C. Wang and G. Xie, “Limit-Cycle-Based Decoupled Design of Circle Formation Control with Collision Avoidance for Anonymous Agents in a Plane,” in IEEE Transactions on Automatic Control, vol. 62, no. 12, pp. 6560-6567, Dec. 2017, doi: 10.1109/TAC.2017.2712758.

Each agent is described as a kinematic point

$${\dot{p}}_i(t)=u_i(t),i=1,2,?,N\text{\hspace{1em}(2)}$$

平面運動，因此 $u_i(t)\in {\mathbb{R}}^2$

Controller consisting two parts
$$u_i(t)=u_i^p(t)f_i(t),i=1,2,?,N\text{\hspace{1em}(8)}$$

Design the first part $u_i^p(t)$ in our controller as a limit-cycle oscillator

$$u_i^p(t)=\lambda \left[\begin{array}{cc}\gamma l_i(t)& ?1\\ 1& \gamma l_i(t)\end{array}\right]{\bar{p}}_i(t),i=1,2,?,N$$

where $\lambda >0,\gamma >0$ are constant, and
$$l_i(t)=r^2?\Vert {\bar{p}}_i(t){\Vert }^2$$

Angular control $u_i^{\alpha }(t)$ as
$$u_i^{\alpha }(t)=\frac{d_{i^{?}}}{d_i+d_{i^{?}}}{\hat{\alpha }}_i(t)$$

$$f_i(t)=c_1+\frac{c_2}{2\pi }{u}_i^{\alpha }(t),i=1,2,?,N$$

已知三邊求角度公式是余弦定理：$\cos A=(b^2+c^2?a^2)/2cb$；cosB=(a平方+c平方-b平方)/2ac；cosC=(a平方+b平方-c平方)/2ab。

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