Simultaneous Optimization of Design and Takeoff Trajectory for an eVTOL Aircraft

Abstract

When designing a vertical takeoff and landing (VTOL) aircraft, it is crucial to consider various operating conditions, including hover, wing-borne cruise, and transition. Takeoff transition and climb phases are critical for vehicle conceptual design because they drive the motor and battery sizing. However, these flight phases are often ignored or simplified in the conceptual design literature due to the complexity of transition dynamics. To address this limitation, we propose simultaneous optimization of VTOL aircraft conceptual design and takeoff trajectory. Simultaneous optimization incorporates the takeoff transition and climb flight dynamics into VTOL design optimization in a fully coupled manner. This paper presents the design-trajectory optimization of lift-plus-cruise and tailsitter UAVs for delivery missions. As a result, simultaneous optimization reduces total energy consumption by 7.4% compared to uncoupled optimization and by 4.8% from iterative sequential optimization in the lift-plus-cruise configuration. It also achieves a 29% energy reduction from uncoupled optimization in the tailsitter configuration. Incorporating takeoff trajectory optimization in UAV sizing enables accurate estimation of takeoff and climb energy, which avoids undersizing or oversizing the battery. Furthermore, simultaneous optimization trades the vehicle weight penalty for a more energy-efficient climb with early transition. These system-level trade-offs can only be captured by simultaneous optimization; they cannot be captured by uncoupled or iterative sequential optimization.

Full text

This is a postprint of the following article:
Shugo Kaneko and Joaquim R. R. A. Martins. Simultaneous Optimization of Design and Takeoff Trajectory for
an eVTOL Aircraft. Aerospace Science and Technology, 2024.
The published article may differ from this postprint and is available at:
https://doi.org/10.1016/j.ast.2024.109617.
Simultaneous Optimization of Design and
Takeoff Trajectory for an eVTOL Aircraft
Shugo Kaneko and Joaquim R. R. A. Martins
Department of Aerospace Engineering, University of Michigan, Ann Arbor, MI, 48109
Abstract
When designing a vertical takeoff and landing (VTOL) aircraft, it is crucial to consider various
operating conditions, including hover, wing-borne cruise, and transition. Takeoff transition and
climb phases are critical for vehicle conceptual design because they drive the motor and battery
sizing. However, these flight phases are often ignored or simplified in the conceptual design literature
due to the complexity of transition dynamics. To address this limitation, we propose simultaneous
optimization of VTOL aircraft conceptual design and takeoff trajectory. Simultaneous optimization
incorporates the takeoff transition and climb flight dynamics into VTOL design optimization in a
fully coupled manner. This paper presents the design-trajectory optimization of lift-plus-cruise and
tailsitter UAVs for delivery missions. As a result, simultaneous optimization reduces total energy
consumption by 7.4% compared to uncoupled optimization and by 4.8% from iterative sequential
optimization in the lift-plus-cruise configuration. It also achieves a 29% energy reduction from
uncoupled optimization in the tailsitter configuration. Incorporating takeoff trajectory optimization
in UAV sizing enables accurate estimation of takeoff and climb energy, which avoids undersizing or
oversizing the battery. Furthermore, simultaneous optimization trades the vehicle weight penalty
for a more energy-efficient climb with early transition. These system-level trade-offs can only be
captured by simultaneous optimization; they cannot be captured by uncoupled or iterative sequential
optimization.
1 Introduction
There has been a growing interest in delivering commercial packages, food, grocery items, and
medical supplies using uncrewed aerial vehicles (UAVs). Drone delivery is faster than truck delivery,
and they can serve isolated locations inaccessible by ground vehicles. Furthermore, UAVs have
the potential to reduce energy consumption for lightweight last-mile delivery [1,2]. This, along
with the electric powertrain’s zero-emission operation, makes UAVs a viable option for lowering the
environmental impact of last-mile delivery. The energy efficiency of delivery UAVs comes from a
significantly higher payload-to-vehicle weight ratio than cars and trucks. This means that drone
delivery reduces the wasteful energy consumed to move the non-payload mass (i.e., vehicle empty
weight).
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Designing an energy-efficient UAV is the key to achieving the full potential of drone delivery. This
is challenging because most delivery UAVs require vertical flight capability, which is far less efficient
compared to wing-borne flight. UAV configurations with a wing are preferable to balance the cruise
efficiency and vertical takeoff and landing (VTOL) capability. For lightweight delivery UAVs, the two
common winged configurations are the lift-plus-cruise and the tailsitter. Lift-plus-cruise UAVs equip
separate propulsion systems (i.e., rotors and motors) for vertical flight and forward flight. Tailsitter
vehicles use the same propulsion system for both flights. Other potential configurations are the
tilt-rotor and tilt-wing. These are popular for larger-scale electric VTOLs (eVTOLs) for advanced
air mobility (AAM) applications but less common for small UAVs. This is mainly because of the
mechanical complexity and weight penalty of the tilt mechanism. For this reason, we focus on the
lift-plus-cruise and tailsitter configurations in this paper.
In VTOL vehicle design, there is an inherent trade-off between vertical flight performance and
wing-borne cruise performance. This means that a vehicle tailored for vertical flight performance is
not efficient in cruise, and vice versa. Therefore, engineers need to appropriately balance the two
contradicting factors to design a satisfactory UAV. Multidisciplinary design optimization (MDO) is
a powerful tool for this non-intuitive task [3, Ch. 13]. MDO finds designs with optimal trade-offs
between different flight phases for the best system-level performance [4–6]. System-level metrics
include the total energy consumption for a delivery mission, UAV takeoff weight, payload weight
fraction, and cost-per-delivery.
This paper aims to improve the VTOL conceptual design methodology by incorporating takeoff
trajectory optimization into vehicle design optimization. Because of the complexity, the takeoff
trajectory and transition flight dynamics are typically not considered in the UAV conceptual design
literature. However, the takeoff and transition dynamics affect the UAV design because they are
power- and energy-intensive operations, which drive the battery and motor sizing. Therefore, there
is a coupling between takeoff trajectory optimization and UAV conceptual design optimization. We
capture this coupling using MDO and improve the VTOL UAV design and system-level performance.
The main goal of this paper is to quantify the gain of considering trajectory optimization in a
VTOL UAV’s conceptual design process. To this end, we compare three design and trajectory opti-
mization approaches. The first approach is conventional optimization, where we optimize the UAV
design without considering the trajectory and then optimize the takeoff trajectory for the given UAV.
We call this approach uncoupled optimization because it cannot address the design-trajectory coupling
mentioned above. The second approach, iterative sequential optimization, repeats the design optimiza-
tion and trajectory optimization sequentially. This partially captures the coupling between design
and trajectory. Finally, the third approach is simultaneous optimization. This concurrently optimizes
the UAV design and trajectory by solving a monolithic optimization problem. We demonstrate that
simultaneous optimization yields the best design by capturing the system-level trade-off that the
iterative sequential optimization cannot fully address.
This paper is composed as follows. Section 2provides a brief review of the relevant literature.
Section 3describes the models of the UAV dynamics, aerodynamics, rotor analysis, and weight
estimation we use in this study. In Sec. 4, we state the design and trajectory optimization problems.
Sec. 5explains the numerical approaches and implementation. Finally, we discuss the optimization
results of the lift-plus-cruise and tailsitter configurations in Sec. 6and Sec. 7, respectively.
2

2 Relevant Literature
Our work lies at the intersection of eVTOL UAV conceptual design optimization and VTOL trajec-
tory optimization. The combination of vehicle design and trajectory optimization is called simultaneous
design and trajectory optimization, and it has been studied for various mechanical and aerospace appli-
cations. For VTOL aircraft, however, simultaneous design and trajectory optimization has not been
well explored in the literature. Most research works only addressed either the vehicle design or flight
trajectory, but not both. Our work aims to fill this gap in the literature.
In this section, we summarize the recent literature on three topics we combine in this paper: eVTOL
UAV conceptual design optimization, eVTOL trajectory optimization, and simultaneous design and
trajectory optimization.
2.1 eVTOL UAV Conceptual Design
For the lift-plus-cruise configuration, Tyan et al. [7] presented a conceptual sizing study that
consists of two steps: initial sizing based on the design requirements and the second resizing iterations
around the selected electric components from the first step. Zhang et al. [8] optimized the electric
propulsion system of a lift-plus-cruise UAV considering the gust rejection. An et al. [9] investigated a
hydrogen-electric lift-plus-cruise UAV design; they provided regression models for UAV component
weight estimations. Finger et al. [10] compared pure-electric and hybrid-electric propulsion systems
for lift-plus-cruise UAVs. Seren and Hornung [11] proposed a multi-fidelity design tool for eVTOL
UAVs. They used their tool to compare the lift-plus-cruise and tilt-rotor configurations.
For the wingless multirotor configuration, Bershadsky et al. [12] presented comprehensive sizing
models and performed conceptual design optimization. Winslow et al. [13] provided regression
models for quadrotor component weight estimation and performed sizing studies by coupling them
to the blade element momentum (BEM) rotor analysis. Vu et al. [14] also proposed regression-based
weight estimation models for electric propulsion system sizing. Budinger et al. [15] presented the
similarity models for multirotor conceptual design based on dimensional analysis and Buckingham’s
theorem. Delbecq et al. [16] performed multirotor sizing optimization using the similarity models by
Budinger et al. [15].
On the tailsitter, Sridharan et al. [17] investigated the quadrotor biplane tailsitter design of various
vehicle sizes. The same authors also performed MDO of the tailsitter UAV [18]. Govindarajan and
Sridharan [19] performed conceptual design optimization of multirotor, tailsitter, and tiltable tricopter
configurations for package delivery.
2.2 eVTOL Trajectory Optimization
Recent literature on VTOL trajectory optimization is summarized in Table 1. The table organizes
the literature based on the VTOL configurations, UAV gross takeoff weight (GTOW), flight phase
considered, and the numerical method. Optimization of transition trajectory has been an active
research topic for various VTOL configurations, including lift-plus-cruise [20–22], tailsitter [23–27],
tiltrotor [28], tiltwing [29–32], and multirotor [33,34]. The scale of the vehicles studied in the literature
ranges from lightweight UAVs [23–27,29,33] to heavier AAM aircraft [20–22,28,30,31,34,35].
Regarding the numerical approach, all works listed in Table 1used direct methods [36], either direct
shooting or direct collocation, to discretize the trajectory.
Most of the literature in Table 1optimized flight trajectory while fixing the aircraft design. The
3

Table 1: Summary of recent VTOL trajectory optimization literature.
Reference Configuration GTOW (kg) Trajectory problem Numerical approach
Park et al. [28] tiltrotor 650 takeoff and landing direct collocation
Orndorff et al. [20] lift-plus-cruise 2000 forward1transition direct shooting
Panish and Bacic [29] tiltwing 6.9forward and backward2transition direct collocation
Orndorff and Hwang [21] lift-plus-cruise 3724 forward transition direct shooting
Delbecq et al. [33] multirotor 50.8vertical climb direct shooting
Wang et al. [22] lift-plus-cruise 600 maneuverability assessment direct collocation
Anderson et al. [23] tailsitter 1.4takeoff and forward transition direct collocation
Chauhan and Martins [30] tiltwing 725 takeoff and forward transition direct shooting
Hendricks et al. [34] multirotor 500 takeoff, cruise, and landing direct collocation
Hendricks et al. [35] tiltwing 6000 climb, cruise, and descent direct collocation
Pradeep and Wei [31] tiltwing 752 backward transition and descent direct collocation
Verling et al. [24] tailsitter 3.0backward transition direct method
Oosedo et al. [25] tailsitter 1.6forward transition direct shooting
Banazadeh and Taymourtash [26] tailsitter 46 forward and backward transition direct shooting
Maqsood and Go [32] tiltwing – forward and backward transition direct shooting
Kubo and Suzuki [27] tailsitter 2.0forward and backward transition direct collocation
1Forward transition: hover-to-cruise 2Backward transition: cruise-to-hover
only exceptions are Delbecq et al. [33] and Hendricks et al. [34,35], who optimized the vehicle design
along with the trajectory.
2.3 Simultaneous Design and Trajectory Optimization.
Simultaneous optimization of vehicle design and trajectory is a class of control co-design (CCD)
optimization. CCD aims to concurrently optimize the physical system design (e.g., UAV design) and
the controller design for dynamic engineering systems [37]. The simultaneous design and trajectory
optimization we study in this paper is categorized as an open-loop CCD because we solve the open-
loop optimal control as a controller design.
On the theory and methodology side, Allison and Herber [38] discussed various problem formu-
lations for CCD optimization and correlated them to the MDO architectures [39;3, Ch. 13].
Herber and Allison [40] compared the nested and simultaneous optimization approaches for
CCD. Hwang et al. [41] proposed a multi-fidelity coupling approach between trajectory analysis and
high-fidelity aerodynamic analysis. Cunis et al. [42] derived the sensitivity of optimal control to be
used in the nested optimization approach. Kaneko and Martins [43] performed benchmark studies
of simultaneous design-trajectory optimization problems that involve a computationally-expensive
dynamics model. Regarding the software, Falck et al. [44] developed Dymos, an open-source trajectory
optimization package built on the OpenMDAO framework [45]. Dymos facilitates CCD optimizations
with analytic derivatives.
On the eVTOL applications, Hendricks et al. [35] applied design-trajectory optimization to NASA’s
urban air mobility (UAM) concept vehicle. They optimized the tiltrotor aircraft design and the climb-
cruise-descent trajectory; however, the transition trajectory was not included in their study. Hendricks
et al. [34] also performed design and trajectory optimization of the electric multirotor configuration.
Their study focused on the coupling between the thermal management system (TMS) design and the
flight trajectory. For lightweight UAVs, Delbecq et al. [33] optimized the design of a wingless multirotor
drone along with its one-dimensional vertical climb trajectory. Mabboux et al. [46] performed the
closed-loop CCD of a multirotor UAV, where they used iterative sequential optimization to optimize
4

the UAV design and its feedback controller.
This work aims to expand the scope of the simultaneous design and trajectory optimization for
eVTOL vehicles. Compared to the above literature, our novel contributions are as follows. First, we
study the lift-plus-cruise and tailsitter configurations, which have not been studied in the eVTOL
CCD literature. Second, our work is the first to include the takeoff transition trajectory of winged
VTOL configurations in design-trajectory optimization. Third, we present a detailed comparison of
simultaneous design-trajectory optimization to uncoupled and sequential optimizations.
3 Models
3.1 UAV Configurations
This paper focuses on the lift-plus-cruise and tailsitter configurations. Figure 1shows notional
vehicle configurations. A lift-plus-cruise aircraft has two separate sets of rotors: one for wing-borne
forward flight and the other for vertical flight. We assume a configuration with one pusher and four
VTOL rotors. This configuration is also called a quadplane. The motors and rotors for the forward
and VTOL propulsion have different designs because each operates under different loading and
inflow conditions. The dedicated propulsion system for each flight mode is advantageous for the
rotor’s aerodynamic efficiency. The main disadvantages are the additional weight of the redundant
propulsion system and the higher aerodynamic drag in cruise due to the VTOL rotors and booms.
A tailsitter uses the same rotors for both vertical and forward flight. We assume a quadrotor
bi-plane configuration [17]. Unlike the lift-plus-cruise configuration, the tailsitter does not carry a
redundant propulsion system. However, tailsitters suffer from lower propeller efficiency because they
must use the same rotors and motors for vertical and forward flights.
3.2 Baseline UAV Specifications
We refer to Wing Aviation LLC’s lift-plus-cruise delivery UAV as a baseline. Table 2summa-
rizes the baseline specifications based on the publicly available data [47]. We will perform design
optimization for the same payload weight, range, and cruise altitude as this reference vehicle.
(a) Lift-plus-cruise (b) Tailsitter
Figure 1: UAV configurations.
5

Table 2: Baseline specification of Wing Aviation LLC’s UAV.
Parameter Value
Takeoff weight 6.4 kg
Payload weight 1.2 kg
Roundtrip range 20 km
Cruise speed 29 m/s
Cruise altitude 45 m
Wing span 1 m
Wing area 0.152 m2
3.3 Vehicle Dynamics
We consider two-degree-of-freedom dynamics in the longitudinal plane. This is a point-mass
model that does not consider rotational degrees of freedom. Figure 2shows a free-body diagram of
the dynamics model.
The equations of motion of a lift-plus-cruise UAV in the horizontal (𝑥) and vertical (𝑦) directions
are
𝑚¤
𝑣𝑥=𝑇𝐹cos(𝜃+𝛽𝐹) − 𝑇
𝑉sin(𝜃+𝛽𝑉) − 𝐷cos 𝛾−𝐿sin 𝛾,
𝑚¤
𝑣𝑦=𝑇𝐹sin(𝜃+𝛽𝐹) + 𝑇
𝑉cos(𝜃+𝛽𝑉) − 𝐷sin 𝛾+𝐿cos 𝛾−𝑚 𝑔 , (1)
where 𝑚is the vehicle mass, 𝑔is the gravitational constant, 𝑣𝑥and 𝑣𝑦are the horizontal and vertical
speed, 𝜃is the body pitch attitude relative to the horizontal plane, and 𝛾is the flight path angle given
by 𝛾=arctan(𝑣𝑥/𝑣𝑦). We assume the zero-wind condition, which makes the airspeed equal to the
ground speed. The body pitch attitude is considered a control input in the point-mass dynamics. The
lift 𝐿and drag 𝐷are computed by an empirical aerodynamic model, and the rotor thrust 𝑇is given by
the blade element momentum (BEM) theory. The subscripts 𝐹and 𝑉denote the forward and VTOL
rotors, respectively. We assume that all VTOL rotors produce the same magnitude of thrust and do
not differentiate between front and rear VTOL rotors. Accordingly, 𝑇
𝑉is the sum of all four rotors’
thrust. When the inflow is not normal to the disk, the thrust direction is not always perpendicular
to the rotor disk. This deflection angle is represented by 𝛽𝐹and 𝛽𝑉for forward and VTOL rotors,
θ
TFβF
TV
βV
L
D
v
γ
mg s
h
Figure 2: Free-body diagram of the longitudinal dynamics model of a lift-plus-cruise configuration.
6

respectively.
We also use the following ordinary differential equation (ODE) to integrate the energy consump-
tion:
¤
𝐸(𝑡)=𝑃𝐹+𝑃𝑉
𝜂motor𝜂ESC
=:𝑃𝐹+𝑃𝑉
𝜂,(2)
where 𝐸is the accumulated energy consumption by the propulsion system, and 𝑃𝐹and 𝑃𝑉are the
power required by forward and VTOL rotors, respectively. The rotor power is computed by the BEM
analysis. We assume a constant motor efficiency of 𝜂motor =0.935 and an electric speed controller
(ESC) efficiency of 𝜂ESC =0.96 [48]. These yield a collective efficiency of 𝜂:=𝜂motor𝜂ESC =0.8976.
The constant efficiency model assumes that the motor and ESC designs are ideally tailored for the
UAV’s operating conditions. Consideration of an efficiency map would increase the practicality of the
results, which remains future work.
For trajectory optimization, we convert Eqs. (1) and (2) into a system of the first-order ODEs:
¤
𝜉=𝑓ODE 𝜉, 𝑢;𝑥design ,(3)
where 𝜉:=[𝑥, 𝑦 , 𝑣𝑥, 𝑣𝑦, 𝐸]is the state vector and 𝑢:=[𝜃,Ω𝐹,Ω𝑉]is the control vector. Besides the
body pitch angle 𝜃, the rotational speed of the forward and VTOL rotors (Ω𝐹and Ω𝑉, respectively)
are the control inputs. The ODE also depends on the UAV design variables 𝑥design, which we detail in
Sec. 4.1.
The tailsitter configuration also follows the same dynamics, except we only have one set of rotors.
Accordingly, we set 𝑇
𝑉=0and 𝑃𝑉=0in Eqs. (1) and (2), respectively.
3.4 Rotor Analysis
We use the BEM rotor analysis to compute the rotor thrust, torque, and power as a function of the
rotor blade geometry, rotational speed, and inflow velocity:
𝑇, 𝛽,𝜏, 𝑃 =𝑓BEM(𝑣𝑥, 𝑣𝑦,𝜃,Ω;𝑥rotor),(4)
where 𝜏is the torque, 𝑃is the power required by the rotors, and 𝑥rotor is the rotor design variables
representing the blade geometry. Rotor design parametrization is detailed in Sec. 4.1.
We use CCBlade [49] as a BEM implementation. CCBlade employs the Prandtl correction for the
tip and hub loss. The hub diameter is assumed to be 15% of the rotor diameter, which is the typical
value of off-the-shelf small rotors [14,50].
We assume two-blade rotors for both forward and VTOL rotors. In the lift-plus-cruise config-
uration, two-blade VTOL rotors are beneficial because they reduce the cruise drag significantly by
placing the blades parallel to the flow [51]. We also use a two-blade propeller for the pusher rotor
and the tailsitter because most off-the-shelf rotors for lightweight drones have two blades [50].
3.5 Aerodynamic Model
An aerodynamic model computes the lift and drag as a function of the flight speed and the angle of
attack. VTOL trajectory optimization requires a post-stall aerodynamic model because the wing may
stall in vertical and transition flight. We use a model developed in the eVTOL trajectory optimization
work by Chauhan and Martins [30].
7

This model combines the airfoil’s pre-stall data (lift coefficient 𝐶𝐿and drag coefficient 𝐶𝐷versus
the angle of attack), a finite-wing correction based on the lifting-line theory, and an empirical post-
stall model developed by Tangler and Ostowari [52]. We assumed a constant wing aspect ratio of 6.6,
which is the aspect ratio of the reference UAV by Wing Aviation LLC, and the Oswald efficiency of 0.8.
We generated the pre-stall airfoil data using XFOIL [53] for the NACA 0012 airfoil at 𝑅𝑒 =200,000.
This roughly equals the cruise Reynolds number of the reference UAV. Although a cambered airfoil is
more practical, we use the symmetric airfoil data because trajectory optimization requires the model
to extend to the negative post-stall angles. The symmetric airfoil data is necessary for this purpose
because the post-stall model by Tangler and Ostowari [52] is only defined for the positive angle of
attack. Figure 3shows the resulting wing aerodynamic model.
For parasite drag, we use the experimental data from Bacchini et al. [51], who performed wind-
tunnel experiments of a lift-plus-cruise configuration at a similar Reynolds number to our study. They
reported a minimum 𝐶𝐷of 0.0397, which includes the drag of VTOL rotors and booms in forward
flight. Accordingly, we adjust the 𝐶𝐷−𝛼curve in Fig. 3to yield 𝐶𝐷,min =0.0397.
We ignore the rotor-wing and rotor-body interactions in the lift-plus-cruise configuration. The
downwash of the VTOL rotors does not affect the wing aerodynamics significantly because the VTOL
rotors are only turned on for slow-speed flight, where the wing aerodynamic force is much smaller
than the weight and rotor thrust. The pusher rotor-fuselage interaction would offer a boundary
layer ingestion benefit at the sacrifice of rotor efficiency due to nonuniform inflow. These require a
higher-fidelity aerodynamic model, which is outside the scope of this work.
For tailsitter, the propeller downwash constantly hits the wings. We capture this interaction using
a simple model based on the rotor-induced velocity. This is a common approach in VTOL conceptual
design optimization and trajectory optimization literature [19,25,27,30,32]. This model computes
the induced velocity using the disk-momentum theory and corrects the wing’s net inflow speed and
angle of attack by adding the induced velocity to the freestream velocity. We assume that the rotors are
placed right in front of the wing’s leading edge; hence, there is no wake contraction. We also assume
that the entire wing is immersed in the propeller downwash, which is a reasonable assumption for
quadrotor bi-plane tailsitters because these vehicles have large propellers relative to the wing span,
as we see in Sec. 7.
0.0
0.5
1.0
CL
0 30 60 90
Angle of attack [deg]
0.0
0.5
1.0
1.5
CD
Figure 3: Wing aerodynamic model.
8

3.6 Weight Estimation
For UAV sizing, we decompose the GTOW into component weights: payload, wing, rotors, motors,
ESCs, battery, and frame. The payload weight is fixed to 1.2 kg. The “frame” weight includes all other
components not explicitly listed here, such as the fuselage, empennage, avionics, payload release
system, and wiring. The weight estimation requires a numerical iteration with respect to GTOW
because the component weights are a function of the GTOW. In this study, we write this implicit
relation as a residual equation
𝑟weight :=𝑊GTOW − (𝑊payload +𝑊wing +𝑊rotor +𝑊motor +𝑊ESC +𝑊battery +𝑊frame)=0,(5)
and impose 𝑟weight =0as an equality constraint in optimization.
We use regression models from the literature to estimate the wing [19,54] and the rotor [9] weight
(kg):
𝑊wing =−0.0802 +2.2854𝑆 , (6)
𝑊rotor,𝐹 =0.670644𝐷2,784
𝐹,(7)
𝑊rotor,𝑉 =0.007281𝑒3.389𝐷𝑉−0.003232 ,(8)
where 𝑆is the wing area (m2), and 𝐷𝐹and 𝐷𝑉are the diameters (m) for the forward and VTOL rotors,
respectively.
We determine the motor weight based on the maximum power and torque required. For torque,
we use the following model from Budinger et al. [15]:
¯𝜏=1.823 𝑊motor
0.305 3.5/3
,(9)
where ¯𝜏is the maximum continuous torque (N-m) and 𝑊motor is the motor’s weight (kg). For the
maximum continuous power, we assume a specific power of 6 kW/kg, which is the average value
of the motor data used to construct the above torque-weight model [15]. The maximum power and
torque required are computed by the BEM analysis on various flight conditions and along the takeoff
trajectory. In addition to the torque and power requirements, thermal considerations could also drive
the motor sizing. A thermal model is not included in this study, and it could be the subject of future
work.
The ESC weight is estimated by a specific power of 31.7 kW/kg [15].
The battery weight is determined based on the energy required for a round-trip delivery mission
and the battery’s specific energy. We assume a 15% reserve energy, a minimum state of charge of
20%, and a 15% loss factor for power transmission (including avionics power) from the battery to
ESC [19]. This gives 𝑊battery =1.15𝐸total/(0.85 ×0.8)𝜌𝑏, where 𝐸total is the total energy consumption
by the propulsion systems and 𝜌𝑒is the specific energy, which we assume to be 175 Wh/kg1.
For the frame weight, we use a constant fraction of 𝑊frame /𝑊GTOW =0.603. This fraction is deter-
mined based on the baseline UAV’s specification and the UAV design optimization. This procedure
is detailed in Sec. 4.1.3.
1Based on Gens ace G-Tech 8000mAh lithium-polymer battery. (Linked webpage: accessed on March 20, 2024)
9

3.7 Mission Profile
We consider a flight mission to consist of the takeoff and climb, cruise, hover, descent, and landing
phases. Figure 4shows the mission profile for a one-way flight. We set a 10 km delivery distance
under the no-wind condition and a 45 m cruise altitude, following Table 2. The cruise speed is a
design variable.
We use this mission profile to compute the total energy consumption, which is our optimization
objective and also drives the battery sizing. The takeoff and climb energy consumption is given by
trajectory optimization. This effectively integrates Eq. (2) along the takeoff trajectory. The cruise
energy is given by multiplying the steady cruise power by the cruise duration. The same applies to
the hover energy, for which we assume a 30-second round-trip duration. The energy for descent and
vertical landing is approximated by 20 seconds of hovering. This assumes the UAV to glide down to
20 m with zero additional power and a 1 m/svertical descent for 20 m. We assume the same mission
profile for outbound and return flights and that the return flight carries the payload back. This gives
a conservative energy estimate in case the UAV fails to release the payload.
This study considers trajectory optimization only for takeoff and climb flight phases but not for
the entire flight mission. This reduces the computational cost without affecting the resulting UAV
design significantly. For takeoff and climb, it is essential to model these flight phases accurately using
dynamic trajectory optimization. This is because these are power and energy-intensive phases, which
drive battery and motor sizing. Although cruise and hover also drive battery sizing, these steady
flight phases do not need dynamic trajectory consideration. Descent and landing flight phases are
dynamic; however, they have a smaller influence on battery sizing and no impact on motor sizing
because they require less energy and power than takeoff and climb. For these reasons, we modeled
cruise, hover, descent, and landing as steady flight phases instead of dynamic trajectories.
Takeoffand climb
Variable time and distance
45 m
Cruise
10 km
Hover (15 sec.)
Descent and landing
(20 sec. hover)
Figure 4: Mission profile used to compute the total energy consumption in simultaneous design-trajectory optimization.
The figure only shows a one-way flight; battery sizing uses the energy for a round-trip flight.
4 Optimization Problems
We optimize the UAV design and takeoff trajectory for minimum energy consumption for the
round-trip mission. This section presents the problem statement of UAV design optimization and
trajectory optimization, respectively. We then discuss the two coupled optimization approaches:
iterative sequential optimization and simultaneous optimization.
4.1 UAV Static Design Optimization
We first consider UAV design optimization without incorporating the takeoff trajectory. We call
this problem static because it does not include dynamic trajectory analysis and is only based on steady
flight analyses.
Table 3provides the problem statement. The UAV design variables consist of GTOW, wing area,
10

Table 3: UAV static design optimization problem.
Function/variable Description
Miminize 𝐸total energy consumption for round-trip delivery mission
by varying design 𝑊GTOW gross takeoff weight
𝑆wing area
𝑣cruise (≤100 mph) cruise speed
¯
𝑃𝐹,¯
𝑃𝑉power rating of forward and VTOL motors
𝐷𝐹,𝐷𝑉rotor diameter of forward and VTOL rotors
𝜙𝐹,𝜙𝑉blade twist distribution of forward and VTOL rotors
𝑐𝐹,𝑐𝑉(≥5mm) blade chord distribution of forward and VTOL rotors
control 𝜃cruise,𝜃climb body pitch attitude at cruise and climb
Ω𝑉,hover,Ω𝑉,OMI VTOL rotor speed at hover and one-motor-inoperative (OMI) hover
Ω𝐹,cruise ,Ω𝐹 ,climb forward rotor speed at cruise and climb
𝑣𝑥,climb (≤100 mph) horizontal speed at climb
subject to design 𝐶𝐿≤0.8lift coefficient in cruise
𝑃OMI ≤𝜂¯
𝑃𝑉motor max power at OMI hover
𝑃climb ≤𝜂¯
𝑃𝐹motor max power at climb
𝜏OMI ≤ ¯𝜏𝑉motor max torque at OMI hover
𝜏climb ≤ ¯𝜏𝐹motor max torque at climb
𝑔(𝐷𝑉, 𝑆) ≤ 0spanwise geometry constraint Eq. (10)
𝑟weight =0weight residual Eq. (5)
𝑇/𝑊GTOW ≥1.4thrust-to-weight ratio at OMI hover
trim ¤
𝑣𝑥=0horizontal equilibrium at cruise and climb
¤
𝑣𝑦=0vertical equilibrium at cruise, climb, and hover
cruise speed, motor power ratings, rotor diameter, and the chord and twist distribution of rotor blades.
The chord and twist distribution is parametrized by B-splines with 8 spanwise control points. For the
lift-plus-cruise configuration, the vertical propulsor (motors and rotors) and the forward propulsor
have different design variables. The component weights of the payload, battery, motors, ECSs,
rotors, and wing also vary, although these parameters are not independent optimization variables. In
total, the static design optimization problem has 46 optimization variables (39 UAV design variables
and 7 steady control variables) for the lift-plus-cruise optimization. The tailsitter optimization has
26 optimization variables (21 for design and 5 for control).
As we discussed in Sec. 3.7, we use trajectory optimization to compute the energy consumption
for the takeoff and climb phases. In static design optimization, however, the takeoff energy must be
provided as an input, or we must make an approximation. As an initial guess, we approximate the
takeoff energy by 60 seconds of hovering, following the sizing method of Govindarajan and Sridharan
[19].
4.1.1 Steady Flight Conditions
The static optimization considers four steady flight conditions: cruise, nominal hover, OMI hover,
and wing-borne climb. The cruise and nominal hover points determine the total energy consumption
based on the mission profile in Fig. 4.
The OMI hover condition drives the VTOL motor sizing. We require a thrust-to-weight ratio 𝑇/𝑊
11

of 1.4, considering a margin on top of 𝑇/𝑊=4/3=1.33 (three rotors supporting the weight instead
of four). This condition gives the maximum power and torque required by the VTOL rotors. These
must be lower than the motor power rating variable and the maximum torque given by Eq. (9). These
are imposed by the power and torque constraints, as shown in Table 3.
Likewise, wing-borne climb requirements size the forward motor. We set the 3 m/srate of climb
(RoC) requirement [7] and impose the motor power and torque constraints for the forward motor.
The horizontal speed at the climb point is an optimization variable.
For the force equilibrium, the optimizer varies the body pitch angle (which gives the angle of
attack) and forward rotor speed to satisfy ¤
𝑣𝑥=¤
𝑣𝑦=0at the cruise and climb points. For the vertical
flight points, we impose ¤
𝑣𝑦=0for nominal hover and 𝑇/𝑊≥1.4for OMI hover by varying the
VTOL rotor speed. We do not distinguish the forward and VTOL motors for the tailsitter, so the same
motor must satisfy both the wing-borne climb and OMI hover requirements.
4.1.2 Design Constraints
We set the maximum cruise lift coefficient of 0.8 to avoid undersizing the wing. We determined
this value based on the cruise speed and wing area of the reference UAV listed in Table 2. The cruise
speed is upper-bounded at 100 mph (44.7 m/s) based on the FAA Part 107 requirement.
We also impose a geometry constraint to limit the VTOL rotor disk size from being too large. The
following spanwise geometry constraint prohibits interference between the VTOL rotors and fuselage:
𝑔(𝑅, 𝑆)=𝐷𝑉+2𝑑+𝑤−p𝑆/𝐴𝑅 ≤0,(10)
where 𝑑is the minimum separation between the fuselage and the rotor tip, 𝑤is the fuselage width, and
𝐴𝑅 is the aspect ratio. Figure 5illustrates this constraint. We use 𝑑=0.05 m [7] and 𝑤=0.15 m based
on the geometry of Wing Aviation LLC’s UAV [47]. This constraint is important for the lift-plus-cruise
configuration because the optimizer increases the VTOL rotor diameter until reaching the geometrical
limit. The optimizer determines the VTOL rotor diameter based on the trade-off between increasing
the rotor diameter to reduce the disk loading, which is more power-efficient, and reducing the rotor
diameter to reduce the rotor weight and torque required. For the lift-plus-cruise configuration, the
power reduction is more significant, and the optimizer prefers the larger rotor diameter.
DV
2
DV
2w dd
pS/AR
Figure 5: Spanwise geometry constraint ensures minimum separation between the rotor tip and the fuselage.
12

We impose the same geometry constraint for the tailsitter, with the wing area 𝑆in Eq. (10) replaced
by 𝑆/2to account for the bi-wing configuration. However, the geometry constraint is not active for
the tailsitter because the rotors are also used for forward flight, which changes the balance of the rotor
size trade-off mentioned above.
4.1.3 Estimation of Frame Weight Fraction
We estimated the frame weight fraction 𝑊frame/𝑊GTOW using this static design optimization and
the baseline UAV’s parameters from Table 2. Here, we manually adjusted the frame weight fraction so
that the static design optimization yields the GTOW of 6.4 kg, which is the weight of the baseline vehi-
cle. As a result of manual iteration, the frame weight fraction of 0.603 gives the 6.4 kg takeoff weight.
We use this weight fraction for the following iterative sequential and simultaneous optimizations.
4.2 Takeoff Trajectory Optimization
Takeoff trajectory optimization minimizes the energy consumption for takeoff and climb by vary-
ing the flight path and the control input history. Here, we first consider fixing the UAV design and
only optimizing the trajectory, as opposed to simultaneous optimization. In this paper, takeoff trajec-
tory consists of vertical takeoff, forward transition, climb to the cruise altitude, and acceleration to the
cruise speed. Table 4summarizes the trajectory optimization problem.
Table 4: Takeoff trajectory optimization problem while fixing the UAV design.
Function/variable Description
Minimize 𝐸𝑓=𝐸takeoff energy consumption for takeoff and climb
by varying time 𝑡𝑓duration
states 𝑥horizontal location
𝑦altitude
𝑣𝑥horizontal speed
𝑣𝑦vertical speed
𝐸accumulated energy consumption
controls 𝜃body pitch attitude
Ω𝑉VTOL rotor speed
Ω𝐹forward rotor speed
subject to defect 𝑟defect =0defect constraints for the ODE (3)
boundary 𝑣𝑥 , 𝑓 =𝑣cruise terminal speed
𝜃𝑓=𝜃cruise terminal body pitch attitude
path 𝑃𝑉≤𝜂¯
𝑃𝑉VTOL motor power upper bound
𝑃𝐹≤𝜂¯
𝑃𝐹forward motor power upper bound
𝜏𝑉≤ ¯𝜏𝑉VTOL motor torque upper bound
𝜏𝐹≤ ¯𝜏𝐹forward motor torque upper bound
takeoff ˜
𝑥≤5,˜
𝑦≥10 vertical takeoff requirement
13

4.2.1 Direct Collocation
We use direct transcription with the Radau collocation method [55] to discretize the trajectory
in time. This method approximates the state trajectory by the piecewise Lagrange polynomial. The
optimizer then drives the state variables at discretization nodes as optimization variables. The
dynamics are imposed by defect constraints, which requires the state rate of change from the ODE (3)
to be equal to the rate of change from the polynomial approximation at collocation nodes, that is,
𝑟defect ≡¤
𝜉ODE −¤
𝜉poly
=𝑓ODE(𝑡 , 𝜉, 𝑞;𝑥wing) − 2
𝑡seg
𝐷𝜉=0,(11)
where 𝑡seg is the segment duration and 𝐷is a differentiation matrix for the Lagrange polynomial [55].
We use 25 third-order segments in this study.
4.2.2 Boundary Conditions and Constraints
The takeoff trajectory starts from the ground at zero velocity and ends when the UAV reaches the
cruise altitude and speed. The initial conditions for state variables are 𝜉0=[𝑥0, 𝑦0, 𝑣𝑥0, 𝑣𝑦0, 𝐸0]=0,
where the subscript 0means the initial state. In addition, we set the body pitch attitude of 𝜃0=0for
the lift-plus-cruise configuration and 𝜃0=90 deg for the tailsitter. The initial conditions are imposed
by eliminating the initial states and pitch attitude from optimization variables; therefore, they do not
appear as boundary constraints.
For the terminal condition, we let the duration 𝑡𝑓and final horizontal location 𝑥𝑓be free variables,
where the subscript 𝑓means the final state. The rest of the conditions are 𝑦𝑓=45 m, 𝑣𝑥𝑓=𝑣cruise,
𝑣𝑦𝑓=0, and 𝜃𝑓=𝜃cruise. The cruise speed and pitch attitude are given by the static UAV design
optimization. We impose the terminal conditions for 𝑦𝑓and 𝑣𝑦𝑓by eliminating the final state variable
from the optimization variable. The terminal cruise speed and pitch attitude are imposed as boundary
equality constraints because 𝑣cruise and 𝜃cruise will be variables in the simultaneous optimization.
We impose the motor’s power and torque limits as path inequality constraints. Throughout the
trajectory, the power required by the motor must be lower than the motor power rating, which the
static design optimization determines. Likewise, the torque required must not exceed the maximum
torque of the motor given by Eq. (9).
We also include a constraint to require vertical takeoff and prohibit horizontal wing-borne takeoff.
For this purpose, we place an obstacle of 10 m height (typical height of a three-story building) at a
5 m horizontal offset from the takeoff point. We then require that an arbitrary discretization point
along the trajectory, (˜
𝑥, ˜
𝑦), must be within the 5 m radius from the takeoff point and above the 10 m
height. This requirement is illustrated in Fig. 6. This is not a rigorous path constraint that completely
avoids the obstacle, and a multimodal path may interfere with the obstacle while satisfying the above
constraint. However, we do not consider these edge cases because such multimodal paths are not
energy-optimal, and the optimizer naturally eliminates those.
14

Building
(˜x, ˜y)
5 m
10 m
Figure 6: Vertical takeoff trajectory is required to clear a representative building.
4.3 Iterative Sequential Optimization
The optimal takeoff trajectory depends on the UAV design variables because (1) the UAV weight and
wing area change the UAV’s equations of motion, (2) cruise speed and attitude define the terminal
conditions for trajectory, (3) the motor sizing variables give the power and torque path inequality
constraints, and (4) rotor design variables affect the thrust, torque, and power computations.
At the same time, the optimal UAV design also depends on the takeoff trajectory because it gives
the energy required for takeoff and climb. The takeoff energy estimation is crucial because takeoff
and climb are energy-intensive operations, and the energy requirement drives the battery sizing.
Therefore, a mutual coupling exists between the UAV design optimization and takeoff trajectory
optimization.
One approach is iterative sequential optimization, which is not the best approach, as we will show
later. Iterative sequential optimization repeats the design optimization and trajectory optimization in
a fixed-point iteration fashion. Figure 7shows the overview of this approach. The design optimization
module gives the optimized design to the trajectory optimization module, and the trajectory module
passes the minimized takeoff energy back to the design module. We consider the iteration to have
converged when the relative differences of the GTOW and total energy between two consecutive
iterations are below 10−3; i.e.,
𝑊𝑗
GTOW −𝑊𝑗−1
GTOW
𝑊𝑗
GTOW
≤10−3and 𝐸𝑗
total −𝐸𝑗−1
total
𝐸𝑗
total
≤10−3,(12)
where the superscript 𝑗denotes the fixed-point iteration count. The fixed-point iteration converges in
five iterations or less for this problem.
4.4 Simultaneous Design and Trajectory Optimization
Another approach to account for the design-trajectory coupling is simultaneous optimization. This
approach combines the UAV design and takeoff trajectory optimizations into one monolithic optimiza-
tion and concurrently finds the optimal design and trajectory. This is shown in Fig. 7in comparison
to iterative sequential optimization, where we solve two separate optimizations. Simultaneous opti-
mization finds a better solution than iterative sequential optimization, as we will show in Sec. 6.
Table 5summarizes the simultaneous optimization problem statement. This problem includes
all optimization variables and constraints from the static design optimization (Table 3) and the fixed-
design trajectory optimization (Table 4). The resulting problem for the lift-plus-cruise configuration
has 657 variables and 1076 constraints; the tailsitter problem has 558 variables and 746 constraints.
15

Optimization
Design
Optimization
Trajectory
UAV design
Takeoffenergy
(a) Iterative sequential optimization
Optimization
Design Trajectory
(b) Simultaneous optimization
Figure 7: Diagrams of iterative sequential optimization and simultaneous optimization. Iterative sequential optimization
repeats design optimization and trajectory optimization in a fixed-point iteration fashion. In simultaneous optimization,
monolithic optimization drives both design and trajectory variables concurrently.
Table 5: Simultaneous design and trajectory optimization problem.
Minimize 𝐸total
by varying UAV design 𝑊GTOW, 𝑆, 𝑣cruise,¯
𝑃𝐹,¯
𝑃𝑉, 𝐷𝐹, 𝐷𝑉,𝜙𝐹,𝜙𝑉, 𝑐𝐹, 𝑐𝑉
steady control 𝜃,Ω𝑉,Ω𝐹, 𝑣𝑥,climb at steady points
trajectory 𝑡𝑓, 𝑥, 𝑦, 𝑣𝑥, 𝑣𝑦, 𝐸, 𝜃,Ω𝑉,Ω𝐹
subject to all constraints in Tables 3and 4
Figure 8shows the computational flow of this problem using the extended design structure matrix
(XDSM) [56].
In the literature, simultaneous optimization with the collocation-based trajectory discretization is
also called direct transcription for co-design [38] or simultaneous formulation [40]. In terms of the MDO
terminology, this approach corresponds to the simultaneous analysis and design (SAND) architecture [3,
39,43]. Rigorously speaking, our approach is not pure SAND because we use a nonlinear solver to
converge the BEM residuals at each time discretization node. At the higher level, our formulation
relies on optimization to converge all the other residuals (defect equations, trim at steady analysis
points, and the weight residual) as equality constraints. This makes the model feed-forward except
for the BEM analysis, as shown in Fig. 8.
16

Optimizer S, D, c, ϕ ¯
PV,¯
PF
Wtotal, S, vcruise,
θcruise,ΩF,cruise
Wtotal, S,
ΩV,hover
Wtotal, S, vx,climb,
θclimb,ΩF,climb
Wtotal, S,
ΩV,OMI
Wtotal, S, vcruise, θcruise ,¯
PV,¯
PF,
tf, x, y, vx, vy, E, θ , ΩF,ΩV
Etakeoff
Wtotal, S,
D, ¯
PV,¯
PF
g(Dv, S)≤0Geometry Rotor geom. Rotor geom. Rotor geom. Rotor geom. Rotor geom.
Motor model ¯τF¯τV¯τV,¯τFWmotors
CL≤0.8,
˙vx= ˙vy= 0 Steady cruise Ecruise
˙vy= 0 Steady hover Ehover
P≤η¯
PF, τ ≤¯τF,
˙vx= ˙vy= 0 Steady climb
P≤η¯
PV, τ ≤¯τV,
T/Wtotal ≤1.4
One-motor-
inoperative hover
Defects, path,
boundary cons. Takeoff trajectory
Etotal Energy Wbattery
rweight = 0 Weight
Figure 8: Problem structure of simultaneous design and trajectory optimization. We adopt the SAND architecture, which makes the problem feed-forward, except for the
rotor BEM analysis. The BEM analyses are performed within each steady flight condition and trajectory optimization block.
17

5 Implementation
We use OpenMDAO [45] as an underlying framework for all optimizations in this paper. Open-
MDAO is a Python-based optimization framework that facilitates the implementation of a multidisci-
plinary model and coupled derivative computation. We also use Dymos [44], a trajectory optimization
library built on top of OpenMDAO. Dymos implements the Radau collocation method with analytic
derivatives. For the BEM analysis, we employ CCBlade [49]. CCBlade is implemented in Julia, and
we use a Julia wrapper2to call it from OpenMDAO. We implemented other models for vehicle sizing
and aerodynamics as OpenMDAO components.
OpenMDAO solves the unified derivatives equation (UDE) [3,57] to compute coupled derivatives.
The UDE is a generalization of the implicit analytic methods (which includes the adjoint method), and
it enables efficient and accurate derivative computations for multidisciplinary systems. In solving
the UDE, we use the functional form representation [3,58] for the BEM analysis to accelerate the
derivative computations. The functional form approach puts the BEM analysis in an OpenMDAO
sub-problem and nests it under the top-level OpenMDAO problem. This decomposes the large UDE
linear system into smaller sub-level and top-level linear systems in an equivalent manner, reducing
the computational cost of solving the linear system. The functional form approach for the UDE and
OpenMDAO is explained by Kaneko [58, Ch. 4].
We also use the total Jacobian graph coloring [45,59] to accelerate the UDE linear solution
process further. The graph coloring exploits the total Jacobian sparsity resulting from the collocation
formulation. This effectively compresses the sparse total Jacobian matrix to reduce the number of
UDE linear solver calls, where each linear solver call computes a row or a column of the compressed
Jacobian [3, Sec. 6.8]. The use of the Jacobian coloring is the reason why we do not apply constraint
aggregation [3, Sec. 5.7] on the path inequality constraints. Constraint aggregation also reduces the
number of UDE linear solver calls by aggregating multiple inequality constraints into one; however,
aggregation cannot be used along with the Jacobian coloring because the aggregated constraint is not
sparse. For our problem, the Jacobian coloring is a more efficient choice for path inequalities than
the constraint aggregation because we use the Jacobian coloring either way for the defect constraints,
which are equality constraints and cannot be aggregated.
Partial derivatives for each OpenMDAO component are computed analytically by hand differentia-
tion, except for the CCBlade BEM components, where we use forward-mode automatic differentiation
in Julia [60].
For optimization, we use SNOPT [61] and IPOPT [62] via the pyOptSparse wrapper [63]. SNOPT
implements sequential quadratic programming, and IPOPT is an interior-point solver. We first run
IPOPT to obtain a reasonable initial guess for trajectory, which is helpful for robust convergence.
IPOPT tends to be more robust than SNOPT for our problem when no good initial guess is available.
Once we obtain a good initial guess, we initialize SNOPT from that solution and solve trajectory and
simultaneous optimizations. All results presented in this paper are obtained by SNOPT with a 10−6
convergence tolerance for both optimality and feasibility.
2https://github.com/byuflowlab/OpenMDAO.jl
18

6 Results: Lift-Plus-Cruise Configuration
We compare three optimization results: uncoupled optimization, iterative sequential optimiza-
tion, and simultaneous optimization. Uncoupled optimization means one iteration of the sequential
optimization. It first optimizes the UAV design by approximating the takeoff energy consumption
with 60 seconds of hovering; then, the fixed-design trajectory optimization follows to compute the
total energy consumption.
6.1 UAV Design
Table 6compares the takeoff weight, cruise speed, and total energy consumption of the three
optimization results. Compared to uncoupled optimization, iterative sequential optimization reduces
energy consumption by 2.7%, and simultaneous optimization achieves a 7.4% reduction. Figure 9
shows the energy consumption breakdown by flight phase.
Uncoupled optimization overestimates the takeoff energy consumption, resulting in oversizing
the battery. The static design optimization approximates the takeoff energy by 60 seconds of hover-
ing, which yields 10.12 Wh consumption per one way. However, the fixed-design takeoff trajectory
optimization gives 6.80 Wh. This difference corresponds to 9.4% of the total energy consumption,
meaning that the uncoupled optimization increased the battery weight by about 10%. This is one of
the reasons the takeoff trajectory optimization should be incorporated in the UAV conceptual design
process: it enables a more accurate estimation of the takeoff energy requirement. Without trajectory
optimization, it is challenging to estimate the takeoff energy due to the complex dynamics.
Iterative sequential optimization achieves the lightest vehicle and the highest cruise speed. The
weight reduction is achieved by having smaller wing, rotors, and motor power ratings, as shown in
Fig. 10. The higher cruise speed enables the smaller wing because we impose the cruise lift coefficient
upper bound of 0.8. The lower vehicle weight contributes to reducing the cruise energy, as shown
in Fig. 9. However, iterative sequential optimization requires more energy in other phases despite
the lower vehicle weight. The energy for hover, descent, and landing (recall that we approximate the
descent and landing by 20 seconds of hovering) is higher because the smaller VTOL rotors increase the
Table 6: Optimized vehicle design variables and total energy consumption for the lift-plus-cruise configuration
Optimization GTOW (kg) Cruise speed (m/s) Energy (Wh)
Uuncoupled (static) 6.400 39.30 63.57
Iterative sequential 5.645 (−11.8%) 44.34 (+12.8%) 61.86 (−2.7%)
Simultaneous 5.904 (−7.8%) 39.31 (+0.0%) 58.86 (−7.4%)
0 20 40 60
Energy consumption [Wh]
Uncoupled
Iterative sequential
Simultaneous
13.6
15.3
11.7
38.2
33.2
35.2
11.8
13.3
12.0
63.6 Wh
61.9 Wh
58.9 Wh
Takeoff
& climb Cruise Hover, descent
& landing
Figure 9: Breakdown of the energy consumption by flight phase. The takeoff and climb energy for uncoupled optimization
is computed by the trajectory optimization, and is different from the 60 seconds hover energy assumption.
19

Top
Front
0 0.5 m
Uncoupled
Iterative sequential
Simultaneous
0.0 0.2 0.4 0.6 0.8
Weight [kg]
Battery
Wing
Rotors
Motors
/ESCs
Component weight
0
0.1
0.2
0.3
Chord / R
VTOL rotors Forward rotor
0.2 0.4 0.6 0.8 1.0
r / R
0
20
40
60
80
Twist [deg]
0.2 0.4 0.6 0.8 1.0
r / R
Figure 10: Comparison of optimized UAV designs between uncoupled static design optimization, iterative sequential
optimization, and simultaneous optimization. The left figure visualizes the wing area and rotor diameter variables. The
upper-right figure shows the component weight. The lower-right figures plot the rotor chord and twist distributions.
hover disk loading, which is less power efficient. The takeoff and climb energy is increased because
of the higher cruise speed, meaning more acceleration is needed in this phase. Iterative sequential
optimization cannot account for this acceleration penalty because it uses a fixed amount of takeoff
energy when optimizing the vehicle design.
Simultaneous optimization can consider this acceleration penalty, reducing total energy con-
sumption by 4.8% compared to iterative sequential optimization. The main difference between the
two results is the cruise speed. Simultaneous optimization decreases the cruise speed compared
to iterative sequential optimization. This requires less acceleration in the takeoff phase, reducing
the takeoff energy by 24% at a small sacrifice in the cruise energy. Simultaneous optimization also
chooses to have larger VTOL rotors and higher motor power ratings. These increase the weight,
but the larger VTOL rotors (hence the lower disk loading) reduce the hovering power. The higher
motor power rating allows an energy-efficient takeoff trajectory, which we discuss in the next section.
Overall, simultaneous optimization found the best design by optimally balancing the advantages and
disadvantages of the component weight and energy efficiency of different flight phases. Iterative
sequential optimization fails to capture this system-level trade-off. This highlights the importance of
simultaneous design-trajectory optimization.
On the rotor blade design, simultaneous optimization results in a shorter chord for the VTOL rotors
than the uncoupled and iterative sequential optimizations. This blade design is slightly tailored
20

toward the rotor-borne forward flight that occurs during the transition, compared to the hover-
optimized blades in uncoupled and iterative sequential optimizations. To demonstrate this point,
we performed a post-optimization BEM analysis of the VTOL rotors in an edgewise forward flight
condition. We set the 20 m/s forward speed, 10 deg tilt angle, and the thrust satisfying 𝑇=𝑚𝑔, which
are representative of the rotor-borne flight experienced by the optimal takeoff trajectories. Table 7
compares the hovering figure of merit (FoM) and 𝑇/𝐹, the ratio of the thrust to the rotor in-plane force.
This ratio is analogous to the lift-to-drag ratio of the lifting surface when the VTOL rotor is seen as a
lifting surface. The VTOL rotor of simultaneous optimization sacrifices the hover figure of merit to
improve 𝑇/𝐹, making rotor-borne forward flight more efficient. Simultaneous optimization fine-tunes
the blade design for the best trade-off between hover and forward flight efficiency. However, for the
overall vehicle design, this rotor design’s trade-off is less significant than the first-order trade-off in
terms of the weight and cruise speed discussed above.
The forward rotor blade designs are almost the same between the three optimizations. The
iterative sequential optimization results in a slightly larger twist and shorter chord to accommodate
the higher cruise speed.
Table 7: VTOL rotor performance in hover and rotor-borne forward flight
Optimization Hover FoM Forward 𝑇/𝐹
Uncoupled 0.661 28.1
Simultaneous 0.614 42.8
6.2 Takeoff Trajectory
Figure 11 compares the takeoff and climb flight path between the three optimizations. We also
show the power and torque time histories in Fig. 12. The power and torque reach the upper bounds,
which are equal to the motor’s power and torque ratings. This shows the significant influence of the
motor sizing variables on the optimal takeoff trajectory.
Simultaneous optimization achieves an energy-efficient climb trajectory by exploiting wing-borne
flight as much as possible. As shown in Figs. 11 and 12, it turns off the VTOL motors and shifts to the
wing-borne flight earlier than the uncoupled and iterative sequential optimizations. This is highly
efficient because the forward thrust in the wing-borne flight is significantly lower than the vertical
thrust for rotor-borne (helicopter-like) flight. To enable the early transition, simultaneous optimization
increases the forward motor’s power and torque ratings, as shown in Fig. 12. Its motor rating is
even higher than the uncoupled optimization despite the GTOW being lighter in the simultaneous
optimization. The high motor rating means a heavy motor, but the benefit of the early transition is
more significant than the weight penalty at the system-level trade-off.
In contrast, the trajectory by iterative sequential optimization is not efficient. It requires the longest
horizontal distance to accelerate to the cruise speed because of its high cruise speed. This directly
increases the energy consumption for the takeoff trajectory. It also remains in rotor-borne flight longer
before gaining sufficient speed for the wing to generate adequate lift. As Fig. 12 shows, the iterative
sequential optimization has the lowest power and torque ratings for the forward motor. This forward
motor is lightweight but takes longer to accelerate, meaning the UAV must rely more on inefficient
rotor-borne flight.
21

0
45
Altitude [m]
Uncoupled Iterative sequential
Simultaneous
Uncoupled
Cruise speed: 39.3 m/s
Takeoff/climb energy: 6.8 Wh
Iterative
sequential
44.3 m/s
7.7 Wh
0 100 200 300 400 500 600 700 800
Horizontal location [m]
Simultaneous
39.3 m/s
5.8 Wh
Figure 11: Optimized takeoff trajectories. The red vectors show the rotor thrust magnitude and direction (VTOL rotors and
a forward rotor combined). The gray aircraft profiles show the body pitch angle. The thrust vectors and body attitudes are
plotted with 2-second intervals.
0
500
1000
1500
VTOL power
Power [W]
0.0
0.1
0.2
0.3
VTOL torque
Torque [N-m]
0 5 10 15 20 25 30
Time [s]
0
200
400
600
Forward power
0 5 10 15 20 25 30
Time [s]
0.0
0.1
0.2
0.3
0.4
0.5
Forward torque
Uncoupled
Iterative sequential
Simultaneous
Figure 12: Time histories of the motor power (left) and the torque (right). The top row corresponds to the VTOL motors,
and the bottom figures are for the forward motor. The dashed lines show the motor power rating (i.e., max continuous
power) and max continuous torque, which are the motor design variables.
22

6.3 Effect of Cruise Speed
We showed that simultaneous optimization reduces energy consumption by lowering the cruise
speed compared to iterative sequential optimization. However, the higher cruise speed might be
preferable from a delivery time standpoint. In this section, we explore the trade-off between the
cruise speed and the energy consumption by performing simultaneous optimizations under fixed
cruise speeds. We still solve the same energy minimization problem stated in Table 5, except we
remove the cruise speed from the design variables.
Figure 13 (upper left) shows how the energy consumption varies with respect to the cruise speed.
Increasing the cruise speed from the optimal solution (39.3 m/s) is accompanied by the energy
penalty. At 44.3 m/s cruise speed (the iterative sequential optimization solution), the total energy
consumption is 3.4% higher than the minimum energy. In the concept of operations shown in Fig. 4,
the 39.3 m/s cruise speed results in 5.2 min delivery flight time, whereas the 44.3 m/s reduces it to
4.7 min.
The vehicle weight decreases as we increase the cruise speed, and it approaches the iterative
sequential optimization solution. However, even at the same 44.3 m/s, simultaneous optimization
resulted in a 1.9% heavier vehicle yet 1.6% less energy consumption than iterative sequential opti-
mization. The difference between these same-speed designs is similar to the previous discussion. The
design by simultaneous optimization has a higher power and torque rating for the forward motor.
This leads to a heavier weight but enables the earlier transition to efficient wing-borne flight. Overall,
the simultaneous optimization reduces the takeoff energy with a small cruise energy penalty. This is
shown in the right plot of Fig. 13.
58
59
60
61
62
Total
energy [Wh]
-1.6%
-4.8%
Iterative
sequential
Simultaneous
Optimal
36 38 40 42 44
Cruise speed [m/s]
5.6
5.7
5.8
5.9
6.0
GTOW [kg]
+1.9%
+4.6%
(a) Total energy and vehicle weight
36 38 40 42 44
Cruise speed [m/s]
5
10
15
20
25
30
35
Energy by
phase [Wh]
Optimal
39.3 m/s
Takeoff & climb
Cruise
Hover
Descent & landing
(b) Energy breakdown by flight phase
Figure 13: Energy consumption and vehicle’s GTOW for various cruise speeds. The upper-left figure shows the trade-off
between energy consumption and cruise speed. The right figure shows the breakdown of energy consumption by each
flight phase.
23

6.4 Effect of Battery Specific Power
So far, we have used the energy requirement to size the battery but have not considered the
battery’s power delivery capability. This is based on the assumption that the UAV equips a lithium-
polymer battery, which is used for the majority of lightweight UAVs [64] and the literature on the UAV
conceptual design [9,14,19]. Lithium-polymer batteries have a high discharge speed3, and the battery’s
maximum power output is not a driving factor for battery sizing. However, power considerations are
essential when using a lithium-ion battery instead, which can be more advantageous for the life-cycle
cost.
In this section, we add a constraint on the battery power output and discuss its implications. The
additional constraint is given as
𝑃𝑉+𝑃𝐹
𝜂≤0.85𝜌𝑝𝑊battery ,(13)
where the left-hand side is the power required for propulsion from Eq. (2), and 𝜌𝑝is the battery’s
specific power. The factor of 0.85 accounts for the power transmission loss, which is consistent with
the assumption made for battery sizing in Sec. 3.6. This constraint is imposed for all steady flight
conditions and trajectories as a path inequality constraint.
Figure 14 shows how the total energy consumption and vehicle weight change as a function of the
battery’s specific power. Here, we solved simultaneous optimizations with three different values of
specific power: 1.2, 1.5, and 2.0 kW/kg. A specific power of 1.2 kW/kg corresponds to the maximum
continuous discharge C-rating of 6.85 under our 175 Wh/kg specific energy assumption. This C-rating
value is approximately the same as existing lithium-ion battery cells4. The values of 1.5 and 2.0 kW/kg
assume further advances in battery technology. The rightmost point in Fig. 14 corresponds to the
results without the battery power constraint, which we discussed in Sec. 6.1 and 6.2. For this case, we
back-calculated the minimum specific power required based on the maximum power discharge and
the battery weight. This yields 3.52 kW/kg of specific power.
The UAV becomes heavier and consumes more energy as we decrease the battery’s specific power,
especially below 2.0 kW/kg. This is because the UAV needs to equip a heavier battery to meet the
power demand in high-power operations (takeoff transition and OMI hover). For a specific power of
1.2 and 1.5 kW/kg, the battery’s state of charge after the round-trip delivery mission is above 20%.
This means that the battery is sized by the power requirement and is oversized in terms of the energy
requirement. This demonstrates the importance of considering the battery’s power output capability
if using a lithium-ion battery.
Figure 15 compares the takeoff trajectories with and without the battery power constraint. With a
specific power of 1.2 kW/kg, the UAV requires a longer horizontal distance to reach the cruise state.
It also relies on inefficient rotor-borne flight for a longer time. These are because of the higher vehicle
weight and the limited power available from the battery.
Figure 16 shows the time history of propulsive power. When not imposing the battery power
constraint, the UAV uses both forward and VTOL motors at full throttle from 4 to 10 seconds. This
enables fast horizontal acceleration while supporting the weight by VTOL rotors, contributing to the
early transition to wing-borne flight. With the power constraint of 1.2 kW/kg specific power, on the
3For example, Gens ace G-Tech 8000mAh lithium-polymer battery, on which our specific energy assumption based, has
a 100 C discharge rate. (Linked webpage: accessed on March 20, 2024)
4Based on the data summarized in https://www.leadsresearchgroup.com/technology-dashboard; retrieved on
March 22, 2024.
24

other hand, it gradually increases the forward motor throttle and decreases the VTOL motor throttle
from 8 to 20 seconds. The UAV cannot operate both motors at full throttle simultaneously because of
the battery power limit. This results in a less efficient transition that uses the VTOL rotors longer.
20.111.48.66.9
Max. discharge C-rate
60
70
80
90
100
Total
energy [Wh] +58.4%
1.2 1.5 2.0 3.5
Battery specific power [kW/kg]
6
7
8
9
10
GTOW [kg]
Figure 14: Energy consumption and takeoff weight for various values of the battery’s specific power. Optimization without
battery power output constraint resulted in requiring a specific power of 3.52 kW/kg.
0
45
Altitude [m]
No battery power output limit With limit
No battery power
output limit
Cruise speed: 39.3 m/s
Takeoff/climb energy: 5.8 Wh
0 200 400 600 800 1000
Horizontal location [m]
With limit
(1.2 kW/kg)
37.0 m/s
10.7 Wh
Figure 15: Optimized takeoff trajectories with and without the battery power constraint.
0 10 20 30 40
Time [s]
0
0.5
1.0
1.5
Power [kW]
No battery power
output limit
With limit
Total power
0 10 20 30 40
Time [s]
VTOL
Forward
Total
Breakdown (no limit)
0 10 20 30 40
Time [s]
VTOL
Forward
Total
Breakdown (with limit)
Figure 16: Propulsive power time histories. Left: comparison of total power. Middle: power breakdown without battery
power constraint. Right: power breakdown with the battery power constraint.
25

7 Results: Tailsitter Configuration
This section presents the optimization results of the tailsitter configuration. We use the same
model, design variables, and constraints as the lift-plus-cruise configuration except for two differences.
First, the tailsitter has only one set of motors, and the same motors satisfy the power and torque
constraints at both the steady climb and OMI hover conditions. Second, we now consider propeller-
wing interaction using a simple model explained in Sec. 3.5.
In practice, a tailsitter has a different parasite drag coefficient and frame weight fraction from
the lift-plus-cruise configuration. Capturing these requires a higher-fidelity aerodynamic model and
a frame structural analysis. We ignore these differences in this work because comparing the two
configurations is not the main goal of this paper. Our main goal and contribution is to clarify the need
to incorporate trajectory optimization into UAV conceptual design for each VTOL configuration.
Table 8shows the GTOW, cruise speed, and total energy consumption as a result of uncoupled
optimization, iterative sequential optimization, and simultaneous optimization. Figure 17 compares
the UAV design variables from the three optimizations. Here, we again assume a lithium-polymer
Table 8: Optimized vehicle design variables and total energy consumption for the tailsitter configuration
Optimization GTOW (kg) Cruise speed (m/s) Energy (Wh)
Uncoupled (static) 9.321 31.33 122.52
Iterative sequential 6.988 (−25.0%) 30.53 (−2.5%) 87.02 (−29.0%)
Simultaneous 7.023 (−24.6%) 29.09 (−7.1%) 86.63 (−29.3%)
Top
Front
0 0.5 m
Uncoupled
Iterative sequential
Simultaneous
0.00 0.25 0.50 0.75 1.00 1.25 1.50
Weight [kg]
Battery
Wing
Rotors
Motors
/ESCs
Component weight
r / R
0
0.1
0.2
Chord / R
Rotor chord and twist
0.2 0.4 0.6 0.8 1.0
r / R
0
25
50
Twist [deg]
Figure 17: Comparison of optimized tailsittter UAV designs between static design optimization, iterative sequential opti-
mization, and simultaneous optimization.
26

battery and omit the battery power constraint (13). Compared to uncoupled optimization, iterative
sequential optimization and simultaneous optimization achieve 29% energy reduction by downsizing
the vehicle. This is because uncoupled optimization significantly overestimates the energy required
for takeoff and climb, hence oversizes the battery. As shown in Fig. 18, the tailsitter climbs and
accelerates to the cruise state quickly with a small energy consumption. This quick transition is
enabled by the high-power motors sized for the OMI hover requirement. Uncoupled optimization,
however, does not compute the takeoff energy by trajectory optimization but approximates it by
60 seconds of hovering. This results in overestimating the takeoff energy and the battery weight.
The difference between simultaneous optimization and iterative sequential optimization is marginal
in the tailsitter configuration, unlike the lift-plus-cruise results. Simultaneous optimization decreases
the total energy consumption by 0.46% by reducing the takeoff and hover energy at a small sacrifice of
the cruise energy, as shown in Fig. 19. This is achieved by having a hover-oriented rotor blade design.
The tailsitter’s rotor design must balance the hover efficiency (near-zero inflow, high loading) and
cruise efficiency (fast inflow, low loading). Simultaneous optimization tailors the design toward the
hover efficiency, which is optimal for overall energy consumption. Iterative sequential optimization
cannot fully capture this hover-cruise trade-off. However, the 0.46% energy difference between the
two optimizations is not significant.
0
45
Altitude [m]
Uncoupled
Cruise speed: 31.3 m/s
Takeoff/climb energy: 5.4 Wh
Iterative
sequential
30.5 m/s
4.0 Wh
0 20 40 60 80 100 120
Horizontal location [m]
Simultaneous
29.1 m/s
3.8 Wh
Figure 18: Optimized takeoff trajectories. The red vectors show the rotor thrust magnitude and direction. The gray aircraft
profiles show the body pitch angle. The thrust vectors and body attitudes are plotted with 1-second intervals.
27

0 25 50 75 100 125
Energy consumption [Wh]
Uncoupled
Iterative sequential
Simultaneous
10.7
8.0
7.7
80.7
53.4
54.1
31.1
25.6
24.8
122.5 Wh
87.0 Wh
86.6 Wh
Takeoff
& climb Cruise Hover, descent
& landing
Figure 19: Comparison of energy breakdown by flight phase for optimized tailsitter UAVs.
Overall, the tailsitter optimization results demonstrate the importance of incorporating takeoff
trajectory optimization in the UAV conceptual design because it enables accurate estimation of the
energy and the battery required. However, unlike the lift-plus-cruise configuration, fully-coupled
simultaneous optimization may not be necessary for the tailsitter configuration.
8 Conclusions
In this work, we presented simultaneous optimization of the conceptual design and takeoff tra-
jectory of eVTOL UAVs. We studied lift-plus-cruise and tailsitter UAVs that deliver a 1.2 kg payload
for a 10 km range. Simultaneous design-trajectory optimization found the optimal vehicle design by
balancing the efficiency of vertical flight, takeoff transition, and cruise for minimum overall energy
consumption.
The UAV sizing model in this study combined a BEM rotor analysis, an empirical post-stall
aerodynamic model, and a component weight build-up. Integrating the BEM analysis into design and
trajectory optimization allowed us to compute the power and torque required for various flight phases
with higher accuracy than the conventional disk-momentum theory. This is crucial for obtaining the
correct rotor design, motor sizing, and battery sizing.
We compared three optimization approaches to minimizing the total energy consumption for
a VTOL flight mission by varying the UAV design and takeoff trajectory. Uncoupled optimization
first performs the UAV design optimization by approximating the takeoff energy with 60 seconds of
hover, then it solves takeoff trajectory optimization while fixing the UAV design. Iterative sequential
optimization repeats the UAV design optimization and takeoff trajectory optimization as a fixed-point
iteration. Finally, simultaneous optimization solves a monolithic optimization problem that concurrently
varies design and trajectory variables.
For the lift-plus-cruise configuration, simultaneous optimization reduced the energy consumption
by 7.4% compared to uncoupled optimization and 4.8% compared to iterative sequential optimization.
Simultaneous optimization chose a lower cruise speed and heavier motors. This increased the vehicle
weight, but it enabled energy-efficient takeoff and climb trajectory that exploits wing-borne flight as
much as possible. Overall, the takeoff energy reduction was more significant than the weight penalty.
This system-level trade-off can only be captured by simultaneous optimization.
For the tailsitter configuration, the iterative sequential optimization and simultaneous optimiza-
tion resulted in a 29% energy reduction compared to the uncoupled optimization. Uncoupled opti-
mization overestimated the energy consumption for takeoff and climb, leading to an oversized battery.
The other two optimization approaches avoided battery oversizing by incorporating takeoff trajectory
optimization. This enables a more accurate estimation of the takeoff energy consumption, which is
otherwise challenging because of the complex dynamics of the takeoff transition. Between simulta-
neous optimization and iterative sequential optimization, there was only a 0.46% difference in total
28

energy consumption.
The results of this work demonstrate the importance of considering takeoff trajectory optimiza-
tion in UAV conceptual design. Furthermore, depending on the VTOL configuration, simultaneous
optimization can find a superior design solution than iterative sequential optimization.
Future work recommendations include considering propeller acoustics and noise constraints,
which are important factors for delivery UAVs and AAM vehicles. Thermal models for motors and
batteries should also be considered because thermal requirements (in addition to the power, torque,
and energy requirements) could drive propulsion system sizing. Using an efficiency map for motor
and ESC efficiency also increases the practicality of the results. Besides improving the UAV design
model, sensitivity studies of the optimized designs with respect to various mission requirements,
particularly the payload weight and range, would offer useful insights into the eVTOL design space.
Acknowledgments
This work was partially funded by Japan Student Services Organization (JASSO) Fellowship
Program. The first author was also partially supported by the Michigan Institute for Computational
Discovery and Engineering (MICDE) fellowship. The authors thank Max Opgenoord for giving
feedback on this work. We also thank Daniel Ingraham for providing an example of the CCBlade’s
OpenMDAO wrapper5.
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