Orientatjon Orientatjon We will defjne orientatjon to mean an - - PowerPoint PPT Presentation

Orientatjon

Orientatjon

• We will defjne ‘orientatjon’ to mean an
• bject’s instantaneous rotatjonal

confjguratjon

• This is the rotatjonal equivalent of positjon
• “Rotatjonal State”

Representjng Positjons

• Cartesian coordinates (x,y,z) are an easy and

natural means of representjng a positjon in 3D space

• There are many other alternatjves such as

Polar notatjon (r,θ,φ)

• Others can be defjned…

Representjng Orientatjons

• Is there a simple means of representjng a 3D
• rientatjon? (analogous to Cartesian coordinates?)
• Not really.
• There are several popular optjons though:

– Euler angles – Rotatjon vectors (axis/angle) – 3x3 matrices – Quaternions – and more…

Euler’s Theorem

• Euler’s Theorem: Any two independent orthonormal

coordinate frames can be related by a sequence of rotatjons (not more than three) about coordinate axes, where no two successive rotatjons may be about the same axis.

• Not to be confused with Euler angles, Euler integratjon, Newton-

Euler dynamics, inviscid Euler equatjons, Euler characteristjc…

• Leonard Euler (1707-1783)

Euler Angles

• This means that we can represent an orientatjon with 3

numbers.

• A sequence of rotatjons around principle axes is called an

Euler Angle Sequence.

• Assuming we limit ourselves to 3 rotatjons without successive

rotatjons about the same axis, we could use any of the following 12 sequences: XYZ XZY XYX XZX YXZ YZX YXY YZY ZXY ZYX ZXZ ZYZ

Euler Angles

• This gives us 12 redundant ways to store an
• rientatjon using Euler angles
• Difgerent industries use difgerent conventjons

for handling Euler angles (or no conventjons)

Euler Angles to Matrix Conversion

• To build a matrix from a set of Euler angles, we just

multjply a sequence of rotatjon matrices together:

=[ c yc z c y sz −s y sxs y cz−cxs z sxs y sz+c xcz sxc y c xs yc z+sx sz cx s ysz−s xcz c xc y]

R x⋅R y⋅R z=[ 1 c x sx −s x cx] ⋅[ c y −s y 1 s y c y ] ⋅[ c z s z −s z c z 1]

Euler Angle Order

• As matrix multjplicatjon is not commutatjve, the
• rder of operatjons is important
• Rotatjons are assumed to be relatjve to fjxed world

axes, rather than local to the object

• One can think of them as being local to the object if

the sequence order is reversed

Using Euler Angles

• To use Euler angles, one must choose one of

the 12 representatjons.

• There may be some practjcal difgerences

between them and the best sequence may depend on what exactly you are trying to accomplish.

• Euler angles can be stored in degrees, radians
• r …

Vehicle Orientatjon

• Generally, for vehicles, it is most convenient to

rotate in roll (z), pitch (x), and then yaw (y)

• In situatjons where there

is a defjnite ground plane, Euler angles can actually be an intuitjve representatjon

x y z

front of vehicle

Orientatjon HPR

Gimbal lock

• Gimbal = device used for

holding a gyroscope

• Illustrates problem with

interpolatjng Euler angles

• Gimbal lock is a basic

problem with representjng 3D rotatjons using Euler angles

Gimbal Lock

• One potentjal problem that Euler Angles can

sufger is the ‘gimbal lock’.

• This results when two axes efgectjvely line up,

resultjng in a temporary loss of a degree of freedom.

• This is related to the singularitjes in longitude

that you get at the north and south poles.

Gimbal Lock

• Roll, pitch, yaw
• Gimbal lock: reduced DOF

due to overlapping axes

Ref: htup://www.fio-emden.de/~hofgmann/gimbal09082002.pdf

Interpolatjng Euler Angles

• One can simply interpolate between the three values

independently

• This will result in the interpolatjon following a

difgerent path depending on which of the 12 schemes you choose

• This may or may not be a problem, depending on

your situatjon

• Interpolatjng near the ‘poles’ can be problematjc
• Note: when interpolatjng angles, remember to check

for crossing the +180/-180 degree boundaries

Euler Angles

• Euler angles are used in a lot of applicatjons, but they

tend to require some rather arbitrary decisions.

• They also do not interpolate in a consistent way (but

this isn’t always bad).

• They can sufger from Gimbal lock and related

problems.

• There is no simple way to concatenate rotatjons.
• Conversion to/from a matrix requires several

trigonometry operatjons.

• They are compact (requiring only 3 numbers).

Axis/Angle

• Euler’s Theorem also shows that any two
• rientatjons can be related by a single rotatjon about

some axis (not necessarily a principle axis)

• This means that we can represent an arbitrary
• rientatjon as a rotatjon about some unit axis by

some angle (4 numbers) (Axis/Angle form)

R = [ θ, ax, ay, az ]

Axis/Angle

a(x,y,z) θ

Axis/Angle to Matrix

• To generate a matrix as a rotatjon θ around an

arbitrary unit axis a:

[

ax

2+cθ(1−ax 2)

axa y(1−cθ)+az sθ axaz(1−cθ)−a y sθ axa y(1−cθ)−az sθ a y

2 +cθ(1−a y 2 )

a y az(1−cθ)+ax sθ axaz(1−cθ)+a ysθ a yaz(1−cθ)−a xsθ az

2+cθ(1−az 2) ]

Axis/Angle Representatjon

• Storing an orientatjon as an axis and an angle uses 4 numbers,

but Euler’s theorem says that we only need 3 numbers to represent an orientatjon

• Mathematjcally, this means that we are using 4 degrees of

freedom to represent a 3 degrees of freedom value

• This implies that there is possibly extra or redundant

informatjon in the axis/angle format

• The redundancy manifests itself in the magnitude of the axis
• vector. The magnitude carries no informatjon, and so it is
• redundant. To remove the redundancy, we choose to

normalize the axis, thus constraining the extra degree of freedom.

• Alternately, we can scale the axis by the angle and compact it

down to a single 3D vector (Rotatjon vector).

Rotatjon Vectors

• To convert a scaled rotatjon vector to a

matrix, one would have to extract the magnitude out of it and then rotate around the normalized axis

• Normally, rotatjon vector format is more

useful for representjng angular velocitjes and angular acceleratjons, rather than angular positjon (orientatjon)

Matrix Representatjon

• We can use a 3x3 matrix to represent an orientatjon

as well

• This means we now have 9 numbers instead of 3,

and therefore, we have 6 extra degrees of freedom

• NOTE: We don’t use 4x4 matrices here, as those are mainly useful

because they give us the ability to combine translatjons.

Matrix Representatjon

• 3 x 3 = 9 DOFs
• Those extra 6 DOFs manifest themselves as 3 scales

(x, y, and z) and 3 shears (xy, xz, and yz)

• If we assume the matrix represents a rigid transform

(orthonormal), then we can constrain the extra 6 DOFs

|a|=|b|=|c|=1 a=b×c b=c×a c=a×b

Matrix Representatjon

• Matrices are usually the most computatjonally

effjcient way to apply rotatjons to geometric data.

• Most orientatjon representatjons ultjmately need to

be converted into a matrix in order to do anything useful (transform verts…)

• Why then, shouldn’t we just always use matrices?

– Numerical issues – Storage issues – User interactjon issues – Interpolatjon issues

Rotatjons

• Euler angles
• Matrix
• Axis / angle
• Rotatjon Vector

Quaternions

Applicatjons of Quaternions

• Used to represent rotatjons and orientatjons of
• bjects in three-dimensional space in:

– Computer graphics – Control theory – Signal processing – Attjtude controls – Physics – Orbital mechanics – Quantum Computjng, quantum circuit design

29

From: http://en.wikipedia.org/wiki/Quaternion

Quaternions

• Quaternions are an interestjng mathematjcal

concept with a deep relatjonship with the foundatjons of algebra and number theory

• Invented by W.R.Hamilton in 1843
• In practjce, they are most useful as a means of

representjng orientatjons

• A quaternion has 4 components

q=[q0 q1 q2 q3 ]

Quaternions

Quaternions (Scalar/Vector)

• Sometjmes, they are writuen as the combinatjon of a

scalar value s and a vector value v where s=q0 v= [q1 q2 q3 ]

q= ⟨s,v ⟩

Quaternions (Imaginary Space)

• Quaternions are an extension to complex numbers
• Of the 4 components, one is a ‘real’ scalar number, and the
• ther 3 form a vector in imaginary ijk space!

q=q 0+iq 1+jq2+kq 3

i2=j2=k 2=ijk=−1 i=jk=−kj j=ki=−ik k=ij=− ji

Unit Quaternions

• For convenience, we will use only unit length quaternions, as

they will be suffjcient for our purposes and make things a litule easier

• These correspond to the set of vectors that form the ‘surface’
• f a 4D hypersphere of radius 1
• The ‘surface’ is actually a 3D volume in 4D space, but it can

sometjmes be visualized as an extension to the concept of a 2D surface on a 3D sphere

|q|=√q 0

2 +q1 2 +q 2 2 +q 3 2=1

Quaternions as Rotatjons

• A quaternion can represent a rotatjon by an angle θ

around a unit axis a:

• If a is unit length, then q will also be unit lenght

q=[cosθ 2 axsin θ 2 aysin θ 2 azsinθ 2 ]

• r

q=⟨cosθ 2 ,asinθ 2 ⟩

Quaternion length

|q|=√q0

2+q1 2+q2 2+q3 2

√cos2θ

2 +ax

2sin 2θ

2 +ay

2 sin2θ

2 +az

2sin 2θ

2

√cos2θ

2 +sin2θ 2 (ax

2 +ay 2 +az 2)

√cos2θ

2 +sin2θ 2 |a|

2=√cos2θ

2 +sin 2θ 2

√1=1

Polar Representatjon

• Remember a 2D unit complex number

cos + i sin = e

• A unit quaternion Q may be writuen as:

Q = (sin uq , cos) = cos + sin uq where uq is a unit 3-tuple vector

• We can also write this unit quaternion as:

Q = e

i uq

Quaternion to Matrix

• To convert a quaternion to a rotatjon matrix:

[

1−2q2

2−2q3 2

2q 1 q 2+2q 0 q 3 2q 1 q 3−2q 0 q 2 2q 1 q 2−2q 0 q 3 1−2q1

2−2q3 2

2q 2 q 3+2q 0 q 1 2q 1 q 3+2q 0 q 2 2q 2 q 3−2q 0 q 1 1−2q1

2−2q2 2 ]

Matrix to Quaternion

• Matrix to quaternion is also possible
• It involves a few ‘if’ statements, a square root,

three divisions, and some other stufg

Spheres

Think of a person standing on the surface of a big sphere (like a planet)

• From the person’s point of view,
• ne can move in along two
• rthogonal axes:

(front/back) and (lefu/right)

• There is no perceptjon of any

fjxed poles or longitude/latjtude, because no matuer which directjon he faces, he always has two

• rthogonal ways to go.
• From his point of view, one might as well be moving on a infjnite

2D plane, however if anyone goes too far in one directjon, he will come back to where he started!

Hyperspheres

• Now extend this concept to moving in the hypersphere
• f unit quaternions.
• The person now has three orthogonal directjons to go.
• No matuer how he is oriented in this space, he can

always go along some combinatjon of forward/backward, lefu/right and up/down

• If he goes too far in any one directjon, he will come

back to where he started.

Hyperspheres

• Now consider that a person’s locatjon on this

hypersphere represents an orientatjon.

• Any incremental movement along one of the
• rthogonal axes in curved space corresponds to an

incremental rotatjon along an axis in real space (distances along the hypersphere correspond to angles in 3D space).

• Moving in some arbitrary directjon corresponds to

rotatjng around some arbitrary axis.

• If you move too far in one directjon, you come back

to where you started (corresponding to rotatjng 360 degrees around any one axis).

Hyperspheres

• Also consider what happens if you rotate a

book 180 around x, then 180 around y, and then 180 around z

• You end up back where you started
• This corresponds to traveling along a triangle
• n the hypersphere where each edge is a 90

degree arc, orthogonal to each other edge

Representjng rotatjons

• A distance of x along the surface of the hypersphere

corresponds to a rotatjon of angle 2x radians.

• This means that moving along a 90 degree arc on the

hypersphere corresponds to rotatjng an object by 180 degrees

• Traveling 180 degrees corresponds to a 360 degree

rotatjon, thus gettjng you back to where you started.

• This implies that q and -q correspond to the same
• rientatjon

Combining rotatjons

Quaternion Dot Products

• The dot product of two quaternions works in the

same way as the dot product of two vectors:

• The angle between two quaternions in 4D space is

half the angle one would need to rotate from one

• rientatjon to the other in 3D space

p⋅q=p0 q0+p 1q1+p 2 q2+p 3q 3=|p||q|cos ϕ

Quaternion Multjplicatjon

• We can perform multjplicatjon on quaternions if we expand

them into their complex number form

• If q represents a rotatjon and q’ represents a rotatjon, then

qq’ represents q rotated by q’

• This follows very similar rules as matrix multjplicatjon (I.e.,

non-commutatjve)

q=q 0+iq 1+jq 2+kq 3 qq'=(q0+iq1+jq2+kq 3)(q0

' +iq1 ' +jq 2 ' +kq 3 ' )

⟨ss'−v⋅v' ,sv '+s' v+v×v⟩

Quaternion Multjplicatjon

• Two unit quaternions multjplied together will result

in another unit quaternion

• This corresponds to the same property of complex

numbers

• Remember that multjplicatjon by complex numbers

can be thought of as a rotatjon in the complex plane

• Quaternions extend the planar rotatjons of complex

numbers to 3D rotatjons in space

Quaternion Conjugate

– Given a quaternion:

• q = w + xi+ yj + zk

– The conjugate of quaternion q is q*, where:

• q* = w – xi – yj – zk

Quaternion Inverse

q

−1 = q∗¿

|q|2 ¿

– Can be used for division – Given a quaternion:

• q = w + xi+ yj + zk

– The inverse of quaternion q is q-1, where:

Linear Interpolatjon

• If we want to do a linear interpolatjon between two points a

and b in normal space Lerp(t,a,b) = (1-t)a + (t)b where t ranges from 0 to 1

• Note that the Lerp operatjon can be thought of as a weighted

average (convex)

• We could also write it in it’s additjve blend form:

Lerp(t,a,b) = a + t(b-a)

Spherical Linear Interpolatjon

• If we want to interpolate between two points
• n a sphere (or hypersphere), we don’t just

want to Lerp between them

• Instead, we will travel across the surface of

the sphere by following a ‘great arc’

Spherical Linear Interpolatjon

• We defjne the spherical linear interpolatjon of

two unit vectors in N dimensional space as:

Slerp (t,a,b)=sin ((1−t)θ ) sin θ a+ sin (tθ ) sin θ b where :θ= cos−1 (a⋅b)

A B C D

Quaternion Interpolatjon

• Remember that there are two redundant vectors in

quaternion space for every unique orientatjon in 3D space

• What is the difgerence between:

Slerp(t,a,b) and Slerp(t,-a,b) ?

• One of these will travel less than 90 degrees while the other

will travel more than 90 degrees across the sphere

• This corresponds to rotatjng the ‘short way’ or the ‘long way’
• Usually, we want to take the short way, so we negate one of

them if their dot product is < 0