三维空间中的旋转--旋转向量

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：

A visualization of a rotation represented by an Euler axis and angle., 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 (itsscalar 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;
    doubler_vec[3]={-2.100418,-2.167796,0.273330};
    doubleR_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]);
    }
}

,, 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).

, 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 , called arotation 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

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 ofA 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. Theeigenvector 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 3Dorthonormal 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.

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

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

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 thehypercomplex numbers satisfying

Quaternion multiplication is performed in the same manner as multiplication ofcomplex 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 theinverse 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 errorsThe quaternion elements vary continuously over the unit sphere in , (denoted by) as the orientation changes, avoidingdiscontinuous jumps (inherent to three-dimensional parameterizations)Expression of the rotation matrix in terms of quaternion parameters involves notrigonometric functionsIt 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

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 usingfloating 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.

where the symbol denotes theouter product. This product of vectors produces two terms: a scalar part from the inner product and abivector 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 theimaginary unit. Geometric algebra uses bivectors in its analogue to the quaternion, therotor, given by , where is a unit bivector that describes theplane of rotation. Because squares to ?1, thepower series expansion of generates thetrigonometric functions. The rotation formula that maps a vector to a rotated vector is then

where is thereverse 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 to. Indeed, the rotation reduces to

exactly as expected. This rotation formula is valid not only for vectors but for anymultivector. 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.

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 thequadrant that the point is in; seeatan2.

When implementing the conversion, one has to take into account several situations: ^{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 thex-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.

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 theRodrigues' rotation formula (with appropriate modification) as follows:

with the

is the cross-product matrix.

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:

The rotation matrix corresponding to the quaternion can be computed as follows:

with the

which gives

or equivalently

from the Euler angles as follows:

Given the rotation quaternion , the x-convention 3-1-3 Euler angles can be computed by

and angle, the quaternion

can be computed by

Given the rotation quaternion , define. Then the Euler axis and angle can be computed by

can be extracted from the derivative of the rotation matrix by the following relation:

The derivation is adapted from ^{consider and differentiate it:}

The derivative of a vector is the linear velocity of its tip. Since A is a rotation matrix, by definition the length of is always equal to the length of, and hence it does not change with time. Thus, when rotates, its tip moves along a circle, and the linear velocity of its tip is tangential to the circle; i.e., always perpendicular to. In this specific case, the relationship between the linear velocity vector and the angular velocity vector is

(seecircular motion andCross product).

By the transitivity of the above mentioned equations,

which implies (Q.E.D.),

can be obtained from the derivative of the quaternion as follows

where is the inverse of.

Conversely, the derivative of the quaternion is