The possibility of using quaternions to describe the position of a gyroscope with a given rotating vector is discussed

Geiger, W.

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N80-l8811 
NASA TECHNICAL MEMORANDUM NASA TM-75731 
NUMERICAL INSTABILITY AND STABILIzATION OF THE KINE~~TIC 
DIFFERENTIAL EQUATION USING QUATERlfIONS 
w. Geiger 
Translation of "Numerische Instabilit1l.t und Stabiliserung
der kinematischen Quaternionendifferentialgleichung", 
Zeitschrift fUr Angewandte Mathematik und Mechanik, 
Vol. 59, No. 5, ¥~y 1979, pp. T 118 - T 120 
(~ASA-TM-75731) NUMERICAL INSTABILITY AND N80-18811 
STABILIZATION OF THE KINEMATIC DTFFERENTIAL 
EQUATION USING QUATERNIONS (National 
A~ronautics and Space Administration) 9 p Unclas 
HC A02/MF A_Ol __________ ... C?~L 12A G3L64 .. !~}_~3!__ 
NATIONAL AERONAUTICS A1~ SPACE ADMINISTRATION 

WASHINGTON, D.C. 20546 NOVEMBER. 1979 

REPRODUCED BY 
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NASA TM-7571 1. 0-11o,' C~~o
 
.eitcAl ri 
 STABiLITY AND STABILIZATION 5.Reporhanie
 
F THE KINEo5 a17 pp.EQ1ATI-N November 1979
 
1SING QUATERNIONS
 
7 Auth sci) 
 8. Performing Orginiotion Report No
 
W. Geiger o. a rg o
 
Institute B for Mechanics, Universiiy wodk uet No
of Stuttgart- n 

I I - Contract ot Grant No. 
9. Perfosming Orgonization N usingd
Adtn s 
 NASw-3199 
-

Leo Kanner Associates 3 Type of Report and Perod CoStred 
Redwood City, California 94063 UTranslation 
12. Sponsoring AgencyNo a Ades 
National Aeronautics and Space Admini; F.Sosrng- .yCd 
1trations, Washingtof , D.C. 20546 ) . Sonso 22. Prec 
15. Supplementary Notes 
Translation of "Numerische Instabilit~t und Stabiliserung
 
der kinematischen Quaternionendifferntialgleichung" ,
 
Zeitschrift ftir Angewandte Mathematik und Mechanik,
 
Vol. 59, No. 5, May 1979, pp. T 11'8 - T 120. (A79-43954) 
16. Abstract 
-Discussion of difficulty involved in describing the 
position of a rigid bo-y or gyroscop@IWl~ l 
rotating vector, including examination of the 
possibility of using quaternions for this purpose. 
17. Key Word, (Seleted bY Autho(S)) 18. D~st ,ut..onSttment 
Unclassified-Unlimited.
 
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Unclassified . Unclassified. 9 
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NUMERICAL INSTABILITY AND STABILIZATION OF THE KINEMATIC
 
DIFFERENTIAL EQUATION USING QUATERNIONS
 
W. Geiger
 
Institute B for Mechanics, University of Stuttgart
 
1. Introduction /T 118*
 
Because of the singularity of the Eulerian angles (frame barrier) 
other coordinates often must be used to describe the position of at 
rigid body or'gyroscope with a given rotating vector wo=,ow(), respectively. 
One possibility is using rotating quaternions. These satisfy the
 
linear differential equation (DGL)
 
od0] -w. -10 1)[0=~ 23] [q:o41 lW, 0 -W3-W,1 qr
oLo W Z-,0o13 q 
The jiare here the normalized coordinates of the rotating vectorjhi
 
in the rigid system. The present text is concerned with this
 
kinematic differential equation using quaternions (KQD).
 
'must
The four coordinates of fulfill the requirement
 
IN1o 92 _ - ,1q3-O (2). 
From KQD follows K 0, thus N = constant is a first integral of
 
KQD1) It is physically stabilized on the boundary.
 
2. Numerical Instability of the Normalized Integral
 
An unsymmetrical force-free gyroscope was employed as an
 
example. Its motion is represented exactly by the Eulerian
 
angles UVj3111. This concerns a periodic nutation-in tof the
 
* Numbers in the margin indicate pagination in the foreign text. 
S)The designation "integral" is also applied in the following for N.
 
1
 
-4004 
figure axis. The Renge-Kutta procedure of the fourth order with
 
constant step H was employed as integration procedure.
 
N 
0 4 8 1 16 20 24 &ZPeriods 
-. 008 -__,, 
-,016 
-Qozo 
Figure 1: Normalized Error N(t)
 
The normalized integral deviated in a nearly linear manner from
 
the desired value. The course (designated by KQD) is given as a
 
function of time, where the first 30 periods of Ljwere calculated
 
with 30 steps for each period. Because of the phyisical stability
 
of the normalized integral it is only a matter of a numerical in­
stability here. The procedural error must always have an effect
 
in one direction. As proof the quaternion ljA+I from the disrupted
 
Taylor series may be calculated after step (j+1-) under the assump­
tion (VI) that the rotating vector may be assumed to be constant
 
during an integration step with the result
 
2 1 4 HIlq,+111 (Hlflu)1q 1 , (H, Go( (,- 232it 
.o=w ),t =jI , 0,1, 2'.... 
where for all possible steps flj>O. Corresponding values apply for
 
procedures of the third and fifth order. The quaternion norm there­
by decreases with each integration step and asymtotically approaches
 
zero.
 
Remarks: 19
 
was achieved
 
a) A rather good approximation of precondition VI 

in the example employed. The value of rotating vector could
 
2 
even further be assumed as always constant (V2).
 
Then
 
- it=t onst.i -I +-i jL( _B,
 
and this function describes the actual course of the normal­
ized integral with a high degree of accuracy.
 
b) If as an especially simple example a permanent rotation is
 
calculated with ;,,=w,=o,,'=-, equations (3) and (4) describe
 
exactly the behavior of the normalized integral.
 
c) The proof resulting from (3) requires in addition to VI
 
only that the matrix Siof KQD is symmetrical and involu­
toric2), so that other differential equations with such 
coefficient matrices will exhibit identical characteristics.
 
WhentheBulerian angle is calculated from the quaternion
 
according to the usual formula
 
COSq q+ (5)1
 
then the normalized error effects a very large error in this angle,
 
for
 
. 

cosg 2 jqj 2 cos#*, hus 7 2 
. . . . 
_ . . . (6 ) . ... 

when iis the exact angle. This causes a numerical singularity in
 
the range of the frame barrier.
 
Independent of these results, suitable steps must be taken to
 
stabilize this integral numerically because of the numerical in­
stability of the normalized integral.
 
3. Stabilisation according to Baumgarte
 
Baumgarte [2 has proposed a gradient stabilisation procadure
 
for systems of usual differential equations of the first order. In
 
the case of the requirement for stabilisation designated here "on
 
average" the differential equation is altered with the aid of known
 
t = -laW ,R =unit matrix 
3
 
first integrals in such a manner that the stabilized differential
 
equation leads to a differential equation for the error in integrals
 
with the result that these are reduced asymtotically on the average
 
when they have deviated from desired value zero through procedural
 
error. Applied to kinematic differential equation using quaternions,
 
the stabilized differential equation (KQDS) becomes
 
= (92- 4kNE) q M2) 
where k is a positive, otherwise arbitrarily selected stabilisa­
tion factor. The normalized integral would have to behave according
 
to
 
N -4N(1 + ) (8). 
Actually, however, the typical course shown in Fig. 1 with k = 0.5 
as an example occurs (designated by KQDS). During the first inte­
gration step a jump in the norm occurs. Thereafter the integral
 
stabiliyes on an erroneous value, the error is therefore not re­
duced asymptotically.
 
Examination of a Renge-Kunge step demonstrates that the series
 
(N.) of the normalized error fulfills the corresponding law of form­
ation
 
N i = Ni- (1 +N,)fl, fi, = ivl, N). (9() 
The exprdssion for filcan no longer be stated explicitly. It is a 
polynominal of the 80th order in N.. The coefficients are in each 
case polynominals in k with parameters H anl.iI. Here V1 is applied 
again. 
N 

II ­
10 01 I. Figure 2: Jump in Norm 
_I 
II o - ­
4 
Since the absolute member ofbY does not disappear, the Value
 
of N, already differs clearly from zero (Fig. 2), although R.4= 0.
 
Stabilisation would probably have no influence, as long as the
 
solution meets the requirement. Because erroneous aids, subsequently
 
used in the averaging, are, however, intentionally applied to the
 
integration procedure, a correct solution via stabilisation is made
 
incorrect.
 
The numerical examination of the convergental behavior of the
 
sequence (Ni.) suceeds, with the application of the additional assump­
tion V2 although this entails a great deal of work. It converges
 
toward a valueNcor demonstrates divergent behavior according to the
 
.
value of the stabilisation factor k ) The result is-pyesented in 
Fig. 2 (logarithmic scaling, step size ana-rotating vector as in 
Fig. 1). There are only two optimal stabilisation factors, in par- /T 120 
ticular the zero positions of order tomaintaii ' j7< JO-_ 
it is necessary to select the stabilisation iactor with a minimum 
of three decimal places exactly equal to one of the two optimal
 
stabilisation factors.
 
The criticism pertaining to the stabilisation procedure
4 )
 
described in 21 may be summarized as follows: a) the accuracy
 
depends to a great extent on the stabilisation factor; b) in some
 
cases stabilisation factors have to be determined very exactly.
 
The search for these definitely entails a great deal of work and
 
these are above all dependent on the problem. They undergo changes
 
with the differential equation supplied at the beginning, commencing
 
conditions, integration procedure and stepsize chosen; c) proof
 
assumptions are questionable from the standpoint of numerics. The
 
asymtotic differential equation applied for the proof of the error
 
in first integrals (here:(8)) describes the behavior of the inte­
grals only when the integration procedure functions with no error.
 
This differential equation, however, is only actually fulfilled
 
'NHJis a zero place of function §,only dependent on N.
 
4)Th& regulation designated here as "individual" stabilisation
 
leads to the same qualitative results..
 
5. 
in the framework of the integration error. Since the commencing
 
condition also remains within this order, it can scarcely describe
 
the numerical behavior of first integrals.
 
4. Stabilisation through Normalisation
 
It has been shown that the difficulties resulting from the
 
numerical instability of the quaternion norm are av6ided completely,
 
When the quaternion is divided by its norm after each integration
 
step. A proof of this procedure will be published separately.
 
6 
REFERENCES
 
I] Magnus, K., Gyroscopes, Theory and Applications, Springer-

Verlag, Berlin-Heidelberg-New York, 1971.
 
[2] 	 Baumgarte, J., "Asymtotic Stabilisation of Integrals in
 
the case of Normal Differential Equations of the
 
First Order," Zeitschrift flir Angewandte Mathematik
 
und Nechanik, 701-704 (1973).
 


Numerical instability and stabilization of the kinematic differential equation using quaternions

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