Maclaurin Hutagalung
Date: 31 July 2008
There are increasingly many researches in the field of underactuated systems, i.e., systems with fewer actuators than the system's degrees of freedom. Several methods have been successful to derive control laws for certain classes of underactuated systems, e.g., (1,2,3,4,5,6). Some of these are based on the widely known result of Sussmann (7).
Sussmann (7) derived a sufficient condition for small-time local controllability (STLC) of a nonlinear system. This work was a generalization of some earlier results (see (7) for details). Many control laws for underactuated systems were proposed based on (7). For example, Lewis and Murray (8) introduced small-time local configuration controllability (STLCC) notion and derived sufficient conditions for simple mechanical systems starting from zero velocity. This result has motivated several authors, e.g., (1,6), to exploit the STLCC conditions. Furthermore, the results in (2,9,10) used the local controllability theorem in (7) and the recursion relation in (11) to design motion algorithms on Lie groups for underactuated systems.
Spacecraft with two thrusters is one example of underactuated systems. Typical problem of spacecraft control is to control its attitude, either to achieve one desired attitude or to follow a desired attitude trajectory. Depending on the spacecraft's actuators, torque may be generated by thrusters, reaction wheels, control moment gyros, etc.
The first general procedure for design and analysis of attitude control systems came in early 70's (12). Based on Meyer (12) and Lie algebraic methods in Hermes (13), Crouch (14) derived necessary and sufficient conditions for controllability of the underactuated spacecraft and constructed an algorithm which locally asymptotically stabilizes the spacecraft around an equilibrium point. The algorithm yields a piecewise constant discontinuous control inputs. Krishnan et al. (15) used the result in (7) to derive a new discontinuous stabilizing feedback control strategy which is simpler than the one in (14).
An article by Samson (16) has shown that continuous, time-varying feedback can stabilize many classes of systems which cannot be stabilized by continuous feedback. This result has triggered many researches on time-varying feedback to solve underactuated spacecraft control problem, e.g. (17,18,19,20,22,23,24). Morin et al. (17) used center manifold theory combined with time-averaging and Lyapunov techniques to construct smooth time-varying feedback that asymptotically stabilizes spacecraft attitude with two controls. Coron and Keraï (18) generalized the case in (17), in which the torque actions are exerted about the principal axes of the inertia matrix of spacecraft. In (19), Morin and Samson improved the polynomial asymptotical convergence rate in (17) and obtained exponential asymptotical convergence rate. While the solution proposed by Coron and Keraï (18) also achieves exponential stabilization, the solution proposed in (19) has a simpler control law.
Tsiotras and Doumtchenko (20) and Tsiotras and Luo (21) provided feasible trajectory generation for an underactuated spacecraft using ideas from differentially flat systems. These feasible trajectories can be used as a reference trajectories in attitude tracking problems. Behal et al. (22) used Lyapunov-based controller and backstepping control techniques for an attitude tracking problem for an underactuated spacecraft. The controller for the kinematic model of an underactuated spacecraft achieved uniformly ultimately bounded tracking provided the initial tracking errors are selected sufficiently small. Morin and Samson (23) used a smooth function, i.e., transverse function approach, to practically stabilize any reference trajectory of configurations with pre-defined precision.
Recent paper by Karami and Sassani (24) proposed a procedure, using Lyapunov and a modified form of the backstepping method, to correct and stabilize the orientation of a spacecraft with only two available control torques subjected to an external disturbance torques. Nordkvist and Bullo (9) extended the results in (2) to compute small amplitude control forces to accelerate, decelarate, or stabilize an underactuated system along a relative equilibrium. This work (9) was also a nice extension of (25), where (25) only considered stabilization.
>From all the methods mentioned above, none of them paid attention to amounts of control effort. Tsiotras and Luo (21,26) proposed a control law that not only stabilizes the kinematics of an underactuated spacecraft but also reduces the control effort. In (21), the authors improved the result in (26) by specifiyng a priori bound for control inputs.
So far, the results presented above deal with the stabilization of the zero equilibrium of the spacecraft attitude. A new result from Casagrande et al. (27) considered the stabilization of both the attitude and the angular velocities of an underactuated non-symmetric spacecraft simultaneously. They used switching control scheme and almost non-increasing (ANI) function to prove global asymptotic stability of the attitude and the angular rates of an underactuated non-symmetric spacecraft.
We see that the development in control theory for single spacecraft has been well established. Now, researchers are more interested in developing control theory for systems with more than one spacecraft working together. There are many advantages for using multiple spacecraft, such as for large-scale measurements and interferometry. Nijsc.ejer (28) tried to give a dynamical control view on synchronization. He showed that observer and feedback control are essential to achieve synchronization between two systems on the basis of partial state measurements of one of the systems. Hanßmann et al. (29) presented reduction theory for two rigid bodies. These rigid bodies are coupled by a control law that depends only on their relative configuration. One advantage that comes from the proposed reduction technique is to reduce dimensionality of the systems, which in turn reduces the complexity of the systems.
In the last decade, the number of researches about spacecraft synchronization has increased, e.g., (30,31,32,33,34). Specifically, Wang et al. (30) and Lawton and Beard (31) designed control strategies based on Lyapunov method. The main difference between their methods is that in (30) they used one spacecraft as a reference, while in (31) they used two closest spacecraft as a reference. Bondhus et al. (32) proposed a controller to synchronize two satellites when angular velocity measurements are not available by using nonlinear observers to estimate the angular velocities. Sarlette et al. (33,34) considered attitude synchronization as a consensus problem on manifolds. In (33), they designed control laws to synchronize identical rigid bodies when the available information between the rigid bodies is restricted to relative orientations and angular velocities. In (34), they focused on states that are far away from the desired equilibrium. This approach is useful for initial deployment of formation or for after strong perturbations to the formation. To the best of our knowledge, there is no result of spacecraft synchronization for the underactuated case.
While there is no result of spacecraft synchronization for the underactuated case, there are some results which consider the case of underactuated multi-link spacecraft, e.g., (35,36). Rui et al. (35) proposed controllers to achieve desired attitude and shape change maneuvers of underactuated multi-body spacecraft using similar approach as in (10). The controllers were derived by assuming zero angular momentum of the system. Ashrafiuon and Erwin (36) extended the results in (35) by proposing a controller that can stabilize a tumbling satellite, which has constant nonzero angular momentum. It seems that there is a connection between synchronization of multiple spacecraft and multi-link spacecraft control problem. We may see a multi-link spacecraft as multiple spacecraft coupled with artificial control inputs between the individuals. An initial result in this topic may be found in (29).
In this research, we adopt a method in (2) and (4) to synchronize attitudes and angular velocities of two underactuated spacecraft without referring to any leader or external reference. Our idea is to use approximate evolution and adopted control law from (2) to bring the final attitudes and angular velocities of two spacecraft close to each other. From that point, we use another adopted control law from (2) to drive the spacecraft such that the differences of attitudes and angular velocities go to zero as time goes to infinity.