TYPE Original Research

PUBLISHED 11 October 2023

DOI 10.3389/fpace.2023.1278726

Trajectory generation based on

OPEN ACCESS power for urban air mobility
EDITED BY
Zhaodan Kong,
University of California, Davis, Russell A. Paielli*
United States
NASA Ames Research Center, Moffett Field, CA, United States
REVIEWED BY
Basman Elhadidi,
Nazarbayev University, Kazakhstan
Sergio Esteban,
Sevilla University, Spain A method of generating trajectories based on power is proposed for Urban Air
*CORRESPONDENCE Taxis. The method is simpler and more direct than traditional methods
Russell A. Paielli, because it does not require a detailed aircraft model or a flight control
russ.paielli@nasa.gov
model. Instead, it allows the user to specify the route, the static
RECEIVED 16 August 2023 longitudinal profile (altitude as a function of distance), and a power model
ACCEPTED 22 September 2023
to determine the progress in time along that profile. The power model can be
PUBLISHED 11 October 2023
determined from a recorded or simulated trajectory of the same aircraft type.
CITATION
Paielli RA (2023), Trajectory generation
This capability allows a trajectory to be generated or reshaped to avoid
based on power for urban air mobility. conflicts while preserving the basic performance characteristics. Net or
Front. Aerosp. Eng. 2:1278726. excess power is defined as the rate of change of mechanical (kinetic and
doi: 10.3389/fpace.2023.1278726
potential) energy, and it is modeled as a function of airspeed. The time steps
COPYRIGHT between discrete points in space along the trajectory are used to yield a
© 2023 Paielli. This is an open-access
article distributed under the terms of the
specified power as a function of airspeed, and they are determined by solving
Creative Commons Attribution License a cubic polynomial at each point. An elliptical profile is used to generate an
(CC BY). The use, distribution or example trajectory. The dependence of trip flight time on various parameters
reproduction in other forums is
permitted, provided the original author(s)
is analyzed and plotted.
and the copyright owner(s) are credited
and that the original publication in this
KEYWORDS
journal is cited, in accordance with
accepted academic practice. No use, eVTOL, trajectory, power, energy, air taxi, simulation
distribution or reproduction is permitted
which does not comply with these terms.

1 Introduction
The developing concept and technology of Urban Air Mobility (UAM) has the potential
to revolutionize short-range air travel in densely populated urban areas (Thipphavong et al.,
2018; Urban Air Mobility, 2020; Goodrich and Theodore, 2021; Hill et al., 2021). Many
airframe developers, both new and old, are developing new electric aircraft models capable of
vertical takeoff and landing (eVTOL) for UAM. Although none of these vehicles are yet
certified by the FAA, the ultimate vision is to have thousands of them in the sky during
commuting hours in major metropolitan areas. The challenges are daunting, however, and
the most critical challenges pertain to safety. Public acceptance will require that catastrophic
accidents be extremely rare.
In the early stages of the development, traffic densities will be low, and the main safety
concerns will be the airworthiness of individual vehicles and keeping them safely in flight and
away from static obstacles. As traffic density increases, however, the safety concerns will shift
to the traffic and the potential for mid-air collisions. The development of an air traffic control
(ATC) system for UAM will certainly require extensive simulation studies, and those studies
will require flight modeling and simulation.
The fidelity of aircraft flight modeling forms a spectrum from low to high. Typically, a
lower level of fidelity is required for modeling air traffic than is required for engineering
development of a particular model of vehicle. Vehicle engineering and control system
design usually requires a detailed model with six or more degrees of freedom, including
controls and actuators, whereas traffic modeling for ATC development and simulation

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testing as well as actual operations (including conflict detection

and resolution) in the field usually only requires a simpler point-
mass model (Zhang et al., 2010; Chatterji, 2020; Shaw-Lecerf et al.,
2020).
This paper proposes an even simpler method of constructing
realistic trajectories without the need for a simulation model of the
aircraft and its flight controls. This method allows the user to
directly specify the route, the static longitudinal profile (altitude as
a function of distance), and a power model to determine the
progress in time along that profile. The power model can be
determined from a recorded or simulated trajectory of the same
aircraft type. This capability allows a trajectory to be “bent” into
any desired shape while retaining the same basic flight
performance characteristics. To the best of the author’s
knowledge, this capability has not been proposed in previous
literature on trajectory generation.
This method of trajectory generation is useful for several
reasons. Aircraft designers are often unwilling to provide
detailed simulation models of their aircraft, but this method
does not require one. And even if such a model is available,
determining the controls to fly a specific trajectory in 4D space
is not simple, but this method does not require that to be done.
Given a sample of a recorded or a simulated trajectory, it can be
used to directly generate a flyable trajectory for conflict resolution.
For many conflicts, this method also facilitates the construction of
a conflict-free path in 3D space using altitude separation, with no FIGURE 1
Top: Plan view of trajectory bounds in the horizontal plane;
dependence on the timing along the path. Bottom: Side view of trajectory bounds in the longitudinal plane.
The methods used in this paper are based in part on the
Trajectory Specification (TS) concept (Paielli and Erzberger,
2019; Paielli, 2022). TS is a method of specifying a trajectory
with explicit tolerances relative to a reference trajectory such that
the position at any given time in flight is constrained to a precisely
defined volume of space as shown in Figure 1. The reference
trajectory is specified as a position in 3D space as a function of
time, where the time steps between discrete points can vary, typically
being larger in steady-state flight than during transients. The
bounding volume at any given time in flight is determined by
tolerances relative to the reference trajectory in all three route-
oriented axes: cross-track, vertical, and along-track. The tolerances
can vary as a function of distance along the route, typically
increasing for departure and decreasing for arrival (for
conventional aviation).
That allowance for varying time steps in TS is used in this
paper to set the specified power level for the airspeed at that time.
The user provides the power function (power as a function of
airspeed) that is appropriate for the aircraft and the flight
FIGURE 2
mission. The time steps between discrete points in 3D space Maximum rate of climb as a function of horizontal airspeed for
are then computed to yield the speed corresponding to the the NASA Reference Quadrotor six-passenger eVTOL aircraft.
required power. The TS software then automatically converts
to a specified constant time step by interpolating between the
varying time steps. The resulting uniform time steps allow for fast
(constant time) lookup of flight variables as a function of time by eVTOL aircraft. Section 3 presents the algorithm for
simple array indexing, followed by interpolation between the two generating the trajectories, and Section 4 analyzes the
bounding points for better accuracy than just choosing the dependence of trip flight time on the various parameters used
nearest point. in generating the trajectory. Finally, conclusions are presented,
The next section discusses the method of generating followed by a brief Appendix A on solving for the roots of a cubic
trajectories based on power for conventional fixed-wing and (third-order) polynomial.

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2 Trajectory generation based on dynamics is common for simulation models that are not high
power fidelity. It can be inaccurate for large turns at high bank angles,
but a simple method will be discussed later to account for the effect
For any aircraft type, steady flight at a given altitude and of bank angle, if necessary. A basic method will also be discussed to
airspeed requires a certain amount of power to maintain. That model the effect of winds.
power ultimately comes from the engines or batteries, and it can be
applied to the vehicle through either direct thrust, propellers, rotors,
or lift, depending on the aircraft type. Any power beyond what is 2.1 Conventional fixed-wing aircraft
needed to maintain steady state that is not wasted as thermal energy
and drag goes into mechanical energy, which is comprised of For a conventional fixed-wing aircraft, the engine power is applied
potential energy due to altitude and kinetic energy due to speed. through 1) the engine thrust and 2) the lift on the wings and other
That “net” or “excess” power is the rate of change of mechanical surfaces. The thrust goes mainly into kinetic energy, but a component of
energy and integrates over time to determine mechanical energy in it also goes into direct lift, depending on the pitch angle of the aircraft at
terms of speed and altitude. any given time. The lift goes into potential energy. The elevator is used
For most aircraft types, the maximum available net power increases to control the pitch angle, which determines the ratio of the engine
with airspeed to some maximum, because it depends on the mass flow power that goes into kinetic and potential energy.
through the engines or rotors, then it decreases as drag increases at The four basic controls of a conventional fixed-wing aircraft are 1)
higher speeds. Maximum available power at takeoff is usually less than it the throttle (or thrust), 2) the elevator, 3) the ailerons, and 4) the rudder.
is in forward flight up to some speed. For example, Figure 2 shows the The throttle and the elevator are for longitudinal control, and the
maximum rate of climb as a function of airspeed for the NASA ailerons and rudder are for lateral control to follow an assigned route. A
Reference Quadrotor six-passenger air taxi at a pressure altitude of simplifying assumption in this paper will be that the aircraft is able to
6,000 ft, as computed by the NASA Design and Analysis of Rotorcraft follow its route exactly, so an explicit model for lateral control is not
(NDARC) program (Johnson, 2015; Silva et al., 2018). The rate of climb needed. For longitudinal control (of a fixed-wing aircraft), an explicit
reaches a maximum of slightly less than 2000 fpm at an airspeed between model can have inputs of throttle and elevator, but a simplified model
50 and 60 knots, and the maximum airspeed is approximately 120 knots. can be based on power, as will now be explained.
To avoid confusion, it should be pointed out that the net or The net or excess power beyond what is needed for steady flight at
excess power referred to here is not based on the maximum available any given speed and altitude can be determined as a function of
power. It is based on the power that is appropriate for the aircraft equivalent airspeed (EAS) or, at lower Mach numbers, “calibrated”
and the mission at any given time, position, and airspeed. In steady- airspeed (CAS). EAS and CAS are based on dynamic pressure and,
state flight, the net power is zero by definition. For climb, it can unlike true airspeed (TAS), implicitly account for the effect of altitude
range from slightly above zero to maximum power minus the power on air density, thereby eliminating the explicit dependence of
needed for steady-state flight at the current airspeed and altitude. aerodynamic forces on altitude. The net power must be split into
For descent it is negative. For reliable flight control, it is usually wise two components: one that drives kinetic energy and another that drives
to maintain a margin below maximum power so that power can be potential energy. The power model therefore requires a kinetic
increased through feedback when necessary to compensate for component and a potential component, both of which are functions
modeling errors, including wind modeling errors in particular. of CAS. Different models can be developed based on different throttle
A net power model for any trajectory can be derived from and airspeed (CAS/Mach) schedules for climb and descent, but that is
recorded or simulated trajectory data by simply computing the rate outside the scope of this paper, which will focus on eVTOL air taxis.
of change of mechanical energy as a function of airspeed and any
other flight variable of interest. The resulting model will depend on
how the aircraft was flown, however, so the trajectory that is used 2.2 eVTOL air taxis
should not have arbitrary maneuvers that would not normally be
used in flight with no other traffic in the airspace. Note also that Many different types of eVTOL air taxi vehicles are currently in
when a new trajectory is generated using this method, the speed can development, including multi-rotor, tilt-rotor, tilt-wing, rotor/push-
be limited at any time in flight if necessary for conflict resolution or prop hybrid, and others. Unlike conventional fixed-wing aircraft,
airspace speed limits (e.g., near vertiports). most of them can climb and descend at any angle. A flight profile
For convenience, power will be expressed in this paper in terms of might be to takeoff and climb vertically for 20 feet, transition to a
power per unit mass divided by gravitational acceleration. This unit has flightpath angle of 10° and climb to the cruise altitude, then reverse
an intuitive meaning because it is the rate at which the vehicle can climb the pattern for descent and landing. This profile determines altitude
vertically, hence it will be expressed in units of feet per minute (fpm). as a function of distance flown, with no reference to speed or time.
Regardless of the units used, however, power varies with airspeed and Such a profile will be referred to here as a static longitudinal profile,
determines both climb rate and acceleration throughout the flight. and it can be converted to a dynamic longitudinal profile (or simply a
The net or excess power determines the longitudinal profile of longitudinal profile) by specifying time as a function of the distance
the flight, which is 1) the horizontal distance flown as a function of along the static profile. The next section provides an algorithm to do
time and 2) the altitude as a function of time or distance. Assuming that conversion based on the power model discussed earlier.
decoupled lateral and longitudinal dynamics, that longitudinal As mentioned earlier, the longitudinal profile can be superimposed
profile superimposed on the route determines the trajectory in on a route to determine a trajectory in 4D space. Determining the
4D space. The assumption of decoupled lateral and longitudinal control settings or values as a function of time that are necessary to fly

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the resulting trajectory for a given vehicle is not easy, but it is not
necessary for a simulation of air traffic or to find a flyable trajectory that
is free of conflicts with other traffic. All that is needed is a realistic
trajectory that is flyable and acceptable to the flight operator. The
aircraft flight control system should then be able to compute the
feedforward control variables as a function of time that are
necessary to track the trajectory, and that will be combined with
feedback to compensate for modeling errors as has been done for
helicopters (Takahashi et al., 2017). The trajectories generated by the
methods in this paper should therefore be usable for actual trajectory
assignments in the field.

3 Trajectory generation algorithm

FIGURE 3
The trajectory generation algorithm presented in this paper is
Static longitudinal profile for an elliptical climb.
composed of four main parts: 1) generating the static longitudinal
profile, 2) converting it to a (dynamic) longitudinal profile by
applying a power model, 3) superimposing that profile onto a
specified route, and 4) adjusting for the effects of winds. The first The descent profile is constructed using the same algorithm that
step is purely geometric, and the second step is kinetic. The is used for the climb segment, with the same or different parameters,
geometric step specifies the shape of the trajectory in 3D space, then it is reversed and shifted along-track to match the end with the
independent of time, and the kinetic step fills in the times to yield a end of the specified route. A steady level cruise segment is then
specified power profile as a function of calibrated airspeed (CAS) inserted between the climb and descent profiles to fill the gap
and any other flight variable of interest, including time, position between them and complete the static longitudinal profile. If the
along the route, altitude, and flightpath angle. The third step is to route is too short for a level cruise segment at the specified altitude,
superimpose the longitudinal profile onto the route, and the final then a lower cruise altitude should be used or the length of the climb
step of adjusting for effects of winds completes the construction of and descent segments should be decreased, or both.
the trajectory in 4D space. Each discrete point of the static longitudinal profile consists
of an along-track distance and an altitude. After the power model
(to be discussed later) is applied, each point will also have a time
3.1 Static longitudinal profile associated with it. The distance and time spacing between points
can vary, and they are arbitrary within a wide range, but a few
As explained earlier, the static longitudinal profile is the altitude as basic considerations apply. The smaller the spacing between
a function of horizontal distance along the route, with no reference to points, the higher the resolution of the trajectory will be (and
speed or time. It is purely geometric. A basic profile that has been the more computation and storage space will be required, of
proposed by industry is to take off and climb vertically to some course). Numerical roundoff error can become significant if the
specified height above the vertipad, then transition to some non- time steps are too small, but with standard 64-bit floating-point
vertical climb angle, say 10°, and climb to the cruise altitude. The arithmetic they would have to be very small for that to become a
methods proposed in this paper can be applied to any reasonable concern. A more detailed description of the construction of the
profile, but an elliptical profile will be used as the main example static longitudinal profile now follows.
because it provides a smooth transition from vertical takeoff to level For an ellipse centered at the origin and aligned with the x and y
flight, which should result in a smooth ride. However, it may not be coordinate axes, the position at any given angle θ is x = a sin θ and
appropriate for all eVTOL aircraft types, depending on the method of y = b sin θ where a and b are the semi-major and semi-minor axes of
transition from vertical to forward flight and possibly other the ellipse. As shown in Figure 3, the ellipse is not centered at the
considerations as well. An optimal or reasonably efficient and origin in this case but is tangent to the vertical axis at the pad
smooth profile can be determined and used for each aircraft type elevation (or at the top of the vertical climb segment if there is one),
and takeoff weight, but that is beyond the scope of this paper. so appropriate offsets are required. The semi-minor axis in this case
The elliptical profile consists of a quarter of an ellipse that is is the height of the ellipse from the pad (or the top of the vertical
vertical at takeoff and transitions to horizontal at the cruise altitude. climb segment) to the cruise altitude. The semi-major axis is the
The elliptical segment can be preceded by a straight vertical segment along-track distance in climb from takeoff to cruise altitude. The axis
at takeoff, or followed by one at landing, to avoid conflicts near the scales in Figure 3 are very different, so the actual shape of the ellipse
vertipad, if necessary. The parameters of the elliptical profile are the is distorted as shown, having a semi-major (horizontal) to semi-
starting and ending altitude and the distance along the route at minor (vertical) axes length ratio of 15.2.
which the cruise altitude is reached. Figure 3 shows an example A function was needed to map from the arc distance along the
where the vertipad elevation is 200 feet, the cruise altitude is ellipse to the along-track and altitude coordinates of the point at that
1,000 feet, and the horizontal distance at the end of climb is distance. A closed-form equation for that mapping is not possible,
2.0 nmi (nautical miles). unfortunately, but a precise numerical mapping was constructed by

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sweeping the angle through the full range from zero to 90 deg in
small angular increments and constructing an interpolated mapping
from the integrated arc distance to the along-track distance and
altitude. The size of the angular increment should be small enough
for high resolution but not so small that numerical roundoff error
becomes an issue. That covers a wide range, and a value of deg/8 was
arbitrarily selected. The details will not be presented here, but the
resulting function was used to step through the arc distance in small
increments to construct the static longitudinal profile as will now be
explained.
As mentioned earlier, the distance between points can vary
and is arbitrary within a wide range. The distance spacing is
perhaps not as critical as the time spacing, but the time spacing is
not known until the power function (to be presented later) is
applied. Hence, determining reasonable distance steps requires a
manual iteration process: distance steps are selected, and if the FIGURE 4
Example altitude change maneuver profile.
resulting time steps are too large, smaller distance steps are tried,
and the procedure is repeated until the time steps are within the
desired range. Note that this manual iteration is only required
once for each power model and static longitudinal profile, not for The normal (vertical) acceleration for this kind of maneuver
every trajectory. Moreover, a new iteration is unlikely to be is approximately an = rs2, where s is the speed and r is the rate of
needed even for a new profile unless it is dramatically increase of the flightpath angle. The value of 10 deg/nmi that was
different than the one for which the iteration was already used for the example shown in Figure 4 yields a normal
done. For the purposes of this paper, time steps in the range acceleration of 0.043 g, which is mild and acceptable for
of approximately 0.1 s to 1.0 s are considered reasonable, with passenger comfort. The threshold for passenger comfort is
smaller time steps preferred in the dynamic segments and larger approximately 0.1 g, but a lower value of approximately 0.05 g
in steady-state. The power of modern computers allows for high is preferable if the traffic situation allows it. Another way to select
resolution at a negligible cost in terms of both computation time a value for this rate is to start with a desired value for normal
and data storage. acceleration and compute the required rate of increase of
For the initial vertical climb segment, the vertical speeds are flightpath angle as r = an/s2. For example, the required rate for
low, so small step sizes in arc distance are needed. Actually, a normal acceleration of 0.05 g and a speed of 130 knots is
several different step sizes were needed, ranging is size from 11.6 deg/nmi.
0.125 ft at takeoff and increasing to 2 ft at an arc distance of The algorithm starts by constructing the profile to the center
600 ft. After that point, the speed is higher, and an increment of altitude, half way between the initial and final altitudes. To realize
10 ft was used. When the power model described in the next symmetry, that first half of the profile is then reflected, first
subsection was applied, these arc distance step sizes yielded the vertically about the center altitude, then horizontally about the
desired range of time steps. end point, to form the second half of the profile. The two-halves
of the profile meet in the center at a flightpath angle that was
3.1.1 Altitude change maneuvers limited to a maximum magnitude of 10 deg in the example shown
A key type of maneuver for conflict resolution is altitude change, in Figure 4. However, that value was never reached in the
usually from flying level at one altitude to flying level at another. example, where the actual maximum flightpath angle was
Figure 4 shows an example of such an altitude transition in level 6.9 deg. Whether the maximum allowed magnitude of the
flight from 1,000 to 1,500 ft. These maneuvers must be smooth flightpath angle is reached or not depends on the parameters
enough for passenger comfort, and they must also be completed of the maneuver.
within a given distance to avoid the conflict. For imminent conflicts,
the distance takes precedence over passenger comfort.
The algorithm used here for this type of maneuver works as 3.2 Power model and longitudinal profile
follows. The flightpath (climb or descent) angle is incremented in
small angular increments (a value of deg/8 was used in the example) As mentioned earlier, a longitudinal profile consists of 1) the
at a constant rate from zero to a specified maximum magnitude, horizontal distance flown as a function of time and 2) the altitude as
which can be in the range of approximately 5–20 deg. A value of a function of time or distance. It is essentially the resulting trajectory
10 deg was used for the example. The rate of increase of the with the route “bent” into a straight line, assuming decoupled lateral
flightpath angle is a constant parameter that should normally be and longitudinal dynamics. It will be constructed here for air taxis by
in the range from approximately 5–30 deg/nmi, depending on the assigning time as a function of arc distance to the static longitudinal
traffic situation and the distance to the conflict to be resolved. A profile constructed above. In other words, a time will be computed as
value of 10 deg/nmi was used for the example, resulting in a total follows to yield a specified net power for each discrete point of the
maneuver length of 1.33 nmi. static longitudinal profile.

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Also as mentioned earlier, the net power at any time is not a
P/m (5)
necessarily based on the maximum power available at the current b
v20
2 − gΔh (6)
flight state. It cannot exceed the maximum power available, of
c
0 (7)
course, and it should leave a margin for controllability. The
maximum power available, and hence the maximum net power d
−(Δd) 22
(8)
available at any time, depends mainly on the airspeed (CAS) at that
This cubic polynomial equation can be solved using the formula
time, but the net power function is essentially a control variable
given in the Appendix A to determine the time increment between
that can be programmed as a function of any flight variable,
points to yield a given level of net power in excess of the power
including time, position along the route, altitude, and flightpath
needed to maintain steady level flight at the current airspeed and
angle. For purposes of this paper, it will be modeled as a function of
altitude. There should be only one positive real root, which will be
CAS only.
the correct solution. Given a sequence of position points in 3D space,
As mentioned in the Introduction, the methods used in this
plugging in the desired power per unit mass and the other relevant
paper are based in part on the Trajectory Specification concept
variables into the equation allows the required time difference
(Paielli, 2022), which allows the time steps of the reference trajectory
between points to be determined.
to vary in size, typically being larger in steady-state flight than
Determining the power model from simulated or recorded
otherwise. That allowance for varying time steps is used here to set
data can be done as follows. The trajectory should be given in
the power level by computing the time step between discrete points
terms of a closely spaced series of points in 4D space. That series
in 3D space that yields the specified power at that time. The TS
can be effectively differentiated with respect to time by back-
software then automatically converts the time steps to a specified
differencing to determine velocity, and the velocity and altitude
uniform time step by linearly interpolating between the varying time
at each point can then be used to determine total mechanical
steps.
energy according to Eq. 1. Net power is then the time derivative of
Once the interpolated points with equal time steps are computed,
energy at each point. CAS can be computed at each point, based
any relevant flight variable as a function of time or along-track
on standard equations, as a function of speed, altitude, and the
distance can be precomputed as an array with uniform time or
wind vector, if applicable. Note that CAS and net power must be
distance steps for fast lookup. If the start time of the trajectory is
determined separately for climb and descent. Once they are
t0, and the time step is Δt, then the array index corresponding to time t
determined, the resulting CAS-power points can be ordered by
is simply (t − t0)/Δt. Because that index value is usually not an exact
CAS value and interpolated to make the CAS steps uniform for
integer, the values of the array at the two closest bounding integer
efficient (constant-time) array access followed by linear
indices are linearly interpolated for better accuracy. This procedure
interpolation (as discussed earlier for time steps). If the
provides a fast (constant-time) lookup and interpolation of the
resulting function is not smooth enough, a smoothing
relevant flight variables as a function of time or distance, making
algorithm or a curve fit (e.g., polynomial) can be used to make
the trajectory model effectively continuous in time and space.
it smoother.
Total mechanical energy is the sum of potential and kinetic
Figure 5 shows the net power vs. CAS for the QEP1 (Shaw-
energy:
Lecerf et al., 2020), a Quadrotor Electric Power single-seat aircraft.
E
mgh + mv2
2 (1) The QEP1 has a rotor diameter of 12.62 ft and an operating gross
weight of 1,428 pounds. The dashed red line is the model target
where m is the vehicle mass, g is gravitational acceleration, h is function to be emulated, which is derived from simulated
altitude, and v is velocity (speed). (This energy does not include the trajectory data in the form of a sequence of data points in 4D
rotational kinetic energy in the rotors, which is not relevant here.) space (t,x,y,z) with a uniform time step of 1.0 s. The solid black line
Now consider the change in the total energy over a single discrete
time step of length Δt. The average power over that time step is P =
ΔE/Δt, so

PΔt
ΔE
mgΔh + mΔv2
2
mgΔh + mv21
2 − mv20
2 (2)

where v0 and v1 and the speeds are the start and end of the interval.
The average speed over the time interval is v = Δd/Δt, where d is the
arc distance flown over the interval (the hypotenuse of the
horizontal and vertical distance between points). Substituting this
expression for v1 yields
2
PΔtm
gΔh + Δd
Δt 2 − v20 2 (3)

Multiplying both sides by (Δt)2 and rearranging yields

P(Δt)3 m + v20
2 − gΔh(Δt)2 − (Δd)2 2
0 (4)

FIGURE 5
This is a cubic polynomial of the form ax3 + bx2 + cx + d = 0 where Example of net power as a function of calibrated airspeed (CAS).
x = Δt and

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FIGURE 6 FIGURE 7
Top: Climb profile, for example, trajectory; bottom: speed. Top: total mechanical energy profile in climb for example
trajectory; Bottom: power profile in climb for example trajectory.

(which obscures much of the red line) represents the resulting net mirror image of this profile. The resulting time steps are in the
power function, which matches the model very closely, verifying desired range from 0.1 s to 1.0 s as discussed earlier. The black
the correctness of the cubic polynomial solution presented above. vertical line marks the top of climb and the start of level flight, and
The slight discrepancies are due to discretization error. The the small red vertical line marks the point at which the steady
descent was modeled as the negative mirror image of this cruise speed is reached.
function but is not shown. The net power at takeoff is Figure 7 shows the resulting energy and power profiles. The top
approximately 300 fpm, and the maximum power is plot shows the potential, kinetic, and total mechanical energy as a
approximately 1,300 fpm at a CAS of approximately 85 knots. function of time (excluding the rotational kinetic energy in the
It is worth noting that the power models discussed in this paper rotors), where energy is represented as altitude equivalent (energy
are actually models of power/weight ratio. They should therefore be per unit mass divided by gravitational acceleration). The point at
scaled by the inverse of the weight if the weight of the aircraft differs which a steady altitude is reached is marked on the potential
significantly from what it was when the trajectory data that the energy (altitude) curve, and the point at which steady speed is
power model is based on was generated. In practice, a weight reached is marked on the kinetic energy curve. The point at which
estimate could be based on the number of passengers (as is steady altitude and speed are both reached (i.e., the later of the two
currently done for airliners). The weight could also serve as a points) is marked on the total energy curve. The symmetry is due to
proxy for the available battery power. If the batteries are not the symmetric modeling of climb and descent power in this
fully charged, the assigned weight could be arbitrarily increased example.
by the ratio of the nominal battery power to the current reduced The bottom plot Figure 7 shows the resulting power profile as a
battery power. function of time during climb. As explained earlier, this power is the
Figure 6 shows the climb profile that results from applying the rate of change of the mechanical energy and is represented here as
power algorithm to the static longitudinal profile shown in Figure 3 altitude rate or vertical speed equivalent (power per unit mass divided
above. The top plot shows altitude as a function of time, and the by gravitational acceleration). The power drops to zero when steady-
bottom plot shows speed as a function of time, where the steady state flight at the cruise speed and altitude is reached. The net power in
cruise speed is 122 knots. The descent profile is essentially the descent is the negative mirror image of this plot but is not shown here.

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3.3 Mapping to a route groundspeed at a given airspeed and vice versa. The cross-wind
effect is more subtle and will not be addressed here, but the effect of
A route is the vertical projection of a trajectory onto the the along-track wind component (headwind or tailwind) can be
geodetic surface of the earth. In the Trajectory Specification (TS) modeled as follows.
concept (Paielli, 2022), a route is specified as a sequence of If a simulated trajectory is used to determine the power
waypoints on that surface and a turn radius associated with each model, it should be simulated with zero winds, and if a
waypoint. The TS algorithm takes those route parameters and recorded trajectory is used, it should preferably be recorded in
constructs a detailed route representation consisting of low-wind conditions. After the new trajectory is generated and
alternating straight and turn segments. All turns are tangent- mapped to a route as explained earlier, the effect of the current
arc or “flyby” turns of constant radius. If two successive wind field during level flight can be modeled by adjusting
waypoints are too close together for the specified turn radius, groundspeed for the headwind or tailwind at each point along
the route is geometrically invalid and is flagged as such by the TS the route. This method is equivalent to adjusting the along-track
software (but is left for the user to correct by changing either the position by the integrated along-track component of the wind
radius or the waypoints). The route representation constitutes a vector along the route.
curvilinear coordinate system in which the coordinates are Nonlevel flight segments can be modified similarly, except
along-track distance and cross-track position, which can be that the adjustment in along-track position should be scaled by
converted to geodetic or locally level Cartesian coordinates the cosine of the flightpath angle (relative to horizontal) at any
and vice versa. given point. Hence there would be no adjustment in a vertical
Recall that the static longitudinal profile was adjusted to the segment, and the adjustment in a level segment would be as
path length of the specified route by adding a steady cruise discussed above. This method preserves the static path of the
segment of the required length between the climb and descent trajectory in 3D space while accounting for the main wind effects
segments. Applying power to determine the time values does not in terms of progress in time along the route.
change the length of the longitudinal profile, so it will still have
the same length as the specified route. It is then superimposed
onto the route by mapping the along-track distance of each
discrete point onto the corresponding position at that distance
on the route (assuming no cross-track error). This has the effect
of “bending” the straight longitudinal profile onto the route,
assuming decoupled lateral and longitudinal dynamics as before.
As mentioned earlier, the simplifying assumption of
decoupled lateral and longitudinal dynamics can be
inaccurate, particularly for large turns at high bank angles
during climb. The bank angle for a coordinated turn is ϕ =
atan(v 2/(rg)), where v is speed, r is the turn radius, and g is
gravitational acceleration. The component of net power that is in
the longitudinal plane is the overall net power scaled by cos ϕ,
the cosine of the bank angle. For a bank angle of 20 deg, for
example, the longitudinal component is 6.0% less than the
overall net power. If the overall net power can be increased
by a factor of 1/(cos ϕ), the effect can be offset, and the
assumption of decoupled lateral and longitudinal dynamics
will be valid. For a bank angle of 20 deg, that requires a 6.4%
increase in net power. If power cannot be increased that much,
which typically happens only in climb, then the net power model
can be adjusted to account for the diminished longitudinal
power during the turn.

3.4 Modeling the effect of winds

A wind model is usually not required for ATC simulation

unless the fidelity of the simulation is intended to be high. To
actually use this method of trajectory generation in the field,
however, the current winds must be accounted for because the
power required in flight depends on the winds. In a cross-wind, the
FIGURE 8
aircraft needs extra power just to stay on course, which is another Top: Flight time as a function of cruise altitude; Bottom: Flight
reason that a margin should be maintained away from maximum time as a function of climb distance.
power. On the other hand, a headwind or tailwind determines the

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Paielli 10.3389/fpace.2023.1278726

required to determine the best values to use for climb and

descent distance based on the traffic scenario.
Figure 9 shows flight time as a function of power, with cruise
altitude as a parameter, for a climb distance of 2.0 nmi. As
before, power is expressed as power per unit mass divided by
gravitational acceleration, in units of fpm. The power level is
varied as a scale factor of the climb power shown in Figure 5,
where a power factor of 1.0 corresponds to that same power
function. The negative power level in descent was left unchanged
because it does not depend on available power. Because power is
actually power per unit mass, the power factor can also be
considered the inverse of a scale factor on aircraft mass or
weight.
Figure 9 shows that flight time can be sensitive to power level,
perhaps even more sensitive than it is to speed for shorter flights.
FIGURE 9 In particular, decreased power levels can significantly increase trip
Flight time as a function of power level. flight time at the higher cruise altitudes. At the lowest shown cruise
altitude of 1,000 ft, however, the sensitivity to power level is
minimal over a wide range. Also, the sensitivity increases with
4 Flight time dependence on cruise altitude and decreases as power/weight ratio increases. And
parameters because power per unit mass depends on weight, the flight time can
also be sensitive to the payload mass, hence more passengers can
The utility and economic viability of Urban Air Mobility will mean significantly longer flight time, particularly for smaller
depend on how much time can it can save compared to driving, so aircraft at higher cruise altitude. Underpowered aircraft could
it is interesting to analyze how various trajectory parameters hamper the UAM business model with flight times that offer no
affect trip flight times. significant time savings over driving.
The top plot of Figure 8 shows how the flight time varies as a
function of the cruise altitude, with climb distance (from takeoff
to top of climb, the semi-major axis of the ellipse) as a parameter, 5 Conclusion
for a route of 20 nmi, using the same power model that was used
in the earlier examples. The dependence is significant, with flight A new method has been developed to generate flyable
time increasing by 3.6 min as cruise altitude is increased from trajectories for eVTOL urban air taxis. The method allows the
1.0 to 2.0 kft with a climb distance of 2 nmi. The time increases by user to directly specify the route, the static longitudinal profile, and
another 4.7 min as cruise altitude is increased again from 2.0 to a power model as a function of airspeed (CAS) and possibly other
3.0 kft, and another 5.4 min from 3.0 to 4.0 kft, for a total increase flight variables. This method has the advantage over previous
of 13.6 min as cruise altitude is increased all the way from 1.0 to methods of not requiring a detailed aircraft model or a model
4.0 kft. The increases in trip flight time with altitude are even of the flight controls.
larger for a smaller climb distance of 1.0 nmi. This plot clearly The trajectory generation algorithm is composed of four main
shows the time advantage of staying at lower altitudes and the parts: 1) generating the static longitudinal profile, 2) converting it to
economic disincentive of flying in the upper regions of the UAM a (dynamic) longitudinal profile by applying a power model, 3)
airspace for this particular aircraft model at least, which may be superimposing that profile onto a specified route, and 4) adjusting
somewhat underpowered. At some point, the time advantage for the effects of winds.
over driving is lost (and the travel time to and from the vertiports A novel method was developed to yield the specified power
must also be accounted for, of course). model as a function of airspeed by solving a cubic polynomial for the
The bottom plot of Figure 8 shows how total flight time varies required time step between discrete points in the longitudinal plane.
as a function of the climb distance, with cruise altitude as a Because the climb trajectory starts vertically and ends
parameter, for the same vehicle as the top plot. At a cruise horizontally, an elliptical profile (a quarter of an ellipse) was
altitude of 1,000 ft, the trip flight time decreases by 0.7 min as used as an example profile shape, which is smoother than other
the climb distance is increased from 1.0 to 2.0 nmi, and it decreases profiles that have been proposed.
by 1.3 min as the climb distance is increased from 1.0 to 4.0 nmi. The dependence of trip flight time on various parameters was
The time reductions are larger at higher cruise altitudes. According analyzed and plotted, showing that flight times are significantly
to this plot, shallower climbs reduce flight time, and the climb longer for higher cruising altitudes, steeper climbs, and lower power
distance should be maximized to minimize flight time. The same levels. These dependencies have implications for the economic
applies for the descent distance, so the shortest flight time requires viability of UAM as an alternative to driving.
the climb and descent to meet somewhere in the middle, with no This method can be used to generate candidate trajectories for
level cruise segment. However, that could reduce airspace capacity conflict resolution, which can then be checked to find the candidate
by blocking altitudes that other flights need, so it cannot always be with the shortest flight time that is free of conflicts. Using advanced
done and perhaps should rarely be done. Further analysis is feedforward/feedback control methods, aircraft should be able to fly

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Paielli 10.3389/fpace.2023.1278726

the resulting deconflicted trajectories to within specified tolerances Acknowledgments

in the field.
The author acknowledges and thanks Chris Silva of NASA Ames
Research Center for running the NDARC (Johnson, 2015) program to
Data availability statement provide the rate-of-climb data that is plotted in Figure 2.

The raw data supporting the conclusions of this article will be

made available by the authors, without undue reservation. Conflict of interest
The author declares that the research was conducted in the
Author contributions absence of any commercial or financial relationships that could
be construed as a potential conflict of interest.
RP: Conceptualization, Methodology, Software, Writing–original
draft.
Publisher’s note
Funding All claims expressed in this article are solely those of the authors
and do not necessarily represent those of their affiliated organizations,
The author(s) declare that no financial support was or those of the publisher, the editors and the reviewers. Any product
received for the research, authorship, and/or publication of that may be evaluated in this article, or claim that may be made by its
this article. manufacturer, is not guaranteed or endorsed by the publisher.

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√
Appendix A: Cubic polynomial roots g ≡  −3 − 1
2 (15)

C1 ≡ 3 h
2 (16)
A cubic polynomial of the form
C2 ≡ g C1 (17)
ax3 + bx2 + cx + d
0 (9)
C3 ≡ g C2 (18)
where a ≠ 0, can be solved using complex mathematics as follows
(from Wikipedia (Cubic equation, 2001)). First, the following terms The three roots for x are then
are defined. xi
−b + Ci + d0
Ci 
(3a), i
1, 2, 3 (19)
d0 ≡ b − 3ac
2
(10) At least one root must be real, and the other two can be
d1 ≡ 2b − 9abc + 27a d
3 2
(11) either real or complex. However, a real root can be computed

to have a tiny imaginary part due to numerical roundoff
z ≡ d21 − 4d30 (12)
error, so the root that has the smallest imaginary magnitude
f ≡ d1 + z (13) should have its imaginary part set to zero and be taken as a real
h ≡ if f ≠ 0 then f else d1 − z (14) root.

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