Almost-global stability without nonlinear-term compensation

Prove almost global asymptotic stability for the constant-reference observer-based distributed control torque in equation \eqref{controlconstantestimate} when the nonlinear term \(\Sigma_i\bar{\omega}_i\) is not compensated.

Background

The paper explains that compensating the nonlinear term Σiωˉi\Sigma_i\bar{\omega}_i is not necessary for the Lyapunov derivative because Σi\Sigma_i is skew-symmetric. However, the compensation is introduced to obtain an autonomous tracking-error subsystem and thereby facilitate the proof of almost global asymptotic stability. Without this compensation, the resulting control torque \eqref{controlconstantestimate} is shown only to guarantee local exponential stability. The authors explicitly leave unresolved whether the same controller also provides almost global asymptotic stability and identify a formal proof as future work.

References

Almost global asymptotic stability with the control torque in controlconstantestimate is not excluded and a formal proof will be part of our future investigations.

controlconstantestimate:

τi=−Γi−kRˉψ(AˉiRˉir)−kωˉωˉi,\tau_i=-\Gamma_i-k_{\bar{R}}\psi(\bar{A}_i\bar{R}_i^r)-k_{\bar{\omega}}\bar{\omega}_i,

— Leader-follower Attitude Synchronization of Rigid-body Systems on SO(3)  (2609.25580 - Li et al., 22 Sep 2026) in Conclusion