*Article* **Motion Planning for an Unmanned Surface Vehicle with Wind and Current Effects**

**Shangding Gu 1,2, Chunhui Zhou 1,\*, Yuanqiao Wen 3,4, Changshi Xiao 1,3,4 and Alois Knoll <sup>2</sup>**

	- Wuhan 430070, China; yqwen@whut.edu.cn 4 Intelligent Transportation Systems Research Center, Wuhan University of Technology, Wuhan 430062, China
	- **\*** Correspondence: chunhui@whut.edu.cn

**Abstract:** Aiming at the problem that unmanned surface vehicle (USV) motion planning is disturbed by effects of wind and current, a USV motion planning method based on regularization-trajectory cells is proposed. First, a USV motion mathematical model is established while considering the influence of wind and current, and the motion trajectory is analyzed. Second, a regularization-trajectory cell library under the influence of wind and current is constructed, and the influence of wind and current on the weight of the search cost is analyzed. Finally, derived from the regularization-trajectory cell and the search algorithm, a motion planning method for a USV that considers wind and current effects is provided. The experimental results indicate that the motion planning is closer to the actual trajectory of a USV in complex environments and that our method is highly practicable.

**Keywords:** motion planning; unmanned surface vehicle (USV); effects of wind and current; regularization-trajectory cell

## **1. Introduction**

The intelligence level of an unmanned surface vehicle (USV) has been improved a lot in recent years; however, it is still challenging to achieve a precise motion planning for USVs in complex environments [1–3]. Due to the particularity of USV navigation environments, it is inevitable that USVs will be affected by the wind, wave, current and other environmental factors in the navigation process. Therefore, it is necessary to consider the impact of environmental factors, where the efficiency and safety of task implementation by the USV need to be taken into account. As far as we know, no previous research has investigated the effects of wind and current on USV motion planning from the perspectives of USV dynamics and cyber-physic systems. In order to fill this gap, we particularly need to figure out how to avoid the adverse influence of wind and current on motion planning for a USV in complex environments, which is one of the key problems that should be solved in the process of USV intelligentization [4].

Although many researchers focus on USV motion and path planning in complex environments [5–13], there are still some problems that need to be solved with respect to the influence of actual wind and current dynamics on USV motion planning. For example, from the perspectives of USV navigation characteristics and cyber-physic systems: How do the wind and current disturb the distance and direction of the USV in the motion planning process? How do the wind and current disturb USV steering in the motion planning process? It is necessary to take these two problems into account during USV navigation. In particular, it is more critical to consider the influence of wind and current on the dynamics of the USV and how the dynamics of the USV changes in the process of motion planning.

**Citation:** Gu, S.; Zhou, C.; Wen, Y.; Xiao, C.; Knoll, A. Motion Planning for an Unmanned Surface Vehicle with Wind and Current Effects. *J. Mar. Sci. Eng.* **2022**, *10*, 420. https:// doi.org/10.3390/jmse10030420

Academic Editor: Dracos Vassalos

Received: 23 February 2022 Accepted: 10 March 2022 Published: 14 March 2022

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The remainder of this paper is organized as follows: an analysis of related works is introduced in Section 2; a major issue and methodology are presented in Section 3; a USV mathematical motion model in terms of wind and current effects is provided in Section 4; the construction of time-varying trajectory cells is presented in Section 5; on the basis of time-varying trajectory cells, regularization-trajectory cells that take into account wind and current effects are also provided in Section 5; a USV motion planning method with respect to wind and current effects is introduced in Section 6; simulation experiments and analyses are introduced in detail in Section 7; the conclusion and outlook of the paper are given in Section 8.

#### **2. Related Works**

Path planning, with the effects of wind and current taken into consideration, needs to be optimized from different perspectives such as the distance of the path, the safe path, and the smooth path. Much research has been performed to help solve the USV motion planning problem with the effects of wind and current.

Singh et al. [5] proposed a method on the basis of the A\* algorithm [14] to solve the USV path planning problem under the influence of the current. This method mainly projects the planning map to a binary electronic map, in which they set the safety distance between the USV and the obstacle as a certain pixel according to the constraint pixels of the electronic map; it then solves the problem of path planning under the influence of the current. To a certain extent, it provides useful inspirations for solving the path planning problem of the current affecting the USV since they considered the path distance and safety of the planned route.

Ma et al. [6] provided a multi-objective optimization method for the influence of fixed current field and time-varying current field USV path planning. In this method, the multiobjective particle swarm optimization algorithm [15] is constructed. The safety, economy, distance, and smoothness of the path are taken as the objective variables, and then the current function [16,17] is constructed as the environmental variables in order to realize path planning for a USV in the time-varying current field environment. They considered the safety, economy, distance, and smoothness of the planning path.

Thakur et al. [7] proposed the state transition model of GPU to simulate a USV's trajectory planning and then realized this trajectory planning in complex sea conditions. Li et al. [18] researched the dynamics and kinematics model of a USV, simplified it according to the characteristics of a USV trajectory, proposed a three degrees-of-freedom motion model for USV navigation, and then further verified the effectiveness of the designed USV trajectory through simulation. However, Thakur and Li et al. [7,18] ignored the specific dynamic constraints of the USV.

Song et al. [8] analyzed a fast marching algorithm [19,20] to solve the route planning problem of a USV facing a fixed current field, and then constructed a double-layer fast marching algorithm to realize a safe and economic route planning for a USV under the influence of a fixed current field. Song et al. [9] improved the fast marching algorithm based on reference [8] and designed the multi-layer fast marching algorithm in order to consider the time-varying current field route planning of USVs. References [8,9] also considered the safety and distance of the planning path. Oren et al. [10] proposed a velocity obstacle method for path planning in response to wave disturbances while the USV is sailing in complex sea conditions. In their algorithm, the waves disturbing the USV are regarded as moving obstacles, and the velocity obstacle method based on probability prediction [21] is constructed to realize USV path planning. This method provides some reference ideas for solving the navigation problems of a USV in complex sea conditions and has certain practical significance. More specifically, with regard to planning behavior, they considered some constraints such as USV size, and they also considered the safety and distance of the planned path. Based on the Voronoi-Visibility roadmap method and the ant genetic algorithm, Niu et al. [11] proposed the optimal energy path planning method for USVs in the environment of the time-varying current field. They established a Voronoi visibility roadmap and combined it with a genetic algorithm to achieve the optimal energy path in a time-varying environment. Experiments indicate that their method's performance is better than other methods.

Subramani et al. [12] developed a path planning method for strong current environments; they mainly considered the influence of a time-varying environment by establishing partial differential equations and then constructed a navigable area in a time-varying environment by using a probability model. The method mainly considers time-varying optimal path planning under the influence of a strong current environment. However, their method cannot guarantee path planning safety since the probability of navigation points cannot be fully computed in a real-time system. The work most related to our study is that of [13]; in our method, regularization-trajectory cells are developed to take into account the effects of wind and current, and we focus more on trajectory analysis with respect to wind and current effects, which can help in fully considering the USV characteristics.

#### **3. A Major Issue and Methodology**

In this section, we will analyse the influence of wind and current on USV motion planning with an experiment, and provide our methodology for USV motion planning considering wind and current effects.

Du et al. [22] developed a trajectory-unit method for USV motion planning that considers the dynamic constraints of the USV. This method works well when USVs are not disturbed by wind and current. However, as shown in Figure 1, the method can not provide good performance in complex environments, it does not consider the impact of wind and current on the dynamics of the USV. If the wind and current disturbances are considered in their method, the planned path will not be spliced. The reasons for this problem are: (1) its search algorithm does not consider environmental factors; (2) its trajectory unit does not consider environmental factors. When the environment changes, the trajectory unit becomes irregular and cannot adapt to complex environments; hence, it cannot effectively make the USV navigate in complex environments.

**Figure 1.** Motion planning algorithm without considering wind and current disturbances.

In order to solve this issue, a USV motion planning method based on the regularizationtrajectory cell is proposed (as shown in Figure 2). More specifically, first, a model is built under the influence of wind and current while considering the dynamics of the USV. Second, a regularization-trajectory cell based on the dynamics of the USV is constructed, taking into account the actual dynamics of the USV in the process of determining how to realize path planning. Third, after computing the impact of wind and current on the dynamics of the USV, an effective algorithm for USV motion planning is proposed to achieve an efficient, safe, and economical path search. In the next section, we will introduce a USV mathematical motion model that will play an important role in USV motion planning.

**Figure 2.** Structure of the motion planning method for USVs based on the regularizationtrajectory cell.

#### **4. A USV Mathematical Motion Model**

In this section, we establish the mathematical model of a USV that is exposed to wind and current disturbances, and the motion characteristics of the USV under the influence of environmental factors such as wind and current are further analyzed.

In the process of establishing the mathematical model of the USV, it is assumed that mainly the forward (*x*), traverse (*y*), and bow (*n*) of the USV are considered.

Therefore, the mathematical model of the USV here is given by Equation (1):

$$\begin{array}{c|c|c} \text{X} & \text{X} & \text{X}\_{I} + \text{X}\_{H} + \text{X}\_{P} + \text{X}\_{R} + \text{X}\_{wind} \\ \text{Y} & = \begin{array}{c|c} \text{X}\_{I} + \text{X}\_{H} + \text{X}\_{P} + \text{X}\_{R} + \text{X}\_{wind} \\ \text{Y}\_{I} + \text{Y}\_{H} + \text{Y}\_{P} + \text{Y}\_{R} + \text{X}\_{R} + \text{N}\_{wind} \end{array} \\ \end{array} \tag{1}$$

In Equation (1), *I*, *H*, *P*, *R*, and *wind* respectively represent the forces (moments) generated by inertia, viscosity, propeller, rudder, and wind. A detailed introduction to the inertial model, viscous model, propeller model, and rudder model of a USV can be found in references [22–25].

#### *4.1. Wind Disturbances*

When the wind on the sea surface changes randomly, wind force disturbances are also random. Here, the wind on the sea surface is assumed to be uniform, *W<sup>A</sup>* denotes the absolute wind speed, *WAD* denotes the direction of the absolute wind speed, *W<sup>R</sup>* denotes the relative wind speed, *WRD* denotes the direction of the relative wind speed, and *V<sup>V</sup>* denotes the USV speed. The absolute wind speed and direction are calculated in a geodetic coordinate system, and the relative wind speed and direction are calculated in a USV coordinate system [26,27].

The relationship between the absolute wind speed, relative wind speed, and USV speed is as follows:

$$W\_{\rm R} = W\_{\rm A} - V\_{\rm V} \tag{2}$$

We map this relationship to the USV-following coordinate system:

$$\begin{aligned} u\_R &= -u - W\_A \cos(W\_{AD} - \phi) \\ v\_R &= -v - W\_A \sin(W\_{AD} - \phi) \end{aligned} \tag{3}$$

where *u<sup>R</sup>* and *v<sup>R</sup>* are the components on the *X* and *Y* axes of the USV-following coordinate system. In the relative coordinate system, when the wind is blowing from the port side of the USV, the relative wind speed *W<sup>R</sup>* and direction *WRD* are positive, and the calculated relative wind speed *W<sup>R</sup>* and direction *WRD* are as follows:

$$\begin{aligned} \mathcal{W}\_{RD} &= \arctan\left(-\frac{\upsilon\_R}{\mu\_R}\right) + \text{sgn}(\pi\_\prime \upsilon\_R) & \mu\_R > 0\\ \mathcal{W}\_{RD} &= \arctan\left(-\frac{\upsilon\_R}{\mu\_R}\right) & \mu\_R < 0 \end{aligned} \tag{4}$$

The relative wind speed value obtained from this formula is:

$$\mathcal{W}\_{\mathcal{R}}^2 = \mathcal{W}\_A^2 + V\_V^2 + 2\mathcal{W}\_A V\_V \cos(\mathcal{W}\_{AD} - \beta) \tag{5}$$

where *β* is the drift angle.

#### 4.1.1. Wind Pressure

The wind pressure and wind ballast calculated from the formula above are as follows. The calculated forces and moments acting on the hull wind can be expressed as:

$$\begin{aligned} X\_{wind} &= 0.5 \rho\_a A\_f W\_R^2 \mathcal{C}\_{wx} (W\_{RD}) \\ Y\_{wind} &= 0.5 \rho\_a A\_s W\_R^2 \mathcal{C}\_{wy} (W\_{RD}) \\ N\_{wind} &= 0.5 \rho\_a A\_s L\_{OA} W\_R^2 \mathcal{C}\_{wy} (W\_{RD}) \end{aligned} \tag{6}$$

In the formula above, *ρ<sup>a</sup>* denotes air density, *LOA* denotes the length of the USV, *A<sup>f</sup>* denotes the forward projection area of the USV's water part, *A<sup>s</sup>* denotes the side projection area of the USV's water part, *Cwx*(*WRD*) denotes the wind pressure coefficient in the x-axis direction, *Cwn*(*WRD*) is the wind pressure moment coefficient in the z-axis direction, and *Cwy*(*WRD*) denotes the wind pressure coefficient in the *y*-axis direction.

The forward projection area and the side projection area need to be calculated in detail according to the general layout of the USV; in the absence of a detailed general layout of the USV, it can be roughly calculated according to reference [28].

In addition, we calculate the wind pressure *Fwind* (the coupling of different wind directions is considered in the calculation of wind pressure):

$$F\_{wind} = 0.5 \rho\_d \mathcal{W}\_R^2 \left( A\_s \sin(\mathcal{W}\_{RD})^2 + A\_f \cos(\mathcal{W}\_{RD})^2 \right) \mathbb{C}\_{wF}(\mathcal{W}\_{RD}) \tag{7}$$

*CwF* in formula (7) is the coefficient of the wind pressure resultant force. According to the wind pressure resultant force, the moment in the *x*-axis, *y*-axis, and *z*-axis is decomposed. By combining these formulas, the relationship of the wind pressure coefficients in each direction can be obtained:

$$\begin{aligned} \mathcal{C}\_{\text{wx}}(\mathcal{W}\_{\text{RD}}) &= \mathcal{C}\_{\text{wF}} \cos \mathcal{W}\_{\text{RF}} \left( A\_s \sin \left( \mathcal{W}\_{\text{RD}} \right)^2 + A\_f \cos \left( \mathcal{W}\_{\text{RD}} \right)^2 \right) / A\_f \\ \mathcal{C}\_{\text{wy}}(\mathcal{W}\_{\text{RD}}) &= \mathcal{C}\_{\text{wF}} \sin \mathcal{W}\_{\text{RF}} \left( A\_s \sin \left( \mathcal{W}\_{\text{RD}} \right)^2 + A\_f \cos \left( \mathcal{W}\_{\text{RD}} \right)^2 \right) / A\_s \\ \mathcal{C}\_{\text{wz}}(\mathcal{W}\_{\text{RD}}) &= \left( 0.5 - \mathcal{x}\_F \right) \mathcal{C}\_{\text{wF}} \sin \mathcal{W}\_{\text{RF}} \left( A\_s \sin \left( \mathcal{W}\_{\text{RD}} \right)^2 + A\_f \cos \left( \mathcal{W}\_{\text{RD}} \right)^2 \right) / A\_s \end{aligned} \tag{8}$$

where *x<sup>F</sup>* is the position point of the pressure resultant force by wind, and *WRF* is the angle of the wind pressure resultant force.

#### 4.1.2. Calculation of Wind Pressure and Moment

Generally, these correlation coefficients are obtained from a wind tunnel test. However, due to the limited experimental conditions and the impossibility of wind tunnel testing for every USV, this paper calculates the correlation coefficients according to the approximate calculation formulas given by A. Iwai and H. Kugumiya, and it is estimated for a general cargo ship [28].

$$\mathcal{L}\_{\text{wF}} = 1.325 - 0.05 \cos(2\mathcal{W}\_{\text{RD}}) - 0.35 \cos(4\mathcal{W}\_{\text{RD}}) - 0.175 \cos(6\mathcal{W}\_{\text{RD}}) \tag{9}$$

$$\mathcal{W}\_{\rm RF} = \left(1 - 0.15\left(1 - \frac{W\_{\rm RD}}{90}\right) - 0.80\left(1 - \frac{W\_{\rm RD}}{90}\right)^3\right) 90 \tag{10}$$

$$\mathbf{x}\_F = (0.291 + 0.0023 \mathbf{W}\_{\rm RD}) L\_{\rm PP} \tag{11}$$

where *LPP* is the length between the perpendiculars of the ship; the wind force disturbances can be obtained according to the correlation coefficients calculated by the formulas above.

#### *4.2. Current Disturbances*

In the actual navigation process, it is assumed that the effect of the current on the ship can make the ship's speed and direction deviate [6]. Therefore, the effect of the current will change the *x*-axis and *y*-axis speed of the USV. To consider current disturbances, velocity is directly superimposed onto the ship's velocity. As shown in Formula (12), where *u* denotes transverse velocity after consideration of the current disturbances, *v* denotes the longitudinal velocity after consideration of the current disturbances, *uUSV* denotes the transverse velocity of the USV, *u<sup>C</sup>* denotes the transverse velocity of the current, *vUSV* denotes the longitudinal velocity of the USV, and *v<sup>C</sup>* denotes the longitudinal velocity of the current.

$$\begin{cases} \boldsymbol{u} = \boldsymbol{u}\_{\text{USV}} + \boldsymbol{u}\_{\text{C}} \\ \boldsymbol{v} = \boldsymbol{v}\_{\text{USV}} + \boldsymbol{v}\_{\text{C}} \end{cases} \tag{12}$$

#### **5. Construction of Time-Varying Trajectory Cells**

*5.1. Analysis of the USV Turning Experiment*

Disturbances by wind and current, dynamic constraints, state constraints, and other characteristics of environmental disturbances need to be considered from the perspectives of USV navigation characteristics and cyber-physic systems. Thus, this paper establishes a USV dynamics model under of wind and current disturbances, and based on a force analysis of the USV, we further analyze how the wind and current affect the navigation state of a USV in detail. The simulation experiments are carried out according to our trajectory analysis. The trajectory of the USV motion state without environmental disturbances, the trajectory of the USV motion state affected by the current, the trajectory of the USV motion state affected by the wind, and the trajectory of the USV motion state affected by wind and current at the same time are shown as follows.

As shown in Figure 3a, there are no environmental disturbances in which the angle of the turning rudder is 12.5°. It is obvious that when there are no environmental disturbances, the results of the USV cycle meet actual needs. As shown in Figure 3b, there are current disturbances of 1 m/s in the *x*-axis direction and 1 m/s in the *y*-axis direction, and in which the angle of the turning rudder is 12.5°; at this time, the turning path is shifted, and with the current disturbances, the turning trajectory becomes irregular. Figure 3c is the result of the turning experiment of a USV, with a wind speed of 2 m/s and a wind direction of true north, and in which the angle of the turning rudder is 12.5°; in this experiment, the turning experiment trajectory of the USV becomes irregular due to the disturbances of

the wind, resulting in an irregular cycle. As shown in Figure 3d, the current velocity is 1 m/s in the *x*-axis direction and 1 m/s in the *y*-axis direction. The turning experiment is conducted with disturbances of wind speed of 2 m/s in the true north direction, in which the angle of turning rudder is 12.5°; in this experiment, the turning trajectory of the USV is irregular because of the superposition effect of the current and wind, and the force direction and force of the USV are constantly changing, which makes it challenging to predict the trajectory.

**Figure 3.** USV turning experiments. (**a**) Turning experiment of a USV without environmental disturbances. (**b**) Turning experiment of a USV with current disturbances. (**c**) Turning experiment of a USV with wind disturbances. (**d**) Turning experiment of a USV with current and wind disturbances.

#### *5.2. Rules of Time-Varying Trajectories*

From these experimental results, when the USV is affected by the wind and current at the same time, the trajectories of the USV show irregular changes. However, the trajectory cell constructed in references [22,23] is a regular trajectory, which is not suitable for the motion planning of a USV with environmental disturbances. Thus, we established a regularization-trajectory cell, which lays the foundation for USV motion planning that takes into consideration the effects of both wind and current.

In this study, the influence of environmental factors is considered, and the rudder angle of the regularization-trajectory cell in this paper needs to be changed at any time through the search algorithm in order to maintain safe USV navigation. Therefore, the rudder angle of the regularization-trajectory cell in this paper is mutable, and it needs to be adjusted according to the environment change. Similar to references [22,23], to facilitate the consideration of the forces and moments on the USV, the following rules need to be considered in the construction of regularization-trajectory cells:

Rule 1: The trajectory cells are divided into 36 categories, and the trajectory distances of each category are equal within a certain error range. Based on this, a regularizationtrajectory cell library is constructed.

Because environmental factors such as wind, wave, and current are changing at every moment, the 36 categories of the regularization-trajectory cells are built according to the search direction of the algorithm. There is a large number of regularization-trajectory cells in each category, that is, each category of regularization-trajectory cells can form a sub-regularization-trajectory cell library, which can provide sufficient reachable areas for USV motion planning.

Rule 2: In order to maintain the continuity of the search path, the motion state of the regularization-trajectory cell at the beginning and at the end is kept stable.

Rule 3: In the case of wind and current disturbances in a certain trajectory cell at a certain time, it is necessary to turn the rudder only once in order to continue with navigation, excluding a rudder's return (steering to counteract the disturbances of the wind and current).

To sum up, the three rules specified in this section will lay the foundation for subsequent motion planning, which can better help to realize motion planning while considering environmental influences and USV navigation characteristics.

#### *5.3. Regularization-Trajectory Cells*

On the basis of the establishment of trajectory rules, in the grid environment, trajectories need to be constantly adjusted with the rudder angle to make the trajectory meet different navigation requirements based on the reachable points, the trajectory cell heading, and the final state of the trajectory cells. In addition, after generating the regularizationtrajectory cells, for the convenience of calculation, the regularization-trajectory cells of the USV are divided into 36 categories, as mentioned above.

#### 5.3.1. Regularization-Trajectory Cell Construction Method

Based on the dynamics model of the USV under the influence of wind and current, the regularization-trajectory cell is constructed on the premise of the regular constraints of the regularization-trajectory cell. First of all, we explore the same rudder angle in different directions, as shown in Figure 4, and search for the reachable points at 0.5° intervals (i.e., change the trajectory cell at 0.5° intervals, and the rudder angle at this time is 15°). It can be seen that the search area of the current point can be covered with full probability by exploring different heading intervals for the same rudder angle; thus, the full probability search can be realized.

**Figure 4.** Reachable points of the rudder angle at 15°; the heading is explored at 0.5° intervals.

According to the above-mentioned total probability exploration of the surrounding nodes, when the fixed trajectory cell [29,30] cannot be spliced, the established regularization-trajectory cell can be used (the trajectory cell changes according to the change of environments).

Figure 5 shows the chart for building a regularization-trajectory cell, which comprises of the USV geometry shape and physical characteristics. First, determine the current environmental information, that is, wind and current will have important disturbances on the trajectory cell; second, by building the uncertain trajectory cell library under the influence of wind and current, judge the characteristics and construct the corresponding trajectory cell; finally, resist the environmental disturbances by steering the rudder, that is to say, regularize the trajectory by changing the rudder angle in order to lay a foundation for the trajectory cell to achieve splice.

**Figure 5.** Construction process of a regularization-trajectory cell.

#### 5.3.2. Building a Regularization-Trajectory Cell Library

In this section, based on real-world environments, assume that the wind speed is 2 m/s and the wind direction is 0° (Figure 6 shows a schematic diagram of ship motion under the influence of wind and current). Assume that the current speed in the *x*-axis direction is 1 m/s and the current speed in the *y*-axis direction is 1 m/s (the calculation of wind speed and current velocity here is based on the absolute wind speed and current speed, that is, the calculation is carried out in the fixed coordinate system).

**Figure 6.** Schematic diagram of ship motion under the influence of wind and current.

Next, the regularization-trajectory cell is constructed on the basis of the above-mentioned method. Figures A1–A3 denote the regularization-trajectory cell library of the abovementioned environment (Figures A1–A3 can be seen from Appendix A).

#### **6. Motion Planning Method**

The trajectory rules and regularization-trajectory cells lay a solid foundation for the following planning method. In this section, based on the regularization-trajectory cell, motion planning for a USV under the influence of wind and current is further realized, which is derived from the A\* algorithm.

First, we analyze the effects of wind and current, and then we generate the regularizationtrajectory cell according to the influencing factors of wind and current. Second, we splice the trajectory cell under the influence of the environment according to the trajectory rules. We further adjust the path search generation value and the regularization-trajectory cells of the A\* algorithm in real time on the basis of the influence of the environments, and we fully consider the motion rules of the USVs under the influence of wind and current. Finally, we construct the motion planning method of the USV under the influence of wind and current on the basis of the search algorithm and the regularization-trajectory cells. Based on this method, not only can the influence of wind and current be considered, but the influence of wave and other complex environments on the path planning of the USV can also be analyzed in more detail in the future.
