Group motion control of multi-agent systems with obstacle avoidance: column formation under input constraints

Abstract

This paper proposes a formation controller for multi-agent systems focusing on the obstacle avoidance and input constraints. In our proposed method, the limitation of translation and angular velocities is adaptably tuned depending on the existence of obstacles, such that the velocities can increase when the robots potentially hit obstacles. Moreover, a potential function of the input velocities that easily satisfies velocity constraints is designed. We also design a formation controller that realizes obstacle avoidance while satisfying the velocity constraints. Mathematical analysis shows the stability of the formation of robots controlled by the proposed model. Simulation results also demonstrate the effectiveness of the proposed method.

Appendix

Here, we derive a necessary condition to satisfy “Approximated constraint”. The feedback parameters with constraint $\lambda _j$ are

$$\begin{aligned} k_{j_{1}}(t) = \frac{\lambda (x_{ij})}{2} \left( \frac{\lambda _ja_j\sqrt{1-\left( \frac{v_i\sin \beta _{ij}}{d_j\lambda _jb_j}\right) ^2} }{|x_{ij}|}- \frac{ v_i\cos \beta _{ij} }{x_{ij}}\right) \end{aligned}$$

(28)

$$\begin{aligned} k_{j_{2}}(t) = \frac{\lambda (y_{ij})}{2} \left( \frac{\lambda _jb_j\sqrt{1-\left( \frac{v_i\cos \beta _{ij}}{\lambda _ja_j}\right) ^2} }{ |y_{ij}| }- \frac{v_i\sin \beta _{ij}}{y_{ij}}\right) . \end{aligned}$$

(29)

If either the value inside the square root of Eq. (28) or that of Eq. (29) is negative, these parameters become complex and thus the control input cannot be calculated. Therefore, the control input is determined only when

$$\begin{aligned} |v_i| \le \lambda _j\min \left\{ \frac{d_jb_j}{\sin \beta _{ij}},\frac{a_j}{\cos \beta _{ij}} \right\} . \end{aligned}$$

(30)

Because $v_i=k_{i_{1}}(t)x_{i-1,i} +v_{i-1}\cos \beta _{i-1,i} +g_i$, the necessary condition becomes

$$\begin{aligned} |g_i| \le \lambda _j\min \left\{ \frac{d_jb_j}{\sin \beta _{ij}},\frac{a_j}{\cos \beta _{ij}} \right\} -\lambda _ia_i,\quad \forall t \ge 0. \end{aligned}$$

(31)

This condition represents the relationship between parameters of robot $R_i$ (leader robot) and those of $R_j$ (follower robot). Eq. (31) requires comparatively larger $\lambda _j$ than $\lambda _i$.

Fig. 7

Relation between possibility of obstacle avoidance and initial distance

Table 1 Result of performance analysis

Fig. 8

a Trajectories of three robots. b Formation error. c Control inputs and elliptic constraints. d Distance between robots and obstacle