
This chapter presents the development of a robotic tail (AeroTail) device design capable of self-righting and balancing on a point-in-plane aerodynamically; it introduces simplistic mathematical models based on the inverted pendulum (IP) and control methods suitable for balancing the proposed AeroTail; and it includes computational simulations and comparison studies to quantify and visualise the advantages of the AeroTail mechanism over traditional reaction-wheel-based (RW) tails under various circumstances.


\section{Mathematical Modelling} \label{sec_theory}

To design and simulate a robot-balancing tail device, a dynamic model of the device and a simplified robot must be established. This model should be general enough to represent most scenarios where the tail is expected to be used. As mentioned in \secref{sec_intro}, studies regarding reaction wheels often build their dynamic models based on the inverted pendulum (IP); its structure can be described as a massed object whose gravity does not align with its contact point with the surroundings. This means that most devices, especially legged robots, which require active balancing, can be well represented by the IP model. The example robot mentioned in \cite{featherstone2017simple} utilises a dynamic model for balancing on a point-in-plane, which simplifies the IP balancing problem to a degree that is suitable for simulation but still representative of wider scenarios. Therefore, the dynamic model for this work will primarily be based on the point-in-plane concept for IP, and further mechanisms will be developed on top of it.

\subsection{The inverted pendulum model}

\begin{figure}
    \centering
    \includegraphics[width=0.75\columnwidth]{pics/pendulum.jpg}
    \caption{The double-rod pendulum system. $\theta_{1}$ is the angle between the vertical hanging position and the leg, while $\theta_{2}$ is that between the leg and the tail in the same direction. The small picture in the top-right corner is an example jumping robot with equivalent components marked in the same colour as in the model.}
    \label{math_model}
\end{figure}

The structure and notation of the double-rod pendulum system are shown in \figref{math_model}. The model consists of three main components: the ``leg'' (blue), the ``body'' (green), and the ``tail'' (orange). The leg and the body are fixed together; they both rotate around Joint 1 which connects a fixed point in the world frame with the leg. Joint 2 connects the mid-point of the tail to the body. Both the leg and the tail are uniform and rigid rods, and the body is a point mass. The masses of the leg, tail and body are $m_1$, $m_2$ and $m_3$ respectively. The leg has a length of $2l_1$ while that of the tail is $2l_2$. In the top right corner of \figref{math_model} is a simple illustration of a monopedal jumping robot. The equivalent components in the double-rod model and the example robot are marked with the same colour. Compared to a realistic robot, the body in the model is largely simplified; Joint 1 can be seen as the contact point between the robot and the ground.

The equation of motion for the system can be simply expressed using the robot dynamics equation:
\begin{equation} \label{robot_dy}
\mathbf{M}(\boldsymbol{\theta})\boldsymbol{\Ddot{\theta}}+\mathbf{C}(\boldsymbol{\theta},\boldsymbol{\dot{\theta}})\boldsymbol{\dot{\theta}}=\boldsymbol{\tau_{g}}(\boldsymbol{\theta})+\mathbf{u}+\mathbf{J^T}(\boldsymbol{\theta})\boldsymbol{\lambda}
\end{equation}
in which $\theta$, $\dot{\theta}$, and $\Ddot{\theta}$ represent the positions, velocities, and accelerations of the pendulums as shown in Fig.\ref{math_model}, the $\mathbf{M}(\boldsymbol{\theta})$ is the mass matrix, $\mathbf{C}(\boldsymbol{\theta},\boldsymbol{\dot{\theta}})\boldsymbol{\dot{\theta}}$ is the Coriolis forces, $\boldsymbol{\tau_{g}}(\boldsymbol{\theta})$ is the gravity vectors, $\mathbf{u}$ represents the control input and 
$\mathbf{J^T}(\boldsymbol{\theta})\boldsymbol{\lambda}$ denotes the sum of external forces. The first three components can be derived using Lagrangian mechanics: 
\begin{equation} \label{lagrange}
    \frac{d}{dt}\frac{\partial T(\boldsymbol{\dot{\theta}})}{\partial\boldsymbol{\dot{\theta}_{i}}}-\frac{\partial U(\boldsymbol{\theta})}{\partial\boldsymbol{\theta_{i}}}=\boldsymbol{\tau_{i}}.
\end{equation}
where $T(\boldsymbol{\dot{\theta}})$ and $U(\boldsymbol{\theta})$ are the kinetic and potential energy of the system, $\boldsymbol{\tau_i}$ is the sum of external forces.

The total kinetic energy of the system is:
\begin{equation} \label{equ_kinetic}
    \begin{split}
        T(\boldsymbol{\dot{\theta}})= & \frac{1}{2}I_1\dot{\theta}_1^2+\frac{1}{2}I_2(\dot{\theta}_2+\dot{\theta}_1)^2+\frac{1}{2}(m_2+m_3)v_2^2 \\
         = & \frac{2}{3}m_1l_1^2\dot{\theta}_1^2+\frac{1}{6}m_2l_2^2(\dot{\theta}_2+\dot{\theta}_1)^2 \\
           & +2(m_2+m_3)l_1^2\dot{\theta}_1^2.
    \end{split}
\end{equation}
where $I_1$, $I_2$, and $I_3$ are the moments of inertia for the leg, the tail, and the motor. 

The total potential energy of the system is:
\begin{equation} \label{equ_potent}
    \begin{split}
        U(\boldsymbol{\theta})= & m_1gy_1+m_2gy_2 \\
         = & -m_1gl_1\cos{\theta_1}-2(m_2+m_3)gl_1\cos{\theta_1}.
    \end{split}
\end{equation}
where $y_1$ and $y_2$ are the displacements of the motor and the centre of mass of the leg from Joint 1 on the $y$-axis. 

Substitute \eqref{equ_kinetic} and \eqref{equ_potent} into the Lagrangian model \eqref{lagrange}, and the external forces can be expressed as a function of the pendulum state. Consequently, the mass matrix in \eqref{robot_dy} can be derived as:
\begin{equation}
    \mathbf{M}(\boldsymbol{\theta}) = 
    \begin{bmatrix}
    \frac{4}{3}m_1l_2^2+4(m_2+m_3)l_1^2 & \frac{1}{3}m_2l_2^2 \\
    \frac{1}{3}m_2l_2^2 & \frac{1}{3}m_2l_2^2
    \end{bmatrix}.
\end{equation}
Note that the mass matrix is independent to position ($\theta$). There are no Coriolis forces involved in the model, thus:
\begin{equation}
   \mathbf{C}(\boldsymbol{\theta},\boldsymbol{\dot{\theta}}) = 
    \begin{bmatrix}
        0 & 0
    \end{bmatrix}^T
\end{equation}
and the gravity vectors can be expressed as:
\begin{equation}
    \boldsymbol{\tau_{g}}(\boldsymbol{\theta}) = 
    \begin{bmatrix}
        gl_1\sin{\theta_1}(m_1+2m_2+2m_3) & 0
    \end{bmatrix}^T
\end{equation}

\subsection{The RW tail version of the model}

When the tail is acting as an RW tail, it has mass, and there are no external forces involved. Since only joint 2 will be actuated, the control input $u$ in \eqref{robot_dy} can be written as:
\begin{equation}
    \mathbf{u} = 
    \begin{bmatrix}
        0 & \tau_{\text{motor}}
    \end{bmatrix}^T
\end{equation}
and the sum of external forces is:
\begin{equation}
    \mathbf{J^T}(\boldsymbol{\theta})\boldsymbol{\lambda} = 
    \begin{bmatrix}
        0 & 0
    \end{bmatrix}^T
\end{equation}

\subsection{The AeroTail version of the model} \label{aero_drag_model}

When the tail is behaving as an AeroTail, there are several changes to the model compared to that of the RW tail scenario. First of all, the tail is assumed to be mass-less in order to eliminate the inertial torque from the mass of the tail; thus, the new expressions for the mass matrix and the gravity vectors are:
\begin{equation}
    \begin{split}
        \mathbf{M}(\boldsymbol{\theta}) = &
    \begin{bmatrix}
    \frac{4}{3}m_1l_2^2+4m_3l_1^2 & 0 \\
    0 & 0
    \end{bmatrix}, \\
    \boldsymbol{\tau_{g}}(\boldsymbol{\theta}) = &
    \begin{bmatrix}
        gl_1\sin{\theta_1}(m_1+2m_3) & 0
    \end{bmatrix}^T.
    \end{split}
\end{equation}
The sum of aerodynamic forces generated by the tail can be seen as a concentrated external torque acting on its centre of mass. Since the tail is theoretically the ``end effector" in the dynamic model, the matrix of the aerodynamic torque ($\tau_{aero}$)is expressed as such:
\begin{equation}
    \mathbf{J^T}(\boldsymbol{\theta})\boldsymbol{\lambda} = 
    \begin{bmatrix}
        -\tau_{\text{aero}} & -\tau_{\text{aero}}
    \end{bmatrix}^T.
\end{equation}
In this scenario, the aerodynamic torque is always equal to the motor torque, and the tail is always rotating at a constant velocity with no acceleration process when the velocity changes from one to another.

Generally, the aerodynamic drag can be calculated using the equation:
\[D=\frac{1}{2}\rho C_{D}v^2A\]
where $D$ is the drag force, $\rho$ is the air density, $ C_{D}$ is the drag coefficient, $v$ is the air speed and $A$ is the frontal area. 
If the aerodynamic drag acts as a torque on a single point, the torque can be calculated using the method provided in \cite{norby2021enabling} with the equation: 
\begin{equation}\label{aero_drag_torque_1}
    \begin{split}
        |\tau_{D}|&=\int_{A} Dl \,dA \\
        &=\int_{L_{1}}^{L_{0}} \frac{1}{2}\rho C_{D}a(\omega_{t} l_{D})^2l \,dl 
    \end{split}
\end{equation}
where $l_{D}$ is the length of the lever on which the drag force exerts, $a$ is the width of the drag inducer, $\omega_{t}$ is the angular velocity of the tail, and $L_{0}-L_{1}$ is the length of the lever that can generate drag assuming the drag-inducing device has a uniform cross-section along its entire length. 

To adapt this equation to the double-rod system, it needs to be modified and simplified. The aerodynamic torques generated by the tail originate from the drag inducers on the tail's ends; thus, the drag generated by a single inducer is considered to act on the endpoint of the tail, regardless of its own geometry, in order to simplify the model. Consequently, Equation \eqref{aero_drag_torque_1} can be rewritten as:
\begin{equation} \label{aero_drag_torque_2}
    \tau_{D}=\frac{1}{2}\rho C_{D}\omega_{t}^2l_{D}^3A 
\end{equation}
where $A$ is the area of one drag inducer. Since there are two of them on each end of the tail, the actual aerodynamic drag generated by the tail is $\tau_{\text{aero}}=2\tau_{D}$.


\subsection{Control method}

\begin{figure}
    \centering
    \includegraphics[width=0.75\columnwidth]{pics/control_diagram.png}
    \vspace{-3mm}
    \caption{The control block diagram of the balancing model. In the diagram, $\alpha$ means acceleration, $\omega$ means velocity and $\theta$ represents angle position. Designation ``1" is that of the body, ``2" is for the tail and ``d" means the desired target input.}
    \label{control_diagram}
\end{figure}

The control method employed for the model is a classic Proportional-Derivative (PD) controller, commonly used for balancing inverted pendulums. \figref{control_diagram} depicts the control block diagram of the model, in which the controller gains are represented by $K_{p}$ (proportional gain) and $K_{d}$ (derivative gain). his controller facilitates the balancing of the robot at a desired angle ($\theta_{d}$) and target velocity ($\omega_{d}$). Given that the proportional term contains the position angle error of the robot body, the derivative term error can be easily derived from the feedback velocity. Consequently, the expression for the summed acceleration input can be formulated as:
\begin{equation}
        \alpha_{in}=K_{p}(\theta_{d}-\theta_{b})+K_{d}(\omega_{d}-\omega_{b}),
\end{equation}
where $\theta_{b}$ and $\omega_{b}$ represent the real-time position angle and velocity of the body, respectively. The integral gain is not required, as there is no noise in the simulation environment. The controller also accounts for gravity and mass, allowing the input to be acceleration rather than torque. As a result, comparative studies in the simulation phase with fixed gains are possible, and altering the mass of components will not affect the robot's result trajectory, provided the gains remain constant.

\section{Simulation studies}

The simulation of the proposed theoretical model is performed using MATLAB. The simulation codes involve iterative calculations employing Euler's method.

\subsection{Tails stand-alone tests} \label{sec_tailstandalone}

An RW tail will be compared to an AeroTail to examine their behaviours under the same conditions. The controlled variables are as follows:
\begin{itemize}
\item Identical motors must be used;
\item The lengths of the tails must be the same.
\end{itemize}

\begin{figure}[t]
    \centering
    \includegraphics[width=0.75\columnwidth]{pics/stand_alone_models.jpg}
    \caption{The tail models for the stand-alone tests. The motors are fixed to the world frame; the RW tail (orange) and the AeroTail (blue) have the same length. The grey arrows point out the torques acting on the tails. In the top-right box is one of the drag-inducers which generates aerodynamic torque when the tail is rotating; it can be seen as a semi-circle prism.}
    \label{only_tail_models}
\end{figure}

\begin{table}
\centering
\caption{Parameters of the tail stand-alone tests}
\vspace{2mm}
\begin{tabular}{lcccl}
\toprule
\vspace{1mm}
Definition & Designation & Parameter\\
\hline 
RW tail mass \rule{0pt}{2.6ex} & $m_I$ & $10$ g  \\ 
%\hline 
AeroTail mass & $m_D$ & varies \\
%\hline
Half-tail lengths & $l$ & $5$ cm \\
%\hline
Rotor mass & $m_R$ & $5$ g \\
%\hline
Motor-torque & $\tau_{\text{motor}}$ & $0.01$ Nm \\
%\hline
Aero-torque & $\tau_{\text{aero}}$ & varies \\
%\hline
Drag coefficient & $C_{D}$ & 2.0 \\
\bottomrule
\end{tabular}
\label{tab_2}
% \vspace{-2mm}
\end{table}

The term ``stand-alone'' in this section refers to both mechanisms operating without the body. \figref{only_tail_models} presents the theoretical models for the tail stand-alone tests. The black circles in the middle of the rods symbolise the motor rotors (the rotor is assumed to be a uniform disc with a diameter of $2$ cm). The effective area of a drag-inducer is equal to the width of the prism multiplied by the diameter of the semi-circle. Parameters for the simulations are detailed in \tabref{tab_2}.



\subsubsection{The behaviour of an AeroTail with mass}

\begin{figure}[t]
    \centering
    \includegraphics[width=0.75\columnwidth]{pics/stand_alone_aero_mass.jpg}
    \caption{The behaviour of the aero-balanced tail with mass. (a) The velocity of the tail versus time. (b) The acceleration of the tail converges to zero.}
    \label{aero_mass}
\end{figure}

The first simulation aims to demonstrate the behaviour of a reaction wheel capable of generating significant aerodynamic drag from its rotation when driven by a constant torque from the motor. In this case, $m_{D}$ is $10$ grams and torque input is $\tau_{\text{motor}}$. The additional $\tau_{\text{aero}}$ is calculated using the aerodynamic model in \secref{aero_drag_model}, assuming the drag acts on the ends of the tail. The drag coefficient is $2.0$, that of the semi-circle prism \cite{norby2021enabling}, and the contact area is set to be $5$ cm$^2$ on each end. The simulation results are shown in \figref{aero_mass}. The acceleration of the tail eventually ceases when it reaches the maximum velocity because the torque provided by the motor equals the torque generated by drag. Unlike pure reaction wheels, which need to keep accelerating to generate torque by inertia, AeroTail can fulfil the same function with a constant velocity.

\subsubsection{Comparison between AeroTail and RW tail}\label{A_2}

\begin{figure}[t]
    \centering
    \includegraphics[width=0.75\columnwidth]{pics/stand_alone_comparison.jpg}
    \caption{The comparison between aero-balanced and reaction-wheel tails. (a) The torque provided by the motor. (b) The aerodynamic torque generated by the AeroTail. (c) \& (d) The velocities of the tails versus time. (e) \& (f) The angles of the tails versus time.}
    \label{aero_rw_comp_s_a}
\end{figure}

To compare the mechanisms, the correcting torque generated by the AeroTail is assumed to be predominantly aerodynamic, thus $m_{D}$ is zero and the tail provides no inertial torque; simultaneously, the RW tail has no aerodynamic property in this section. \figref{aero_rw_comp_s_a} illustrates the comparison between the AeroTail (right) and RW tail (left), with all other conditions remaining the same. Because the AeroTail is mass-less and the inertia of the rotor is negligible, its angular velocity nearly instantaneously reaches its maximum. The net energy consumption is also calculated as the total energy output from the motor. During the test, the AeroTail used $4.53$ J, while the RW tail consumed $11.99$ J.

\subsection{Scenario tests} \label{sec_scenario_test}

For scenario tests, the tails will be used to balance a robot body based on the model in \secref{sec_theory}. This implies that, unlike the stand-alone tests, the motor output during balancing will be controller-driven using the control method mentioned in the same section.

\subsubsection{Example test}
An example test is conducted to demonstrate the characteristics of the tails' performances. The robot (using the previous dynamic model and control method in \secref{sec_theory}) needs to be balanced to an upright position from a $90{^\circ}$ tilt within $3$ seconds; the mass of the body is 20 grams, the RW tail is $10$ grams, and the AeroTail is mass-less. The gains for the PD control are also determined in the example test. An additional requirement stipulates that there should only be one positive and one negative overshoot for the body position before convergence, with an error allowance of $1{^\circ}$ plus or minus. The results of the example test are shown in \figref{fig_example_test}. The tail's peak velocity is $355$ rad/s with a net energy consumption of $4.05$ J. The body positions of both tails follow identical trajectories as expected, and the RW tail experiences significantly more rotations than its counterpart. The tail velocity in (d) appears to be as smooth as other graphs because it is solely dependent on the motor torque and there is no acceleration process when the motor torque changes; it also converges to zero after the motor torque does with some delay.

\begin{figure}[t]
    \centering
    \includegraphics[width=0.75\columnwidth]{pics/Example_test.jpg}
    \caption{Results from the example tests (RW tail on the left and AeroTail on the right): (a) \& (d) The velocity of the tail reaches a constant as the system converges. (b) \& (e) The position of the tail versus time. (c) \& (f) The body angle reaches the target up-right position ($\pi$).}
    \label{fig_example_test}
\end{figure}

Following the example tests, specific metric comparisons with controlled variables will be conducted. Four tests in total will be performed, and the controller gains will remain constant throughout all of them.

\subsubsection{Test 1}
The first test compares the peak motor output torque of both mechanisms with different robot body mass ($m_3$). \figref{fig_test results} (a) displays the change in peak motor torques as the robot body mass increases from $20$ g to $70$ g. The peak torques of both the RW tail and AeroTail rise linearly, with the RW tail's graph generally higher and exhibiting a steeper slope than the AeroTail's. To maintain constant peak velocity, the RW mass must increase with the body mass; thus, as the body mass increases, the peak torque of the RW tail grows faster than that of the AeroTail. On average, AeroTail can achieve a $33.2\%$ reduction of peak torque input. This test demonstrates that the AeroTail offers an advantage in terms of motor torque output requirements and the potential to reduce the robot's total mass.

\subsubsection{Test 2}
The second test compares the peak motor velocities of both tails with varying correction angles (angles required for righting from the initial to the target position). As shown in \figref{fig_test results} (b), the AeroTail's peak velocity remains relatively stable within the $200$ to $400$ rad/s range as the correction angle increases, whereas the RW tail's peak velocity triples. The AeroTail's ability to reach an ultimate velocity when exerting torque causes its peak velocity to fluctuate less when the correction angles differ significantly. This test indicates that the AeroTail has a lower requirement for peak motor velocity, consistently lower than that of the RW tail; particularly at a correction angle of $120{^\circ}$, for which the AeroTail manage to achieve a peak velocity reduction of $72.8\%$.

\subsubsection{Test 3}
The third test compares the energy consumption of both mechanisms with varying tail masses (for the RW) as a percentage of the robot's body mass. The AeroTail's graph in \figref{fig_test results} (c) is included for comparison purposes only, as it does not have mass. The AeroTail's energy consumption remains constant at $7.42$ J under initial conditions (identical to the example tests), while the RW tail's energy consumption dramatically decreases as the tail mass increases from $20\%$ to $80\%$ of the body, intersecting the AeroTail's graph at $60\%$. This test reveals that the RW tail can outperform the AeroTail in some metrics even with the mass disadvantage. However, to match the AeroTail's performance, the RW tail mass must constitute $60\%$ of the body, which is impractical if the robot needs to perform functions other than balancing.

\subsubsection{Test 4}
The fourth test compares the energy consumption of both tails with different correction angles. In \figref{fig_test results} (d), the AeroTail's graph is closer to that of the RW tail, with a less pronounced curvature, resembling the results of Test~2. This indicates that the RW tail's energy consumption is more susceptible to changes in correction angles, leading to the shapes of the graphs: the outcomes for the tails are similar at very small and very large correction angles but increasingly divergent in the middle range. As with Test 2, the difference in energy consumption of the tails is greatest at a correction angle of $120{^\circ}$. For most angles, the AeroTail consumes less energy to balance, rendering it superior in this metric.

\begin{figure}[t]
    \centering
    \includegraphics[width=0.7\columnwidth]{pics/Test_results .jpg}
    \caption{The results from the 4 simulation studies: (a) Peak motor output torque versus robot body mass of RW tail and AeroTail (purple); (b) Peak velocity versus correction angle of both tails. (c) Energy consumption versus tail masses of the tails as percentages of the robot. (d) Energy consumption versus correction angles of the tails.}
    \label{fig_test results}
\end{figure}


\subsection{AeroTail length sensitivity study}

We have established in previous sections (Equation \eqref{aero_drag_torque_2}) that the aerodynamic drag generated is a cubic function of the half length of the AeroTail assuming the drag concentrates on its ends. This means the torque generating capability and balancing performance of AeroTail is highly sensitive to its length. This study examine the effects of scaling the AeroTail's length on the required peak motor velocity and energy consumption. The simulation setup copies that of the example test in the previous section, namely, balancing the inverted pendulum from a horizontal to a upright position.
In this case, the controlled variables are initial body angle, mass, convergence criteria, and controller gains. Therefore, the torque pattern and convergence time scale are constant throughout the study and the performance is only judged according to required peak motor velocity and energy consumption. The tail half length (\(l_{2}\)) is scaled from $40\%$ to $200\%$ of the original ($5$cm) with an interval of $20\%$, and the results can be found in \figref{l_sensitivity}. As we can see, both peak velocity and energy consumed drop drastically as the tail length increases. To visualise the trends of result data points, best fitting curves (cubic) are added to the figures using the POLYFIT function in MATLAB. It can be deducted from the results that using longer AeroTails dramatically reduces both the peak required motor velocity and energy consumed to achieve the same task. Therefore, maximising the length of the AeroTail with constraints applied should be used as a general strategy for mechanical design in the next chapter.


\begin{figure}
    \centering
    \includegraphics[width=0.99\linewidth]{pics/l_sensitivity.jpg}
    \caption{AeroTail length sensitivity study: (a) Peak motor velocity required, and (b) Energy consumption versus tail length scaling.}
    \label{l_sensitivity}
\end{figure}



\section{Chapter summary}

This chapter introduced the AeroTail, a robotic tail device design capable of self-righting and balancing on a point-in-plane aerodynamically. The tail mechanism's dynamic model is based on the classic IP with additional aerodynamic features, and a suitable control method for such a device was chosen accordingly. Computational simulations using MATLAB visualised and quantified the behaviour of the tails with both aerodynamic and inertial effects under various conditions and scenarios. Comparative studies with controlled variables were conducted to evaluate their performances, illustrating their respective advantages and limitations. The most significant advantage of the AeroTail over the RW tail is its lack of mass, allowing it to maintain a low increase in peak motor torque output as the robot body mass increases. Conversely, the RW tail experiences dramatically high energy consumption with a low tail-to-body mass ratio, while the AeroTail does not require mass to function. In terms of both peak motor velocity and energy consumption, the AeroTail outperforms the RW tail when the correction angle is between the horizontal and $140{^\circ}$, with the differences in results peaking at $120{^\circ}$. However, the RW tail has a relative advantage at very small correction angles or near the vertically hanging position. 

This chapter only introduced the theoretical concept and initial simulation of the AeroTail balancer; more details on the mechanical design of the AeroTail, especially of the drag-inducing devices on the ends of the tail body, are yet to be decided. Furthermore, experimental validation is needed to prove that this concept is indeed functional in a realistic scenario. Therefore, in the next chapter, a thorough optimisation study for the drag-inducers is introduced, and experiments are conducted to test the real-world performance of the AeroTail.

%Future work will involve the design and construction of a physical tail device to conduct similar experiments and validate the points made in this paper. This device will utilize the same theoretical model and parameters used in the simulations. Results from physical experiments will better represent the tails' performance in realistic scenarios, and noise can be easily added to simulate the external turbulence encountered during stabilization.
