6.1 Introduction
One of the key features of the rigid body dynamics problems that we will shortly examine is the presence of a variable axis of rotation. This is one of the reasons for the richness of phenomena in rigid body dynamics. It is also a reason why this subject is intimidating. To quote the mechanician Louis Poinsot (1777–1859), from [172], “. . . if we have to consider the motion of a body of sensible shape, it must be allowed that the idea which we form of it is very obscure.” In this chapter, several representations of rotations are discussed that will enable us to establish both a clear picture of rigid body motions and straightforward proofs of several major results. To this end, many results on two key kinematical quantities for rigid bodies, rotation tensors and their associated angular velocity vectors, are discussed in considerable detail.
The subject of rotations in rigid body dynamics has a wonderful history, a wide range of interesting results, and an impressive list of contributors. Here, however, space limits the presentation of only the handful of results that are most relevant to our purposes. From a historical perspective, much of what is presented was estab-lished by Leonhard Euler (1707–1783) in his great works on rigid body dynamics that started to appear in the 1750s. The foundations Euler established were built upon by such notables as Cayley, Gauss, Hamilton, and Rodrigues in the early part of the 19th century. Despite the wealth of their results, the subject of rotations remains an active area of discovery (and some unfortunate reinvention) to this day.
Our exposition makes extensive use of tensors, and the background required for these quantities is provided in the Appendix. Following related developments in continuum mechanics, tensors are an invaluable tool for providing concise expla-nations of many important results. They are not universally invoked in dynamics texts, and the interested reader is invited to compare our treatment with those in the textbooks cited in the bibliography.
We start this chapter with a discussion of a rotation in which the axis of rotation is fixed, and this example is used to introduce several of the key concepts in this
163
chapter. The example also serves to illuminate many of the generalizations that are required for characterizing more elaborate rotations. To this end, our exposition then turns to proper-orthogonal tensors in Section 6.3, and many of their interesting features are shown there. Then Euler’s representation of a rotation tensor is presented, and we examine his important theorem that any proper-orthogonal tensor is a rotation tensor. This result has many consequences for rigid body dynamics. One of the intriguing aspects of rotation tensors is the vast range of representations. Here the Euler angle representation is emphasized, but sufficient material is present for the interested reader to examine many of the other represen-tations such as the Euler–Rodrigues symmetric parameters (often synonymous with quaternions).
Our principal reference for this chapter is the authoritative review by Shuster [196]. This material is supplemented with the elegant relative angular velocity vector that was proposed by Casey and Lam [29], the dual Euler basis [160], and a proof of Euler’s theorem by Guo [83]. In the interests of presenting a unified treatment, our notation differs from these references in several places, but the differences are easily deciphered once the material in this chapter has been comprehended.
6.2 The Simplest Rotation
To motivate many of our later developments, we start with the simplest case of a rotation about a fixed axis p3through an angle θ= θ(t). This example should be fa-miliar to you from many different venues. To describe this rotation, we consider the action of this rotation on this set of orthonormal right-handed basis vectors
p1,p2,p3
. As shown in Figure 6.1, we suppose that these vectors are transformed to the set{t1,t2,t3} by the rotation.
Using a matrix notation, we can represent the transformation from the set of basis vectors
p1,p2,p3
to{t1,t2,t3} as
⎡
⎢⎣ t1 t2
t3
⎤
⎥⎦ = R
⎡
⎢⎣ p1 p2
p3
⎤
⎥⎦ , (6.1)
θ θ θ
p1 p2
t1
t2
p3= t3
Figure 6.1. The transformations of various basis vectors in-duced by a rotation about p3through an angle θ.
6.2 The Simplest Rotation 165
It is easy to see that the matrix R in this equation has a determinant of+1 and that its inverse is its transpose: R−1= RT. That is, the matrix R is proper-orthogonal.
By differentiating (6.1) with respect to time, we find that
⎡ be introduced that has the useful property that
˙tk= ˙θ p3× tk (k= 1, 2, 3) . (6.3) You should notice how the vector ˙θ p3can be inferred from the components of ˙RRT. It is convenient to use a tensor notation to describe the rotation we have been discussing. In particular, we can write (6.1) in the form∗
tk= Rpk (k= 1, 2, 3) , (6.4)
where R is the tensor
R= cos(θ) (p1⊗ p1+ p2⊗ p2)− sin(θ) (p1⊗ p2− p2⊗ p1)+ p3⊗ p3. It is left as an exercise to verify that this representation of the rotation is equivalent to (6.1).†Indeed, because
I− p3⊗ p3= p1⊗ p1+ p2⊗ p2, p3= p1⊗ p2− p2⊗ p1,
∗ Background on tensor notation can be found the Appendix.
† The definition of the tensor product needed to establish this equivalence can be found in (A.1) in the Appendix. The alternating tensor is defined in Section A.7 [see, in particular, (A.11).]
we can express the tensor R entirely in terms of the axis of rotation p3and the angle of rotation θ:
R= cos(θ) (I − p3⊗ p3)− sin(θ)p3+ p3⊗ p3. (6.5) As we shall see later, this representation naturally leads to the general form of a tensor that represents a rotation about an arbitrary axis through an arbitrary angle of rotation. It is left as another exercise to show that RT= R−1and that det(R)= 1.
Differentiating (6.4), we find that
˙tk= ˙Rpk+ ˙R 0
˙pk
= ˙R R./01Ttk
=pk
=RR˙ T tk.
Some straightforward computations are used to show that
R˙ = − ˙θ sin(θ) (p1⊗ p1+ p2⊗ p2)− ˙θ cos(θ) (p1⊗ p2− p2⊗ p1) , RR˙ T= ˙θ (−p1⊗ p2+ p2⊗ p1) .
With the help of these results, we conclude that
˙tk=RR˙ T tk
= ˙θ p3× tk (k= 1, 2, 3) .
As expected, this result is in agreement with (6.3). You might have already noticed that ˙θ p3= ˙θ t3is the axial vector of ˙RRT.
We have demonstrated how several results for a familiar rotation can be ex-pressed by using a tensor notation. This notation will prove to be very useful next when we wish to examine more complex rotations. To this end, we need to answer several questions. The first among them is this: What is the representa-tion for a rotarepresenta-tion about an arbitrary axis? Once this is known, the quesrepresenta-tion of whether its time derivative leads to a vector such as ˙θp3 then presents itself. The answers to these questions were first formulated by Euler in the 1750s. However, numerous alternative representations for his solutions have appeared since then, and we shall leverage several of these representations in the remainder of this chapter.
6.3 Proper-Orthogonal Tensors
To proceed with our treatment of rotations, we appear to regress somewhat and discuss proper-orthogonal tensors. This discussion will lay the foundations for the possibility of three-parameter representations of rotations and subsequently the ex-istence of angular velocity vectors. It is also a starting point for several investigations on experimental measurements of rotations. We recall that a proper-orthogonal
6.3 Proper-Orthogonal Tensors 167
second-order tensor R is a tensor that has a unit determinant and whose inverse is its transpose:
RTR= RRT= I, det(R)= 1. (6.6)
The first of these equations implies that there are six restrictions on the nine com-ponents of R. Consequently, only three comcom-ponents of R are independent. In other words, any proper-orthogonal tensor can be parameterized by three independent parameters.
The tensors of interest here are second-order tensors, any one of which has the representation
R=
3 i=1
3 k=1
Rikpi⊗ pk.
Let us now consider the transformation induced by R on the basis vectors p1, p2, and p3. We define
t1= Rp1=
3 i=1
Ri1pi,
t2= Rp2=
3 i=1
Ri2pi,
t3= Rp3=
3 i=1
Ri3pi.
You should notice that, by using the vectors ti, R has the representation R= t1⊗ p1+ t2⊗ p2+ t3⊗ p3.
We now wish to show that{t1,t2,t3} is a right-handed orthonormal basis. First, let us verify the orthonormality:
ti· tk= Rpi· Rpk= RTRpi· pk
= pi· pk= δik.
Hence the vectors ti are orthonormal. To establish right-handedness, we use the definition of the determinant [see (A.6)]:
[t1,t2,t3]= [Rp1,Rp2,Rp3]
= det(R)[p1,p2,p3]
= (1)(1) = 1.
Hence{t1,t2,t3} is a right-handed orthonormal basis.∗
∗ It is left as an exercise to verify that this result holds for the example of a simple rotation presented in (6.1).
A proper-orthogonal tensor also has a rather unusual representation. To arrive at it, we embark on a series of manipulations:
R= RRRT= R
In summary, we have the following representations for R:
R=
Notice that the components of R for the first two representations are identical, and the handedness of{p1,p2,p3} is transferred without change by R to {t1,t2,t3}.
We observed that the components Rikof R are equal to tk· pi. As this product is equal to the cosine of the angle between tkand pi, each Rikis often referred to as a direction cosine. Consequently the matrix [Rik] is often known as the direction co-sine matrix. Clearly, the nine angles whose coco-sines are tk· piare not all independent, for if they were then [Rik] would have nine independent components and this would contradict the requirement RTR= I. Indeed, as we shall shortly see, it is possible to arrive at three independent angles to parameterize R, but these angles are not all easily related to the angles between piand tk.
6.4 Derivatives of a Proper-Orthogonal Tensor
Here we consider a proper-orthogonal tensor R that is a function of time: R= R(t).
Consider the derivative of RRT: d dt
RRT
= ˙RRT+ R ˙RT.
However, ˙I= O, so the right-hand side of the preceding equation is zero. Hence, RR˙ T= −R ˙RT= −
RR˙ TT
.
In other words, ˙RRTis a skew-symmetric second-order tensor. We define
R= ˙RRT,
in part because this tensor appears in numerous places later on. The tensorRis known as the angular velocity tensor (of R).
6.4 Derivatives of a Proper-Orthogonal Tensor 169
The skew-symmetry of ˙RRTallows us to define the angular velocity vectorωR: ωR= −1
2,RR˙ T -.
As mentioned in the Appendix,ωR× a = ( ˙RRT)a for all vectors a.∗The most com-mon example of the calculation of an axial vector arises when we consider the mo-tion of a rigid body rotating about the E3 direction. In this case, we will see later that the skew-symmetric tensor
R= (E2⊗ E1− E1⊗ E2) . Consequently, we can compute that
[R]= −2E3.
We conclude that the axial vector ofR is the angular velocity vector E3. It also useful to check that
( (E2⊗ E1− E1⊗ E2)) a= E3× a for all vectors a.
In a similar manner, we can also show that RTR is a skew-symmetric tensor and˙ define an angular velocity tensor0Rand another angular vectorω0R:
0R= RTR,˙ ω0R = −1
2[RTR].˙ (6.7)
At a later stage, the reader should be able to show that Rω0R= ω and R0RRT=
R. A key to establishing one of these results is the identity, which holds for all orthogonal Q,
, QBQT
-= det (Q) Q ( [B]) .
Notice how this identity simplifies when Q is a proper-orthogonal tensor.
Corotational Derivatives
Recall that, for a proper-orthogonal tensor R, we have the representation
R=
∗ In Section A.7 of the Appendix, several examples featuring the calculation of the axial vectorωRof a given skew-symmetric tensorRare presented.
If we now considerRtk, we find a familiar result:
˙ti= Rti = ωR× ti. It is left as an exercise to show the less familiar result
˙ti= R(ω0R× pi).
It is important to note that if ti are defined by use of a proper-orthogonal tensor R and a fixed basis pi, then their time derivatives can be expressed in terms of the angular velocity vector of the rotation tensor and the basis vectors ti.
Given any second-order tensor A and any vector a, we have the following rep-resentations:
If we assume that a is a function of time, then
˙
Similarly, if we assume that A is a function of time, then A˙ =
The derivativesA ando a are known as the corotational derivatives (with respect too R) of A and a, respectively. They are the respective derivatives of A and a if the vectors tiare constant:
A=o
Using the recently established expression fora, we observe thato
˙a=ao+ ωR× a.
6.5 Euler’s Representation of a Rotation Tensor 171
Similarly, for any second-order tensor A, we have A˙ =Ao + RA− AR.
The terms involving the angular velocity vectors and tensors in these expressions are the result of the orthonormal vectors tichanging with time.
We shall subsequently use several distinct corotational derivatives, and we wish to do so without introducing a laborious notation. To this end, it will be explicitly stated which rotation tensor the corotational derivative pertains to in situations in which confusion is possible.
6.5 Euler’s Representation of a Rotation Tensor
Leonhard Euler defined a rotation by using an angle of rotation φ and an axis of rotation r.∗Using notation introduced over a century later by Gibbs,†Euler’s rep-resentation for a tensor that produces this rotation can be written as
R= L(φ, r) = cos(φ)(I − r ⊗ r) − sin(φ)(εr) + r ⊗ r, (6.8) where r is a unit vector and φ is a counterclockwise angle of rotation. We refer to (6.8) as Euler’s representation of a rotation tensor and use the function L to prescribe the rotation tensor associated with an axis and angle of rotation. The three independent parameters of R are the angle of rotation and the two independent components of the unit vector r. The reason r is known as the axis of rotation lies in the fact that it is invariant under the action of R: Rr= r. We shall shortly examine the role of φ.
It is interesting to examine some of the features of representation (6.8). To this end, we define an orthonormal basis {p1,p2,p3} with p3= r. Using this basis, we have
I= p1⊗ p1+ p2⊗ p2+ p3⊗ p3, r⊗ r = p3⊗ p3,
I− r ⊗ r = p1⊗ p1+ p2⊗ p2,
−(εr) = p2⊗ p1− p1⊗ p2. Consequently we can write
R= L(φ, r = p3)= cos(φ)(p1⊗ p1+ p2⊗ p2)
+ sin(φ)(p2⊗ p1− p1⊗ p2)+ p3⊗ p3.
∗ This representation can be seen in Section 49 in one of Euler’s great papers on rigid body dynamics from 1775 [56]. There, he provides expressions for the components of the tensor R in terms of an angle of rotation φ and the direction cosines p, q, r of the axis of rotation. In our notation, r= pE1+ qE2+ rE3.
† See Section 129 of [231].
Using the identity (a⊗ b)T= b ⊗ a, we find
RT= L (φ, r = p3)= cos(φ)(p1⊗ p1+ p2⊗ p2)
+ sin(φ)(p1⊗ p2− p2⊗ p1)+ p3⊗ p3.
It is now easy to check that RRTis the identity tensor, and the details are left to the reader. Next, we examine the determinant of R:
det(R)= det
⎡
⎢⎣
cos(φ) − sin(φ) 0 sin(φ) cos(φ) 0
0 0 1
⎤
⎥⎦
= 1.
In conclusion, RTR= I and det(R) = 1. Thus R is a proper-orthogonal tensor. We shall examine the converse of this statement shortly.
Composition of Rotations
Suppose we have a rotation through an angle φ1about an axis r1and we follow this by a rotation through an angle φ2about an axis r2; then the tensor
Rc= L (φ2,r2) L (φ1,r1)
represents the composite rotation. It is left as an exercise to show that Rcis a proper-orthogonal tensor. As can be shown later by Euler’s theorem, this implies that Rcis a rotation tensor. Thus the composition of two rotations is also a rotation.
As tensor multiplication is noncommutative, the order in which we consider the composition is important. In general,
L (φ2,r2) L (φ1,r1)= L (φ1,r1) L (φ2,r2) .
The exceptions arise when r1 r2or one of the angles of rotation is 0. The fact that rotations are sequence dependent will play a major role later on in the definition of Euler angle sequences.
Euler’s Formula
We now examine the action of R on a vector a. As shown in Figure 6.2, the part of a that is parallel to r is unaltered by the transformation, whereas the part of a that is perpendicular to r is rotated through an angle φ counterclockwise about r. To see this, it is convenient to decompose a:
a= a⊥+ a, where
a⊥ = a − (a · r)r = (I − r ⊗ r)a, a= (a · r)r = (r ⊗ r)a.
6.5 Euler’s Representation of a Rotation Tensor 173
Ra
r a
φ a
a⊥ Ra⊥
Figure 6.2. The transformation of a vector a by the rotation tensor R= L (φ, r).
With the assistance of this decomposition, we now compute that Ra= L(φ, r)a
= (cos(φ)(I − r ⊗ r) − sin(φ)(εr) + r ⊗ r) a
= cos(φ)(I − r ⊗ r)a − sin(φ)(εr)a + (r ⊗ r)a
= cos(φ)(I − r ⊗ r)a + sin(φ)r × a + (r · a)r,
= cos(φ)a⊥+ sin(φ)r × a⊥+ a. (6.9) Noting that cos(φ)a⊥+ sin(φ)r × a⊥is a rotation of a⊥about r, we obtain the desired conclusion. The final expression for Ra in (6.9) is known as Euler’s formula.
Remarks on Euler’s Representation
Euler’s representation (6.8) is unusual in several respects. First, you should notice that
L(φ, r)= L(−φ, −r).
This implies that there are two different representations for the same rotation ten-sor. Second, as R is a rotation tensor, R−1= RT, and this leads to two easy repre-sentations for these tensors:
R−1 = RT= L(−φ, r) = L(φ, −r).
In words, the inverse can be calculated by either reversing the angle of rotation or inverting the axis of rotation. Another peculiarity is that L(φ= 0, r) = I holds for all vectors r. Finally, we note that Euler’s representation is used to define other representations of rotation tensors in this book.
6.5.1 Calculating the Axis and Angle of Rotation
Given a rotation tensor R, it is a standard exercise to calculate the axis of rotation r and the angle of rotation θ associated with this tensor. First, one is normally
presented with the matrix components of R with respect to a basis, say{p1,p2,p3}:
If we compare this representation with (6.8), we find that
Rik= ((cos(θ)(I − r ⊗ r) − sin(θ)(εr) + r ⊗ r) pk)· pi
Expanding the expressions for the components of R, we find the matrix represen-tation
Notice that this matrix can be expressed as the sum of a symmetric and skew-symmetric matrix. The skew-skew-symmetric part is the only part that changes when we transpose the tensor.
To determine the angle of rotation, we calculate the trace of R:
cos(θ)= 1
2(tr(R)− 1) = 1
2(R11+ R22+ R33− 1) . (6.11) Looking at the skew-symmetric part of R, we find that
⎡
To verify the calculated value of r, we could examine the eigenvectors of R: The eigenvector corresponding to the unit eigenvalue should be parallel to the axis of rotation r.
It is interesting to notice that, if R=3
i=1ti⊗ pi, then r has the same compo-nents with respect to both sets of basis vectors:
r=
6.5 Euler’s Representation of a Rotation Tensor 175
The proof of this result is based on the observation that R has the same components with respect to the bases pi⊗ pk and ti⊗ tk. That is, R=3
i=1
3
k=1Rikpi⊗ pk=
3
i=1
3
k=1Rikti⊗ tk.
AN EXAMPLE. As an example, suppose that the components of a rotation tensor R are
⎡
⎢⎢
⎣
R11 R12 R13 R21 R22 R23
R31 R32 R33
⎤
⎥⎥
⎦ =
⎡
⎢⎣
0.835959 −0.283542 −0.469869 0.271321 0.957764 −0.0952472
0.47703 −0.0478627 0.877583
⎤
⎥⎦ .
We can compute the angle of rotation θ of this tensor by using (6.11):
cos(θ)= 1
2(0.835959+ 0.877583 + 0.957764 − 1) .
That is, θ= 33.3161◦. We can calculate the axis of rotation r by using (6.12):
r= 0.043135p1− 0.861981p2+ 0.505103p3
= 0.043135t1− 0.861981t2+ 0.505103t3.
In writing this result, we are emphasizing that the components of r in the basis
p1,p2,p3
and{t1,t2,t3} are identical.
The Associated Angular Velocity Vector
Given Euler’s representation (6.8) we assume that R= R(t). This implies, in gen-eral, that φ= φ(t) and r = r(t). We now seek to establish representations for ωR.
As a preliminary result, we note that, because r is a unit vector, r· ˙r = 0. In addition, ˙ = 0 and ˙I = 0. Now, starting from (6.8),
R= L(φ, r) = cos(φ)(I − r ⊗ r) − sin(φ)(εr) + r ⊗ r, we differentiate to find
R˙ = − ˙φ sin(φ)(I − r ⊗ r) − ˙φ cos(φ)(εr) + (1 − cos(φ))(˙r ⊗ r + r ⊗ ˙r) − sin(φ)(ε˙r).
To proceed further, we define a right-handed orthonormal basis{t1,t2,t3}, such that t3= r at a given instant in time. Then, at the same instant in time,
r = (t1⊗ t2− t2⊗ t1),
˙r= at1+ bt2, r× ˙r = at2− bt1,
˙r = a(t2⊗ t3− t3⊗ t2)+ b(t3⊗ t1− t1⊗ t3). (6.13)
In these expressions, a and b are the scalar components of ˙r. Using (6.13) along with some manipulations, we find that
RR˙ T= − ˙φ(t1⊗ t2− t2⊗ t1)+ (a(1 − cos(φ))
+ bsin(φ))(t1⊗ t3− t3⊗ t1)+ (−b(1 − cos(φ)) + a sin(φ))(t3⊗ t2− t2⊗ t3)
= − ˙φt3− (a(1 − cos(φ)) + bsin(φ))t2− (−b(1 − cos(φ)) +a sin(φ))t1.
With the assistance of (6.13), we can now write the desired final result:
R= ˙RRT= − ˙φr − (1 − cos(φ))(r × ˙r) − sin(φ)˙r.
The associated angular velocity vector is
ωR= ˙φr + sin(φ)˙r + (1 − cos(φ))r × ˙r. (6.14) If r is constant, then the expression for the angular velocity vector simplifies con-siderably. It is interesting to note that a constantωR does not necessarily imply a constant r. However, as shown in [161], it is usually possible to choose a fixed ba-sis
p1,p2,p3
where R=3
k=1tk⊗ pkso that a constant angular velocity implies a constant ˙φand a constant r.
6.6 Euler’s Theorem: Rotation Tensors and Proper-Orthogonal Tensors
A tensor Q is proper-orthogonal if, and only if,
QTQ= I, det(Q)= 1.
From our discussion of rotation tensors, it follows that a rotation tensor is a orthogonal tensor. However, is the converse true? In other words, is every proper-orthogonal tensor a rotation tensor? The affirmative answer to this question is known as Euler’s theorem.
Our approach to presenting the proof of Euler’s theorem is adapted from a wonderful paper by Zhong-heng Guo [83]. One of the key results needed is to re-call that we can define two of the invariants, IA= tr(A) and IIIA= det(A), of any second-order tensor A by using the scalar triple product:
[Aa, b, c]+ [a, Ab, c] + [a, b, Ac] = IA[a, b, c], [Aa, Ab, Ac]= IIIA[a, b, c], where a, b, and c are any three vectors (see Section A.6).
For a proper-orthogonal tensor, the fact that det(Q)= 1 implies that this tensor has an eigenvalue λ= 1. There are several ways to see this, but first consider
QT(Q− I) = I − QT.
6.6 Euler’s Theorem: Rotation Tensors and Proper-Orthogonal Tensors 177
Taking the determinant of both sides and using the fact that det(QT)= 1, one finds that
det (Q− I) = (−1)3det (Q− I) . It follows that
det (Q− I) = 0, (6.15)
and thus Q has an eigenvalue of 1.
and thus Q has an eigenvalue of 1.