1. Introduction
CubeSats have seen an increase of popularity in recent years. Born as an educational platform inside university courses [
1], in the last decade they became a low-cost high reward platform to perform in-orbit technology development [
2], scientific missions [
3], and interplanetary exploration [
4]. The increasing popularity of this kind of platform has generated interest on developing miniaturized technologies to extend their capabilities.
One of the key areas of interest is providing them with maneuverability, momentum control, and orbit adjustment. Traditionally, propulsion systems have been utilized for attitude control and for delta-V maneuvers; however, most CubeSats do not include on-board propulsion systems due to volume, mass and safety limitations. For this reason, alternative solutions are under investigation for orbital and re-entry applications, among others. One of the possible solutions is based on the concept of exploiting forces and torques that are usually considered disturbances. Similar to the idea of using the magnetic field of the Earth for attitude control; atmospheric drag can be exploited in Low Earth Orbit (LEO) by using variable shape devices to modify the experienced aerodynamic acceleration and torque.
Changes in orbital velocity enable the capability of performing maneuvers in the same orbital plane, like formation flying or rendezvous and docking, without using on-board propulsion systems. The introduction of differential drag for formation flying control dates back to 1989, when the Clohessy–Wiltshire (CW) linear equations for relative motion between two spacecraft were used to design an algorithm to control the relative in-plane motion [
5]. to regulate the states with a discrete input. An improved solution based on the Schweighart–Sedwick (SS) linear equations for relative motion that include the
perturbation were introduced in [
6]. In [
7], the generation and use of lift to enable out-of-plane maneuvering capability was implemented. A Lyapunov-based control strategy was used in [
8] to achieve spacecraft rendezvous using differential drag. In [
9], the attitude of a spacecraft was used to change the experienced drag instead of dedicated actuators for drag surfaces. An adaptive sliding mode strategy was used in [
10] to control the relative dynamics using a continuous differential drag input. In [
11], a constrained least squares problem is formulated to find the best achievable set of individual inputs to control a set of spacecraft consisting of multiple chasers and a single target under mutual constraints and actuator saturations. Drag-based algorithms and techniques have been successfully demonstrated in orbit by the ORBCOMM constellation of satellites [
12] to save propellant in thruster-based formation keeping maneuvers, and the Planet Labs satellites constellation [
13] for propellant-less phasing maneuvers along the same orbit.
Variable shape devices consisting of several independent surfaces can be installed on the spacecraft, locating the center of pressure at distances with respect to the Center of Mass (CoM) such that significant torques can be applied. This feature allows exploiting aerodynamic forces for attitude control maneuvers. In [
14], a variable shape device has been used to control the attitude for an earth observation satellite mission; while in [
15], an on-line parameter estimation procedure has been used to improve the performance in the presence of environmental and spacecraft uncertainties. In [
16], several independent surfaces have been used to control the attitude of a spacecraft also during the re-entry phase, using a direct force control method in substitution of the classic bank angle modulation control.
Variable shape devices can also be used to de-orbit satellites from Low Earth Orbit [
17]. They can also be used to perform a controlled re-entry maneuver to target a precise location on the ground [
18], and be able to sustain the aero-thermodynamic loads [
19] typical of a deployable capsule re-entry profile [
20].
The University of Florida Advanced Autonomous Multiple Spacecraft (ADAMUS) laboratory has designed the Drag Maneuvering Device (DMD) [
21]. The capabilities provided by the DMD have been previously studied for controlled re-entry, as well as independent orbital and attitude maneuvering applications. In this paper, relative orbit and attitude adaptive controllers are integrated to perform a propellant-less roto-translational maneuver involving DMD-equipped CubeSats. The capability of changing the aerodynamic torques as well as the gravity gradient torque are considered in the attitude dynamics, whereas the effect of differential drag is exploited in the spacecraft relative dynamics. The adaptive controllers compensate for uncertainties in environmental and physical parameters such as atmospheric density, drag/lift coefficients, location of the Center of Mass (CoM) and inertia matrix. Uniformly ultimately bounded convergence is obtained through Lyapunov-based stability analysis for the integrated roto-translational system.
The paper is organized as follows:
Section 2 presents the translational and attitude spacecraft dynamics,
Section 3 shows the individual adaptive controllers and Lyapunov-based stability result when considering attitude-orbit coupling.
Section 4 and
Section 5 present results from a simulation example of two-spacecraft along-orbit formation with simultaneous attitude control, and concluding remarks, respectively. The foremost contributions of this paper are:
Design and verification through numerical simulation of an adaptive controller capable of simultaneously achieving three-axis attitude stabilization and in-plane relative maneuvering.
Compensation for physical and environmental uncertainties such as drag/lift coefficients, atmospheric density and CoM location.
Guaranteed ultimately bounded stability through Lyapunov-based analysis in the presence of uncertainties and perturbations.
4. Simulation Results and Discussion
The simulation scenario for this maneuver considered a set of two identical DMD-equipped CubeSats (i.e., one chaser and one target) with their physical parameters shown in
Table 1. The translational states were propagated for each spacecraft independently using Equations (5)–(7) and transforming to the LVLH relative states to compute (35) and (36). The atmospheric density used for attitude and orbit propagation, and considered unknown for the controllers, was obtained by using the NRLMSISE-00 model. The nonlinear attitude dynamics (i.e., Equation (
12)) were simultaneously propagated and coupling with the translational dynamics was explicitly included in the computation of the attitude-dependent cross-sectional area, and the analytical models for the drag and lift coefficients [
33].
The spacecraft were required to perform a phasing maneuver while controlling their attitude to achieve the desired orientation. The desired orientation (i.e.,
and
) was computed to be equivalent to a regulation maneuver with respect to the LVLH frame. For visualization purposes, the initial and desired orientations were expressed in 3-2-1 Euler angle representation with respect to the LVLH frame and are shown in
Table 2 and
Table 3 for both spacecraft, where
are rotations with respect to
and
, respectively. The initial orbit was identical for both spacecraft and its orbital elements are shown in
Table 4. The objective for the relative controller was to perform an along-orbit formation maneuver where the desired inter-spacecraft separation was 4 km. To achieve the along-orbit maneuver, a modified reference frame that had a position offset with respect to the LVLH with origin at the CoM of the chaser spacecraft is considered. A desired user-defined along-orbit distance
was specified, which can be expressed as an offset in true anomaly
with respect to that of the chaser spacecraft as
where
is the true anomaly of chaser’s orbit, then the orbital elements for the origin of the new reference frame are the same as those of the chaser but adding
to the true anomaly and the control objective remains to regulate
to zero.
The control and adaptation laws in (27), (30), (35) and (36) could be computed every time step by each spacecraft since they did not require any iterative algorithm. The function minimization problem in (37) was solved every 30 s by each spacecraft to produce a new set of DMD lengths that considered the required total cross-sectional areas and environmental torques. This update rate for the DMD lengths could be reduced; however, 30 s provided a practical balance between the transient response and the computational demands. In the numerical simulation, the function minimization algorithm was implemented using the fmincon command in MATLAB and the average time required to obtain a solution was 0.3152 s using a Windows laptop, 2.7 GHz quad core Intel Core i7 processor, and 16 GB RAM. Since the relative maneuvering control law computed the cross-sectional required by the chaser, the target spacecraft was tasked with the only objective of attitude control and to broadcast its ECI states and cross-sectional area. The control gains are presented in
Table 5 along with the weights
and
, where
is defined as
Figure 4 presents the relative states of the target expressed in the LVLH frame with origin at the CoM of the chaser spacecraft. In this simulation, the chaser required approximately 55 h to enter the ultimate bounds of
and
meters along the directions
and
, respectively, as compared to the convergence to a circle of 10 m radius around the desired position obtained with the individual, decoupled counterpart in [
29]. The resulting residual error was expected due to the ultimately bounded stability result obtained in the coupled attitude-orbit case. The uncertain parameters for the relative maneuvering controller are presented in
Figure 5, where dynamic variation of the estimates to compensate for the uncertainties can be observed.
Figure 6 shows the actual and desired quaternion and
Figure 7 presents the actual and desired angular velocities. Note that although the required orientation was fixed with respect to LVLH, the desired
was time-varying and
was different from zero; therefore, the controller was really performing a tracking maneuver.
Figure 8 shows the resulting orientations in Euler angle representation for the target and chaser. Although the orientation of both spacecraft had the same initial conditions, and the control parameters were also the same, the resulting orientation had different transients and ultimate bounds. This behavior can be explained by the fact that the chaser spacecraft was the only one in charge of satisfying the required cross-sectional area for the relative orbit maneuver. The only goal of the target was to regulate its attitude and to broadcast its cross-sectional area and ECI states to the chaser so that it could compute (
33). While the target required 4.8 h to enter the ultimate bounds of
and
degrees of error in roll, pitch and yaw, respectively, the chaser required 43 h to enter the ultimate bounds of
and
degrees of error in roll, pitch and yaw, respectively. The resulting ultimate bounds showed an increase with respect to the individual, decoupled counterpart in [
30] of
and
in roll, pitch and yaw, respectively. The observed increase was expected due to the additional requirement of achieving a specific cross-sectional areas for the relative orbit maneuver.
The required control inputs are presented in
Figure 9 for the target and chaser. Saturation due to the physical limitations of the DMD lengths was applied in the simulation. The DMD lengths reach saturation levels at the beginning of the maneuver where compensation for the initially large errors was required. Significant differences in the DMD length profiles could be observed between target and chaser. The chaser spacecraft required changing the total cross-sectional and simultaneously generate the required torques using only the DMD surfaces, which can be observed in
Figure 9. The DMD surfaces, in the case of the chaser spacecraft, were significantly more active than those of the target. This behavior was especially noticeable when the requirements of the relative orbit controller were demanding (i.e., required cross-sectional area at or near saturation levels), the changes between maximum and minimum area could be observed during the first 40 h. To avoid instantaneous and fast changes of the DMD lengths due to the 30 s interval between computations, the points were joined with splines and passed through a low-pass filter with cut-off frequency
of 0.017 Hz. The maximum resulting deployment rate required from an actuator during the peak of control demand was 2.5 m per minute, as illustrated in
Figure 10.
The parameter estimates for the attitude controller of the target are presented in
Figure 11 the parameters associated with the aerodynamic drag, and
Figure 12 those associated with the aerodynamic lift, respectively. Similarly, in the case of the chaser,
Figure 13 shows the estimated parameters associated with aerodynamic drag and
Figure 14 presents those associated with aerodynamic lift, respectively. The estimates were adjusted by the adaptive update law to compensate for the uncertainties. Some estimates exhibited a diverging trend as compared to others, this behavior can be explained by the fact that the ultimately bounded results obtained from both controllers ensured convergence of the states but not necessarily of the estimates. Inside the ultimate bound, an increase of the Lyapunov function could indicate that the norm of the estimates was increasing, generating the behavior observed in some estimates. Although this behavior did not affect the stability result, in case the estimates kept growing, the continuous projection algorithm used in the adaptive update laws would keep them bounded. However, during the 70 h of simulation, these bounds were never reached.
To evaluate the influence that the applied inputs may have on the flexible DMD surfaces, the first natural frequencies of a fully deployed DMD surface, modeled as a catilevered beam, were computed using SolidWorks and are presented in
Table 6. Since the attitude dynamics were faster than the translational dynamics, the Fast Fourier Transform (FFT) of the applied torques were computed for both spacecraft and are presented in
Figure 15. The range of frequencies of the applied torques were reasonably below the first natural frequency of the DMD surface.
5. Conclusions
The obtained results have shown that it is feasible to perform simultaneous orbital and attitude maneuvers by using the Drag Maneuvering Device as the only control means in conditions where physical and environmental parameters are uncertain.
In particular, adaptive controllers for orbital and attitude maneuvers were integrated to perform roto-translational maneuvers in the presence of uncertainty in atmospheric density, drag/lift coefficients, location of the Center of Mass and inertia matrix. The uniformly ultimately bounded convergence of the attitude error and relative orbit states is guaranteed by the Lyapunov-based stability analysis. Validation through numerical simulation of a phasing maneuver with simultaneous attitude control requirement results in ultimate errors below 60 m and 6.5 degrees for the translational and attitude maneuvers, respectively.
The algorithms developed together with the Drag Maneuvering Device could be particularly useful for providing small platforms, such as CubeSats, with propellant-less translational and attitude maneuverability. In-orbit inspection and servicing are envisioned as future applications for this technology.
Future work will explicitly include the influence of flexible bodies on the spacecraft dynamics and will be considered in the controller design. Studies on how to incorporate compensation for time-varying uncertain parameters, as well as efforts on relaxing the requirement of using numerical algorithms to solve for the DMD lengths, are considered of great importance to improve the obtained stability result. Opportunities for improving the distribution of the control effort between chaser and target without necessarily centralizing the control algorithm have also been identified as topic of interest for future research efforts.