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# 3D orientation - PowerPoint PPT Presentation

3D orientation. Rotation matrix Fixed angle and Euler angle Axis angle Quaternion Exponential map. 3 translational DOFs. 48 rotational DOFs. Joints and rotations. Rotational DOFs are widely used in character animation. Each joint can have up to 3 DOFs. 1 DOF: knee. 2 DOF: wrist.

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Presentation Transcript

• Rotation matrix

• Fixed angle and Euler angle

• Axis angle

• Quaternion

• Exponential map

48 rotational DOFs

Joints and rotations

Rotational DOFs are widely used in character animation

Each joint can have up to 3 DOFs

1 DOF: knee

2 DOF: wrist

3 DOF: arm

• Homogeneous coordinates (review)

• 4X4 matrix used to represent translation, scaling, and rotation

• a point in the space is represented as

• Treat all transformations the same so that they can be easily combined

translation

matrix

new point

old point

new point

scaling matrix

old point

X axis

Y axis

Z axis

A series of transformations on an object can be applied as a series of matrix multiplications

: position in the global coordinate

: position in the local coordinate

• In order to “move things”, we need both translation and rotation

• Interpolating the translation is easy, but what about rotations?

• Finding the most natural and compact way to present rotation and orientations

• Orientation interpolation which result in a natural motion

• A closed mathematical form that deals with rotation and orientations (expansion for the complex numbers)

• A general rotation can be represented by a single 3x3 matrix

• Length Preserving (Isometric)

• Reflection Preserving

• Orthonormal

• Angles used to rotate about fixed axes

• Orientations are specified by a set of 3 ordered parameters that represent 3 ordered rotations about fixed axes, ie. first about x, then y, then z

• Many possible orderings, don’t have to use all 3 axes, but can’t do the same axis back to back

• A rotation of 10,45, 90 would be written as

• Rz(90) Ry(45), Rx(10) since we want to first rotate about x, y, z. It would be applied then to the point P…. RzRyRx P

• Problem occurs when two of the axes of rotation line up on top of each other. This is called “Gimbal Lock”

• A 90 degree rotation about the y axis essentially makes the first axis of rotation align with the third.

• Incremental changes in x,z produce the same results – you’ve lost a degree of freedom

• Phenomenon of two rotational axis of an object pointing in the same direction.

• Simply put, it means your object won't rotate how you think it ought to rotate.

• How about interpolating each entry of the rotation matrix?

• The interpolated matrix might no longer be orthonormal, leading to nonsense for the in-between rotations

Example: interpolate linearly from a positive 90 degree rotation about y axis to a negative 90 degree rotation about y

Linearly interpolate each component and halfway between, you get this...

• Easily composed?

• Interpolate?

• Rotation matrix

• Fixed angle and Euler angle

• Axis angle

• Quaternion

• Exponential map

• Angles used to rotate about fixed axes

• Orientations are specified by a set of 3 ordered parameters that represent 3 ordered rotations about fixed axes

• Many possible orderings

• A rotation of Rz(90)Ry(60)Rx(30) looks like

• Same as fixed angles, except now the axes move with the object

• An Euler angle is a rotation about a single Cartesian axis

• Create multi-DOF rotations by concatenating Euler angles

• evaluate each axis independently in a set order

Y

X

Euler angle vs. fixed angle

• Rz(90)Ry(60)Rx(30) = Ex(30)Ey(60)Ez(90)

• Euler angle rotations about moving axes written in reverse order are the same as the fixed axis rotations

z

z

3

/2

1

1

y

y

y

1

1

x

x

x

x’

x’

Gimbal Lock (again!)

• Rotation by 90o causes a loss of a degree of freedom

• Phenomenon of two rotational axis of an object pointing in the same direction.

• Simply put, it means your object won't rotate how you think it ought to rotate.

A Gimbal is a hardware implementation of Euler angles used for mounting gyroscopes or expensive globes

Gimbal lock is a basic problem with representing 3D rotation using Euler angles or fixed angles

When two rotational axis of an object point in the same direction, the rotation ends up losing one degree of freedom

• Easily composed?

• Interpolate?

• What seems to be the problem?

• A general rotation is a combination of three elementary rotations: around the x-axis (x-roll) , around the y-axis (y-roll) and around the z-axis (z-roll).

y

π

x

x

x-roll π

y

y

y

z

z

x

π

x

x

z-roll π

y-roll π

π

z

z

z

Euler angles interpolation

R(0,0,0),…,R(t,0,0),…,R(,0,0)

t[0,1]

R(0,0,0),…,R(0,t, t),…,R(0,, )

Euler Angles Interpolation Unnatural movement !

• Rotation matrix

• Fixed angle and Euler angle

• Axis angle

• Quaternion

• Exponential map

• Find a parameterization in which

• a simple steady rotation exists between two key orientations

• moves are independent of the choice of the coordinate system

• Represent orientation as a vector and a scalar

• vector is the axis to rotate about

• scalar is the angle to rotate by

• Any orientation can be represented by a 4-tuple

• angle, vector(x,y,z) where the angle is the amount to rotate by and the vector is the axis to rotate about

• Can interpolate the angle and axis separately

• No gimbal lock problems!

• But, can’t efficiently compose rotations…must convert to matrices first!

• (,n) defines an angular displacement of  about an axis n

• Can avoid Gimbal lock. Why?

• Can interpolate the vector and the scalar separately. How?

• Rotation matrix

• Fixed angle and Euler angle

• Axis angle

• Quaternion

• Exponential map

4 tuple of real numbers:

scalar

vector

Same information as axis angles but in a different form

• Extend the concept of rotation in 3D to 4D.

• Avoids the problem of "gimbal-lock" and allows for the implementation of smooth and continuous rotation.

• In effect, they may be considered to add a additional rotation angle to spherical coordinates ie. Longitude, Latitude and Rotation angles

• A Quaternion is defined using four floating point values |x y z w|. These are calculated from the combination of the three coordinates of the rotation axis and the rotation angle.

• Solution to "Gimbal lock"

• Instead of rotating an object through a series of successive rotations, a quaternion allows the programmer to rotate an object through a single arbitary rotation axis.

• Because the rotation axis is specifed as a unit direction vector, it may be calculated through vector mathematics or from spherical coordinates ie (longitude/latitude).

• Quaternions interpolation : smooth and predictable rotation effects.

Quaternion to Rotation Matrix

• Extension of complex numbers

• 4-tuple of real numbers

• s,x,y,z or [s,v]

• s is a scalar

• v is a vector

• Same information as axis/angle but in a different form

• Can be viewed as an original orientation or a rotation to apply to an object

• The conjugate and magnitude are similar to complex numbers

• Quaternions are non commutative

q1 = (s1,v1) q2 = (s2,v2)

q1*q2 = (s1s2 – v1.v2 , s1v2 + s2v1 + v1x v2)

• inverse:

• unit quaternion:

To rotate a vector, v using quaternion math

• represent the vector as [0,v]

• represent the rotation as a quaternion, q

• Rotation of P=(0,r) about the unit vector n by an angle θ using the unit quaternion q=(s,v)

but q=(cos½θ, sin½θ•n) where |n|=1

Quaternions as Rotations

Unit quaternion

Multiplication

Conjugate

Inverse

the unit length quaternion

If is a unit quaternion and

Quaternion Rotation

proof: see Quaternions by Shoemaker

If and are unit quaternion

the combined rotation of first rotating by and then by is equivalent to

Quaternion interpolation

• Interpolation means moving on n-D sphere

• Now imagine a 4-D sphere for 3-angle rotation

1-angle rotation can be represented by a unit circle

• Moving between two points on the 4D unit sphere

• a unit quaternion at each step - another point on the 4D unit sphere

• move with constant angular velocity along the great circle between the two points on the 4D unit sphere

Direct linear interpolation does not work

Linearly interpolated intermediate points are not uniformly spaced when projected onto the circle

Spherical linear interpolation (SLERP)

Normalize to regain unit quaternion

• It’s easiest to express rotations in Euler angles or Axis/angle

• We can convert to/from any of these representations

• Choose the best representation for the task

• input:Euler angles

• interpolation: quaternions

• composing rotations: quaternions, orientation matrix

• Rotation matrix

• Fixed angle and Euler angle

• Axis angle

• Quaternion

• Exponential map

• Represent orientation as a vector

• direction of the vector is the axis to rotate about

• magnitude of the vector is the angle to rotate by

• Zero vector represents the identity rotation

• No need to re-normalize the parameters

• Fewer DOFs

• Good interpolation behavior

• Singularities exist but can be avoided

• Choose the best representation for the task

• input:

• joint limits:

• interpolation:

• compositing:

• rendering:

Euler angles

Euler angles, quaternion (harder)

quaternion or exponential map

quaternions or orientation matrix

orientation matrix ( quaternion can be represented as matrix as well)