Let us first introduce some definitions. A k-dimensional parametrization of the group of rotation SO(3) is the process of finding a subset Uk of Euclidean space Rk, and an
5. The magnetometers provide the measurement of the geomagnetic direction, but the accelerometers only measure the gravity direction during non-accelerated motion.
6. The involvement of the GPS measurement is in turn a price to pay.
Recalls on attitude parametrizations 99 associated smooth function γ :Uk −→SO(3) such that around each point po ∈ Uk, γ is locally subjective (i.e. γ send any neighborhood ofpo ∈Uk into a neighborhood ofγ(po) in SO(3)). A parametrization is 1-1 if the function γ is a global diffeomorphism. It is 2-1 if the functionγ is a local diffeomorphism and for anyR ∈SO(3) there exist two and only two elements u1, u2 ∈ Uk such that γ(u1) = γ(u2) =R. A parametrization is singular if there exists an elementR ∈SO(3) such that the solution to the equationγ(u) =R is not well-posed (i.e. there does not exist any solution or there exists an infinity of solutions).
Studies about the rotation group SO(3) started in the eighteenth century. As a mat-ter of fact, the problem of parametrization of the group of rotation of the Euclidean 3D-space has been of interest since 1776, when Euler first showed that this group is a three-dimensional manifold. A rotation matrix has nine scalars components. However, it is possible to represent an element of the group of rotation by a set of less than nine parameters, and three is the minimum number of parameters needed for this. Neverthe-less, it was shown that no three-dimensional parametrization can be 1-1 (Stuelpnagel, 1964). Previously in 1940 Hopf showed that no four-dimensional parametrization can be 1-1, and that a five-dimensional parametrization can be used to represent the rota-tion group in a 1-1 global manner. However, the greatest inconvenience of Hopf’s five-dimensional parametrization concerns the nonlinearity of the associated differential equa-tions (seee.g.(Stuelpnagel, 1964)). On the other hand, four-dimensional parametrizations (see e.g. (Stuelpnagel, 1964), (Robinson, 1958)), alike the quaternions parametrization, only represent the rotation group in a 2-1 way. Nevertheless, although the quaternion parametrization is not 1-1, no difficulty arises for practical purposes because the trans-formation of a unit quaternion to SO(3) is everywhere a local diffeomorphism. Hereafter, the Euler angles and the quaternion parametrizations are recalled and discussed. These parametrizations, and many others, are also discussed in the survey papers (Stuelpnagel, 1964), (Shuster, 1993).
3.2.1 Euler angles parametrization
A number of three-dimensional parametrizations can be found in the literature (see e.g.(Stuelpnagel, 1964)), among which the Euler angles parametrization is most widely-used. As a matter of fact, the definition of the Euler angles depends on the problem to be solved and on the chosen systems of coordinates. The definition adopted here is the definition commonly used in the aerospace field for which the Euler angles φ, θ, ψ cor-respond to the parameters of roll, pitch, and yaw (see e.g. (Stuelpnagel, 1964), (Mayer, 1960)). The corresponding Euler angles are defined by
⎧⎪
withri,j the component of rowiand columnj of the rotation matrixR. The Euler angles allows to factorizeR into a product of three matrices of rotation about three axes of the
body frame, as
From Eq. (3.1) and Eq. (3.6), a direct calculation gives (see e.g. (Stuelpnagel, 1964),
(Mayer, 1960)) ⎧ well-defined. Therefore, the Euler angles constitute a parametrization of the rotation group, except at points (on the subset) corresponding to θ = ±π/2. Furthermore, when θ =
±π/2, φ˙ and ψ˙ are not well-defined either. The problem of singularities is a weakness of the Euler angles parametrization and, as a matter of fact, of all three-dimensional parametrization techniques.
3.2.2 Quaternion parametrization
Compared to three-dimensional parametrizations, four-dimensional parametrizations al-low to avoid singularities. The earliest formulation of the four-dimensional parametriza-tion, as pointed out in (Robinson, 1958), was given by Euler in 1776. Earlier in 1775, he stated that in three dimensions, every rotation has an axis. This statement can be refor-mulated as follows (see e.g. (Robinson, 1958), (Palais and Palais, 2007) for the proof) Euler’s theorem: For any R∈SO(3), there is a non-zero vector v satisfying Rv = v.
This theorem implies that the attitude of a body can be specified in terms of a rotation by some angle about some fixed axis. It also indicates that any rotation matrix has an eigen-value equal to one. A number of four-dimensional parametrizations can be found in the literature (see e.g. (Robinson, 1958), (Shuster, 1993), (Borgne, 1987)) such as the Euler parameters, the quaternion parameters, the Rodrigues parameters, and the Cayley-Klein parameters. Here, only the quaternion parameters are presented.
The quaternions were first invented by Hamilton in 1843 (Hamilton, 1843), (Hamilton, 1844), and further studied by Cayley and Klein. A unit quaternion has the form
q =s+ir1+jr2+kr3,
where s, r1, r2, r3 are real numbers satisfying s2+r21+r22+r32 = 1, called constituents of the quaternion q; and i, j, k are imaginary units which satisfy
i2 =j2 =k2 =−1, ij =−ji=k, jk=−kj =i, ki=−ik=j.
In the literature the quaternionq can be represented in a more concise way asq= [s, r], with s ∈ R the real part of the quaternion q and r = [r1, r2, r3] ∈ R3 its pure part or
Recalls on attitude parametrizations 101 imaginary part. The quaternions are not commutative, but associative, and they form a group known as the quaternion group where the unit element is 1[1,0] and the quaternion product associated with this group is defined by
s
The transformed rotation matrix R is uniquely defined from the unit quaternion q as follows
This relation is equivalent to Rodrigues’ rotation formula
R=I3+ 2s S(r) + 2S(r)2, (3.8) with I3 the 3×3 identity matrix. Besides, if the rotation matrix R is represented by a rotation by an angleθabout a fixed axis specified by a unit vector u∈R3, then the above equation is equivalent to
R=I3+ sinθ S(u) + 2 sin2(θ/2)S(u)2
On the other hand, converting a rotation matrix to a quaternion is less direct. Using Eq. (3.7) one obtains
There always exists at least one component of the unit quaternion q different from zero. Once this component is identified, the quaternion can be directly deduced from the relations given in Eq. (3.9). Note that only two values of the quaternionq correspond to the rotation matrixR, and that they have opposed signs. For example, iftr(R)=−1, then
s=±1 2
1 +tr(R), S(r) = R−R 4s .
It matters now to express the time-derivative of the quaternionq. One has
⎡
The quaternion parametrization involves four parameters (i.e. only one redundant parameter) and is free of singularities. The associated differential equation is linear in q. Furthermore, the structure of the quaternion group is, by itself, of great interest. In view of the above remarks, one may wonder why the Euler angles parametrization is still widely used especially for aircraft control. Robotists and automaticians working on nonlinear control and estimation are not fond of the Euler angles parametrization due to the associated singularities and increased nonlinearity of the representation. On the contrary, engineers and automaticians used to linear control techniques for aircrafts find reasons to defend this parametrization. For instance, most of aircrafts are designed and controlled in such a way that the pitch and roll Euler angles are limited to small values, away from singularities. When applying linear control techniques based on first order approximations, the Euler angles parametrization equally allows for the decoupling of the altitude control and the longitudinal control. Historical and security issues may also impede the replacement of existing linear control techniques by any nonlinear control technique without costly and long-term validation procedures. Such a debate never ends, and changing habits is never simple.