Distributed Continuous-Time Optimization on the Special Orthogonal Group SO(3) over Tree Interaction Graphs

Abstract

We study continuous-time distributed optimization on the three-dimensional rotation group over undirected tree graphs, where agents with heterogeneous local costs seek a common attitude minimizing the geodesically strongly convex sum of their costs on a geodesically convex operating ball. We propose a nonsmooth protocol with fixed gains that combines local Riemannian gradient descent with signum-based consensus feedback computed from the Lie-group logarithms of relative rotations between neighboring agents. Since geodesic strong convexity alone does not guarantee invariance, as demonstrated by a certified counterexample, we impose an inward-pointing boundary condition and prove that every Filippov solution with all agents initialized in the ball remains there. We construct a nonsmooth intrinsic disagreement Lyapunov function and derive a uniform dissipation bound through a cluster-contraction argument exploiting the tree structure. With bounded local gradients, this bound gives an explicit gain-ratio condition under which we establish exact finite-time consensus for every such solution and obtain a settling-time estimate requiring no structural graph constant beyond the number of agents. After consensus, we identify the sliding dynamics as a scaled Riemannian gradient flow of the summed objective. When the minimizer lies in the interior of the ball, the dynamics converge to it exponentially in geodesic distance. Simulations illustrate our theoretical results.