Fully distributed control for task-space formation tracking of nonlinear heterogeneous robotic systems

Abstract

This paper studies task-space formation tracking problem of nonlinear heterogeneous robotic systems involving external disturbances, kinematic and dynamic uncertainties, where the cases with both single and multiple time-varying leaders are considered. To solve the aforementioned nonlinear control problem, several novel fully distributed control algorithms, in which no global information is employed, are developed for the nonlinear systems under directed communication topologies. Based on the proposed control algorithms, the control process is classified into two parts, namely the task-space formation tracking of master robots with a single leader and that of slave robots with multiple leaders. By invoking Barbalat’s lemma and input-to-state stability theory, the sufficient criteria for the asymptotic convergence of the task-space formation tracking errors are established. In addition, the obtained results are extended to formation-containment and consensus problems in similar nonlinear cases. Finally, numerical examples are provided to illustrate the validity and advantages of the main results.

Appendix

Appendix A. Proof of Lemma 1. By Assumption 2, it is clear that $L_3$ is invertible. Based on the definition of Laplacian matrix $\mathcal{L}$, one obtains ${L_2}{1_m} + {L_3}{1_{n - m}} = {0_m}$, i.e., $- L_3^{ - 1}{L_2}{1_m} = {1_{n - m}}$ since $L_3$ is invertible. Thus, it gives that $-{L_3^{ - 1}{L_2}}$ is nonnegative, and each row sums equal to one.

Appendix B. Proof of Theorem 1. Here we prove the theorem by contradiction. First we replace ${\beta _i}$ in (11a) with a constant ${{{\bar{\beta }}}_i}$, which satisfies ${{\bar{\beta }}_i} > {\mathcal{L}_\infty }\left\| {{\dot{x}_0}} \right\|$. Then, by similar proofs of Theorem 3.1 given in [3] and Lemma 3 given in [10], one can easily obtain $x_i - x_0 = 0$ in finite time. If we assume that $x_i - x_0 = 0$ cannot be achieved in finite time such that it gives

$$\begin{aligned} {\beta _{i}(t)} < {{{\bar{\beta }}}_i}, \end{aligned}$$

(26)

While ${{{\dot{\beta }}}_i}$ in (11b) satisfies ${{{\dot{\beta }}}_i} \ge 0$, which reveals that ${\beta _{i}}$ continuously increases such that there must exist a finite time satisfying

$$\begin{aligned} {\beta _{i}(t_1)} > {{{\bar{\beta }}}_i}, \end{aligned}$$

(27)

By comparing with (26) and (27), it is contradictory such that it follows that $x_i - x_0= 0$ can be obtained in finite time, and it ends the proof.

Appendix C1. The details derivative of $V_{1i}(t)$. Differentiating (16) with respect to time yields

$$\begin{aligned} \dot{V}_{1i}(t)= & {} \frac{1}{2} s_i^T{{\dot{H}}_i}(q_i){{s}_i} + s_i^T{H_i}(q_i){{\dot{s}}_i} + {\tilde{\vartheta }}_{di}^T\Lambda _{di}^{-1}{{\dot{{\tilde{\vartheta }}}}_{di}} \\&+ {\tilde{\mu }}_{i}^T\Lambda _{\mu i}^{-1}{{\dot{{\tilde{\mu }}}}_{i}}, \end{aligned}$$

Then, by invoking (15), Properties 2 and 4, one obtains

$$\begin{aligned} \dot{V}_{1i}(t)= & {} \frac{1}{2}s_i^T{{\dot{H}}_i}({q_i}){s_i} + s_i^T\left( \phantom {\left. - {K_i}{s_i}- {Y_{di}}{{{\tilde{\vartheta }}}_{di}} + {{{\tilde{\mu }} }_i} \right) } - {C_i}\left( {{q_i},{{\dot{q}}_i}} \right) {s_i} \right. \\&\left. - {K_i}{s_i}- {Y_{di}}{{{\tilde{\vartheta }}}_{di}} + {{{\tilde{\mu }} }_i} \right) \\&+ \,{\tilde{\vartheta }} _{di}^T\Lambda _{di}^{ - 1}{\Lambda _{di}}Y_{di}^T{s_i} - {\tilde{\mu }} _i^T\Lambda _{\mu i}^{ - 1}{\Lambda _{\mu i}}{s_i}\\= & {} \frac{1}{2}s_i^T\left( {{{\dot{H}}_i}({q_i}) - 2{C_i}\left( {{q_i},{{\dot{q}}_i}} \right) } \right) {s_i} - s_i^T{K_i}{s_i} \\&- s_i^T{Y_{di}}{{\tilde{\vartheta }} _{di}} + s_i^T{{\tilde{\mu }} _i} + {\tilde{\vartheta }} _{di}^TY_{di}^T{s_i} - {\tilde{\mu }} _i^T{s_i}\\= & {} - s_i^T{K_{i}}{{s}_i} \le 0, \end{aligned}$$

Appendix C2. The details derivative of $V_{2i}(t)$. Firstly, differentiating (17) with respect to time yields

$$\begin{aligned} {{\dot{V}}_{2i}}(t) = {{\tilde{x}}_i^T}{\dot{{\tilde{x}}}_i} + {\tilde{\vartheta }}_{ki}^T\Lambda _{ki}^{-1}{{\dot{{\tilde{\vartheta }}}}_{ki}} - \frac{1}{\alpha _1}({J_i s_i})^T{J_i}{s_i}, \end{aligned}$$

Then, substituting ${J_i}{s_i}={\dot{{\tilde{x}}}_i} + {\alpha _1}{{\tilde{x}}_i} - {Y_{ki}}{{{\tilde{\vartheta }}}_{ki}}$ derived from (13a), and ${{\dot{{\tilde{\vartheta }}}}_{ki}} = - {\Lambda _{ki}}Y_{ki}^T({\frac{1}{{\alpha _1}}{\dot{{\tilde{x}}}_i} + 2{{\tilde{x}}_i}})$ yields

$$\begin{aligned} {{\dot{V}}_{2i}}(t)= & {} -\frac{1}{{2{\alpha _1}}}\left( {{\dot{{\tilde{x}}}_i} + {\alpha _1} {\tilde{x}}_i} \right) ^T \left( {{\dot{{\tilde{x}}}_i} + {\alpha _1} {\tilde{x}}_i} \right) -\frac{\alpha _1}{2}{\tilde{x}}_i^T{{\tilde{x}}_i}\\&-\frac{1}{2{\alpha _1}}{\dot{{\tilde{x}}}_i^T}{\dot{{\tilde{x}}}_i}- \frac{1}{\alpha _1}(Y_{ki}{\tilde{\vartheta }}_{ki})^T{Y_{ki}}{{{\tilde{\vartheta }}}_{ki}}\\&-\frac{1}{{\alpha _1}}{\tilde{\vartheta }}_{ki}^T Y_{ki}^T{\dot{{\tilde{x}}}_i}, \end{aligned}$$

Based on the standard basic inequalities, one obtains $-\frac{1}{{\alpha _1}}{\tilde{\vartheta }}_{ki}^T Y_{ki}^T{\dot{{\tilde{x}}}_i} \le \frac{1}{2{\alpha _1}}{\dot{{\tilde{x}}}_i^T}{\dot{{\tilde{x}}}_i} + \frac{1}{{2{\alpha _1}}}(Y_{ki}{\tilde{\vartheta }}_{ki})^T{Y_{ki}}{{{\tilde{\vartheta }}}_{ki}}$ such that

$$\begin{aligned} {{\dot{V}}_{2i}}(t)\le & {} -\frac{1}{{2{\alpha _1}}}\left( {{\dot{{\tilde{x}}}_i} + {\alpha _1} {\tilde{x}}_i} \right) ^T \left( {{\dot{{\tilde{x}}}_i} + {\alpha _1} {\tilde{x}}_i} \right) -\frac{{\alpha _1}}{2}{\tilde{x}}_i^T{{\tilde{x}}_i}\\&-\frac{1}{2{\alpha _1}}{\dot{{\tilde{x}}}_i^T}{\dot{{\tilde{x}}}_i} -\frac{1}{{{\alpha _1}}}(Y_{ki}{\tilde{\vartheta }}_{ki})^T{Y_{ki}}{{{\tilde{\vartheta }}}_{ki}}\\&+\frac{1}{2{\alpha _1}}{\dot{{\tilde{x}}}_i^T}{\dot{{\tilde{x}}}_i} + \frac{1}{{2{\alpha _1}}}(Y_{ki}{\tilde{\vartheta }}_{ki})^T{Y_{ki}}{{{\tilde{\vartheta }}}_{ki}}\\\le & {} -\frac{1}{{2{\alpha _1}}}\left( {{\dot{{\tilde{x}}}_i} + {\alpha _1} {\tilde{x}}_i} \right) ^T \left( {{\dot{{\tilde{x}}}_i} + {\alpha _1} {\tilde{x}}_i} \right) \\&-\frac{{\alpha _1}}{2}{\tilde{x}}_i^T{{\tilde{x}}_i}- \frac{1}{{2{\alpha _1}}}(Y_{ki}{\tilde{\vartheta }}_{ki})^T{Y_{ki}}{{{\tilde{\vartheta }}}_{ki}} \le 0, \end{aligned}$$