[Paper] Bottom following preview controller of autonomous underwater vehicle

Summary

This article describes a solution to the bottom-following problem of an underwater vehicle based on the echo sounder's evaluation of the terrain features in front of the underwater vehicle. The method used treats the problem as a discrete-time path tracking control problem, in which the factory's easily defined error state space model is augmented with bathymetric (ie depth) preview data. The segmented affine parameter correlation model is used to describe the linearization error dynamics of the underwater robot in a set of predetermined working areas. For each region, the state feedback and control problem of the affine parameter-dependent system is proposed and solved by linear matrix inequalities (LMIs). The resulting nonlinear controller is implemented as a gain scheduling controller using the D-method. The simulation results of the underwater robot on the vertical plane based on the nonlinear dynamic model are given and discussed.

Index terms—discrete time system, linear parameter change system, preview control, tracking, transient response, underwater vehicle

Introduction

This article describes a solution to the design problem of the bottom following controller of an autonomous underwater vehicle, which explicitly considers the characteristics of the water depth in front of the vehicle measured by two echo sounders. The method used belongs to the category of pre-control theory. The proposed solution was evaluated by simulating a prototype Invent underwater robot model (see Figure 1), which was built and operated by the Polytechnic Institute of Lisbon, Portugal.

Preview control algorithms have been widely used to improve closed-loop performance. When future information about the environment reflected in information about future commands and interference is available, a feedback compensator with limited bandwidth can be used to obtain closed-loop performance. A series of papers on the application of linear quadratic preview control theory in vehicle active suspension design can be found in the literature.

Special emphasis should be placed on Tomizuka’s pioneering work, in which the optimal preview control problem was formulated and solved, and the impact of different preview lengths on the performance of the entire suspension system was discussed. Prokop and Sharp [2] proposed an alternative method that includes incorporating disturbances or reference dynamics into the design model, and then solving the resulting linear quadratic control problem. Recently, Takaba [3] used linear matrix inequalities to solve the foreseeable robust servo mechanism design problem for polyhedral uncertain systems.

For the design of linear control system, this paper uses a discrete-time state feedback foreseeing controller synthesis algorithm. Among the methods studied in this paper, the results given in documents [3]-[6] are used to develop a linear matrix inequality (LMI)-based predictive controller synthesis algorithm for affine parameter-dependent systems. For large preview intervals, the technique proposed in this paper leads to LMI optimization problems involving a large number of variables. In order to overcome this limitation, an alternative algorithm is proposed that uses the specific structure of the enhanced preview system to calculate the required feedforward gain matrix.

In this paper, for a limited number of discrete-time object models dependent on piecewise affine parameters, a linear state feedback predictive controller is synthesized. Each of these models is composed of discrete equivalents of the generalized error linearization of each underwater robot's work area determined by well-defined boxes in the parameter space (defined by the total speed and angle of attack of the aircraft). The error space adopted is consistent with the solutions given in [7]–[9], and includes an important directivity coefficient that takes the current vehicle direction into consideration in the definition of the line speed error. The author applies a similar technique to the problem of gyroplane terrain tracking, see [10] for details.

The final realization of the nonlinear gain scheduling controller uses the D-method described in this section, which guarantees the basic linearization characteristics and eliminates the need to feed forward state variables and input values ​​during fine-tuning. A key issue in the design of sensor-based bottom following control system is the calculation of bottom height data measured by sonar. In this article, the technology employed uses the geometry of the sensor to effectively establish the contour of the seabed in front of the vehicle.

The organization is as follows. The third section introduces the nonlinear model of the vertical plane dynamics of the Invent underwater vehicle. The third section elaborates on the bottom following problem and briefly introduces the path-related error space used to describe vehicle dynamics. The fourth section states the preview control issues. Section 5 describes the method used for linear controller design, where LMI synthesis technology is applied to affine parameter-dependent systems. Section VI introduces the reconstruction technique used to construct the reference path based on the measurement results of the sonar profiler. Section 7 focuses on the realization of the nonlinear bottom following controller of the underwater robot. Finally, in the eighth section, the simulation results obtained by using the vertical plane nonlinear dynamic model are given.

Vehicle dynamics

Here, we describe the dynamic model of the underwater robot on the vertical plane. See [12] and [13] for details. The vehicle is 4.5 m long, 1.1 m wide and 0.6 m high. It is equipped with two main thrusters (propellers and nozzles) for cruising and fully moving surfaces (rudder, bow plane and stern plane) for steering and diving on the horizontal and vertical planes, respectively. The symbols used and the structure of the vehicle model are standard [12], [14].

The variables u and w represent surge and heave speed, and $\theta$、$q$、$x$ and$z$ represents pitch, pitch rate, inertial position and depth respectively. Symbol$d_b$And $d_s$Respectively represent the bow and stern plane deflection. Using this symbol and ignoring the stable roll mode, the dynamics of the underwater robot on the vertical plane can be written in a concise form.


(1), (3) and (5) describe the surge, heave and pitch motions respectively. , $X_{(.)},{WITH}_{(.)}And{M}_{(.)}$Is the hydrodynamic derivative term, ${with}_{CB}$Indicates the stability distance. Equations (2), (4) and (6) capture vehicle kinematics. The values ​​of hydrodynamic parameters are shown in [12] and [13]. Variables m, L, W, B and $I_{and}$They are the mass, length, weight, buoyancy and moment of inertia around the axis of the vehicle, $\rho$ is the density of water.

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