I Introduction
Ia Motivation
The ability to fly has given rise to the use of unmanned aerial vehicles (UAVs) in several applications such as surveillance [1], mapping [2], delivery [3], and search and rescue missions [4]. A fundamental limitation of most UAVs is their flight time – they must land when their energy source is depleted. There is a growing demand for higher endurance and range in UAVs with their increasing usage in the research, commercial, and industrial settings.
IB Literature review
Innovative approaches have been explored to increase the flight time of UAVs. Broadly, we can classify the approaches into two types: assisted and unassisted.
Assisted methods typically involve the use of fixed stations or mobile vehicles for the replacement or the charging of energy sources. Battery swapping at a fixed ground station has been presented in [5, 6, 7], and on a mobile ground base has been shown in [8]. Flying replacement batteries to a multirotor using other multirotors is discussed in [9].
Unassisted methods typically involve increasing mechanical or electrical efficiency or the use of optimizationbased methods over objectives such as flight time or range. One such approach is exploiting the efficiency of a fixedwing and hovering ability of a multirotor by converting them into a hybrid aerial vehicle [10, 11]. Vehicle structure manipulation by tilting rotors to increase efficiency is shown in [12]. An online strategy for optimizing efficiency by altering flight parameters (e.g. speed) over a trajectory is presented in [13]. The usage and analysis of a solarpowered UAV, which can potentially fly for a really long time, are discussed in [14]. However, such a vehicle requires a large wingspan, is sensitive to weather conditions, and is not suitable for indoor settings.
An unassisted approach which is largely unexplored for UAVs is the staging of energy sources. A possibility of ejecting depleted energy sources is mentioned in [15], which presents a modular drone consisting of drone cells, which can potentially be staged. Energy sources such as batteries are used widely on UAVs, primarily because they are easy to recharge and reuse, and are nonpolluting, making them a good candidate for indoor settings. Electric powertrains also have fewer moving parts, making maintenance easy. Batteries, however, have a clear disadvantage as opposed to combustion fuels, in the sense that the consumed portion still remains as a mass on the vehicle. This has a significant impact because in the case of electric vehicles, battery mass accounts for a notable fraction of the total mass. We take inspiration from multistage rockets [16, 17, 18] which are designed for missions that would otherwise require a much larger singlestage rocket. Staging reduces the overall mass and reduces the amount of fuel required. We note that rocket staging is about discretely staging the structure, not the fuel.
IC Contributions of this work
In this paper, we explore applications of the energy storage staging concept to multirotors. The specific contributions of this paper with respect to previous work are outlined in the following paragraphs.
In Section II, we present a flight time vs. energy storage mass analysis for multirotor aerial vehicles. We then consider two specific optimal staging design problems for discrete energy storage stages. First, given a certain number of energy source stages of different masses, we prove that the optimal order of using these is from the heaviest first to the lightest last. Second, given an energy source mass budget, we show the optimal way to partition it into a given number of stages to maximize flight time. We also consider the effect on flight time for the case of continuously staged energy sources such as combustion fuels or reaction engines, and conclude that combustion driven propellers have a fundamental limit on flight time that reaction engines do not.
In Section III, we discuss the design of an experimental vehicle (a quadcopter) whose batteries can be staged. Fig. 1 shows the quadcopter ejecting its first stage, and hovering using its second stage.
In Section IV, we predict the flight times from the staging analysis for a singlestage case vs. a twostage case for our quadcopter. We then experimentally demonstrate that by consuming batteries successively in discrete stages and detaching the used stages, an increase in flight time is observed. The predicted and experimental flight times are compared to validate the staging analysis.
We note that discarding batteries in the open can be costly and dangerous as impact to batteries might set them on fire and damage surroundings. Care must be taken to safely land the ejected stages (e.g. by using a parachute) so that they can be reused, and so they do not become a hazard to society.
Ii Staging Analysis
In this section, we present an analysis of how using an energy source in stages is beneficial for the flight time performance of hovering multirotors. A brief review is first given of the actuator disk model for power consumption of a propeller; this model is then applied to quantify the gain in flight time by discarding the used energy storage stages inflight. This is then compared to two cases where an infinite number of stages exist: first when the energy source is the fuel combusted (and then exhausted) in an engine that drives rotors, and secondly where the rotors themselves are replaced by rocket motors.
Iia Rotor power consumption and flight time
For an idealized (“actuator disk”) model of a propeller that is not translating in the ambient air, the aerodynamic power required may be computed [19] as
(1) 
where is the thrust produced, is the density of the air, and is the area swept by the rotor. This derivation assumes that the flow is inviscid, incompressible, and follows from applying conservation of mass and energy to a control volume containing the propeller. Note that translating propellers have more complex relationships, see e.g. [13].
The actual power drawn from the energy source to the actual system will include additional losses, including aerodynamic losses which may be captured in a propeller’s figure of merit [19] as well as losses in transmission of power to the propeller (e.g. gears or electric resistance). We make the simplifying assumption that these losses are all proportional to the power drawn, so that the actual power consumption of a propeller producing thrust may be captured by
(2) 
where the constant is a function of the propeller size, ambient air density, propeller figure of merit, and powertrain efficiency. Note that a similar relationship can be derived from the mechanical power required to drive the propeller shaft under the assumption that the thrust is proportional to the rotational speed of the propeller squared, and that the propeller torque is proportional to the propeller thrust [12].
For a symmetric quadcopter of mass to hover, each propeller must produce a force equal to a quarter of the vehicle’s weight, and the total power consumption is
(3) 
The local acceleration due to gravity is given by . Let be the specific energy of the energy source, so the energy budget for a given energy source mass is . Then, at constant total mass , a vehicle with energy energy source can hover for a flight time of
(4) 
where is a vehiclespecific constant, and is specifically independent of the vehicle mass.
IiB Discrete energy storage stages
A primary disadvantage of batteries for energy storage is that the storage mass does not decrease as the chemical energy is depleted (unlike, for example, a combustion engine). From (3), it is clear that the power consumption would be reduced if the total mass of the vehicle could be reduced, for example by ejecting parts of the vehicle’s battery as it is depleted. We will assume that all the energy storage stages have equal specific energy (energy per unit mass).
For a vehicle with battery stages, where the battery has mass , the total mass of the vehicle at stage is given by , where is the dry mass of the vehicle, i.e. mass of all components of the vehicle excluding energy storage (but including payload, etc.). In particular, at stage , the mass has been ejected. The flight time for the stage can be computed from (4) as
(5) 
with total flight time over all stages
(6) 
This equation shows that achievable flight time is directly proportional to the specific energy and depends on the stage masses in a nonlinear fashion.
IiB1 Equal staging
For a vehicle with dry mass , and total energy storage mass split equally over stages, the flight time can be computed from (4) (after some manipulation) as
(7) 
Fig. 2 shows plots of (normalized) flight time vs. the ratio of total energy storage mass () to total initial vehicle mass (), for a different number of stages. The case of equal staging is plotted using solid curves.
The plots in Fig. 2 and the results from equal staging can be used in the following ways. For a given energy storage mass, one can fix the xaxis value, and decide the number of stages they want based on their flight time requirements. On the other hand, the choice for designers might be flight time for their vehicle. In that case, they can fix the yaxis value, and then choose the number of stages based on the mass constraints.
IiB2 Optimal staging order
We consider the case of having a series of energy storage stages, of known, fixed but different masses, with the only design variable being the order in which to stage them. We prove that the optimal flight time is achieved by staging in order of decreasing mass, so that the heaviest stage is depleted and discarded first.
The proof follows by contradiction: assume that an optimal staging sequence is given by where for some value the stage is lighter than the following stage , i.e. . Let represent the total flight time, computed with (6) for this sequence.
Let represent a modified staging order, which only changes the order of the original th stage with th, and let represent the associated total flight time. We note that the first and the stages continuing after are identical. We compare the total flight time. We denote the shorthand as the total mass of the vehicle after discarding both the stages and . Then the difference in flight times is
(8)  
(9) 
where we define for convenience. We note that
(10)  
(11) 
so that we can relate the average slope of as follows
(12)  
(13) 
Rearranging this, and using gives
(14) 
Substituting this in (9) gives . However, this contradicts the assumption that is the optimal flight time, and therefore cannot be an optimal staging sequence. Thus, the optimal staging sequence has , proceeding from the heaviest stage first to lightest stage last.
IiB3 Optimal mass partitioning
Given a dry mass , an energy storage mass budget , and a total number of stages to be used, it is of interest to find the optimal stage masses that add up to , to maximize the flight time.
This is equivalent to maximizing in (6) over the decision variables , under the inequality constraints and the equality constraint .
We define new decision variables, , so that . The objective is defined as,
(15) 
This can then be formulated as a constrained optimization problem as follows,
(16)  
s.t.  (17)  
(18)  
(19) 
Solving the above problem using the KKT conditions [20], we obtain a solution in the form of simultaneous nonlinear equations shown below,
(20)  
(21)  
(22) 
Even for the simplest case of , this requires the roots of a fifthdegree polynomial, for which no closedform expression exists. Nonetheless, the simultaneous equations can be solved numerically. Plots for a few sample values are shown in Fig. 2 as dashed lines to compare with the equally staged case (solid lines). We see that for energy storage mass fraction from , there is little difference (less than 1%) between equal and optimal staging. But for fractions beyond the difference increases to more than 5% for a small number of stages ().
From the discrete staging analysis, we conclude that staging can be beneficial in the following ways. First, for the same amount of energy, we can increase the flight time as compared to a singlestage vehicle. This is to be expected because as the multistage vehicle ejects stages, its mass and consequently power consumption reduces. Since the total energy is the same in both cases, the flight time of the multistage vehicle is higher. Second, to achieve the same flight time, a multistage vehicle will be lighter and more compact, by a similar argument. This makes it safer and more maneuverable. Third, in the case of any multistaged UAV, the inertia reduces with every stage ejected, thereby increasing the agility of the vehicle after each ejection.
IiC Continuous staging
A vehicle powered by combustion, where the combustion products are exhausted, can be described as a limiting case of the above, with continuous staging. We consider two forms of this: using a combustion engine to provide power to rotors, and a reaction (e.g. rocket) engine.
IiC1 Internal combustion engines
When burning (and exhausting) fuel to perform the mechanical work of driving propellers, the remaining mass of the energy storage will evolve at a rate proportional to the power consumption,
(23) 
where we’ve used (3) and again assumed a symmetric quadcopter at hover. Solving this, and substituting , and gives the total flight time as
(24) 
This flight time value can also be derived by taking the limit as in (7).
Note that the flight time is, as may be expected, monotonically increasing in the initial fuel mass . However, there exists a natural upper limit to achievable flight time, even for arbitrarily large quantities of fuel:
(25) 
This reveals a fundamental limit of rotor propulsion. Even under the best case of continuous staging, achievable flight time has an upper limit no matter how much fuel is used. for a specific vehicle (fixed ), energy source (fixed ), fixed environment and fixed dry mass. In practice, other constraints would act to limit achievable performance, e.g. thrust and structural limitations.
We also note that as the dry mass is reduced, the flight time increases, with the limit as
IiC2 Reaction engines
Another method of achieving continuous staging exists is reaction engines (such as rocket motors), where propellant is expelled at high velocity, producing thrust in response to the momentum flux leaving the engine. A simple analysis, assuming that fuel is expelled at a constant exhaust velocity , gives a thrust force as a function of the fuel mass flow rate as
(26) 
For a quadcopterlike rocket ship, with four rocket engines symmetric around the center of mass, we have that
(27) 
Solving this, and setting again , the achievable hover flight time is
(28) 
This result is very closely linked to the Tsiolkovsky rocket equation [21] which describes the achievable velocity change as a function of fuel burnt, with the same functional form.
We note that this achievable flight time will grow unbounded with increasing fuel mass . Thus, rocket propulsion does not suffer from the same fundamental limitation as aerodynamic propulsion. This suggests that, for certain extreme design specifications, it may be preferable to create a rocketpropelled multirotorstyle robot, rather than using aerodynamic propulsion. Notable is also that, for sufficiently large gravity environments, a rocketpropelled system may again beat an aerodynamically propelled system, as appears in the denominator with a smaller exponent.
Iii Experimental Hardware Design
In this section, we explain the design of the quadcopter used in our experiments, the battery staging mechanism, and the battery switching circuit.
Iiia Vehicle design
The quadcopter is designed to have enough payload capacity for carrying useful sensors such as surveillance cameras, or environmental sensors. Its dry mass is , and it can generate a maximum thrust of . Its arm length is , and it uses four diameter propellers. We stack a battery switching circuit on top of the quadcopter for staging. The quadcopter is powered by two batteries, the firststage battery placed at the bottom of the quadcopter that can be ejected when depleted, and the secondstage battery placed at the center of the quadcopter which always stays onboard. Copper plates mounted at the bottom serve as input leads from the firststage battery. Both the first and second batteries are 3S lithium polymer (LiPo) batteries, weighing each. The staging mechanism adds an additional to the first battery. So the quadcopter weighs when carrying only the second battery, and when carrying both batteries. Fig. 1 shows a picture of the quadcopter.
IiiB Staging mechanism
The staging mechanism is shown in Fig. 3. The mechanism is placed on the lower section of the quadcopter frame. In order for the firststage battery to engage with the mechanism, we have an attachment on one end of the battery. This attachment has a bolt, and two springloaded contacts, which are electrically connected to the output leads of the firststage battery. On the side of the quadcopter frame, a ”detaching” motor is installed, which has a nut fixed on its shaft. Next to the motor are two copper plates that serve as input leads from the first battery. Including all the components, the staging mechanism adds mass to the quadcopter.
To connect the first battery, we engage the bolt from the battery with the nut on the motor shaft. This connects the springloaded contacts to the input leads. To disconnect the battery, we command the motor to unscrew the bolt. When the bolt is completely disengaged from the nut, the battery loses support and thus drops.
IiiC Battery switching circuit
The design of our battery switching circuit is inspired by [9]. Since our system is flying, we cannot afford to cut the power supply when switching from the first battery to the second battery. The two batteries need to be connected in parallel for some time to achieve this. We connect diodes in series with each of the batteries to avoid reverse currents due to the voltage difference between the batteries.
A normally closed relay is connected in series with the first battery. By opening the switch, we can draw power from the first battery even when it is at a lower voltage than the second battery. This is necessary to completely consume the first battery, before starting the use of the second battery. The relay coil is connected across the first battery input leads in series with a MOSFET. The gate terminal of the MOSFET is connected to a GPIO pin on the flight controller to control the relay. Fig. 4 shows a schematic diagram of the battery switching circuit. The circuit weighs .
Iv Experimental Validation
We validate the staging analysis by first predicting the flight times for our quadcopter with and without staging. We then conducting hovering flight experiments for singlestage and twostage cases. The flight times in the two cases are compared to each other to show that flight time for the multistage case is higher. The experimental flight times are also compared with the predictions from the analysis and are shown to match well, validating the analysis.
Iva Flight time prediction from analysis
We use the results from equal staging (Section IIB1) since we use the same battery for both stages. From the datasheet of the batteries that we use, the specific energy value is . The value of is determined empirically from flight experiments on the quadcopter, which is . Using Eq. (7) with as dry mass and as total battery mass, we predict flight times of for the singlestage case, and for the twostage case.
IvB Experimental setup
The quadcopter used in our experiments is localized via sensor fusion of a motion capture system and an onboard rate gyroscope [22]. Experimental data from the motion capture system, voltage sensor, and the current sensor is logged via radio for postprocessing. We control the quadcopter using a cascaded position and attitude controller shown in Fig. 5.
IvC Demonstration
The benefit of staging batteries is demonstrated by conducting two types of experiments: one without staging (singlestage), and the other with staging (two stages). The flight experiments can be seen in the video attachment.
In the singlestage experiments, the quadcopter hovers using the two batteries until both the batteries are completely discharged. A battery is considered completely discharged when it reaches per cell. We do two experiments in this category:

Both batteries are consumed simultaneously.

First and second batteries are consumed successively.
The quadcopter hovered for with simultaneous consumption, and with successive consumption.
In the twostage experiment, the quadcopter initially hovers using the first battery. Once the first battery is discharged, the second battery is connected to the quadcopter by closing the relay in the switching circuit. The first stage is then detached by activating the detaching motor in the staging mechanism. Hovering continues until we completely consume the second battery. Fig. 7 shows a sequence of snapshots from the twostage experiment. The quadcopter hovered for a total of in this experiment.
IvD Discussion
The plots of input voltage and power vs. time for the experiments are shown in Fig. 6. We see the characteristic LiPo battery discharge curve [23] in each of the voltage vs. time plots.
We compare the singlestage successive consumption experiment with the twostage experiment. The first discharge curve shows the complete consumption of the first battery, and the next one shows the complete consumption of the second battery. Since the quadcopter has the same total mass in both experiments when consuming the first battery, we expect that this discharge would take approximately the same time in both cases. The voltage vs. time plot shows that this is indeed the case. However, during the consumption of the second battery, the quadcopter has different masses in the two experiments. In the twostage experiment where we detach the first battery, the quadcopter mass is lower. We expect the power consumption to be lower and consequently the flight time to be higher for this case. The voltage and power vs. time plot confirms that this is true.
The power vs. time plot shows that power input to the quadcopter remains approximately constant to hover continuously, as long as its mass is not changing. For the singlestage experiments, we see that this is true for the entire duration of the experiment. For the twostage experiment, we see that the power input remains constant until the first battery is completely discharged. Once we detach the first stage, the power consumption settles to a lower value needed to hover at the reduced mass.
An important observation here is energy consumption (calculated from the power vs. time plot) is approximately the same for all experiments. This is to be expected since we have the same total energy in all cases. The energy consumption in the three experiments (singlestage simultaneous, singlestage successive, and twostage) respectively are , , and . We note that the values mentioned here are the output energy utilized by the quadcopter. In the singlestage successive consumption experiment and twostage experiment, since we are using one battery at a time, the effective internal resistance is the same, which results in similar electrical efficiency. In the singlestage simultaneous consumption experiment however, we are using two batteries in parallel which, in essence, halves the internal resistance. This increases the electrical efficiency, and thus the quadcopter is able to utilize more output energy for the same input energy. Consequently, we get a slightly higher flight time as compared to the singlestage successive consumption experiment. To be fair in comparing staged and unstaged cases, we use the higher flight time () for the singlestage case.
We see that the flight time predictions match well with the experimental values, with less than 2% difference. Part of the reason for this accuracy is the empirical determination of the parameters. However, it does validate that the assumptions used in the staging analysis are, in fact, realistic.
In addition, we compute the battery mass required for a singlestage vehicle to give us the same flight time as is obtained with two stages. Using Eq. (7) with , , , and and same as before, we calculate . So a singlestage vehicle would need an additional of battery to achieve the same flight time as a twostage vehicle. Even considering the additional mass added by the staging mechanism and the switching circuit, we save or about 26% of the dry mass by using just one additional stage.
We list a few shortcomings of staging energy sources to wrap this section up. First, since we are ejecting energy sources inflight which are potentially hazardous, care needs to be taken to land them safely to avoid polluting the environment, and to avoid any damage to the energy source or its surroundings. Secondly, these energy sources are reusable and it would be a waste to simply discard the ejected stages. So careful planning is necessary for the recovery of these ejected stages. Lastly, we see that adding a stage requires the design of a mechanism to eject that stage, which adds to the mass and complexity of the quadcopter. Although the ejection mechanism does involve detailed design, we saw in the last paragraph that the additional mass of the mechanism plus the circuit is well compensated by the flight time and mass benefits of staging.
V Conclusion and future work
Va Conclusion
In this paper, we have introduced the concept of staging energy sources for UAVs, specifically considering multirotors for our analysis. The idea is to discard the energy sources that cannot supply power anymore. This reduces the mass of the vehicle, thereby reducing power consumption. This, in turn, increases the overall flight time of the vehicle.
We presented a model to predict the flight time of a multistage multirotor with given physical parameters related to the energy source and power consumption. We then analyzed two specific cases of optimal staging. First, given energy storage stages of fixed, known masses, we proved that flight time can be maximized by staging them in order of decreasing mass, with the heaviest stage being depleted and ejected first. Second, given an energy storage mass budget, finding the optimal way of partitioning it into stages to maximize the flight time. This problem does not have a closedform solution. Numerical results were presented and compared with the equal staging case. From the discrete staging analysis, we conclude that the choice of number of stages matters more than optimally partitioning the stages.
We also presented an analysis for continuous staging for two cases: internal combustion engines driving propellers, and reaction engines. We observed that there is a fundamental limit on flight time when using I.C. engines, even when using an unlimited amount of fuel, which does not exist for reaction engines. We also noticed that the dependence on gravity was different for the two engines. So in particular cases, e.g. higher gravity environments (other planets or moons), reaction engines could be a preferred choice over aerodynamic propulsion for longer flight times.
We designed a twostage quadcopter with a staging mechanism to detach a used battery inflight, and conducted hovering flight experiments for two cases: singlestage and twostage. Flight times were predicted from the analysis for the singlestage and twostage case, with the twostage case showing a 20% higher flight time for the same battery mass. Experimental flight times matched the predictions well (within 2%) with observed flight times of for the singlestage case and for the twostage case. This validates the staging analysis
We note that the potential gains of using more stages are even higher, with certain flight times not achievable by a singlestage vehicle. Since higher flight times can be achieved via staging without adding as much battery mass as required for a single stage, usually none or minimal changes to the vehicle’s structure and powertrain would be required. Moreover, the inertia of the vehicle reduces with every battery ejected, thereby increasing its agility after each ejection.
VB Limitations
Limitations of the staging approach include the requirement of careful ejection, landing, and recovery of the energy source, and intricate design of the staging mechanism. This is to minimize damage to the energy source as well as its surroundings. We saw that the shortcomings were compensated by the flight time benefit by comparing masses of singlestage and twostage vehicles that have similar flight times.
VC Extensions
Future work may include the use of additional stages to further increase in flight time, or to decrease mass for the same flight time as the lower stage versions. This would entail a more intricate design for ejection of each stage individually and seamless switching between the usage of these energy sources. Another extension could be exploring the design and control of a multirotor using reaction engines as thrusters. This would require careful design to have actuation in the desired degrees of freedom. A third extension could be the planning of locations for stage ejections in an urban setting, so that collection of these ejected stages can be localized and potentially automated.
Acknowledgment
We gratefully acknowledge financial support from NAVER LABS and Code42 Air.
The experimental testbed at the HiPeRLab is the result of contributions of many people, a full list of which can be found at hiperlab.berkeley.edu/members/.
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