Dynamical Pose Estimation

The paper’s Theorem 9 states that for point cloud registration under the DAMP dynamics there are exactly four equilibria, a unique one minimizes the potential energy, the other three differ from it by a π rotation, and those three are locally unstable; hence DAMP converges to the optimal equilibrium. The main text asserts convergence to an equilibrium by damping/Lyapunov theory and defers the detailed proof to the Supplementary Material, which carries out an SVD-based analysis of the stationarity condition and a direct energy-decrease argument around the spurious equilibria . The supplement derives equilibrium conditions, identifies the optimal solution R? = U+V+^T, shows the four diagonal-sign solutions for the rotation via the symmetry of R̄S′, proves the π separation, and establishes local instability of the three non-optimal equilibria via a small-rotation (Rodrigues) perturbation that strictly lowers potential energy in appropriate directions . The candidate solution mirrors this structure: it decouples translation/rotation, reduces to maximizing tr(RM), uses the SVD to obtain four sign patterns constrained by det, identifies the unique maximizer, and then uses an energy/Lyapunov–LaSalle argument to show convergence to equilibria and instability of the others. Notational differences aside (paper uses U+, V+, S′; the model uses U, V, diag(±1)), the arguments are equivalent in substance. Therefore, both are correct and follow substantially the same proof.