We have just introduced a first kind of specifying change of orientation. Let s call it Axis-Angle.

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1 2.1.5 Rotations in 3-D around the origin; Axis of rotation In three-dimensional space, it will not be sufficient just to indicate a center of rotation, as we did for plane kinematics. Any change of orientation in 3 D defines an axis of rotation. This axis was given in case of plane kinematics: It was always perpendicular to the plane in question. Now, the axis can have any direction in space. This fact alone opens up entirely new concepts and techniques in kinematics, especially also as to intuition and mental representation of the geometry of motion. How many parameters define a rotation in 3-D? How many parameters are necessary to characterize a change of orientation (a rotation) in 3-D? Of course, as before, we will first have an angle of rotation. But now the direction of the rotation axis needs also to be specified. A direction of an axis in 3-D space is fixed by only two parameters. These could be, as an example, latitude and longitude of a point on the surface of a sphere. The axis is then defined as going from the center of the sphere through the point of given latitude and longitude. Together, this gives three parameters, two for the axis direction and one for the angle of rotation around this axis. Alternatively, we could consider the axis given by a vector v = [x a, y a, z a ] T i.e. three parameters, defining the rotation axis and the angle of rotation around this axis. This gives a total of four parameters. So there is an excess of one parameter: The excess parameter is the length of the vector, which does not have any meaning for the definition of our rotation. This length could be normalized ( v = x a 2 + y a 2 + z a 2 = 1 ), such an additional constraint on the four parameters removes one freedom, so we are again at the minimal number of three for the definition of a change in orientation. Different ways to express a change of orientation in 3-D We have just introduced a first kind of specifying change of orientation. Let s call it Axis-Angle. In practice many other ways to specify this information are used, so let us introduce them here. In the context of flight dynamics, the three angles roll, pitch and yaw are used to define orientation. They are sometimes also called Tait-Bryan angles or Bryan Angles (George H Bryan, ). These angles can also be used in robotics. In French, the names of these same angles are have familiar names in the navigation context: roulis, tangage, lacet, in German they are called Roll, Nick Gier Winkel. The particularity of this set of angles (again, three parameters to define an orientation) is, that they are defined with respect to a direction of a vehicle motion, or, in case of a robot, with respect to the approach direction.

2 Furthermore, they can either be defined with respect to a fixed direction independent of the instantaneous vehicle orientation (e.g. in navigation), or with respect to body-fixed axes (as in flight dynamics or robotics), this should be specified from the context. Be careful not to confuse the two situations! This leads us to a further generalization, which is linked to the work of Leonard Euler. While Newton formulated the laws of motion for point masses, it is Leonard Euler who clarified the dynamics of rigid bodies by generalizing the equations of motion for rigid body dynamics. He thus had to describe the orientation of a rigid body in space. He studied specifically the gyroscope and it is in this context that the three Euler Angles were first introduced. They are: Precession, nutation and self-rotation. Euler being one of the most prolific mathematician of all times, there is a large collection of Eulersomething. Orientation in space can also be represented by the Euler parameters, which are NOT the Euler angles! We will define the Euler parameters in a short moment, they are equivalent to the quaternions discussed a bit later. Let us first connect all this to the 2-D rotations and give some examples. Let us start with a straightforward generalization from 2-D (plane) kinematics to 3-D kinematics. We have introduced previously rotation as a matrix used in the transformation v (after) = R(ϑ) v (before) where v (before) and v (after) are the vectors indicating a point on an object before resp. after rotation (active transformation, fixed coordinate frame). The 2x2 orthogonal matrix R(ϑ) contains the rotation by angle ϑ as seen in section How do we now apply this formalism when we have to specify a rotation axis? We know the answer for rotations about the (vertical) z-axis for rotations in the x-yplane: Such a rotation matrix, now 3x3, has to leave the z-coordinate values unchanged. The (x-y) coordinates are transformed as before. Thus we obtain: R z θ = cos θ sin θ 0 sin θ cos θ Exercise 5: Give the rotation matrices R z (ϑ), R x (ϑ) and R y (ϑ) for rotations about the z-, x- and y- axis. Hint: Use the transform of the unit vectors e x, e y, e z as in equation (8) with the direction cosines. It is more than useful to draw a figure for each case! R x (ϑ) =? R y (ϑ) =? Now for the core problem in this context. Try the following: Exercise 6: Rotate an object a) 90 around the z-axis, then 90 around the y-axis b) 90 around the y-axis, then 90 around the z-axis

3 This is done mathematically by composition of the rotations, i.e. by matrix multiplications. What do you observe? Exercise 7: Solve the previous question graphically, by rotating the object (a sheet with a 1 printed, to define orientation): Can you identify the rotation axis and angle equivalent to the two rotations about the z- and y-axis? How can we find this axis in the matrix products resulting from 6a) and 6b)? A point on the rotation axis will not be affected by the rotation. We therefore have the identity Rv = v for any point v on the rotation axis. The rotation axis is therefore the eigenvector of the rotation matrix corresponding to the eigenvalue 1. The angle can then be found by projecting a transformed vector on the plane normal to the axis. Example: Matlab command [V,d]=eig(A) gives eigenvectors in matrix V and eigenvalues in d: i i i 0 0 A = V = i i 0.57 d = i i i These calculation can be simplified and expressed in the following formula: Transformation from direction cosine matrix to angle-axis representation: The angle ϑ of rotation matrix R is then given by the following expressions:! R = a d g b e h c f i cos(ϑ ) = 1 2 (tr(r) 1) sin(ϑ ) = ± 1 2 ( f h)2 + (g c) 2 + (b d) 2 where the trace operator tr(r) is defined as the sum of the diagonal elements of R. The rotation axis is given by 1 2sin(ϑ ) f h g c b d (10) Note that this formula breaks down for rotation angle zero! Worse, this means that numerical condition will deteriorate as the angle approaches zero, which is physically a

4 completely normal situation. This behavior is undesirable for the control of robots. It will be a strong incentive to find other representations for change of orientation than angle/axis. Exercise 8: Find the direction cosine matrix (the rotation matrix) for a rotation about an angle ϑ around the axis [0, sin(ϕ), cos(ϕ)] T Hint: Use composition of rotations around known coordinate axes. Of course, this could be done more easily by an inverse of formula (10), i.e. by finding a formula rotating directly a vector when rotation axis and angle are given. Let us assume that the rotation axis k is given as a vector of length one: k=[x,y,z] T and let the rotation angle be ϑ. Any vector v may then be transformed by decomposing it into a component parallel to the rotation axis and a component perpendicular to it. While the parallel component remains unchanged, the rotated perpendicular component can be found by a combination of vector cross-product and trigonometric functions. The final result is! v rot =! v cosϑ + (! k! v)sinϑ +! k(! k! v)(1 cosϑ ) This formula was first published by Benjamin Olinde Rodriguez, a French mathematician who lived from 1795 to By applying this formula to the base vectors of a coordinate system, one obtains the matrix form of the Rodriguez equation, which gives the rotation matrix (direction cosines) from an angle-axis representation. Given axis k=[x,y,z] T with x 2 + y 2 + z 2 = 1 ϑ R = (1 cosϑ ) xx xy xz xy yy yz xz yz zz + cosϑ sinϑ 0 z y z 0 x y x 0 (11) with x 2 + y 2 + z 2 = 1. With rotation matrices we use 9 parameters for the change of orientation which are interconnected for the three degrees of freedom of orientation. With the angle-axis representation as introduced above, we run into problems for small rotation angles. This why mathematicians in early 19 th century where searching for a way to represent orientation in 3-D space by an analog of complex numbers, after Jean-Robert Argand ( ) from Geneva had popularized the complex plane allowing a very elegant representation of plane rotation. After many unsuccessful attempts to generalize the complex numbers with one real and two imaginary components, William Rowan Hamilton famously succeeded in 1843 based on work by Gauss and Euler, by introducing quaternions. It is worth noting that this work is was motivated by the representation of orientation in 3-D space.

5 Quaternions, Euler Parameters, Olinde-Rodriguez Parameters While Euler Parameters (also called Rodriguez Parameters) are strictly concerned with kinematics, quaternions have acquired a far greater significance as generalized complex numbers. The «trick» in Hamilton s breakthrough is two-fold. 1) He recognized that a generalization to of complex numbers to three dimensions is not possible, that he must jump directly to four dimensions, i.e. a real part and three imaginary parts. 2) Furthermore, he also recognized, that he must abandon the law of commutatitvity of multiplication. (remember, this was many years before introduction of matrices and vectors). Based on this, the new «hypercomplex» numbers, called «quaternions» could be introduced, according on the following algebraic rules : There is a real part and three imaginary parts, with units i, j, k. We have. i 2 = j 2 = k 2 = ijk = 1. (11a) and the non-commutative ij = k= ji jk = i = kj ki = j = ik (11b) In analogy to complex numbers where a number z is a pair of scalars a and b according to z = {a,b} = a + bi we now have for the quaternion Q a quadruplet of real numbers (scalars) λ 0, λ 1, λ 2, λ 3 Q = { λ 0, λ 1, λ 2, λ 3 } = λ 0 + i λ 1 + j λ 2 + k λ 3 (11c) With this, we have all necessary algebraic definitions. In modern notation, the three imaginary parts are often grouped as a 3x1 vector λ = [λ 1, λ 2, λ 3 ]: Q = { λ 0, λ 1, λ 2, λ 3 } ={ λ 0, λ } How is spatial orientation coded in quaternions? This is where the four Euler Parameters introduced in the 18 th century by Euler are used: λ 0 is the cosine of half of the rotation angle ϑ : λ 0 = cos(ϑ/2) (11d) The direction of the rotation axis is given by the vector part λ = [λ 1, λ 2, λ 3 ] As we use four parameters for three degrees of freedom, we may introduce a constraint. It is essential. The constraint is that the modulus (the absolute value) of the quaternion is equal to one ( unitary quaternion ): λ λ λ λ 3 2 = 1 (11e)

6 This means that the length of vector λ = [λ 1, λ 2, λ 3 ] is restricted to sin(ϑ/2): λ = sin(ϑ/2) leading to Q = 1 (11f) These rules lead to the following composition of quaternions (and therefore of rotations): Q M Q L = { µ 0, µ} { λ 0, λ} = { µ 0 λ 0 µ T λ, µ 0 λ + λ 0 µ + µ λ } (11g) With this, we can now solve problems such as exercises 6, 7 or 8 much more easily. Exercise 8b) Find the rotation axes of 67 with the help of quaternions Finally, it is useful to have equations translating from direction cosines to quaternions and vice-versa: 2(λ λ 2 1 ) 1 2(λ 1 λ 2 λ 0 λ 3 ) 2(λ 1 λ 3 + λ 0 λ 2 ) R = 2(λ 1 λ 2 + λ 0 λ 3 ) 2(λ λ 2 ( 2 ) 1 2(λ 2 λ 3 λ 0 λ 1 )( 2(λ 1 λ 3 λ 0 λ 2 ) 2(λ 2 λ 3 + λ 0 λ 1 ) 2(λ λ 2 3 ) 1 ( (11h)! R = r 11 r 12 r 13 r 21 r 22 r 23 r 31 r 32 r 33 λ = 1 2 sgn(r 32 r 23 ) r 11 r 22 r sgn(r 13 r 31 ) r 22 r 11 r sgn(r 21 r 12 ) r 33 r 22 r λ 0 = 1 2 r 11 + r 22 + r (11i)