Abstract
In this technical note, we solve the control problem
of rigid bodies with only quaternion measurements for all initial
rotations and angular velocities. The proposed solution is based on
the theory of cascades using any switching certainty equivalence
controller satisfying certain assumptions along with an in the
large hybrid observer. The equilibrium point of the proposed
observer in closed loop with the rigid body dynamics is proven
to be κ-exponentially stable in the large i.e., we prove that the
equilibrium point is stable and that the error states converge
exponentially fast towards the origin for all initial rotations and
angular velocities. Until now, stability results for quaternion-based
observers have typically only been valid for a bounded set of initial
conditions. To overcome this issue, our observer design is based
on dynamic scaling and switching logic. Furthermore, we show
that the origin of the proposed switching certainty equivalence
controller in closed loop with the hybrid observer is asymptotically
stable in the large for all available initial conditions associated with
the quaternion space. Simulation results for the proposed scheme
are presented with the particular case of the PD+ controller,
revealing that all states converge as expected from our theoretical
findings.
of rigid bodies with only quaternion measurements for all initial
rotations and angular velocities. The proposed solution is based on
the theory of cascades using any switching certainty equivalence
controller satisfying certain assumptions along with an in the
large hybrid observer. The equilibrium point of the proposed
observer in closed loop with the rigid body dynamics is proven
to be κ-exponentially stable in the large i.e., we prove that the
equilibrium point is stable and that the error states converge
exponentially fast towards the origin for all initial rotations and
angular velocities. Until now, stability results for quaternion-based
observers have typically only been valid for a bounded set of initial
conditions. To overcome this issue, our observer design is based
on dynamic scaling and switching logic. Furthermore, we show
that the origin of the proposed switching certainty equivalence
controller in closed loop with the hybrid observer is asymptotically
stable in the large for all available initial conditions associated with
the quaternion space. Simulation results for the proposed scheme
are presented with the particular case of the PD+ controller,
revealing that all states converge as expected from our theoretical
findings.