Adaptive Attitude Synchronization Control of Spacecraft Formation with Adaptive Synchronization Gains

T HIS Note is concerned with the time adaptation of the synchronization gains used in the attitude synchronization control problem. The problem under consideration assumes that a network of N spacecraft can exchange information through an appropriate communication topology and the objective is to synchronize the spacecraft via the appropriate choice of the control inputs so that each spacecraft has identical orientation. This synchronization objective is quantified via the pairwise difference of the orientation states, chosen here as the modified Rodrigues parameters [1,2]. The contribution in this Note follows earlier work [3,4], but with the difference that it does not parameterize the plant parameters and does not attempt to use adaptation of the parameters to design the control laws. Instead, the synchronization gains, used for synchronization penalty, are adapted. The idea behind this time adaptation of the synchronization gains is that the gains used to penalize the pairwise mismatch of the orientation states among the spacecraft should be proportional to the difference between them. This idea of using adaptive adjustment of the synchronization gains was first examined in [5–9] for general finite dimensional systems and in [10] for a class of infinite dimensional systems. In essence, one adjusts the strengths of the interconnections among network nodes in accordancewith adaptive policies. These strengths are represented by the nonzero elements of the time-varyingLaplacianmatrix associated with the networked systems [5]. Before considering the adaptation of the synchronization gains, we revisit the case of constant synchronization gains to explore the optimization of appropriate performance metrics that produce optimal constant synchronization gains. We consider two cases in which the local weights (synchronization gains) of the synchronization signalmay differ for each spacecraft (termed the node-dependent gains or vertex-dependent gains) ormay differ for each of its communicating neighbors (termed the edge-dependent gains). In essence, the third case, serving as the default case, is the one considered in earlier work [3]. The control design and stability for both cases of these constant synchronization gains are examined and an algorithm for obtaining these gains in proportion to the initial pairwise difference of spacecraft states is presented. Additionally, it is shown that the node-independent case (default case) and the node-dependent gains case are special cases to the more general edge-dependent synchronization gain case presented here. The simple optimization of these synchronization gains aims at reducing the transients of the rotational kinetic energy while at the same time trying to improve synchronization. For the latter, an appropriate measure of spacecraft attitude agreement, termed the deviation from themean, is used as the metric for examining success of synchronization. The adaptive adjustment of the synchronization gains leads to combined control and adaptation laws and their stability and convergence properties must be examined. The specifics of the proposed adaptive modification assume that the synchronization signal contains a fixed synchronization gain but the torque inputs for each spacecraft areweighted by a local adaptive synchronization gain. The requisite adaptation laws for the synchronization signals are extracted using Lyapunov redesign methods. The problem formulation along with a summary of established results on attitude dynamics, attitude synchronization, and graph theory are presented in Sec. II. The proposed modifications to the attitude synchronization problem that include constant edgedependent gains is presented in Sec. III, and some guidelines for the selection of the edge-dependent gains satisfying some optimality criteria are summarized in Sec. IV. The main result on the use of adaptive edge-dependent synchronization gains in the control torques is presented in Sec. V, and numerical studies that provide further insight on both the optimality and the adaptation of the synchronization gains are given in Sec. VI. Conclusions with future work follow in Sec. VII.