1. Introduction:

2. Preliminary Studies

2.1. Studies about the Drone Energy Consumption

There are different types of drones defined according to their power supply.

Figure 2. Classification of the drones according to power supply (the energy supply of the actual drone used in the article is colored by red).

Figure 2. Classification of the drones according to power supply (the energy supply of the actual drone used in the article is colored by red).

In the case of hybrid systems, an Energy Management Strategy (EMS) system usually regulates and controls the efficient energy consumption. These EMSs can be based on different strategies, such as: Rule based systems; Intelligent based systems (Fuzzy or NN); Optimization based; etc. Classic electric UAVs, which operate only in motorized mode (where there is no recharging the battery) lack these systems, and energy savings has to be ensured by another approach. This is why energy efficient path planning was examined and improved in this project.

The energy consumption of the drone can be defined in two ways:

• practically: by direct measuring of the amps consumed by various parts of the drone.
• theoretically: with calculations of the energy consumptions of the parts.

For energy consuming calculations the following relations must be considered as a minimum. Engine power can be calculated based on forces, torques and moments on the x/y axis. Generally, Newton’s Laws are used to calculate forces and moments. See Figure 3 for a better understanding of the equations.

In general, the flight lift forces (F₁ – F₄) and M_X, M_y moments can be calculated as follows:

$$F_i=k_f\times {\omega }_i^2$$

(1)

$$M_x=\left(F_3-F_4\right)\times L$$

(2)

$$M_y=(F_1-F_2)\times L$$

(3)

where “L” is the distance between 2 rotors.

The operation’s conditions are given by the following relations:

• Hovering motion: Equilibrium conditions for hovering: mg=F₁+ F₂+ F₃+ F₄; and all moments = 0;

Equation of motion:

m=F₁+ F₂+ F₃+ F₄ - mg; m=0

(4)

2.

Rise/fall motion: Conditions for operations: mg < F₁+ F₂+ F₃+ F₄ = rise; mg > F₁+ F₂+ F₃+ F₄ = fall; and all moments = 0;

Equation of motion:

m=F₁+ F₂+ F₃+ F₄ - mg; m>0

(5)

3.

Yaw motion: Conditions for hovering mg = F₁+ F₂+ F₃+ F₄ ; and all moments ≠ 0;

Equation of motion:

(mass * linear acceleration) m*a = F₁+ F₂+ F₃+ F₄ - mg;

(6)

(I_ZZ * angular acceleration around ”Z”) I_ZZ*α_Z=M₁+ M₂+ M₃+ M₄

(6)

4.

Pitch / Roll motion: Conditions for hovering mg < F₁+ F₂+ F₃+ F₄ ; moments ≠ 0;

Equation of motion:

(mass * linear acceleration) m*a = F₁+ F₂+ F₃+ F₄ - mg;
(I_XX * angular acceleration around ”X”) I_XX*α_x=(F₃ - F₄)×L

(7)

For the precise description of the forces and exact modelling of the drone, the rigid-body dynamical relations in 3D environment must be known. Initially, the reference coordinate system is set up along with the related movements. Afterward, the following effects on the drone are considered:

• aerodynamic forces: friction and drag of propellers rotating in air
• secondary aerodynamic effects: blade flapping, ground effect, local flow fields
• inertial counter torques: gravitation, acting at the center, affect the rotation of propellers.
• gyroscopic effect: change in the orientation of the drone body and plane rotation of propellers

The dynamic equation can then be formulated as follows (in essence, this is rigid body dynamics extended to 3D environment):

$$\left[\begin{array}{c}\overrightarrow{F}\\ \tau \end{array}\right]=\left[\begin{array}{cc}{m}_3& 0_3\\ 0_3& \overrightarrow{I_3}\end{array}\right]\left[\begin{array}{c}\overrightarrow{a}\\ \propto \end{array}\right]+\left[\begin{array}{c}\omega \times m\overrightarrow{v}\\ \omega \times \overrightarrow{I_3}\omega \end{array}\right]$$

(8)

where: v – linear velocity (this is the origin linear velocity of the drone which will be modified by the wing velocity, see eq. (17) v_resulting), ω - angular velocity, α - angular acceleration, I - moment of inertia, a - linear acceleration, m - mass, F - total force, τ - total torque. The technical terms used earlier as well as energy consumptions concepts, are detailed in Figure 4.

2.2. The Energy Evaluation

Many ideas and calculations (see equations (9)-(12)) have been applied to the energy calculations from [21], maybe with a slightly different interpretation. Generalizing, the drone’s energy consumption can be summarized as follows:

E_total= E_processors+ E_motors + E_{communication} + E_peripherials

(9)

where, E_(anything)= P_(anything)*t

Some constant and very low energy for (E_{communication} + E_peripherials) is assumed. The energy consumed by the processor(s), is specified depending on control system complexity:

$$E_{processors}=\sum _{i=1}^{nr. ofproc.}{\int }_{t0}^{t1}{P}_{processor\left(i\right)}\left(t\right)*dt$$

(10)

Furthermore, the energy consumed by motors:

E_motors= E_takeoff(a,w)+ E_hover(a,w,t)+ E_move(w,s,t,ρ)

(11)

where the dominant energy is:

E_move(w,s,t,ρ)= E_drag(d,ρ,k_drag,S_eff)+ E_kinetic(w,s)

(12)

This E_move(w,s,t,ρ) is dominant and vital in different atmospheric conditions (this will be modelled later in the space model). Having completed the calculations for a particular case, the authors created the following “average energy map” of the drone, as seen in the diagram below (Phantom 4 drone pro/pro+ series).

Figure 5. The generalized energy consumption of drone (hover – blue, move – orange, idle – blue, peak - orange).

Figure 5. The generalized energy consumption of drone (hover – blue, move – orange, idle – blue, peak - orange).

These calculations were based on a “Phantom 4 Pro” drone was used. For energy calculations the simple math relations were used, where voltages are known and currents were measured: P= U*I = E/t. For thrust force calculations see equations (1-3), above. The two different colors of bars represent the “Hover/Move” energy of motors, and the “Idle/Peak” energy of the “Powerful” and “Weakly” loaded processors.

3. Environment Description and Work Space Partitioning

4. Creating the Polygonal Graph over the Work Space

Figure 7. Creation of the graph-like, (topological) map of the space, where the center points of the FS blocks are connected. Where the connecting line crosses (or touches) the occupied block (see red line) the connection is not possible.

Figure 7. Creation of the graph-like, (topological) map of the space, where the center points of the FS blocks are connected. Where the connecting line crosses (or touches) the occupied block (see red line) the connection is not possible.

4.1. The Weighting Mechanism of the Graph, Calculations of the Weights of Nodes and Edges

Let the graph “G” be defined as follows: G=〈n,e〉, where n(i) – are the nodes of the graph, and e(j) - are the edges of the graph. The weighting of the graph is divided into two phases. In phase 1, the edges are weighted related to the length (l(j)) of the edge e(j), then, in phase 2, the energy demand of the node (E(i)) is calculated using the previously defined mathematical expressions.

4.1.1. The Energy Demand Calculation of the Node

To evaluate the energy demand of the node, the following conditions are taken into account: the orientation of the drone (see: yaw/pitch/roll –equations of motion (6), (7)) [22]; altitude changing (see: rise/fall –equations of motion (5)); air resistance and wind directions (see: E_move – calculation, eq.: (11)). But basically in E_motors eq.: (10), all these calculations can be found. Consequently, the final energy demand of graph-node (i) can be calculated:

$$E_{\left(i\right)}=E_{motors}+f\left(yaw, pitch, roll\right)+f\left(rise, fall, hover\right)$$

(13)

Figure 8. Illustration of weight calculations along the path in 3 different situations, see nodes n(i). The drone is represented as a point and orientation, see the blue full arrow line. The wind direction is represented with dashed arrow line.

Figure 8. Illustration of weight calculations along the path in 3 different situations, see nodes n(i). The drone is represented as a point and orientation, see the blue full arrow line. The wind direction is represented with dashed arrow line.

Description of n(1): This is the node where the drone starts from, accelerates to the desired velocity, reaches the desired altitude (rise), and changes its orientation by an angle of α(1) (rotate around x-axis). After this, the drone will cover the distance l(1) in the given orientation at the given altitude. The tailwind is considered in E_motors equation.

Description of n(2): In this node the drone maintains the altitude, it only has to change its orientation around z-axis by an angle of α(2). In E_motors equation a crosswind has to be considered. After this, the drone will cover the distance l(2).

Description of n(3): In this node the drone has to change its altitude (drop to the given altitude), change the orientation by the angle α(3) around the z-axis, and the headwind (contra wind) is considered in E_motors equation.

4.1.2. The Final Weighting Calculation of the Path

There are many routes between the START and GOAL positions. Let the index of these paths be k={1,…,p} . Then the final weighting of the selected (k^th) path can be calculated as:

$${sumE}_{\left(k\right)}=\sum _{i=1}^n{E}_{\left(i\right)}+\sum _{j=1}^m{l}_j$$

(14)

The final execution path can be chosen based on the minimum cost function (C_f(min)), where:

C_fmin = min(sumE_k)

(15)

5. Possible Graph-Search Procedures, Surveying and Comparison the Effectiveness of Different Optimal Path Planning Methods.

6. Calculation the Resulting Velocity and Energy Demand of the Drone

(17)

This calculated energy is and must be considered, and added to the final calculation of the node’s energy demand.

7. Completing the Algorithm and Flowchart of the Process

Figure 11. The flow-chart of the entire process.

Figure 11. The flow-chart of the entire process.

8. Results and Analysis

• In Scenario 1, the tailwind is acting in the direction of the velocity vector of the drone. The energy consumption of the drone is minimal, the path faces the target position almost directly, see Figure 12. The starting position S=[1,1,0] and the goal position coordinates G=[11,13,10], v_wind=0,75*v_drone
• In Scenario 2, the drone’s velocity vector and the wind velocity vector were at 90 degrees. It is clear from the Figure 13, while the drone is flying between the houses, near to ground in leeward, it does not have much effect on the trajectory planning, but in an open field it does. The gliding effect can be partially observed. The wind vs. drone velocity ratio remains the same.
• In Scenario 3 the total headwind affects the drone, the two velocity vectors are opposite to each other. The ratio is: v_vind=0,75*v_drone. In this instance two different starting points were considered. S_L=[6,3,0] starting between the buildings (Figure 14, left), and S_R=[6,0,0]. The goal positions are identical in both instances. The significant difference between the routes becomes apparent only in the open area, namely, the gliding effect is much more prominent, compared to the previous case.

Figure 12. The calculated optimal trajectory from the START (left bottom corner) to the GOAL positions in a tailwind situation.

Figure 12. The calculated optimal trajectory from the START (left bottom corner) to the GOAL positions in a tailwind situation.

Figure 13. The calculated optimal trajectory from the START (left bottom corner) to the GOAL positions in a crosswind situation.

Figure 13. The calculated optimal trajectory from the START (left bottom corner) to the GOAL positions in a crosswind situation.

Figure 14. The calculated optimal trajectory from the START (left bottom corner) to the GOAL positions in a headwind situation.

Figure 14. The calculated optimal trajectory from the START (left bottom corner) to the GOAL positions in a headwind situation.

Figure 15. The sum of calculated optimal trajectories from the START (left bottom corner) to the GOAL positions.

Figure 15. The sum of calculated optimal trajectories from the START (left bottom corner) to the GOAL positions.

9. Conclusion

Essentially, a modified A* algorithm was used, because in the classic algorithm the edges were weighted and the optimal path was identified based on a cost function. In this modification, not only the edges but also the nodes were weighted, and the algorithm stepped from one node to the next, selecting the branch to the next node based on the actual weight (which is calculated and evaluated) at each node. A good example can be seen in Figure 14 (see the yellow path), where the drone flew in headwind and “sailed” between the nodes to achieve minimum energy use. Basically, this is a highly complex system, therefore it was divided into sub-programs (procedures) and their operations were interconnected and/or interlinked. The system relied on two Databases (DBs): 1. “MeteoCloud” – where the IoT data of the meteo stations was uploaded. The wind map for the selected Working Space was created from this DB; 2. is the “PointCloud” of geographical data of the selected WS. After partitioning the WS, the points were represented by the centre-points of the cubes, and these points were also the nodes of the graph (of course, only in free-spaces). These points (nodes) were in fact vectors (arrays) containing coordinates and calculated weights, .

10. Future Works

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