## Abstract

This work describes a design optimization framework for a rolling-flying vehicle consisting of a conventional quadrotor configuration with passive wheels. For a baseline comparison, the optimization approach is also applied for a conventional (flight-only) quadrotor. Pareto-optimal vehicles with maximum range and minimum size are created using a hybrid multi-objective genetic algorithm in conjunction with multi-physics system models. A low Reynolds number blade element momentum theory aerodynamic model is used with a brushless DC motor model, a terramechanics model, and a vehicle dynamics model to simulate the vehicle range under any operating angle-of-attack and forward velocity. To understand the tradeoff between vehicle size and operating range, variations in Pareto-optimal designs are presented as functions of vehicle size. A sensitivity analysis is used to better understand the impact of deviating from the optimal vehicle design variables. This work builds on current approaches in quadrotor optimization by leveraging a variety of models and formulations from the literature and demonstrating the implementation of various design constraints. It also improves upon current ad hoc rolling-flying vehicle designs created in previous studies. Results show the importance of accounting for oft-neglected component constraints in the design of high-range quadrotor vehicles. The optimal vehicle mechanical configuration is shown to be independent of operating point, stressing the importance of a well-matched, optimized propulsion system. By emphasizing key constraints that affect the maximum and nominal vehicle operating points, an optimization framework is constructed that can be used for rolling-flying vehicles and conventional multi-rotors.

## Introduction

Mobile robots promise advancements in many fields, with recent research focusing on exploration and search and rescue capabilities [1]. Commercially, mobile robots offer avenues for revolutionizing delivery [2], inspection [3], surveillance, and emergency response [4]. Despite rapid advancements in capabilities and applications, improvements in power management are needed to expand the operating time and range of mobile robots. Achieving these improvements requires addressing the fundamental energetic costs and tradeoffs inherent to robot locomotion modalities.

Traditionally, mobile robots have relied upon a single locomotion modality, such as rolling, flying, walking, or swimming. Each modality can be further subdivided; flight can be achieved using fixed wings, flapping wings, lighter-than-air structures, rotary wings, or combinations of these configurations. Advantages and limitations are inherent to each mode and configuration. For example, fixed wing flight is less maneuverable than rotary wing flight but is better suited for covering long distances at high speeds. To address the fundamental limitations of unimodal locomotion, robots capable of multi-modal transportation are being developed. These vehicles are designed with complementary modes to better operate in multiple complex environments. To this end, vehicles capable of flying-crawling [5–7], flying-swimming [8,9], and rolling-flying-swimming [10] have been developed. Some of these vehicles rely on transforming or reconfigurable mechanisms [11]. Others have a fixed configuration and rely on sharing actuators to achieve bimodal locomotion [12,13].

Mobility and energy efficiency are of particular importance for exploratory robots in unstructured environments. A mobile robot must be able to traverse rough or evolving terrain for long distances and durations. The vehicle must also be maneuverable and operate at a variety of speeds: lower speeds for high-resolution data collection and higher speeds for enhanced deployment. Currently, multi-rotor flying vehicles are very mobile, as indicated by their six-dimensional configuration space. However, as evidenced by their high cost-of-transport [14], multi-rotor flying vehicles are not particularly energy efficient due to the constant high-power consumption required to stay aloft. Commercially available quadrotors, such as the DJI Mavic Pro, have a maximum 30 min operating time at steady operating conditions. Researchers have proposed methods of improving unimodal operating endurance; for example, a 15% reduction in power consumption was obtained by using a single large rotor to provide lift with smaller rotors providing maneuverability [15]. Alternatively, others have focused on optimizing specific components [16] to improve subsystem efficiency and reduce vehicle mass.

In contrast to flying vehicles, rolling vehicles tend to have very low cost-of-transport [14], but the tradeoff is decreased mobility. To leverage the low cost-of-transport of a rolling vehicle, while maintaining the mobility of a flying vehicle, researchers have developed several rolling-flying vehicles (RFVs) [10,11,17–19], including the initial micro-aerial-vehicle (MAV) scale, rotor-propelled hybrid terrestrial/aerial quadrotor (HyTAQ) invented by Kalantari and Spenko [20]. The RFV configuration under consideration here consists of a quadrotor suspended between two passive wheels, which is one of the HyTAQ variants patented by Kalantari and Spenko [21]. This vehicle is capable of flight in the same fashion as a conventional quadrotor, but can also roll on its two wheels, with the propulsive force supplied by its propellers. Appropriately combining these modalities can produce a bimodal vehicle capable of energy-efficient rolling under normal operation, but with the ability to fly when necessitated by the environment or task at hand [22]. If successfully executed, such a configuration offers high mobility and maneuverability along with an energy-efficient locomotion capability. The energetic analysis and power minimization of such a vehicle have been considered previously [1,22], where it was shown that in contrast to conventional quadrotors in flight, the RFV's angle-of-attack and forward velocity are independent of one another during rolling operation; angle-of-attack can, therefore, be used to minimize power consumption during rolling transit. This power minimization capability is a key difference in comparison with other rolling-flying bimodal approaches. Simulations reveal that the power consumption is dependent upon complex interactions between aerodynamics, electromechanics, terramechanics, and rigid body dynamics [1].

The RFV prototype design detailed in Ref. [1] was somewhat ad hoc, with components and parameters selected based on availability and approximate sizing rules. This paper formalizes a detailed design process involving modeling, parameterization, and simulation, using a framework that is general enough to apply to both conventional quadrotors and RFVs. The design process for an RFV differs from that of a conventional quadrotor because the RFV must operate over a large angle-of-attack range and at operating conditions that vary from a conventional quadrotor's nominal near-hover state. Optimal quadrotor design has a variety of approaches, depending on the design goal and subsequent formulation. Many approaches iterate through a component database until a design meets some constraint, or use heuristic optimization [23–27], often taking spatial constraints into account. Alternative approaches parameterize components using statistical regression [28–31] and are able to predict off-the-shelf vehicle mass to within ±5% [31] and flight time to within ±5.4% [29]. Others, instead of parameterizing and optimizing the vehicle, focus on modeling the propulsion system [32–34]. Finally, Internet-based tools for performance estimation also exist.^{2} The approach taken here is most similar to Refs. [35,36], where the component mass and parameter correlations are used as in Refs. [28,29,31] and are combined with first-principles models. However, key differences exist. Because of the desire for energetically efficient performance at nominal operating conditions that vary from those of conventional multi-rotors, the first-principles models in this paper must be valid for a large angle-of-attack range and formulated in a manner conducive to RFV parameterization. matlab simulations utilize the parameterized models to evaluate a design's range and size, allowing a multi-objective genetic algorithm (MOGA) to create Pareto frontiers of maximum-range and minimum-size RFVs. The Pareto-optimal designs are used to better understand design variable relationships, demonstrate the importance of constraints, and explore subsystem interactions.

This paper first details the multi-physics models and system parameterization. Next, the MOGA implementation is described, and the resulting design trends are investigated. A comparison of rolling-optimized and flying-optimized RFVs is presented alongside conventional quadrotors, with key differences noted. A case study for an optimized design lends further insight into a well-matched propulsion system. Finally, the sensitivity of the vehicle range to changes in design variables is investigated.

## Rolling-Flying Vehicle Modeling and Parameterization

There are numerous approaches to quadrotor modeling and design, generally differing based on the intended application. For example, designing for maximum thrust-to-weight ratio, maximum flight time, or minimum size is only a small subset of potentially useful design objectives. Depending on the specific application, the design process, parameterization, and optimization are formulated differently. The modeling approach taken here has similarities and differences to other approaches in the literature, so that the vehicle can be parameterized in a way that makes the simulation and comparison of optimal quadrotors and RFVs tractable. For example, many quadrotor design algorithms implement different hover-based momentum theory or blade element momentum theory (BEMT) solutions from the helicopter literature, which inherently assume propellers undergoing small deviations from a nominal horizontal rotor plane. However, because the RFV must operate with a comparatively large angle-of-attack range and at many different operating points, the BEMT implementation offers a general formulation that allows for oblique flow, as in Refs. [1,37].

This section formulates the modeling and parameterization of the RFV such that a heuristic design tool, in this case a multi-objective genetic algorithm, can be utilized to create optimal designs. To this end, the RFV is conceptualized as a multi-rotor with attached wheels, where the quadrotor pitch can be controlled independently of the wheel motion. To perform the detailed design of the vehicle, first the free body diagram and mechanics model for the RFV and a conventional quadrotor are described. Next, the independent physical dimensions and component measures are parameterized for use in simulation. The key vehicle subsystems for parameterization are the vehicle geometric model, brushless DC motor model, propeller model, battery model, and vehicle mass model.

### Free Body Diagram and Terramechanics.

*α*, flying up an incline with constant climb angle,

_{P}*θ*, as shown in Fig. 1(a). Forces acting on the quadrotor include the gravitational force,

*W*, parasitic drag,

*F*, net propeller thrust,

_{D}*T*, and net propeller in-plane force,

_{sys}*F*(i.e., the force acting normal to the thrust vector in the propeller plane, caused by the differing airspeeds acting on the advancing and retreating blades of the propeller). At a given steady-state velocity,

_{H,sys}*v*, and incline angle, the quadrotor's angle-of-attack and net thrust force are entirely prescribed because the thrust force must maintain equilibrium with the vehicle weight and drag. However, when considering the RFV shown in Fig. 1(b), the vehicle weight is at least partially offset by the ground normal force,

*F*, allowing the vehicle angle-of-attack and thrust to be controlled independently. This allows for the computation of an optimal angle-of-attack, where the vehicle thrust serves to propel the vehicle and to partially unload the wheels, therefore reducing rolling resistance losses [22]. The RFV mechanics model used here is nearly identical to that developed in Ref. [1], where the rolling resistance is conceptualized as the vehicle continuously rolling up a small step [38], and where vehicle velocity, incline angle, and the step height are assumed to be operating condition parameters. The rolling resistance,

_{N}*F*, is

_{R}*β*is a nondimensional terrain parameter dictated by the wheel diameter,

*D*, and the effective terrain step size,

*x*. For the rolling case when

_{R}*α*> 0, the propeller thrust serves to decrease the normal force and thus reduce the rolling resistance of the vehicle. The parasitic drag force scales with airframe planform area,

_{P}*A*

_{0}, angle-of-attack, and velocity, and is computed as

*ρ*is the air density and

*F*is the dynamic-pressure-normalized drag force obtained using experimental data from Ref. [39]. Power consumption is related to the required thrust using an oblique flow BEMT aerodynamic model and an electromechanics model. The in-plane propeller force is also computed using the BEMT aerodynamic model. The RFV energetic modeling and power minimization are detailed in Ref. [1].

_{q}(A_{0}, α_{P})The propeller-driven vehicle configuration is chosen over a direct-drive vehicle for several reasons. Although a direct-drive approach may offer energetic benefits because the drive motor and gearing can match the load to an efficient drive motor operation, the RFV can also operate efficiently by using a portion of its thrust to unload the wheels, thus reducing rolling resistance and reducing required power [1]. The direct-drive configuration also inherently forces a discrete roll/fly decision to be made. In contrast, the propulsion-driven vehicle removes the requirement of a discrete roll/fly decision by changing its angle-of-attack such that power consumption is minimized, as described in Ref. [1]. Finally, whereas the RFV is towed by the rotors, a direct-drive approach is traction driven and, therefore, subject to terrain limitations due to wheel slip. A two-wheeled direct-drive configuration is analogous to a pendulum-driven robot, such as the GroundBot [40]. Due to the mechanics of using a displaced pendulous mass in traction-driven operation, unimodal pendulum-driven robots are limited to traversing a maximum slope of approximately 30 deg with limited acceleration. The two-wheeled direct-drive configuration requires two additional motors and a distally located center of mass for optimal operation. A four-wheeled direct-drive vehicle might be simpler to control than the RFV but will generally require a mobility-reducing tank-drive steering, or a steering linkage with differential. Furthermore, the four-wheeled direct drive will either require four additional motors, a drivetrain with a single motor, or a transforming mechanism allowing the drive and flight-propulsion motors to be shared. Because the RFV uses the same actuators for flying and rolling, it is expected to have an inherently lower mass than an RFV with a four-wheeled direct-drive or a two-wheeled direct-drive configuration, allowing for improved flight characteristics.

### Vehicle Geometry.

*D*and are spaced one diameter apart. The propeller locations are constrained such that the propeller discs (i.e., the two-dimensional swept area formed by a revolution of the propeller) are contained within the outer dimensions of the wheels. The distance from the center of one propeller to its neighbor,

*s*, can take on a range of values, with a maximum value of

*s*constrained by the propeller disc impinging upon the wheel and a minimum value of

_{max}*s*determined by the propellers impinging upon one another. To avoid adverse aerodynamic performance, the center of each propeller is constrained to be a minimum distance of $2rP$ units away from the edge of the neighboring propeller blade or wheel [41,42], where

_{min}*r*is the radius of the propeller.

_{P}*s*and

_{min}*s*are given by

_{max}*h*, of the propeller disc plane above the wheel axle can similarly take on a range of values, with a minimum value of

*h*equal to the motor height, and a maximum value of

_{min}*h*is computed as

_{max}*x*and

_{s}*x*. The propeller center-to-center distance scale factor,

_{h}*x*, and the propeller plane height scale factor,

_{s}*x*, are used to define the propeller location in terms of their constraints, where

_{h}*b*, is

### Motor Model.

*ω*, is related to the phase voltage,

*V*, the winding resistance,

_{s}*R*, the winding current,

*I*, and the back-EMF constant,

*k*, by

_{e}*τ*, is proportional to the current and the back-EMF constant, such that

*I*is the no-load current associated with overcoming internal friction and losses. The brushless DC motor performance is parameterized using the back-EMF constant, the winding resistance, and the no-load current. Generally, motors of the size and scale typically used for quadrotors are marketed according to their

_{0}*k*rating, such that an unloaded motor will produce a no-load speed approximately equal to the

_{V}*k*rating multiplied by the applied voltage. Assuming nominal units, the

_{V}*k*rating is related to the back-EMF constant by

_{V}*k*is generally specified in rpm/V, whereas

_{V}*k*is specified in V/(rad/s). The electrical power consumed by the motor is

_{e}*k*and

_{V}*R*are used as the independent design variables. Ampatis and Papadopoulos [28] aggregated a data set of maximum torques for a line of MAV scale motors and correlated this data to characteristic motor lengths. The characteristic motor length

*l*is a nonphysical dimension equal to the cube root of motor volume and is correlated to

_{motor}*k*rating and resistance by

_{V}Some approaches in the literature also attempt to relate *k _{V}* and

*R*such that they are dependent on each other; however, this approach over-constrains the motor design. Two motors with identical

*k*ratings can have different winding resistances (and therefore, different dimensions and masses). This is evident when comparing commercially available motors; many sizes of motors exist with identical

_{V}*k*ratings but different resistances.

_{V}*I*, can be computed using the maximum sustainable motor torque,

_{max}*Q*, as parameterized in Ref. [28], where

_{max}The voltage required to produce this current is a function of the torque–speed operating point. For a motor-propeller system, the load is computed by a steady-state propeller model, which is described later in this section.

### Battery Model.

*S*, battery voltage,

*V*, battery mass,

_{batt}*m*, battery capacity,

_{batt}*C*, and stored battery energy,

_{batt}*E*. The battery is assumed to be a Lithium-Polymer (LiPo) type common in MAV designs due to its high specific energy density and high discharge rates. A LiPo cell has a nominal voltage of 3.7 V, so the battery voltage is related to the number of cells by

_{batt}The relationships between battery energy, battery capacity, battery voltage, and mass are shown in Fig. 3. Note that stored battery energy is proportional to battery mass. This steady-state analysis assumes nominal 80% maximum battery depth of discharge (indicating that the usable battery energy is 80% of the value computed in Eq. (20)) and a 3.7 V cell voltage.

### Battery-Motor Subsystem Interaction.

The motor and battery may restrict the vehicle's maximum thrust due to the available, tolerable, or required current and voltage. The subsystems are well-matched when the maximum battery current and voltage balance the motor's required voltage and tolerable current at the maximum operating point. For a given motor mass, many feasible *k _{V}* and

*R*combinations exist; those with lower relative

*k*ratings will require more voltage (but less current) to produce a given torque–speed (

_{V}*τ*–

*ω*) operating point than a pair with higher relative

*k*rating. The battery voltage and discharge current are dictated by the battery mass,

_{V}*S*-rating (i.e., the number of cells in series), and

*C*-rating (i.e., the discharge rating, equal to the battery capacity divided by the maximum discharge current). If the battery cannot supply enough voltage or current for a motor's

*k*value, then the desired

_{V}-R*τ*–

*ω*operating point cannot be reached and thrust will be limited. To more concretely understand how this information is used in designing the subsystem parameters, consider an example vehicle with 60 g motors and a 100 g battery operating at

*τ*= 0.075 Nm and

*ω*= 1400 rad/s. Nominal values for LiPo batteries are assumed such that the voltage per cell is 3.7 V and the battery internal resistance is 0.005 Ω per cell. Each motor is assumed to operate at the same point, such that the motor current is one-quarter of the battery current. To define the current-limited case, a

*C*-rating of 40C is assumed, implying a maximum discharge current. The power system design process is illustrated in Fig. 4. The dashed curve indicates possible

*k*-

_{V}*R*values for the given motor mass. The shaded area denotes infeasible

*k*-

_{V}*R*values. Using the

*τ*–

*ω*operating point, the voltage and current required to drive the motor can be computed for all

*k*-

_{V}*R*pairs, and therefore the required battery voltage and minimum

*C*-rating are known for each

*k*-

_{V}*R*pair. The

*k*-

_{V}*R*pairs that utilize full battery voltage to create

*τ*–

*ω*are indicated by the black markers, where each marker corresponds to a different

*S*-rating. The lighter markers define the battery-current-limited case for each

*S*-rating. The solid lines connecting the markers represent

*k*-

_{V}*R*pairs that can produce

*τ*–

*ω*while operating somewhere between maximum battery voltage and maximum battery current. As shown in Fig. 4, there may be multiple sets of

*k*pairs and

_{V}-R*S*-ratings that are capable of driving the

*τ*–

*ω*operating point.

### Propeller Model.

The propeller model provides a method of relating the motor brake power to the propeller thrust. Many propeller models are formulated for either the axial flow case (e.g., an airplane in level flight) or the near-hover case (e.g., a rotorcraft with zero freestream velocity). Because the RFV angle-of-attack can be anywhere between these two configurations, propulsion in oblique flow (i.e., when the incoming air velocity is neither normal nor parallel to the propeller disc plane) must be accounted for. Oblique flow considerably impacts propulsive forces as compared with conventional axial or near-hover operation. At low freestream velocities, as the propeller transitions from an axial configuration (*α _{P}* = 0 deg) to a transverse configuration (

*α*= 90 deg) at constant angular velocity, the increasing flow angle serves to slightly increase thrust and decrease the required torque [37]. Qualitatively, this is because as the propeller approaches the horizontal configuration, it more closely resembles a static thrust (zero freestream inflow) condition. At increased airspeeds, the literature shows that the propeller-induced power decreases due to the increased propeller inflow [46]. Additional in-plane forces, present due to differences in relative airspeed on the retreating and advancing propeller blades, also become significant. The in-plane forces are the cause of

_{P}*p*-factor during traditional fixed wing aircraft climb-out, and partially necessitate the cyclic pitch control required in helicopter flight. Although not always observable at low velocities, the model described here is of sufficient fidelity to demonstrate these effects [1].

Propeller modeling is generally based on momentum theory, which uses the conservation of momentum of the airstream passing through the propeller disk area to relate the thrust of a propeller to the mechanical power required to turn the propeller. More complex models, such as the BEMT used in this paper, more accurately quantify propulsion by using propeller geometric and aerodynamic data. BEMT considers the lift and drag generated by differential spanwise cross sections of the propeller blade. Each spanwise cross section is modeled as an airfoil using the geometric and aerodynamic data, and a local air velocity vector is used to compute the differential lift and drag contributions to the thrust produced and the torque required to drive the propeller. By integrating the differential elements along the blade radius and along the annulus formed by a blade rotation, the thrust and torque of a propeller can be related to the angular and forward velocity. BEMT methods that account for oblique flow generally vary in how the induced velocity is computed. Different techniques are reviewed in Ref. [47]; for implementing BEMT in the RFV model, induced velocity is formulated as in Ref. [48]. This model assumes that the induced velocity does not vary with azimuthal location, simplifying computations considerably with less than 2% differences in thrust and torque values [1] when compared with more complex models [49]. Detailed formulation of the BEMT model used for the RFV, including the derivatives used for computationally efficient convergence via the Newton-Raphson method, is described in their entirety in Ref. [1]. Other propeller modeling approaches include computational tools such as QPROP [50], vortex lattice methods [37], and computational fluid dynamics (CFD). BEMT is chosen in lieu of these options because QPROP is not formulated for oblique flow, vortex lattice methods are prone to wake instabilities [37], and CFD has high setup and computational costs, making implementation in the vehicle simulations impractical.

*λ*is the pitch length,

*r*is the radial blade section station, and

*γ*is the aerodynamic pitch angle. The propeller blade is assumed to have an optimal taper ratio as described in Ref. [51], where the local chord length,

*c*(

*r*), is related to the propeller radius, the chord tip length,

*c*, and the radial station,

_{tip}*r*, by

The chord length is limited to a maximum value to ensure a pragmatic shape for manufacturing, as in Ref. [33]. Winslow et al. tested a variety of propellers and showed experimentally that blades with thin, cambered airfoil section shapes provide the best performance for small-scale MAV propellers. The recommended NACA 6504 airfoil shape is, therefore, chosen to represent the propeller blade section airfoil. The propeller blade geometry is described in Fig. 5.

Due to its small size, a MAV propeller operates in a low Reynolds number (Re < ∼100,000) regime, which has been shown to reduce airfoil performance due to laminar separation bubble formation [52]. This low Reynolds number effect is often neglected in the MAV literature but limits the thrust output and propulsive efficiency. At a moderate Reynolds number, airfoil lift and drag coefficients are nearly constant; however, at a low Reynolds number, the lift and drag coefficients are functions of the Reynolds number. To account for performance degradation, a process as in Ref. [53] is implemented: NACA 6504 lift and drag polars are computed using XFOIL [54] for a range of Reynolds numbers from Re = 10,000 to Re = 200,000, *C _{l}* and

*C*values are extracted from the lift and drag polars, and the

_{d}*C*and

_{l}*C*data are interpolated as functions of Reynolds number and blade section angle-of-attack. The BEMT computation can then locally compute the Reynolds number and angle-of-attack for the current blade section and refer the appropriate

_{d}*C*and

_{l}*C*values [55].

_{d}### Mass Model.

*b*(as determined earlier) and length of

*L*. The maximum battery length is determined by balancing the center of mass of the four rotors and the battery such that the resulting center of mass lies on the vehicle axis of rotation

_{batt}*x*, is introduced as a design variable, where

_{B}*ρ*= 2113 kg/m

_{LiPo}^{3}, is computed using the specific energy determined for LiPo batteries in Ref. [56]. Quadrotor airframes, defined here as the vehicle structure not including the wheels, are correlated to the propeller radius and battery mass [31] using the function

*m*= 0.6 kg for all vehicle sizes. This value was determined by weighing components and sensors, such as control electronics and a small LIDAR, which could represent one RFV implementation [1]. The RFV wheel mass is parameterized by multiplying the wheel solidity

_{avionics}*σ*by the wheel area and the wheel material area density

*ρ*, and adding a constant value to represent the wheel hub hardware

_{wheel}*m*

_{hub}Values of *σ* = 0.4, *ρ _{wheel}* = 1.4 kg/m

^{2}, and

*m*= 0.05 kg are used in this study. The wheel solidity and hub mass are computed using CAD models and physical prototyping. The wheel material used is DragonPlate (ALLRed & Associates Inc, Elbridge, NY), a stiff, lightweight composite material comprised of a balsa wood inner core laminated with thin carbon fiber sheets on either side.

_{hub}### Range and Endurance Evaluation.

*k*rating, motor resistance, propeller diameter, propeller pitch length, and three scale factors (the propeller plane height scale factor, propeller spacing scale factor, and battery length scale factor). Using the design variables and subsystem models, a vehicle's rolling range,

_{V}*z*, and hover endurance,

*t*, are computed as

_{hover}*P*is the power consumed by a single motor at the rolling operating point and

_{e,rolling}*P*is the power consumed by a single motor at the hover operating point.

_{e,hover}To assess the validity of the subsystem models, the parameterized models are implemented in matlab. Using manufacturer specifications^{3} [57], and published values [29] to supply propeller pitch length, propeller diameter, motor *k _{V}* rating, motor resistance, battery weight, and gross takeoff weight (GTOW), the predicted hover endurances of four quadrotors are computed. The quadrotors represent a spectrum of available sizes, with GTOWs ranging from 80 g to 1375 g. The system model successfully predicts all four vehicles’ hover endurances to within 9.6%, with a mean absolute error of 6.3%, as shown in Table 1. The GTQ-Mini calculated endurance includes published specifications for the power consumed by onboard electronics (48 W), leading to an endurance computation within 20 s of the reported value.

## Methods—MOGA Problem Formulation and Vehicle Optimization

A MOGA is implemented to heuristically determine optimal vehicle configurations using the system model. Because larger vehicles can use larger rotors, and larger rotors improve rotor efficiency [41], a tradeoff is expected between vehicle range and vehicle size. Pareto-optimal frontiers are, therefore, used to characterize the tradeoffs between vehicle range and vehicle size. To illuminate the fundamental differences between a conventional, flight-only quadrotor and the RFV, three classes of optimal vehicle designs are generated. First, a Pareto frontier of rolling-range-optimized (yet flight-capable) vehicles is created. For this vehicle class, the Pareto frontier axes are vehicle size and rolling range. Next, a Pareto frontier of hover endurance -optimized (yet rolling capable) RFVs is created. For this vehicle class, Pareto frontier axes are vehicle size and hover time. The hover-optimized RFVs are also evaluated for their rolling range, and the rolling-optimized RFVs are likewise evaluated for hover endurance. Finally, a conventional quadrotor class is implemented using the described parameterization framework, but with zero-wheel mass. In this case, the Pareto frontier also uses axes of vehicle size and hover time. For the flying-optimized and rolling-optimized RFVs, the vehicle size is the wheel diameter. For the conventional quadrotor, the vehicle size is equal to the minimum diameter wheel that could be used to turn the conventional quadrotor into an RFV.

**x**= [

*k*]), subject to the bounded design variables in Eq. (35) and the constraints in Eq. (36):

_{V}, R, λ, r_{P}, x_{B}, x_{H}, x_{S}, DThe fitness function *f*_{1}(**x**) is equal to the vehicle size (i.e., minimize vehicle size). To maximize range, the fitness function *f*_{2}(**x**) = −*s* is used (i.e., minimize the negative of range), or to maximize hover endurance, *f*_{2}(**x**) = −*t*_{hover} is used (i.e., minimize the negative of hover endurance). In Eq. (35), the maximum and minimum design variable values are based on a survey of commercially available components [28,29,31]. The scale factor *x _{B}* ensures that the battery is sized so that the vehicle center of mass is located on the wheel axis of rotation. The scale factors

*x*and

_{h}*x*ensure that the vehicle is parameterized such that the geometry is feasible. Finally, nonlinear constraints are shown in Eq. (36), where

_{s}*ω*is the maximum rotor speed,

_{max}*TW*is the minimum thrust-to-weight ratio,

_{min}*ω*is the rotor speed required to produce

_{TW}*TW*,

_{min}*τ*(

*ω*) is the torque required to operate the propeller at

_{TW}*ω*, and

_{TW}*V*(

*ω*) and

_{TW}*I*(

*ω*) are the voltage and current required to drive the rotor at

_{TW}*ω*. To ensure sufficient maneuverability during flight, a minimum thrust-to-weight ratio of 2.0 is imposed in the first constraint [58]. The final three constraints represent the physical limitations that can prevent a vehicle from satisfying the minimum thrust-to-weight ratio: either (1) the maximum motor torque is insufficient to generate enough rotor speed, and therefore the propeller cannot create enough thrust; (2) the battery cannot supply enough voltage or current as described previously, and thus is unable to drive the motor at the required torque–speed operating point; or (3) the maximum propeller tip speed is aerodynamically limited by an imposed Mach limit of 0.4, as in Ref. [33]. Hypothetically, a propeller could mechanically fail due to loading or vibration before reaching the imposed maximum speed; however, researchers have demonstrated carbon fiber propellers at this scale capable of this angular velocity [33], so the aerodynamic limitation is imposed in lieu of a propeller structural limitation. All designs are checked for feasibility as part of the MOGA algorithm.

_{TW}Using these design variables, parameterized models, and constraints, the MOGA is implemented using the matlab*gamultiobj* function [59] via the matlab Global Optimization Toolbox [60]. The MOGA uses a controlled elitist approach to maintain population diversity while favoring the most fit designs, helping to ensure that a global minimum is found. The default options with adaptive feasible mutation, intermediate crossover, and tournament selection were found to provide satisfactory performance. Once the MOGA converges as dictated by a prescribed tolerance, a local single-objective constrained optimization solver is used to refine the final solution (matlab*fmincon*). Because the local optimization solver is single objective (as compared with the multi-objective GA), it is sequentially run for discrete vehicle size values between 35 and 75 cm at 5 cm intervals. For each vehicle size, the local optimization solver is seeded with the MOGA Pareto frontier design at that diameter. The local optimization solver slightly improves the Pareto-optimal designs, generally improving the solution less than 1%. The hybrid approach uses the MOGA to ensure a global minimum is found and uses the single-objective solver to home in on the optimum design more quickly than is possible with only a heuristic MOGA. Although the exact amount was not measured, the most noticeable computation time reduction came from vectorizing the innermost-nested BEMT loops. Simulations were executed on the North Carolina State University High-Performance Computing cluster [61].

## Results

Optimal designs are presented in this section as Pareto frontier curves. Each point on the Pareto frontier represents a different Pareto-optimal design. Figure 6(a) depicts the Pareto-optimal frontier for a rolling-optimized RFV of maximum rolling range versus vehicle size, optimized for operating conditions of *v* = 5 m/s, *x _{R}* = 25 mm, and

*θ*= 0 rad (referred to as the rolling-optimized RFV). Supplementary curves in Fig. 6(a) show the rolling range associated with an RFV optimized for the hover operating condition (hover-optimized RFV), and the flying range of a wheel-less, flying-optimized conventional quadrotor operating at a velocity of 5 m/s (conventional QR). At small vehicle sizes, the rolling-optimized RFV and hover-optimized RFV have the same maximum range. As size increases, the rolling-optimized RFV range begins to outperform the hover-optimized RFV range. Both RFV cases have significant range advantages over the conventional quadrotor. All three curves show that maximum range increases as vehicle size increases, verifying the expected tradeoff between size and range. As the vehicles become larger, propellers and motors increase in size (becoming more efficient and allowing for greater thrust production), which allows for larger battery sizes. Figure 6(b) depicts the maximum hover endurance of the same designs in Fig. 6(a). Although the rolling-optimized RFV shows a rolling range advantage over the hover-optimized RFV and the conventional quadrotor, this advantage comes at a sacrifice of flying endurance. As will be shown in the “Discussion” section, this is due to differences in the propulsion system and, in the case of the conventional quadrotor, not having to hover with additional weight from the wheels. Pareto-optimal RFV configurations show ranges that are two to three times greater than the Pareto-optimal conventional quadrotor configurations, at the cost of an 18–25% reduction in hover endurance.

The designs represented by distinct points in Fig. 6 are next broken down into their respective design variables. The associated propeller diameters and pitch lengths are shown in Fig. 7. The black dashed lines represent constraints of the design space, either imposed as bounds on the design variables or implicitly based on spatial constraints. The Pareto-optimal solutions show nearly identical propeller diameters for the rolling-optimized, hover-optimized, and conventional cases. The diameter upper bound shown in Fig. 7(a) is for a propeller plane height collocated with the vehicle axis of rotation; however, if the propeller plane height is non-zero, the maximum diameter is less than that shown in Fig. 7(a), as determined by the spatial constraints. The propeller diameters show little variation between configurations, with a maximum difference of 0.8 cm. In contrast to similarities in propeller diameters, the optimal pitch lengths for the rolling-optimized case show higher values than the other cases, with a 4.3 cm difference in the large vehicles’ pitch lengths. Rationale for this observation is provided in the “Discussion” section.

Figure 8 shows the Pareto-optimal motor masses, *k _{V}* ratings, and resistances. The RFV motors are heavier than the conventional quadrotor motors, indicating greater maximum torques. All configurations favor low

*k*motors, with mass differences resulting primarily from changes in resistance. This is contrary to many published models that assume mass is inversely proportional to

_{V}*k*without considering resistance. Whether considering the RFV or a conventional quadrotor, both

_{V}*k*and resistance must be considered to design vehicles with maximum range.

_{V}Figure 9 shows the Pareto-optimal propeller plane height and the Pareto-optimal center-to-center propeller distance. All cases show nearly identical spatial configurations, an important result that supports using a vehicle with a fixed mechanical design, a modular battery, a modular propulsion system, and removable wheels to adapt to user needs. The battery size scale factor *x _{B}* has a unity value for all cases, implying that the MOGA maximizes battery size for the optimal vehicle configurations.

## Discussion

### Constraint Boundaries.

The MOGA maximizes performance by finding designs that *simultaneously* reach the constraint boundaries imposed by the vehicle geometry, the prescribed thrust-to-weight ratio, and the dependent electromechanical and aerodynamic constraints, as shown in Fig. 10. Suboptimal designs that do not satisfy these relationships can improve performance by increasing battery size, using larger rotors to increase thrust, or using larger motors to increase the maximum torque. The 45 cm and 50 cm conventional quadrotor designs do not reach the angular velocity constraint boundary; this potentially indicates that these designs are not fully optimized, and a slight endurance increase can be realized by computing additional GA generations.

### Nominal Design Case Study.

To understand the relationships between design variables and vehicle systems, the rolling-optimized and hover-optimized RFVs with 60 cm vehicle size are considered in a nominal case study. The propulsion system design variables are investigated to demonstrate how a well-matched propeller and motor improve system performance. Table 2 shows the *k _{V}* rating, resistance, propeller diameter, and pitch length for these vehicles.

*PL*is a measure of how effectively a propeller produces thrust with respect to the required motor power, and is computed as

_{e}This efficiency measure is useful in rotorcraft analysis because it allows the computation of a static (i.e., zero velocity) operating point [62], as compared with traditional efficiency which is trivially equal to zero in the static case. The electrical power loading for the hover and rolling cases as functions of thrust produced by a single rotor are shown in Fig. 11. The curves are generated using the Pareto-optimal motor parameters. The solid line indicates the power loading at the optimal rotor pitch. The dashed lines indicate off-design pitch values. The vertical dashed lines indicate the thrust required at the rolling or hover design points. The endpoint of each curve represents the maximum thrust of the configuration, as dictated by a constraint (either voltage, current, or aerodynamic). Figure 11(a) shows that the MOGA selects a propeller pitch that maximizes the electrical power loading at the thrust required for rolling operation. This propeller pitch is suboptimal for the hover operating point. Similarly, in Fig. 11(b), the propeller pitch that maximizes electrical power loading at the hover operating point is selected by the MOGA. This propeller pitch value is suboptimal when the hover-optimized vehicle is in a rolling configuration. Although not considered in detail here, using a variable pitch propeller with collective pitch control could allow the propulsion system to operate at a maximum electrical power loading, regardless of the required thrust.

Both designs use the minimum allowed *k _{V}* value. Although multiple combinations of

*k*and

_{V}*R*are possible for a given motor mass, the optimal vehicles use the lowest feasible

*k*value for the optimal motor mass, as it requires the least current and, therefore, reduces the electronic speed controller mass. A future implementation of the optimization model could use motor mass as a design variable (with a minimum

_{V}*k*assumed) in lieu of

_{V}*k*and

_{V}*R*, reducing the dimension of the optimization.

### Parameter Sensitivity.

The local sensitivity of the objective function to changes in the propulsion system design variables is computed using numerical partial derivatives. The partial derivatives are computed using a finite-difference symmetric-midpoint-quotient method. A negative sensitivity value indicates that the objective function (range) will increase by decreasing the design variable, whereas a positive sensitivity value indicates that the objective function will increase by increasing the design variable. This method is applied for every design variable at each vehicle size, in 5 cm increments. Sensitivities to motor parameters are shown in Fig. 12 as functions of vehicle size. Vehicle range can be improved by decreasing the motor *k _{V}* rating or by decreasing the motor resistance; however, decreasing either of these values will increase motor mass, potentially driving the design infeasible due to an insufficient thrust-to-weight ratio. This further demonstrates the necessity of including component constraints in the model formulation. Vehicle performance is relatively insensitive to changes in motor resistance at smaller vehicle sizes; however, as the vehicle size increases, the magnitude of the sensitivity to changes in

*k*and resistance increases.

_{V}Sensitivities to propeller parameters are shown in Fig. 13 as functions of vehicle size. Vehicle range can be improved by increasing the blade radius; however, this could lead to spatially infeasible designs. Future work should experimentally determine if the merits of increasing the blade radius outweigh any adverse aerodynamic effects associated with close propeller spacing. Figure 13(b) shows that for 35 cm and 40 cm sized vehicles, further reducing the propeller pitch will increase range—note that optimum design variable values rest on the lower constraint bound. This suggests that custom low-pitch propellers can offer performance benefits to the smaller vehicles. For design sizes greater than 40 cm, increasing the propeller pitch will increase the vehicle range; however, this will also require more torque, which will overload the motors at the maximum thrust condition. The complexities of propulsion system interactions and constraints must be considered when designing optimum range vehicles.

## Conclusion

This work describes the systematic design optimization of a rolling-flying vehicle using a multi-objective genetic algorithm to optimize a parameterized multi-physics model. By emphasizing key constraints that affect the maximum and nominal vehicle operating points, an optimization framework is constructed that can be used for RFVs and conventional multi-rotors. A low Reynolds number BEMT aerodynamic model is used to parameterize the propeller in conjunction with a three-phase brushless DC motor model. The optimization yields a better understanding of the interaction between design variables and system performance by demonstrating the link between key geometric, aerodynamic, and electromechanical constraints on the system. Although discussed in the context of RFVs, the methodology and conclusions also apply to conventional quadrotors. The resulting optimized vehicles show that the ranges and flight endurances of rolling-optimized and hover-optimized RFVs are similar, with more significant differences for larger vehicles. Both RFV configurations result in ranges that are two to three times greater than a conventional quadrotor, at the cost of an 18–25% reduction in hover endurance. Using electrical power loading, the relationships between propeller parameters and system performance are investigated, demonstrating that the MOGA selects parameters that maximize electrical power loading at the required operating thrust. Finally, sensitivities to changes in the optimum design variables are examined, allowing the designer to understand where to concentrate design effort. This work provides a baseline understanding of the desired components in a high-range multi-rotor vehicle. Future work will leverage this work to study effects due to being constrained to commercial off-the-shelf products and investigate vehicle performance over a range of operating conditions.

## Footnotes

See Note 3.

See Note 3.

## Conflict of Interest

There are no conflicts of interest.

## Funding Data

U.S. Army Research Office and the U.S. Army Special Operations Command (Contract No. W911-NF-13-C-0045; Funder ID: 10.13039/100000183).

## Nomenclature

*b*=battery tray side length (m)

*h*=height of propeller disc plane above axle (m)

*r*=propeller blade radial station (m)

*s*=distance from center-to-center of two neighboring props (m)

*v*=vehicle velocity (m/s)

*z*=range (m)

*D*=wheel diameter (m)

*I*=winding current (A)

*N*=number of rotors

*R*=winding resistance (Ω)

*S*=number of battery cells

*c*=_{tip}tip chord length

*h*=_{min}minimum height of the propeller disc plane above axle (m)

*h*=_{max}maximum height of the propeller disc plane above axle (m)

*k*=_{e}back-EMF constant (V/(rad/s))

*k*=_{V}motor constant (rpm/V)

*l*=_{batt}battery length (m)

*l*=_{motor}motor characteristic length (m)

*m*=_{airframe}airframe mass (kg)

*m*=_{avionics}avionics mass (kg)

*m*=_{batt}battery mass (kg)

*m*=_{ESC}electronic speed controller mass (kg)

*m*=_{motor}motor mass (kg)

*m*=_{prop}propeller mass (kg)

*m*=_{total}total vehicle mass (kg)

*m*=_{wheel}wheel mass (kg)

*r*=_{P}propeller radius (m)

*s*=_{min}minimum propeller center-to-center distance (m)

*s*=_{max}maximum propeller center-to-center distance (m)

*t*=_{hover}hover endurance time (s)

*x*=_{h}propeller plane height scale factor

*x*=_{s}propeller center-to-center distance scale factor

*x*=_{B}battery height scale factor

*x*=_{R}terrain step size (m)

*A*=_{0}vehicle planform area (m

^{2})*C*=_{batt}battery capacity (Ah)

*C*=_{d}blade section airfoil drag coefficient

*C*=_{l}blade section airfoil lift coefficient

*E*=_{batt}battery energy (J)

*F*=_{D}parasitic drag force (N)

*F*=_{H,sys}system in-plane propeller force (N)

*F*=_{N}normal force (N)

*F*=_{R}rolling resistance (N)

*F*=_{q}dynamic-pressure-normalized parasitic drag force (m

^{2})*I*=_{0}no-load current (A)

*I*=_{max}current required to produce

*Q*(A)_{max}*P*=_{e,hover}power required for hover operation (W)

*P*=_{e,rolling}electrical power required for rolling operation (W)

*P*=_{e,sys}system electrical power (W)

*Q*=_{max}maximum motor torque (Nm)

*T*=_{sys}system thrust (N)

*V*=_{batt}battery nominal voltage (V)

*V*=_{s}winding voltage (V)

*c(r)*=chord length as a function of radial station (m)

*PL*=_{e}electrical power loading (N/W)

*TW*=_{min}minimum thrust-to-weight ratio

*α*=_{P}propeller angle-of-attack (rad)

*β*=nondimensional terrain parameter

*γ*=aerodynamic pitch angle (rad)

*θ*=incline angle (rad)

*λ*=propeller pitch length (in.)

*ρ*=air density (kg/m

^{3})*ρ*=_{LiPo}LiPo battery density (kg/m

^{3})*ρ*=_{wheel}wheel material area density (kg/m

^{2})*σ*=wheel solidity

*τ*=motor torque (Nm)

*ω*=rotor angular velocity (rad/s)

*ω*=_{max}maximum rotor angular velocity (rad/s)

*ω*=_{TW}rotor speed required to produce required thrust-to-weight ratio (rad/s)