Rotation formalisms in three dimensions
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In geometry, various formalisms exist to express a rotation in three dimensions as a mathematical transformation. In physics, this concept extends to classical mechanics where rotational (or angular) kinematics is the science of describing with numbers the purely rotational motion of an object. The orientation
of an object at a given instant is described with the same tools, as it
is defined as an imaginary rotation from a reference placement in
space, rather than an actually observed rotation from a previous
placement in space.According to Euler's rotation theorem the general displacement of a rigid body (or three-dimensional coordinate system) with one point fixed is described by a single rotation about some axis. 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 three orthogonal unit vectors fixed to its body (representing the three axes of the object's local coordinate system). The basic problem is to specify the orientation of these three unit vectors, and hence the rigid body, with respect to the observer's coordinate system, regarded as a reference placement in space.
Contents
Formalism alternatives
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])- 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 above properties are equivalent to:
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.
Euler axis and angle (rotation vector)
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 :
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
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.
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
Inspection shows that the quaternion parametrization obeys the following constraint:
- with
- 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
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:Similarly, the Gibbs representation can be expressed as follows:
Modified Rodrigues parameters (MRPs) can be expressed in terms of Euler axis and angle by:
Cayley–Klein parameters
This section requires expansion. (September 2013) |
Higher dimensional analogues
Conversion formulae between formalisms
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:
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.
Rotation matrix ↔ Euler axis/angle
|
It has been suggested that Rotation matrix#Conversion from and to axis-angle be merged into this section. (Discuss) Proposed since September 2013. |
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:
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:
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 from the Euler angles as follows:
Euler axis/angle ↔ quaternion
Given the Euler axis and angle , the quaternionConversion formulae for derivatives
Rotation matrix ↔ angular velocities
The angular velocity vector can be extracted from the derivative of the rotation matrix by the following relation:For any vector consider and differentiate it:
- (see circular motion and Cross product).
Quaternion ↔ angular velocities
The angular velocity vector can be obtained from the derivative of the quaternion as follows:[5]Conversely, the derivative of the quaternion is
Rotors in a geometric algebra
|
It has been suggested that this section be merged into Geometric algebra#Rotations. (Discuss) Proposed since September 2013. |
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
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
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.
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