Quaternion-Based Coordinated Control of a Subsea ... - CiteSeerX

1995 IEEE Conference on Decision and Control ( New Orleans, LA )

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Quaternion-Based Coordinated Control of a Subsea Mobile Manipulator with Only Position Measurements Fernando Lizarralde



Dept. of Electrical Engineering COPPE/Federal University of Rio de Janeiro 21945/970 - Rio de Janeiro - BRAZIL e-mail: [email protected]

John T. Wen

Center for Advanced Technology in Automation and Robotics Dept. of Electrical, Computer and Systems Eng. Rensselaer Polytechnic Institute - Troy, NY 12180 e-mail: [email protected] Liu Hsu

Dept. of Electrical Engineering COPPE/Federal University of Rio de Janeiro 21945/970 - Rio de Janeiro - BRAZIL e-mail: [email protected]

Abstract

A control scheme for the coordinated control of a Subsea vehicle/manipulator system is presented. The unit quaternion representation is used in a singularity free representation of the attitude. The control scheme is formulated in a 12 d.o.f. task space and only position measurements are required. The performance is evaluated by simulation with a realistic model of a commercial underwater vehicle, which includes the full dynamic interaction between vehicle and manipulator.

1 Introduction

An underwater vehicle equipped with a manipulator is used in several intervention tasks, e.g., cutting, welding, valve manipulation, maintenance and construction of underwater structures, etc. The manual control of this multi-body system is a long and stressing task. The operating period and the performance for a given task are limited due to operator fatigue. An option is an automatic combined control in order to keep a desired position and orientation of the tool. Several works have addressed the problem. In [17] an ecient dynamic simulation algorithm was developed for the complete system. A detailed model of the underwater vehicle-manipulator system was also proposed by Schjlberg and Fossen [20]. In [8], an adaptive macro-micro control based on the Slotine-Li scheme [21] was developed. In [6, 15] the related problem of a spacecraft/manipulator was considered. A manipulator mounted in a wheeled platform was considered by Yamamoto and Yun [22]. In all those control strategies the velocity measurement is required. However this assumption may not always be satis ed (e.g., the manipulator cannot be equipped with tachometers). For manipulator control, the justi cation for the use of velocity-free  Supported by

the Brazilian Research Council (CNPq).

PD schemes has been presented recently [1, 2, 11]. In [14], a velocity-less PD for the attitude control problem, which use an unit quaternion representation, was also proposed. In this paper, we show that a PD-based control strategy can be extended to the case where vehicle/joints velocities are not available. The velocity is replaced by the use of a non-linear lead lter. Global asymptotic stability is shown by using the Invariance Set Theorem [12]. The main complication comes from the fact that the angular velocity cannot be integrated to an equivalent position variable. We show that in the case of the unit quaternion representation, this diculty can be overcome by noting that the columns of the representation Jacobian are of unit length and mutually orthogonal. The redundancy created for combining the two structures (underwater vehicle and manipulator) is solved de ning a preferred con guration for the manipulator [23]. Thus, the manipulator is controlled to perform a manipulation task, and the vehicle is controlled to bring the manipulator to this preferred con guration. This paper is organized as follows. In Section 2, the unit quaternion representation is presented. Section 3 presents the mathematical modeling of the complete system. The control strategy and the global asymptotic stability are shown in Section 4. Section 5 presents some simulation results and the paper is closed with some conclusions.

2 Unit quaternion representation

There are many possible parameterizations of the manifold SO(3) on which the orientation evolves. There are 9 parameters in the attitude matrix R, subject to 6 constraints imposed by the orthogonality. For manipulation, analysis, and implementation reasons, frequently it is simpler to use other representations. The minimal

number of parameters needed to represent R is three with no constraint. Some common minimal representation are: Euler angles, Gibbs vector, unit equivalent axis/angle, and vector quaternion. Minimal representations are only locally one-to-one and onto mapping of the attitude matrix, and there are always singular orientations. The minimal number of parameters that can globally represent attitude without singularities is four, with one constraint equation. The unit quaternion (or Euler parameters) is a popular nonsingular four-parameter representation due to their desirable computational properties [10]. On the other hand, a common representation of the orientation for underwater remote vehicles is the Euler angles, the well-known quantities roll-pitch-yaw. This representation is not de ned for pitch angle  = 90 degrees. However, during practical operations with marine vehicles the parameter region  = 90 degrees is not likely to be entered. Thus, unit quaternion could be a questionable choice, however, in this paper we are interested in the control of the manipulator end-eector, which could assume any posture depending on the task. Thus, a singularity-free representation for the orientation, as unit quaternion, is totally justi ed. Consider two orthonormal right-handed coordinate frames: the inertial (world) coordinate, E I = [eI1; eI2; eI3 ], and body coordinate (attached to a rigid body), E B = [eB1 ;eB2 ; eB3 ]. De ne the 3  3 attitude matrix as R = E I E B where (
) denotes the adjoint operation. The unit quaternion representation of the attitude matrix R is de ned by:

qT = [q0 q1 q2 q3] = [q0 qTv ]

In general q both represent R and this sign ambiguity can be resolved by using the kinematic equation below: (6) q_ = 12 E(q)!;   T E(q) = q0I ?+q(vq ) (7) v

where ! is the angular velocity in body coordinates. The Jacobian E (q) satis es the following important properties: E(q)T E(q) = I 33; E(q)T q = 0; (8) Consequently, from (7) and (8), the angular velocity can be obtained from q_ as: ! = 2ET (q)q_ (9)

3 Mathematical Modeling

In this Section the system dynamic and kinematic equations of motion are discussed.

3.1 Kinematic equations

The kinematic model describes the geometrical relationship between motions in two dierent frames. In what follows, we describe the kinematic of each one of the subsystems.

3.1.1 Vehicle kinematic

The transformation matrix J B 2