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Unit Quaternion-Based Output Feedback for the Attitude Tracking Problem

机译：基于单位四元数的姿态跟踪输出反馈

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In this note, we propose a quaternion-based dynamic output feedback for the attitude tracking problem of a rigid body without velocity measurement. Our approach consists of introducing an auxiliary dynamical system whose output (which is also a unit quaternion) is used in the control law together with the unit quaternion representing the attitude tracking error. Roughly speaking, the necessary damping that would have been achieved by the direct use of the angular velocity can be achieved, in our approach, by the vector part ${mathtilde {q}}$ of the error signal between the output of the auxiliary system and the unit quaternion tracking error. The resulting velocity-free control scheme guarantees almost globalIn this paper we use the term almost global (see, for instance, ) to indicate that the boundedness of the states is guaranteed over $BBQ _{u},times, BBQ _{u} ,times, BBR ^{3}$ , where $BBQ _{u}$ is the set of unit-quaternion. Furthermore, the closed-loop system has four equilibrium points (three repeller equilibria and one attractor) that are mathematically different but represent the same physical attitude of the rigid body. All trajectories starting in $BBQ _{u} ,times, BBQ _{u} ,times, BBR ^{3}$–except the three repeller equilibria— will converge to the unique attractor equilibrium. For more details about the topological obstruction to continuous global stabilization of rotational motion, the reader is referred to .- - asymptotic stability which is as strong as the topology of the motion space can permit. In the regulation case, our control law is a pure quaternion feedback (i.e., consisting of two terms that are vector parts of unit-quaternion), and hence, the control torques are naturally bounded by the control gains. Simulation results are provided to show the effectiveness of the proposed control scheme.
机译：在本说明中，我们针对没有速度测量的刚体的姿态跟踪问题提出了基于四元数的动态输出反馈。我们的方法包括引入一个辅助动力学系统，该系统的输出（也是单位四元数）与表示姿态跟踪误差的单位四元数一起用于控制律。粗略地讲，在我们的方法中，可以通过辅助系统的输出之间的误差信号的矢量部分$ {mathtilde {q}} $来实现通过直接使用角速度可以实现的必要阻尼。和单位四元数跟踪误差。最终的无速度控制方案保证了几乎全局性。在本文中，我们使用术语“几乎全局性”（例如）来表示，在$ BBQ _ {u}，时间，BBQ _ {u}上，保证了状态的有界性，times，BBR ^ {3} $，其中$ BBQ _ {u} $是单位四元数的集合。此外，闭环系统具有四个平衡点（三个推斥极平衡点和一个吸引子），它们在数学上是不同的，但代表的刚体的物理姿态相同。除三个推斥极均衡外，所有以$ BBQ _ {u}，时间，BBQ _ {u}，时间，BBR ^ {3} $开始的轨迹将收敛到唯一的吸引子平衡。有关旋转运动连续全局稳定的拓扑障碍的更多详细信息，请读者参阅---渐近稳定性，其强度与运动空间的拓扑结构所允许的强度一样强。在调节情况下，我们的控制定律是纯四元数反馈（即由两个单位四元数的矢量项组成），因此，控制转矩自然受控制增益的限制。仿真结果表明了该控制方案的有效性。

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