I. Introduction

Marine vehicles have several applications, they are widely used in recreative activities —such as fishing—, in military applications, and as a means of transport. The term includes ships, semi-submersibles, submarines, Remotely Operated Vehicles [ROV], Autonomous Underwater Vehicles [AUV], torpedoes and other propelled and energized structures able to navigate (Fossen, 2011).

The so-called Unmanned Surface Vehicles [USV] have as a main advantage the fact of not requiring human intervention, allowing them to perform missions with higher efficiency. Every vehicle in this category has a movement control system to operate autonomously; this system is composed of guidance, navigation, and control systems. In this research paper, our interest is in the guidance system since its main objective is to provide the state of the necessary references —position, speed, and acceleration— to the controller in a continuous way to ensure the vehicle always goes through the precise path (Lekkas, 2014).

The path following without temporary restrictions is a movement control scenario where the control objective is to converge and follow a geometric path previously defined without considering temporary restrictions. The guidance and control solutions proposed for this scenario are several (Breivik & Fossen, 2009), where an important number of works employ the guidance strategy known as Line of Sight [LOS] (Breivik & Fossen, 2009; Fossen, 2002).

The Automation, Robotics, and Sensing Group [GARP, Grupo de Automatización, Robótica y Percepción] of the Universidad Central Marta Abreu de Las Villas (Cuba) works in the study of marine vehicles, specifically in modelling, control, and guidance topics. Former research works performed by the group demonstrate that the LOS strategy is not capable to ensure the precision during the path following with sea currents; hence, an addition of an integral action in the guidance law is proposed to counteract the effect of these currents during the path following process using the HRC-AUV submersible vehicle (Hernández, Valeriano, & Fernández, 2016; Valeriano, Hernández, & Hernández, 2015; Miranda, Valeriano, & Hernández, 2017).

One of the current research interests of the group is focused on the development of the Krick Felix robotic ship; the objective regarding this vehicle is to have a mockup where control, guidance, and navigation algorithms can be tested to extend the maneuverability possibilities during several missions. The Krick Felix is an underactuated vehicle because it has less control inputs than degrees of freedom to control. This, together with the non-linearity of its dynamics and to the effects of the sea currents makes necessary to use a guidance solution within it to ensure precision during the following of straight paths.

II.Dynamic Model

TheKrickFelix (Figure1) is a scale model of a Hamburg harborship; for its movement, it has an electric engine, a coupled propeller, and the necessary equipment to make it a robotic ship. The main geometric specifications of the ship —which have incidence in the modelling and control of it— are shown in Table 1.

Figure 1
Krick Felix

Table 1
Geometric, physical, and inertial data of the KrickFelix

Traditionally, the translation and position movements in a marine vehicle are represented using several expressions, which entails complicated models formed by hundreds of elements. The maneuvers achieved with marine vehicles are related with 6 degrees of freedom movements, which are determined by the independent movements and by the rotations specifying the vehicle position and orientation (Fossen, 2011).

In Table 2 we present the standard nomenclature —SNAME notation (1950)— employed in the movement description of marine vehicles, where the yawing is the most important degree of freedom in the design of direction controllers.

Table 2
Geometric, physical, and inertial data of the KrickFelix

The movement of a marine vehicle can be described through two coordinate systems (see Figure 2): one regarding the vehicle and the other inertial relative to ground.

Figure 2
Coordinate systems

The first of them is a mobile coordinate system called boat system. The origin of this system (Ob) is in the vehicle’s center of gravity. The axis x0, y0, and z0 coincide with the main inertial axis.

For surface vehicles, only the horizontal plane is considered by using a model with three degrees of freedom. As per the boats float (z ≈ 0) and they are lengthy and laterally metacentric (ϕ = θ = ϕ = θ ≈ 0), the heave, roll, and pitch dynamics are discarded. The speed, position, forces, and moments vectors defining the movement of a ship in the three degrees of freedom are described by equations (1), (2) y (3) (Fossen, 2011)

(1)

(2)

(3)

where,

υ represents the linear and angular speed vector with coordinates in the vehicle system;

η is the position and orientation vector with coordinates in the reference system fixed in the ground; and

τ is used to represent the forces and moments acting in the vehicle over its coordinate system.

The kinematic equations are expressions that relate the derivative of the position and speed in a rigid body. Such equations can be expressed in matrices (see (4) and (5)) by using the Euler angles transformations (Fossen & Ross, 2006):

(4)

where,

R(η) is the transformation matrix of the boat system to the inertial reference system, defined as follow:

(5)

The general model of a marine vehicle is influenced by the forces and hydrostatic and hydrodynamic moments, effects mathematically expressed through the so-called added mass and damping terms, and through the restoring forces. Considering this, the non-linear equation describing the movement of a USV is expressed as equation (6) indicates (Fossen, 2002; Fossen, 2011; Fossen, 1994).

(6)

The sea currents are one of the main alterations affecting the marine vehicles during the navigation. For this reason and for our study, we considered them as external perturbations acting on each one of the axis in the vehicle coordinate system (x0,y0,z0 ). Consequently, pursuing the objective to reach a larger precision in the modelling of the Krick Felix, the relative terms to the sea currents were added to equation (6), resulting in equation (7).

(7)

where,

M=MRB+MA is the inertia matrix including the added masses;

C=CRB(υr )+CA(υr) is the matrix including the Coriolis terms of the rigid body and the added masses;

D(υr) is the damping matrix;

g(η) is the vector of gravitational forces and buoyancy;

τ=[τX,τy,τn ]T represents the vector of forces and moments of the control inputs; and

υr is the relative speed of the vehicle regarding the sea currents.

In the case of vehicles navigating through the surface, the g(η)vector is considered as zero. The structure of each one of these matrices and vectors for the Krick Felix vehicle has been determined by the GARP in previous researches (Valeriano, García, & Balanza, 2017).

The vr term represents the relative speed of the vehicle regarding the sea currents and it is defined (8) as follow.

(8)

where,

v is the vehicle vector speed defined in equation (1); and

vc = [uc, vc, 0]T is the vector representing the speed of the sea currents relative to the origin of the vehicle system, by considering that those currents do not generate rotational movements in the boat.

The components uc and vc relative to the inertial system can be calculated through the speed module of the currents (VC) and through the direction angle they have (βc ) (see (9) and (10).

(9)

In order to obtain the components relative to the Ob, it is necessary to apply the coordinate transformations in function of the Euler angles (Fossen, 2011), as shown in (11).

(11)

After, applying trigonometric identities, it is possible to develop equations (12) and (13) as follow:

(12)

(13)

The expressed model in equation (7) is the most accurate mathematical representation of the vehicle and it is the one employed by us in the simulation to assess the performance of the guidance scheme.

III. Linear Model for the Lateral Subsystem

In order to achieve in a surface vehicle maneuvers in environments where the sea perturbations are constant, it is not very practical to use a detailed model of the ship. Consequently, Equation (7) is not useful for the design of conventional and uncoupled controllers in a USV. Nevertheless, there is an alternative consisting of dividing the system in subsystems with low interaction. The objective of this option is to achieve faster simulation speeds than with the real-time maneuvers, which are especially important for dynamic prediction purposes (Fossen, 2002; Sutulo, Moreira, & Guedes, 2002). This decomposition is possible due to the geometric properties and the high symmetry grade that the employed vehicle has.

The two subsystems conforming the three degrees of liberty model of the Krick Felix and their corresponding state variables are:

• the lateral subsystem, employed for the maneuvers relative to drive the vehicle (state variables: v, r y ψ); and

• the speed subsystem, used for the design of the speed controller (state variable: u).

From the uncoupled expressions describing the surge, sway, and yaw of the boat, it is possible to obtain the following movement linear equations (Chávez, Picado, & Steller, 2005): speed (14), sway (15) and (16).

(14)

(15)

(16)

Equation (14) —longitudinal movement— can be uncoupled from the other two —transversal movement and yaw— by supposing that the ship speed and the thrust are constant.

In this research, the lateral subsystem is the one with larger interest, since it is the one employed for the design of the direction controller; this latter receives the references from the guidance system. As per the origin of the coordinate system of the Krick Felix ship is located in the center of gravity, the term XG is zero. Considering these elements, the dynamics of the vehicle for the lateral subsystem can be described from equations (17) and (18).

(17)

(18)

The kinematic transformation relation corresponding to the term by considering the operation conditions of the vehicle is defined by Equation (19).

(19)

The expressions corresponding to Y (20) and N (21) for this subsystem are:

(20)

(21)

The linear model in space-state representing the dynamics of the lateral subsystem is provided by (22).

(22)

where,

Yv and Nr are the force and movement causedby the added masses;

Yv and Nr have negative values, since they are the damping force and moment; and

b2 and b3 are gains generating the force and moment triggeredby the rudder.

The second-order Nomoto model is a widely employed alternative for control system developers to obtain low-order models in the lateral subsystem (Fossen, 2011; Moreno, Besada, López, Chaos, Aranda, & Cruz, 2015). From this model, it is possible to obtain the transfer function between the yaw angle and the horizontal deflection angle of the rudder as follow (23):

(23)

IV. Heading Controller

The main objective of this controller type is to follow the references supplied by the guidance algorithm (Valeriano et al., 2015). The direction controller selected in this work is from the P-D type, since the derivative action provides an additional phase range, increasing its robustness (Jalving & Storkersen, 1995).

Equation (23) describes —in a simplified manner— the lateral dynamic of the Krick Felix. From this transfer function, the controller is adjusted. We present the control signal as follow (24):

(24)

where,

Kp and Kd represent the proportional and derivative gain, respectively; and

the ψd value is employed to denote the desired direction angle, which comes from the guidance algorithm in function of the desired path to follow.

The adjustment is performed using the poles and zeros location technique in the tool. The obtained gain values are Kp = 10 and Kd = 1.

V. I-LOS Controller

For the case of the path following feature, where only spatial restrictions are important, the objective of the guidance law is focused on the vehicle convergence to the straight path; hence, path following errors associated to the vehicle convergence to the road arose, as Figure 3 shows. Let xe(t) the path following error through the road and ye(t) the path following error perpendicular to the road. Their expressions (25, 26) are shown below.

Figure 3
Main variables of the guidance algorithm based on the look-ahead distance

(25)

(26)

In order to counteract the deviation triggered by the sea currents during the path following of straight roads, it is necessary to define the following control objective (27):

(27)

We propose to use a controller with integral action to achieve this objective by taking the lateral sliding angle caused by the sea currents as a small perturbation of low variation, i.e., βr ≈ 0.

It is necessary to define the following (28) to employ these variants:

(28)

By adopting the mentioned consideration, equation (28) can be rewritten as follow (29):

(29)

The desired direction value must coincide with the curse angle χ(ye), which is calculated through equation (30).

(30)

where,

(31)

(32)

where,

χp is the tangent angle to the road;

χr(ye) is a correction angle that ensures the vehicle speed is pointed to the path where the boat goes; and,

ki = kykp, being ky > 0 a design parameter,

(33)

(34)

The Δ parameter is associated to the look-ahead distance, which is the distance existing between the projection of the vehicle position in the road and the point of the road where the boat goes. This parameter can be variable or constant and its selection is associated to the vehicle length —two to six times that length—. For the case of this research, we considered this value as constant.

To consider that a point in the road has been surpassed, we propose a procedure to do it, which is associated to an acceptance circle established for each point in the road. Its radio would be R(k+1) > 0 for the k+1 point; hence, the criterium to change the vehicle direction towards the next point is defined by equation (35):

(35)

Consequently, to producea point change, it is necessary that the vehicle position is within the acceptance sphere, making it required to define a R(k+1) value.

VI. Simulation Results

To validate the proposed method, it is indispensable to define the desired path for the vehicle. This path is composed of two points with coordinates (x,y) as Table 3 shows. It is important to focus that the vehicle starts from the position[x,y]T = [20,5]T with a heading angle = 0°.

Table 3
Points in the path

To recreate the typical operation conditions of the vehicle in our simulations, we used the following data: speed of the sea currents, VC = 0.1 m⁄s; and direction angle of the sea currents, βc = 10°. Similarly, the turn speed of the Krick Felix engine was set to 600 rpm and the sampling period for the simulations is T = 0.01s. During the simulations, we also set the controller gain value, ky = 0.5 m⁄s. Once these parameters are set, we assessed the simulation results for several Δ values and we confirmed that the best response was obtained with Δ = 2. This because of the fact that this value is within the range of two and six times the length of the vehicle; allowing that the perpendicular error to the road is quickly stabilized to zero and without oscillations, as the reader might observe in Figure 4.

Figure 4
Perpendicular error to the road

Once the parameters are adjusted to make that the error caused by the sea currents tends to zero, it is necessary to check that the vehicle can perform the following of the defined path, as Figure 5 shows.

Figure 5
Path following for Δ=2

The set-up effectiveness is assessed via simulations by varying the sea currents speed influencing the vehicle. The objective of these tests is to demonstrate that the adjusted I-LOS controller can minimize the ye(t) error when several sea currents speeds are present. The simulations were performed under the same conditions described for the controller set-up by considering the speeds of the sea currents, [VC] as 0.1, 0.6,and1m⁄s, respectively.

The result of the performed tests is presented in Figure 6, where the reader might see that, for the elevated values of the sea current speeds (i.e., ye(t)), the error grows; nonetheless, the integral action in all the cases makes it to tend to zero.

Figure 6
Perpendicular error to the road with variations in the sea current speeds

VII. Conclusions

The I-LOS guidance scheme ensures the vehicle convergence to the path. With the inclusion of an integral action inside the guidance law, we ensure a reduction in the perpendicular path following error caused by the sea currents effect within it.

The mathematical structure defined in the I-LOS controller is easy to be implemented and it also considers the windup effect, caused by an integral action. The controller gain calculation depends mainly of the selection in the look-ahead distance, which is considered constant and dependent of the vehicle length.

The simulation results indicate the viability of using this scheme type in the Krick Felix robotic boat.

References

Breivik, M. & Fossen, T. (2009). Guidance laws for autonomous underwater vehicles. In: A. V. Inzartsev (Ed.), Underwater vehicles (pp. 51-76). Vienna, Austria: InTech.

Chávez, J., Picado, A., & Steller, J. M. (2005). Aplicaciones de control en barcos [thesis]. Universidad de Costa Rica: San José.

Fossen, T. (1994). Guidance and control of ocean vehicles. Chichester, UK: John Wiley & Sons.

Fossen, T. (2002). Guidance, navigation, and control of ships, rigs and underwater vehicles. Trondheim, Norway: Marine Cybernetics.

Fossen, T. & Ross, A. (2006). Nonlinear modelling: Identification and control of UUVs. In: G.N. Roberts & R. Sutton (Eds.), Advances in unmanned marine vehicles (pp. 13-42). London, UK: IET

Fossen, T. (2011). Handbook of marine craft hydrodynamics and motion control. Chichester, UK: John Wiley & Sons.

Hernández, A., Valeriano-Medina, Y. & Fernández, J. (2016). Control I-LOS considerando la distancia lookahead constante para el seguimiento de camino curvos en AUV [2016 Convención Científica de Ingeniería y Arquitectura, La Habana, Cuba].

Jalving, B. & Storkersen, N. (1995). The control system of an autonomous underwater vehicle. Modeling, Identification and Control, 15, 107-117.

Lekkas, A. M. (2014). Guidance and path-planning systems for autonomous [thesis]. NTNU: Ålesund, Norway. Available at: https://brage.bibsys.no/xmlui/handle/11250/261188

Miranda, L., Valeriano-Medina, Y. & Hernández, A. (2917). Sistema de guiado desacoplado en 3D para el vehículo HRC-AUV [2017 Convención Internacional 2017 Universidad Central ‘’Marta Abreu” de Las Villas. XVII Simposio de Ingeniería Eléctrica].

Moreno, D., Besada, E., López, J. A., Chaos, D., Aranda, J. & Cruz, J. M. (2015). Identificación de un modelo no lineal de un vehículo marino de superficie usando regresión simbólica. In Actas de las Jornadas de Automática 2015 (pp. 850-855). Bilbao, España: IFAC.

Sutulo, S., Moreira, L. & Guedes, C. (2002). Mathematical models for ship path prediction in maneuvering simulation systems. Ocean Engineering, 29(1), 1-19. doi:https://doi.org/10.1016/S0029-8018(01)00023-3

Valeriano-Medina, Y., García, D. & Balanza, C. (2017). Modelado dinámico del barco de pequeño porte Krick Felix. [2017 Convención Internacional 2017 Universidad Central ‘’Marta Abreu” de Las Villas . XVII Simposio de Ingeniería Eléctrica].

Valeriano-Medina, Y., Hernández-Julián, A. & Hernández, L. (2015). Controlador I-LOS para el seguimiento de caminos en línea recta de un vehículo autónomo subacuático. Revista de Ingeniería Electrónica, Automática y Comunicaciones, 36(2), 15-28.

Notes

Author notes

Additional information

Cómo citar: Miranda, L., Valeriano-Medina, Y., & Camero, Á. (2018). Guidance scheme for paths without temporary restrictions using the Krick Felix model boat under the influence of sea currents, Sistemas & Telemática, 16(46), 59-70. doi:10.18046/syt.v16i46.3088