A note on “Fractional-order adaptive backstepping control of robotic manipulators in the presence of model uncertainties and external disturbances”

Abstract

This article extends the finite-time control and stabilization of the paper “fractional-order adaptive back stepping control of robotic manipulators in the presence of model uncertainties and external disturbances” to voltage level control. In practical applications, bounded voltage is applied to the robot joint motors. This fact is presumed in designing the controller and analyzing the stability. It is proved that the controlled system is robust and BIBO stable. The uniformly boundedness of the joint angle tracking error is also validated utilizing the Lyapunov lemma. The presented approach is less conservative than the work presented by the aforementioned article, since it does not require any knowledge of the manipulator dynamic parameters and their upper bound. The experimental outcomes validate the acceptable functioning of the presented controller.

Appendix 1

The proof is similar to [15]. Let us consider (5) in SISO form. If both sides of (5) are multiplied by $I_{a}$, the following power equation is obtained.

$$I_{a} v(t) = L_{a} I_{a} \dot{I}_{a} + R_{a} I_{a}^{2} + k_{b} I_{a} \dot{\theta }$$

(35)

The following is a translation of Eq. (35). The electrical power indicated by $I_{a} v(t)$ is used by the motor to provide the mechanical power expressed by $k_{b} I_{a} \dot{\theta }$ in (35). The term $R_{a} I_{a}^{2}$ stands for the loss power in the winding. The power $L_{a} I_{a} \dot{I}_{a}$ is the magnetic energy derivative. Based on (35), it is concluded that for $t \ge 0$:

$$\int\limits_{0}^{t} {I_{a} v(\tau )d\tau } = \int\limits_{0}^{t} {L_{a} I_{a} \dot{I}_{a} d\tau } + \int\limits_{0}^{t} {R_{a} I_{a}^{2} d\tau } + \int\limits_{0}^{t} {k_{b} I_{a} \dot{\theta }d\tau }$$

(36)

With $I_{a} (0) = 0$, (36) is

$$\int\limits_{0}^{t} {I_{a} v(\tau )d\tau } = 0.5L_{a} I_{a}^{2} + R_{a} I_{a}^{2} t + \int\limits_{0}^{t} {k_{b} I_{a} \dot{\theta }d\tau }$$

(37)

Since $R_{a} I_{a}^{2} t \ge 0$ and $L_{a} I_{a}^{2} \ge 0$

$$\int\limits_0^t {{k_b}{I_a}\dot \theta d\tau } \le \int\limits_0^t {{I_a}v(\tau )d\tau }$$

(38)

So, (39) gives the upper bound of mechanical energy.

$$\int\limits_{0}^{t} {k_{b} I_{a} \dot{\theta }d\tau } = \int\limits_{0}^{t} {I_{a} v(\tau )d\tau }$$

(39)

Hence, it is clear that at the mechanical energy upper bound (40) holds.

$$k_{b} \dot{\theta } = v(t)$$

(40)

So, $\dot{\theta }$ is bounded as follows.

$$\left| {k_{b} \dot{\theta }} \right| \le \left| {v(t)} \right|$$

(41)

Based on remark 1 and Assumption 3, we have

$$\left| {\dot{\theta }} \right| \le \underline{{k_{b} }}^{ - 1} \xi_{u} \triangleq \xi_{{\dot{\theta }}}$$

(42)

where $\xi_{{\dot{\theta }}}$ stands for the maximum value of the actuator joint velocity vector. Considering (42) in SISO form one have

$$L_{a} \dot{I}_{a} + R_{a} I_{a} = \omega$$

(43)

where

$$\omega = v(t) - k_{b} \dot{\theta }$$

(44)

is limited where $v(t)$ and $\dot{\theta }$ are limited as presented in remark 1, and (42), respectively. As a result, the input $\omega$ in (43) is bounded. Regarding the Routh–Hurwitz criterion, the stability of the linear differential Eq. (43) is confirmed. The boundedness of the input $\omega$ leads to the output $I_{a}$ boundedness by $\xi_{I}$. Based on (43), it is clear that

$$L_{a} \dot{I}_{a} = \omega - R_{a} I_{a}$$

(45)

Therefore, the boundedness of $\omega$ and $I_{a}$ results in the boundedness of $\dot{I}_{a}$ by $\xi_{{\dot{I}}}$. Furthermore, boundedness of $\dot{q}$ can be concluded from definition $\dot{q} = r\dot{\theta }$, Assumption 3 and Eq. (42).