1. Rotational Kinematics
AOE 5204
Description of attitude kinematics using reference frames, rotation matrices, Euler parameters, Euler angles, and quaternions
Recall the fundamental dynamics equations
For both equations, we must relate momentum to kinematics ~ f = d t p g h

2. Translational vs Rotational Dynamics
AOE 5204
Linear momentum = mass  velocity
d/dt (linear momentum) = applied forces
d/dt (position) = linear momentum/mass
Angular momentum = inertia  angular velocity
d/dt (angular momentum) = applied torques
d/dt (attitude) = “angular momentum/inertia” ~ g = d t h ~ f = d t p

3. Translational Kinematics
AOE 5204
Newton’s second law can be written in first-order state-vector form as _ ~ r = p m f H e r , ~ p i s t h l n a m o u d b y k c q ; . = _
Thus, the kinematics differential equation allows us to integrate the velocity to compute the position
The kinetics differential equation allows us to integrate the applied force to compute the linear momentum
In general, ~ f = ( r ; p )

4. Translational Kinematics (2)
AOE 5204 C o n s i d e r ~ f = m a x p t l : Ä 1 , 2 3 E q u v y _
We need to develop rotational equations equivalent to the translational kinematics equations

5. Reference Frame & Vectors
AOE 5204 ~ v = 1 ^ i + 2 3
Reference frame is dextral triad of orthonormal unit vectors
Vector can be expressed as linear combination of the unit vectors
Must be clear about which reference frame is used ^ i 3 v 3 F i n ^ 1 ; 2 3 o ~ v v 1 v 2 ^ i 1 ^ i 2

6. Orthonormal
AOE 5204
Orthonormal means the base vectors are perpendicular (orthogonal) to each other, and have unit length (normalized)
This set of 6 (why not 9?) properties can be written in shorthand as
We can stack the unit vectors in a column “matrix”
And then write the orthonormal property as ^ i 1 = 2 3 ^ i j = 1 f 6 o r n ^ i o = 1 2 3 T n ^ i o T = 2 4 1 3 5

7. Dextral, or Right-handed
AOE 5204
Right-handed means the ordering of the three unit vectors follows the right-hand rule
Which can be written more succinctly as
Or even more succinctly as ^ i 1 = ~ 2 3 ^ i j =
" k 8
< : 1 f o r ; a n e v p m u t , 2 3 d h w s ( . y c ) n ^ i o T = 8
< : ~ 3 2 1 9 ;

8. Skew Symmetry
AOE 5204
The [] notation defines a skew-symmetric 3  3 matrix whose 3 unique elements are the components of the 3  1 matrix []
The same notation applies if the components of the 3  1 matrix [] are scalars instead of vectors
The “skew-symmetry” property is satisfied since n ^ i o = 8
< : ~ 3 2 1 9 ; a = 2 4 1 3 5 ) ( a ) T =

9. Vectors
AOE 5204
A vector is an abstract mathematical object with two properties: direction and length
Vectors used in this course include, for example, position, velocity, acceleration, force, momentum, torque, angular velocity
Vectors can be expressed in any reference frame
Keep in mind that the term “state vector” refers to a different type of mathematical object -- specifically, a state vector is a column matrix collecting all the system states

10. Vector, Frame, and Matrix Notation
AOE 5204 S y m b o l M e a n i g ~ v c t r , s h j w d ^ 1 ; 2 3 u f F x p R E !

11. Vectors Expressed in Reference Frames
AOE 5204 T h e s c a l r , v 1 2 n d 3 t o m p f ~ x i F . u w b S y = ^ ; - j k g ~ v = 1 ^ i + 2 3 ^ i 3 v 3 ~ v v 1 v 2 ^ i 1 ^ i 2

12. Vectors Expressed in Reference Frames (2)
AOE 5204
Frequently we collect the components of the vector into a matrix
When we can easily identify the associated reference frame, we use the simple notation above; however, when multiple reference frames are involved, we use a subscript to make the connection clear. Examples: v = 2 4 1 3 5
Note the absence of an overarrow v i d e n o t s c m p F b

13. Further Vector Notation
AOE 5204
Matrix multiplication arises frequently in dynamics and control, and an interesting application involves the 3  1 matrix of a vector’s components and the 3  1 “matrix” of a frame’s base vectors*
We frequently encounter problems of two types: ~ v = [ 1 2 3 ] 8
< : ^ i 9 ; T n o = v T i n ^ o f g b G i v e n a d b , t r m h u o f F w s p c G i v e n t h a u d o f F b w r s p c , m
* Peter Hughes coined the term “vectrix” to denote this object

14. Rotations from One Frame to Another
AOE 5204 S u p o s e w k n t h c m f v r ~ i F b , d a I l ( ) ; x q y = T ^ R W

15. Rotations (2) ² P r o b l e m i n d : K w g v , t W h a R = I f k j u
AOE 5204 P r o b l e m i n d : K w g v , t W h a R T = I f k j u s y 3 x ( . 2 ) ^ c p 1 + ?

16. Rotations (3) ² C o n s i d e r j u t f h q a v l c m p R : ^ = b + g
AOE 5204 C o n s i d e r j u t f h q a v l c m p R : ^ 1 = b + 2 3 g x w , ; U T y D M ( )

17. Rotations (4) ² A n o t h e r w a y d s c i b R v m p f F x , l u T g
AOE 5204 A n o t h e r w a y d s c i b R v m p f F x , l u T g ^ = 1 . S k q

18. Rotations (5) ² A s u m i n g w e h a v c o p t d R , j l r q y T = f
AOE 5204 A s u m i n g w e h a v c o p t d R , j l r q y T = b f - x I ^ 1 ! W F

19. Rotations (6) S u m a r y o f R t i n N ² x , h l d e c s T v p F : =
AOE 5204 S u m a r y o f R t i n N x , h l d e c s T v p F b : = g ^ 2 4 1 3 5

20. Rotations (7) M o r e P p t i s f R a n c ² S = , ( h d y m x ) u : O
AOE 5204 M o r e P p t i s f R a n c S 1 = T , ( h d y m x ) u : O g v l + E b ; . F

21. Rotational Kinematics Representations
AOE 5204
The rotation matrix represents the attitude
A rotation matrix has 9 numbers, but they are not independent
There are 6 constraints on the 9 elements of a rotation matrix (what are they?)
Rotation has 3 degrees of freedom
There are many different sets of parameters that can be used to represent or parameterize rotations
Euler angles, Euler parameters (aka quaternions), Rodrigues parameters (aka Gibbs vectors), Modified Rodrigues parameters, …

22. Euler Angles
AOE 5204
Leonhard Euler ( ) reasoned that the rotation from one frame to another can be visualized as a sequence of three simple rotations about base vectors
Each rotation is through an angle (Euler angle) about a specified axis C o n s i d e r t h a f m F b u g E l , 1 2 3 T ^ x R = ( ) R i = 3 ( 1 ) 2 4 c o s n 5 v

23. Visualizing That “3” Rotation
AOE 5204 R i = 3 ( 1 ) 2 4 c o s n 5 v ^ i 2 ^ i 2 ^ i 1 1 Ã ^ i 3 a n d r e o u t f h p g 1 ^ i 1

24. Euler Angles (2) ² T h e s c o n d r t a i b u ^ x , g l µ f m F R = (
AOE 5204 T h e s c o n d r t a i b u ^ 2 x , g l f m F R = ( ) p - j w . E q R i = 2 ( ) 4 c o s n 1 3 5 v t h e r a m x f g F , w d l u y z . T p

25. Euler Angles (3) ² T h e t i r d o a n s b u ^ x , g l µ f m F R = ( )
AOE 5204 T h e t i r d o a n s b u ^ 1 x , g l 3 f m F R = ( ) R b i = 1 ( 3 ) 2 4 c o s n 5 N t e h a r m d g l f x , w - v F ^ i 3 ^ b 3 ^ b 2 3 Ã ^ i 1 a n d b r e o u t f h p g 3 ^ i 2

26. Euler Angles (4) W e h a v p r f o m d 3 - 2 1 t i n F R = ( µ ) 4 c s
AOE 5204 W e h a v p r f o m d 3 - 2 1 t i n F R b = ( ) 4 c s 5 + l y u g x . , 7 : 8 6 9 C w E

27. Euler Angles (5) T o e x t r a c E u l n g s f m i v , q h w : R = 2 4
AOE 5204 T o e x t r a c E u l n g s f m i v , q h w : R b = 2 4 1 3 + 5 8 7 6 9 C y ) Q d k - . S

28. Euler Angles (6) W e j u s t d v l o p a 3 - 2 1 r i n , b h ² T c f ¯
AOE 5204 W e j u s t d v l o p a 3 - 2 1 r i n , b h T c f = E g q ( ) w m y ? !

29. Euler Angles (7) ² T h e t r s i m p l o a n c R ( µ ) = 2 4 ¡ 3 5
AOE 5204 T h e t r s i m p l o a n c R 1 ( ) = 2 4 3 5
(1,0,0) in 1st row and column
Cosines on diagonal, Sines on off-diagonal, negative Sine on row “above” the 1st row
(0,1,0) in 2nd row and column
Cosines on diagonal, Sines on off-diagonal, negative Sine on row “above” the 2nd row
(0,0,1) in 3rd row and column
Cosines on diagonal, Sines on off-diagonal, negative Sine on row “above” the 3rd row

30. Roll, Pitch and Yaw
AOE 5204
Roll, pitch and yaw are Euler angles and are sometimes defined as a sequence and sometimes defined as a sequence
What’s the difference? T h e 3 - 2 1 s q u n c ( w d i a r l ) t o R b = 4 + 5 y g , p

31. Roll, Pitch, Yaw (2) Note that the two matrices are not the same
AOE 5204
Note that the two matrices are not the same
Rotations do not commute
However, if we assume that the angles are small (appropriate for many vehicle dynamics problems), then the approximations of the two matrices are equal 3 - 2 1 S e q u n c o s a d i ) R b 4 5 w h r t y g l , p 1 - 2 3 S e q u n c o s a d i ) R b 4 5 w h r t l g , p y R b i 1 2 4 3 5 ) R b i 1 2 4 r o l p t c h y a w 3 5 ( R b i 1

32. Perifocal Frame Earth-centered, orbit-based, inertial
AOE 5204
Earth-centered, orbit-based, inertial
The P-axis is in periapsis direction
The W-axis is perpendicular to orbital plane (direction of orbit angular momentum vector, )
The Q-axis is in the orbital plane and finishes the “triad” of unit vectors
In the perifocal frame, the position and velocity vectors both have a zero component in the W direction

33. A One-Minute Course on Orbital Mechanics
AOE 5204
^ K
Equatorial plane n w
^ I W i
Orbital plane
^ J
^ n
Orbit is defined by 6 orbital elements (oe’s): semimajor axis, a; eccentricity, e; inclination, i; right ascension of ascending node, W; argument of periapsis, w; and true anomaly, n

34. Inertial Frame to Perifocal Frame
AOE 5204 U s e a 3 - 1 q u n c R i g h t o f d b ^ K : ( ) , r I v l W m A p ! P R 3 ( ) = 2 4 c o s i n 1 5 ! R p i = 3 ( ! ) 1 2 4 c s + 5

35. Orbital Reference Frame
AOE 5204
Orbital Reference Frame
Same as “roll-pitch-yaw” frame, for spacecraft
The o3 axis is in the nadir direction
The o2 axis is in the negative orbit normal direction
The o1 axis completes the triad, and is in the velocity vector direction for circular orbits
In the orbital frame, position and velocity both have zero in the o2 direction, and position has zero in the o1 direction as well
You will find the rotation from perifocal to orbital easier to visualize if you make yourself two reference frames

36. Perifocal Frame to Orbital Frame
AOE 5204 U s e a 2 - 3 q u n c 7 b o t ^ Q : R ( ) , r W P h d i 9 v g m l N y R 2 ( 7 ) = 4 1 3 5 9 c o s i n R o p = 2 ( ) 3 9 7 4 s c 1 5

37. Inertial to Perifocal to Orbital
AOE 5204 R p i = 3 ( ! ) 1 2 4 c s + 5 o 9 7 u w h e r

38. An Illustrative Example
AOE 5204 I n a i e r t l f c m , E h - o b g s p d v y : ~ = 6 + 1 ; J 5 K [ k ] 2 / T ( u ) 4 3 9 8 ! W R 7 w F

39. An Illustrative Example (continued)
AOE 5204 W e c a n u s = 1 3 : 9 8 t o m p R 2 4 6 5 h l i y g , r w d ! + 7 T v F [ ; ]

40. Differential Equations of Kinematics
AOE 5204
Given the velocity of a point and initial conditions for its position, we can compute its position as a function of time by integrating the differential equation
We now need to develop the equivalent differential equations for the attitude when the angular velocity is known
Preview:
d/dt (attitude) = “angular momentum/inertia” _ ~ r = v _ = 2 4 s i n 3 c o 1 5 ! S

41. Euler Angles and Angular Velocity
AOE 5204 O n e f r a m - t o i , j u s w d v l p g c 3 2 1 F b ^ h T y ~ ! = _ W x : [ ] K

42. Euler Angles and Angular Velocity (2)
AOE 5204 2 - r o t a i n f m F b u ^ h g T e l v c y w s p ~ ! = _ W x , d : [ ] K

43. Euler Angles and Angular Velocity (3)
AOE 5204 1 - r o t a i n f m F b u ^ h g 3 T e l v c y w s p ~ ! = _ W x , d : [ ] K

44. Adding the Angular Velocities
AOE 5204 A n g u l a r v e o c i t s d k : ~ ! b = + W h x p f m ; , w T y 3 1 j F [ _ ] ) R 2

45. Complete the Operation
AOE 5204 C a r y o u t h e m i x l p c n s d ! b = + 2 4 _ 3 1 5 S ( )
What happens when 2  n/2, for odd n ?
What happens when the Euler angles and their rates are “small”?

46. Kinematic Singularity in the Differential Equation for Euler Angles
AOE 5204
For this Euler angle set (3-2-1), the Euler rates go to infinity when cos q2  0
The reason is that near q2 = p/2 the first and third rotations are indistinguishable
For the “symmetric” Euler angle sequences (3-1-3, 2-1-2, 1-3-1, etc) the singularity occurs when q2 = 0 or p
For the “asymmetric” Euler angle sequences (3-2-1, 2-3-1, 1-3-2, etc) the singularity occurs when q2 = p/2 or 3p/2
This kinematic singularity is a major disadvantage of using Euler angles for large-angle motion
There are attitude representations that do not have a kinematic singularity, but 4 or more scalars are required

47. Linearizing the Kinematics Equation
AOE 5204 ! b i = S ( ) _ 2 4 3 s n 1 c o + 5 I f t h e E u l r a g d m , :
Exercise: Repeat these steps for a sequence and for a symmetric sequence.

48. Refresher: How To Invert a Matrix
AOE 5204 S u p o s e y w a n t i v r h m x A , l j T f 1 = C c d b g ( ) + k - O L U E N : I ;

49. Matrix Inversion Application
AOE 5204 L e t u s i n v r h m a x S ( ) = 2 4 1 c o 3 5 I w d , g y : p l - f b k P N

50. Matrix Inversion Application (2)
AOE 5204 2 4 c o s 3 i n 1 5 M u l t p y r w b = a d . + D v e m f g ,

51. Euler’s Theorem T h e m o s t g n r a l i f d b y w ¯ x p u . ² , c E
AOE 5204 T h e m o s t g n r a l i f d b y w x p u . , c E H e r t h b l a c k x s v o f i n - m , d u y w = [ p 2 ] T g 4 W ?

52. Euler’s Theorem (2) A n y r o t a i m x c b e p s d f © : R = 1 + ( ¡
AOE 5204 A n y r o t a i m x c b e p s d f : R = 1 + ( ) T S g v w h l u , C k 2

53. Extracting a and  from R, and Integrating to Obtain a and 
AOE 5204
Extracting a and  from R, and Integrating to Obtain a and  G i v e n a y r o t m x w c p u h E l s d g : = 1 2 ( R ) T W b ? O k q f _ ! S ,

54. Quaternions (aka Euler Parameters)
AOE 5204
Quaternions (aka Euler Parameters) D e n a o t h r 4 - p m s f v i b l u d : q = 2 c T 3 1 x E , . C y k w g

55. Quaternions (2) R ( ¹ q ) a n d : = ¡ ¢ 1 + § p t r c e 5 D i ® l u o
AOE 5204
Quaternions (2) R ( q ) a n d : = 2 4 T 1 + p t r c e 3 5 D i l u o s _ ! Q N h k m g y w -

56. Typical Problem Involving Angular Velocity and Attitude
AOE 5204
Typical Problem Involving Angular Velocity and Attitude
Given initial conditions for the attitude (in any form), and a time history of angular velocity, compute R or any other attitude representation as a function of time
Requires integration of one of the sets of differential equations involving angular velocity

57. Spherical Pendulum Problem
AOE 5204 U s e a r o t i n f m F u - g w E l , d Á T h ( 3 ) b ^ x c 2 v : _ = +
Illustration from Wolfram Research Mathworld
mathworld.wolfram.com/SphericalCoordinates.html

58. Linearization for Small 
AOE 5204 A s w i t h E u l e r a n g , f q y d m o - . I = ( 2 ) 4 1 H c R T + C p x v W ; ? S : b

59. Other Attitude Representations
AOE 5204
We have seen direction cosines, Euler angles, Euler angle/axis, and quaternions
Two other common representations
Euler-Rodriguez parameters
Modified Rodriguez parameters p = a t n 2 R 1 + T ( ) _ ! = a t n 4 R 1 + T ( ) 2 _

60. Typical Problem Involving Angular Velocity and Attitude
AOE 5204
Typical Problem Involving Angular Velocity and Attitude
Given initial conditions for the attitude (in any form), and a time history of angular velocity, compute R or any other attitude representation as a function of time
Requires integration of one of the sets of differential equations involving angular velocity