Robert Falck and Leon Gefert

Glenn Research Center, Cleveland, Ohio

A Method of Efficient Inclination Changes for

Low-Thrust Spacecraft

NASA/TM—2002-211871

October 2002

AIAA–2002–4895
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Robert Falck and Leon Gefert

Glenn Research Center, Cleveland, Ohio

A Method of Efficient Inclination Changes for

Low-Thrust Spacecraft

NASA/TM—2002-211871

October 2002

National Aeronautics and

Space Administration

Glenn Research Center

Prepared for the

Astrodynamics Specialist Conference and Exhibit

cosponsored by the American Institute of Aeronautics and Astronautics and

the American Astronautical Society

Monterey, California, August 5–8, 2002

AIAA–2002–4895
Available from

NASA Center for Aerospace Information

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National Technical Information Service

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This report is a formal draft or working

paper, intended to solicit comments and

ideas from a technical peer group.

This report contains preliminary

findings, subject to revision as

analysis proceeds.

Available electronically at http://gltrs.grc.nasa.gov
ABSTRACT

The evolution of low-thrust propulsion

technologies has reached a point where such systems 

have become an economical option for many space

missions. The development of efficient, low trip time 

control laws has received an increasing amount of

attention in recent years, though few studies have

examined the subject of inclination changing

maneuvers in detail. A method for performing

economical inclination changes through the use of an 

efficiency factor is derived from Lagrange’s planetary 

equations. The efficiency factor can be used to regulate 

propellant expenditure at the expense of trip time. Such 

a method can be used for discontinuous-thrust transfers 

that offer reduced propellant masses and trip-times in 

comparison to continuous thrust transfers, while

utilizing thrusters that operate at a lower specific

impulse. Performance comparisons of transfers

utilizing this approach with continuous-thrust transfers 

are generated through trajectory simulation and are

presented in this paper.

NOMENCLATURE

a semi-major axis

e eccentricity

i inclination

ω argument of periapsis

ν true anomaly

Isp specific impulse

p spacecraft propulsive power

ηp spacecraft power subsystem efficiency

m0 initial spacecraft mass

mp spacecraft propellant mass expenditure

MET mission elapsed time

t time

r radius

n mean motion

W magnitude of orbit-normal thrust vector

ηΔi efficiency of inclination change in given orbit

χ minimum magnitude of ηΔi to enable thrusting

INTRODUCTION

The evolution of low-thrust propulsion

technologies has reached a point where such systems 

have become an economical option for many space

mission applications. The development of efficient

control laws has received an increasing amount of

attention in recent years, though few studies have

examined the subject of inclination changing

maneuvers in detail. 

The topics covered in this paper are the results of 

trajectory design work in which large changes in

inclination were required for orbits of moderate

eccentricity. The resulting methods allow the tradeoff 

between trip time and propellant mass to be quantified, 

thus giving a mission designer a convenient way of

managing mission performance. There is a wide array 

of applications, including transfers between low-Earth 

orbit, the Moon, and other planets, where a significant 

change in inclination is likely to be required at some 

point in the mission.

Using Lagrange’s planetary equations as a starting 

point, an efficiency factor can be derived that

normalizes the time-rate of change in inclination at a 

given point in the orbit by the maximum value of di/dt 

in the orbit. Previous work along these lines has been 

conducted for in-plane orbital transfers utilizing

discontinuous-thrust1

 and for continuous-thrust orbit

raising maneuvers incorporating inclination changes2

. 

Methods for managing inclination through the

efficiency factor for both continuous-thrust and

discontinuous-thrust cases will be explained in the

following sections of this paper.

BEHAVIOR OF di/dt

From Lagrange’s planetary equations, the time-rate 

of change in inclination can be written3

:

22 1

cos( )

na e

Wr

dt

di

−

ω+ν = (1)

A METHOD OF EFFICIENT INCLINATION CHANGES FOR LOW-THRUST SPACECRAFT

Robert Falck*

 and Leon Gefert

National Aeronautics and Space Administration

Glenn Research Center

Cleveland, Ohio 44135

*Case Western Reserve University, Student Engineer

NASA/TM—2002-211871 1
Using the expression for radius in Keplerian elements, 

Equation 1 can be written

(1 cos( ))

1 cos( ) 2

+ ν

− ω+ ν = na e

W e

dt

di (2)

Plots of the magnitude of di/dt, from Equation 2, with 

respect to true anomaly and eccentricity are shown in 

Figures 1 and 2. A value of 1.0 is used for W, n, and a.

Figure 1: Abs(di/dt) vs. True Anomaly And 

Eccentricity, ω=0ο

Figure 2: Abs(di/dt) vs. True Anomaly and 

Eccentricity, ω=0ο

Note that the zeros always occur on the antinodal 

lines and the behavior is sinusoidal at an eccentricity of 

zero. As eccentricity is increased the behavior deviates 

from the sinusoidal case due to the presence of the

radius term in the numerator of Equation 1. This

suggests that sinusoidal inclination control laws

developed in the past are not well suited for orbits of 

significant eccentricity.

CONTROL LAW APPLICAIONS

Two control-laws are considered here. The first is 

a continuous-thrust case in which orbit-normal thrust is 

scaled by the magnitude of di/dt. Excess thrust is 

diverted to the in-plane components of thrust to control 

semi-major axis or eccentricity as the mission requires. 

The second control law completely suspends orbitnormal thrust when di/dt is arbitrarily deemed to be 

unreasonably low. In this case the Isp may be lowered, 

if possible, to reduce the time required to perform the 

inclination change. The result is a maneuver requiring 

less propellant mass to achieve a trip time equal to that 

of the continuous-thrust case, as will be demonstrated 

later in this paper.

Continuous-Thrust Control Law Derivation

The effectiveness of performing a change in

inclination at a given point in an orbit may be gauged 

by comparing the value of di/dt per unit of orbit-normal 

thrust at a given point in the orbit with the maximum 

value of di/dt as a function of true anomaly. The

resulting efficiency factor (ηΔi) may be used to scale the 

out of plane thrust.

The first step is to determine the extrema of

Equation 2. The sign of di/dt is ignored as only the 

magnitude of the extrema is of interest. Thrust is 

assumed to be considerably low such that all elements 

aside from true anomaly are constant through a single 

orbital period. Equation 2 can be effectively simplified 

by removing all terms that are assumed to be constant, 

giving equation 3:

Const

e

v

dt

di ⋅ + ⋅ ν

ω⋅ − ω⋅ ν = (1 cos )

coscos sinsin

(3)

Taking the derivative of Equation 2 with respect to 

ν and solving for the zeros yields the extrema:

ν −= ( ω )− ω − sin sin( ) 1 e (4)

ν = ( ω ) −ω + ° − sin sin( ) 180 1 e (5)

The absolute maximum magnitude of di/dt can then 

be determined by comparing the results when these

values are used in Equation 3. Now that di/dtmax is 

known, the ratio of di/dt at the current true anomaly to 

di/dtmax can be used to quantify the efficiency of orbitnormal thrusting in changing the inclination of an orbit. 

The constant terms will cancel, and can therefore be 

neglected. The efficiency of orbit normal thrusting to 

achieve an inclination change can then be defined as 

follows:

Abs(di/dt)

Abs(di/dt)

NASA/TM—2002-211871 2
hΔi =

di

dt

di

dt max

= cosw cosv − sinw sin n

di

dt max (1+ e cosn)

 (6)

Figures 3 through 5 show values of ηΔi mapped 

onto various orbits. Note that the magnitude of di/dt is 

much larger near apoapsis for an orbit of significant 

eccentricity. 

Figure 3: Orbit-Normal Thrust Vectors and Efficiency 

of Inclination Change, ω=0ο

, e=0.5

Figure 4: Orbit-Normal Thrust Vectors and Efficiency 

of Inclination Change, ω=90ο

, e=0.5

Figure 5: Orbit-Normal Thrust Vectors and Efficiency 

of Inclination Change, ω=0ο

, e=0.0

Discontinuous-Thrust Control Law Derivation

As has previously been demonstrated for in-plane 

control laws by Gefert and Hack1

, lower trip times and 

propellant mass expenditures may be achieved with 

thrusters of lower Isp by coasting during portions of the 

orbit where thrust has relatively little effect. In the case 

of an inclination change, ηΔi can be used to quantify the 

effectiveness of orbit-normal thrust in changing a

spacecraft’s inclination at any given point.

Orbit-normal thrusting will be suspended when ηΔi

drops below some threshold value, henceforth referred 

to as χ. In the extreme cases, χ=1.0 will result in 

infinitesimally short bursts of thrust at the true anomaly 

of di/dtmax, given by Equation 2 and/or Equation 3. The 

other extreme, χ=0.0, will result in continuous orbitnormal thrusting. The value of χ that will provide the 

desired tradeoff between trip time and propellant

expenditure will, of course, vary from mission to

mission. Figure 6, below, illustrates regions of

thrusting and coasting for an eccentric orbit.

Figure 6: Thrusting and Coasting Regions for e=0.25, 

ω = 45°, and χ=0.5

For this case the values of true anomaly at which 

ηΔi

 rises above or drops below χ are desired, so that 

thrusting can be suspended and resumed at the

appropriate points in the orbit. It is assumed that

di/dtmax is known from Equations 3 through 5. The 

values of true anomaly which satisfy the following 

equation are sought:

ηΔ χ==

dt max

di

dt

di

i (7)

Using the reduced definition of di/dt given by

Equation 3, Equation 7 becomes

(1 cos ) max

coscos sinsin

dt

di

e

χ= ⋅ + ν

ω ν − ω ν (8)

NASA/TM—2002-211871 3
Solving for ν yields the following:

sign( ) ( ) YY

Y

X 1

2

1 tan

1

cos −− ⋅ − 















+

ν = (9)

and

⋅ ( ) ( )+− ° 















+−

ν = −− tan 180

1

cos 1

2

1 sign YY

Y

X (10)

where

e

dt

di

dt

di

X

ω χ− ⋅⋅

χ⋅

=

max

max

cos( )

 (11)

and

e

dt

di

Y

ω χ− ⋅⋅

ω =

max

cos( )

sin( ) (12)

Again, since only the magnitude of di/dt is of

interest here, Equations 9 and 10 must be solved for 

both positive and negative values of χ. Two or four 

solutions will be generated, depending on the threshold 

value and the eccentricity of the orbit. The boundaries 

between the thrusting and coasting regions in Figure 6 

were determined using this method.

COMPUTATIONAL RESULTS

 Performance comparisons between the

discontinuous-thrust control law discussed above, and 

optimal continuous-thrust inclination changes are

shown on the following pages. Three cases are

considered below, to give the reader some indication of 

the benefits available with discontinuous-thrust 

inclination control in various situations.

Optimal continuous-thrust cases are generated

using SEPSPOT4

, a low-thrust trajectory optimization 

program that uses an orbital averaging integration

method, and uses calculus of variations to find

minimum-time transfers. Discontinuous-thrust control 

laws were generated via SNAP, a RK78 trajectory 

propagation tool used under development at NASA

Glenn Research Center. The discontinuous-thrust 

examples below were integrated in Cartesian

coordinates. All tests were run neglecting j2 and higher 

order gravity effects, SEPSPOT and SNAP would both 

be expending propellant only to achieve the desired 

inclination change. The spacecraft specifications for 

the following examples are as follows:

p = 75 kWe

ηp = 0.70

m0= 10000 kg

Example 1: Circular Orbit Inclination Change

In the example below the case of an inclination 

change of a circular orbit is examined. Since the radius 

remains constant throughout an orbit, Equation 3

reduces to a simple cosine function. 

The purpose of this example is to show whether a 

discontinuous-thrust inclination change control law can 

provide any significant propellant mass or trip time 

benefits over the continuous-thrust case at low

eccentricities. 

Initial Orbit:

a = 42161 km

e = 0.000

i = 28.45°

Ω = 0°

ω = 0°

Final Orbit:

a = 42161 km

e = 0.000

i = 51.6°

Ω = 0°

ω = 0°

Table 1: SEPSPOT Results from Example 1

Isp

(sec)

MET

(days)

mp

(kg)

3800 76.1 497

3500 70.0 539

3000 59.7 626

Table 2: Discontinuous-thrust Results for Example 1

χ 3800 sec 3500 sec 3000 sec

 MET mp MET mp MET mp

 (days) (kg) (days) (kg) (days) (kg)

0.0 78.2 511 71.8 553 61.3 642

0.1 78.7 481 72.3 521 61.7 605

0.2 79.8 456 73.4 494 62.7 574

0.3 82.2 433 75.6 469 64.5 545

0.4 85.6 414 78.7 448 67.2 521

0.5 90.7 395 83.3 429 71.0 499

0.6 98.6 379 90.6 411 77.3 477

0.7 110.3 364 101.2 395 86.2 460

0.8 131.1 351 120.6 381 102.8 443

0.9 182.0 339 167.6 367 144.6 427

As expected, trip time increases and propellant 

mass expenditure decreases in the discontinuous-thrust 

cases, shown in Table 2.

NASA/TM—2002-211871 4
400

450

500

550

600

650

700

55.0 65.0 75.0 85.0 95.0

MET (days)

Propellant Expenditure (kg)

3800sec

3500sec

3000sec

SEPSPOT

Figure 7: Trip Time vs. Propellant Mass Comparisons 

for Continuous-Thrust and Discontinuous-Thrust 

Inclination Changes in a Circular Orbit.

Figure 7, above, shows tradeoff in trip time and 

propellant expenditure in graphical form. The results of 

the SEPSPOT continuous-thrust cases (see Table 1) are 

shown for comparison.

For a given Isp, increasing the value of χ can only 

reduce propellant mass compared to the continuousthrust case at the expense of trip time. However,

reducing specific impulse and using an appropriate

value for χ can achieve savings in both time and 

propellant consumption.

In this example, χ=0.0 yields results that are

inferior to the SEPSPOT results in both trip time and 

propellant mass. This discrepancy is a result of the fact 

that SEPSPOT optimizes only continuous-thrust cases. 

The solutions it finds utilize a portion of the

spacecraft’s thrust to increase the semi-major axis of 

the orbit, thereby increasing the magnitude of di/dt 

throughout the orbit. The relationship between di/dt 

and semi-major axis can be explained by starting with 

Equation 1 and the definitions of radius and mean 

motion5

. Equation 1 can be rewritten as

di

dt = W 1− e2 cos(w + n)m

a

( ) 1+ e cos(n)

 (13)

Only the average change in di/dt throughout an orbit is 

of interest, thus terms involving true-anomaly will be 

neglected. Also, since di/dt is proportional to the orbitnormal thrust magnitude, W, the term is dropped. The 

relationship between the average value of di/dt and 

semi-major axis can then be expressed as

di

dt avg

∝ m

a

1− e2 ( ) (14)

Figure 8 shows the relative value of di/dt, from 

Equation 14, throughout a SEPSPOT-generated transfer 

from Example 1. 

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

0 20 40 60 80

MET (days)

di/dt (km/s)

41000

42000

43000

44000

45000

46000

47000

48000

49000

50000

a (km)

di/dt SEPSPOT

semi-major axis (km)

Figure 8: Relative Effect of Semi-Major Axis Change 

on di/dt 

SEPSPOT achieves an increase in di/dt of

approximately 10% by increasing the semi-major axis 

for a portion of the transfer. The discontinuous-thrust 

cases run here use no in-plane thrust component to 

modify semi-major axis. Increased performance from 

the discontinuous-thrust case might be realized by

directing thrust in-plane for di/dt inefficient portions of 

the orbit. However, the combination of the reduction of 

Isp and suspension of thrust throughout portions of the 

orbit can more than compensate for the difference in 

performance, even without changing the semi-major 

axis. 

Example 2: Eccentric Orbit Inclination Change, ω= 0°

The second example mission compares continuous and 

discontinuous-thrust inclination changes in a highly 

elliptic orbit. The maneuver transfers a spacecraft from 

an inclination of 28.45° to a final inclination of 63.4°. 

Other orbital elements are not changed, and the semimajor axis and eccentricity of the orbit are the same as 

those for a Molniya orbit. Continuous-thrust and

discontinuous-thrust results are shown in Tables 3 and 

4, respectively.

Increasing χ

NASA/TM—2002-211871 5
Initial Orbit:

a = 26600 km

e = 0.73

i = 28.45°

Ω = 0°

ω = 0°

Final Orbit:

a = 26600 km

e = 0.73

i = 63.4°

Ω = 0°

ω = 0°

Table 3: SEPSPOT Results for Example 2

Isp

(sec)

MET 

(days)

mp 

(kg)

3800 53.9 352

3500 49.5 382

3000 42.3 444

Table 4: Discontinuous-thrust Results for Example 2

χ 3800 sec 3500 sec 3000 sec

MET mp MET mp MET mp

(days) (kg) (days) (kg) (days) (kg)

0.0 58.0 379 53.3 411 45.6 478

0.1 58.4 352 53.7 382 45.8 443

0.2 59.6 328 54.8 354 46.8 412

0.3 60.7 316 55.8 342 47.7 398

0.4 62.5 304 57.6 330 49.2 384

0.5 65.3 293 60.1 318 51.3 370

0.6 69.7 283 64.2 307 54.8 357

0.7 76.8 274 70.7 297 60.3 345

0.8 90.1 264 82.7 287 70.7 334

0.9 122.6 255 112.6 277 96.2 323

For this case, a discontinuous-thrust inclination 

control law using only orbit-normal thrust can come 

close to the performance of SEPSPOT, as Figure 9 

shows. Since da/dt is at its maximum value at periapsis 

and di/dt is at its maximum at apoapsis, SEPSPOT is 

able to increase the semi-major axis of the orbit and 

boost the magnitude of di/dt while incurring almost a 

minimal gravity loss penalty. However, even with 

purely orbit normal thrust the discontinuous control law 

can come close to matching the performance of

SEPSPOT. 

If χ is significantly large such that the spacecraft is 

not thrusting out-of-plane at periapsis, a tangential

thrust component or “perigee-hold1

” steering law can 

be introduced to modify the semi-major axis without 

taking any thrusting time away from the orbit-normal 

component. Greater efficiency can certainly be

achieved through the use of a more complex control 

law that blends both in-plane and orbit-normal thrust in 

an effective manner. 

250

300

350

400

450

500

40 50 60 70 MET (days)

Propellant Mass (kg)

3800sec

3500sec

3000sec

SEPSPOT

Figure 9: Trip Time vs. Propellant Mass Comparisons 

for Continuous-Thrust and Discontinuous-Thrust 

Inclination Changes in a Highly Elliptical Orbit, ω=0o

Example 3: Eccentric Orbit Inclination Change, ω= 90°

For thoroughness, the inclination change

performed in the previous example is reexamined for 

the case where ω=90°. All other orbital elements are 

the same as those in the previous example. The results 

for continuous-thrust and discontinuous-thrust cases are 

shown in Tables 5 and 6, respectively.

Table 5: SEPSPOT Results for Example 3

Isp

(sec)

MET

(days)

mp

(kg)

3800 148.3 969

3500 136.0 1057

3000 115.5 1211

Table 6: Discontinuous-thrust Results for Example 3

χ 3800 sec 3500 sec 3000 sec

 MET mp MET mp MET mp

 (days) (kg) (days) (kg) (days) (kg)

0.0 144.4 943 132.3 1018 112.1 1175

0.1 145.9 890 133.7 961 113.5 1110

0.2 148.8 842 136.5 912 115.8 1051

0.3 153.5 801 140.8 867 119.6 1002

0.4 160.1 767 147.0 830 125.3 958

0.5 170.0 735 156.5 791 133.5 912

0.6 185.3 704 169.9 763 145.6 879

0.7 210.5 676 192.8 733 164.0 850

0.8 263.5 646 244.9 698 218.9 786

0.9 397.9 571 363.9 615 316.4 737

Increasing χ

NASA/TM—2002-211871 6
600

700

800

900

1000

1100

1200

1300

110 125 140 155 170 185 200

MET (days)

Propellant Expenditure (kg)

3800sec

3500sec

3000sec

SEPSPOT

Figure 10: Trip Time vs. Propellant Mass Comparisons 

for Continuous-Thrust and Discontinuous-Thrust 

Inclination Changes in a Highly Elliptical Orbit, ω=90o

The results shown in Figure 10 indicate that the

discontinuous-thrust control scheme is outperforming 

SEPSPOT at values of χ=0.0 even at the same specific 

impulse. It is likely that since the magnitudes of both 

da/dt and di/dt are small near apoapsis, the fact that 

SEPSPOT is constrained to using continuous-thrust 

essentially wastes propellant during the apoapsis dwell. 

Even greater reductions in propellant and trip time can 

be achieved by reducing the specific impulse. The

discontinuous-thrust control law is clearly preferable to 

continuous-thrust for inclination changes in orbits

where periapsis occurs at or near the antinodes.

CONCLUSIONS

Using the discontinuous-thrust control law for

inclination changes can reduce both the propellant load 

and the trip time required for an inclination-changing 

maneuver. While the performance benefits of such a 

method are dependent upon the argument of periapsis, 

it effectiveness has been demonstrated for both circular 

and highly elliptical orbits.

While the discontinuous-thrust control law

presented here is capable of outperforming continuousthrust, calculus of variations solutions, like those

generated by SEPSPOT, they are at the same time 

relatively simple. Through the use of this method, the 

mission designer has a significant amount of control in 

being able to modify the trip time for an inclination 

change maneuver to achieve an acceptable propellant 

expenditure, or vice versa.

 

Increasing χ

REFERENCES 1

 Gefert, L.P., Hack, K.J.., Low-Thrust Control Law 

Development for Transfer from Low Earth Orbits to 

High Energy Elliptical Parking Orbits, Paper Number 

AAS 99-410, August 1999.

2

 Kluever, C., Simple Control Laws for Low-Thrust 

Orbit Transfers, Paper Number AAS 98-203. 

Spaceflight Mechanics 1998, Volume 99, Part II, 

American Astronautical Society, 1998.

3

 Roy, A.E., Orbital Motion, Adam Hilger, Bristol, 

Philadelphia, and New York, 1991, p. 192

4

 Edelbaum, T.N., Malchow, H.L., Sacket, L.L., Solar 

Electric Geocentric Transfer with Attitude Constraints: 

Analysis, NASA CR 134927, August 1975. 

5

 Bate, R.R., Mueller, D.D., White, J.E., Fundamentals 

of Astrodynamics, Dover Publications, New York, 

1971, pp. 20, 185.

NASA/TM—2002-211871 7
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October 2002

NASA TM—2002-211871

AIAA–2002–4895

E–13553

WU–800–9B–20–00

13

A Method of Efficient Inclination Changes for Low-Thrust Spacecraft

Robert Falck and Leon Gefert

Low-thrust; Inclination; Spacecraft; SEPSPOT; SNAP

Unclassified -Unlimited

Subject Category: 13 Distribution: Nonstandard

Prepared for the Astrodynamics Specialist Conference and Exhibit cosponsored by the American Institute of Aeronautics

and Astronautics and the American Astronautical Society, Monterey, California, August 5–8, 2002. Robert Falck, NASA

Glenn Research Center, Student Engineer, and Leon Gefert, NASA Glenn Research Center. Responsible person, Robert

Falck, organization code 7820, 216–433–2295.

The evolution of low-thrust propulsion technologies has reached a point where such systems have become an economical

option for many space missions. The development of efficient, low trip time control laws has received an increasing

amount of attention in recent years, though few studies have examined the subject of inclination changing maneuvers in

detail. A method for performing economical inclination changes through the use of an efficiency factor is derived front

Lagrange’s planetary equations. The efficiency factor can be used to regulate propellant expenditure at the expense of trip

time. Such a method can be used for discontinuous-thrust transfers that offer reduced propellant masses and trip-times in

comparison to continuous thrust transfers, while utilizing thrusters that operate at a lower specific impulse. Performance

comparisons of transfers utilizing this approach with continuous-thrust transfers are generated through trajectory simulation and are presented in this paper.