Path planning for underwater glider under control constraint

Abstract

Energy saving is important for the underwater gliders because they have limited onboard battery power. A novel optimal three-dimensional path planning method is introduced in this article, which has low energy consumption and is useful for the path planning of underwater gliders. The method is derived using optimization theory. In the planning process, the constrained control does not pass the zero point to obtain the turning path formed by the unidirectional arc. Then, the method is extended to three dimensions, which includes depth parameter. For the more complicated median-depth path, glider’s glide distance is increased by extending the turning radius, in order to achieve the end state. Compared with the traditional three-dimensional Dubins path, the simulation results indicated that the proposed optimal path leads to the glide with a lower energy consumption.

Introduction

Underwater glider is a type of buoyancy-driven, fixed wings underwater vehicle. It can run a long time in the water, sampling and detecting in certain water area. ¹ The capability of long running time and low energy consumption is essential to achieve the set task in a large area. To reduce energy consumption and improve running range by reasonable planning of the movement and glide path, the low-energy motion control of the glider is generally achieved by reducing the operating frequency of the glider’s actuators and the operating time of the control system.^2,3 The glider paths’ type needs to be considered in the path planning.

The steady glide path of the glider includes the zigzag movement in the longitudinal plane and the spiraling movement in the three-dimensional (3D) space. ⁴ The path length of the glider is long in the longitudinal plane, and the boundary condition of the path planning is mainly from the environment.^5,6 A glider needs steering movement when adjusting its heading or performing a special task. Energy consumption and steering time need to be considered in the path planning of the movement. Excessive heading adjustment method will waste energy and time; meanwhile, a great 3D path planning for glider can further reduce energy consumption. An optimal turning path planning method for underwater gliders is presented in this article.

Dubins paths are the shortest path between two oriented points, which contain arc segments of minimum turning radius and straight-line segment. ⁷ Dubins paths are obtained based on geometrical methods in accordance with the optimization principle. Based on Pontryagin’s minimum principle (PMP), the optimum of Dubins paths is validated, and the form of path and the position of trajectory transformation point are fully explained using the singular arc theory. ⁸ Dubins paths have been widely used in the shortest path planning on many kinds of vehicles.^9–12 Dubins paths are also applied on underwater gliders. Mahmoudian et al. ¹³ planned an underwater Dubins path especially for gliders by taking into account the control rate limit. Cao et al. ¹⁴ planned a glider energy optimization 3D Dubins path of the glider by considering the energy consumption of the executive parts. When the glider gliding in a steady state, it no longer consumes energy and is able to move continuously. This character is different from wheel-type robots and aircraft which need continuous driving. For which, the shortest path is the optimal path of energy consumption. The glider has steady state glide characteristics, which are similar to those of satellites. The optimal Hohmann orbit method for satellite fuels only requires the actuator to act twice. ¹⁵ Unlike the satellite’s unidirectional orbit, the glider’s turning orbit has two directions, either clockwise or counter-clockwise. It consumes more energy to ensure the control of orbit transfer between two different rotational directions in one control period. Referring the form of the satellite orbit, the position of the glider’s turning actuator is constrained so that the glider’s steering orbits are co-steered during a complete control period.

The dynamics model of underwater glider is the basis of motion analysis, and the model is a typical multi-body dynamics system, which adjusts the body posture by means of internal actuators rigidly connected to the hull. Leonard and Graver ¹⁶ first modeled a glider dynamics based on classical mechanic method. Considering the different driving modes, a variety of dynamics models of glider can be obtained.^17–19 The 3D dynamic equations of underwater gliders are nonlinear, and the variety and the quality of the parameter increase its difficulty of gaining the solution. Mahmoudian ²⁰ used the perturbation theory to obtain the analytical solution of steady state glide. Cao et al. ²¹ obtained the approximate semi-analytical solution by simplifying dynamic equations. Zhang et al. ¹⁷ proposed an iterative logic to obtain steady state solutions. Fan and Woolsey ²² used the Trust-region-dogleg to obtain numerical solution. In this study, the particle swarm optimization (PSO) algorithm is used to obtain the solution which can avoid to get a local optimal solution.

In this study, the optimal path of the glider is proved with PMP theory, and the construction method of the path is given. As the two-dimensional (2D) path is extended to the 3D space, the depth of the path needs to be considered. ²³ The proposed path is extended to 3D space, and the application under different depths is investigated.

Optimal path of underwater glider

Model description

Consider a glider that moves in an inertial, horizontal plane at a unit constant forward speed and in some direction relative to a reference frame fixed in the plane. The glider performs a straight-line motion when the turn actuator of the underwater glider is at natural zero point, and the glider performs a turning motion when the turn actuator is on non-zero position. If the glider moves in the Dubins paths, which contain two arc segments of minimum turning radius, more energy is needed, and it is a disadvantage for glider’s endurance capacity.

For a further energy saving, the rotary actuator of the glider is limited only to move on one side of the zero position in one control period. In other words, in this study, the rotary actuator’s movement range is $0^0<\mu \le {\mu }_{+max}(\mathrm{or}{\mu }_{-max}\le \mu <0^0)$ , where ${\mu }_{+max}$ is the maximum clockwise angle of the actuator, and ${\mu }_{-max}$ is the minimum clockwise angle of the actuator. The coordinate system of the glider is defined as shown in Figure 1. $[i,j,k]$ is inertial coordinate. $[x,y,z]$ is body coordinate system, and the direction of the coordinate system is shown by the arrow in the figure. The direction of rotation follows the right-hand rule, that is, the clockwise rotation around the coordinate axis is positive and the counter-clockwise rotation is negative. $m_p$ is the mass of the altitude adjustment mechanism, and $\mu$ is the angle around x-axis. $\mu$ is zero when $m_p$ at the initial zero position.

Figure 1.

Underwater glider coordinate system.

The above constructions on the control indicate that the heading rate is always greater than zero or constant less than zero in a control period. The rotation period of the rotating mechanism is very small compared with the glider’s turning period. Therefore, in this study, the limitation of the control rate is neglected, and the action of the rotating mechanism can be completed instantaneously.

The kinematic model of the underwater glider includes the glide angle and the roll angle in a form similar to the kinematics of the aircraft. In order to simplify the analysis, in the discussion of this article, the kinematic model of the aircraft is used to express the motion of the glider ²⁴

$$\begin{array}{c}\stackrel{·}{x}=V\cos \psi \cos \gamma \hfill \\ \stackrel{·}{y}=V\sin \psi \cos \gamma \hfill \\ \stackrel{·}{z}=-V\sin \gamma \hfill \\ \stackrel{·}{\psi }=\frac{g}{V}\tan \varphi \hfill \end{array}$$

(1)

where V is glide speed, $\psi$ is yaw angle, $\gamma$ is glide path angle, $\varphi$ is roll angle, and g is acceleration of gravity. The control constraint is $|\varphi |\le {\varphi }_{max}$ , $|\gamma |\le {\gamma }_{max}$ .

Optimal path of underwater glider

In 2D case, when the glide angle is 0°, the depth is not included in the equation of motion, and the heading rate is chosen as the control variable. The kinematics equation can be simplified from equation (1) as

$$\begin{array}{c}\stackrel{·}{x}=V\cos \psi \hfill \\ \stackrel{·}{y}=V\sin \psi \hfill \\ \stackrel{·}{\psi }=\mathrm{ku}\hfill \end{array}$$

(2)

where u is the offset of gliders’ turning actuator respect to the zero point, and k is a constant.

Set the unit glide speed. According to the optimization principle, the shortest path problem for a glider can be described as

$$\begin{array}{c}\mathrm{minimize}\underset{0}{\overset{T}{\int }}1\mathrm{dt}\hfill \\ \mathrm{subject\; to}\hfill \\ \begin{array}{cc}\stackrel{·}{x}(t)={[\cos \psi (t),\sin \psi (t),u(t)]}^T\hfill & \mathrm{system\; dynamics}\hfill \\ \multicolumn{1}{c}{x(0)={[x_i,y_i,{\psi }_i]}^T=X_i}& \mathrm{Initial\; configuration}\hfill \\ \multicolumn{1}{c}{x(T)={[x_f,y_f,{\psi }_f]}^T=X_f}& \mathrm{final\; configuration}\hfill \\ \multicolumn{1}{c}{u(t)\in (0,u_{max}]\mathrm{or}u(t)\in [-u_{min},0)}& \mathrm{input\; constraint}\hfill \\ \multicolumn{1}{c}{T\in (0,\infty )}& \mathrm{final\; time}(\mathrm{to\; be\; minimized})\hfill \end{array}\hfill \end{array}$$

(3)

Control constraint means that the glider can only turn in one direction during a control period. Based on the PMP, the Hamiltonian function of the system is

$$H(x,\lambda ,u)={\lambda }_x\cos \psi +{\lambda }_y\sin \psi +{\lambda }_{\psi }u+1$$

(4)

The adjoint system is

$$\begin{array}{l}{\stackrel{.}{\lambda }}_x=-\frac{\partial H}{\partial x}=0,{\stackrel{.}{\lambda }}_y=-\frac{\partial H}{\partial y}=0,\\ {\stackrel{.}{\lambda }}_{\psi }=-\frac{\partial H}{\partial \psi }={\lambda }_x\sin \psi -{\lambda }_y\cos \psi \end{array}$$

(5)

To minimize equation (4), have

$$u^{*}=\mathrm{argmin}H(x,\lambda ,u)=\mathrm{argmin}{\lambda }_{\psi }u$$

(6)

Therefore, the minimum value of the above equation depends on the sign of ${\lambda }_{\psi }$ , which

$$\{\begin{array}{c}{\lambda }_{\psi }<0\Rightarrow u^{*}=u_{max}\hfill \\ \multicolumn{1}{c}{{\lambda }_{\psi }>0\Rightarrow u^{*}=u_{min}}\end{array}$$

(7)

When ${\lambda }_{\psi }=0$ , the trajectory of control variable u enters the singular arc, and transition state of u can be obtained by solving equation (4). When ${\lambda }_{\psi }=0$ and ${\stackrel{·}{\lambda }}_{\psi }=0$ , $\psi$ should be constant according to equation (5). However, $\stackrel{·}{\psi }\ne 0$ according to equation (1), and $u\ne 0$ has been set, thus $\psi$ is not a constant. Therefore, the singular arc does not exist in one control period.

From the above analysis, the optimal turning motion control of the underwater glider is the bang–bang control problem. The paths consist only of the maximum and minimum turning radius arc, which converts state between two arcs at most one time in one control period. An underwater glider moves at minimum turning radius when $u^{*}=u_{max}$ and moves at maximum turning radius when $u^{*}=u_{min}$ . Since the actuator can move infinitely close to zero, the glider’s maximum turning radius approaches infinity, so, no value can be obtained. The determination of the maximum turning radius of a glider will be discussed in the next section.

From the above discussion, only six forms of path may be obtained, which are $L_{mina}{L}_{maxb}$ , $L_{maxa}{L}_{minb}$ , $R_{mina}{R}_{maxb}$ , $R_{maxa}{R}_{minb}$ , $L_{maxa}$ , and $R_{maxa}$ , where L and R is the left and right directions, respectively, and the subscripts min and max are the minimum turning radius and the maximum turning radius, respectively.

Geometric construction of optimal path

The shortest path of the underwater glider consists of two arcs of different turning radii. According to the Dubins paths construction method, set the initial and final states as $X_i=(x_i,y_i,{\psi }_i)$ and $X_f=(x_f,y_f,{\psi }_f)$ , respectively. The notations used in the algorithm are illustrated in Figure 2.

• Step 1. Determine the state vector. From the given data, find the position vectors r and the velocity vectors V.

• Step 2. Determine the minimum turning radius circle. Drawing two circles with minimum turning radius tangent to the velocity vector V at each end point, respectively, thus four circles $r_{\mathrm{ci}+}$ , $r_{\mathrm{ci}-}$ , $r_{\mathrm{cf}+}$ , and $r_{\mathrm{cf}-}$ are obtained. To ensure the continuity of the path established in the following steps, two circles on the same side of the vector V are selected, one for each end point. For example, $S_p$ and $T_p$ are two end points, there are two circles $r_{\mathrm{ci}+}$ and $r_{\mathrm{ci}-}$ with minimum radius of point $S_p$ , and two circles $r_{\mathrm{cf}+}$ and $r_{\mathrm{cf}-}$ of point $T_p$ . $r_{\mathrm{ci}+}$ and $r_{\mathrm{cf}+}$ or $r_{\mathrm{ci}-}$ and $r_{\mathrm{cf}-}$ can be connected.

• Step 3. Determine the maximum turning radius curve. As previously discussed, the path between the end points is composed of at most two kinds of arcs with different turning radii. Drawing a large circular arc tangent to the two circles selected in step 2 from either end point, the two circles are connected by the large arc. The connected two circles of minimum radius tangent to vector V for each end point in step 2 should be on the same side of each V to ensure the continuity. That is only one circle for each end point is selected. For example, circles $r_{\mathrm{ci}+}$ and $r_{\mathrm{cf}+}$ are connected by two large circular arcs from $S_p$ and $T_p$ , respectively, as shown in Figure 2.

• Step 4. Determine the optimum path. Comparing all the available paths, the path 2 is shorter, and it is selected as the optimum path.

Figure 2.

Construction of optimal path of gliders.

The glider optimal path is longer than the Dubins paths because the straight-line segment is no longer included. As shown in Figure 3, path 2, labeled blue in the figure, is the glider optimal path and path 3, labeled green, is Dubins path.

Figure 3.

Glider optimal path and Dubins path: path 2 is glider optimal path (blue) and path 3 is Dubins path (green).

The Dubins paths require that the distance between the starting point and the end point more than twice longer than the turning radius. Likewise, the application conditions of the glider optimal path proposed in this article are that the distance between the starting point and the ending point is more than twice longer than the turning radius.

Optimal path in 3D space

Because it contains information such as depth and glide angle, the motion path in 3D case is more complicated than in 2D case. Following M Owen and RW Beard, gliders’ glide depth is divided into three types: low depth, median depth, and high depth. ²⁵ The three cases are briefly described as below.

The depth between the start and end point is said to be low depth, if

$$|z_d-z_s|\le {\mathrm{L}}_{2\mathrm{D}\mathrm{glider}}\left(R_{min}\right)\tan {\gamma }_{max}$$

(8)

where ${\mathrm{L}}_{2\mathrm{D}\mathrm{glider}}(R_{min})$ is the length of 2D glider paths, and ${\gamma }_{max}$ is the maximum glide angle. The depth is said to be medium depth, if

$$\begin{array}{l}{\mathrm{L}}_{2\mathrm{D}glider}\left(R_{\min }\right)\tan {\gamma }_{\max }<|z_d-z_s|\\ \le \left[{\mathrm{L}}_{2\mathrm{D}glider}\left(R_{\min }\right)+2\pi R_{\min }\right]\tan {\gamma }_{\max }\end{array}$$

(9)

The depth is said to be high depth, if

$$|z_d-z_s|>[{\mathrm{L}}_{2\mathrm{D}\mathrm{glider}}\left(R_{min}\right)+2\pi R_{min}]\tan {\gamma }_{max}$$

(10)

Low-depth glider path

At low depth, the glider can reach its target at an unsaturated glide angle. The Hamiltonian function of equation (1) can be constructed as

$$H={\lambda }_{x}x+{\lambda }_{y}y+{\lambda }_z{u}_z+{\lambda }_{\psi }{u}_{\psi }+1$$

(11)

where

$$\lambda \left(t\right)=\left(\begin{array}{c}{\lambda }_x\hfill \\ \multicolumn{1}{c}{{\lambda }_y}\\ \multicolumn{1}{c}{{\lambda }_z}\\ \multicolumn{1}{c}{{\lambda }_{\psi }}\end{array}\right)=\left(\begin{array}{c}{c}_1\hfill \\ \multicolumn{1}{c}{c_2}\\ \multicolumn{1}{c}{c_3}\\ \multicolumn{1}{c}{c_{1}y-c_{2}x+c_4}\end{array}\right)$$

(12)

The control law on the optimal path is

$$u_z=\mathrm{sgn}\left(c_3\right)\mathrm{if}{c}_3\ne 0$$

(13)

$$u_z\in [u_{zmin},u_{zmax}]\mathrm{if}{c}_3=0$$

(14)

$$u_{\theta }=\mathrm{sgn}(c_{1}y-c_{2}x+c_4)\mathrm{if}{c}_{1}y-c_{2}x+c_4\ne 0$$

(15)

$$u_{\theta }\in [u_{\theta min},u_{\theta max}]\mathrm{if}{c}_{1}y-c_{2}x+c_4=0$$

(16)

According to equation (14), $u_z$ can vary between the upper and lower limits if $c_3=0$ . The optimal glide angle is

$${\gamma }^{*}={\tan }^{-1}\left(\frac{|z_d-z_s|}{L_{2\mathrm{D}\mathrm{glider}}\left(R_{min}\right)}\right)$$

(17)

The optimal glide path length is

$$L_{3\mathrm{D}\mathrm{glider}}(R_{min},{\gamma }^{*})=\left(\frac{L_{2\mathrm{D}\mathrm{glider}}\left(R_{min}\right)}{\cos {\gamma }^{*}}\right)$$

(18)

High-depth glider path

At high depth, the Dubins paths are too short to reach the target, and the path needs to be extended to the end by increasing the helical path. Corresponding to $c_3\ne 0$ in equation (13), the glider should glide at the maximum glide angle. According to the discussion in section “Optimal path of underwater glider,” the glider optimal path consists of a minimum turning radius arc and a larger turning radius arc. To save energy, the helical path is increased in the minimum turning radius segment. For example, if the glider starts at the arc of minimum turning radius, add a helical path at the start point; if the glider starts at the maximum turning radius arc, add a helical path at the end point. To ensure that the glider can still accurately reach the end point, appropriate adjustment of the minimum turning radius should be made. Define k as the required number of spirals, then have

$$\begin{array}{c}[{\mathrm{L}}_{2\mathrm{D}\mathrm{glider}}\left(R_{min}\right)+2\pi k{R}_{min}]\tan {\gamma }_{max}\le |z_d-z_s|\hfill \\ \le [{\mathrm{L}}_{2\mathrm{D}\mathrm{glider}}\left(R_{min}\right)+2\pi (k+1)R_{min}]\tan {\gamma }_{max}\hfill \end{array}$$

(19)

and

$$k=\lfloor \frac{1}{2\pi R_{min}}\left(\frac{z_d-z_s}{\tan {\gamma }_{max}}\right)-L_{2\mathrm{D}\mathrm{glider}}\left(R_{min}\right)\rfloor$$

(20)

where $\lfloor x\rfloor$ means rounded down. The adjusted minimum turning radius is

$$[{\mathrm{L}}_{2\mathrm{D}\mathrm{glider}}\left(R_{min}^{*}\right)+2\pi \mathrm{k}{R}_{min}^{*}]\tan {\gamma }_{max}=|z_d-z_s|$$

(21)

The length of path is

$$L_{3\mathrm{D}\mathrm{glider}}(R_{min}^{*},{\gamma }_{max})=\left(\frac{L_{2\mathrm{D}\mathrm{glider}}\left(R_{min}^{*}\right)}{\cos {\gamma }_{max}}\right)$$

(22)

Median-depth glider path

In the median-depth case, the Dubins paths are not sufficient to reach the target; if one spiral is added, the path is too long. Now, the path is extended to the end by extending the path. Owen and Beard extended the path by increasing the arc path. However, for gliders, another added arc means the glider needs to transfer orbit twice, and the actuator needs to move twice, which is not conducive to gliders’ endurance. In this article, the path is extent by increasing the minimum turning radius. The radius of turn after the increase is

$${\mathrm{L}}_{2\mathrm{D}\mathrm{glider}}\left(R_{min}^{*}\right)\tan {\gamma }_{max}=|z_d-z_s|$$

(23)

The path length is

$$L_{3\mathrm{D}\mathrm{glider}}(R_{min}^{*},{\gamma }_{max})=\left(\frac{L_{2\mathrm{D}\mathrm{glider}}\left(R_{min}^{*}\right)}{\cos {\gamma }_{max}}\right)$$

(24)

Case study

In this section, numerical analysis for the above three depth cases are studied.

For the three cases, set the minimum turning radius of the vehicle to 20 m, and in the north–east coordinate system, the initial point coordinates and direction were (20, 60) m and 10° north east, respectively. The terminal point coordinates and direction were (250, 250) m and 171° north east, respectively.

For the low-depth case, set the initial and terminal depths as −20 m and −70 m, respectively. The numerical results show that the maximum turning radius is 377.72 m, the optimal path length is 343.85 m, the glide angle is −8.27°, and the 3D path length is 347.46 m. The result path is shown in Figure 4, in which the green square marks the starting point and the red circle marks the ending point.

Figure 4.

Low-depth glider paths.

For the high-depth case, set the initial and terminal depths as −20 and −1500, respectively. The iterative numerical results show that the minimum turning radius after fine tuning is 20.3 m, and the maximum turning radius is 379.27 m. The 2D optimization path length is 344.37 m, moving at the maximum glide angle of −60°. At the end of the path, four complete spiral movements are added to the terminal point, and the length of 3D path is 1709.10 m. The path is shown in Figure 5.

Figure 5.

High-depth glider paths.

For the median-depth case, set the initial and terminal depths as −20 and −700, respectively. The iterative numerical calculation shows that the minimum turning radius after adjustment is 45.05 m, the maximum turning radius is 616.72 m, the length of optimal 2D path is 392.48 m, the glide angle of −60°, and the length of optimal 3D path is 784.97 m. The path is shown in Figure 6.

Figure 6.

Median-depth gliding paths.

Simulation and discussion

In this section, the optimal method used on underwater glider is discussed based on the analysis above.

Steady state solution

The dynamic equation of the underwater glider includes nine parameters. Generally, parameters such as speed, glide angle, and heading rate are given to obtain the parameters as pitch angle, axial position of the slider, slide angle, angle of the actuator, body roll angle, and net buoyancy.

In this study, thermal Slocum glider is the research object, which is roll-yaw inverse coupling, during the downward gliding, the glider moves clockwise with the maximum turning rate around the z-axis when the turning actuator move about the x-axis to the clockwise largest position; otherwise, the glider will rotate anti-clockwise around the z-axis with the maximum turning rate. Parameters are defined as the same as in the literature. ⁴ The steady state gliding dynamic model is shown in Appendix 1.

The minimum turning radius of the Slocum was set to 20 m, and in the north–east coordinate system, the coordinates and direction of initial point were set to (20, 60, −20) m and 10° north east, respectively. The coordinates and direction of terminal point were set to (250, 250, −100) m and 171°north east, respectively. The iterative numerical calculation shows that the maximum turning radius is 377.72 m, the length of optimal 2D path is 343.85 m, the glide angle is −13.0975°, and the length of optimal 3D path is 353.03 m. The glide speed was set at 0.725 m/s and the angle of attack was set at 4.3°, in which configuration the glider reaching the maximum lift-to-drag ratio. When glider spirals, its parameters such as the speed, yaw rate, and turning radius need to satisfy the following equation

$$V\cos (\xi )=\stackrel{·}{\psi }R$$

(25)

where $\xi$ is gliding angle, $\stackrel{·}{\psi }$ is heading rate, and R is turning radius. According to the given speed and glide angle, the heading rate corresponding to the turning radius of 20 m is 0.035 rad/s and that of 377.72 m is 0.0019 rad/s.

Set $V=0.725\mathrm{m}/\mathrm{s}$ and $\xi =-13.0975\circ$ . The axial position of the rotary actuator $r_{p_x}$ , roll angle of the rotary actuator $\mu$ , glider roll angle $\varphi$ , and net bouncy $m_0$ can be calculated at different heading rates.

As shown in Table 1, due to the coupling of the dynamic parameters, there are slight differences in $r_{p_x}$ and $m_0$ between the two cases. In this article, the small differences in $r_{p_x}$ and $m_0$ are neglected for that the main research object is the rolling actuator.

Table 1.

Underwater glider steady state solution in different heading rates.

$\stackrel{·}{\psi }(\mathrm{rad}/\mathrm{s})$	$r_{\mathrm{px}}(m)$	$\mu (\circ )$	$\varphi (\circ )$	$m_0(\mathrm{kg})$
0.0019	0.0129	−1.1768	0.6963	0.5581
0.0035	0.0096	−23.5557	12.7099	0.5410

Result analysis

As shown in Figure 7, the solid curve is the optimal path proposed in this article, and the glider is initially gliding at a straight line with no roll angle. First, the actuator rotates −1.1768° and glided 308.81 m by the turning radius of 377.72 m, then, rotates −22.3789° to −23.5557° and glided 44.22 m by the turning radius of 20 m, and finally, back to initial zero position by rotate 23.5557°. The total rotation angle of the actuator is 47.1114° and the total path length is 353.03 m. The dashed curve is the 3D Dubins path. At the beginning of the path, the actuator rotates −23.5557°, glided 12.54 m by 20-m turning radius, then, rotates 23.5557° to reach the straight-line state and glided 227.41 m, then, the actuator rotates −23.5557°, glided 44.20 m by 20-m turning radius. To achieve the end state, the rotary actuator rotation 23.5557° back to zero initial position. In the path, the total rotation angle of the actuator is 94.2228°, and the total path length is 334.15 m.

Figure 7.

Comparison between optimal glider path and 3D Dubins path: the solid line curve is the optimal path $R_{maxa}{R}_{minb}$ and the dashed curve is the 3D Dubins path $R_a{L}_b{R}_c$ .

The path energy consumption was set as

$$P=\frac{S}{V}\times k\sum \mathrm{\Delta }\mu$$

(26)

where $\sum \mathrm{\Delta }\mu$ is the total actuator movement, in $\mathrm{rad}$ ; k is the power consumption coefficient, in $\mathrm{W}/\mathrm{rad}$ , and set to unit in this study. $S/V$ is the trajectory time, in s. The energy consumption of the optimized 3D glider path is 400.38 J, and 3D Dubins path energy consumption is 757.94 J. We can see that the energy consumption of the 3D Dubins path is 1.9 times over that of the optimized path.

The energy consumption of the Dubins car path is not always greater than the energy consumption of the optimal glider path. The energy consumption of Dubins car path is larger, mainly because of the movement of the actuator is more frequent. For typical Dubins car path consisting of two arcs and one straight line, the actuator executes four times in the path. For the proposed optimal glider path, the actuator executes two times in the path. When the length of the Dubins car path is less than half the length of the optimal glider path, the energy consumption of the Dubins car path will be smaller.

Conclusion

This article presents an optimal path planning method for underwater glider based on Dubins paths. The optimal path can be optimized by adjusting some restrictions based on optimization theory. The path can be composed by the maximum and the minimum turning radius arcs by constraint actuator’s movement position. The proposed method can be extended to a 3D space containing depth parameters. For the complex median-depth path, the method of extending the turning radius can effectively increase the gliding distance of the glider, so as to obtain the path optimization method to construct the shortest path. The application of the optimal path to the underwater glider is studied. And the results indicated that it has a lower energy consumption rate compared with the traditional 3D Dubins path.