This paper proposes a new type of control laws for free rigid bodies. The start point is the dual quaternion and its characteristics. The logarithm of a dual quaternion is defined, based on which kinematic control laws can be developed. Global exponential convergence is achieved using logarithmic feedback via a generalized proportional control law, and an appropriate Lyapunov function is constructed to prove the stability. Both the regulation and tracking problems are tackled. Omnidirectional control is discussed as a case study. As the control laws can handle the interconnection between the rotation and translation of a rigid body, they are shown to be more applicable than the conventional method.

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This paper presents a new formulation for control of spatial rigid bodies. After revealing the geometric structure of dual quaternions, logarithmic feedback are utilized to derive control laws in both kinematic design and dynamic design. The regulation problem and the tracking problem are tackled respectively. Comparison with conventional methods together with simulation results are given to show the effectiveness of the new method.control laws, together with some simulation results. The paper is concluded by Section 6.II. MATHEMATICAL PRELIMINARIES This section presents a brief review of unit quaternion and dual quaternion. Readers are referred to [3] and [9] for more details.

It is difficult to achieve accurate distribution of reactive power based on conventional droop control due to the line impedance mismatch in an islanded microgrid. An adaptive virtual impendence method based on consensus control of reactive current is proposed in this paper. A distributed control structure without the central controller has been established. In this structure, each distributed generation unit (DG) is an independent agent, one-way communication is used between the adjacent DGs, and the reactive power sharing is equivalent to a problem of reactive power current consensus. It has been proven that the system is asymptotically stable under the proposed control strategy. When the adjacent DG's reactive power is not proportionally distributed, the current weight error term will generate a virtual impedance correction term through the proportional-integral controller based on the reactive current consensus control strategy, thus introducing adaptive virtual impedance to eliminate mismatches in output impedance between DGs. Reactive power auto-proportional distribution can be achieved without knowing the line impedance. At the same time, the power control loop is simplified and the virtual impedance compensation angle is employed to compensate the decreased reference voltage magnitude and varied phase angle due to the introduction of the virtual impedance, so the stability of the system can be improved. Finally, the correctness and effectiveness of the proposed strategy are verified by modeling analysis and microgrid simulations.

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