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

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

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 )

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

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

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 = ½ 1 f 6 o r ± n ^ i o = 1 2 3 T n ^ i o ¢ T = 2 4 1 3 5

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 ;

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 = ¡

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

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 µ E !

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

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

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

Rotations from One Frame to Another AOE 5204 ² 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 ±

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 ² 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 + ?

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 ² 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 ( )

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 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 ^ ¢ = 1 . S k q

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 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 = b f - x I ^ ¯ ¡ 1 ! W F

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 ² x , h l d e c s T v p F b : = g ^ ¢ 2 4 1 3 5 £ ¤

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 ² S ¡ 1 = T , ( h d y m x ) u : O g v l + E b © ; . F

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, …

Euler Angles AOE 5204 Leonhard Euler (1707-1783) 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 ² C o n s i d e r t h a f m F b u g E l , µ 1 2 3 T ¯ ^ x R = ( ) R i = 3 ( µ 1 ) 2 4 c o s n ¡ 5 v

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

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 ² T h e s c o n d r t a i b u ^ 2 x , g l µ f m F R = ( ) p - j w . ¯ E q R i = 2 ( µ ) 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

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 ² T h e t i r d o a n s b u ^ 1 x , g l µ 3 f m F R = ( ) R b i = 1 ( µ 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 µ 3 Ã ^ i 1 a n d b r e o u t f h p g µ 3 ^ i 2

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 = ( µ ) 4 c s ¡ 5 + l y u g x . , ¼ 7 : 8 6 9 C w E

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 µ 1 ¡ 3 + 5 8 7 6 9 C y ¯ ) ¼ Q d k - . S

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 ² T c f ¯ £ = E g q ( ¡ ) w m y ? ­ !

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 ² T h e t r s i m p l o a n c R 1 ( µ ) = 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

Roll, Pitch and Yaw AOE 5204 Roll, pitch and yaw are Euler angles and are sometimes defined as a 3-2-1 sequence and sometimes defined as a 1-2-3 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 µ ¡ + 5 y g , p

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 µ ¼ 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 µ ¼ a d i ) R b 4 ¡ 5 w h r t l g , p y R b i ¼ 1 ¡ 2 4 µ 3 5 £ ) R b i ¼ 1 ¡ 2 4 r o l p t c h y a w 3 5 £ ( R b i ¼ 1 ¡ µ £

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

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

Inertial Frame to Perifocal Frame AOE 5204 ² 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

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

Perifocal Frame to Orbital Frame AOE 5204 ² 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 º R 2 ( 7 ± ) = 4 1 ¡ 3 5 9 º c o s i n R o p = 2 ( ¡ º ) 3 9 ± 7 4 s c 1 5

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

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 ± ­ ! º W R 7 w F £

An Illustrative Example (continued) AOE 5204 W e c a n u s º = 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 [ ; ] £

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 _ µ = 2 4 s i n 3 c o ¡ 1 5 ! S

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 ² 3 2 1 F b ^ ´ h µ T y ~ ! = _ W x : [ ] K

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

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

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

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 _ µ 3 ¡ 1 5 S ( ) What happens when 2  n/2, for odd n ? What happens when the Euler angles and their rates are “small”?

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

Linearizing the Kinematics Equation AOE 5204 ! b i = S ( µ ) _ 2 4 3 ¡ s n 1 c o + 5 I f t h e E u l r a g d m , ¼ : Exercise: Repeat these steps for a 1-2-3 sequence and for a symmetric sequence.

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 ;

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

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

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 . ² , 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 © ¼ 4 W ?

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

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 , ¼

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 ¹ g ¤

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 ( ¹ 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 -

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

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

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 ) ¼ 4 1 H c R ¡ T ¢ + £ C p x v µ W ; ? S : b ¤

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 ( £ ¡ ) _ ! ¾ = a t n © 4 R 1 + T £ ( ¡ ) 2 ¤ _ · ¸

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