Avoiding Gimbal Lock - 3D

Avoiding Gimbal Lock
o place an object in 3D space,its transformation proper-
is a pivoted device,most often a ring, which rotatesaround a single
ties need to be specified:position, scale,and orientation. The
axis. By mounring a gimbal inside another one, the inner ring ro-
first two attributes are easily definable by three numbers for
tatesaround an additional axis, increasingrhe degreesoffreedom
each. The meaning of the x, y, and z positions and scaleparamerers
by one. Defining the orientation of an objecr with Euler anglesis
are easyto understand,visualize,manipulare, and animate for art-
like putting it inside a virtual three-gimbaldevice,and then rotar-
ists. However, that is not the casefor orientation.
ing each ring by the correspondingangle. (Once again, the order
Using an xyz rriplet (three angular values) to manipulare an
of rotations and the coordinate frame axesmust be agreed upon.)
object'sorientation may become impossible-for ir.rstance,
during
some configurations of the rhree angles,such as gimbal lock-and
lead to major problems when animating rhesevalues.Gimbal lock
is a phenomenon known for a long rime, and it has causedsevere problems long before compurer graphicsemerged.According
to NASA documents on the Apollo spaceprogram, pilots had to
keep a closeeye on the Gimbal Lock warning light while maneuvering the spacecraft in order to avoid unwanted and dangerous
malfi-rnctioningin the guidanceand control sysrems.
The orientation, or angular position, of a rigid object has three
degreesof freedom. By holding a camera in our hands, for example, we can pan left and right, tik it up and down, or roll it
without changing the point of interest.The mosr common and intuitive way to define theseartributesis rhe useof Euler angles:The
orientation is represented by three consecutive rotations around
the main axesof a reference frame. However, the order of the rotational axes is something the industry has never agreed on, so
The outer ring can represenrthe rilt, the middle rir.rgthe pan, and
it is essentialto supply this information if we transfer animation
data using Euler angles.Each major 3D application has a way to
the innermost ring the roll. However, most of the publications on
Euler anglesrefer to the three attributes as yaw pirch, and roll, as
changethe order ofrotations.
used in aerospaceapplications.
Using three gimbals, it is possiblero consrrucra physicaldevice,
a gimbal sysrem,basedon the principle of Euler angles.A gimbal
Having full control over the three degreesof freedom, one could
conclude that Euler angles (or gimbal sysrems,in general) are the
perfect way to describe orientation. Unfortunately, rhere are configurations wherein we lose one degree of freedom: the gimbal
lock. In this state, one of the girnbal rings is rotated such that it
aligns perfectly with another. In this siruation, the enrire range of
rotations is unreachable, and we may need to first re-orient the
locked gimbal in order to rorare the ring arbitrarily. If the angles
are near the gimbalJock stare, the gimbal system becomes unsrable, as even small rotations (or round-off errors of the numerical
representation) may yield unexpected results.
rol
J u n e2 0 0 9
'T?awprertr-rutmtrffiffi
It is fairly easyto get closeto gimbal lock
'We
simplyneedto rotatethe'middle" angle,
or gimbal, by 90 degrees.Becausenryoofthe
rotational axeshavebecomealigned,chang-
ing gimbals, Euler anglesare the industry
qmnot rotatefreely,therebyriskingthe loss
ing the first or the last anglein the chain re- standardfor describingrotations.
Real-world gimbal systems have been
sultsin a rotation around the sameaxis(see
ofreference.
imagesabove).The only way to leavethis
known and usedfor a very long time now.
ing the magazineand begin watching the
locked sate is to adjust the middle rotation
The gyroscope--invented nearly 200 years
againand move it out of the dangerzone.
ago-is a fine example:At the heart of this
1995 movieApolb I 3 farvring Tom Hanl6
(and some great visual effects). You will
deviceis a pladorm,with oneor morespin-
be surprisedhow much they talk, or yell,
GimbalLod<in CG
This may be a good dme to finish read-
ning wheelsor disla mounted in gimbals. about avoiding gimbal lock and losing the
Computer animators controlling charac- The largeattgular momentum of the spin-
referenceof the Inenial MeasurementUnit.
ters' body pans or any kind ofobject us-
ning elementsforces the central platform
The IMU of the Apollo space program
ing only three angularvaluesshould try to
to remain almost perfecdyfixed, regardless was a three-gimbalrystem,and pilots were
avoid gimbal lock. Investipte your setup
of the motion of the devicet'thell," asthe
required to navigatethe spacecraftso they
beforeyou stan to animate,and changethe
gimbalslet it rotatefreely.By installing sen-
did not approachthe lO-degreedangerzone
order of the axesso the middle axis is the
sors to detect the rotation of eachgimbal,
around gimbal lock. The curent atdrudeof
leastone used.Ifdl three degreesoffree-
we can measure the relative orientation
(or anitude) of any object as it relatesto
the vehide was displayedon the Flight Direcor Anitude Indicator,or the "eight bdl,"
aspilots refered to it. Ifthe indicatorentered
help. If readabiliry and human interac-
a fixed reference$zstem;That is how the
Inenid MeasurementUnit (IMU) of air-
tion are not important, such as under the
craft, spacecraft,and watercraft, including
I80-degree poles, and t}le stable member
hood of game engines, then quaternions
guided missiles,works. The three gimbals
and rontional matricesare usedinsteadof
lost its attitude reference,the gimbalshad to
be re-alignedin-fight againststarreferences.
Euler angles.Quaternions arefour-dimen-
hold the central stableplatform-widr the
spinning glsms115-v/ithout introducing
sional representadons,ideal for compudng
any notable external torque, whic} would
smooth transitionsbetweenposes.\7hile
force the pladorm to loseits origind orien-
gimbal lock never happens when using
tation. That is where gimbal lock becomes 1-i crew members,who were losing oxygen
quaternions,they arehard to readand con-
a real problem: By losing one degreeof
and electricalpower in a crippled spacecraft
ceptualize.In spite of all the issuesof lock-
freedom near this state,the fixed platform
at the sametime.
dom are needed,hierarchical transformations (parenting an additional node) may
fie red dangerzone centeredat yaw 0 and
So,the nexttime you arefighting gimbal
lock in a 3D animation packageor game
engine,dont panic.Just rhink of theApolb
June2009 't:'i;t'+! | 11