



This chapter introduced a theoretical continuously variable transmission gearbox that can be incorporated into series-elastic actuators for robotic jumping, a mathematical models for the hopping robot powertrain, including the CVT gearbox and a simplified model for an electric motor powered prismatic actuator suitable for iterative simulation, and a comparative simulation studies to examine the performances of both the C-SEA and regular SEA in various scenarios. The advantages and limitations of the mechanism are analysed. Following that is the mechanical design section for jumper prototypes utilising this novel actuator.


\begin{figure*}
    \centering
    \includegraphics[width=\linewidth]{pics/hopper_powertrain.jpg}
    \caption{A simplified structure of a hopping robot with C-SEA (a) and the isolated powertrains. Scenario (b) is the powertrain with a CVT and (c) is for a conventional SEA.}
    \label{hopperpowertrain}
\end{figure*}

The structure of the chapter is as follows. In Section.~\ref{sec_meth}, how to approach the analysis of a CVT-enhanced SEA is discussed; the structure of the hopping robot's powertrain is defined and the design criteria are specified. Section~\ref{sec_mathmodel} introduces the mathematical models of the powertrain, the CVT gearbox, and the electric motor powered prismatic actuator; the constants that are later used in the simulation are also calculated from the design criteria. In Section.~\ref{sec_sim_cfd}, two simulations are done: simulation 1 examines the general behaviours of the mechanisms when being powered by an ideal actuator; simulation 2 uses the full motor model to evaluate the effects on the height performance by specific variables. And finally, in \secref{sec_proto} are the mechanical design for CVT and jumper prototypes.

\section{Methodology for designing the C-SEA} \label{sec_meth}

To simulate a CVT-enhanced SEA robotic jumping, the advantages of having a mechanism between the elastic element and the end-effector with variable MA need to be analysed and understood. First of all, by aligning the MA profile and the spring's linear behaviour, the end-effector can produce almost constant force output, which reduces the peak ground reaction force. At the same time, having a high MA at the beginning of a stroke allows the spring to be fully compressed, thus it stores maximum energy to be released during the remainder of the stroke \cite{plecnik2017design}. The first advantage improves structural integrity and reduces the material requirements for the leg mechanism, and the second one enables the robot to achieve higher jumps. This varying MA ``gearbox" can be in the form of a CVT gearbox that can produce a precise MA profile that exactly matches the behaviour of the spring. 


Generally in simulation, the structure of the device needs to be simplified enough so it can be mathematically defined. As shown in Fig.~\ref{hopperpowertrain} (a), the hopper consists of a body and a leg mechanism for locomotion, with the powertrain contained inside the body. In a realistic scenario, the powertrain only provides angular motion; then, it is converted into linear motion by a rack-and-pinion mechanism. However, to avoid any complex dynamic model, the powertrain of the jumper is isolated and linearised. In Fig.~\ref{hopperpowertrain} (b) and (c), the mass M represents the robot body and $M_{2}$ is the mass of moving components in a motor; the corresponding behaviours of each section of the powertrain are denoted by the letters on the side. In the linearised model, the motion of the robot body relative to the ground is simply represented by that of the mass M with respect to the surroundings. Since the primary objective of utilising the C-SEA is to maximise jump height, there should be a target height for the robot to reach, a fixed body mass and a maximum stroke length for the end-effector. In this case, the design criteria for the robot with a body mass of $100$ g is to jump to $1$ m with a stroke length ($x_{max}$) of $0.2$ m. The parameters are averages from past light-weight jumping robots like Salto \cite{plecnik2017design}, EPFL jumper \cite{kovavc2011epfl} and MSU jumper \cite{zhao2013msu}; the numbers are adjusted to be mathematically convenient. The variable $v_{max}$ is defined as the velocity of the robot body at the lift-off (the end of a stroke where the end-effector is no longer touching the ground) which directly determines the height the jump can reach. In this model, only the one dimensional vertical motion of the robot is concerned. However, balancing mechanisms such as AeroTail \cite{sun2023aerotail} can be easily added to allow locomotion due to its structural simplicity. 

\section{Mathematical modelling}\label{sec_mathmodel}

Following the assumptions made in Section.\ref{sec_meth}, mathematical models for the powertrain and the CVT gearbox can be derived. Some initial conditions that will be useful in the simulation are also provided by the models. Additionally, a model for a realistic electric motor is provided in this section.

\subsection{Powertrain and CVT models}

The characteristics of CVTs compared to regular gearboxes with variable gear ratios is that a CVT can adjust gear ratio in a smooth range instead of fixed steps. Therefore, how the gear ratio changes is very flexible and open to imagination. For the simplicity of modelling, the CVT gearbox has linear inputs and outputs in the forms of displacement and force instead of angles and torques. Still using the powertrain structure in Fig.\ref{hopperpowertrain} (b) and assuming the spring used has perfectly linear behaviour, the input force ($F_{in}$) can be calculated using:
\begin{equation}\label{eq_fin}
    F_{in} = k(x_{u} - x_{in})
\end{equation}
where k is the spring constant, $x_{u}$ is the displacement of the actuator and $x_{in}$ is the input displacement into the gearbox. The gear ratio ($Gr(x_{in})$) of the CVT is set to be a linear function of the input displacement: 
\begin{equation}\label{eq_gr}
        Gr(x_{in}) =  G_{0} - R \cdot x_{in} 
\end{equation}
where $G_{0}$ and $R$ are constants according to the properties of the CVT gearbox. In this case, the mechanical advantage ($M_{A}$) of the gearbox is simply the inverse of the gear ratio:
\begin{equation}\label{eq_ma}
    M_{A} = \frac{1}{Gr(x_{in})} = \frac{1}{G_{0} - R \cdot x_{in}}.
\end{equation}
Therefore, the output displacement ($x_{out}$) and the output force ($F_{out}$) from the gearbox can also be calculated: 
\begin{equation}\label{eq_xout}
    \begin{split}
        x_{out} = & \int Gr(x_{in})\,dx_{in} =  G_{0}x_{in} - \frac{1}{2}Rx_{in}^{2} \\
        F_{out} = & F_{in}\cdot M_{A} = \frac{k(x_{u} - x_{in})}{G_{0} - R \cdot x_{in}}
    \end{split}
\end{equation}
and the dynamic behaviour of the powertrain can be simulated using Euler method in later sections.

\subsection{Constants calculation}\label{constants}

Using the design criteria and assumptions made in Section.\ref{sec_meth}, constants from equations \eqref{eq_fin} to \eqref{eq_xout} can be calculated and used as known values later on in the simulations. Given the stroke length $x_{max}$, target height $h_{t}$ and mass of the robot body $m_{r}$, the ideal spring constant k, and target $v_{max}$ can be calculated using the conservation of mechanical energy:
\begin{equation}\label{jump_eq}
    \begin{split}
        U_{1} = & \, U_{2} =  U_{3} \\
          mgh_{t} = &  \, \frac{1}{2}k(x_{max})^{2} = \, \frac{1}{2}m_{r}(v_{max})^{2}
    \end{split}
\end{equation}
where \(U_{1}\) is the potential energy from height, \(U_{2}\) is that from the compressed spring and \(U_{3}\) is the kinetic energy of the mass at the end of the stroke. In this model, drag and damping are assumed to be negligible. Thus, the ideal spring constant can be calculated as $49.05$ N/m; the target $v_{max}$ is $4.43$ m/s; and the fully compressed spring produces a force of $9.81$ N (denoted by $F_{0}$). To generate a constant output force, this condition has to be true: 
\begin{equation}
    \frac{dF_{out}}{dx_{in}} = 0,
\end{equation}
and thus we get relations:
\begin{equation}
    \frac{F_{0}}{G_{0}} = \frac{k}{R}
\end{equation}
where \(F_{0}\) and k are known. If we assume the total output stroke length is the same as the input one, the remaining two constants can be calculated: \(G_{0} = 2\) and \(R = 10\). When the CVT consists of conical rollers, R directly reflects the slope of the cones; when it does allow the gearbox to have a mechanical advantage profile that produces constant output force, it is named the ``critical gear ratio constant". For clarity and realism, the R used in the simulation is $9.9$, slightly smaller than the critical gear ratio constant. 

\subsection{Motor model}\label{sec_motor}

Various previous works have used a standard model for electric motors \cite{beckerle2016series,pentzer2012investigation}. The linearised version of the equations is illustrated below: 
\begin{equation}\label{motor1}
\begin{split}
     V = & L\frac{dI}{dt} + RI + kv \\
     I = & \frac{F + k_{m}v}{k}
\end{split}
\end{equation}
where L is the terminal inductance, R is the winding resistance and \(k\) is the motor's back EMF constant with a unit of $N\cdot A^{-1}$ (newton over ampere); V, I, F are the voltage input, current and the motor force output respectively. Compared to their original counterpart, several modifications have been made. Motor constant $k$ no longer has a unit of $Nm/A$ since it is linearised and describes the relation between the input voltage and the $v$ (unit m/s) instead of the angular velocity $\omega$ (unit $s^{-1}$). At the same time, the motor torque output $T_{m}$ is changed to simply force $F$. From \eqref{motor1}, the expression of the motor's force output as a function of its velocity can be derived:
\begin{equation}\label{motor2}
    F = (\frac{k_{m}}{k})V - k_{m}v
\end{equation}
where v is the motor velocity and \(k_{m}\) is a constant known as the viscous damping coefficient. It can be calculated using:
\begin{equation}
    k_{m} = \frac{kI_{0}}{v_{0}}
\end{equation}
where \(v_{0}\) and \(I_{0}\) are the no load velocity and current. Since the input voltage is assumed to be constant, \eqref{motor2} can be further simplified to: 
\begin{equation}\label{motor3}
    F = F_{0} - k_{m}v
\end{equation}
where \(F_{0}\) is the ``stall force" of the motor (same as the maximum spring force).


\section{Simulation}\label{sec_sim_sea}

Two simulation studies are done in this section. Simulation 1 is to showcase the general behaviour of the C-SEA with a single end-effector stroke intending to propel the robot body to the target height. The output accelerations and velocities are compared with their SEA counterparts in specific scenarios. Simulation 2 studies the factors that can affect the $v_{max}$ achieved by the actuators in question and the results from various input combinations are compared and analysed. The simulations in this study are exclusively done in MATLAB using the Euler method with a time step of $0.1$ millisecond. 

\subsection{Simulation 1}\label{sec_sim_1}

An important characteristic of the C-SEA is producing reduced and almost constant output force, and this is enabled by the gear ratio being a linear function of the input displacement. To showcase this feature, a kinematic simulation of the CVT gearbox is done. The governing equation of the behaviour is based on \eqref{eq_xout} with some modifications. We assume the system is not actuated but simply powered by a compressed spring and only the analytic outputs of the gearbox are being examined (masses are ignored); thus in this case, $x_{u}$ is a fixed value of $x_{max}$. If we plug in the constants from Section.~\ref{constants}, the behaviour of the system in a single stroke is plotted with respect to input displacement $x_{in}$ in Fig.~\ref{sim_1_kinematic}. The output force is kept nearly constant and only drops at the end of the stroke with a peak value of $4.91$N, half of the peak input force of $9.81$N. 

\begin{figure}[t]
    \centering
    \includegraphics[width=0.75\columnwidth]{pics/Simulation_1_kinematics.jpg}
    % \vspace{-4mm}
    \caption{The results from the kinematic simulation. The input force ($F_{in}$), mechanical advantage (MA), end-effector position ($x_{out}$) and output force ($F_{out}$) are plotted against the input displacement ($x_{in}$).}
    \label{sim_1_kinematic}
\end{figure}

\begin{figure}[t]
    \centering
    \includegraphics[width=0.75\columnwidth]{pics/Simulation_1_dynamics.jpg}
    % \vspace{-4mm}
    \caption{The results from the dynamic simulation. The output force ($F_{out}$), acceleration and velocity along with the actuator velocity are plotted against time. Since the mass remains unchanged, $F_{out}$ and the acceleration have the same shape. }
    \label{sim_1_dynamic}
\end{figure}

The next step is to simulate the full dynamic behaviour of both the C-SEA and the regular SEA according to the powertrain structure in Fig~\ref{hopperpowertrain} and the mathematical model in Section~\ref {sec_mathmodel}. The powertrain is placed along the vertical direction with the full effect of gravity. $m_{2}$ is set to be $10$g and the actuator is assumed to generate constant force with no limit on velocity. The maximum extent of $x_{u}$ is constrained to $x_{max}$ and the lift-off happens when the output displacement reaches $x_{max}$. The results are demonstrated in Fig.~\ref{sim_1_dynamic} where the x-axis is now time. The C-SEA experiences almost constant output acceleration thus linear velocity and ground reaction force throughout the stroke with a peak value around half of its SEA counterpart. The results also show that the stroke have two stages, the charging and the releasing, in a continuous manner which sees end-effector starting to displace as the spring is being compressed. 

\begin{table}[t]
\centering
\caption{Summary of Simulation 1 results}
%\vspace{3mm}
\begin{adjustbox}{max width=\linewidth}
\begin{tabular}{lccccl}
\toprule
\vspace{1mm}
Scenarios & $v_{max}$(m/s) \quad & Time(s) \quad & Height(m) \quad & Peak force(N)\\
\hline 
Ideal scenario \rule{0pt}{2.6ex} & $4.43$ & -- & $1.000$ & $9.81$ \\
%
C-SEA actuated & $3.93$ & $0.117$ & $0.787$ & $4.91$ \\
%
SEA actuated & $3.97$ & $0.091$ & $0.803$ & $9.81$ \\
\bottomrule
\end{tabular}
\end{adjustbox}
\label{tab_sim}
% \vspace{-2mm}
\end{table}

The summarised results can be found in Table~\ref{tab_sim}. The system with C-SEA takes $0.117$ s to reach a velocity of $3.93$ m/s while the one without CVT takes $0.091$ s to achieve a velocity of $3.97$ m/s. C-SEA takes longer to reach the end velocity but $v_{max}$ and thus jump height is only slightly affected ($0.787$ m for C-SEA and $0.803$ m for SEA). Therefore, without the CVT, a regular SEA would have produced $99.8\%$ more peak ground reaction force while reducing the time duration by $22.1\%$. However, this simulation result still cannot reflect a realistic scenario since the actuator, if being powered by an electric motor, simply cannot generate perfectly constant force output while having no velocity limitation. The maximum velocity reached by the actuator, as shown in Fig.~\ref{sim_1_dynamic}, is a whopping $13.87$ m/s which is an unrealistically big number. To make the simulation more realistic, the mathematical model for the electric motor introduced in Section.~\ref{sec_motor} should be included in the overall model. 



\subsection{Simulation 2}

\begin{figure}[h]
    \centering
    \includegraphics[width=0.75\columnwidth]{pics/Simulation_2.jpg}
    % \vspace{-4mm}
    \caption{The results of simulation 2. The input force from spring ($F_{in}$), output force ($F_{out}$), actuator velocity and the end-effector velocity are plotted against time. Since there is no gearbox in SEA, its ($F_{in}$) and ($F_{out}$) are identical here.}
    \label{sim_2_dynamic}
\end{figure}

As discussed in Section.~\ref{sec_meth}, the second advantage arises from the initial high MA at the beginning of the stroke and a lower MA at the end, which allows for more thorough compression of the spring, thereby storing greater energy for achieving higher jumps. The rationale behind it is that prolonged charging stages can lead to premature expansion of the spring, resulting in incomplete compression when energy is released. However, the results from Section.~\ref{sec_sim_1} show that the conventional SEA still outperforms in terms of jump height. This largely attributes to the assumption that the ideal actuator produces constant force with unlimited velocity, which is unrealistic and causes the charging stage to be shorter than if an electric motor is powering the actuator.

When considering an actuator powered by a linear electric motor, the time required to compress the spring is influenced by two factors: $m_{2}$ and $k_{m}$. The first factor is self-explanatory, as larger masses are harder to accelerate according to Newton's second law. The second factor determines how much the force output from the motor decreases as it speeds up. If the actuator has a linear behaviour, the charging will take longer; this consequently causes the spring to prematurely expand. When using the C-SEA, this premature expansion would be mitigated by the high MA at the beginning of the stroke, thus allowing the robot to reach a better height. 


To prove the addition of the motor model in Section.~\ref{sec_motor} does affect the performance in the manner described above, the general behaviour of the system with the new model is simulated. The $k_{m}$ is set to be $2$ so that when the stall force is $F_{0}$, the no-load velocity of the actuator is $4.9$ m/s slightly larger than the required $v_{max}$; everything else remains unchanged from the setup of the dynamic Simulation 1. The results can be found in Fig.~\ref{sim_2_dynamic}. As expected, the peak force output from C-SEA is still way less than the input even though the constant section is shorter compared to Simulation 1; SEA still takes less overall time to complete the stroke. A major difference from the previous simulation is that the peak force output now is only slightly higher than its C-SEA counterpart which is precisely caused by the spring in SEA not being fully compressed during the charging stage. Consequently, the $v_{max}$ achieved by C-SEA this time is $3.81$ m/s, exceeding the $3.57$ m/s of the SEA. This proves the notion mentioned previously: when using a realistically modelled actuator with linear behaviour, the C-SEA still manages to compress the spring fairly well during the charging stage which is now longer, while the conventional performs poorly. 

To further study the effects of $m_{2}$ and $k_{m}$ on $v_{max}$ for both C-SEA and SEA, combinations of the variables from wide ranges are used to produce a 3D plot of the results (in Fig.~\ref{motorplot}). $M_{2}$ has a range from $1$ g to $100$ g; the lower bound is a number small enough to represent the lowest possible value for the mass of the moving component in the actuator but still easy enough to simulate, while the upper bound is the mass of the robot body which is impossible to exceed. At the same time, $k_{m}$ ranges from $0$ to $5$. When \(k_{m} = 0\), \(F = F_{0}\); this represents the scenario in simulation 1 where the actuator produces constant force. Lastly, the end-effector fails to complete a full stroke when $k_{m}$ exceeds $5$, thus the upper bound. The values for the variables are distributed in an exponential manner throughout the ranges, and the results are plotted in 3D. Two best-fit surfaces are generated for both the C-SEA and the SEA using the ``polyfit23" function. The contours of the surfaces are also shown in Fig.~\ref{motorplot}. The red dotted line divides the contours in two regions, most $m_{2}$ and $k_{m}$ combinations to the top-right of the red line, see higher $v_{max}$ for C-SEA. This indicates that for motors with both larger $m_{2}$ and $k_{m}$, the C-SEA usually have a clear advantage over the regular SEA.

\begin{figure}
    \centering
    \includegraphics[width=0.99\linewidth]{pics/motor_model_plot.jpg}
    % \vspace{-4mm}
    \caption{The 3D plots for the results from simulation 2 are on the top and their corresponding contours are also shown on the bottom. The endpoints of the red dotted line have Cartesian coordinates $[0, 0.1]$ and $[3, 0.001]$ in the contour plots. The dotted line divides the regions according to whether C-SEA or SEA has the better performance.}
    \label{motorplot}
\end{figure}


\section{CVT sensitivity study}

In the previous simulations, the mass and friction of the CVT mechanism are assumed to be negligible, which is not realistic in practice. The loss brought by the extra weight and additional friction of CVT can inevitably compromise the performance of the jumper; this study aims to examine to what extend they do. The typical NG gear style CVT has a mass of $10$g and an efficiency of $98\%$ (according to spur gear data).This study manipulate these parameters to see their effects on peak ground reaction force and jump height reached. All other parameters are identical to the previous simulations.

\subsubsection{Peak GRF vs. CVT mass}

The CVT mass factor is introduced as a parameter representing how many folds the CVT mass is increased ($2$ means the current CVT mass is twice as the original). The CVT mass is treated as extra load to the body (originally $100$g). The results in \figref{cvt_sensitivity} (a) show that when the factor is increased from $1$ to $10$, the peak GRF of C-SEA increases $0.29\%$ from $4.881$N to $4.895$N, which is almost negligible (even though jumper height does suffer from the weight gain). When the same additional mass is added to the SEA jumper, the peak ground reaction force rises $21.02\%$ to $6.405$N. Notice that when a CVT of typical mass is included, the peak GRF of SEA is still $4.95\%$ higher than that of C-SEA. It is safe to concluded that the mass of CVT have little effect on the peak GRF produced.  

\subsubsection{Jump height vs. CVT mass}

The increment of CVT mass factor influences the jump height performances differently. Since the SEA does not include a CVT component, the performance is constant at $0.543$m and is only used for comparison with that of C-SEA. The results in \figref{cvt_sensitivity} (b) show that the height reached by the C-SEA decreases with CVT mass and the overtake happens when factor equals $2.7$. At typical CVT mass, the C-SEA jumps $16.98\%$ higher than SEA reaching $0.625$m.

\subsubsection{Jump height vs. efficiency}

Due to the asymmetric geometry of the NC gears, modelling the exact Coulombic and viscous friction of them would be difficult. Instead, we simply represent CVT loss using an general efficiency $\eta$, so that \(CVT_{Out}=\eta\cdot CVT_{In}\) where ``Out" and ``In" refer to the output and input powers of the CVT. The height performance is then calculated from the CVT work output. \figref{cvt_sensitivity} (c) shows that as $\eta$ varies from $100\%$ to $60\%$, the jump height performance for C-SEA drops linearly while that of SEA stays constant (due to the absence of CVT). The overtake happens at $77\%$ efficiency, meaning any value above that would result in C-SEA having a stronger performance. At typical efficiency, the C-SEA jumps $26.60\%$ higher than the SEA jumper.


\begin{figure}
    \centering
    \includegraphics[width=0.99\linewidth]{pics/CVT_sensitivity.jpg}
    \caption{The CVT sensitivity study: (a) Peak GRF vs. CVT mass factor, (b) Jump height vs. CVT mass, and (c) Jump height vs. efficiency. Best fit curves are added on top of the data points using POLYFIT. The typical values and overtake points are indicated using dotted vertical lines.}
    \label{cvt_sensitivity}
\end{figure}



\section{Summary for the simulation study}

In conclusion, the C-SEA manages to generate nearly constant and lower ground reaction force than the regular SEA sacrificing the time taken to accelerate with minimal loss on jump height. The reduced ground force is beneficial to the structural integrity and means less material requirements for the end-effector. At the same time, when actuated by a realistic motor, C-SEA has a clear advantage in most cases with higher $m_{2}$ and $k_{m}$ outperforming the SEA in terms of achieving high jumps because of the C-SEA's ability to compress the spring more thoroughly even when the charging stage is extended. 

In the following sections, the findings in the paper will facilitate the development of a physical CVT gearbox that can be integrated with an SEA for robotic jumping and a prototype of such C-SEA enabled hopping robot will be constructed. Next steps for this research include: choosing a CVT type, parameter optimization for the CVT, designing and fabricating the hopping robot as a testing platform, developing a control strategy for such a robot and conducting comparative experiments. Promising CVT options include Evan’s variable speed counter-shaft and the ratcheting CVT; the former boasts a simple and reliable structure, while the latter excels in transmitting impressive torque with minimal loss \cite{benitez2004infinitely}.


\section{The jumping mechanism prototype} \label{sec_proto}

The simulation studies have demonstrated the advantages of the C-SEA over conventional SEA. The next step of the research to design and construct a C-SEA jumping mechanism prototype to test the real-world performance and to validate the findings from the simulation studies. There are several design criteria and challenges facing the prototype design task. Compromises will be have to be made to bypass the problems in the prototyping process. 

\begin{itemize}
    \item The parameters of the jumper prototype will be based on the values used in simulation. However, it is difficult to make them exact matches. Thus, the experiment conditions with alter accordingly to fit the real parameters while the C-SEA and the SEA can still be compared fairly.
    \item The CVT gearbox is only conceptual in the simulations. In order to make it practical, a specific mechanism with required functionality needs to be chosen and parametrically tuned to be incorporated into overall powertrain of the jumper.
    \item The jumper is to be actuated by an electric motor. The motor choice and control method should be suitable so that the tasks in the simulation can be best replicated in practical scenarios. 
    \item Since the jumper is very simplistic with a single degree of freedom leg mechanism, there needs to be an experiment platform supporting the jumper. At the same time, the platform also host sensors to collect data from the experiments.
\end{itemize}

This section will include the mechanical design process of jumper, the CVT gearbox, the motor control electronics and methods. 


\subsection{CVT gearbox design}

The section details the process of choosing a CVT type that fits the criteria in the simulation section and the construction of the jumping robot prototype. There are several requirements of the CVT design that are uncommon among the more conventional types.
\begin{itemize}
    \item The gear ratio of the CVT should be a linear function of the displacement of the end-effector. 
    \item The gear ratio should vary being lead by the internal mechanical properties of the gearbox so that the general structure of the device is kept simple. 
    \item The gearbox should have positive engagement so large torque can be transmitted.
\end{itemize}

The non-circular (NC) gears are chosen to be the basis of this new CVT for its simplicity, compatibility. Even though NC gears and conventional CVT are usually separately fields of study, some researches over the years do consider them as a type of CVT that have fixed gear ratio profiles. Generally speaking, NC gears when being used in a transmission scenario, do fulfill the definition of CVT (although they cannot be consider IVTs) and they do fit every criteria of the ideal CVT design. Moreover, the simple structure of the NC gears poses a huge advantage over other CVT types, making it the ideal choice for this application. Furthermore, NC gears have been used in robotic jumping but never in combination with a SEA. Therefore, developing a NC gear-based C-SEA does have scientific significance and novelty. 

There are two steps in the NC gear design, namely, determining the pitch curve and then generating the teeth profile. For the first step, the pitch curve (\(r(\theta)\)) is usually expressed by the radius (the distance from the center of rotation to the pitch line of the gear) as a function of the rotation angle $\theta$ in the polar coordinate. The pitch curve is determined by the required gear ratio profile. Knowing the pitch curve, there are online tools to generate the suitable teeth profile so the gears can be modeled and manufactured. 

\begin{figure}
    \centering
    \includegraphics[width=0.8\columnwidth]{pics/NC_gears_demo.png}
    \caption{The SW assembly that shows the rotation of the gear pair with correctly meshed teeth.}
    \label{assembly}
\end{figure}

The online tool is on a website called OTVINTA which generate the teeth profile providing the pitch curve in the form of a script. Then, the script can be exported to Blender to draw a sketch of the gear profiles. Finally, the sketch can be converted into solid bodies in Solidworks so that further modelling can be done. For example, if we assume the expression of the pitch curve is:
\begin{equation}
    r = 1 + \frac{1}{\pi}\theta,
\end{equation}
the sketch for the gear profile is then generated in blender. The sketch then was exported to Solidworks to create this assembly of the gear pair shown in \figref{assembly}. 

\subsubsection{Pitch curve generation}

Generally, the pitch curve of the driving gear in a NC gear pair is \(r_{1}(\theta)\) and that of the driven gear can be expressed as:
\begin{equation}
    r_{2}(\theta) = A - r_{1}(\theta)
\end{equation}
where A is the distance between the two CoRs of the gear pair. Then the gear ratio of such gear pair can be calculated as:
\begin{equation}\label{gr1}
    Gr(\theta) = \frac{r_{1}(\theta)}{r_{2}(\theta)} = \frac{r_{1}}{A - r_{1}}.
\end{equation}
The required gear ratio profile is known from Section \ref{sec_mathmodel}, equation \eqref{eq_gr} and its polar expression is:
\begin{equation}\label{gr2}
    Gr(\theta) = G_{0} - R\theta
\end{equation}
where $G_{0}$ and R are constants. Equations \eqref{gr1} and \eqref{gr2} can be combined and re-arranged into:
\begin{equation}\label{r1}
    r_{1}(\theta) = \frac{A(G_{0} - R\theta)}{1 + (G_{0} - R\theta)}.
\end{equation}
In Section \ref{sec_mathmodel}, in order to calculate the values for $G_{0}$ and R, other assumptions have to be made, namely, the full extension of motor and leg are both $0.2$m. However, the rotational nature of the new model here means the range of motion for both NC gears are simply a full rotation (from $0$ to $2\pi$). In this case, $G_{0}$ and R are calculated to be $2$ and $\frac{1}{\pi}$ respectively. 
Now equation \eqref{r1} becomes:
\begin{equation}\label{r1_2}
    r_{1}(\theta) = \frac{A(2 - \frac{1}{\pi}\theta)}{(3 - R\theta)}.
\end{equation}
If we than plug in \(A=3cm\), we get the polar expression for the pitch curve of the driving gear. The resultant NC gear set will be used for the CVT component for all jumper prototype versions.

Now we have derived the theoretical pitch curve for the NC gear set, one more step needs to be taken before implementation. The values of $G_{0}$ and R cannot be  $2$ and $\frac{1}{\pi}$ exactly in practice because this would cause the gear ratio to reach $0$ at full leg extension, which is mathematically invalid. The physical representation of a $0$ gear ratio would be the gears being stuck and we will try to avoid that. During the theoretical calculation of $G_{0}$ and R, a relation between them is met: 
\begin{equation} \label{GR_relation1}
    G_{0}=2\pi \cdot R
\end{equation}
which is not actually a hard requirement. This relation ensures the resultant ground reaction force to be completely constant throughout the leg stroke. However, this does not need to be rigidly true for the advantages of having a CVT gearbox to mostly embody. In this case, we can rewrite \eqref{GR_relation1} into:
\begin{equation}
    G_{0} = (2\pi + \rho_{\kappa}) \cdot R
\end{equation}
where \(\rho_{\kappa}\) is the the value that curves this relation. When \(\rho_{\kappa}\) is a positive real number, the gear ratio would never be $0$ thus solving the mathematical invalidity in the theoretical model. \figref{pitch_test} shows the output torque from the CVT gearbox with respect to the input rotation under three different \(\rho_{\kappa}\) values. We have established that a complete constant output would be impossible, a very large \(\rho_{\kappa}\) value is also problematic. When \(\rho_{\kappa} = 5\), the peak output torque is $0.225$Nm, only achieving a reduction of $40.9\%$; there is also significant work loss (work done can be obtained from the areas under the curves). Through trail and error, we found a reasonable value of $0.13$ for \(\rho_{\kappa} = 5\), and we manage to produce a output curve that is largely constant and only dropping at the end. In this case, the $G_{0}$ and R values use are $1.96$ and $0.306$ respectively. 

\begin{figure}
    \centering
    \includegraphics[width=0.99\linewidth]{pics/NC_gear_pitch_test.png}
    \caption{The plots of output torque from the CVT gearbox against the rotation angle of the driving gear.}
    \label{pitch_test}
\end{figure}


\subsubsection{Teeth profile design}

When trying to implement the NC gear pairs into a gearbox configuration, we realised the gears often loose grip when the leg is at full extend. After carefully examining the gear models and contact testing, we concluded that the models are topologically sound, but rather, the problem is with gear meshing. When under load, the gears cause the surrounding structures to deform, which causes the teeth to slip and loose contact. This deformation is small enough to be handled by circular gears but is problematic to NC gears due to their acute positioning. To solve this, there are two ways to improve gear meshing: pitch angle and gear size. As the pitch curve and the overall gear size is know, increasing the gear size means decreasing the teeth number. Based on this idea, we conducted a qualitative gear meshing comparison test.

The NC gear sets in question are shown in \figref{nc_gear_mesh} and they have various pitch angles (PA) and numbers of teeth (TN). In the online tool for NC gears, there are two options for the pitch angle $20^{\circ}$ and $14.5^{\circ}$ but the number of teeth can be freely customised. The set originally used was a PA$20$TN$30$ (a pitch angle of $20^{\circ}$ and $30$ teeth), thus for TN of $30$, only the PA $14.5^{\circ}$ is made. Another $6$ sets are made with both PA options and TN ranging from $15$ to $25$. They are tested on regular jumper frames with an applied torque of $0.318$ Nm with is the highest torque the gear set need to endure. The results shows that $14.5^{\circ}$ PA generally have a better grip then $20^{\circ}$ and increasing the size of teeth do improve contact. However, have too few teeth also have drawbacks, namely, the accuracy starts to drop and unwanted vibration begins to appear when TN is reduced to $15$. Sets PA$14.5$TN$30$, PA$14.5$TN$25$, and PA$20$TN$25$ all failed (loose contact during) the test. Therefore, PA$14.5$TN$20$ are chosen to the optimal NC gear set for the jumper. 

\begin{figure}
    \centering
    \includegraphics[width=0.9\linewidth]{pics/NC_gear_mesh_test.jpg}
    \caption{All the NC gear teeth configurations in the meshing test. ``PA" stands for pitch angle and ``TN" stands for teeth number. Material colours are irrelevant.}
    \label{nc_gear_mesh}
\end{figure}

\subsection{Jumper prototypes}

\subsubsection{Jumper prototype Version 0}

The general structure of the jumper powertrain have already been mention in the previous sections (\figref{hopperpowertrain} (a)). To summaries, the system is powered by an electric motor which is usually of rotational nature; the CVT gearbox, which consists of a NC gear pair, also transmits rotation and torque. This means the most sensable option for the elastic element, which is between the motor and the CVT, is a torsion spring. Then, the angular output from the CVT gearbox is converted to linear motion by a rack-and-pinion mechanism. \figref{jumper_ver.0} shows the overall structure of the jumper prototype. The positions of the main components (motor, spring, CVT gearset, and the rack-and-pinion mechanism) in the jumper powertrain are shown in the figure. The rack is fixed on the leg which is confined by two leg supports with rollers to a 1 degree-of-freedom linear motion. The jumper also contains mounting position for encoders in order to sense the rotation angle and speed of both the motor and the pinion. All the components are sandwiched between two main frames. See Appendix \ref{app_supp} for more details of Jumper Ver.0 (figures with zoomed in views). Ver.0 is the conceptual design of the jumper prototype in which the mechanical details reflecting real-life components are not illustrated. They will be introduced in Ver.1.

\begin{figure}
    \centering
    \includegraphics[width=0.75\linewidth]{pics/Jumper_ver0.png}
    \caption{The conceptual mechanical design of a C-SEA jumping device. The main components in the powertrain are pointed out in the figure. More detailed illustrations are included in the Appendix.}
    \label{jumper_ver.0}
\end{figure}

\subsubsection{Jumper prototype Version 1}

The Ver.1 jumper (in \figref{jumper_ver.1}) includes realistic mechanical details making it manufacturable in practice. This version does not have either motor or encoders and only the mechanical components are included (the spring casing is fixed on the frame). It can only perform single jump by a push-and-release action. The main purpose is to test the compatibility of the  the gear sets in the powertrain. The leg support system are now functional with grooved rollers and track added to guide and restrict the leg motion. Ball bearings are used at the connection between shafts and frames. A gearbox support frame is added to side of the main frame for extra strength and compactness. All components are 3D printed except screws, washers, nuts, and bearings. See Appendix \ref{app_supp} for the constructed Ver.1 jumper. 

The testing results show that the leg reaction force at any given spring compression is smaller than the theoretic value while the general trend of the measured values matches the theoretic value (\figref{prototype_tests}). This is caused by a mismatch of the converter gearset resulting the spring not being compressed fully. At the same time, the leg motion is way slower than expected meaning the friction in the roller-track leg support system is too high. 

\begin{figure}
    \centering
    \includegraphics[width=0.75\linewidth]{pics/Jumper_ver1.png}
    \caption{The Ver.1 of the jumper, which includes realistic mechanical details. This figure highlights the main changes made to the design compared to the conceptual Ver.0 design.}
    \label{jumper_ver.1}
\end{figure}


\subsubsection{Jumper prototype Version 2}

In order to solve the aforementioned issues and to overall improve the jumper design, several modifications have been made since. 
\begin{itemize}
    \item The converted gears are tuned to the correct sizes which enable the spring to be fully compressed.
    \item The roller-track leg support system is replaced by a guide rod. Instead if having 3 rollers on each side of a leg with rectangular cross-section, the leg now consists of a steel guide rod and the rack parallel to each other; and there are linear ball bearings acting as the connection between the guide rod and the support frames.
    \item The elastic element have been updated from a joint two-spring mechanism to a single spring with desired constant. The casings are modified to fit the new design. 
    \item The material used for the 3D printed gears used to be Tough 2000 resin which is sticky and thus can induced unwanted friction. The material has been changed to PLA, which have a smoother surface; this alternation has improve the gear efficiency. 
    \item The gears and shafts used to be single parts since they used the same material. Since the gears now use a different material which is not suitable for shafts, a new connection method is implemented. Hexagon connector nuts are used between shafts and gears to transmit torque and rotation and they secured using superglue. 
\end{itemize}
See Appendix \ref{app_supp} for the constructed Ver.2 jumper. The next step is to install the motor and encoder to complete the jumper design. 

\begin{figure}
    \centering
    \includegraphics[width=0.75\linewidth]{pics/Jumper_ver2.png}
    \caption{The Ver.2 of the jumper. This version includes new features that solves the previous issues and made improvements overall. The modifications are highlighted.}
    \label{jumper_ver.2}
\end{figure}


\subsubsection{Jumper prototype Version 3}

The Ver.3 of the jumper is a motor-actuated device unlike the previous versions which can only perform push-and-release actions. The motor and the gearbox are installed onto the jumper frame together as an integrated module. The gearbox has a $2$ step speed reduction; the rotation is transmitted from the motor at the bottom to the output gear at the top. The gear ratio of the gearbox is $10:1$ which is realised by the first $3$ gears and the last output gear is added as a connector for distance management. The output from the gearbox is connected to the spring casing which is no longer fixed on the frame. Magnetic encoders are installed as well; one is on the back of the motor and the other is behind the pinion, measuring their respective positions and velocities. More closed in view of the encoder boards and the gearbox-motor module are included in Appendix \ref{app_supp}. Additionally, mounting frames are added to the back of the jumper. The mount include casings for linear bearings that are compatible with the experiment platform. 

\begin{figure}
    \centering
    \includegraphics[width=0.99\linewidth]{pics/Jumper_ver3.png}
    \caption{The Ver.3 of the jumper prototype. The motor with reduction gearbox has been added to the jumper. On the right side of the figure is a close up view of the integrated motor-gearbox module. Mounting frames are also added to the back on the jumper to be mounted onto the experiment platform.}
    \label{jumper_ver.3}
\end{figure}

\begin{figure}
    \centering
    \includegraphics[width=0.99\linewidth]{pics/Jumper_ver3_real.png}
    \caption{The manufactured Ver.3 of the jumper prototype. The picture on the left is an overview and the right one is a more zoomed in view of the jumper from the top.}
    \label{jumper_ver.3_real}
\end{figure}

\subsubsection{Measured GRF vs. Theoretical values}

This section includes the plots of measured GRF values with respect to leg extension compared with the theoretical values. Results for both Ver.1 and Ver.3 are included in \figref{prototype_tests}. The results for Ver.1 reveals a converter sizing problem that has been discussed in the Ver.1 section; Ver.3 has solved the issue by modifying the converters to the correct sizes. The results for Ver.3 indicates the measured data roughly coincide with the theoretical values confirming the successful modification. Notice that the theoretical data corresponds to these shown by \figref{sim_1_kinematic} in \ref{sec_sim_1}. Since the smooth action cannot be guaranteed by Ver.3, the dynamics results like in \figref{sim_1_dynamic} cannot be obtained in the current version. These tests will be done as a part of future work.

\begin{figure}
    \centering
    \includegraphics[width=0.8\linewidth]{pics/measured_vs_theory.jpg}
    \caption{Caption}
    \label{prototype_tests}
\end{figure}


\subsubsection{Actuation}

The system is actuated by a three-phase brushless direct-current (BLDC) motor with a KV constant of $1800$. The motor is controlled by a DengFOC motor drive and a ESP32 microcontroller board. The motor control code used is Arduino based field-oriented control (FOC) algorithm. The pictures of these devices can be found in \figref{gearbox_motor_real} in Appendix~\ref{app_supp}.


\subsection{Experiment platform }

The experiment platform is shown in \figref{ex_plat_jump} in Appendix \ref{app_supp}, which consists of the supporting frame, the guide rods and the sensors. The jumper is constrained by two aluminium guide rods that only allows it to move vertically up and down. There is a load cell underneath the robot leg to measure the ground reaction force during a jump, and the ultrasonic distance sensor on the top measures the height of the jump. 


\section{Chapter summary}\label{sec_conclu}

This chapter introduced a CVT enhanced SEA jumping mechanism. The simulation results shows that the C-SEA manages to produce nearly constant ground reaction force than the regular SEA and almost halves the peak ground reaction force sacrificing the time taken to accelerate. The reduced ground force is beneficial to the structural integrity and means less material requirements for the end-effector; it could also be useful when the robot needs to locomote on soft surfaces that might be unstable under high pressure. At the same time, when actuated by a realistic motor, C-SEA has a clear advantage. It outperforms the SEA in terms of achieving high jumps because of the C-SEA's ability to compress the spring more thoroughly even when the charging stage is extended. The mechanical design for the CVT gearbox also yielded great results with NC gear set being chosen for the task. The NC gear set have varying gear ratio lead by its own internal mechanical properties and positive engagement for better torque transmission. The jumper prototyping process took three iterations with the final results fit the design criteria. 

Following the jumper mechanism design, we will dive into the full robot integration in the next chapter. The compatibility, functionality and agility of the jumping robot incorporating the C-SEA leg and AeroTail balancer will be demonstrated in the form of simulation studies. New dynamic model and control strategy will be introduced to fit the planar jumping robot. Its advantages over other conventional counterparts will also be showcased. 