
After the general direction and research questions are defined, the literature review ensues. The literature review initially focused on the past examples and methods of designing jumping robotic legs using linkage systems since my study was inspired by the Salto robot design which uses a Stephenson-II linkage \cite{plecnik2017design}. The goal was to find or design a linkage system and a corresponding optimization method that allows a jumping robot to outperform Salto on the jumping agility metric. However, as the research progressed on, it was clear that there are many other straight-line mechanisms used on robotic jumpers and hopper. Therefore, the possible solutions that are way more simplistic than linkage were largely explored as well. Many designs for jumping robots includes elastic elements for the storage of energy either for timed release or for power modulation. Past examples of this feature being incorporated usually takes the form of serial or parallel elastic actuators for powering the jumpers. Then, the potential strategies for utilising aerodynamics in robotic balancing are included with past example of robots using reaction wheel or tail-like devices but combining them with aerodynamic drag inducing structures. Finally, gliding has been considered as a promising additional function for the robot. Existing wing designs for gliding robots were reviewed as well as potential height boosting mechanisms to enable gliding. Inspiration for optimal wing shape design was drawn from past and existing sailplanes since their wings were designed to perform long and stable glides. At the end of this section, the gaps in the current literature will be explored and they serve as justifications to the existence of this research.

%Since gliding is beyond the scope of this project, the related literature review will be put into \textbf{Appendix.\ref{app_glide}}.  


\section{Robotic Jumping}

\subsection{Bioinspiration}

The Galago, for example, can jump to a 1.74$m$ height in a single jump producing 191$W$ of power in the process and maintain an average 1.5$m$ height in continuous jumps \cite{aerts1998vertical}. It is able to do that due to its tissue composition having power modulation ability thus amplifying the power during jumping. It is said that 60\% percent of the power comes from the bi-articular calf muscle which is behaving elastically, storing and releasing energy during jumps. The existence and structure of the tendon-muscle complex in Galago’s leg effectively amplifies the power by 15-fold. Bullfrogs can jump to a distance of 1.3$m$ with $31.5^{\circ}$ of angle and with power of 193$W$ on average \cite{astley2013chasing}. Cuban tree frogs can reach a jumping distance of 0.95$m$ with a body-mass-specific power of 91$W/kg$ during the takeoff period \cite{marsh1994jumping}. The geckos can utilize the circular movements of its tail to control yaw and pitch as it descends when hopping between trees \cite{jusufi2008active}. This is also an inspiration of the pitch controlling tail on the Salto robot design.  

\subsection{Straight line mechanisms }

\subsubsection{Prismatic actuators and rimless wheels}

Maarten J. van den Broek \cite{van2019fast} claims that bipedal running gaits may be self-stable if they rely on the intrinsic system dynamics to attenuate deviations. A virtual pendulum model can be applied in robotic design to achieve passive stabilization. Planar elliptical runner was compared with planar spring-loaded inverted pendulum model with a trunk (TSLIP). The HexRunner and the planar elliptical runner are two candidates for the experiment and unfortunately, the planar elliptical runner was chosen. Thus, the HexRunner has no detailed description and all information were gathered from a related video. Another example of bipedal running robots that utilize a straight-line system is the Cassie robot \cite{apgar2018fast} \cite{abate2018mechanical}. 

In a video introducing the HexRunner robot claims that the dynamics and energetics of the runner does fit the criteria of a running robot rather a wheel because of the fact that it has a “flight phase” (in which no leg is touching the ground). However, the concept of having unactuated compliance legs around a rotating platform generally falls under the umbrella term “rimless wheels”, and because of the unactuated legs, they cannot perform jumping. Sebastian Sanchez et al. \cite{sanchez2020differential} define the rimless wheel, or a wheel without a rim the simplest example of a legged robot. The device in this paper is torso-actuated, which is also the actuation method for HexRunner.

For a rimless wheel inspired robot with potential jumping ability, individual legs must be actuated. J. Blake Jeans et al. \cite{jeans2009impass} presented a walking robot (IMPASS) with spokes which expand and contract (thus actuated) as the methos of locomotion. A spoke can actuate by expanding and contracting and behave as a robotic leg is, in natural, a prismatic joint. Nicholas Paine et al.\cite{paine2012new} introduced the most desirable design so far. Although the device was designed for high frequency motions, the structure of the prismatic series elastic actuator could be used in robotic legs for jumping. There are also some examples of engineers using the rimless wheel configuration as the basis for developing bipedal walking gaits \cite{bhounsule2016dead} \cite{asano2009asymptotically} \cite{asano2012stability}. 

\subsubsection{Rack-and-pinion, ball screw and ServoBelt}

An online article was found about a bipedal walking robot designed by SCHAFT. Although there was no published paper behind it, but an extremely similar design SLIDER \cite{wang2020design} \cite{wang2022fast} was found. Slider is a walking robot, thus the most valuable feature to my research is its prismatic actuator called ServoBelt linear drive. ServoBelt is said to have a light-weight, low cost, high precision and high maximum load comparing to other more traditional prismatic joint transmissions like lead/ball screws or rack and pinion drives. 

There were concerns about the speed of linear actuation that a ball screw can provide being not sufficient for jumping. After going through some websites of companies that manufactures “high speed” linear actuators, this conclusion is confirmed to some degree. In fact, most existing linear actuators in machinery or those I found online for sale use either ball/lead screw or pneumatics/hydraulics as their actuation method. Ball screws and pneumatic cylinders in general are used for slow speed actuation with high load (lifters and jacks) or high-frequency back and forth motion (jackhammers and example in \cite{paine2012new}). Any actuator capable of extending a suitable distance for jumping does so in an insufficiently slow manner. 

The actual problem with the lead/ball screw for jumping actuation is that the linear motion when landing should be easily transferred into the elastic element, but the ball screw gives the motor too much mechanical advantage, which cause the kinetic energy cannot be efficiently absorbed by the elastic component. Therefore, the elastic component should be placed between the ball screw and end-effector but not between the motor and the screw (rotational spring) if one is to be used. 

In the papers \cite{sheikh2013towards} and \cite{sheikh2013towards}, the hopper design does exactly that. The passive springs and the rack-and-pinion actuator are two separate components connected in series and the spring is right between the rack and end-effector. The robot also has another degree of freedom allowing it to hop forwards. The paper \cite{sayyad2007single} demonstrates and discusses many monopod-hopper designs that have largely the same design philosophy. The monopods seem to be a trend in the 80s and 90s and there are sufficient previous examples for me to explore \cite{raibert1984experiments} \cite{rad1993design} \cite{wei2000design}. The structure of their actuators is very similar to the one in previously mentioned paper but they actually utilize pneumatic cylinders and ball screws as their transmission. However, this is not evidence of the ball screw and pneumatic cylinder being absolutely suitable for my research because the hoppers are not mean to be jump high but just enough to locomote. Still, the examples presented by A. Sayyad et al. seem to contradict the conclusion I made through searching for in-service linear actuators online. This uncertainty and lack of details will lead to the formation of my future plans.

\subsubsection{Linkage systems}

The Stephenson-II six-bar structure is a popular choice for linkage-based straight-line mechanism. There are two papers detailing the design process for Salto’s leg design. In the first paper \cite{haldane2016power}, D. W. Haldane et al. presented the energetics and the test results of the Salto leg and the reasoning behind expanding the six-bar linkage into an eight-bar one. This paper focus on the fact that the serial elastic nature of the leg transmission enables the robot to use power modulation in order to generate impressive peak power for jumping. The Stephenson-II six-bar linkage was used on the prototype for Salto but it was replaced by eight-bar linkage with most other features unchanged to tackle the body rotation problem. A second paper \cite{plecnik2017design} was found later but it predates the first one and explains the selection and design process for the original six-bar linkage used on the Salto prototype. Comparing to the first paper, this paper mainly focuses on the design and optimization of the linkage system. It is clear that the main reason that the six-bar linkage was chosen is its ability for the trajectory of the end-effector/ground reaction force to pass through the center of mass as a straight line with only one degree of freedom. The largest contribution of this paper is the extensive research the authors did on the various types of linkage system could be fit for related tasks and the huge number of referenced papers on the basic design ideas of jumping robots.

For the Stephenson linkage system specifically, the paper  \cite{plecnik2017design} provides a detailed design process of the Salto leg which utilizes said system. One of the steps involves using a function generator that produce an atlas of linkage designs that fulfills the same end effector trajectory. An optimal arrangement will then be chosen from the variations given. This generator is introduced and demonstrated in the paper \cite{plecnik2016design}. This method is useful for exploring the possible designs of linkage arrangements given the task and design criteria. Further explanation of the same approach can be found in the papers \cite{plecnik2016computational} \cite{plecnik2016kinematic} \cite{ting1996classification}. The next step for the Stephenson linkage is to learn the optimization methods that are available and commonly used to improve the parameters of a linkage to fit specific tasks. 

Another option is the Hoeken’s four bar linkage. One typical example using this mechanism is the tactical hazardous operations robot (THOR) \cite{knabe2014inverted} \cite{lee2014design}. The actuator for its knee joint utilizes an inverted Hoeken’s four bar linkage \cite{lu2014new} \cite{rodriguez2013effector}. The Hoeken’s linkage usually translates rotation into linear motion. By inverting it, linear motion generated by a ball screw motor can be turned into the rotations of the knee joint. This device is able to keep the mechanical advantage high and constant for the motor at any given angle of the knee joint. An interesting feature of the device is the use of cantilever beam as elastic element \cite{orekhov2013configurable} (this design can also be found on a similar robot SAFFiR \cite{lahr2016biologically}). The bending of the cantilever beam provides some compliance and flexibility for the system. This design is a further look into the realm of linkage systems; it will be put aside for the moment since the main focus is prismatic actuation. 

\subsection{Incorporation of elastic elements on jumping robots}

\subsubsection{Series-elastic and parallel-elastic actuators}

Since power modulation is an essential part of achieving required power for high jumping agility, its necessary condition of having a serial elastic actuator is also vital. Martin grimmer et al. \cite{grimmer2012comparison} explained the trade-offs between series- and parallel-elastic actuators (SEAs and PEAs), and the conclusions are listed. PEA can reduce peak power better than SEA by pre-loading the spring in the flight phase. This loading needs additional energy so that the ER increase in comparison to the SEA. The SE+PEA concept can further decrease the PP and ER are less than PEA but higher than SEA. Mimicking human ankle joint behavior in running and walking at higher speeds is still challenging for single motor devices. Both SEAs and PEAs were used in the design of Eleg \cite{roozing2021efficient} \cite{roozing2018design}, which is a three-degree-of-freedom compliantly actuated robotic leg that also uses a bio-inspired biarticulated tendon. Biarticulated muscles commonly found in biological systems saw increased adoption into robots, but the combination with elastic compliance was rare. The experiments in the paper proved that the energy consumption when jumping reduced by 68\% by using a biarticulated energy-storage branch.

\subsubsection{Robots with elastic elements}

From the examples of jumping robots searched from mechanism types, it is found that huge amount of jumping robot designs involves a mechanical trigger-and-release system to use the energy pre-stored in springs for locomotion (examples: Grillo \cite{li2012jumping}, Flea Catapult \cite{noh2012flea}, ParkourBot \cite{degani2014design}). A very innovative jumper design that uses a combination of rubber bands and carbon fibre ``bows" as its elastic elements was found \cite{hawkes2022engineered}. This robot is designed to maximise the specific energy of jumping and it manages to reach a height of 30 meters, which is higher than any engineered jumper before. There is even a jumping mechanism design combining combustion and a spring element \cite{wang2014design}. This is useful in terms of demonstration when combined with a miniature robot that usually has a simplistic structure. However, they are incapable of power modulation within a single jump and being fine-tuned in control. The most practical structure for a robotic leg capable of jumping is still the linkage system. 

\subsubsection{Power modulation}

In the same paper \cite{plecnik2017design}, there two most important design criteria for the leg design that enables power modulation. The required behaviors of the leg include possessing low mechanical advantage at the top of the stroke and that its mechanical advantage defines a constant ground force for the remainder of stroke. These requirements reveal more insights of power modulation. Low mechanical advantage essentially multiplies the weight of the robot (even though the force acting on the linkage is high; it does not expand much) allowing the motor to operate near its stall torque and to transfer energy into the series elastic spring element. Once the leg extends past the low region of mechanical advantage, the increased one (in high region) triggers high-power energy transfer from the spring into the linkage and thus vertical motion of the jumper. At the same time, the linkage should transfer the unwinding spring torque into a constant force at the end-effector by its altering mechanical advantage during the expanding process.

\subsection{Continuously variable transmission}

Series-elastic actuation (SEA) is frequently used in robots designed for continuous hopping \cite{csomay2023nonlinear,shin2022elastic} due to its ability to reduce energy consumption and peak torque requirements \cite{beckerle2016series}. During continuous hopping, the elastic element can absorb and store the energy from the last jump as the robot lands and releases it into the next jump. However, previous studies on robots that use SEA often emphasize their agility rather than jump height. This preference stems from the common use of parallel-elastic actuators (PEA) to achieve high jumps for their ability to preload the spring during the flight phase \cite{grimmer2012comparison}. Nonetheless, PEAs do not have the aforementioned benefits of energy conservation and torque reduction, making an SEA with enhanced jump capability the ideal actuator for hopping robots.

To optimise jump height while retaining the advantages of an SEA, the Salto robot provides a solution to improve the power modulation ratio (ratio between peak power output from a mechanism and that of its actuator) by having a linkage system with variable mechanical advantage (MA) as its leg mechanism \cite{plecnik2017design}. However, the design process of a linkage system requires complex kinematic tuning to produce a desired MA profile. In contrast, a continuously variable transmission (CVT) can produce the same MA profile with much simpler structures. Furthermore, while the Salto robot does manage to improve the peak power output of the jumping mechanism, this enhancement does not directly contribute to the jump height, which is determined by the upward velocity of the robot body at the moment of leaving the ground. Therefore, the exact manner in how having varying MA increases the jump height of SEA-actuated hoppers needs to be further researched. 

Previous studies have also incorporated CVT in robotics \cite{kim2002design,kernbaum2017ultra}, but they often focus on wheeled robots instead of legged ones. Some studies utilise CVTs with compliance as the method of changing MA \cite{matsushita2009development,nobaveh2023compliant} but hardly any combine them with a SEA. One study investigates a CVT-SEA combination \cite{mooney2013continuously}, mainly focusing on the control strategy for such actuators instead of specific scenarios for jumping application. Hence, the concept of CVT-enhanced SEA is novel for robotic jumping and necessitates mathematical modeling and simulation to evaluate its performance.


\subsection{Actuation Paradigms comparison}

The actuation paradigm adopted by a legged robot strongly influences its ability to generate large transient forces, manage, and exchange energy efficiently. This is particularly critical for jumping robots, where energy is rapidly injected, stored, and released over short time scales. Broadly, three actuation strategies are commonly employed in legged robotics: conventional high-ratio geared drives, quasi-direct drive (QDD), and series elastic actuation (SEA).

Conventional high-ratio drives, typically realised using harmonic, planetary, or cycloidal gear trains, amplify motor torque through large reduction ratios \cite{plecnik2017design,van2019fast}. This approach enables high continuous joint torque with compact motors and has been widely used in early legged robots. However, the high reduction ratio increases reflected inertia and friction while reducing backdrivability and force bandwidth. As a result, impact loads and rapid force transients are transmitted directly through the drivetrain, making this paradigm less suitable for highly impulsive behaviours such as jumping, where robustness and efficient energy exchange are critical.

Quasi-direct drive actuation has emerged as a dominant paradigm in modern agile legged robots, particularly for locomotion involving continuous ground contact \cite{apgar2018fast}. By combining high-torque-density motors with low transmission ratios, QDD systems achieve high force bandwidth, backdrivability, and accurate torque control. These properties are highly effective for dynamic interaction and locomotion tasks such as running and trotting. However, for jumping behaviours, the low reduction ratio places stringent demands on the motor and structure during peak loads and impacts. Moreover, because QDD systems typically lack an explicit elastic element, their ability to mechanically store and reuse energy across jump cycles is limited, and energy must instead be supplied and dissipated primarily through the motor and controller.

Series elastic actuation introduces a compliant element between the motor and the load, enabling mechanical energy storage and release, impact mitigation, and force modulation through spring deflection \cite{paine2012new}. These characteristics align naturally with the dynamics of jumping, where energy is accumulated during a charging phase and rapidly released to generate lift-off. By allowing a significant portion of the mechanical energy to be stored elastically, SEA can reduce peak motor power requirements and electrical energy consumption, particularly during repeated or continuous jumping motions.In this thesis, the continuously variable transmission enhanced SEA (C-SEA) is considered a special case of SEA in which an additional gearbox is introduced to modulate the effective transmission ratio. This extension preserves the fundamental energy storage and impact-mitigation benefits of SEA while enabling improved jumping agility. A qualitative comparison of these actuation paradigms is summarised in Table~\ref{tab:actuation_comparison}.

\begin{table}[h]
\centering
\caption{Qualitative comparison of actuation paradigms for dynamic legged robots.}
\vspace{1mm}
\label{tab:actuation_comparison}
\renewcommand{\arraystretch}{1.3}
\begin{tabular}{p{2.5cm} p{4.5cm} p{4.5cm} p{2.5cm}}
\hline
\textbf{Paradigm} & \textbf{Key advantages} & \textbf{Key limitations} & \textbf{Suitability for jumping} \\
\hline
High-ratio geared drive
&
\begin{tabular}{@{}l@{}}
High torque; \\ 
Compact motor sizing; \\ 
Mature technology 
\end{tabular}
&
\begin{tabular}{@{}l@{}}
Low backdrivability; \\ 
High friction; \\ 
No compliance; \\ 
\end{tabular}
&
Poor
\\
\hline
Quasi-direct drive (QDD)
&
\begin{tabular}{@{}l@{}}
High force bandwidth; \\ 
Backdrivable; \\ 
Accurate torque control\\ 
\end{tabular} 
&
\begin{tabular}{@{}l@{}}
Loads borne by motor; \\ 
Limited energy storage; \\ 
High electrical demand\\ 
\end{tabular} 
&
Poor
\\
\hline
SEA / C-SEA
&
\begin{tabular}{@{}l@{}}
Energy modulation; \\ 
Impact mitigation; \\ 
Reduced motor effort\\ 
\end{tabular} 
&
\begin{tabular}{@{}l@{}}
Mechanically complex; \\ 
Lower bandwidth \\ 
\end{tabular} 
&
Well suited
\\
\hline
\end{tabular}
\end{table}


\section{Aerodynamic Robotic balancing} 

Contemporary machines and devices, such as drones, satellites, legged robots, rockets, and underwater vehicles, necessitate active re-orientation and attitude error correction to maintain balance. These machines perform critical tasks, including telecommunications, inspection, cargo delivery, and access to non-human-friendly environments. To fulfil their intended functions, adherence to specific payload requirements and limitations is paramount. Thus, it is essential that additional, yet necessary payloads, such as balancing mechanisms, are minimised.

\subsection{A novel strategy to balance}

Various balancing methods exist for these devices and vehicles, with reaction wheels being the most widely used approach. Employing inertial forces to provide balancing torques, reaction wheels are particularly prevalent in satellite applications, where external forces are challenging to obtain in space \cite{carre1999analysis,ismail2010study}. In fact, the use of reaction wheels on satellites is common enough that there is a report by W. Bialke and E. Hansell in 2017 \cite{bialke2017newly} explaining a systemic reaction wheel failure in currently serving satellites. Even though most studies on said topic focus on the dynamic analysis and mechanical innovations, new ideas for controlling the reaction wheel are also being introduced. For example, a fault tolerant control for satellites is presented in this paper \cite{jin2008fault}. Fault tolerance means the system is able to maintain performance and stability even with unexpected faults in the system. By using the proposed dynamic model and time-delay control, the attitude error of the satellite can be reduced to $10\%$ of the original value in around $110$ seconds.

In robotics, reaction wheels have also been utilised for balancing and re-orientation purposes. Research in this area frequently employs dynamic models based on the inverted pendulum \cite{meyer2009design,bartuccelli2002stability,reist2010simulation}, with a primary focus on control. One such study~\cite{featherstone2017simple} introduces a dynamic model for balancing robots on a point-in-plane and demonstrates an example robot that utilises a reaction wheel-style crossbar for balance. The Cubli \cite{hofer2023one}, a cube-shaped robot with three reaction wheels controlling each axis, is capable of jumping up and balancing on a corner. Another popular method for robotic balancing is utilising propellers facing a perpendicular plane thus generating torque from their thrust. Notable examples include Salto~\cite{haldane2017repetitive}, the slackliner robot~\cite{kim2021bipedal}, and LAWCDR~\cite{huang2022land}. Although these solutions offer some advantages, they suffer from the additional mass and structural complexity that could be otherwise reserved for their primary functions. Furthermore, unlike satellites, terrestrial robots have ample means to generate external forces for balance, and therefore, using inertial balancing as the default option may not be well justified.

Inspiration for alternative balancing mechanisms can be found in nature, where animals utilise their tails for balance during activities such as jumping and running due to evolutionary pressures \cite{shield2021tails}. Researchers have investigated the balancing role of tails and appendages in animal behaviours, including running, jumping, and self-righting \cite{young2021stabilizing,schwaner2021future}. Others have developed dynamic models and control systems to integrate biomimetic tail mechanisms into robots \cite{briggs2012tails}. A study by \cite{siddall2021mechanisms} found that an increased tail-to-body length ratio in geckos results in a significant rise in turning rate, but the benefits of a longer tail rapidly diminish with a larger ratio. An increase of tail-to-body length ratio from $0.93$ to $1.8$ results in a $50\%$ rise in turning rate. However, the benefits of longer tail diminishes rapidly when the length ratio is above $2$.Thus, animal tails can be used effectively in a reaction-wheel manner, albeit with limitations.

Certain animal tails possess surface properties, typically due to fur or feathers, which enable them to induce aerodynamic drag when moving. The Cheetah's tail inspires a simplified tail device that swings in a conical motion during turns \cite{patel2015conical}, achieving $70\%$ more lateral acceleration with a tail than without one. Wind tunnel tests quantify the aerodynamic forces on the tail, revealing that the fur nearly doubles the effective frontal area \cite{patel2016quasi}. The aerodynamic effects contribute to the angular impulse imparted by the tail onto the cheetah’s body. In these experiments, aerodynamic torque surpasses inertial torque as the tail accelerates at around $0.05$ s, albeit with diminishing increments thereafter. The results show that $26\%$ of the pitch angular impulse is induced aerodynamically even though the tail only takes $2.5\%$ of the body mass \cite{fisher2015state}. A study by \cite{fukushima2021inertial} demonstrates that a squirrel's bushy tail, despite comprising just $3\%$ of the body mass, has a projected area that can reach around $55\%$ of the body size in a side view during jumping. The aerodynamic torque produced by the tail is smaller but comparable to the inertial torque.

These animal-related studies suggest that although animals can balance using their tails in a purely inertial manner, tails with aerodynamic properties significantly enhance the generated righting torque. In some cases, the aerodynamic torque induced by the tail even exceeds that produced by its inertia, suggesting that a tail's torque-generating capability could be derived entirely from its aerodynamic properties, rather than its mass. Consequently, it is theoretically possible to design a mass-less robot-balancing tail that relies solely on its aerodynamic nature to generate righting torques, thus addressing the problem of unjustified additional payload from reaction wheel mass.

\subsection{Aerodynamic drag maximisation} \label{sec_intro}

Balancing and re-orientation are integral functions for a multitude of machines and devices currently in operation. Various approaches exist for these functions, with the reaction wheel being the most prevalent. This mechanism utilises inertial forces to generate balancing torques. Contemporary studies often showcase robots maintaining equilibrium through reaction wheels; for instance, the Cubli \cite{hofer2023one}--a cube-shaped robot--can jump and balance on a corner, thanks to three reaction wheels that control each axis. Interestingly, this method is mirrored in nature. Animals, particularly those adept at jumping and running, often exploit the inertial properties of their tails for balance and re-orientation, a trait developed through evolutionary pressure \cite{shield2021tails}. A notable example is geckos, where a larger tail-to-body length ratio correlates with improved turning rates \cite{siddall2021mechanisms}. Consequently, biomimetic research endeavours to transplant these animalistic traits into robotic systems \cite{briggs2012tails}.

In certain animals, tails induce aerodynamic drag, attributed to features like feathers or fur. For instance, the fur on a cheetah's tail significantly increases its frontal area, enhancing aerodynamic drag \cite{patel2016quasi, patel2015conical}. This drag force contributes to the angular impulse of the tail, thereby assisting in lateral acceleration during turns. Similarly, squirrels' bushy tails exhibit remarkable aerodynamic properties, with the drag-induced torque rivalling that generated by inertia \cite{fukushima2021inertial}. These insights suggest the feasibility of a purely aerodynamic, mass-less tail device for robot balancing, circumventing the added payload issue inherent in reaction wheels. This notion inspired our preceding work, AeroTail, a simulated investigation into a tail device that achieves balance aerodynamically on a point-in-plane, employing a dynamic model akin to the inverted pendulum \cite{sun2023aerotail}. In this study, the drag-inducing elements of the AeroTail are termed ``drag-inducers" (DIs). However, the specific design of these DIs was not finalised. This leads to the pivotal question: what constitutes the optimal design for a drag-inducing device in an aerodynamically balanced tail?

Traditionally, aerodynamic drag is calculated using the drag equation \cite{hoerner1965practical}:
\begin{equation}\label{aero_drag}
    D =\frac{1}{2}\rho C_{d}v^2A
\end{equation}
where $D$ represents the drag force, $\rho$ is the fluid density, $C_{d}$ denotes the drag coefficient, $v$ signifies the flow velocity and $A$ is the object's effective area. Within this equation, the drag coefficient and the effective area are specific to the object, or in this context, the drag-inducing device. The DI's effective area is constrained by its surroundings or the overarching geometry of the device to which it is attached, while the drag coefficient is shaped by the DI's form and can vary considerably. Hoerner's seminal work on fluid dynamic drag includes wind tunnel experiments for numerous standard shapes \cite{hoerner1965practical}. These experiments provided data for both 3D shapes and 2D profiles sandwiched between wind tunnel walls. Remarkably, a hollow semi-sphere facing the wind yields an impressive drag coefficient, a design employed in parachutes and anemometers. The hollow semi-cylinder also exhibits a high $C_{d}$ of $2.3$, the highest among all tested shapes, and has been utilized in robotic self-righting devices \cite{norby2021enabling}, demonstrating a more practical $C_{d}$ of $2.0$. A further study \cite{aziz2008online} corroborated these results and extended the range of shapes examined.

However, these solutions are not exhaustive. The aforementioned wind tunnel tests predominantly focused on common, intuitive shapes, lacking a numerical framework for quantifying and optimizing shape features in specific scenarios. Numerous studies have sought to address this by developing mathematical models for irregular shapes \cite{roostaee2022developing}, aiming to determine their drag coefficients. While most research concentrates on minimizing drag \cite{abolpour2022novel} for applications like vehicles and aircraft, others have devised analytical solutions for irregular, sphere-like shapes \cite{michaelides2023drag}. However, these methods have limited applicability for cup-like shapes. Additionally, the wind tunnel results, typically from controlled environments, may not accurately represent real-world performance. For example, the 2D ``prism" shapes in the sources \cite{hoerner1965practical, aziz2008online} (detailed comparison in \tabref{tab1}) yielded unrealistic drag coefficients achievable only by infinitely long objects, a limitation absent in practical scenarios. Therefore, it's prudent to undertake computational fluid dynamic simulations on specific shapes to ascertain the most effective design for the DI. Moreover, optimizing both the highest $C_{d}$ and the effective area of the DI are crucial and distinct aspects of this problem, each significantly affecting the DI's induced drag. Hence, a methodology to maximize the DI's frontal area within given constraints is also essential.


\section{Control strategies for jumping robots}

Among the aforementioned example of past robot designs, many have the ability to jump but they do not necessarily use jumping as their primary method of locomotion. This could be due to their inability to achieve continuous jumping, or simply, the fact that walking or running align more with their design criteria. However, even being visually different, there are intrinsic similarities between the dynamics of walking, running, and jumping so much so researcher and engineers often use the same model to describe them. The spring-loaded inverted pendulum model (SLIP) was first introduced in $1989$ by R.Blickhan to model the dynamics of running and jumping behaviours in animals \cite{blickhan1989spring}. He observes that animals’ centre of mass during running behaves like a point mass bouncing on a spring leg; the leg compliance leads to energy exchange between kinetic and potential forms. This provides a simple, predictive template for analysing gait mechanics for running and jumping. \cite{geyer2006compliant} confirms the similarities in the energy exchange patterns of running and walking and provides biomechanical justifications for SLIP-based robot designs. 


Various past robot design have incorporated the SLIP model in their control strategies. \cite{raibert1984experiments} introduced a 3D hopper which is able to achieve controlled locomotion on a surface plane without physical support. This robot also utilises a simple controller that enables it to achieve horizontal jumping velocity control \cite{schwind1995control}. Some designs also adds an extra torso joint between the leg and the body, creating a mismatch between the centre of mass of the body and root of the leg; and this helps the robot to balance in the flight phase. For example, in \cite{poulakakis2007formal} and \cite{poulakakis2009spring}, the authors developed a two-level control method: the continuous level for torso stabilization and an event-based level for step-to-step locomotion. This method bridges the non-linear control theory framework with Raibert’s empirical controller. 

Other methods are also used for the balancing and orientation control for the robot's flight phase. For example, \cite{yim2018precision} introduced the Salto robot which utilises a reaction wheel directly acting on the centre of mass of the robot body to provide control torque; no torse joint is needed in this case. They developed a deadbeat foot placement controller for the robot and they concluded that the attitude error is the main limiting factor to the precision of high robotic jumping. Similarly, authors of \cite{hoseinifard2024standing} use two arm-like mechanism to maintain the balance of their robot's flight phase. The principle and dynamics of the arms are identical to reaction wheels and only difference is that they have smaller range of motion. This robot implemented a hybrid reinforcement Learning (DDPG) and PD control method which is more robust than traditional strategies. 

Some works even include the torque element in the stance phase control, directly implementing a reaction wheel (flywheel) into the SLIP model itself. For example, \cite{xiong2020sequential} introduces the Flywheel-SLIP model which see the flywheel also actively generating torque on during the stance phase; thus, both the vertical and rotational momentum are taking to account in their stance dynamics. Similarly, authors of \cite{hamzaccebi2015enlarging} propose the torque-enhanced active SLIP by adding a hip torque actuator. This work also develops a partial feedback linearization method that allows closed-form stance solutions and greatly expands the stability region of apex return map. However, this model is not without limitations. For one, the stance phase is relatively short compared to the flight when high jumps are achieved meaning the flywheel has a smaller window to take affect; also, the motion of the robot body is way more susceptible to gravity on the ground which could pose high requirement to the torque actuator's capacity. Therefore, working during the stance phase could greatly undermines the effectiveness of the flywheel when compare to its ``natural habitat", the flight phase. 

Past robot designs have expanded the reaction wheel method into 3D version and we often see robots that use $3$ reaction wheels placed diagonally to each other to achieve 3D jumping locomotion. For example, the REx hopper \cite{bokserrex} and the ARCHER robot \cite{ambrose2022creating} use exactly that configuration. The former design also implements model predictive control with a lumped-mass model and a hybrid floating-body formulation. This is the most popular way of achieving 3D locomotion for monopedal jumping robots. However, the obvious drawback is the added weight from the reaction wheels. Since they require inertia to function, one can already significantly increase the overall mass of the robot, let alone three. 

Many researcher also try to improve or optimise the SLIP model itself for either faster calculations or more versatile controlled task options. For instance, \cite{shen2016piecewise} and \cite{kilic2023novel} both made attempts to simplify the SLIP model by introducing approximation methods that are faster than the regular numerical integration. The former one develops a piecewise-linear, time-invariant SLIP approximation that provides closed-form solutions for stance and stride maps; the latter one introduces a sine-wave approximation for horizontal ground reaction forces and is able to handle large, asymmetric steps, which is difficult for regular passive SLIP. Researches have also been done to expand the theoretical versatility and generality of SLIP-based control. \cite{han20223d} extends the originally planar SLIP model into a 3D version and proposed a 3D hybrid feedback control methods for quadrupeds. \cite{lu2024generalized} defines generalised SLIP which is parametrised to cover many SLIP variants including rolling foot, two-segment, and torque-driven; it can be used as a robot leg morphology design guideline. \cite{chen2020underactuated} takes a step further and introduces the Wheeled-SLIP model which added wheel dynamics to the end of robot legs. It demonstrates a LQR-based task-space whole-body controller with disturbance observers.


\section{Gaps in the current literature and conclusion}

The many research results being covered above look promising in their own right, but there are much to discuss and explore and find the gaps for this research to fit in. Jumping agility is a new concept that brings the bioinspired robots and their organism counterparts closer. The common practice for any bionic engineering is to only consider a limited aspect of inspiration, which is the reason that most jumping robot only focus on one metric. With the concept of jumping agility combining both height and frequency, the robots are practically competing on the same level with natural organisms. In order to maximise this metric, energy storage branch using elastic elements in a jumping mechanism becomes the logical solution; even though having the storing and releasing processes to be present in a single stroke is still novel. This design needs to be enabled by the power modulation feature, which describes when a mechanism has a mechanical advantage (MA) profile that changes within a single stroke. This is the reason that linkage was chosen as the leg transmission for the Salto robot for its built-in MA profile. However, a changing but predictable mechanical advantage can be realised using other methods. The linkage system, at the same time, is not without its faults. For instance, a linkage with a complex structure capable of straight-line motions for its end-effector is only approximating the trajectory; in other words, they are hardly true straight lines. This alone would put emphasis on excessive calculations and simulations that can be otherwise avoided. 

For aerodynamic robotic balancing, we found that although animals can balance using their tails in a purely inertial manner, tails with aerodynamic properties significantly enhance the generated righting torque. In some cases, the aerodynamic torque induced by the tail even exceeds that produced by its inertia, suggesting that a tail's torque-generating capability could be derived entirely from its aerodynamic properties, rather than its mass. Past robot design that does incorporate aerodynamics usually use tail like structure that has one end (instead of its centre of mass) attached to the body to be balanced, which is structurally very different from the reaction wheel. We have yet to see a balancer that can replace the reaction wheel without significantly changing the robot structure or the dynamic model. Furthermore, alternative aerodynamic solutions like propellers also suffer from the limitations of weight and compactness. Additionally, we found the aerodynamic drag coefficient data from past studies are usually from unrealistic wind tunnel conditions or simulations in less than universal scenarios. They are not useful in terms of determining the best drag-maximising method in a robotic balancing scenario.

In terms of the jumping control strategy, most past jumping robot design utilise SLIP or SLIP-based variants for dynamic model and controller. Even though most of them works well for their intended purpose, there is still room for improvements especially for the leg mechanism structure. Dynamic models used in past robot designs do include a spring element for the leg but most of the them use motor driven joints at the end, only few actually incorporate SEAs in the leg mechanism. Moreover, none of the past designs have integrated gearbox specifically between the spring element and the end-effector meaning in order to control a robot with such leg structure, a new variant of the SLIP must be developed to suit the job. Furthermore, none of the balancer choice, whether it be torso joint, arms, or reaction wheels, utilise aerodynamics as the source of torque input; novel control strategies could be introduced for this application.


In this section, the literature regarding jumping robots, their natural organism counterparts and potential leg mechanisms were reviewed; research were also done to find the best strategies to utilise aerodynamic drag for robotic balancing and to maximise the drag on a single device. It is clear that there are many gaps in current literature where further improvements can be made. This research will try to improve upon the limitations of past designs and ideas; it will also serve as an organic and logical progression of them in order to achieve innovations. 


%The EPFL jumper in seen as a positive example in the previous section but there are many limitations can be exploited and aspects to be improved. Firstly, the gliding ability does not cover a distance and thus cannot be viewed as an actual means of locomotion, but rather a parachute-like method with extended trajectory. Even though the wings of the glider does generate lift, it is so poorly controlled that it cannot maintain a period of level flight with constant speed which is needed to cover a considerable distance. The ease of control brought by the level flight phase retains accuracy that cannot be otherwise provided. A controllable and accurate glide trajectory also improves the ease of landing which is an important quality that a potential inspection machine should possess. Moreover, the EPFL jumper wings lack a proper airfoil profile, which is essential to even enter the constant speed gliding phase. 

%Another gap to consider is the design of foldable wings. Most foldable wings designs are for flight control, and the nature of their folding mechanism can be included under the umbrella term "wing morphing". By altering the structure of wings thus morphing their surface, a shift in the aerodynamic force generated can be achieve. Yet as a jump-glider with two relatively separate functions, the folding function's main purpose is for the wings to be tucked away during jumping. The innovative point about the mechanism should be to fold the wing as compactly as possible (redution in volume or aerodynamic drag, for example, would be valid metrics to consider) in order to reduce unwanted air resistance during jumping.

%The research on the wingtip devices and inspirations from birds have resulted in some finding about the nature of winglets and their functionality. Even though most existing airplanes today have upward pointing winglets, they do not have the best aerodynamic performance even when examined on the exact functions they are designed to fulfil. Downward pointing winglet, which are rare in aircrafts, have better performance in reducing the drag and increasing lift generated by the wings. Thus, using them in gliders seem to be beneficial especially when their sizes are confined by parameters including weight of the device and strength of the materials available. Furthermore, it is extremely rare to see airplane wing design to include both dihedrals and downward pointing winglets. However, there is no evidance so far stating this is not feasible. It could be worthwhile to try and combine the advantages of both downward winglets and dihedral effect onto a single wing design.