Model-based motion control for underwater vehicle-manipulator systems with one of the three types of servo subsystems

Abstract

This paper deals with a motion control scheme for underwater robots equipped with manipulators. In general, for each robot, its manipulator is directly driven by electric motors with position sensors such as encoders, whereas its robot body is propelled by marine thrusters with position sensors such as inertial measurement units. It has been pointed out in the literature that for this type of underwater robots, its robot body control is more challenging than its manipulator control, because the robot body has much larger inertia, and many more inaccurate position sensors and actuators than the manipulator. Therefore, it may be practically difficult to accurately control the motion of the robot body, even if an advanced control scheme with good performance is implemented in the controller of the robot body. In this paper, we develop a model-based motion controller for the manipulator under the condition that the robot body is independently controlled by a motion controller with poor performance. Its features are to design the manipulator controller in consideration of the dynamics of the robot body including the marine thrusters as well as those of the manipulator, and to be applicable to underwater robots equipped with one of the three types of servo systems (i.e., a voltage-controlled, a torque-controlled, and a velocity-controlled servo system). Some stability properties of the control system were theoretically ensured in the stability analysis. Furthermore, these results were supported by those obtained in numerical simulations.

Appendices

Appendix 1: Derivation of (5)

For simplicity, we express $\exp (\cdot )$ as ${e}^{(\cdot )}$ in this appendix. Substituting the second equation of (3) into the time derivative of the first equation of (3), we have

$$\begin{aligned} \dot{f}_{V}(\cdot )=\, & {} [\dot{T}(\cdot )-T(\phi _{V})C_{1}D(v)] C_{3}D(v)v(t) \nonumber \\&+T(\phi _{V})C_{2}C_{3} D(v)\tau _{V}(t). \end{aligned}$$

(90)

In the derivation of (90), we use the differential relation $d\{ D(v)v(t)\} /dt =2D(v)\dot{v}(t)$. Substituting (90) into the time derivative of (2), we obtain

$$\begin{aligned}&d\{ M_{V}(\phi ) \ddot{x}_{V}(t) \} /dt +d\{ M_{VM}(\phi ) \ddot{q}(t)\} /dt \nonumber \\&\qquad +d\{ n_{V}(\cdot )+d_{V}(t)\} /dt \nonumber \\&\quad =[ \dot{T}(\cdot )-T(\phi _{V})C_{1} D(v)] C_{3}D(v)v(t) \nonumber \\&\qquad +T(\phi _{V})C_{2}C_{3}D(v)\tau _{V}(t). \end{aligned}$$

(91)

Replacing the symbol t by the symbol $\bar{\tau }$ in (91), then multiplying both its sides by ${e}^{-\rho (t-\bar{\tau })}$, and then integrating both its sides from 0 to t with respect to $\bar{\tau }$, we have

$$\begin{aligned} s_{V1}(t)+s_{V2}(t)+s_{V3}(t)=s_{V4}(t) \end{aligned}$$

(92)

$$\begin{aligned} s_{V1}(t)= & {} \int _{0}^{t} {e}^{-\rho (t-\bar{\tau })} [ d\{ M_{V}(\phi (\bar{\tau })) \ddot{x}_{V}(\bar{\tau }) \} /d\bar{\tau } ] d\bar{\tau } \nonumber \\ s_{V2}(t)= & {} \int _{0}^{t} {e}^{-\rho (t-\bar{\tau })} [ d\{ M_{VM}(\phi (\bar{\tau })) \ddot{q}(\bar{\tau })\} /d\bar{\tau } ] d\bar{\tau } \nonumber \\ s_{V3}(t)= & {} \int _{0}^{t} {e}^{-\rho (t-\bar{\tau })} [ d\{ n_{V}(\phi (\bar{\tau }),\dot{\varphi }(\bar{\tau })) \nonumber \\&+d_{V}(\bar{\tau }) \} /d\bar{\tau } ] d\bar{\tau } \nonumber \\ s_{V4}(t)= & {} \int _{0}^{t} {e}^{-\rho (t-\bar{\tau })} \{ [ \dot{T}(\phi _{V}(\bar{\tau }),\dot{\phi }_{V}(\bar{\tau })) \nonumber \\&-T(\phi _{V}(\bar{\tau }))C_{1} D(v(\bar{\tau }))] C_{3}D(v(\bar{\tau }))v(\bar{\tau }) \nonumber \\&+T(\phi _{V}(\bar{\tau }))C_{2}C_{3}D(v(\bar{\tau }))\tau _{V}(\bar{\tau }) \} d\bar{\tau }. \end{aligned}$$

(93)

It should be noted that this operation is equivalent to applying the stable first-order filter $1/(s+\rho )[\, \cdot \,]$ to (91). Applying the integration by parts to the first equation of (93), we have

$$\begin{aligned} s_{V1}(t)=\, & {} M_{V}(\phi )\ddot{x}_{V}(t)-{e}^{-\rho t} M_{V}(\phi (0))\ddot{x}_{V}(0)\nonumber \\&-\rho {e}^{-\rho t} \int _{0}^{t} {e}^{\rho \bar{\tau }} M_{V} (\phi (\bar{\tau })) \ddot{x}_{V}(\bar{\tau })d\bar{\tau }. \end{aligned}$$

(94)

Reapplying the integration by parts to (94), we obtain

$$\begin{aligned} s_{V1}(t)=\, & {} M_{V}(\phi )\ddot{x}_{V}(t)-\rho M_{V}(\phi )\dot{x}_{V}(t)\nonumber \\&+{e}^{-\rho t} [ - M_{V}(\phi (0))\ddot{x}_{V}(0)+\rho M_{V}(\phi (0))\dot{x}_{V}(0)] \nonumber \\&+\rho \int _{0}^{t} {e}^{-\rho (t-\bar{\tau })} [ \rho M_{V} (\phi (\bar{\tau })) \nonumber \\&+\dot{M}_{V}(\phi (\bar{\tau }),\dot{\phi }(\bar{\tau })) ] \dot{x}_{V}(\bar{\tau }) d\bar{\tau }. \end{aligned}$$

(95)

In a way similar to the derivation of (95), we can rewrite the second equation of (93) as

$$\begin{aligned} s_{V2}(t)=\, & {} M_{VM}(\phi )\ddot{q}(t)-\rho M_{VM}(\phi )\dot{q}(t)\nonumber \\&+{e}^{-\rho t} [ - M_{VM}(\phi (0))\ddot{q}(0)+\rho M_{VM}(\phi (0))\dot{q}(0)] \nonumber \\&+\rho \int _{0}^{t} {e}^{-\rho (t-\bar{\tau })} [ \rho M_{VM} (\phi (\bar{\tau })) \nonumber \\&+\dot{M}_{VM}(\phi (\bar{\tau }),\dot{\phi }(\bar{\tau })) ] \dot{q}(\bar{\tau }) d\bar{\tau }. \end{aligned}$$

(96)

Applying the integration by parts to the third equation of (93), we have

$$\begin{aligned} s_{V3}(t)=\, & {} n_{V}(\phi ,\dot{\varphi })+d_{V}(t)\nonumber \\&-{e}^{-\rho t} [ n_{V}(\phi (0),\dot{\varphi }(0))+d_{V}(0)] \nonumber \\&-\rho \int _{0}^{t} {e}^{-\rho (t-\bar{\tau })} [ n_{V}(\phi (\bar{\tau }),\dot{\varphi }(\bar{\tau }))+d_{V}(\bar{\tau }) ] d\bar{\tau }. \end{aligned}$$

(97)

Substituting (93) (the fourth equation) and (95) to (97) into (92), and then considering the fact that the first-order filters (7) have the solution

$$\begin{aligned} w_{VF1}(t)=\, & {} {e}^{-\rho t} w_{VF10} \nonumber \\ w_{VF2}(t)= & {} -\rho \int _{0}^{t} {e}^{-\rho (t-\bar{\tau })} d_{V}(\bar{\tau }) d\bar{\tau } \nonumber \\ z_{VF}(t)= & {} \int _{0}^{t} {e}^{-\rho (t-\bar{\tau })} \{ \rho [ \rho M_{V}(\phi (\bar{\tau })) \nonumber \\&+\dot{M}_{V}(\phi (\bar{\tau }),\dot{\phi }(\bar{\tau })) ] \dot{x}_{V}(\bar{\tau })+\rho [ \rho M_{VM}(\phi (\bar{\tau })) \nonumber \\&+\dot{M}_{VM}(\phi (\bar{\tau }),\dot{\phi }(\bar{\tau })) ] \dot{q}(\bar{\tau }) -\rho n_{V}(\phi (\bar{\tau }),\dot{\varphi }(\bar{\tau })) \nonumber \\&-[ \dot{T}(\phi _{V}(\bar{\tau }),\dot{\phi }_{V}(\bar{\tau })) \nonumber \\&-T(\phi _{V}(\bar{\tau }))C_{1}D(v(\bar{\tau }))] C_{3} D(v(\bar{\tau })) v(\bar{\tau }) \nonumber \\&-T(\phi _{V}(\bar{\tau }))C_{2}C_{3}D(v(\bar{\tau }))\tau _{V}(\bar{\tau })\} d\bar{\tau }, \end{aligned}$$

(98)

we obtain the model (5). In this derivation, we use (6).

Appendix 2: Derivations of (21) and (23)

For simplicity, we express $\exp (\cdot )$ as ${e}^{(\cdot )}$ in this appendix. Using the first equation of (6), we can write the norm of $z_{C}(t)$ as

$$\begin{aligned} \Vert z_{C}(t) \Vert\le & {} c_{N1}+(c_{M1}+c_{M2}) \rho \Vert \dot{\varphi }(t) \Vert +c_{N2} \Vert \dot{\varphi }(t) \Vert ^{2} \nonumber \\&+\Vert z_{VF}(t) \Vert . \end{aligned}$$

(99)

In the derivation of (99), we utilize (17) (the first and second inequalities), (19) (the first inequality), and the inequalities $\Vert \dot{q}(t) \Vert \le \Vert \dot{\varphi }(t) \Vert$ and $\Vert \dot{x}_{V}(t) \Vert \le \Vert \dot{\varphi }(t) \Vert$. Using the solution of the third differential equation of (7) [i.e., the third equation of (98)], we can write the norm of $z_{VF}(t)$ as

$$\begin{aligned} \Vert z_{VF}(t) \Vert\le & {} \bar{c}_{VF1} \int _{0}^{t} {e}^{-\rho (t-\bar{\tau })} [ \rho +\rho ^{2} \Vert \dot{\varphi } (\bar{\tau }) \Vert \nonumber \\&\quad +\rho \Vert \dot{\varphi } (\bar{\tau }) \Vert ^{2}+\Vert \dot{\varphi } (\bar{\tau }) \Vert \Vert v(\bar{\tau }) \Vert ^{2} +\Vert v(\bar{\tau }) \Vert ^{3} \nonumber \\&\quad +\Vert v(\bar{\tau }) \Vert \Vert \tau _{V}(\bar{\tau }) \Vert ] d\bar{\tau } \end{aligned}$$

(100)

where $\bar{c}_{VF1} \in R^{+}$ is given by

$$\begin{aligned} \bar{c}_{VF1}= & {} \max \{ c_{N1},c_{M1}+c_{M2},c_{M4}+c_{M5}+c_{N2}, \nonumber \\&c_{T2}c_{V1}\Vert C_{3} \Vert , c_{T1}c_{V1}^{2}\Vert C_{1} \Vert \Vert C_{3} \Vert , c_{T1}c_{V1}\Vert C_{2}C_{3} \Vert \}. \nonumber \\ \end{aligned}$$

(101)

In the derivation of (100), we utilize (17) (the first, second, fourth, and fifth inequalities), (19) (the first inequality), (20), and the inequalities $\Vert \dot{x}_{V}(t) \Vert \le \Vert \dot{\varphi }(t) \Vert$ and $\Vert \dot{q}(t) \Vert \le \Vert \dot{\varphi }(t) \Vert$. In view of the fact that the time integral in (100) is equivalent to the solution of the first-order differential equation (22), we can rewrite the inequality (100) as

$$\begin{aligned} \Vert z_{VF}(t) \Vert \le \bar{c}_{VF1} z_{QF}(t). \end{aligned}$$

(102)

Applying (102) to (99), we have

$$\begin{aligned} \Vert z_{C}(t) \Vert\le\, & {} c_{N1}+(c_{M1}+c_{M2}) \rho \Vert \dot{\varphi }(t) \Vert +c_{N2} \Vert \dot{\varphi }(t) \Vert ^{2} \nonumber \\&+\bar{c}_{VF1} z_{QF}(t). \end{aligned}$$

(103)

Furthermore, applying (103) to the norm of $z_{Q}(t)$ [given by the second equation of (12)], we obtain

$$\begin{aligned} \Vert z_{Q}(t) \Vert\le & {} c_{M2}c_{M7}c_{N1}+c_{N3}+c_{M2}c_{M7}(c_{M1} \nonumber \\&+c_{M2}) \rho \Vert \dot{\varphi }(t) \Vert + (c_{N4}+c_{M2}c_{M7}c_{N2}) \Vert \dot{\varphi }(t) \Vert ^{2}\nonumber \\&+c_{M2}c_{M7}\bar{c}_{VF1}z_{QF}(t), \end{aligned}$$

(104)

and we can directly obtain (21) from (104). In the derivation of (104), we utilize (17) (the second and seventh inequalities) and (19) (the second inequality).

In view of the condition that the disturbances $d_{V}(t)$ and $d_{M}(t)$ are bounded, we obtain the inequalities

$$\begin{aligned} \Vert d_{V}(t) \Vert \le c_{D1},\; \Vert d_{M}(t) \Vert \le c_{D2} \end{aligned}$$

(105)

where $c_{D1}$ and $c_{D2}$ are positive constants. Using the second equation of (6), we can write the norm of $w_{C}(t)$ as

$$\begin{aligned} \Vert w_{C}(t) \Vert \le c_{D1}+\Vert w_{VF1}(t) \Vert +\Vert w_{VF2}(t) \Vert . \end{aligned}$$

(106)

In the derivation of (106), we utilize the first inequality of (105). Using the solution of the first differential equation of (7) [i.e., the first equation of (98)], we can write the norm of $w_{VF1}(t)$ as

$$\begin{aligned} \Vert w_{VF1}(t) \Vert \le \Vert w_{VF10} \Vert . \end{aligned}$$

(107)

Similarly, using the solution of the second differential equation of (7) [i.e., the second equation of (98)], we can write the norm of $w_{VF2}(t)$ as

$$\begin{aligned} \Vert w_{VF2}(t) \Vert \le c_{D1}. \end{aligned}$$

(108)

In the derivation of (108), we utilize the first inequality of (105). Applying (107) and (108) to (106), we obtain

$$\begin{aligned} \Vert w_{C}(t) \Vert \le 2c_{D1}+\Vert w_{VF10} \Vert . \end{aligned}$$

(109)

Furthermore, using the third inequality of (12), we can write the norm of $w_{Q}(t)$ as

$$\begin{aligned} \Vert w_{Q}(t) \Vert \le c_{D2}+2c_{D1}c_{M2}c_{M7}+c_{M2}c_{M7}\Vert _{VF10} \Vert , \end{aligned}$$

(110)

and we can directly obtain (23) from (110). In the derivation of (110), we utilize (17) (the second and seventh inequalities), (105) (the second inequality), and (109).