## Resolved Acceleration Control for Underwater Vehicle Manipulator Systems Continuous and Discrete Time Approach

Shinichi Sagara

Kyushu Institute of Technology Japan

### 1. Introduction

Underwater robots, especially Underwater Vehicle-Manipulator Systems (UVMS), are expected to have important roles in ocean exploration (Yuh, 1995). Many studies about dynamics and control of UVMS have been reported (Maheshi et al., 1991; McMillan et al., 1995; McLain et al., 1996; Tarn et al., 1996; Antonelli & Chiaverini, 1998; McLain et al., 1998; Antonelli et al., 2000; Sarkar & Podder, 2001). However, there are only a few experimental studies. Most of the control methods of UVMS have been proposed based on the methods of Autonomous Underwater Vehicles. In these control methods, the desired accelerations and velocities of the end-tip of the manipulator are transformed to the desired manipulator's joint accelerations and velocities only use of the kinematic relation, and the computed torque method with joint angle and angular velocity feedbacks are utilized. In other words, the control methods use errors consisting of task-space signals of vehicle and joint-space signals of manipulator. Therefore, the control performance of the end-effector depends on the vehicle's control performance.

We have proposed continuous-time and discrete-time Resolved Acceleration Control (RAC) methods for UVMS (Yamada & Sagara, 2002; Sagara, 2003; Sagara et al., 2004; Sagara et al., 2006; Yatoh & Sagara, 2007; Yatoh & Sagara, 2008). In our proposed methods, the desired joint values are obtained by kinematic and momentum equations with feedback of task-space signals. From the viewpoint of underwater robot control, parameters and coefficients of hydrodynamic models are generally used as constant values that depend on the shape of the robots (Fossen, 1994). Our proposed methods described above can reduce the influence of the modelling errors of hydrodynamics by position and velocity feedbacks. The effectiveness of the RAC methods has been demonstrated by using a floating underwater robot with vertical planar 2-link manipulator shown in Figure 1.

In this chapter, our proposed continuous-time and discrete-time RAC methods are described and the both experimental results using a 2-link underwater robot are shown. First, we explain about a continuous-time RAC method and show that the RAC method has good control performance in comparison with a computed torque method. Next, to obtain higher control performance, we introduce a continuous-time RAC method with disturbance compensation. In practical systems digital computers are utilized for controllers, but there is no discrete-time control method for UVMS except our proposed methods. Then, we address discrete time RAC methods including the ways of disturbance compensation and avoiding singular configuration.

Fig. 1. Vertical type 2-link underwater robot

2. Modelling

The UVMS model used in this chapter is shown in Figure 2. It has a robot base (vehicle) and an n-DOF manipulator.

Manipulator

Fig. 2. Model of underwater robot with n-link manipulator

Manipulator

Fig. 2. Model of underwater robot with n-link manipulator

The symbols used in this chapter are defined as follows:

w : number of joints

Xj: inertial coordinate frame

Xi: link i coordinate frame (i = 0, 1, 2, • • •, n; link 0 means the vehicle)

1 Ri: coordinate transformation matrix from X to Xj pe: position vector of the end-tip of the manipulator with respect to Xj

Pi: position vector of the origin of Xi with respect to Xj

Vi: position vector of the center of gravity of link i with respect to Xj

</>l: relative angle of joint i y/0 : roll-pitch-yaw attitude vector of Zq with respect to Zi yfe: roll-pitch-yaw attitude vector of the end-tip of the manipulator with respect to Zi ®0 : angular velocity vector of Zq with respect to Zi me: angular velocity vector of the end-tip of the manipulator with respect to Zi

( : relative joint angle vector (= [( • • • (n )

iki: unit vector indicating a rotational axis of joint i (= [0 0 1]) mi: mass of link i

1Ma : added mass matrix of link i with respect to Zi 1 Ii: inertia tensor of link i with respect to Zi 11a : added inertia tensor of link i with respect to Zi

Xq : position and orientation vector of Zq with respect to Zi (= [vq ]T )

xe: position and orientation vector of the end-tip with respect to Zi (= [pT ^J ]T)

Vq : linear and angular vector of Xq with respect to Zi (= [Vq aT ]T)

Ve: linear and angular vector of the end-tip with respect to Zi (= [pJ a>T ]T )

li: length of link i ag : position vector from joint i to the center of gravity of link i with respect to Zi a-fr : position vector from joint i to the buoyancy center of link i with respect to Zi

Di : width of link i

Vi : volume of link i p : fluid density

Cd : drag coefficient of link i g: gravitational acceleration vector Ej : j x j unit matrix

~ : tilde operator stands for a cross product such that ra = r x a 2.1 Kinematics

First, from Figure 2 a time derivative of the end-tip position vector pe is

On the other hand, relationship between end-tip angular velocity and joint velocity is expressed with n

From Equations (1) and (2) the following equation is obtained:

where

0 e3

Next, let n and f be a linear and an angular momentum of the robot including hydrodynamic added mass tensor i Ma and added inertia tensor i1 a of link i. Then n = Z Mjri,

where MT = miE3+1 Ri iMa i RI and IT =IR (iIi +iI ) iRI. Here, linear and angular velocities of the center of gravity of link i are described as ri = ro + ®0

Therefore, the following equation is obtained from Equations (4)-(7):

where

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