Modelling and Control Design of a Non-Collaborative UAV Wireless Charging System

This study proposes an analytical model of a WPT system with three orthogonal transmitter coils organised to produce a concentrated and controlled omnidirectional magnetic field suited for charging a moving, rotating load, providing maximal energy transfer without receiving end feedback. In order to create a realistic 3D WPT simulation system and a precise controller design, the mutual coupling values in terms of the receiver angular positions are modelled using the Ansys software. In using the established model of the 3DWPT system, an extremum seeking control (ESC) is used to maximize the power transfer utilizing the input power as an objective function assigned with specified parametric values defining the WPT model. The output power transmitted by the sending-end coils to a load of a moving UAV rotating in orbit is displayed. According to simulation results, when the receiver UAV speed is close to 2250 deg/s, the controller can accomplish a maximum power transfer of 2.6w in almost 1ms.


Introduction
WPT is an attractive battery-less solution for the Internet of Things (IoT), where the network of devices collects to share data wirelessly. WPT reduces the size of sensors and electrical devices by eliminating the physical connection between the transmitters and the receiver. It allows charging a wide variety of devices in both industrial and quotidian realms [1]. It is a game-changing technology for automating the recharging process for unmanned aerial vehicles (UAVs) or drones, particularly those employed in agriculture [2], crowd management [3], traffic monitoring in big cities [4], and disaster management [5]. The major limitation of WPT technology based on inductive coupling is the inability to maintain a high rate of wireless power transmission efficiency. In order to overcome these challenges, numerous coil-based topologies have been investigated.
A three orthogonal coils topology was proposed to pick up wireless power in robot applications. The primary and secondary coils were designed to fit inside a robot elbow to achieve a maximum coupling coefficient when the robot arm rotates [6]. A multi-coil structure WPT energy harvester was developed to alleviate the misalignment issue for mobile devices [7]. A cubic magnetic dipole coils arrangement was also proposed in [8] for omnidirectional and multi-load power transmission. Another study showed a 3D wireless charging cylinder with two orthogonal coils that can generate a spinning magnetic field to charge numerous loads [9]. The system of multiple bowl-shaped transmitting coils reported by the authors in [10] can create a uniform magnetic field inside the charging region to compensate for the misalignment issue. The authors developed a re-designed terminal receiver structure in [11] to improve the efficiency of an omnidirectional WPT system using The contribution of this paper is also presented in the implementation of the extremum seeking control technique for a 3DWPT system with circular coils, where the multi-parameter controller can maximise the input power of the system by controlling two parameters at the same time (θ, β) with an unknown reference function of the moving receiver. Eventually, the proposed control ESC technique can maximise the power transfer regardless of the position variation and velocity of the load.
Eventually, the proposed control ESC technique can maximise the power transfer regardless of the position variation and velocity of the load. The novel developed model considers the receiver's continuous rotational movement at a variable speed by expressing the input power of the 3DWPT system in terms of the receiver position angles.
Furthermore, the extremum seeking control technique was employed to maximise the power transfer when the receiver rotates at high velocity. The simulation results of this work are validated with those experimental results in [16]. The proposed model can simulate an average angular speed of 2250 deg/s, and the designed controller can provide maximum power delivery with a response time of less than 1ms. The remaining parts of the paper are structured as follows: Section 2 contains an analysis of the parameters of the three-dimensional WPT system, including the derivation of the relationship between the input power and the location of the mobile receiver angles. Section 3 explains extremum seeking control (ES) algorithms and the mobile receiver tracking process as the result of Section 2. Section 4 concludes the paper with a quantified conclusion of this research.

System Modelling
The 3DWPT system architecture is based on an orthogonal topology with three identical circular sending coils T x , T y , and T z , as shown in Figure 1. By using an orthogonal structure for the transmitter side, the mutual coupling between the three sending coils can be cancelled.
The contribution of this paper is also presented in the implementation of tremum seeking control technique for a 3DWPT system with circular coils, wh multi-parameter controller can maximise the input power of the system by con two parameters at the same time (θ, β) with an unknown reference function of the receiver. Eventually, the proposed control ESC technique can maximise the power regardless of the position variation and velocity of the load.
Eventually, the proposed control ESC technique can maximise the power regardless of the position variation and velocity of the load. The novel developed considers the receiver's continuous rotational movement at a variable speed by exp the input power of the 3DWPT system in terms of the receiver position angles.
Furthermore, the extremum seeking control technique was employed to m the power transfer when the receiver rotates at high velocity. The simulation re this work are validated with those experimental results in [16]. The proposed mo simulate an average angular speed of 2250 deg/s, and the designed controller can maximum power delivery with a response time of less than 1ms. The remaining the paper are structured as follows: Section 2 contains an analysis of the parameter three-dimensional WPT system, including the derivation of the relationship betw input power and the location of the mobile receiver angles. Section 3 explains ext seeking control (ES) algorithms and the mobile receiver tracking process as the r Section 2. Section 4 concludes the paper with a quantified conclusion of this resea

System Modelling
The 3DWPT system architecture is based on an orthogonal topology with thr tical circular sending coils T x , T y , and T z , as shown in Figure 1. By using an orth structure for the transmitter side, the mutual coupling between the three sending c be cancelled.
positions of the receiver.
The contribution of this paper is also presented in the implementation of the extremum seeking control technique for a 3DWPT system with circular coils, where the multiparameter controller can maximise the input power of the system by controlling two parameters at the same time (θ, β) with an unknown reference function of the moving receiver. Eventually, the proposed control ESC technique can maximise the power transfer regardless of the position variation and velocity of the load.
Eventually, the proposed control ESC technique can maximise the power transfer regardless of the position variation and velocity of the load. The novel developed model considers the receiver's continuous rotational movement at a variable speed by expressing the input power of the 3DWPT system in terms of the receiver position angles.
Furthermore, the extremum seeking control technique was employed to maximise the power transfer when the receiver rotates at high velocity. The simulation results of this work are validated with those experimental results in [16]. The proposed model can simulate an average angular speed of 2250 deg/s, and the designed controller can provide maximum power delivery with a response time of less than 1ms. The remaining parts of the paper are structured as follows: Section 2 contains an analysis of the parameters of the three-dimensional WPT system, including the derivation of the relationship between the input power and the location of the mobile receiver angles. Section 3 explains extremum seeking control (ES) algorithms and the mobile receiver tracking process as the result of Section 2. Section 4 concludes the paper with a quantified conclusion of this research.

System Modelling
The 3DWPT system architecture is based on an orthogonal topology with three identical circular sending coils Tx, Ty, and Tz, as shown in Figure 1. By using an orthogonal structure for the transmitter side, the mutual coupling between the three sending coils can be cancelled. We assume that the receiving coil is rotating in a spherical surface trajectory around the transmitter with a fixed radius (r = 0.3 m). In order to employ the resonant inductive WPT mechanism, a serial capacitor and resistor are added for each transmitting coil circuit.
The magnetic fields Bx, By, and Bz are produced, correspondingly, by the currents flowing in each transmitting coil, which are defined as ix, iy, and iz. The three transmitting end coils' resulting current vector I and magnetic field vector B are related, as deduced in [19]. We assume that the receiving coil is rotating in a spherical surface trajectory the transmitter with a fixed radius (r = 0.3 m). In order to employ the resonant in WPT mechanism, a serial capacitor and resistor are added for each transmitting coi The magnetic fields B x , B y , and B z are produced, correspondingly, by the c flowing in each transmitting coil, which are defined as i x , i y , and i z . The three trans end coils' resulting current vector I and magnetic field vector B are related, as d in [19]. We assume that the receiving coil is rotating in a spherical surface trajectory Γ around the transmitter with a fixed radius (r = 0.3 m). In order to employ the resonant inductive WPT mechanism, a serial capacitor and resistor are added for each transmitting coil circuit.
The magnetic fields B x , B y , and B z are produced, correspondingly, by the currents flowing in each transmitting coil, which are defined as i x , i y , and i z . The three transmitting end coils' resulting current vector I and magnetic field vector B are related, as deduced in [19]. Using Biot-Savart law, we can express the generated summation of the three orthogonal coil's magnetic fields using Equation (2): We can deduce from Equation (2) that the components of the resultant vectors of the magnetic field, B, and the current, I, have the same direction as the unity vectors of the Cartesian plane. As a result, the magnetic field can be directed without the transmitter moving. By tuning the current amplitudes of the sending end, we can enable the generation of a directed magnetic field towards any desired point in the Cartesian plane. As illustrated in Figure 2a, we define θ as the magnetic resultant polar angle and β as the azimuthal magnetic angle in spherical coordinates. As shown in Figure 2b, α and φ are the polar and the azimuthal angles, respectively, of the receiver's position in the spherical coordinates as well.
Using Biot-Savart law, we can express the generated summation of the three orthogonal coil's magnetic fields using Equation (2): We can deduce from Equation (2) that the components of the resultant vectors of the magnetic field, B, and the current, I, have the same direction as the unity vectors of the Cartesian plane. As a result, the magnetic field can be directed without the transmitter moving. By tuning the current amplitudes of the sending end, we can enable the generation of a directed magnetic field towards any desired point in the Cartesian plane. As illustrated in Figure 2a, we define θ as the magnetic resultant polar angle and β as the azimuthal magnetic angle in spherical coordinates. As shown in Figure 2b, α and φ are the polar and the azimuthal angles, respectively, of the receiver's position in the spherical coordinates as well. The three transmitting coils (Tx, Ty, and Tz) of the WPT system produce the resultant magnetic and electric vectors B and I represented by the angular direction θ and β in the spherical coordinates. The relationship between the electric resultant vector I and the currents ix, iy, and iz flowing in the transmitting coils (Tx, Ty, and Tz), respectively, is described by Equation (3).
Here, we realise that controlling θ and β allows the targeting of any moving load in 3D space by pointing the resulting magnetic and electric field vectors at any point in the (x, y, z) plane. The electric circuit diagram of the 3D WPT system in Figure 3 is used to derive the electric mathematical model of the system. The three transmitting coils (T x , T y , and T z ) of the WPT system produce the resultant magnetic and electric vectors B and I represented by the angular direction θ and β in the spherical coordinates. The relationship between the electric resultant vector I and the currents i x , i y , and i z flowing in the transmitting coils (T x , T y , and T z ), respectively, is described by Equation (3).
Here, we realise that controlling θ and β allows the targeting of any moving load in 3D space by pointing the resulting magnetic and electric field vectors at any point in the (x, y, z) plane. The electric circuit diagram of the 3D WPT system in Figure 3 is used to derive the electric mathematical model of the system.  The electric currents flowing in the sending coils Tx, Ty, and Tz are replaced by i1, i2, and i3, respectively, throughout the entire rest of the article. The system's electric parameters are related mathematically using Equation (4).
The inverter blocks in Figure 3 generate alternating voltages U1, U2, and U3, which cause the transmitting coils (Tx, Ty, and Tz) to exhibit the formation of the AC currents i1, i2, and i3, respectively. At the same time, the receiving load experiences an induced current i4. The three parameters, M13, M23, and M33, related to the coils Tx, Ty, and Tz concerning the receiver, are used to quantify the mutual coupling between the transmitting end and the receiving end. The self-inductances (L1, L2, L3), resistors (R1, R2, R3), and capacitances (C1, C2, C3) of the three transmitting loops are set to be equal for mathematical simplicity reasons. By changing (3) into (4) to become (5), we can introduce the angular direction parameters Theta and Betha as well as the resulting electrical vector I in one set of system equations, allowing us to manipulate the magnetic field resultant vector using the electric currents. The electric currents flowing in the sending coils T x , T y , and T z are replaced by i 1 , i 2 , and i 3 , respectively, throughout the entire rest of the article. The system's electric parameters are related mathematically using Equation (4).
The inverter blocks in Figure 3 generate alternating voltages U 1 , U 2 , and U 3 , which cause the transmitting coils (T x , T y , and T z ) to exhibit the formation of the AC currents i 1 , i 2 , and i 3 , respectively. At the same time, the receiving load experiences an induced current i4. The three parameters, M 13 , M 23 , and M 33 , related to the coils T x , T y , and T z concerning the receiver, are used to quantify the mutual coupling between the transmitting end and the receiving end. The self-inductances (L 1 , L 2 , L 3 ), resistors (R 1 , R 2 , R 3 ), and capacitances (C 1 , C 2 , C 3 ) of the three transmitting loops are set to be equal for mathematical simplicity reasons. By changing (3) into (4) to become (5), we can introduce the angular direction parameters Theta and Betha as well as the resulting electrical vector I in one set of system equations, allowing us to manipulate the magnetic field resultant vector using the electric currents.
As we can notice, the load current reaches its positive maximum when both the rotating magnetic field angles θ • and β • are directed to 45 • , as shown by the yellow contour in Figure 4. When the magnetic field angles are directed to the opposite position of the receiver (225 • ), the direction of the load current will be reversed-resulting in a negative pick as it is showing the blue contour.
As we can notice, the load current reaches its positive maximum wh tating magnetic field angles θ° and β° are directed to 45°, as shown by the in Figure 4. When the magnetic field angles are directed to the opposite receiver (225°), the direction of the load current will be reversed-resultin pick as it is showing the blue contour. Since the motion of the receiver is in a continuous trajectory, it is nece late the mathematical relationship between the continuous mutual coupli the system and the spherical coordinates functions of the receiver position φ(t). The three mutual coupling receiver trajectory functions in relation to th M14(α, φ), M24(α, φ), and M34(α, φ) are obtained using Ansys software Since the motion of the receiver is in a continuous trajectory, it is necessary to formulate the mathematical relationship between the continuous mutual coupling functions of the system and the spherical coordinates functions of the receiver position angles α(t) and φ(t). The three mutual coupling receiver trajectory functions in relation to the sending coils M 14 (α, φ), M 24 (α, φ), and M 34 (α, φ) are obtained using Ansys software finite elements calculations, as shown in Figure 5. Figure 6 shows a few positions of the receiver from the overall simulated trajectory. calculations, as shown in Figure 5. Figure 6 shows a few positions of the receiver from the overall simulated trajectory.

Model Benchmarking
As we can notice from Figure 6, the starting point of the receiver movement is represented in pink colour with the spherical coordinates of (r = 0.3m, α = 0°, φ = 0°). By fixing r and changing α and φ with a 1° step from 0 to 360°, we have 360 calculated values for each mutual coupling parameter (the black coils represent the positions occupied by the receiver when moving in trajectory), which is sufficient to fit accurately in a continuous function M (α, φ). Figure 6 represents the obtained mutual coupling values in terms of the calculations, as shown in Figure 5. Figure 6 shows a few positions of the receiver from the overall simulated trajectory.

Model Benchmarking
As we can notice from Figure 6, the starting point of the receiver movement is represented in pink colour with the spherical coordinates of (r = 0.3m, α = 0°, φ = 0°). By fixing r and changing α and φ with a 1° step from 0 to 360°, we have 360 calculated values for each mutual coupling parameter (the black coils represent the positions occupied by the receiver when moving in trajectory), which is sufficient to fit accurately in a continuous function M (α, φ). Figure 6 represents the obtained mutual coupling values in terms of the

Model Benchmarking
As we can notice from Figure 6, the starting point of the receiver movement is represented in pink colour with the spherical coordinates of (r = 0.3m, α = 0 • , φ = 0 • ). By fixing r and changing α and φ with a 1 • step from 0 to 360 • , we have 360 calculated values for each mutual coupling parameter (the black coils represent the positions occupied by the receiver when moving in Γ trajectory), which is sufficient to fit accurately in a continuous function M (α, φ). Figure 6 represents the obtained mutual coupling values in terms of the receiver angular positions α and φ. To measure the performance of the proposed model and benchmark the results, the system parameters used for the Ansys software simulation are the ones in [16], as shown in Table 1. In Table 2, the mutual coupling values between the T z coil and the receiver M z -L 2 imply that Ref. [16] considered the receiver to be only present in the (x, y) plane. In contrast, in Table 3, the receiver's movement occurs in a 3D trajectory. Table 2. Mutual coupling values Ref. [16]. As we can see, both results are in close agreement, and the small existing inequality is due to the difference in the orbit and the planes. In the first case, when θ = 45 • , the receiver is coupled equally only with Tx and Ty in the 2D plane [16]. In this work, the receiver is simultaneously coupled equally with the three coils for the same position. The remaining two cases of the two tables are when the receiver is coupled with one coil at a time.

The 3D WPT System Input Power in Terms of the Receiver Angles
By employing the attained mutual coupling angular functions of the system in the current load Equation (6), it is possible to establish a mathematical relationship that describes the power variation of the system in terms of the angular functions of the receiver trajectory. By substituting (7)- (9) in (6), the current load formula becomes: We define the power load formula as follows: The amplitude of i 4 is defined by Equation (12) The load power angular Equation (13) is obtained by substituting Equations (12) into (11).
As it is showing in Figure 7, when the receiver is placed at α = φ = 45 • and the magnetic field resultant is directed to the receiver position θ = β = 45 • , the load power reaches its maximum for the 3D WPT.
As it is showing in Figure 7, when the receiver is placed at α = φ =45°and field resultant is directed to the receiver position θ = β = 45°, the load pow maximum for the 3D WPT. Because of the receiver mobility, it is impractical to use the load powe for the controlling process, therefore deriving the input power expression i The input power is defined as follows: Because of the receiver mobility, it is impractical to use the load power as feedback for the controlling process, therefore deriving the input power expression is a must.
The input power is defined as follows: where By putting all the ohmic resistors of the coils equal to R and substituting (12) into (15) we have: Equation (16) incorporates all the important parameters of the 3DWPT system. By introducing the angular parameters of the receiver trajectory (α, φ) and the magnetic field angles (θ, β) to the input power expression, we have established a comprehensive model able to simulate the system power with relation to the receiver dynamics for a particular trajectory.
The following input power curves are plotted for different receiver angular positions in Cartesian and spherical coordinates using the developed 3DWPT angular model as elaborated previously.
Using (16), we can simulate the input power curve when the receiver is fixed and the magnetic field resultant is rotating across the sending end, as shown in Figure 7.
When the receiver is located on the top of the transmitting coils, facing the T z coil, in this case, the maximum transferred input power is directed through the Z-axis, as shown in Figures 8 and 9.
By putting all the ohmic resistors of the coils equal to R and substituting we have: Equation (16) incorporates all the important parameters of the 3DWP introducing the angular parameters of the receiver trajectory (α, φ) and the m angles (θ, β) to the input power expression, we have established a compreh able to simulate the system power with relation to the receiver dynamics fo trajectory.
The following input power curves are plotted for different receiver ang in Cartesian and spherical coordinates using the developed 3DWPT angu elaborated previously.
Using (16), we can simulate the input power curve when the receiver is magnetic field resultant is rotating across the sending end, as shown in Figu When the receiver is located on the top of the transmitting coils, facing this case, the maximum transferred input power is directed through the Z-a in Figures 8 and 9.  From Figure 10, it is noticeable that when the receiver is placed at th position, the power transfer reaches the maximum of 2.5 W towards the rece In this position, the receiver is coupled equally with the three transmitters significant rise in the input power. The spherical plot in Figure 11 shows the input power pattern when From Figure 10, it is noticeable that when the receiver is placed at the 45 • angular position, the power transfer reaches the maximum of 2.5 W towards the receiver direction. In this position, the receiver is coupled equally with the three transmitters, resulting in a significant rise in the input power. From Figure 10, it is noticeable that when the receiver is placed at the 45° position, the power transfer reaches the maximum of 2.5 W towards the receiver d In this position, the receiver is coupled equally with the three transmitters, resul significant rise in the input power. Figure 10. The input power plot for the 3D WPT system when the receiver is placed at α = The spherical plot in Figure 11 shows the input power pattern when apply tating magnetic field on a fixed receiver position. It is noticeable that the symmet input power peaks is due to the omnidirectional nature of the 3DWPT. The spherical plot in Figure 11 shows the input power pattern when applying a rotating magnetic field on a fixed receiver position. It is noticeable that the symmetry in the input power peaks is due to the omnidirectional nature of the 3DWPT.
Sensors 2022, 22, x FOR PEER REVIEW Figure 11. The input power spherical plot for the 3D WPT system when the receiver is pla In Figures 12 and 13, the receiver is fully coupled with Tx coil only, due to w input power drops from 2.5 W to 1.4 W compared to the central position in Figur maximum input power flow is from the Tx coil to the receiver across the Y-axis. Figure 12. The input power plot for the 3D WPT system when the receiver is placed at α = Figure 11. The input power spherical plot for the 3D WPT system when the receiver is placed at In Figures 12 and 13, the receiver is fully coupled with Tx coil only, due to which the input power drops from 2.5 W to 1.4 W compared to the central position in Figure 11; the maximum input power flow is from the Tx coil to the receiver across the Y-axis.
Sensors 2022, 22, x FOR PEER REVIEW Figure 11. The input power spherical plot for the 3D WPT system when the receiver is pla In Figures 12 and 13, the receiver is fully coupled with Tx coil only, due to w input power drops from 2.5 W to 1.4 W compared to the central position in Figur maximum input power flow is from the Tx coil to the receiver across the Y-axis. By comparing Figure 14 with Figure 10 we can conclude that the 3DWPT s genuine omnidirectional because the receiver will pick the same amount of pow it is placed in symmetric positions (α = φ = 45°, α = φ = 135°).  By comparing Figure 14 with Figure 10 we can conclude that the 3DWPT system is genuine omnidirectional because the receiver will pick the same amount of power when it is placed in symmetric positions (α = φ = 45 • , α = φ = 135 • ). By comparing Figure 14 with Figure 10 we can conclude that the 3DW genuine omnidirectional because the receiver will pick the same amount o it is placed in symmetric positions (α = φ = 45°, α = φ = 135°). By observing Figures 11 and 15, we can understand that both the receiver spherical positions when α = φ = 45 • , α = φ = 135 • produce the same input power of 2.5 W. However, the flow direction of the input power is focused on the 135 • axis in the 3D plan. The spherical plots of the input power in Figures 9, 11, 13 and 15 are repres a dumbbell shape with two maxima. Depending on the position of the receiver resultant magnetic angles θ and β in the spherical coordinates, the power transf maximised through the directional control of the magnetic field resultant toward act position of the receiver. Table 4 in the benchmarking paper Ref. [16] shows t ured input power values when the receiver is coupled with one transmitting co is similar to the cases when the receiver is positioned at α = φ = 0° and α = φ = 90 research. By comparing Figures 8 and 12 with Table 4, we can notice that the designe in this paper is in close accordance with the measured value of the benchmarkin Since this paper proposes a non-collaborative control technique, all the feedback receiver is eliminated. By monitoring the input power measurement, we can s receiver position indirectly and use it to update the controller.
The plots in Figures 16 and 17 show that the power transfer efficiency is u when the magnetic resultant points to the exact position of the receiver and 0% w magnetic resultant is in a phase difference of 90° with the receiver. Thus, contro magnetic field resultant is a must to maintain power maximisation.
The spherical plots of the input power in Figures 9, 11, 13 and 15 are represented as a dumbbell shape with two maxima. Depending on the position of the receiver and the resultant magnetic angles θ and β in the spherical coordinates, the power transfer can be maximised through the directional control of the magnetic field resultant towards the exact position of the receiver. Table 4 in the benchmarking paper Ref. [16] shows the measured input power values when the receiver is coupled with one transmitting coil, which is similar to the cases when the receiver is positioned at α = φ = 0 • and α = φ = 90 • for this research. By comparing Figures 8 and 12 with Table 4, we can notice that the designed model in this paper is in close accordance with the measured value of the benchmarking paper. Since this paper proposes a non-collaborative control technique, all the feedback from the receiver is eliminated. By monitoring the input power measurement, we can sense the receiver position indirectly and use it to update the controller.
The plots in Figures 16 and 17 show that the power transfer efficiency is up to 80% when the magnetic resultant points to the exact position of the receiver and 0% when the magnetic resultant is in a phase difference of 90 • with the receiver. Thus, controlling the magnetic field resultant is a must to maintain power maximisation.         Figure 19 represents the maximum input power for the 3D WPT system when the receiver is moving according to the 8-shape continuous trajectory. The curve was simulated by rendering the functions of the magnetic resultant θ(t) and β(t), equal to the receiver angular continues trajectory α(t) and φ(t). As is depicted in Figure 20 by a blue and yellow colour, the input power fluctuates between two maxima of 1.3 W and 2.5 W, respectively.   Figure 19 represents the maximum input power for the 3D WPT system when the receiver is moving according to the 8-shape continuous trajectory. The curve was simulated by rendering the functions of the magnetic resultant θ(t) and β(t), equal to the receiver angular continues trajectory α(t) and φ(t). As is depicted in Figure 20 by a blue and yellow colour, the input power fluctuates between two maxima of 1.3 W and 2.5 W, respectively.  Figure 19 represents the maximum input power for the 3D WPT system when the receiver is moving according to the 8-shape continuous trajectory. The curve was simulated by rendering the functions of the magnetic resultant θ(t) and β(t), equal to the receiver angular continues trajectory α(t) and φ(t). As is depicted in Figure 20 by a blue and yellow colour, the input power fluctuates between two maxima of 1.3 W and 2.5 W, respectively.   The developed model in this work can generate the input power function at any given point from the chosen trajectory . As shown in Figure 20a, the overall input power spherical plot for the trajectory is forming a butterfly shape. It is noticeable that the moving receiver can pick up power at any given point from ; however, it is necessary that the magnetic resultant angles are always equal to the receiver's angular position regardless of its movement.
As we can notice from Figure 20b, the input power has a linear relationship with the load power, hence both the input power and the load power have the same variation behaviour with a different amplitude.
From this section, we conclude that if we can design a control action that can track magnetically the receiver position across the trajectory , it is possible to ensure an autonomous maximised power transfer for a moving load in a 3D trajectory without feedback.

Extremum Seeking Scheme for the Multi-Parameter System
According to (13), maximising the load power through the input power is possible. The only available feedback from the transmitter is the input power to maximise the power transfer efficiency. Thus, taking it as the objective function ( ) The continuously differentiable function can be approximated locally by Equation (18) as detailed in [20]: The developed model in this work can generate the input power function at any given point from the chosen trajectory Γ. As shown in Figure 20a, the overall input power spherical plot for the Γ trajectory is forming a butterfly shape. It is noticeable that the moving receiver can pick up power at any given point from Γ; however, it is necessary that the magnetic resultant angles are always equal to the receiver's angular position regardless of its movement.
As we can notice from Figure 20b, the input power has a linear relationship with the load power, hence both the input power and the load power have the same variation behaviour with a different amplitude.
From this section, we conclude that if we can design a control action that can track magnetically the receiver position across the trajectory Γ, it is possible to ensure an autonomous maximised power transfer for a moving load in a 3D trajectory without feedback.

Extremum Seeking Scheme for the Multi-Parameter System
According to (13), maximising the load power through the input power is possible. The only available feedback from the transmitter is the input power to maximise the power transfer efficiency. Thus, taking it as the objective function f (γ) for the extremum seeking controller (ESC) as follows: By putting f (γ) = P in , we have: The continuously differentiable function can be approximated locally by Equation (18) as detailed in [20]: where f is a C 2 function, γ is the input and, γ * is the optimal value that renders the output equal to the extremum f * . The objective is to minimise the difference between the unknown optimal input value and the current value so that γ it converges to γ * which makes f (γ) = f * . P l×l = P T < 0 is a gain matrix, γ = [γ 1 . . . .γ l ] T and l indicates the number of the input parameters to the plant, in our case l = 2 (the rotating magnetic field angles θ, β), In the controller design for the 3DWPT system, we are using a multiparameter extremum seeking control with two inputs (θ, β) and one output, the input power P in . In order to simulate the receiver dynamics in the 3DWPT plant, the functions of the mutual coupling M 14 (α, φ), M 24 (α, φ), and M 34 (α, φ) are set to be varied during the tracking process of the controller. It is possible to simulate various movement patterns and velocities according to the defined trajectory in Figure 19 simply by selecting the desired signals of the receiver position angles α and φ in the spherical coordinates. Figure 21 illustrates the proposed scheme for the system simulation and control implementation.
where f is a 2 C function, γ is the input and, γ * is the optimal value that renders the output equal to the extremum * f . The objective is to minimise the difference between the unknown optimal input value and the current value so that γ it converges to γ * which is a gain matrix, 1 .... In the controller design for the 3DWPT system, we are using a multiparameter extremum seeking control with two inputs (θ, β) and one output, the input power Pin. In order to simulate the receiver dynamics in the 3DWPT plant, the functions of the mutual coupling M14(α, φ), M24(α, φ), and M34(α, φ) are set to be varied during the tracking process of the controller. It is possible to simulate various movement patterns and velocities according to the defined trajectory in Figure 19 simply by selecting the desired signals of the receiver position angles α and φ in the spherical coordinates. Figure 21 illustrates the proposed scheme for the system simulation and control implementation. The controller starts to search for the optimum input γ * values that maximise the objective function regardless of the mobility behaviour of the receiver. Since the power transfer is omnidirectional, the magnetic field angles (θ, β) control process is run only in the first half of the sphere of the spherical coordinates. Consequently, it will cover the other remaining half. The simulation results obtained by running a Simulink-based design scheme are shown in Figure 21, which accurately tracks the maximum input power with a small-time response for both steady-state and continuous rapid movement of the receiver.

The Closed-Loop Response When Using ESC for a Continuous Trajectory and Constant Velocity
For the receiver to achieve the proposed trajectory in Figure 18, α and φ must be a ramp signal with a constant slope from 0° to 360°, as represented with dashed lines in Figures 22 and 23. The controller starts to search for the optimum input γ * values that maximise the objective function regardless of the mobility behaviour of the receiver. Since the power transfer is omnidirectional, the magnetic field angles (θ, β) control process is run only in the first half of the sphere of the spherical coordinates. Consequently, it will cover the other remaining half. The simulation results obtained by running a Simulink-based design scheme are shown in Figure 21, which accurately tracks the maximum input power with a small-time response for both steady-state and continuous rapid movement of the receiver.

The Closed-Loop Response When Using ESC for a Continuous Trajectory and Constant Velocity
For the receiver to achieve the proposed trajectory in Figure 18, α and φ must be a ramp signal with a constant slope from 0 • to 360 • , as represented with dashed lines in Figures 22 and 23.  The ESC controller tracks the values of the receiver angles accurately by re the magnetic felid angles θ and β to be equal to α and φ, as it is shown in the abov which results in the maximisation of the input power, as shown in Figure 24.  The ESC controller tracks the values of the receiver angles accurately by reg the magnetic felid angles θ and β to be equal to α and φ, as it is shown in the above which results in the maximisation of the input power, as shown in Figure 24. The ESC controller tracks the values of the receiver angles accurately by regulating the magnetic felid angles θ and β to be equal to α and φ, as it is shown in the above graph, which results in the maximisation of the input power, as shown in Figure 24 Figure 24 shows that the input power fluctuates with a constant frequency due to the constant velocity of the receiver.

The Closed-Loop Response When Using ESC for a Continuous Trajectory with Intermittent Movement
In this section, we simulate an intermittent motion for the receiver dynamics. We use the following angular signal for the receiver trajectory block as input for α and φ; the signal consists of a movement at a fixed angular speed of 31.4 rad/s from 0 s to 0.1 s. After that, the receiver takes a static position from 0.1 s to 0.2 s. The movement sequence is repeated, as shown in Figures 25 and 26. The black dashed line represents the trajectory, as mentioned earlier, of the receiver in the spherical coordinates, while the green and blue signals represent the searched magnetic angles θ and β, respectively.

The Closed-Loop Response When Using ESC for a Continuous Trajectory with Intermittent Movement
In this section, we simulate an intermittent motion for the receiver dynamics. We use the following angular signal for the receiver trajectory block as input for α and φ; the signal consists of a movement at a fixed angular speed of 31.4 rad/s from 0 s to 0.1 s. After that, the receiver takes a static position from 0.1 s to 0.2 s. The movement sequence is repeated, as shown in Figures 25 and 26. The black dashed line represents the trajectory, as mentioned earlier, of the receiver in the spherical coordinates, while the green and blue signals represent the searched magnetic angles θ and β, respectively.

The Closed-Loop Response When Using ESC for a Continuous Trajectory with Inte Movement
In this section, we simulate an intermittent motion for the receiver dynamics the following angular signal for the receiver trajectory block as input for α and φ; t   When applying an interment movement on the receiver, we can notice that th parameter extremum seeking control technique can preserve an accurate trackin maximum power under instantaneous movement transitions, as shown in Figure   Figure 27. The maximised input power response in the closed-loop versus the calculat power when the receiver moves intermittently.
The receiver accelerated movement is reflected in the input power graph, w frequency of the input power signal is increased in the time interval: 0 s to 0.1 s. Figure 28 shows the tracking process of the maximum input power, as we no controller can reach the optimum in a small-time response of 1ms with a small When applying an interment movement on the receiver, we can notice that the multiparameter extremum seeking control technique can preserve an accurate tracking of the maximum power under instantaneous movement transitions, as shown in Figure 27. When applying an interment movement on the receiver, we can notice that th parameter extremum seeking control technique can preserve an accurate trackin maximum power under instantaneous movement transitions, as shown in Figur The receiver accelerated movement is reflected in the input power graph, w frequency of the input power signal is increased in the time interval: 0 s to 0.1 s. Figure 28 shows the tracking process of the maximum input power, as we n controller can reach the optimum in a small-time response of 1ms with a smal state error. The receiver accelerated movement is reflected in the input power graph, where the frequency of the input power signal is increased in the time interval: 0 s to 0.1 s. Figure 28 shows the tracking process of the maximum input power, as we notice the controller can reach the optimum in a small-time response of 1ms with a small steadystate error. In Figure 29, the difference between the open-and closed-loop input power is quite remarkable; the controller will continuously track the maximum available power for each receiver position.

The Closed-Loop Response When Using ESC for a Continuous Trajectory with an Accelerated Movement
This section examines the effect of the receiver's accelerated movement on the 3DWPT system in the closed loop. By increasing the ramp slope of the receiver angular input signals α and φ, one can increase the velocity and make it variable. From 0 (s) to 0.2 (s), the receiver rotates from 0° to 135° with a velocity of 675 deg/s. After that, the velocity is increased to 2250 deg/s, as shown in dashed lines in Figures 30 and 31. In Figure 29, the difference between the open-and closed-loop input power is quite remarkable; the controller will continuously track the maximum available power for each receiver position. In Figure 29, the difference between the open-and closed-loop input power is quite remarkable; the controller will continuously track the maximum available power for each receiver position.

The Closed-Loop Response When Using ESC for a Continuous Trajectory with an Accelerated Movement
This section examines the effect of the receiver's accelerated movement on the 3DWPT system in the closed loop. By increasing the ramp slope of the receiver angular input signals α and φ, one can increase the velocity and make it variable. From 0 (s) to 0.2 (s), the receiver rotates from 0° to 135° with a velocity of 675 deg/s. After that, the velocity is increased to 2250 deg/s, as shown in dashed lines in Figures 30 and 31.

The Closed-Loop Response When Using ESC for a Continuous Trajectory with an Accelerated Movement
This section examines the effect of the receiver's accelerated movement on the 3DWPT system in the closed loop. By increasing the ramp slope of the receiver angular input signals α and φ, one can increase the velocity and make it variable. From 0 (s) to 0.2 (s), the receiver rotates from 0 • to 135 • with a velocity of 675 deg/s. After that, the velocity is increased to 2250 deg/s, as shown in dashed lines in Figures 30 and 31.  The green signal in Figure 30 represents the optimum value of the polar angle θ searched by the controller, which maximises the input pow see, the value of θ always oscillates around the exact receiver trajectory α, accurate magnetic tracking for the moving target with small angular erro Figure 32.  The green signal in Figure 30 represents the optimum value of the polar angle θ searched by the controller, which maximises the input pow see, the value of θ always oscillates around the exact receiver trajectory α, accurate magnetic tracking for the moving target with small angular erro Figure 32. The green signal in Figure 30 represents the optimum value of the magnetic field polar angle θ searched by the controller, which maximises the input power. As we can see, the value of θ always oscillates around the exact receiver trajectory α, which ensures accurate magnetic tracking for the moving target with small angular error, as shown in Figure 32. Similarly, in the closed loop, the magnetic field azimuthal angle β tracks the receiver position φ precisely, as shown in blue colour from Figure 31. Correspo the tracking angle error of β fluctuates between −4° and 4°in Figure 33, which is co a minor value due to the calculation error present in the model. The calculated input power in the black dashed line in Figure 34 shows how power range alternates from partially coupled to a fully coupled receiver under erated movement at high velocity (simulation time 0.3 s). When the closed-loop E action is applied, the red curve represents the maximised input power searche controller. When the acceleration occurs at 0.3 s, the frequency of the input pow increase due to the rise in the receiver velocity. Similarly, in the closed loop, the magnetic field azimuthal angle β tracks the angular receiver position φ precisely, as shown in blue colour from Figure 31. Correspondingly, the tracking angle error of β fluctuates between −4 • and 4 • in Figure 33, which is considered a minor value due to the calculation error present in the model. Similarly, in the closed loop, the magnetic field azimuthal angle β tracks the receiver position φ precisely, as shown in blue colour from Figure 31. Correspo the tracking angle error of β fluctuates between −4° and 4°in Figure 33, which is co a minor value due to the calculation error present in the model. The calculated input power in the black dashed line in Figure 34 shows how t power range alternates from partially coupled to a fully coupled receiver under erated movement at high velocity (simulation time 0.3 s). When the closed-loop E action is applied, the red curve represents the maximised input power searche controller. When the acceleration occurs at 0.3 s, the frequency of the input pow increase due to the rise in the receiver velocity. The calculated input power in the black dashed line in Figure 34 shows how the input power range alternates from partially coupled to a fully coupled receiver under an accelerated movement at high velocity (simulation time 0.3 s). When the closed-loop ES control action is applied, the red curve represents the maximised input power searched by the controller. When the acceleration occurs at 0.3 s, the frequency of the input power signal increase due to the rise in the receiver velocity. From Figure 31, it is observable that the receiver will pick up the maximum regardless of its position or speed under the closed-loop control. Hence the maxi of the power transfer for the 3D orthogonal coils system is achieved for rapidly acc mobility. The developed controller is able to attain a maximum input power ou the 3D WPT system in a very small-time response and almost insignificant static Figure 35 shows the values of the tracking error, which is less than 0.1 w. It i able that when the receiver velocity increased in the last region of the simulation frequency of the error picks rises as well.  Table 5 summarises the limitations as well as the contribution of each develo trol method for the 3dWPT system. Both references [15,16] do not take into consi the mobility of the receiver, neither the continuous trajectory or the velocity. From Figure 31, it is observable that the receiver will pick up the maximum power regardless of its position or speed under the closed-loop control. Hence the maximisation of the power transfer for the 3D orthogonal coils system is achieved for rapidly accelerated mobility. The developed controller is able to attain a maximum input power output for the 3D WPT system in a very small-time response and almost insignificant static error. Figure 35 shows the values of the tracking error, which is less than 0.1 w. It is noticeable that when the receiver velocity increased in the last region of the simulation time, the frequency of the error picks rises as well. From Figure 31, it is observable that the receiver will pick up the maximum regardless of its position or speed under the closed-loop control. Hence the maxi of the power transfer for the 3D orthogonal coils system is achieved for rapidly ac mobility. The developed controller is able to attain a maximum input power ou the 3D WPT system in a very small-time response and almost insignificant static Figure 35 shows the values of the tracking error, which is less than 0.1 w. It able that when the receiver velocity increased in the last region of the simulation frequency of the error picks rises as well.  Table 5 summarises the limitations as well as the contribution of each develo trol method for the 3dWPT system. Both references [15,16] do not take into consi the mobility of the receiver, neither the continuous trajectory or the velocity.  Table 5 summarises the limitations as well as the contribution of each developed control method for the 3dWPT system. Both references [15,16] do not take into consideration the mobility of the receiver, neither the continuous trajectory or the velocity. In Figure 36, we have encapsulated the modelling steps of the receiver trajectory along with the 3DWPT transmitter. First, we define a trajectory for the drone using angular positions; after that, the dynamic model will generate the angular mutual coupling functions for the used trajectory. After that, mutual coupling drone's dynamics is implemented in the objective function of the input power (electric model of the 3DWPT).  Furthermore, the ES control loop is also presented. The value of the gradient change for the plant is probed by the sinusoidal perturbation signal that is fed into the system's measured output.
K is the learning rate which determines the convergence speed to the maximum, whereas the high pass filter is used to reduce the static error as well as ensure the stability of the closed-loop response.

Conclusions
This paper presented a design control and modelling for a three-dimensional wireless power transfer system with dynamic charging for mobile loads. Modelling an extensive WPT plant system incorporating the receiver mobility in a 3D trajectory will contribute to a relabel control design for the magnetic tracking concept. Based on the obtained model, an extremum seeking controller is integrated to optimise the input power function under high-speed receiver motion. The controller was tested at a variable velocity receiver trajectory, including the accelerated and intermittent movement of the load. In the closed loop, the input power converged with the calculated maximum of 2.6 W in less than 1ms for the studied 8-shaped trajectory. When the receiver rotates, the maximum power is guaranteed for an angular velocity of 2250 deg/s. In theory, the control method is suitable for intelligent applications such as drones and smart sensors. The ESC is effective in response time and steady-state error, providing a maximum power transfer regardless of the receiver's position, trajectory, or speed.