Active manipulation of a tethered drone using explainable AI

3.1. General system dynamics

Figure 1 shows a schematic diagram of a drone of mass $$ m $$ tethered to a ground robot. $$ W $$ and $$ B $$ are the world frame of the system and the body frame of the drone, respectively. The diagram also presents different forces acting on both the drone and the ground rover at equilibrium condition in which the drone is moving at constant speed with the magnitude and the direction of motion same as that of the ground rover. The drone is assigned symbol ‘$$ o $$’ while the ground rover is assigned symbol ‘$$ i $$’. At equilibrium, the control of the drone is designed to align the pitch axis of its body frame to the direction of motion of the ground vehicle. Thus, it does not need to roll or yaw in this state (the desired roll and yaw angles, $$ \phi^d_o $$=$$ \psi_o^d $$=0). In the equilibrium condition, the drone only pitches by angle $$ \theta^{eq}_o $$ (equilibrium pitch angle of the drone) to create the required tension in the tether and the motion of the drone. In order to compute the desired pitch angle at equilibrium or hovering position, we perform the following analysis.

Overall, three forces act on the drone in the tethered position, viz., the tether tension ($$ T $$), the drone actuation force or thrust force due to rotors ($$ F_o^t $$), and the weight of the drone ($$ mg $$), respectively, where $$ g $$ is the acceleration due to gravity. It should be noted that $$ F_o^w $$ denotes the force due to wind disturbance which will be considered later in sections of wind disturbance analysis. Now, the force acting on the drone $$ F^a_o = \begin{bmatrix} F^a_x & F^a_y & F^a_z \end{bmatrix}^T $$ (in world frame $$ W $$) due to actuation of the rotors is given by:

where $$ F^t_o = (f_1 + f_2 + f_3 + f_4) $$ is the thrust force (in frame $$ B $$) created by the drone which corresponds to the summation of rotor forces ($$ f_1 $$, $$ f_2 $$, $$ f_3 $$, $$ f_4 $$), and $$ R_B^W $$ is the rotation matrix to go from body ($$ B $$) to world frame ($$ W $$) (see Figure 1). We use the Euler sequence of $$ Z-X-Y $$ to model the rotation matrix of the system in the world frame. The rotation matrix $$ R_B^W $$ is:

where $$ c(.) $$ and $$ s(.) $$ are the cosine and sine terms, respectively. Upon analysis of the free body diagram, at equilibrium, we have:

By linearizing Equation 2, we can compute the desired pitch angle ($$ \theta_o^{eq} $$) to maintain a desired tension ($$ T^{eq}=\begin{bmatrix} T^{eq}_x & T^{eq}_y & T^{eq}_z \end{bmatrix}^T $$) in tether and desired angle ($$ \alpha^{eq} $$) of tether with the ground. Thus, $$ \theta_o^{eq} $$ can be computed using:

To compute the hovering rotor speed $$ \omega_o^h $$ of the drone, we compute the total equilibrium force $$ F_o^{eq} $$ acting on the drone as follows:

The hovering speed $$ \omega_o^h $$ of the drone is,

Here, $$ k_f = 2.2\times10^{-4} $$ is the chosen motor constant (^[39]). The next step consists of writing the equations of motion of the drone and the ground rover. The translational and rotational equations of motion of the tethered drone are given by,

where $$ r_o $$ is the position of the drone in world frame ($$ W $$). $$ I $$ and $$ L $$ are the mass moment of inertia and the arm length of the drone, respectively. $$ M_1 $$, $$ M_2 $$, $$ M_3 $$, and $$ M_4 $$ are the moments created by the four rotors of the drone. $$ \eta_o = \begin{bmatrix} \phi_o & \theta_o & \psi_o \end{bmatrix}^T $$ is the orientation of the drone. It should be noted that a small angle approximation is assumed in the entire paper. Thus, angular velocity and angular acceleration of the drone are $$ \dot{\eta} $$ and $$ \ddot{\eta} $$, respectively. We then compute the net desired pitch angle $$ \theta_o^d $$ and yaw angle $$ \psi_o^d $$ as follows:

where,

where $$ F^m = \begin{bmatrix} F^m_x & F^m_y & F^m_z \end{bmatrix}^T $$ is the measured force at the point of contact between the drone and the tether. Here, $$ \theta_o^f $$ and $$ \psi_o^f $$ are the pitch and yaw commands (additional to being applied at equilibrium) to enable the drone to follow the ground robot. It is worth noting that we calculate this sensed force mathematically for simulation analysis, while for experiments, we use a force sensor using load cells. $$ func_{1} $$ and $$ func_{2} $$ are the proposed fuzzy logic functions that evaluate the desired pitch $$ \theta_o^f $$ and yaw $$ \psi_o^f $$ angles based on the contact forces and their rates. The net desired pitch and yaw angles, $$ \theta_o^d $$ and $$ \psi_o^d $$, are then commanded to the attitude controller of the drone thus making the drone coordinate with the ground robot based on the measured forces. The ground rover actively manipulates the drone without requiring to know the three-dimensional position of the drone. Such techniques are especially beneficial for GPS denied environments and planetary explorations. Note that we assume that the tether is elastic and there is no separately implemented controller for the vertical $$ Z_W $$ axis of the drone. The drone is made to operate at hovering speed $$ \omega_o^h $$ and the desired orientation that allows it to obtain its position along vertical $$ Z_W $$ direction based on the tether length.

3.2. System dynamics in presence of wind disturbance

This section formulates dynamics of the system considering force due to wind disturbance. Referring to Figure 1, we first write the general nonlinear form of dynamical equations of the drone as:

where, $$ p_o = \begin{bmatrix} r_o^T & \eta_o^T \end{bmatrix}^T $$ is the position and orientation vector of the drone in the world fame $$ W $$. $$ M_o $$, $$ C_o $$, $$ g_o $$, and $$ u_o $$ are the symmetric positive definite inertia matrix, damping coefficient matrix, gravity force term, and the input to the drone, respectively. Note that $$ \dot{M}_o - 2C_o $$ is skew-symmetric for a certain $$ C_o $$. Now, for a given arbitrarily defined velocity $$ \nu_o $$, we can write

where $$ F_o^w $$ is the force due to wind which we propose to estimate using an adaptive control strategy as presented in the next section. The input $$ u_o $$ provides asymptotic stability in wind disturbance and makes the drone cooperate with the ground robot; it is obtained using:

$$ \hat{F}_w $$ is the estimated force due to wind, $$ T^d $$ is the desired tension required to be maintained in the tether for force-based coordination between the drone and the ground rover and $$ K_o> 0 $$ is the feedback gain matrix. $$ s_o $$ is the velocity error of the drone and is obtained as^[40,41]:

Here, $$ \dot{r}_{or} $$ and $$ \dot{\eta}_{or} $$ are the reference linear and angular velocities of the drone. Similarly, we can write a simple general form of dynamical equations of the ground robot as,

where, $$ p_i = \begin{bmatrix} r_i^T & \eta_i^T \end{bmatrix}^T $$ is the position ($$ r_i $$) and orientation ($$ \eta_i $$) vector of the ground robot in the world fame $$ W $$. $$ M_i $$ is the symmetric positive definite inertia mass matrix of the ground robot. Since this paper does not focus on developing a lower-level controller for the ground robot, it is assumed that $$ F_i^d = F_i $$; i.e., its actual actuating force matches the desired actuating force of the ground robot. In other words, the ground robot actuators and controller perform perfectly in producing the desired force. The velocity error of the ground robot is $$ s_i = \begin{bmatrix} \dot{r}_i^T & \dot{\eta_i}^T \end{bmatrix}^T - \begin{bmatrix} \dot{r}_{ir}^T & (\dot{\eta_{ir}})^T \end{bmatrix}^T $$. Where $$ \dot{r}_{ir} $$ and $$ \dot{\eta}_{ir} $$ are the reference linear and angular velocities of the drone. We use the control law for $$ \dot{p}_{ir} = \begin{bmatrix} \dot{r}_{ir}^T & \dot{\eta}_{ir}^T \end{bmatrix}^T $$ as,

where $$ p_i^d $$ is the desired goal position and orientation of the ground robot. Note that we obtain the desired tension in the tether during wind disturbance as:

where $$ k_o $$ is a proportional constant.

5.1. Fuzzy logic force feedback controller

Based on our earlier work^[11,42], we propose a fuzzy logic-based force feedback controller to control the pitch and yaw angles of the drone based on the tether forces and their rates acting on the drone due to the motion of the ground rover. Initially, before the ground robot starts moving, the drone goes into its equilibrium position with orientation as $$ \eta_o = \begin{bmatrix} 0 & \theta_o^{eq} & 0 \end{bmatrix}^T $$. This orientation maintains an equilibrium tension $$ T^{eq} = \begin{bmatrix} T^{eq}_x & 0 & T^{eq}_z \end{bmatrix}^T $$ in the tether and also makes the tether inclined to the ground at an angle $$ \alpha^{eq} $$ (see Equations 4 to 8). After reaching this equilibrium position, the ground robot starts navigating towards its goal location. The next step consists of computing the desired pitch $$ \theta_o^f $$ and yaw $$ \psi_o^f $$ angles of the drone based on contact forces and their rates thus making the drone coordinate with the rover via force feedback. Consequently, when the rover moves along the $$ X_W $$ direction, the drone pitches accordingly, while when the rover moves along the $$ Y_w $$ axis, it creates a force on the drone along the $$ Y_w $$ axis making the drone yaw with an objective of aligning its body pitch direction ($$ X_B $$) with the current direction of motion of the rover. Finally, the net desired pitch $$ \theta_o^d $$ is calculated and commanded to the attitude controller of the drone.

This section presents the fuzzy-based approach used to find $$ \theta_o^f $$ and $$ \psi_o^f $$. Two fuzzy controllers are implemented for pitching and yawing actions, respectively. For calculating $$ \theta_o^f $$, two inputs to the fuzzy logic force feedback controller are considered: ⅰ) force $$ F^m_x - T^{eq}_x $$, and ⅱ) rate of force change $$ \dot{F}_x^m $$. Since the first input of the fuzzy logic controller for pitch axis $$ F^m_x - T^{eq}_x $$ is proportional to the sensed force along the $$ X_W $$ axis by the drone, we call this controller a force feedback controller. Five membership functions are considered for both the inputs and the output of this controller. The membership functions are negative large (NL), negative medium (NM), Zero (Z), positive medium (PM), and positive large (PL). Table 1 summarizes the fuzzy rule base. The rule base is built using human intuition which can be understood as follows. Let us consider an example when the input $$ F^m_x - T^{eq}_x $$ is PL and another input $$ \dot{F}_x^m $$ is NL. In such a scenario, a PL first input $$ F^m_x - T^{eq}_x $$ means that high force is acting on the drone due to motion of the rover and the drone is supposed to move faster with a large pitch value. However, the second NL input to the fuzzy controller $$ \dot{F}_x^m $$ means that the rate of contact force change is large in the opposite manner; thus, the contact force is expected to increase in magnitude in a negative direction, which means that the drone should pitch negatively. The net effect of the two inputs is that the drone should not pitch at all; thus, the desired pitch angle (output) $$ \theta_o^f $$ should then be Z. Similarly, when $$ F^m_x - T^{eq}_x $$ is PM and $$ \dot{F}_x^m $$ is PL, $$ \theta_o^f $$ is chosen as PL. The drone thus compromises between the applied force and its rate of change. This approach is termed the minimization principle of contact forces, where, in order for the drone to move cooperatively with the ground rover, the drone must minimize the contact forces acting at its end due to motion of the ground rover^[11]. The fuzzy logic controller provides interpretability by providing a linguistic representation where output (control action) is related to inputs (force and rate of change of force) via an intuitive rule base. Figures 2 and 3 show the membership functions and control surface of the fuzzy logic-based force feedback controller along pitch direction while Table 1 shows the fuzzy rules used in simulations. It should be noted that the membership functions were chosen to be Gaussian to maintain continuity and enable gradual transition from one membership function to another during control; this avoids peaks/spikes in output signal.

Figure 2. Simulations: Membership functions for fuzzy logic force feedback controller in pitch direction.

Figure 3. Simulations: Control surface of the fuzzy logic force feedback controller along pitch direction.

The yaw motion controller is similar to the pitch axis controller except that the first input is $$ F_y^m $$. The yaw axis figures are not indicated here for brevity purposes. The outputs of fuzzy-based force feedback controllers along pitch and yaw axes are used to obtain the net desired attitude commands using Equations 11 and 12. The net desired attitude commands are used by the attitude controller of the drone to control the rotor speeds using Proportional-Derivative controllers (see^[39,43] for more details). Figure 4 illustrates the control architecture of the system with fuzzy logic force feedback controller for a tethered drone, and more details are provided.

Figure 4. Fuzzy logic force feedback controller: Control architecture of the entire system.

5.2. Proportional controller for wind disturbance

We now present another controller for the tethered drone for wind disturbance analysis. Once the input $$ u_o $$ is computed, the desired pitch $$ \theta_o^f $$ and yaw $$ \psi_o^f $$ angles based on forces are then calculated using a proportional controller as

and

where $$ k^p_{t, \theta} $$ and $$ k^p_{t, \psi} $$ are the proportional gains. Please note that the symbols used in all sections have been kept the same to maintain consistency. The wind analysis control part of this paper is different from the fuzzy logic control part presenting the active control of the tethered drone without presence of wind. The wind analysis section is just proposed as an additional preliminary work, and in the future, we plan to apply a fuzzy scheme for it.

5.3. Attitude controller of drone

Based on the computed $$ \phi_o^d $$, $$ \theta_o^d $$ and $$ \psi_o^d $$, we compute the deviations from $$ \omega_0^h $$ for different rotors used to navigate the drone along different directions. These deviations $$ \Delta\omega_o^\phi $$, $$ \Delta \omega_o^\theta $$ and $$ \Delta \omega_o^\psi $$ are computed by implementing proportional-derivative (PD) controllers as follows,

where, $$ k^p_{\eta, \phi} $$, $$ k^p_{\eta, \theta} $$ and $$ k^p_{\eta, \psi} $$ are the proportional gains and $$ k^d_{\dot{\eta}, \dot{\phi}} $$, $$ k^d_{\dot{\eta}, \dot{\theta}} $$ and $$ k^d_{\dot{\eta}, \dot{\psi}} $$ are the derivative gains of the PD controller. The desired individual rotor speeds of the drone are then computed based on $$ \Delta\omega_o^\phi $$, $$ \Delta \omega_o^\theta $$ and $$ \Delta \omega_o^\psi $$ using a control allocation matrix^[39].

6.1. Simulations

Fuzzy logic force feedback controller

Simulation setup and parameters - A drone of mass $$ m = 1kg $$ is tethered to a ground rover of mass $$ 3kg $$. The desired tension in the tether is chosen as $$ T^d = T^{eq} = 4N $$, while the desired angle of the tether with the ground is $$ \alpha^d = \alpha^{eq} = 70^{\circ} = 1.22 rad $$. The desired pitch angle at equilibrium position is $$ \theta_o^{eq} = 5.75 ^{\circ} = 0.1 rad $$. The ground robot is required to reach the goal location of ($$ -5m, 5m $$). The tether is chosen to have stiffness $$ K = 20 N/m $$, damping $$ C = 1 Nsec/m $$, and length $$ l = 5m $$. Other parameters used in simulation are: $$ L = 0.12m $$, $$ k_\eta^p = \begin{bmatrix} 100 & 100 & 100 \end{bmatrix}^T $$, $$ k^d_{\dot{\eta}} = \begin{bmatrix} 20 & 20 & 20 \end{bmatrix}^T $$, $$ k^p_g = \begin{bmatrix} 0.4 & 0.4 \end{bmatrix}^T $$, $$ k^d_g = \begin{bmatrix} 1 & 1 \end{bmatrix}^T $$. The fuzzy membership functions were chosen to be Gaussian to maintain continuity and enable gradual transition from one membership function to another during control; this avoids peaks/spikes in the output signal.

Results - Figure 5 shows the tension change in the tether and the angle $$ \alpha $$ made by the tether with the ground. It can be seen that the drone tries to maintain the desired values of tension and angle with the ground. Figure 6 presents the inputs and outputs generated by the fuzzy controller for both pitch and yaw axes. The top row demonstrates the inputs and output for the pitch controller, and the bottom row indicates the inputs and output for the yaw controller. Figure 7 exhibits the position of the ground robot and the drone during the simulation period. It can be seen that both the ground rover and the drone stably reach their specified goal locations starting from their initial points. Please note the slight offset between desired and actual position values along the $$ Z_W $$ direction. This is expected since we do not implement any controller along the vertical $$ Z_W $$ axis of the drone. The rightmost column of Figure 7 shows the net desired pitch and yaw angles ($$ \theta_o^d $$ and $$ \psi_o^d $$) commanded to the attitude controller of the drone. It can be observed that the attitude controller of the drone stably tracks the desired values.

Figure 5. Simulations: Graph showing the angle ($$ \alpha $$) the tether makes with the ground and the tether tension ($$ T $$).

Figure 6. Simulations: Graph showing the inputs and outputs of fuzzy logic force feedback controllers for pitch (top row) and yaw angles (bottom row). Input 1 is $$ F^m_x - T^{eq}_x $$, Input 2 is $$ \dot{F}_x^m $$, Input 3 is $$ F_y^m $$ and Input 4 is $$ \dot{F}_y^m $$.

Figure 7. Simulations: Graph showing the desired and actual position of the ground rover and the drone. The left column shows the ground robot position (actual and desired). The middle column shows the drone position (actual and desired). The rightmost column shows the net desired and actual roll, pitch, and yaw angles.

Proportional controller for wind disturbance analysis

Simulation setup and parameters - For simulation study of the system under wind disturbance, we chose the following system parameters. The mass of a drone is taken as $$ m = 1kg $$, while the ground rover has a mass of $$ 3kg $$. The desired tension in the tether is chosen as $$ T^d = T^{eq} = 4N $$, while the desired angle of the tether with the ground is $$ \alpha^d = \alpha^{eq} = 70^{\circ} = 1.22 rad $$. The desired pitch angle at equilibrium position is $$ \theta_o^{eq} = 5.75 ^{\circ} = 0.1 rad $$. The ground robot is required to reach the goal location of ($$ -5m, 3m $$). The tether is chosen to have stiffness $$ K = 20 N/m $$, damping $$ C = 1 Nsec/m $$, and length $$ l = 5m $$. A constant wind force of 6N along the $$ X_W $$ direction and 1N along the $$ Y_W $$ direction is applied at t=4 sec and persists till the end of the simulation. Other parameters are: $$ L = 0.12m $$, $$ k_\eta^p = \begin{bmatrix} 100 & 100 & 100 \end{bmatrix}^T $$, $$ k^d_{\dot{\eta}} = \begin{bmatrix} 20 & 20 & 20 \end{bmatrix}^T $$, $$ k^p_{t, \theta} = 0.01 $$, $$ k^p_{t, \psi} = 0.01 $$, $$ k^p_g = \begin{bmatrix} 1 & 0.1 \end{bmatrix}^T $$, $$ k^d_g = \begin{bmatrix} 1 & 1 \end{bmatrix}^T $$, $$ K_o = diag\begin{pmatrix} 0.01 & 0.01 & 0 \end{pmatrix} $$, $$ \Gamma = diag\begin{pmatrix} 0.22 & 1 & 0 \end{pmatrix} $$.

Results - The results for this subsection are shown in Figures 8 and 9. It can be seen from Figure 8 that both the ground robot and the drone reach their respective goal locations in presence of wind disturbance. The vertical motion along the $$ Z_W $$ axis is not controlled; hence, slight offset is seen between the desired and actual altitude values. The attitude controller of the drone also tracks its desired orientation well. Figure 9 shows that the adaptive controller computes the estimate of the unknown wind disturbance effectively. Adaptive estimate of wind force is then used to control the tethered drone, as presented in Section 5. Figure 10 further presents the stable flight of the tethered drone in windy environments. The green dashed line indicates the trajectory of the drone, while the red denotes the path of the ground robot. The blue and the red arrows depict the lateral wind forces acting on the drone. It should be noted that the wind disturbance is assumed to be along the lateral $$ X_WY_W $$ plane.

Figure 8. Simulations: Graph showing the position of the drone and the ground robot under wind disturbances. The left column shows the ground robot position (actual and desired). The middle column shows the drone position (actual and desired). The rightmost column shows the net desired and actual roll, pitch, and yaw angles.

Figure 9. Simulations: Graph showing the estimate of the constant unknown wind force/disturbance on the drone using the proposed adaptive control strategy.

Figure 10. Simulations: Tethered drone flying in a simulated environment with wind disturbance.

6.2. Experiments to validate the fuzzy logic controller

For experimental validation of the fuzzy logic controller, TurtleBot 3 Burger^[44] and Crazyflie 2.1^[45] were used as the ground robot and quadrotor drone, respectively. The drone was tethered to the turtle bot by a tether. Figure 11 shows the system of the Crazyflie tethered to the TurtleBot. We use Raspberry Pi (Rpi) as an onboard computer on the TurtleBot which sends commands to the Crazyflie via a Radio attached to the Rpi. Two inexpensive noisy load cells (attached to the ground rover) are employed to measure forces in the $$ X_WY_W $$ plane. A/D amplifying converters are also applied along with the load cells. Notably, since tension in tether is equal and opposite at the two ends, i.e., the drone and the ground rover, this allows us to attach the force sensor to the ground rover instead of the drone. It may be noted that this requires properly taking care of the directions of the forces when controlling the drone based on them. Further, the onboard computer, Rpi, gathers load cell data, applies a Kalman filter for the data, and then implements the proposed fuzzy logic force feedback controller. The fuzzy controller then generates the desired pitch and yaw angles and then uses them to calculate the net desired pitch and yaw angles. The net desired pitch and yaw angles are then commanded to the attitude controller of the Crazyflie. It should be noted that the Crazyflie is made to fly at constant thrust. The parameters used in experiments for the Crazyflie platform are $$ thrust = 40000 $$ and $$ \theta_o^{eq} = 4 ^{\circ} = 0.07 rad $$.

Figure 11. Experiments: Schematic diagram of the system used in experiments.

Figures 12 and 13 demonstrate the membership functions and control surface of the fuzzy logic force feedback controller used in experiments. The fuzzy rules are similar to those used for simulations and, thus, are not indicated in this subsection. Two cases are evaluated in the experiments: ⅰ) Case 1: Straight line motion and ⅱ) Case 2: Circular trajectory tracking. The results along the $$ X_W $$ or pitch axis for Case 1 are shown in Figure 14. It can be seen that the fuzzy logic controller effectively generates the pitch output based on forces and their rates ($$ F^m_x - T^{eq}_x $$ and $$ \dot{F}_x^m $$). Moreover, the attitude tracking controller of the Crazyflie tracks the desired pitch angle well. The black dashed line marked on the bottom plot of Figure 14 shows the equilibrium pitch angle $$ \theta_o^{eq} $$ for Case 1 about which the net desired pitch angle $$ \theta_o^d $$ varies. Figure 15 presents the results for yaw control for Case 1 based on forces and their rates ($$ F^m_y $$ and $$ \dot{F}_y^m $$). The yaw rate controller of the Crazyflie shows successful tracking of the desired yaw rate. It should be noted that, given hardware constraints, we decided to command the yaw rate to the Crazyflie instead of the desired yaw angle based on force feedback. It can be seen from these results that the drone can be actively manipulated by the ground rover as if the ground robot is flying the drone akin to a kite. Figures 16 and 17 exhibit similar results for Case 2 when the ground robot is made to move along a circular path performing the trajectory tracking. These results show that irrespective of whether the ground robot moves in a straight line or tracks a trajectory, the drone can coordinate with it via forces and their rates.

Figure 12. Experiments: Membership functions for fuzzy logic force feedback controller in pitch direction.

Figure 13. Experiments: Control surface of the fuzzy logic force feedback controller along pitch direction.

Figure 14. Experiments (Case 1: Straight line motion): Graph showing the inputs - forces and their rates ($$ F^m_x - T^{eq}_x $$ and $$ \dot{F}_x^m $$) -acting on the drone along the pitch direction. The bottom plot shows the desired net pitch angle $$ \theta_o^d $$ and the actual pitch angle $$ \theta_o $$. The black dashed line on the bottom plot shows marks the desired equilibrium pitch angle $$ \theta_o^{eq} $$.

Figure 15. Experiments (Case 1: Straight line motion): Graph showing the inputs - forces and their rates ($$ F^m_y $$ and $$ \dot{F}_y^m $$) -acting on the drone for yaw control. The bottom plot shows the desired yaw rate $$ \dot{\psi}_o^d $$ and the actual yaw rate $$ \dot{\psi}_o $$.

Figure 16. Experiments (Case 2: Circular trajectory tracking): Graph showing the inputs - forces and their rates ($$ F^m_x - T^{eq}_x $$ and $$ \dot{F}_x^m $$) -acting on the drone along the pitch direction. The bottom plot shows the desired net pitch angle $$ \theta_o^d $$ and the actual pitch angle $$ \theta_o $$. The black dashed line on the bottom plot shows marks the desired equilibrium pitch angle $$ \theta_o^{eq} $$.

Figure 17. Experiments (Case 2: Circular trajectory tracking): Graph showing the inputs - forces and their rates ($$ F^m_y $$ and $$ \dot{F}_y^m $$) - acting on the drone for yaw control. The bottom plot shows the desired yaw rate $$ \dot{\psi}_o^d $$ and the actual yaw rate $$ \dot{\psi}_o $$.

The results in this section validate the performance of the proposed approach for an actively manipulated tethered drone.

Acknowledgments

We would like to acknowledge the University of Cincinnati's Space Research Institute for Discovery and Exploration (SRIDE) for supporting this research in the form of SRIDE Fellowship to the first author of this paper for the years 2022-23.

Authors' contributions

Contributed to the proposed methodology, simulation and experimental validation, and draft writing: Barawkar S

Provided regular feedback on all aspects of research, and assisted in writing the manuscript (The second author is the faculty advisor who investigated and supervised the first author): Kumar M

Availability of data and materials

Not applicable.

Financial support and sponsorship

The University of Cincinnati's Space Research Institute for Discovery and Exploration (SRIDE) funding support was utilized for this research in the form of SRIDE Fellowship to the first author of this paper for the years 2022-23.

Conflicts of interest

All authors declared that there are no conflicts of interest.

Ethical approval and consent to participate

Not applicable.

Consent for publication

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Copyright

© The Author(s) 2024.