Lagrange modeling and navigation based on quaternion for controlling a micro AUV under perturbations

Highlights

• The dynamic model uses quaternions for the angular position.

• Lagrange method is modified to implement the quaternion formulation.

• The LOS guidance algorithm is also modified to integrate the quaternion formulation.

Abstract

This paper addresses the modeling of an autonomous underwater vehicle using quaternion formulation for angular position description and Lagrange method to compute the equations of motion. As the four parameters are dependent and generate a constraint, Lagrange multipliers are used with Baumgarte method to solve and stabilize the system. The dynamic model includes underwater effects like added mass and inertia, hydrodynamic damping, buoyancy and propeller forces. Moreover, a quaternion-based line of sight guidance algorithm is derived to avoid any use of trigonometric function and compute directly the orientation error of the underwater vehicle and the desired attitude in terms of quaternions. Motion control is achieved with a quaternion-based adaptive sliding mode controller rejecting model uncertainties and water current. The simulation results, where the vehicle follows a sequence of way-points including vertical diving motion demonstrate that the proposed guidance algorithm and motion control are deeply relevant both in terms of effectiveness and robustness for this particular type of vehicle and orientation formulation.

Introduction

Autonomous Underwater Vehicles (AUV) have an essential role in the offshore industry and ocean exploration. These vehicles can reach places where human intervention is dangerous or impossible. They usually perform tasks like visual inspection or more complex actions such as pipelines maintenance, mine-clearing operations or search and rescue [1].

To ensure precise and effective navigation, AUVs integrate navigation, guidance and control systems as inertial sensors for orientation, barometers for depth measure, acoustic systems for absolute/relative positioning and accelerometers. Besides, these vehicles are subject to unexpected dynamics due to the environment such as water currents [2], [3], tether cable forces or obstacles. Thus, the control needs to overcome uncertainties in the model geometry, mass, damping, actuation forces and also noise in the signals. When operating, the control should be robust again all these external and internal perturbations to guarantee precise navigation. For micro AUVs, as their mass is inferior to 4.5 kg [4], the effect of disturbances is greater than for classical heavier vehicles. This constraint requires control for this category of vehicles to especially have properties for disturbance rejection.

AUVs are generally modeled as 6 degrees of freedom (DOF) systems. The classical denomination for the degrees of freedom is: surge, sway, heave, roll, pitch and yaw. An overview of dynamic modeling methods for underwater vehicles can be found in [5]. Most of the models in literature use Newton/Euler approach to derive the equations of motion [6], [7] with Euler angles to define the vehicle orientation. However, this classical formulation contains singularities that limit the vehicle range of motion. Typically, in the roll/pitch/yaw formulation of the cosine matrix (Tait–Bryant sequence of rotations), the pitch angle $\theta$ should be limited to $-\frac{\pi }{2}<\theta <\frac{\pi }{2}$ to avoid numerical instability of the model.

This singularity in the cosine matrix can be prevented using a quaternion formulation to define the angular position of the vehicle in the reference frame. In [8], [9] this method is applied to a 6-DOF AUV model with a PD control for attitude. The authors include the quaternion formulation into the classical Newton–Euler approach to derive the equations of motion. Quaternion feedback control gives similar performances for tracking trajectory compared to Euler angles but without the use of trigonometric function or singularity in the cosine matrix. However, no perturbation is applied to the system in these analyses.

The Lagrange method is well adapted for complex dynamic systems. Nonetheless, this formulation needs a state vector $\underline{q}$ of independent coordinates and the 4 terms of the quaternion are dependent. To apply quaternion formulation into Lagrange derivation of the equations of motion, Lagrange multipliers are used as summarized in [10] to maintain the unit norm constraint of the quaternion parameters. New methodologies to avoid using Lagrange multipliers have been derived in [11], however, no strict comparison has been done on complex mechanical systems in terms of precision and computation efficiency. Furthermore, the majority of studies applying the quaternion formulation to Lagrange derivation of the equations of motion focus on general equations for 6-DOF bodies or spacecraft.

Attitude control and guidance of autonomous vehicles using quaternion formulation is a key area for systems that can describe complex tridimensional trajectories in space. The line of sight (LOS) algorithm is a widely used method to drive the velocity control of a vehicle in space and make it follow a fully defined trajectory or way-points [12]. The intersection between the look-ahead distance and the path allows computing the LOS vector which is the target direction for the vehicle to follow. This look-ahead distance parameter is crucial and has to be tuned properly, small values make the vehicle more dynamic on his heading changes but can lead to oscillations in its trajectory [13]. On the contrary, a high look ahead-distance makes the vehicle heading changes smoother but is not relevant for close range environments.

To control the AUV orientation, the angular error is generally computed in the horizontal plane and also in the vertical plane for slender body types AUVs with rudders [14], [15]. Nevertheless, as a high-level task, the different LOS guidance algorithms still need the computation of angular errors with trigonometric functions and do not avoid the inherent singularities due to Euler angles. For surface vessels, this singularity is not problematic since the pitch angle remains small [16]. However, when it comes to AUVs this statement is no longer true, especially for torpedo-shaped vehicles.

In [17] and [18], the horizontal and vertical angular errors are transformed into a quaternion error vector assuming the vehicle roll angle is not to be controlled. Nonetheless, this method still decouples the orientation error in the different planes and needs the computation of trigonometric functions.

In [19], the desired quaternion and cosine matrix are computed using directly the LOS vector. A 6-DOF AUV model derived via Newton–Euler method follows a straight line in the presence of constant ocean current and sensor noise. Nevertheless, the trajectory proposed to illustrate the navigation algorithm interest is limited.

This manuscript relies on proposing a complete solution for modeling and navigation of micro underwater vehicles using the Lagrange method and quaternions for orientation. The model includes added masses and inertia, quadratic damping, actuator forces, buoyancy effect, realistic random water currents, and parametric uncertainties. The contribution is, alongside with the solving of the constrained Lagrange equations, the derivation of a LOS guidance algorithm directly in the quaternion formulation. No trigonometric function or angular error is computed to keep the system free from singularities in the rotation matrix and allow the vehicle to take any desired orientation in space. For motion control of the AUV, a quaternion-based adaptive sliding mode controller is proposed. Finally, the feasibility and efficiency of the proposed guidance algorithm coupled to the motion controller are tested in a simulation where the underwater vehicle follows a sequence of way-points, including vertical diving maneuver in presence of realistic internal and external disturbances.

In Section 2, the modeling method for the underwater vehicle using Lagrange and the quaternion formulation is presented. The next Section 3 develops the ASM controller on the dynamic model. Then, Section 4 addresses the quaternion-based LOS guidance algorithm. Section 5 exposes the simulations results and the associated analysis.