dynamics matrix $\tilde{\mathcal{A}}=\mathcal{A}+\alpha \mathbf{I}$ in the enlarged system (29a) are guaranteed to be negative ($\tilde{\mathcal{A}}$ becomes Hurwitz) by subtracting the discount factor $\alpha$ from each of the eigenvalues of $\mathcal{A}$.

The effectiveness of the SDRE technique can be assessed by its extensive application in a wide variety of nonlinear control problems, ranging from aerospace [99,115,121,128,131,132,133] to industrial [112,122,134,135] and biological [136,137] systems. It can effectively handle input and state constraints [138] as well as disturbance rejection [118,127,139], paving the way for applications in control for chaos suppression on Micro- and Nano-electromechanical systems (MEMS and NEMS) [140,141], for non-ideal vibration in structures for energy harvesting mechanisms [142,143,144] and for synchronization of chaotic systems [124,126,129,145], among others. A thorough description of SDRE techniques, the related theories, and some examples can be found in [39,97,109,146].

4. Numerical Results

In what follows, we will apply the SDRE suboptimal control scheme to hover a spacecraft or make it track a reference trajectory in the disturbed environment around some of the NEAs depicted in Table 1. All the errors involved in the calculations are of the order of ${10}^{-14}$ for absolute and relative errors, as defined in the MATLAB^® environment that we used for the calculations, which means our numerical integrations imparted, for instance, position errors of at most ${10}^{-9}$ m (maximum distance $\sim {10}^5$ m) as happens in the case of trajectories around 16 (Psyche).

To show the effectiveness and flexibility of the SDRE suboptimal control scheme, we will initially use body-fixed hovering to maintain a spacecraft around the stable and unstable relative equilibria for the asteroid (16) Psyche. Then, we will apply this strategy to hover the equilibrium points of the main body, ALPHA, of the triple NEA system 2001SN263. The next task will be to track a reference orbit within the asteroid triplet as a strategy for an approaching phase mission of the asteroid system.

4.1. Hovering a Single-Body Symmetric Asteroid

To tackle the proposed problem of body-fixed hovering a spacecraft around a symmetric and regularly spinning asteroid, let us put the problem stated by Equation (1) in a planar ($z=\dot{z}=T_z=0$) state-dependent form.

Defining state $X={\left[x\dot{x}y\dot{y}\right]}^T$ and control $U={\left[u_x{u}_{\dot{x}}{u}_y{u}_{\dot{y}}\right]}^T$, we may adopt a planar version of Equation (1) and obtain the standard/direct SDC parameterization by writing

$$\dot{X}=A\left(X\right)X+B\left(X\right)U,$$

(32)

in which, from Equation (1) and [75], we obtain

$$A\left(X\right)=\left[\begin{array}{cccc}0& 1& 0& 0\\ A_{21}\left(X\right)& 0& 0& 2\omega \\ 0& 0& 0& 1\\ 0& -2\omega & A_{43}\left(X\right)& 0\end{array}\right],B\left(X\right)=\left[\begin{array}{cccc}0& 0& 0& 0\\ 0& 1& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 1\end{array}\right],$$

(33)

$$\begin{array}{cc}\hfill A_{21}\left(X\right)& ={\omega }^2-{\displaystyle \frac{\mu }{r^3}}+{\displaystyle \frac{3\mu r_0^2{C}_{20}}{2r^5}}+{\displaystyle \frac{6\mu r_0^2{C}_{22}}{r^5}}-{\displaystyle \frac{15\mu r_0^2{C}_{22}}{r^7}}(x^2-y^2),\hfill \\ \\ \hfill A_{43}\left(X\right)& ={\omega }^2-{\displaystyle \frac{\mu }{r^3}}+{\displaystyle \frac{3\mu r_0^2{C}_{20}}{2r^5}}-{\displaystyle \frac{6\mu r_0^2{C}_{22}}{r^5}}-{\displaystyle \frac{15\mu r_0^2{C}_{22}}{r^7}}(x^2-y^2),\hfill \end{array}$$

(34)

r is the distance of the spacecraft to the asteroid, $\mu$ is the asteroid’s gravity constant, $C_{20}$ and $C_{22}$ are the harmonic coefficients of the asteroid’s gravity field [76,78], and $r_0$ is a reference distance, here assumed to be the synchronous radius defined in (2). We used the discount factor $\alpha =-{10}^{-6}$, through the examples shown here. As can be seen in Figure 6, this value was enough to guarantee negative eigenvalues in the modified dynamics matrix in Equation (29a), thus avoiding numerical instabilities in the determination of control (30), while imparting marginal changes in the cost or performance index (29b).

4.1.1. Body-Fixed Hovering Around (16) Psyche

We now present the results from simulations for body-fixed hovering around the relative equilibrium points for the asteroid (16) Psyche shown in Table 1 and Figure 2. For this case we used $C_{20}=-0.0320$, $C_{22}=0.0069$ [76,78] and, from Equation (2), the reference synchronous radius is $r_0\sim$ 215 km, well outside the $\sim 140\mathrm{km}$-radius Brillouin circumference enveloping Psyche’s equator.

The evolution of the uncontrolled and controlled planar trajectories for a spacecraft in the vicinity of the stable equilibrium point $(x_s,y_s)=(0,\mathrm{213,712.06})$ in the $Oxy$ effective potential (3) for (16) Psyche are shown in Figure 7 (left). The two hovering maneuvers shown in the left picture were performed using (constant) control parameters $Q\left(X\right)={10}^{-3}diag(1,1,4,1)$ and $R\left(X\right)={10}^{5}diag(4,1,4,1)$, for the lower trajectory (dotted curve), meaning that errors in the x position coordinate are penalized 4 times less than errors in the y coordinate, while control efforts are equally penalized within the position variables. For the upper trajectory (dashed), we used $Q\left(X\right)={10}^{-4}diag(3,1,3,1)$ and $R\left(X\right)={10}^{9}diag(1,1,1,1)$, meaning a $10\times$ lower penalty for overall state errors and ${10}^4\times$ higher penalty for control effort. The corresponding (dashed, dotted) $U_{v_y}$ control efforts for these cases are shown in the picture at right. From these results, one can see the great flexibility the SDRE approach affords mission designers in trading control effort and convergence of the controlled trajectory, ranging from a faster (∼0.5 h) and almost direct trajectory (dashed) to a slower convergent (∼1 h) one (dotted) with ∼20 times lower cost (area delimited by the dashed and dotted curves at right).

In Figure 8, we present the trajectories and control-effort results for hovering around the unstable equilibrium point $(x_u,y_u)=(\mathrm{221,891.42},0)$ (red star), by this time adopting control strategies with different loads for the control-penalty matrices R and Q for two different maneuvers. In this case, we used $Q\left(X\right)={10}^{-5}diag(4,1,4,1)$ and $R\left(X\right)={10}^{9}diag(1,1,1,1)$ for the lightly controlled trajectory (dashed curve) while only $2\times$ smaller $Q\left(X\right)={10}^{-5}diag(2,1,2,1)$ and $1000\times$ smaller control-penalty index $R\left(X\right)={10}^{6}diag(1,1,1,1)$ for the tightly controlled trajectory (dotted curve).

4.1.2. Body-Fixed Hovering Around 2001SN263-ALPHA

We now present the simulation results for body-fixed hovering around the main body (ALPHA) of the triple NEA System 2001SN263. From the physical parameters of Table 1 we have $C_{20}\sim$$0.0193$, $C_{22}\sim$$0.0031$ [76,78] and $r_0\sim$$1480\mathrm{m}$. It must be observed that this synchronous radius is only marginally outside the body’s equatorial circumference, imposing strong limitations on the radial motion of a spacecraft approaching either the stable or the unstable relative equilibria as they are only a few tenths of a meter away from the body’s surface (see Figure 5). For this maneuver, we used the control parameters $Q\left(X\right)={10}^{-4}diag(5,1,5,1)$ and $R\left(X\right)={10}^{6}diag(1,1,1,1).$ The higher values for the control-effort penalty matrix, compared to those used for (16) Psyche, are intended to lower the control reaction in the highly disturbed force field around this smaller body, even though it is fainter. The evolution of the controlled and uncontrolled planar trajectories for a spacecraft in the vicinity of the stable equilibrium point $(x_e,y_e)=(0,1450.75)$ of the Effective Potential for 2001SN263-ALPHA is shown in Figure 9.

The evolution of the x-coordinate and x-control effort for this maneuver are shown in Figure 10.

We may note here that, considering the differences in the control cost coefficient matrix R, the smaller fuel consumption for the station using hovering/active control around this lighter body as compared to the same maneuver around the heavier (16) Psyche, is a characteristic already mentioned by [42] for the spacecraft NEAR-Shoemaker orbiting the massive NEA (433) Eros.

4.2. Hovering Within the Triple NEA System (153591) 2001SN263

The triple NEA system (153591) 2001SN263 was chosen as the target for the ASTER mission, the first proposed Brazilian Deep-Space Mission [40]. This system describes a heliocentric orbit with $1.99\mathrm{AU}$ semi-major axis, $0.48$ eccentricity, $6.7\mathrm{deg}$ inclination and a pericenter distance of $1.04\mathrm{AU}$ [147], such that it presents Earth minimum orbit intersection distance (MOID) of $0.05061\mathrm{VAU}$, marginally out of the PHA’s category (MOID $<0.05\mathrm{AU}$). Such orbital characteristics make this asteroid system an easily accessible target for a space mission.

The ASTER mission will consist of two different remote-sensing phases. During the first phase, the spacecraft will be in a hovering position at approximately 40 km from the primary body. The second phase will consist of a terminator orbit phase with very low demand for maneuvers [63]. Proper specification of approximation/hovering orbits near such small body systems is mandatory for ensuring the success of such visiting missions. This task usually involves the search for stable trajectories that can be used by a spacecraft to observe the triplet from a mid-range distance, with minimum fuel expenditure, aiming at data acquisition for better environmental understanding and model updates. There may then follow a search for a trajectory closer to the bodies, eventually internal to the moonlets’ orbits, or ever closer parking on a natural or artificial equilibrium point within the system.

4.2.1. System Environment

The primary (largest central body) of the system 2001SN263, named Alpha, is a spheroidal body measuring ∼$2.8\times 2.7\times 2.9\mathrm{km}$ (($x,y,z$) axis) and equivalent diameter of ∼$2.5\mathrm{km}$, with density $\sim 1.13{\mathrm{g}/\mathrm{cm}}^3$, mass $\sim 917.47\times {10}^{10}\mathrm{kg}$ and sidereal rotation period ∼$3.4\mathrm{h}$. The larger moonlet, Beta, is a slightly elongated body (∼$0.7\times 1\times 0.6\mathrm{km}$) and equivalent diameter of $\sim 0.8\mathrm{km}$, with density $\sim 1.0{\mathrm{g}/\mathrm{cm}}^3$, mass $\sim 24.04\times {10}^{10}\mathrm{kg}$ and sidereal rotation period $\sim 13.4\mathrm{h}$. The smaller moonlet, Gamma, is almost spherical with an equivalent diameter of $\sim 0.4\mathrm{km}$, density $\sim 2.3{\mathrm{g}/\mathrm{cm}}^3$, mass $\sim 9.77\times {10}^{10}\mathrm{kg}$ and sidereal rotation period $\sim 16.4\mathrm{h}$. Beta’s orbit around Alpha has a semi-major axis equal ∼$16.6\mathrm{km}$, orbital period of ∼$149\mathrm{h}$, eccentricity $0.015$ and is almost aligned with Alpha’s equatorial plane. The orbit of Gamma has a semi-major axis of ∼$3.8\mathrm{km}$, eccentricity $0.016$, an inclination of $\sim 15\mathrm{deg}$ relative to Alpha’s equatorial plane and an orbital period of ∼$16.5\mathrm{h}$, which makes the body’s rotation almost tidally synchronized with its orbital motion [74,147].

As the system is composed of bodies with similar physical characteristics, such as size and density, and are close to each other, a spacecraft traveling in their neighborhood will experience perturbations arising from all the bodies. This will be decisive in determining the regions of stability and instability inside and outside the system, a feature whose understanding and characterization are essential for the designing of rendezvous mission aiming to explore this NEA system [59,63,86,148].

Ref. [43] mapped long-lasting orbits around the main body of the triplet. The map was built from a grid of initial conditions concerning semi-major axis and eccentricity. The orbits were numerically integrated for a period of 500 days, double the time planned for the ASTER mission. The model includes perturbations from the oblateness of Alpha, the gravity field of the two secondary bodies, the Sun, the Moon, the asteroids Vesta, Pallas, and Ceres, and all the planets of the Solar System. These perturbations were considered individually to identify their relative strength. The results showed that Gamma is the main perturbing body, followed by Beta, 10 times smaller, and the group Sun–Mars–Alpha’s oblateness, with perturbations 1000 times smaller than that of Gamma. The other bodies have perturbations ${10}^7$ times smaller. The results also showed that circular and polar orbits are less perturbed when compared to elliptical and equatorial ones. Regarding the semi-major axis, an internal orbit is the best choice, followed by a larger external orbit. The inclination of the orbit plays an important role in their model, so it is necessary to make adequate orbit selection to attain minimum disturbance.

4.2.2. Tracking Long-Lasting Orbits Within the System

We now consider the case of observing one or more of the bodies in the triplet from trajectories that may not be connected to the relative equilibria engendered within the triplet. Admitting that the spacecraft will navigate well inside ALPHA’s Hill radius $R_H\sim 180\mathrm{km}$, only the forces due to the gravitational interaction within the triplet members will be considered. Following [43], other external perturbations were not incorporated as their effects, four orders of magnitude smaller than the internal forces, will not impact the results for the adopted two-month time span.

In this case, we will use the model called “Processing Inclined Bi-Elliptical 4BP with Radiation Pressure—PIBEPRP” developed by [61], in which a Restricted Four-Body Problem, consisting of a spacecraft orbiting the triple system 2001SN263, is separated in two interacting Two-Body Problems and also including the Solar Radiation Pressure and the precession engendered by the oblateness of the central body in the noncoplanar orbits of the two moonlets. Given the coordinates $(x,y,z)$ of the spacecraft with respect to the central body (ALPHA), $(x_{*},y_{*},z_{*})$ the coordinates of asteroid $“{\ast }^{”}$ related to an inertial reference and $r_{*}$ the distance from the spacecraft to the asteroid $“{\ast }^{”}$, we define the respective distances

$$\begin{array}{cc}\hfill r& =\sqrt{x^2+y^2+z^2}\hfill \\ \hfill r_{\alpha }& =\sqrt{{(x-x_{\alpha })}^2+{(y-y_{\alpha })}^2+{(z-z_{\alpha })}^2}\hfill \\ \hfill r_{\beta }& =\sqrt{{(x-x_{\beta })}^2+{(y-y_{\beta })}^2+{(z-z_{\beta })}^2}\hfill \\ \hfill r_{\gamma }& =\sqrt{{(x-x_{\gamma })}^2+{(y-y_{\gamma })}^2+{(z-z_{\gamma })}^2},\hfill \end{array}$$

(35)

and the equations governing the motion of the satellite are written as [61]

$$\begin{array}{cc}\hfill \ddot{x}& =-{\mu }_{\alpha }{\displaystyle \frac{(x-x_{\alpha })}{r_{\alpha }^3}}-{\mu }_{\beta }{\displaystyle \frac{(x-x_{\beta })}{r_{\beta }^3}}-{\mu }_{\gamma }{\displaystyle \frac{(x-x_{\gamma })}{r_{\gamma }^3}}+\hfill \\ \hfill & -{\displaystyle \frac{3{\mu }_{\alpha }{J}_{2\alpha }{r}_{\alpha }^2}{2r^5}}\left(1-{\displaystyle \frac{5z^2}{r^2}}\right)·x+A_x\hfill \\ \hfill \\ \hfill \ddot{y}& =-{\mu }_{\alpha }{\displaystyle \frac{(y-y_{\alpha })}{r_{\alpha }^3}}-{\mu }_{\beta }{\displaystyle \frac{(y-y_{\beta })}{r_{\beta }^3}}-{\mu }_{\gamma }{\displaystyle \frac{(y-y_{\gamma })}{r_{\gamma }^3}}+\hfill \\ \hfill & -{\displaystyle \frac{3{\mu }_{\alpha }{J}_{2\alpha }{r}_{\alpha }^2}{2r^5}}\left(1-{\displaystyle \frac{5z^2}{r^2}}\right)·y+A_y\hfill \\ \hfill \\ \hfill \ddot{z}& =-{\mu }_{\alpha }{\displaystyle \frac{(z-z_{\alpha })}{r_{\alpha }^3}}-{\mu }_{\beta }{\displaystyle \frac{(z-z_{\beta })}{r_{\beta }^3}}-{\mu }_{\gamma }{\displaystyle \frac{(z-z_{\gamma })}{r_{\gamma }^3}}+\hfill \\ \hfill & -{\displaystyle \frac{3{\mu }_{\alpha }{J}_{2\alpha }{r}_{\alpha }^2}{2r^5}}\left(3-{\displaystyle \frac{5z^2}{r^2}}\right)·z+A_z\hfill \end{array}$$

(36)

in which ${\mu }_{*}=\mathit{G}{M}_{*}$, $M_{*}$ is the mass of asteroid “∗”, $\mathit{G}=6.674287\times {10}^{20}{\mathrm{km}}^3{\mathrm{kg}}^{-1}{\mathrm{s}}^{-2}$ is the universal gravitational constant and the vector $(A_x,A_y,A_z)$ stands for the accelerations imparted to the spacecraft, comprising control accelerations and the disturbances that may considered in the model.

Figure 11 and Figure 12 depict representative characteristics of the system geometry and dynamics. Figure 11 shows a “long-lasting” orbit of a spacecraft (black) with the initial position in direct opposition to the secondary body Beta (the stars mark initial positions). The time span corresponds to 5 orbits of Beta (around 30 days). Figure 12 shows the disrupting case of a spacecraft’s orbit with initial position $\pi /5$ ahead of Beta escaping after a time span of 15 days. Figure 13 presents the respective satellite-asteroids distances, showing the two close approaches of the spacecraft with Beta prior to escaping from the system in the second case (b).

We made then use of the SDRE control to make a spacecraft track a long-lasting trajectory shown in Figure 14, in which we may consider the spacecraft as a co-orbiting satellite of Beta as it occupies the same orbit 180deg shifted from the moonlet, similar to the case shown in Figure 11. In Figure 14, we also show the effect of considering different penalty matrices for tracking reference trajectories. For the first case, we used $Q={10}^{-6}diag(1,1,1,1,1,1)$ and $R={10}^{8}diag(1,1,1,1,1,1)$. For the second case, the adopted coefficients are $Q={10}^{-4}diag(1,1,1,1,1,1)$ and $R={10}^{6}diag(1,1,1,1,1,1)$, meaning a higher cost/penalty $P{I}_{pds}$ (see Equation (28)) for higher values state errors (higher values of $\parallel \mathbf{X}\parallel$ in Equation (29b)), thus imparting faster convergence $\parallel \mathcal{X}\parallel \to 0$, while allowing lower cost/penalty (higher values for $\parallel \mathcal{U}\parallel$), thus allowing higher fuel expenditure.

Figure 15 shows the control effort for the referred associations of penalty matrices. Here, also, one can observe the large amplitude of choices afforded by SDRE through the selection of the matrix coefficients in Equation (29b), resulting in trajectories converging almost $10\times$ faster (thicker curve in Figure 14), but with more than $100\times$ cost.

As an intermediate case, we present in Figure 16 a maneuver considering a trajectory starting some kilometers away from Alpha and targeting the stable equilibria $\sim 100\mathrm{m}$ above the asteroid’ surface. For this case we used $Q\sim {10}^{-2}$ and $R\sim {10}^9$.

5. Conclusions

In this work, motivated by the challenging astrodynamic environment, as quoted by Scheeres [36] “among the most extreme found in the Solar System”, we perform a qualitative assessment on how the State-Dependent Riccati Equation (SDRE) suboptimal control formulation would behave in the highly nonlinear and strongly perturbed environment within multi-body NEA systems, to the authors’ knowledge a never before undertaken. We made a qualitative assessment on the suitability of the implementation of SDRE in body-fixed hovering and reference-trajectory tracking of a spacecraft in the highly nonlinear environment around Near-Earth Objects (NEOs) [38], a class of small and faint objects of the Solar System with high relevance either in scientific, economic, or planetary defense-related aspects, targets of various rendezvous missions executed/planned by different space agencies [149]. As stated before, we intended to prove the suitability of using the SDRE approach in the highly disturbed environment around Small Solar System Bodies. For real missions, it is necessary to make deeper studies. In our numerical results, the absolute and relative errors are of the order of ${10}^{-14}$, as defined in the MATLAB^® environment we used for the calculations, which means our numerical integrations imparted, for instance, position errors of at most ${10}^{-9}$ m (maximum distance $\sim {10}^5$ m, see Figure 8).

In our results, the great flexibility offered by the state-dependent (SD) factorization in the specification of the control-effort or state-error coefficient matrices, trading off the convergence of the controlled trajectory against fuel consumption, already known in many fields as cancer treatment, satellite or robotic formation dynamics, or in energy harvesting, confirm that this control approach is also highly effective when addressing the hovering of a spacecraft around NEO, as we demonstrated in the cases of body-fixed hovering around the NEAs (16) Psyche and the much smaller main body of the triple system (153591) 2001SN263, or in the highly stringent case of trajectory tracking in the highly nonlinear environment within the triple NEA system (153591) 2001SN263, all envisaging the best possible trade-off between time of convergence and fuel consumption, a central aspect in such heavenly missions.

This effectiveness can be observed particularly in Figure 7, which depicts control effort and time of convergence of the controlled trajectories around (16) Psyche ranging from a half-time convergence with ∼20 times lower cost. Figure 14 and Figure 15 show convergence and control cost cases for (153591) 2001SN263’s depicting a trajectory converging 10 times faster but with up to 100 times higher cost.

As a future extension to this work, we suggest tackling the inertial hovering approach around the Lagrange points of the triplet 2001SN263 and a sensitivity analysis considering the influence of uncertainties engendered by errors in the physical parameters of the bodies [79,150]. We also intend to perform an analysis/determination of the basins of attraction for the system, a significant aspect considering closer observation orbits or touch-and-go and descent maneuvers.