Robust decentralized force/position fault-tolerant control for constrained reconfigurable manipulators without torque sensing

Abstract

A major technical challenge in decentralized fault-tolerant control for a torque sensorless constrained reconfigurable manipulators is associated with decentralized control of the constraint force. This paper proposed a novel decentralized force/position control method based on the motor-side and link-side position measurements along with harmonic drive model for each joint module of constrained reconfigurable manipulators. In addition, a modified sliding mode controller is designed to guarantee force/position tracking performance, and the unknown non-affine actuator fault in independent subsystem can be compensated by adding an adaptive compensation term to the sliding mode controller. The stability of closed-loop system is analyzed using the Lyapunov method. Finally, simulations are performed to verify the advantages of the proposed method.

Appendix: Joint torque estimation

Considering the constrained reconfigurable module manipulators that consists of n-modules, each module provides an independently rotating joint with harmonic drive transmission. The graph in Fig. 13 [36] depicts the main components of harmonic drives. The wave generator (WG) is connected to a motor, the circular spline (CS) is connected to the joint base, and the flexspline (FS) is sandwiched in between CS and WG and connected to the joint output.

Owing to flexibility of FS and WG, a kinematic representation of a harmonic drive is illustrated in Fig. 14. Therefore, FS and WG torsion are defined as follows:

$$\begin{aligned} \varDelta {\theta _{fi}}= & {} {\theta _{foi}} - {\theta _{fsi}},\end{aligned}$$

(53)

$$\begin{aligned} \varDelta \theta _{gi}= & {} {\theta _{goi}} - {\theta _{gsi}}, \end{aligned}$$

(54)

where ${\theta _{fsi}}$ and ${\theta _{foi}}$ refer to the angular position at the FS gearside (gear-toothed circumference) and the FS angular position at the load side which is measured using the link-side encoder, respectively. ${\theta _{ goi}}$ and ${\theta _{gsi}}$ denote the positions of the WG outside part (ball bearing outer rim) and the center part (WG plug), respectively.

The total torsional angle of harmonic drive can be described by:

$$\begin{aligned} \varDelta {\theta _i} = {\theta _{foi}} + \frac{{{\theta _{gsi}}}}{{{\lambda _i}}}. \end{aligned}$$

(55)

By adding and subtracting the terms ${\theta _{fsi}}$ and ${\theta _{ goi}}$ to (55), one obtains:

$$\begin{aligned} \varDelta {\theta _i}= & {} {\theta _{foi}} - {\theta _{fsi}} + \left( {{\theta _{fsi}} + \frac{{{\theta _{goi}}}}{{{\lambda _i}}}} \right) - \left( {\frac{{{\theta _{goi}}}}{{{\lambda _i}}} - \frac{{{\theta _{gsi}}}}{{{\lambda _i}}}} \right) \nonumber \\= & {} \varDelta {\theta _{fi}} - \varDelta {\theta _{gi}}/{\lambda _i} + {\theta _{\mathrm{err}i}}, \end{aligned}$$

(56)

where ${\theta _{\mathrm{err}i}}$ denotes the kinematic error of harmonic drive transmission. Based on the assumption that the relative motion between the WG output and the FS input can be ignored, we can conclude that ${\theta _{\mathrm{err}i}} = 0$ [36]. The FS and WG torque can be modeled as:

$$\begin{aligned} {\tau _{fi}}= & {} {L_f}\varDelta {\theta _{fi}}, \end{aligned}$$

(57)

$$\begin{aligned} {\tau _{ gi}}= & {} {L_g}\varDelta {\theta _{gi}}, \end{aligned}$$

(58)

where ${L_f}$ and ${L_g}$ are the stiffness of FS and WG, respectively.

Because the torsional angle is described a function of the torque, the local elastic coefficient increases as ${\tau _{fi}}$ increases. Therefore, one can define the local elastic coefficient ${L_{fk}}$ as:

$$\begin{aligned} {L_{fk}} = \frac{{\mathrm{d}{\tau _{fi}}}}{{\mathrm{d}\varDelta {\theta _{fi}}}} = {L_{fo}}\left( {1 + {{\left( {{a_f}{\tau _{fi}}} \right) }^2}} \right) , \end{aligned}$$

(59)

where ${L_{fo}}$ and ${a_f}$ are constants to be determined. If ${L_{fo}} \ne 0$ , then the FS torsion can be counted as:

$$\begin{aligned} \varDelta {\theta _{fi}} = \int _0^{{\tau _{fi}}} {\frac{{\mathrm{d}{\tau _{fi}}}}{{{L_{fk}}}}} = \frac{{\arctan \left( {{a_f}{\tau _{fi}}} \right) }}{{{a_f}{L_{fo}}}}. \end{aligned}$$

(60)

Note that the harmonic drive deformation deformation range drops down to zero sharply at rated torque, which implies that the stiffness of WG increase sharply. In order to replicate the hysteresis shape of this stiffness property, the local elastic coefficient of WG can be modeled as:

$$\begin{aligned} {L_{gk}} = {L_{go}}{\mathrm{e}^{{a_g}\left| {{\tau _{gi}}} \right| }}, \end{aligned}$$

(61)

where ${L_{ go}}$ and ${a_{g}}$ are constants to be determined. If ${L_{go}} \ne 0$ , then the WG torsional angle can be calculated using the following relation:

$$\begin{aligned} \varDelta {\theta _{gi}} = \int _0^{{\tau _{gi}}} {\frac{{\mathrm{d}{\tau _{gi}}}}{{{L_{gk}}}}} = \frac{{sign\left( {{\tau _{gi}}} \right) }}{{{a_g}{L_{go}}}}\left( {1 - {\mathrm{e}^{ - {a_g}\left| {{\tau _{gi}}} \right| }}} \right) . \end{aligned}$$

(62)

Finally, by substituting the FS and WG deformation given in  (60) and  (62) into  (56), the total deformation of the harmonic drive can be expressed as:

$$\begin{aligned} \varDelta {\theta _i} = \frac{{\arctan \left( {{a_f}{\tau _{fi}}} \right) }}{{{a_f}{L_{fo}}}} - \frac{{sign\left( {{\tau _{gi}}} \right) }}{{{a_g}{\lambda _i}{L_{go}}}}\left( {1 - {\mathrm{e}^{ - {a_g}\left| {{\tau _{gi}}} \right| }}} \right) . \end{aligned}$$

(63)

Then,

$$\begin{aligned} {\tau _{fi}}= & {} \frac{1}{{{a_f}}}\tan \left( {{a_f}{L_{fo}}\left( {\varDelta {\theta _i}} + \frac{{sign\left( {{\tau _{gi}}} \right) }}{{{a_g}{\lambda _i}{L_{go}}}}\right. } \right. \nonumber \\&\left. \left. \left( {1 - {\mathrm{e}^{ - {a_g}\left| {{\tau _{gi}}} \right| }}} \right) \right) \right) , \end{aligned}$$

(64)

where the WG torque ${\tau _{gi}}$ can be approximated by the motor torque command.

By the following formula, one can get the constrained torque, which is obtained by the constrained force on the end-effector of manipulator:

$$\begin{aligned} {\tau _{ci}} = {\tau _{fie}} - {\tau _{fio}}, \end{aligned}$$

(65)

where $\tau _{fio}$ denotes the joint torque which is obtained in free space, $\tau _{fie}$ denotes the total joint torque in the constrained space. The total joint torque $\tau _{fie}$ and/or joint torque in free space $\tau _{fio}$ is directly obtained from formula  (64) under the condition of constrained space and free space, respectively.