Practical robust nonlinear PD controller for cable-driven parallel manipulators

Abstract

This paper presents a model-free robust nonlinear PD (R-NPD) controller for cable-driven parallel manipulators (CDPMs) in joint space. Generally, in various mechanical manipulators and in particular CDPMs for fast and high-precision tracking, a precise dynamic model is required. However, the dynamic model of the robot is always contaminated with uncertainties such as nonlinear and time-varying parameters as well as external disturbances. For this purpose, in the proposed controller structure, the time-delay estimation (TDE) technique is used to indirectly use the robot dynamics into the control structure without need of its prior knowledge. Furthermore, a nonlinear PD controller is designed in joint space in such a way that the robot can track the reference trajectory quite fast and accurate, without the need for any auxiliary sensors. The stability of the closed-loop system has been examined through Lyapunov direct method, and it has been shown that tracking error remains uniformly ultimately bounded. Finally, to demonstrate the effectiveness of the proposed controller, simulations and experiments have been performed on two different categories of CDPMs, whose results show that the proposed control scheme outperforms modified TDE control method in practice.

Appendices

Appendix A

To illustrate the boundedness of the TDE error introduced in (21), let us rewrite the dynamics as follows

$$\begin{aligned} \mathbf {\pmb {\varepsilon }} = \varvec{u}_r - \ddot{\varvec{x}}. \end{aligned}$$

(55)

Multiply both sides of by $\varvec{M}_T(\varvec{x})$ and substitute it in (11) and (13)

$$\begin{aligned}&\varvec{M}_T(\varvec{x}) \mathbf {\pmb {\varepsilon }} = \varvec{M}_T(\varvec{x}) (\varvec{u}_r - \ddot{\varvec{x}}) \nonumber \\&= \varvec{M}_T(\varvec{x}) \varvec{u}_r + \varvec{C}_T(\varvec{x}, {\dot{\varvec{x}}}){\dot{\varvec{x}}} + \varvec{g}_T(\varvec{x}) + \varvec{N}_T(\varvec{x}, {\dot{\varvec{x}}}) \nonumber \\&\quad - \varvec{u}_T \nonumber \\&= \varvec{M}_T(\varvec{x}) \varvec{u}_r + \varvec{C}_T(\varvec{x}, {\dot{\varvec{x}}}){\dot{\varvec{x}}} + \varvec{g}_T(\varvec{x}) + \varvec{N}_T(\varvec{x}, {\dot{\varvec{x}}}) \nonumber \\&\quad - {\bar{\varvec{M}}} \varvec{u}_r - {\hat{\varvec{h}}}(\varvec{x}, {\dot{\varvec{x}}}, \ddot{\varvec{x}}). \end{aligned}$$

(56)

Referring to (15), it is clear that

$$\begin{aligned} \begin{aligned}&{\hat{\varvec{h}}}(\varvec{x}, {\dot{\varvec{x}}}, \ddot{\varvec{x}}) = \bigg ((\varvec{M}_T(\varvec{x}) - {\bar{\varvec{M}}}) \ddot{\varvec{x}} \bigg ) \bigg |_{t - \varDelta t} + \\&\bigg (\varvec{C}_T(\varvec{x}, {\dot{\varvec{x}}}){\dot{\varvec{x}}} \bigg ) \bigg |_{t - \varDelta t} + \varvec{g}_T(\varvec{x}) \bigg |_{t - \varDelta t} + \varvec{N}_T(\varvec{x}, {\dot{\varvec{x}}}) \bigg |_{t - \varDelta t}.\nonumber \end{aligned}\!\!\!\!\!\\ \end{aligned}$$

(57)

Rewrite (56) according to (57)

$$\begin{aligned} \varvec{M}_T(\varvec{x}) \mathbf {\pmb {\varepsilon }}&= (\varvec{M}_T(\varvec{x}) - {\bar{\varvec{M}}}) \varvec{u}_r&\\ \nonumber&\quad - \bigg ((\varvec{M}_T(\varvec{x}) - {\bar{\varvec{M}}}) \ddot{\varvec{x}} \bigg ) \bigg |_{t - \varDelta t} + \mathbf {\pmb {\varOmega }}, \end{aligned}$$

(58)

in which

$$\begin{aligned} \mathbf {\pmb {\varOmega }}&= \varvec{C}_T(\varvec{x}, {\dot{\varvec{x}}}){\dot{\varvec{x}}} - \bigg (\varvec{C}_T(\varvec{x}, {\dot{\varvec{x}}}){\dot{\varvec{x}}} \bigg )\bigg |_{t - \varDelta t} + \varvec{g}_T(\varvec{x}) \nonumber&\\ \nonumber&\quad - \varvec{g}_T(\varvec{x}) \bigg |_{t - \varDelta t} + \varvec{N}_T(\varvec{x}, {\dot{\varvec{x}}}) - \varvec{N}_T(\varvec{x}, {\dot{\varvec{x}}}) \bigg |_{t - \varDelta t}.\nonumber \\ \end{aligned}$$

(59)

Given $\varvec{N}_T(\varvec{x}, {\dot{\varvec{x}}})$ defined in (6), $\mathbf {\pmb {\varOmega }}$ may be divided into two (continuous and discontinuous) parts as follows

$$\begin{aligned} \mathbf {\pmb {\varOmega }}_{con}&= \varvec{C}_T(\varvec{x}, {\dot{\varvec{x}}}){\dot{\varvec{x}}} - \bigg (\varvec{C}_T(\varvec{x}, {\dot{\varvec{x}}}){\dot{\varvec{x}}} \bigg )\bigg |_{t - \varDelta t} \nonumber&\\ \nonumber&\quad + \varvec{g}_T(\varvec{x}) - \varvec{g}_T(\varvec{x}) \bigg |_{t - \varDelta t} + \varvec{F}_v {\dot{\varvec{x}}} - \varvec{F}_v {\dot{\varvec{x}}} \bigg |_{t - \varDelta t}&\\ \nonumber&\quad + \mathbf {\pmb {\tau }}_{dis} - \mathbf {\pmb {\tau }}_{dis} \bigg |_{t - \varDelta t},&\\ \mathbf {\pmb {\varOmega }}_{dis}&= \varvec{F}_c({\dot{\varvec{x}}}) - \varvec{F}_c({\dot{\varvec{x}}}) \bigg |_{t - \varDelta t}, \end{aligned}$$

(60)

$\mathbf {\pmb {\varOmega }}_{dis}$ is discontinuous and has an upper bound $\rho$ at velocity reversal. In addition, if boundedness and continuous condition of real value of $\mathbf {\pmb {\varOmega }}_{con}$ is verified, then $\mathbf {\pmb {\varOmega }}_{con} = O(\varDelta t^2)$ [38]. Therefore, $\mathbf {\pmb {\varOmega }}$ is bounded by $\rho + O(\varDelta t^2)$ while $\varDelta t$ is sufficiently small.

Add and subtract $\varvec{M}_T(\varvec{x})$ to (58) to reach:

$$\begin{aligned} \varvec{M}_T(\varvec{x}) \mathbf {\pmb {\varepsilon }}&= (\varvec{M}_T(\varvec{x}) - {\bar{\varvec{M}}}) \varvec{u}_r - (\varvec{M}_T(\varvec{x}) - {\bar{\varvec{M}}}) \ddot{\varvec{x}} \bigg |_{t - \varDelta t}\nonumber&\\ \nonumber&\quad + (\varvec{M}_T(\varvec{x}) - \varvec{M}_T(\varvec{x}) \bigg |_{t - \varDelta t}) \ddot{\varvec{x}} \bigg |_{t - \varDelta t} + \mathbf {\pmb {\varOmega }}.\nonumber \\ \end{aligned}$$

(61)

Use delayed (55) to write:

$$\begin{aligned} \varvec{M}_T(\varvec{x}) \mathbf {\pmb {\varepsilon }}&= \big (\varvec{M}_T(\varvec{x}) - {\bar{\varvec{M}}}\big ) \mathbf {\pmb {\varepsilon }} \bigg |_{t - \varDelta t}&\nonumber \\ \nonumber&\quad - \big (\varvec{M}_T(\varvec{x}) - {\bar{\varvec{M}}}\big ) \big (\varvec{u}_r - \varvec{u}_r \bigg |_{t - \varDelta t}\big )&\\ \nonumber&\quad + \big (\varvec{M}_T(\varvec{x}) - \varvec{M}_T(\varvec{x}) \bigg |_{t - \varDelta t}\big ) \ddot{\varvec{x}} \bigg |_{t - \varDelta t} + \mathbf {\pmb {\varOmega }}.\nonumber \\ \end{aligned}$$

(62)

The above equation shows a discrete-time system in the state-space form with state variable vector $\mathbf {\pmb {\varepsilon }}$ as follows

$$\begin{aligned} \mathbf {\pmb {\varepsilon }}(k) = \varvec{A}(k) \mathbf {\pmb {\varepsilon }}(k-1) + \varvec{B}(k) \varvec{u}_k(k) + \mathbf {\pmb {\nu }}(k), \end{aligned}$$

(63)

in which

$$\begin{aligned}&\varvec{A}(k) = \varvec{B}(k) =\varvec{I}- \varvec{M}_T^{-1}(\varvec{x}) {\bar{\varvec{M}}},&\\ \nonumber&\varvec{u}_k(k) = \varvec{u}_r - \varvec{u}_r\bigg |_{k-1},&\\ \nonumber&\mathbf {\pmb {\nu }}(k) = \varvec{M}_T^{-1}(\varvec{x})\bigg (\big (\varvec{M}_T(\varvec{x}) - \varvec{M}_T(\varvec{x}) \bigg |_{k-1} \big ) \ddot{\varvec{x}} \bigg |_{k-1} + \mathbf {\pmb {\varOmega }}\bigg ), \end{aligned}$$

(64)

where $\varvec{u}_k(k)$ and $\mathbf {\pmb {\nu }}(k)$ are bounded, if the chosen sampling time $\varDelta t$ is sufficiently small. Moreover, the above-mentioned system is asymptotically stable if and only if $\Vert \varvec{A}(k)\Vert < 1$, meaning that ${\bar{\varvec{M}}}$ must satisfy the following condition

$$\begin{aligned} \Vert \varvec{I}- \varvec{M}_T^{-1}(\varvec{x}) {\bar{\varvec{M}}}\Vert < 1. \end{aligned}$$

(65)

Appendix B

In this appendix, we examine that the Lyapunov function candidate selected in Sect. 2 is positive definite, for all $t>0$. To this end, let us restate Lemma 1 of [34] as follows

Lemma 1

Consider the continuous diagonal matrix

$$\begin{aligned} \varvec{K}_p(\varvec{e}) = diag(k_{p1}(e_1), \dots , k_{pn}(e_n)), \end{aligned}$$

Assume that there exist class K functions $\alpha _i(\cdot )$ such that

$$\begin{aligned} \xi k_{pi}(\xi ) \ge \alpha _i(|\xi |), \quad \forall \xi \in {\mathcal {R}}, \quad and \quad i = 1, \dots , n \end{aligned}$$

then

$$\begin{aligned} \int _{0}^{\varvec{e}} \mathbf {\pmb {\chi }}^T \varvec{K}_p(\mathbf {\pmb {\chi }}) d\mathbf {\pmb {\chi }} > 0, \quad \forall \varvec{e}\ne \mathbf {\pmb {0}}, \end{aligned}$$

and

$$\begin{aligned} \int _{0}^{\varvec{e}} \mathbf {\pmb {\chi }}^T \varvec{K}_p(\mathbf {\pmb {\chi }}) d\mathbf {\pmb {\chi }} \rightarrow \infty \quad as \Vert \varvec{e}\Vert \rightarrow \infty . \end{aligned}$$

$\square$

Since $k_{pi}(e_i)$ is defined in this paper as follows

$$\begin{aligned} k_{pi}(e_i) = {\left\{ \begin{array}{ll} k_{pi}|e_i|^{\alpha _p - 1}, &{} |e_i| > \delta _p \\ k_{pi} \delta _p^{\alpha _p - 1}, &{} |e_i| \le \delta _p \end{array}\right. } \end{aligned}$$

(66)

define class K functions $\alpha _i(\cdot )$ as

$$\begin{aligned} \alpha _i(|e_i|) = {\left\{ \begin{array}{ll} \beta _i |e_i|^{\alpha _p}, &{} |e_i| > \delta _p \\ \beta _i |e_i| \delta _p^{\alpha _p - 1}, &{} |e_i| \le \delta _p \end{array}\right. } \end{aligned}$$

(67)

if $k_{pi}>\beta _i > 0$, then $k_{pi}(e_i) \ge \alpha _i(|e_i|)$, and therefore, according to the above-stated lemma the function $\int _{0}^{\varvec{e}} \mathbf {\pmb {\chi }}^T \varvec{K}_p(\mathbf {\pmb {\chi }}) d\mathbf {\pmb {\chi }}$, and the Lyapunov function $V(\varvec{e}, {\dot{\varvec{e}}})$ is positive definite in $\varvec{e}\in {\mathcal {R}}^n$.