Misconceptions about gyroscopic stabilization

Patric M€uller,a) Achim Sack, and Thorsten P€oschelInstitute for Multiscale Simulation, Friedrich-Alexander-Universit€at Erlangen-N€urnberg, Erlangen, Germany

(Received 2 August 2019; accepted 21 December 2019)

A spinning gyroscope resists small torques in directions perpendicular to its axis, that is, the angularorientation of a body rigidly attached to a gyroscope is stable against rotation around certain axes.Since the angular orientation of a body is described by three angles (e.g., Euler angles), one mightconclude that it is possible to stabilize the orientation of an object against rotation using acombination of three gyroscopes spinning around non-collinear axes. We perform experiments underconditions of weightlessness to demonstrate that systems of coupled gyroscopes cannot arrest theangular orientation of free-floating objects, in contradiction to a widespread myth about gyroscopicstabilization, based on the above arguments. VC 2020 American Association of Physics Teachers.

https://doi.org/10.1119/10.0000517

I. INTRODUCTION

Spinning flywheels or gyroscopes can stabilize the orienta-tion of objects. This property is exploited in a variety ofapplications including children’s toys, marine gyrostabilizers,gyrocompasses, and many more. Apart from being of utter-most practical importance, gyroscopes fascinate physicistsfor their interesting behavior (Fig. 1). The sometimes coun-terintuitive dynamics of gyroscopes are frequently among themost difficult subjects encountered by new students in phys-ics. This is evidenced by a vast body of literature on illustrat-ing and teaching gyrodynamics, see, e.g., Refs. 1–19.

The mathematics of the dynamics of bodies in contactwith gyroscopes is far from trivial, and apparently simplephenomena turn out to be very complicated. For example,the stability of a bicycle seems to be related to the conserva-tion of angular momentum of the front wheel, but the mathe-matical description is rather involved20,21 and even the factitself is not without contradiction.22 The stabilizing effect ofa gyroscope can be nicely demonstrated under conditions ofweightlessness such that the position of the center of mass isconserved, see Fig. 2. When the astronaut kicks the gyro-scope slightly, it translates in space, while its axis of rotationis conserved. Similar to the spinning wheel of a bicycle,angular momentum defines an axis whose orientation isstable. However, just as for the bicycle wheel, the object isfree to rotate around this axis.

A common misconception among students, graduates, andsometimes even professors23 results from the following con-sideration: assume we wish to fix the position of an object inthree-dimensional space. This requires fixing three scalarquantities (the coordinates) along three non-collinear unitvectors. Thus, an apparatus capable of fixing an object inspace must provide three invariant vectors and a mechanismto keep the object at certain positions along these vectors.The angular orientation of a solid body is also determined bythree quantities (e.g., the Euler angles); therefore, the samearguments should apply. Because of its angular momentum,a gyroscope defines a unit vector, that is, the direction of itsangular momentum and a scalar quantity, that is, the absolutevalue of the angular momentum. Therefore, an arrangementof three spinning gyroscopes with non-collinear axes shouldallow for fixing the angular orientation of a solid body.

This is, however, incorrect. The error is due to a misunder-standing of the nature of angular momentum. Nevertheless,the above argument can be found in many variations. A

prominent example for the popularity of this misconceptionis again provided by a movie from the International SpaceStation.24 There, NASA astronaut Don Pettit uses portableCD-players as gyroscopes. First, he demonstrates that thespinning disc inside the CD-player stabilizes one axis. Inorder to fix the orientation of a flashlight, he then combinesthree CD-players such that the axes of rotation of the spin-ning discs are perpendicular to each other.

The experiment presented in the movie24 was not gearedtowards a systematic quantitative investigation and may becriticized from a scientific point of view. For example, theastronaut did not perturb the flashlight to test the desired sta-bility. On the contrary, the flashlight rotates slowly even ifnot perturbed. Therefore, we repeated the experiment shownby the astronaut24 under conditions of microgravity aboardan airplane performing parabolic flights. In contrast to thedemonstration on the ISS, our experiment allows for record-ing the time series of the angular orientation of the free-floating device and, thus, a quantitative analysis. Subsequentdata analysis reveals that three gyroscopes cannot stabilizethe orientation of an object in space, in agreement withtheory.

In order to understand the experiment, we first introducethe basic elements of gyro-dynamics and then present theexperimental data.

II. GYRO-DYNAMICS

Saying that spinning flywheels have the tendency to retaintheir orientation because angular momentum is conserved iscertainly oversimplifying. We wish to explain clearly, inmathematical terms, what is meant by this phrase. But first,let us briefly comment on the need for a mathematical analy-sis: while in many cases, plausible arguments may help tounderstand the physics behind observed phenomena, webelieve that there are systems where a deep understandingcan be better (or only) achieved through the language ofmathematics. We also believe that gyroscopic motionbelongs to this class of systems where “scientific literacy” isa necessary precondition for a fundamental understanding.For more discussion on this subject, also regarding the teach-ing of physics courses, see a recent paper by Carl Mungan.25

The evolution of the angular orientation of a rotating solidbody is described by Euler’s equations (see, e.g., Ref. 26 fora derivation) as follows:

175 Am. J. Phys. 88 (3), March 2020 http://aapt.org/ajp VC 2020 American Association of Physics Teachers 175

Hxx _xx þ ðHzz #HyyÞxzxy ¼ Tx ;

Hyy _xy þ ðHxx #HzzÞxxxz ¼ Ty ;

Hzz _xz þ ðHyy #HxxÞxyxx ¼ Tz;

(1)

where Hxx, Hyy, and Hzz are the principal moments of inertiaof the rotating body, ~x ¼ ðxx;xy;xzÞ is its vectorial angular

velocity, and ~T ¼ ðTx; Ty; TzÞ is an external torque acting on

the rotating body. Note that ~x and ~T are expressed in therotating body-fixed reference frame.

Euler’s equations simplify when we assume that the gyro-scope is an axisymmetric body with respect to the z-axissuch that Hxx ¼ Hyy ¼ H? and Hzz ¼ Hk and moves in the

absence of external torque, ~T ¼~0,

H? _xx þ Hk #H?! "

xzxy ¼ 0 ;

H? _xy þ H? #Hk! "

xxxz ¼ 0 ;

Hk _xz ¼ 0:

(2)

Suppose the flywheel rotates around its principal inertia axis,z, (its symmetry axis) at angular velocity ~x ¼ ð0; 0;x0

z Þ. Tostudy the stability of the rotational motion, we assume a

small initial perturbation~d0¼ ðd0

x ; d0y ; d

0z Þ such that the rotation

axis and the symmetry axis no longer coincide, ~x¼ ðd0

x ; d0y ;x

0z þ d0

z Þ. A stability analysis of Euler’s equations

(2) (see, e.g., Refs. 27–29) quantifies the temporal evolution ofthis perturbation,

H? _dx þ Hk #H?! "

x0z dy ¼ 0 ;

H? _dy þ H? #Hk! "

x0z dx ¼ 0 ;

Hk _dz ¼ 0:

(3)

From the last equation follows

dz ¼ const: (4)

Since both the first and second equations do not contain dz,the system does not react to a change of the rotational veloc-ity, xz, except that it spins slower or faster; the other twocomponents of ~x are not affected. The reaction of the systemto d0

x and d0y is different: we solve the second equation for dx

and insert its time derivative, _dx, into the first equation toobtain

€dy þðHk #H?Þ2

H2?

x0z

! "2dy ¼ 0: (5)

This equation describes a harmonic oscillation of dy, andan equivalent equation can be derived for dx. This meansthat the initial perturbations d0

x and d0y do not grow in time

but oscillate around zero, which implies that the rotationalmotion of the gyroscope around its symmetry axis isstable.

To see how this oscillation affects the vector of rotation,~x, we go back to Euler’s equations (2) and obtain for thecomponent parallel to the symmetry axis

xz ¼ const & xk: (6)

Multiplying the first two equations of (2) by xx and xy,respectively, and adding the equations yield

_xxxx þ _xyxy ¼ 0; (7)

which can be written as

d

dtðx2

x þ x2yÞ ¼ 0; (8)

and, thus,ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffix2

x þ x2y

q& x? ¼ const: (9)

With c & xzðHk #H?Þ=H?, Euler’s equations (2) read

_xx ¼ #cxy; (10)

Fig. 1. Niels Bohr and Wolfgang Pauli in 1954, debating about a toy gyro-scope at the inauguration of the new Institute of Physics at Lund.

Fig. 2. ESA astronaut Tim Peake demonstrating the stability of a gyroscopeagainst small torques aboard the ISS in weightlessness. [Snapshot from amovie (Ref. 39) where ESA astronaut Tim Peake shows how gyroscopes canbe used to keep the spacecraft stable during his six-month Principiamission.]

176 Am. J. Phys., Vol. 88, No. 3, March 2020 M€uller, Sack, and P€oschel 176

_xy ¼ cxx: (11)

Therefore,

€xx ¼ #c2xx; (12)

with the solution

xxðtÞ ¼ c1 cos ðctÞ þ c2 sin ðctÞ: (13)

By means of Eq. (10), we find

xyðtÞ ¼ c1 sin ðctÞ # c2 cos ðctÞ: (14)

If we choose the zero of time such that for t¼ 0, xy ¼ 0, weobtain c2 ¼ 0, and with Eq. (9),

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffix2

xðtÞ þ x2yðtÞ

q¼ c1 ¼ x?: (15)

The final result is then

~xðtÞ ¼x? cos ðctÞx? sin ðctÞ

xk

0

B@

1

CA; (16)

that is, the gyroscope rotates around its symmetry axis, z, atconstant velocity c. The corresponding angular momentumreads

~L ¼H?H?Hk

0

B@

1

CA ' ~x ¼H?x? cos ðctÞH?x? sin ðctÞ

Hkxk

0

B@

1

CA: (17)

At first glance, this result may seem surprising since, obvi-ously, angular momentum is not constant, although we arestudying a torque free system. But recall that Euler’s equa-tions and hence our entire derivation are formulated in therotating, body-fixed frame of reference. Angular momentumis indeed conserved in the inertial laboratory frame.

In the laboratory frame, where ~L ¼ const, the symmetryaxis of the gyroscope, z, describes a cone whose axis coin-cides with the space-fixed orientation of the angular momen-tum, ~L, with the angular frequency. From Eqs. (17) and (16),we find

~L ¼ H?~x þ ðHk #H?Þxzez; (18)

where ez is a unit vector whose direction is given by thesymmetry axis of the gyroscope. From Eq. (18), we see that~L; ~x, and ez are all in one plane and that the angles between~L; ~x, and ez are invariant in time. Therefore, we concludethat ~x rotates along the surface of a cone whose symmetry

axis is defined by ~L (see Fig. 3).

III. STABILITY OF SYSTEMS OF GYROSCOPES:EXPERIMENTS IN WEIGHTLESSNESS

A. Two common misconceptions

Consider a spinning flywheel whose angular momentumpoints along the direction of the symmetry axis. As shown inSec. II, the spatial orientation of such a flywheel is stable

against perturbations. That is, if we hit the gyroscope moder-ately, its axis of rotation will stay close to the axis of rotationof the unperturbed system. To be more precise, the axisof rotation will be slightly deflected with respect to theunperturbed axis and starts to rotate around the axis of thespace-fixed angular momentum, shaping the surface of acone (see Fig. 3).

Therefore, since a single gyroscope can stabilize the angu-lar orientation of one axis and the spatial orientation of anythree-dimensional object is determined by angles aroundthree axes, it seems reasonable to fix the angular orientationof the object in space by three gyroscopes each stabilizingone of the axes. This, however, implies two misconceptionsabout the nature of angular momentum.

The first misconception concerns the number of indepen-dent axes: if the gyroscope is rigidly attached to an object,one axis of the object is stabilized, defined by the angularmomentum of the gyroscope. The object can, however, stillbe rotated around this stabilized axis without any resistanceexerted by the gyroscope. This rotation could be suppressedby a second gyroscope rigidly attached to the object suchthat it stabilizes rotations around any other axis, differentfrom the first gyroscope. Therefore, as a special case, a com-bination of two flywheels rigidly attached to an object suchthat the directions of their angular momenta are perpendicu-lar to one another would resist rotations around any axis.This reveals the first error: although the orientation is deter-mined by three angles, two flywheels (instead of three)should be sufficient to fix the angular orientation of an objectin space. It turns out, however, that the latter argument isincorrect due to the second misconception.

The law of conservation of angular momentum states thatthe total (vectorial) angular momentum is conserved, pro-vided the net external torque is zero. This implies that thetotal (vectorial) value of angular momentum inherent in abody is a property of the body itself and is not related to theinternal processes inside the body. Thus, independent of theconstitution of the bodies, there is always one vectorial

Fig. 3. Angular momentum, ~L, axis of rotation, ~x, and axis of symmetry ofthe gyroscope, ez, are all in a plane. Consequently, ez describes circles cen-tered by the invariant vector of angular momentum ~L (see the text forexplanation).

177 Am. J. Phys., Vol. 88, No. 3, March 2020 M€uller, Sack, and P€oschel 177

quantity—its angular momentum—which is conserved, irre-spective of the number of gyroscopes attached to this body.Instead, the total angular momentum of the body resultsfrom the vectorial sum of the angular momenta of all its con-stituents, ~M ¼ ~M1 þ ~M2 þ ' ' '. Consequently, an array ofgyroscopes attached to an object has the same stabilizingeffect as a single gyroscope with angular momentum ~M. Inparticular, this means that the angular orientation cannot befixed in space using an array of gyroscopes spinning at con-stant velocities.

Although this argument seems to be almost trivial, theopposite is claimed on many occasions, even in prominentpublications.24

B. Experimental setup

To test the conjecture of multi-gyroscopic stabilization,we performed experiments with an array of three flywheelsrigidly combined such that their axes are pairwise perpendic-ular. The apparatus is sketched in Fig. 4. The edge length ofthe cubic arrangement is 12 cm, and the weight of the wholeapparatus is 1:0 kg. The resulting moment of inertia withrespect to rotations around the diagonal of the whole setup is10:6( 103 g ' cm2. Each of the flywheels is made from poly-carbonate, has a diameter of 80 mm, is 6 mm thick, and hasa mass of 36:19 g. The resulting moment of inertia withrespect to the symmetry axis is 34 g ' cm2. Each flywheel isdriven at a constant rate of 10000 rpm by a motor of the typeused for model helicopters. A MEMS (microelectromechani-cal sensor) is used to measure the orientation of the setup asa function of time. An integrated microcontroller operatesthe motors at constant speed of rotation and records the datafrom the MEMS. Curiously, the MEMS device itself uses asubmillimeter reference gyroscope30 to measure the rate atwhich its orientation changes—that is, the components of theangular velocity ðxx;xy;xzÞ at which the experimental setuprotates around the axes x, y, and z of the body-fixedCartesian coordinate frame.

To study the dynamics of the experimental setup in the free-floating, torque-free regime (see Eq. (2)), the experiments were

performed aboard an aircraft performing parabolic flights,31

where special flight maneuvers allow for conditions of weight-lessness for intervals of approximately 22 s. Usually, theexperiments are fixed to the floor of the airplane. Since techni-cally both the airplane and the apparatus follow parabolic tra-jectories, the apparatus would be weightless, provided theairplane performs an exact parabola. However, due to atmo-spheric influences and other perturbations, the motion of theaircraft is not ideal. One can decrease the deviations fromweightlessness considerably by performing the experiment infree-floating mode. Here, the experimentalist releases the appa-ratus that floats decoupled from the airplane and, thus, feels(almost) perfect weightlessness. Any imperfections of the tra-jectory of the aircraft lead to a motion of the free-falling appa-ratus with respect to the airplane. Figure 5 shows a snapshotfrom a movie of the free-floating experiment. The full movie isavailable as the supplementary material.32 Using free-floatingexperiments, one can achieve weightlessness up to a high preci-sion, however, only for a short period of typically 5–10 s, whilethe duration of the parabola performed by the aircraft is about22 s. We recorded the angular velocity ðxx;xy;xzÞ of the arrayof spinning flywheels during such free-floating intervals.

C. Experimental results

The movie of the parabolic flight experiment (see mov-ie1.mp4 of the supplementary material32) illustrates the mis-conceptions described in Sec. III A: despite the threeperpendicular flywheels, the experimental setup does notretain its orientation. Let us analyze the motion of the appa-ratus quantitatively using the recorded angular velocityðxx;xy;xzÞ as a function of time, shown in Fig. 6.

The highlighted intervals in Fig. 6 indicate periods of free-floating where the apparatus moves in the absence of gravity.During these intervals, the components of the angular veloc-ity in the body-fixed reference frame reveal periodic oscilla-tions, in agreement with Eq. (16). In the time intervalsbetween the highlighted intervals, the experimentalist rein-stalled the apparatus in the center of the aircraft’s cabin to

Fig. 4. Experimental setup. (Left): a cube with three perpendicular flywheels. The padding along the edges is due to the operation on a parabolic flight cam-paign. (Right): sketch of the array of gyroscopes.

178 Am. J. Phys., Vol. 88, No. 3, March 2020 M€uller, Sack, and P€oschel 178

allow for a new free-floating interval. Here, the measuredvelocity components fluctuate irregularly.

The angular velocity, ðxx;xy;xzÞ in the co-moving refer-ence system is, however, a rather unintuitive quantity since itdescribes the rate at which the experimental setup rotatesaround a set of body-fixed axes whose orientations vary intime themselves. For a more intuitive characterization, weconsider rather the orientation of the apparatus in the inertiallaboratory system. The question “what is the laboratory sys-tem in the case of our experiment?” is delicate. In an inertialsystem, a force-free body moves at constant velocity. In the

presence of gravity, this can only be achieved by switchingto a coordinate system in free fall. Despite performing para-bolic flight maneuvers, the aircraft is not in perfect free fall,first due to atmospheric perturbations and second since thecabin of the aircraft rotates around the axis perpendicular tothe main direction of the cabin and perpendicular to gravity.Therefore, a coordinate system attached to the airplane can-not be used as an inertial reference frame. Our experimentalapparatus that is floating inside the aircraft is, however,freely falling to a very-good approximation. Therefore, wedefine our laboratory system such that its origin is attachedto the center of mass of the free-floating apparatus. Its coor-dinate axes are perpendicular to one another and do notrotate. Consequently, in this coordinate system, a force-freebody33 would perform a straight trajectory at constant veloc-ity, that is, this system qualifies as an inertial system.

The time-dependent orientation of the apparatus in thelaboratory frame can be obtained from the momentary angu-lar velocity ~x ¼ ðxx;xy;xzÞ expressed in the body-fixed ref-erence frame (x, y, z), as delivered by the MEMS sensors:for a short time interval, dt, the body-fixed coordinate systemrotates around its axes by ~u ¼ ~x dt. Starting from an arbi-trary initial orientation, the orientation of the apparatus inthe laboratory frame can be obtained by successively inte-grating the angular velocities and correspondingly rotatingthe initial reference frame.

The transformation between the body-fixed frame and thelaboratory frame (that is, the rotation of the apparatus aroundthe axes x, y, and z) can be described by the matrices

Mx ¼1 0 0

0 cos ux #sin ux

0 sin ux cos ux

0

B@

1

CA;

My ¼cos uy 0 sin uy

0 1 0

#sin uy 0 cos uy

0

BB@

1

CCA;

Mz ¼cos uz #sin uz 0

sin uz cos uz 0

0 0 1

0

B@

1

CA:

(19)

For sufficiently small rotation angle ~u, we can approximatecos u ) 1 and sin u ) u, and, additionally, the sequence ofapplication of the matrices is unimportant.27,34 We choosethe sequence x, y, z and define the general rotation matrix

M ¼ MxMyMz: (20)

Therefore, the orientation at time tðnÞ & tð0Þ þ ndt can beexpressed by a rotation matrix

M tðnÞð Þ ¼ Mð0Þ 'Mð1Þ… MðnÞ; (21)

where MðkÞ represents the rotation ~uðkÞ #~uðk#1Þ during theinterval ðtðk#1Þ; tðkÞÞ. The direction of the rotation axis at timetðnÞ, expressed in the laboratory frame, is then given by theunit vector eðnÞ, which is the eigenvector of MðnÞ correspond-ing to the unit eigenvalue.27

Figure 7 shows the motion of the apparatus, representedby the vector e during the parabolic flight, based on the mea-sured angular velocities presented in Fig. 6. When freely

Fig. 6. (Color online) Components of the angular velocity, xx, xy, and xz,in a body-fixed reference frame as functions of time, recorded during a zero-gravity parabola. The highlighted (labeled) intervals indicate four periods offree floating. The colors and labels agree with those in Fig. 7.

Fig. 5. Snapshot of the free-floating experiment. While the array of fly-wheels is freely floating inside the airplane, the experimentalist himself isfastened on the floor of the airplane by a belt.

179 Am. J. Phys., Vol. 88, No. 3, March 2020 M€uller, Sack, and P€oschel 179

floating, the vector e moves approximately on the surfaceof a cone whose symmetry axis is defined by the vector oftotal angular momentum of the apparatus. For each free-floating interval, the vector of total angular momentum wasdetermined by fitting a cone to the momentary positions ofe. These vectors of angular momentum are shown byarrows whose colors (labels) correspond to the colors(labels) used in Fig. 6. The blue cone (label c) correspondsto the third free-floating interval and degenerates almost toa disc. The direction of e at time tðnÞ is marked by a dot onthe unit sphere. The dots corresponding to free-floatingperiods appear circular around one of the colored (labeled)arrows. Dots outside the intervals of free floating are dis-tributed randomly on the unit sphere and are shown in graycolor. The left panel of movie2.mp4 of the supplementaryvideo material32 shows an animated version of these datawhere additionally the momentary axis of rotation is indi-cated by a red arrow. The right panel of the movie showsthe synchronized experiment as performed during the para-bolic flight. Gray arrows show the momentary axes of theco-moving reference frame. A slow motion version of thevideo is available as movie3.mp4 of the supplementarymaterial.32

The representation of the measured data shown in Fig. 7and Ref. 32 reveals immediately the misconception dis-cussed in the Introduction and in Sec. III A: although theapparatus contains three flywheels with perpendicularly ori-ented rotation axes, the device rotates around a body-fixedaxis, which in turn rotates around the space-fixed axis of thetotal angular momentum. The latter axis is conserved during

each free-floating period due to the absence of externaltorques.

IV. SUMMARY

The orientation of spinning flywheels is robust against per-turbations. This means that a single gyroscope that is rigidlyattached to an object is able to stabilize one axis of the object.A popular misconception suggests that the orientation of anobject can be arrested completely by mounting a second gyro-scope. While the first gyroscope stabilizes one axis of theobject, the second gyroscope stabilizes the remaining rotationaldegree of freedom. Despite its popularity, this mechanism isspoiled because if several gyroscopes are combined rigidly toone mechanical system, their angular momenta add up to onetotal angular momentum. As illustrated by our experimentalresults, only this total angular momentum is conserved and, justlike for a single gyroscope, this total angular momentum onlydefines and stabilizes one axis of the object, whereas the objectis still able to rotate around this axis. Even though the ISS astro-naut24 uses three perpendicular gyroscopes, he only accom-plishes stabilization of one axis. If the stabilized axis coincideswith the direction of the light beam, this can be sufficient tohold a flashlight because the non-stabilized rotation around anaxis parallel to the direction of the light beam is unimportantfor the function of the flashlight. In this case, however, themost efficient spin stabilization is achieved when arranging thegyroscopes collinear rather than perpendicular because (in thiscase) the vectorial summation of the individual angularmomenta leads to the largest total angular momentum.

With this paper, we aimed to demonstrate that the na€ıveconcept of gyroscopic stabilization using an array of gyro-scopes as explained, e.g., in Ref. 24, does not work asexpected. For an effective gyroscopic stabilization, the gyro-scopes should be actively driven either by controlling theirindividual angular velocity or by controlling the directions oftheir axis of rotation. There is a wide range of applications ofsuch stabilizations from shipping35 to astronautics.36 Contraryto passive stabilization, multiple gyroscopes in orthogonalarrangement are reasonable in the case of active stabilizationbecause a single gyroscope can determine, at most, two of thethree degrees of freedom of an arbitrary rotation in threedimensions. The Hubble space telescope, for example, isequipped with six (partially redundant) gyroscopes (see, e.g.,Ref. 37). These gyroscopes are part of the Hubble point con-trol system. By active driving, angular momentum can betransferred between the gyroscopes and the remaining parts ofthe telescope. The resulting torques are applied to point thetelescope in a desired direction. Similar techniques are used ininertial navigation systems in general (see, e.g., Ref. 38).

ACKNOWLEDGMENTS

The authors thank the German Aerospace Center, DLR,for financial support through Grant No. 50 WM 1752-GRANADA. The presented data were recorded during 5parabolic trajectories of the 31th DLR parabolic flightcampaign in Bordeaux.

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Fig. 7. (Color online) Orientation of the apparatus in microgravity during aparabolic flight. The white arrows show the momentary axes of the co-moving reference frame, and the red arrow without label shows the momen-tary axis of rotation, e. For each instant of time, tðnÞ, the position of e ismarked by a dot on the unit sphere. During the intervals of free floating, thetotal angular momentum of the apparatus is conserved. For each free floatinginterval specified in Fig. 6, the angular momentum is shown by an arrow,marked by the same color and label as in Fig. 6. The direction of e at timestðnÞ is marked by a dot on the unit sphere. The dots corresponding to freefloating periods appear circular around one of the colored (labeled) arrows.Dots corresponding to values of e outside the intervals of free floating arescattered randomly on the unit sphere and are shown in gray color.

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Cole, Belmont, 2004).30N. Yazdi, F. Ayazi, and K. Najafi, “Micromachined inertial sensors,” Proc.

IEEE 86, 1640–1659 (1998).31F. Haber and H. Haber, “Possible methods of producing the gravity-free

state for medical research,” J. Aviation Med. 21, 395–400 (1950).32See the supplementary material at https://doi.org/10.1119/10.0000517 for

movies of the parabolic flight experiments. movie1.mp4 shows the arrayof perpendicular flywheels under conditions of weightlessness. Despite thethree perpendicular flywheels, the apparatus does not retain its orientation.It rather precesses around its space-fixed total angular momentum. Thisobservation can be quantified by measuring the vectorial angular velocityand orientation of the device. movie2.mp4 compiles the video of the para-bolic flight experiment (right part of the movie) and a synchronized visual-ization of the quantitative measurement data (left part of the movie). Thethree gray arrows in the left part indicate the current orientation of theapparatus, the red arrow illustrates the axis around which it currentlyrotates. For each instant of time the orientation of the axis of rotation (thetip of the red arrow) is marked by a gray dot. During the free-floating inter-vals, the dots are colored: green dots correspond to the first, yellow dots tothe second, blue dots to the third, and orange dots to the fourth free-floating period. During the free-floating intervals, the space-fixed angularmomentum can be obtained by fitting cones to the corresponding datapoints. These fitting cones are drawn together with the resulting directionof the total angular momentum. The blue cone, corresponding to the thirdfree-floating interval, almost degenerates to a circle. movie3.mp4 is a slowmotion version of movie2.mp4.

33By “force-free” we mean that the body does not experience any other forceexcept the gravitational force of Earth.

34M. J. Benacquista and J. D. Romano, Classical Mechanics (Springer,Basel, 2018).

35T. W. Chalmers, The Automatic Stabilisation of Ships (Chapman and Hall,London, 1931).

36N. S. Bedrossian, S. Bhatt, W. Kang, and I. M. Ross, “Zero-propellantmaneuver guidance,” IEEE Control Syst. Mag. 29, 53–73 (2009).

37NASA Space Telescope, “Hubble’s pointing control system”, <https://www.spacetelescope.org/about/general/gyroscopes/> (2018).

38W. T. Thomson, Introduction to Space Dynamics (Dover, New York,1986).

39European Space Agency, “Gyroscopes in space”, <http://www.esa.int/spaceinvideos/Videos/2016/03/Gyroscopes_in_space> (2016).

181 Am. J. Phys., Vol. 88, No. 3, March 2020 M€uller, Sack, and P€oschel 181