Synchronization Approach to Formation Control of Mobile Robots from the Cluster Space Perspective

Abstract

In this paper, a synchronization approach to solve the formation control problem for three differential-drive wheeled mobile robots from a cluster space perspective is studied. In order to solve the individual trajectory tracking task of each mobile robot while maintaining a desired formation, besides the controller at cluster level, another inner robot controller is implemented. With this, the robustness of formation keeping to perturbations is increased and motions of all mobile robots are synchronized. Based on Lyapunov theory, it is demonstrated that the proposed synchronization strategy guarantees that both the tracking errors and the synchronization errors of all mobile robots asymptotically converge to the origin. Achieving this, the cluster errors asymptotically converge to zero as well. Furthermore, the robots orientation errors exponentially converge to the origin. The control laws are experimentally validated, using a set of TurtleBot3 mobile robots; these results show the efficiency of the proposed scheme.

Appendices

Appendix A: Proof of lemma 1

As can be shown in (39), with k_qi > 0, the orientation error of each differential wheeled mobile robot exponentially converges to zero. Therefore this implies that $q_{i} \rightarrow q_{id}, \ C_{i} \rightarrow \cos \limits (q_{id}), \ S_{i} \rightarrow \sin \limits (q_{id}),$ with i = 1, 2, 3. After a transient, the terms $r_{x}(\dot {x}_{id},\dot {y}_{id},S_{i}, C_{i})$ and $r_{y}(\dot {x}_{id},\dot {y}_{id},S_{i}, C_{i})$ given in (42) and (43), respectively, satisfy the nonholonomic constraint (3), that is

$$r_{x} \rightarrow \dot{y}_{id}[\sin(q_{id}) \cos(q_{id})]-\dot{x}_{id}\sin^{2} (q_{id})=0,$$

$$r_{y} \rightarrow \dot{x}_{id}[\sin(q_{id}) \cos(q_{id})]-\dot{y}_{id}\cos^{2} (q_{id})=0.$$

Thereby, the terms r_x,r_y are vanished signals of the system (44)-(46) due to their convergence to zero only depends on the orientation errors e_qi, with i = 1,...,n.

Due to the convergence to zero of the orientation errors is independent of the tracking errors, and under the condition that the desired orientation for all mobile robots is the same, that is q_id = q_d, and ω_id = ω_d, the system (44)-(45) can be written as the following subsystem

$$ \dot{e}_{xi} = -k {\Delta} x_{i} C^{2} -k {\Delta} y_{i} SC, $$

(47)

$$ \dot{e}_{yi} = -k {\Delta} y_{i} S^{2} -k {\Delta} x_{i} CS, $$

(48)

where $C = \cos \limits (q_{d}), \ S = \sin \limits (q_{d})$, with i = 1, 2, 3. Although (47)-(48) is a time-varying linear system, in this particular case, it is possible to use a globally defined change of coordinates to obtain an equivalent time-invariant system, [36]. The new state variables are given by

$$ e_{li} = {\Delta} x_{i} C + {\Delta} y_{i} S, $$

(49)

$$ e_{pi} = -{\Delta} x_{i} S + {\Delta} y_{i} C, $$

(50)

with i = 1,...,n. Geometrically, e_li and e_pi are the tracking error on the X_i and Y_i axes, of the mobile reference frame of the i th robot, respectively. Thus, the tracking errors dynamics on the X_i and Y_i axes of the i th mobile robot is given by

$$\dot{e}_{l1} = e_{p1}\omega_{d} -\frac{k}{n}\mu e_{l1} -\frac{k}{n}\bar{\mu} (e_{l2}+e_{l3}),$$

$$\dot{e}_{l2} = e_{p2}\omega_{d} -\frac{k}{n}\beta e_{l2} -\frac{k}{n}\bar{\mu} e_{l1} -\frac{k}{n} \bar{\beta} e_{l3},$$

$$\dot{e}_{l3} = e_{p3}\omega_{d} -\frac{k}{n}\beta e_{l3} -\frac{k}{n}\bar{\mu} e_{l1} -\frac{k}{n} \bar{\beta} e_{l2},$$

$$ \dot{e}_{pi} = -e_{li}\omega_{d}, $$

(51)

with i = 1, 2, 3. Taking the Lyapunov function candidate given by

$$ V(e_{li},e_{pi}) = \frac{1}{2} \sum\limits_{i=1}^{n} (e^{2}_{li} + e^{2}_{pi}), $$

(52)

with n = 3, it can be easily seen that the derivate with respect to time of V (e_li,e_pi) is given by

$$ \dot{V}(e_{li},e_{pi}) = [e_{l1} \ e_{l2} \ e_{l3}] \left( -\frac{k}{n} \right) \boldsymbol{\mathrm{S}_{\mathrm{G}}} [e_{l1} \ e_{l2} \ e_{l3}]^{T}, $$

(53)

where S_G is given by (32). Since S_G is a positive definite matrix, according to (33), $\dot {V}(e_{li}, e_{pi})$ is negative definite. However, due to the terms e_pi are not presented in $\dot {V}$, guaranteeing that $\dot {V}$ is negative definite implies that the system (51) is only stable.

Note that $\ddot {V}(e_{l},e_{p})$ is bounded, due to Assumption 2 is satisfied; hence $\dot {V}(e_{l},e_{p})$ is uniformly continuous. Considering Barbalat’s lemma [37], yields $\dot {V_{3}}(e_{l},e_{p}) \rightarrow 0$ as $t \rightarrow \infty$ and consequently $e_{li} \rightarrow 0$ with i = 1, 2, 3. From the first third equations of (51) and under the assumption that ω_d≠ 0, it is clear that $e_{pi} \rightarrow 0$, therefore the tracking errors of all mobile robots converge to zero. Hence, the origin of the system (44)-(46) is asymptotically stable.

Finally, due to S_G matrix is nonsingular, guaranteeing that e_xi,e_yi converge to zero implies that Δx_i,Δy_i converge to zero as well. In a similar manner, since K matrix is nonsingular (16), also the cluster error vector (24) converge asymptotically to zero. $\blacksquare$

Appendix B: Proof of lemma 2

By using a methodology similar to lemma 1, under the condition that all robots orientations are different each other, after a transient, the system (44)-(46) can be reduced to the time-varying linear subsystem given by

$$ \dot{e}_{xi}(t) = -k g_{ci}(t) {\Delta} x_{i} -k {\Delta} y_{i} g_{i}(t), $$

(54)

$$ \dot{e}_{yi}(t) = -k g_{si}(t) {\Delta} y_{i} -k {\Delta} x_{i} g_{i}(t), $$

(55)

where

$$g_{ci}(t) = \cos^{2}[q_{id}(t)], \ \ \ g_{si}(t) = \sin^{2}[q_{id}(t)],$$

$$ g_{i}(t)=\sin[q_{id}(t)]\cos[q_{id}(t)] =\frac{1}{2}\sin[2q_{id}(t)], $$

(56)

with i = 1, 2, 3. The previous functions satisfy

$$ 0 \leq g_{ci}(t) \leq 1, \ \ 0 \leq g_{si}(t) \leq 1, \ \ |g_{i}(t)| \leq 1/2, \ \ \forall t \geq 0. $$

(57)

The g_i(t) functions are continuously differentiable and from Assumption 2, their time derivatives satisfy

$$ -\omega_{max} \leq \dot{g}_{i}(t)=\omega_{id}(t) \cos[2q_{id}(t)] \leq \omega_{max}, \ \ \forall t \geq 0, $$

(58)

with i = 1, 2, 3. Consider the robots tracking errors vector given as e = [e_x1,e_y1,e_x2,e_y2,e_x3,e_y3]^T, and the Lyapunov function candidate given by

$$ V(t,e)= \sum\limits_{i=1}^{n} \left( [1+g_{i}(t)] \left( e^{2}_{xi}+e^{2}_{yi} \right) \right). $$

(59)

It can be easily seen that

$$ \frac{1}{2} \sum\limits_{i=1}^{n} \left( e^{2}_{xi}+e^{2}_{yi} \right) \!\leq\! V(t,e) \!\leq\! \frac{3}{2} \sum\limits_{i=1}^{n} \left( e^{2}_{xi}+e^{2}_{yi} \right), \ \forall e \in \mathbb{R}^{2n}. $$

(60)

Hence, V (t,e) is positive definite, decrescent, and radially unbounded [37]. The derivative of V (t,e) along the trajectories of the system is given by

$$ \begin{array}{ll} \dot{V}(t,e) = & \sum\limits_{i=1}^{n} ( 2[1+g_{i}(t)]e_{xi} \dot{e}_{xi}\\ & + 2[1+g_{i}(t)]e_{yi} \dot{e}_{yi} +\dot{g}_{i}(t)[ e^{2}_{xi} + e^{2}_{yi}] ). \end{array} $$

(61)

In order to guarantee that $\dot {V}(t,e)$ is negative definite, the following inequality must be satisfied

$$\dot{V}(t,e) \leq -W_{3}(e),$$

where W₃(e) is a time-invariant positive definite function. Substituting (54) and (55) into (61), and using the lower limits of the inequalities (57) and (58), yields

$$ \begin{array}{ll} -W_{3}(e) & = \left( \frac{k}{2} \right) \sum\limits_{i=1}^{n} \left( e_{xi}{\Delta} y_{i} + e_{yi} {\Delta} x_{i} -\omega_{max}(e^{2}_{xi}+e^{2}_{yi})\right)\\ & = -e^{T} Q_{3} e, \end{array} $$

(62)

where Q₃ =

$$ \left( \begin{array}{cccccc} \omega_{max} & \frac{-k\mu}{2n} & 0 & \frac{-k\bar{\mu}}{2n} & 0 & \frac{-k\bar{\mu}}{2n}\\ & & & & & \\ \frac{-k\mu}{2n} & \omega_{max} & \frac{-k\bar{\mu}}{2n} & 0 & \frac{-k\bar{\mu}}{2n} & 0\\ & & & & & \\ 0 & \frac{-k\bar{\mu}}{2n} & { \omega_{max}} & { \frac{-k\beta}{2n}} & {0} & {\frac{-k\bar{\beta}}{2n}} \\ & & & & & \\ \frac{-k\bar{\mu}}{2n} & 0 & {\frac{-k\beta}{2n}} & {\omega_{max}} & {\frac{-k\bar{\beta}}{2n}} & {0} \\ & & & & & \\ 0 & \frac{-k\bar{\mu}}{2n} & {0} & {\frac{-k\bar{\beta}}{2n}} & {\omega_{max}} & {\frac{-k\beta}{2n}} \\ & & & & & \\ \frac{-k\bar{\mu}}{2n} & 0 & {\frac{-k\bar{\beta}}{2n}} & {0} & {\frac{-k\beta}{2n}} & {\omega_{max}} \end{array} \right), $$

(63)

with n = 3. Note that, the required Q₃ matrix for n = 2 is the one which is composed of the last 4 columns and 4 rows of (63). In order to obtain a time-invariant positive definite function (W₃(e)), the Q₃ matrix must be positive definite. Thus, it is necessary to analyzed the properties of Q₃ to present the corresponding interval of the gains which compose Q₃.

Again, if $\bar {\mu }= \bar {\beta } = 0$ implies that a synchonization among the mobile robots is not required, obtaining 3 uncoupled systems. Due to ω_max is a parameter that depends on the desired trajectory, it is easy to assign a large enough value to ω_max in order to guarantee that Q₃ is positive definite. Actually, the ω_max parameter provides the convergence speed of the system [37].

From the second principal minor of Q₃ matrix, in order to guarantee that its determinant is positive, the following inequality must be satisfied

$$ \omega_{max} > k\mu /(2n), \ \text{with} \ n=3. $$

(64)

Hence, to omit the ω_max parameter, take into account the following assignation

$$ \omega_{max} = Lk\beta /(2n), $$

(65)

with μ = β (33), where L > 1 is a scaling factor, which cannot be so large due to it is just required for satisfying the inequality in (64). Notice that Q₃ is a symmetric matrix composed of the gains $k, \mu , \beta \in \mathbb {R}^{+}$, $\bar {\mu } = \bar {\beta } \neq 0$, then the eigenvalues of Q₃ are all real values. Taking into account (33) and (65), the eigenvalues of Q₃ are given by

$$ \begin{array}{c} \lambda_{1} = \lambda_{2} = [\bar{\beta}k+(L-1)\beta k]/(2n),\\ \lambda_{3} = \lambda_{4} = [-\bar{\beta}k+(L+1)\beta k]/(2n),\\ \ \ \lambda_{5} = [2\bar{\beta}k+(L+1)\beta k]/(2n),\\ \ \ \lambda_{6} = -[2\bar{\beta}k+(1-L)\beta k]/(2n). \end{array} $$

(66)

The solution that satisfies that all eigenvalues of Q₃ are positive, taking $\bar {\beta }>0$, is

$$ \bar{\beta} < \frac{L-1}{2}\beta. $$

(67)

Otherwise, taking $\bar {\beta }<0$, there are 2 solution intervals that satisfy that all eigenvalues of Q₃ are positive, due to the less interval depends on the parameter L, those are

$$ -(L-1)\beta < \bar{\beta}, \ \text{with} \ L \leq 3; \ \ \ -\frac{L+1}{2}\beta < \bar{\beta}, \ \text{with} \ L \geq 3. $$

(68)

As was mentioned, it is required that L is large enough to satisfy (18), thereby the interval with L ≥ 3 is neglected.

In order to consider positive and negative values to the $\bar {\beta }$ gain and to guarantee that Q₃ matrix is positive definite, it is necessary to take the union of the intervals (67) with (68) considering L ≤ 3, that is

$$ \bar{\beta} \in (-[L-1]\beta,0) \cup (0,[L-1]\beta/2), \ \text{where} \ 1 < L \leq 3. $$

(69)

In a similar manner to $\bar {\mu }$. The S_G matrix can be positive definite by following the inequality given in (33). Therefore, using L = 2 in the solution for the time-varying case in (69), the solution interval is given by

$$ \bar{\beta} \in (-\beta,0) \cup (0,\beta/2). $$

(70)

So, considering L = 2 is a proper assignation. By doing a similar procedure for the time-varying system, with n = 2, the solution interval is given by

$$ -(L-1)\beta < \bar{\beta} < (L-1)\beta, \ \text{with} \ L>1. $$

The result of the time-invariant system, with n = 2, is given by

$$ -\beta < \bar{\beta} < \beta. $$

Again, considering L = 2 is a proper assignation.

With this result it can be concluded that there is a solution, by choosing proper values of the gains $k, \mu , \beta , \bar {\mu }, \bar {\beta }$, and the parameter ω_max, which makes that Q₃ is a positive definite matrix. Hence, the origin of the system (44)-(46) is asymptotically stable.

Finally, due to S_G matrix is nonsingular, the robots synchronization errors (31) asymptotically converge to zero as well. Since K matrix is nonsingular (16), also the cluster error vector (24) converge asymptotically to zero. $\blacksquare$