【Paper】2017_Limit-Cycle-Based Decoupled Design of Circle Formation Control with Collision Avoidance

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

p˙i(t)=ui(t),i=1,2,⋯,N(2)\dot{p}_i(t) = u_i(t), \quad i = 1,2,\cdots,N \tag{2}p˙i(t)=ui(t),i=1,2,⋯,N(2)

平面运动，因此 ui(t)∈R2u_i(t) \in \mathbb{R}^2ui(t)∈R2

Controller consisting two parts
ui(t)=uip(t)fi(t),i=1,2,⋯,N(8)u_i(t) = u_i^p(t) f_i(t), \quad i = 1,2,\cdots,N \tag{8}ui(t)=uip(t)fi(t),i=1,2,⋯,N(8)

Design the first part uip(t)u^p_i(t)uip(t) in our controller as a limit-cycle oscillator

uip(t)=λ[γli(t)−11γli(t)]pˉi(t),i=1,2,⋯,Nu^p_i(t) = \lambda \left[\begin{matrix} \gamma l_i(t) & -1 \\ 1 & \gamma l_i(t) \end{matrix}\right]\bar{p}_i(t), \quad i = 1, 2, \cdots, Nuip(t)=λ[γli(t)1−1γli(t)]pˉi(t),i=1,2,⋯,N

where λ>0,γ>0\lambda>0, \gamma > 0λ>0,γ>0 are constant, and
li(t)=r2−∥pˉi(t)∥2l_i(t) = r^2 - \|\bar{p}_i(t)\|^2li(t)=r2−∥pˉi(t)∥2

Angular control uiα(t)u^\alpha_i(t)uiα(t) as
uiα(t)=di−di+di−α^i(t)u^\alpha_i(t) = \frac{d_{i^-}}{d_i + d_{i^-}} \hat{\alpha}_i(t)uiα(t)=di+di−di−α^i(t)

fi(t)=c1+c22πuiα(t),i=1,2,⋯,Nf_i(t) = c_1 + \frac{c_2}{2\pi} u^\alpha_i(t), \quad i = 1,2,\cdots,Nfi(t)=c1+2πc2uiα(t),i=1,2,⋯,N

已知三边求角度公式是余弦定理：cos⁡A=(b2+c2−a2)/2cb\cos A=(b^2+c^2-a^2)/2cbcosA=(b2+c2−a2)/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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【Paper】2017_Limit-Cycle-Based Decoupled Design of Circle Formation Control with Collision Avoidance

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