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三维空间中的旋转--旋转向量

2015-04-06 17:37 288 查看

处理三维旋转问题时，通常采用旋转矩阵的方式来描述。一个向量乘以旋转矩阵等价于向量以某种方式进行旋转。除了采用旋转矩阵描述外，还可以用旋转向量来描述旋转，旋转向量的长度（模）表示绕轴逆时针旋转的角度（弧度）。旋转向量与旋转矩阵可以通过罗德里格斯（Rodrigues）变换进行转换。

OpenCV实现Rodrigues变换的函数为

int cvRodrigues2( const CvMat* src, CvMat* dst, CvMat* jacobian=0 );

src为输入的旋转向量（3x1或者1x3）或者旋转矩阵（3x3）。

dst为输出的旋转矩阵（3x3）或者旋转向量（3x1或者1x3）。

jacobian为可选的输出雅可比矩阵（3x9或者9x3），是输入与输出数组的偏导数。

旋转向量的物理意义为：

Euler axis and angle (rotation vector)

A visualization of a rotation represented by an Euler axis and angle.

Main article: Axis angle
From Euler's rotation theorem we know that any rotation can be expressed as a single
rotation about some axis. The axis is the unit vector (unique except for sign) which remains unchanged by the rotation. The magnitude of the angle is also unique, with its sign being determined by the sign of the rotation axis.

The axis can be represented as a three-dimensional unit vector

,
and the angle by a scalar

.

Since the axis is normalized, it has only two degrees of freedom. The angle
adds the third degree of freedom to this rotation representation.

One may wish to express rotation as a rotation vector, a non-normalized three-dimensional vector the direction of which specifies the axis, and the length of which is

The rotation vector is in some contexts useful, as it represents a three-dimensional rotation with only three scalar values (its scalar
components), representing the three degrees of freedom. This is also true for representations based on sequences of three Euler angles (see below).

If the rotation angle

is zero, the axis is not uniquely defined. Combining two successive rotations,
each represented by an Euler axis and angle, is not straightforward, and in fact does not satisfy the law of vector addition, which shows that finite rotations are not really vectors at all. It is best to employ the rotation matrix or quaternion notation,
calculate the product, and then convert back to Euler axis and angle.

验证代码如下：

#include <stdio.h>

#include <cv.h>

void main()

{

int i;

double r_vec[3]={-2.100418,-2.167796,0.273330};

double R_matrix[9];

CvMat pr_vec;

CvMat pR_matrix;

cvInitMatHeader(&pr_vec,1,3,CV_64FC1,r_vec,CV_AUTOSTEP);

cvInitMatHeader(&pR_matrix,3,3,CV_64FC1,R_matrix,CV_AUTOSTEP);

cvRodrigues2(&pr_vec, &pR_matrix,0);

for(i=0; i<9; i++)

{

printf("%f\n",R_matrix[i]);

}

}

Rotation formalisms in three dimensions

From Wikipedia, the free encyclopedia

Jump to: navigation, search

In geometry, various formalisms exist to express a rotation in
three dimensions as a mathematicaltransformation. In physics, this
concept extends to classical mechanics where rotational (or angular) kinematicsis
the science of describing with numbers the purely rotational motion of an object.

According to Euler's rotation theorem the general displacement of a rigid
body (or three-dimensionalcoordinate system) with one point fixed is described by a rotation about
some axis. This allows the use of rotations to express orientations as a single rotation from
a reference placement in space of the rigid body (or coordinate system). Furthermore, such a rotation may be uniquely described by a minimum of three parameters. However, for various reasons, there are several ways to represent it. Many of these representations
use more than the necessary minimum of three parameters, although each of them still has only three degrees
of freedom.

An example where rotation representation is used is in computer vision, where an automated observer
needs to track a target. Let's consider a rigid body, with an orthogonal right-handed
triad

,
and

of unit vectors fixed to its body (representing the three axes of the object's coordinate
system). The basic problem is to specify the orientation of this triad, and hence the rigid body, in terms of the reference coordinate system (in our case the observer's coordinate system).

[hide]

1 Rotation
matrix
2 Euler
axis and angle (rotation vector)
3 Euler
rotations
4 Quaternions
5 Rodrigues
parameters
6 Cayley–Klein
parameters
7 Higher
dimensional analogues
8 Rotors
in a geometric algebra
9 Conversion
formulae between formalisms

9.1 Rotation
matrix ↔ Euler angles
9.2 Rotation
matrix ↔ Euler axis/angle
9.3 Rotation
matrix ↔ quaternion
9.4 Euler
angles ↔ quaternion
9.5 Euler
axis/angle ↔ quaternion

10 Conversion
formulae between derivatives

10.1 Rotation
matrix ↔ angular velocities
10.2 Quaternion
↔ angular velocities

11 See also
12 References
13 External
links

[edit] Rotation
matrix

Main article: Rotation matrix
The above mentioned triad of unit vectors is also called a basis.
Specifying the coordinates (scalar components) of this basis in its current (rotated)
position, in terms of the reference (non-rotated) coordinate axes, will completely describe the rotation. The three unit vectors

and

which
form the rotated basis each consist of 3 coordinates, yielding a total of 9 parameters. These parameters can be written as the elements of a 3 × 3 matrix

,
called a rotation matrix. Typically, the coordinates of each of these vectors are arranged along a column of the matrix (however, beware that an alternative definition of rotation matrix exists and is widely used, where the vectors coordinates
defined above are arranged by rows[1])

The elements of the rotation matrix are not all independent – as Euler's rotation theorem dictates, the rotation matrix has only three degrees of freedom. The rotation matrix has the following properties:

A is a real, orthogonal
matrix, hence each of its rows or columns represents a unit vector.
The eigenvalues of A are

where i is the standard imaginary unit with the property i2 = −1

The determinant of A is +1, equivalent to the product of its eigenvalues.
The trace of A is

,
equivalent to the sum of its eigenvalues.

The angle

which appears in the eigenvalue expression corresponds to the angle of the Euler axis
and angle representation. The eigenvector corresponding with the eigenvalue of 1 is the accompanying
Euler axis, since the axis is the only (nonzero) vector which remains unchanged by left-multiplying (rotating) it with the rotation matrix.

The above properties are equivalent to:

which is another way of stating that

form a 3D orthonormal
basis. Note that the statements above constitute a total of 6 conditions (the cross product contains 3), leaving the rotation matrix with just 3 degrees of freedom as required.

Two successive rotations represented by matrices

and

are
easily combined as follows:

(Note the order, since
the vector being rotated is multiplied from the right). The ease by which vectors can be rotated using a rotation matrix, as well as the ease of combining successive rotations, make the rotation matrix a very useful and popular way to represent rotations,
even though it is less concise than other representations.

[edit] Euler
axis and angle (rotation vector)

,
and the angle by a scalar

[edit] Euler
rotations

Euler rotations of the Earth. Intrinsic (green),Precession (blue)
andNutation (red)

Main article: Euler angles#Euler rotations
The idea behind Euler rotations is to split the complete rotation of the coordinate system into three simpler constitutive rotations, called Precession, Nutation,
and intrinsic rotation, being each one of them an increment on one of the Euler
angles. Notice that the outer matrix will represent a rotation around one of the axes of the reference frame, and the inner matrix represents a rotation around one of the moving frame axis. The middle matrix represent a rotation around an intermediate
axis called line of nodes.

Unfortunately, the definition of Euler angles is not unique and in the literature many different conventions are used. These conventions depend on the axes about which the rotations are carried out, and their sequence (since rotations are not commutative).

The convention being used is usually indicated by specifying the axes about which the consecutive rotations (before being composed) take place, referring to them by index (1, 2, 3) or letter (X, Y, Z). The engineering and robotics communities typically use
3-1-3 Euler angles. Notice that after composing the independent rotations, they do not rotate about their axis anymore. The most external matrix rotates the other two, leaving the second rotation matrix over the line of nodes, and the third one in a frame
comoving with the body. There are 3×3×3 = 27 possible combinations of three basic rotations but only 3×2×2 = 12 of them can be used for representing arbitrary 3D rotations as Euler angles. These 12 combinations avoid consecutive rotations around the same axis
(such as XXY) which would reduce the degrees of freedom that can be represented.

Therefore Euler angles are never expressed in terms of the external frame, or in terms of the co-moving rotated body frame, but in a mixture. Other conventions (e.g., rotation
matrix or quaternions) are used to avoid this problem.

[edit] Quaternions

Main article: Quaternions and spatial
rotation
Quaternions (Euler symmetric parameters) have proven very useful in representing rotations due to several advantages
above the other representations mentioned in this article.

A quaternion representation of rotation is written as a normalized four-dimensional vector

In terms of the Euler axis

and angle

this vector's elements are expressed as follows:

The above definition follows the convention as used in (Wertz 1980) and (Markley 2003). An alternative definition used in some publications defines the "scalar" term as the first quaternion element, with the other elements shifted down one position. (Coutsias
1999), (Schmidt 2001)

Inspection shows that the quaternion parametrization obeys the following constraint:

The last term (in our definition) is often called the scalar term, which has its origin in quaternions when understood as the mathematical extension of the complex numbers, written as

with

and where

are the hypercomplex
numbers satisfying

Quaternion multiplication is performed in the same manner as multiplication of complex numbers,
except that the order of elements must be taken into account, since multiplication is not commutative. In matrix notation we can write quaternion multiplication as

Combining two consecutive quaternion rotations is therefore just as simple as using the rotation matrix. Remember that two successive rotation matrices,

followed
by

, are combined as follows:

We can represent this quaternion parameters in a similarly concise way. Please note the inverse ordering of quaternion multiplication when compared to matrix multiplication.

Quaternions are a very popular parametrization due to the following properties:

More compact than the matrix representation and less susceptible to round-off errors
The quaternion elements vary continuously over the unit sphere in

, (denoted by

)
as the orientation changes, avoiding discontinuous jumps (inherent to three-dimensional
parameterizations)
Expression of the rotation matrix in terms of quaternion parameters involves no trigonometric
functions
It is simple to combine two individual rotations represented as quaternions using a quaternion product

Like rotation matrices, quaternions must sometimes be re-normalized due to rounding errors, to make sure that they correspond to valid rotations. The computational cost of re-normalizing a quaternion, however, is much less than for normalizing a 3 × 3 matrix.

[edit] Rodrigues
parameters

Main article: Euler–Rodrigues
parameters
See also: Rodrigues' rotation formula
Rodrigues parameters can be expressed in terms of Euler axis and angle as follows:

This has a discontinuity at 180° (π radians): each vector, r, with a norm of π radians represent the same rotation as −r.

Similarly, the Gibbs representation can be expressed as follows:

A rotation g followed by a rotation f in Gibbs representation has the form

The Gibbs vector has the advantage (or disadvantage, depending on context) that 180° rotations cannot be represented. (Even using floating
point numbers that include infinity, rotation direction cannot be well-defined; for example, naively a 180° rotation about the axis (1, 1, 0) would be

,
which is the same representation as 180° rotation about (1, 0.0001, 0).)

Modified Rodrigues parameters (MRPs) can be expressed in terms of Euler axis and angle by:

The modified Rodrigues parameterization shares many characteristics with the rotation vector parametrization, including the occurrence of discontinuous jumps in the parameter space when incrementing the rotation.

[edit] Cayley–Klein
parameters

See definition at Wolfram Mathworld

[edit] Higher
dimensional analogues

See also: Rotations
in 4-dimensional Euclidean space

[edit] Rotors
in a geometric algebra

The formalism of geometric algebra (GA) provides an extension and interpretation of the quaternion method.
Central to GA is the geometric product of vectors, an extension of the traditional inner and cross
products, given by

where the symbol

denotes the outer
product. This product of vectors

produces two terms: a scalar part from the inner product
and a bivector part from the outer product. This bivector describes the plane perpendicular to what the cross product of
the vectors would return.

Bivectors in GA have some unusual properties compared to vectors. Under the geometric product, bivectors have negative square: the bivector

describes
the

-plane. Its square is

.
Because the unit basis vectors are orthogonal to each other, the geometric product reduces to the antisymmetric outer product –

and

can
be swapped freely at the cost of a factor of −1. The square reduces to

since
the basis vectors themselves square to +1.

This result holds generally for all bivectors, and as a result the bivector plays a role similar to the imaginary
unit. Geometric algebra uses bivectors in its analogue to the quaternion, the rotor, given by

,
where

is a unit bivector that describes the plane
of rotation. Because

squares to −1, the power
series expansion of

generates the trigonometric
functions. The rotation formula that maps a vector

to a rotated vector

is
then

where

is
the reverse of

(reversing the order of the vectors in

is
equivalent to changing its sign).

Example. A rotation about the axis

can
be accomplished by converting

to its dual bivector,

,
where

is the unit volume element, the only trivector (pseudoscalar) in three-dimensional
space. The result is

.
In three-dimensional space, however, it is often simpler to leave the expression for

,
using the fact that

commutes with all objects in 3D and also squares to −1. A rotation of the

vector
in this plane by an angle

is then

Recognizing that

and that

is
the reflection of

about the plane perpendicular to

gives
a geometric interpretation to the rotation operation: the rotation preserves the components that are parallel to

and
changes only those that are perpendicular. The terms are then computed:

The result of the rotation is then

A simple check on this result is the angle

. Such a rotation should map
the

.
Indeed, the rotation reduces to

exactly as expected. This rotation formula is valid not only for vectors but for any multivector. In addition, when
Euler angles are used, the complexity of the operation is much reduced. Compounded rotations come from multiplying the rotors, so the total rotor from Euler angles is

but

and

.
These rotors come back out of the exponentials like so:

where

refers to rotation in the original coordinates. Similarly for the

rotation,

.
Noting that

and

commute
(rotations in the same plane must commute), and the total rotor becomes

Thus, the compounded rotations of Euler angles become a series of equivalent rotations in the original fixed frame.

While rotors in geometric algebra work almost identically to quaternions in three dimensions, the power of this formalism is its generality: this method is appropriate and valid in spaces with any number of dimensions. In 3D, rotations have three degrees
of freedom, a degree for each linearly independent plane (bivector) the rotation can take place in. It has been known that pairs of quaternions can be used to generate rotations in 4D, yielding six degrees of freedom, and the geometric algebra approach verifies
this result: in 4D, there are six linearly independent bivectors that can be used as the generators of rotations.

[edit] Conversion
formulae between formalisms

It has been suggested that Rotation matrix#Conversions be merged into
this article or section. (Discuss) Proposed
since October 2009.

[edit] Rotation
matrix ↔ Euler angles

The Euler angles

can be extracted from the rotation matrix

by
inspecting the rotation matrix in analytical form.

Using the x-convention, the 3-1-3 Euler angles

and

(around
the

and
again the

-axis) can be obtained as follows:

Note that

is equivalent to

where
it also takes into account the quadrant that
the point

is in; see atan2.

When implementing the conversion, one has to take into account several situations:[2]

There are generally two solutions in (−π, π]3 interval. The above formula works only when

is
from the interval [0, π)3.
For special case

shall
be derived from

.
There is infinitely many but countably many solutions outside of interval (−π, π]3.
Whether all mathematical solutions apply for given application depends on the situation.

The rotation matrix

is generated from the Euler angles by multiplying the three matrices
generated by rotations about the axes.

The axes of the rotation depend on the specific convention being used. For the x-convention the rotations are about the

and

axes
with angles

and

,
the individual matrices are as follows:

This yields

Note: This is valid for a right-hand system, which is the convention used in almost all engineering and physics disciplines.

[edit] Rotation
matrix ↔ Euler axis/angle

If the Euler angle

is not a multiple of

,
the Euler axis

and angle

can
be computed from the elements of the rotation matrix

as follows:

Alternatively, the following method can be used:

Eigen-decomposition of the rotation matrix yields the eigenvalues 1, and

.
The Euler axis is the eigenvector corresponding to the eigenvalue of 1, and the

can be computed
from the remaining eigenvalues.

The Euler axis can be also found using Singular Value Decomposition since it is the normalized vector spanning the null-space of the matrix

.

To convert the other way the rotation matrix corresponding to an Euler axis

and
angle

can be computed according to the Rodrigues'
rotation formula (with appropriate modification) as follows:

with

the 3 × 3 identity
matrix, and

is the cross-product matrix.

[edit] Rotation
matrix ↔ quaternion

When computing a quaternion from the rotation matrix there is a sign ambiguity, since

and

represent
the same rotation.

One way of computing the quaternion

from the rotation
matrix

is as follows:

There are three other mathematically equivalent ways to compute

. Numerical inaccuracy can
be reduced by avoiding situations in which the denominator is close to zero. One of the other three methods looks as follows:[3]

The rotation matrix corresponding to the quaternion

can
be computed as follows:

with

the 3 × 3 identity matrix, and

which gives

or equivalently

[edit] Euler
angles ↔ quaternion

Main article: Conversion
between quaternions and Euler angles
We will consider the x-convention 3-1-3 Euler Angles for the following algorithm. The terms of the algorithm depend on the convention used.

We can compute the quaternion