Here, we discuss several alternatives to the popular Euler angle parameterization of a rotation tensor. These representations include the Euler-Rodrigues and quaternion parameterizations, the Rodrigues representation, Cayley’s representation, and the exponential map.

Contents

## The Euler-Rodrigues and quaternion parameterizations

The use of four Euler-Rodrigues symmetric (or Euler symmetric) parameters to parameterize a rotation dates to Euler [1] in 1771 and Rodrigues [2] in 1840 [3, 4, 5]. We denote these parameters by the pair , where is a scalar and is a vector. In 1843, Hamilton [6] made his discovery of quaternion multiplication, and shortly afterwards Cayley [7] published results showing how quaternions could be used to parameterize a rotation. The historical development of these parameterizations features some of the greatest mathematicians of the 18th and 19th centuries. These figures include Euler, Gauss [8], and Cayley, among others. The topic also provides a fascinating introduction to this period. For further details, we refer the reader to the works [3, 4, 5, 9, 10].

The Euler-Rodrigues symmetric parameters and satisfy the constraint

(1)

Let the pair , where is a scalar and is a vector, denote a unit quaternion, which has unit norm:

(2)

Thus, a unit quaternion can be used to define a set of Euler-Rodrigues symmetric parameters, and vice versa: and . In the four-dimensional space parameterized by the components of a quaternion, the set of all unit quaternions defines a unit sphere referred to as the 3-sphere . The parameters and can be used to define a rotation about an axis through an angle using the identifications

(3)

The resulting representation of the rotation tensor , which transforms the fixed basis into the corotational basis , is given by

(4)

Examining the tensor components , one finds that the components are quadratic functions of the four parameters and :

(5)

Consequently, representation (4) is free of trigonometric functions and singularities, making it attractive from a numerical implementation standpoint. In addition, the fact that unit quaternions lie on the 3-sphere and can be used to represent rotations is exploited in computer graphics to interpolate animations (see [11]). Examining (5) in closer detail,

(6)

where each of the nine 4 4 matrices are symmetric and proper-orthogonal. For example,

(7)

The matrices play a role in constructing stiffness matrices [12] and in establishing identities for the derivatives of the components of the matrix [13].

In terms of and , the angular velocity vector associated with has the well-known representation

(8)

It is interesting to note that the time derivative of and its corotational rate are simply related [12]:

(9)

To verify this result, recall that because . Computing and then using the identity confirms (9). With the help of (9), some lengthy but straightforward manipulations of (8) reveal that has the equivalent representation

(10)

For completeness, we note that the angular velocity , defined as the axial vector of , can be obtained from according to

(11)

Using (10) and the identities , (9), and for any vectors and , it follows that has the representation

(12)

As discussed in [14, 15], this representation is convenient to use when computing Lagrange’s equations of motion for rigid bodies. Lastly, taking the components of representation (10) for leads to

(13)

where

(14)

The matrix has the generalized inverse

(15)

That is, , where is the 4 4 identity matrix. Thus, using , (13) can be rearranged to establish the following differential equations for the parameters and in terms of :

(16)

Given measurements for and the initial orientation and of a body, these differential equations can be integrated to determine and , and hence the body’s orientation .

## Rodrigues’ representation

The Rodrigues representation of a rotation about an axis through a counterclockwise angle features the vector

(17)

This vector, which dates to 1840, is sometimes called the Gibbs or Rodrigues vector [2, 5, 16]. Clearly, the vector is not a unit vector. Indeed, when , and is undefined when rad. Consequently, if varies through rad, then we cannot use the forthcoming Rodrigues representation. With the assistance of the identities

(18)

one can verify that

(19)

Recall that for Euler’s representation of a rotation, the rotation tensor and associated angular velocity vector are given by, respectively,

(20)

Substituting for and in terms of , we obtain Rodrigues’ representation for and :

(21)

The components of with respect to an orthonormal basis, say, , are readily obtained by substituting . The simplicity of the Rodrigues representation of is remarkable.

## Cayley’s representation

In 1846, Cayley [17] introduced what is now known as the Cayley transform of a second-order skew-symmetric tensor :

(22)

He showed that the Cayley transform of is a proper-orthogonal tensor, and hence a rotation tensor^{1}. The transform is often invertible. If , then

(23)

provided is invertible. It is interesting to note that

(24)

Thus, the inverse of a rotation is obtained by setting . Given a skew-symmetric tensor , we denote the representation defined by the Cayley transform of as the Cayley representation of a rotation:

(25)

The angular velocity tensor associated with the Cayley representation is particularly elegant:

(26)

A remaining issue is a physical interpretation of . As a skew-symmetric tensor has an axial vector, it is natural to suspect that should be related to the axis and angle of a rotation. On pages 121-122 of [17], Cayley showed that this was indeed the case. In fact, he pointed out that the axial vector of is none other than the Rodrigues vector [2]:

(27)

Therefore,

(28)

Just as Rodrigues’ representation cannot accommodate a rotation by 180, neither can the Cayley representation. For modern applications of Cayley’s results to elasticity, see [18] and references therein. A recent application of Cayley’s representation has been found by Mladenova and Mladenov [19], who used the representation to determine how any rotation can be decomposed into the product of three rotations about three prescribed axes.

## The exponential map

According to Pfister [20], the relationship between the exponential map of a skew-symmetric tensor and a rotation dates to Gibbs [21, Article 177] in the 1880s. This representation features in many applications, including robotics [22], computer graphics [23], and simulations of the motions of elastic rods [24]. As emphasized in [24], the exponential map is also useful when considering infinitesimal rotations. To start the discussion, it is convenient to define a rotation vector , where is the axis of the desired rotation and is the desired counterclockwise angle of rotation. We then define a skew-symmetric tensor

(29)

that has the following properties:

(30)

Now, given a tensor , the exponential map is defined as [20, 25]

(31)

Consequently,

(32)

Repeatedly invoking the identities (30) and identifying the power series of with and , one arrives at a series of results that are equivalent to Euler’s representation of a rotation tensor:

(33)

Consequently, the exponential mapping of a skew-symmetric tensor is a proper-orthogonal tensor, and thus a rotation tensor:

(34)

As discussed in [20], the inverse of this map can be used to identify the logarithm of a rotation tensor with a skew-symmetric tensor.

## Notes

- To explore why the Cayley transform can lead to a rotation tensor, the following identities are useful:

(35)

for any second-order skew-symmetric tensor . The easiest way to verify these identities is to substitute numerical values for and and then calculate the quantities involved. Then, with the help of (35), a direct computation shows that

(36)

and

(37)

Therefore, the Cayley transform of a skew-symmetric tensor is proper orthogonal and thus can be identified with a rotation tensor.

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