EE 565: Position, Navigation and Timing

Transcription

1 EE 565: Position, Navigation and Timing Navigation Mathematics: Angular and Linear Velocity Kevin Wedeward Aly El-Osery Electrical Engineering Department New Mexico Tech Socorro, New Mexico, USA In Collaboration with Stephen Bruder Electrical and Computer Engineering Department Embry-Riddle Aeronautical Univesity Prescott, Arizona, USA February 6, 2018 Kevin Wedeward, Aly El-Osery (NMT) EE 565: Position, Navigation and Timing February 6, / 19

2 Lecture Topics 1 Review 2 Introduction to Velocity 3 Derivative of Rotation Matrix and Angular Velocity - Approach I 4 Derivative of Rotation Matrix and Angular Velocity - Approach II 5 Properties of Skew-symmetric Matrices 6 Propagation/Addition of Angular Velocity 7 Linear Position, Velocity and Acceleration Kevin Wedeward, Aly El-Osery (NMT) EE 565: Position, Navigation and Timing February 6, / 19

3 Review z b r bc z c translation between frames {a} and {c}: y b y c r ac = r ab + r bc x b r ab r ac x c z a y a x a Kevin Wedeward, Aly El-Osery (NMT) EE 565: Position, Navigation and Timing February 6, / 19

4 Review z b r bc z c translation between frames {a} and {c}: y b y c r ac = r ab + r bc x b r ab r ac z a x c written wrt/frame {a} y a r a ac = r a ab + r a bc x a = r a ab + C a b r b bc Kevin Wedeward, Aly El-Osery (NMT) EE 565: Position, Navigation and Timing February 6, / 19

5 Introduction to Velocity Given relationship for translation between moving (rotating and translating) frames what is linear velocity between frames? r a ac = r a ab + C a b r b bc Kevin Wedeward, Aly El-Osery (NMT) EE 565: Position, Navigation and Timing February 6, / 19

6 Introduction to Velocity Given relationship for translation between moving (rotating and translating) frames what is linear velocity between frames? r a ac = r a ab + C a b r b bc Why is C a b 0 in general? r ac a d r a ac = d ( ) r a ab + C b a r b bc = r ab a + C b a r b bc + C b a r bc b Kevin Wedeward, Aly El-Osery (NMT) EE 565: Position, Navigation and Timing February 6, / 19

7 Introduction to Velocity Given relationship for translation between moving (rotating and translating) frames what is linear velocity between frames? r a ac = r a ab + C a b r b bc r ac a d r a ac = d ( ) r a ab + C b a r b bc = r ab a + C b a r b bc + C b a r bc b Why is C a b 0 in general? Recoordinatization of r b bc is time-dependent. Kevin Wedeward, Aly El-Osery (NMT) EE 565: Position, Navigation and Timing February 6, / 19

8 Introduction to Velocity Given relationship for translation between moving (rotating and translating) frames what is linear velocity between frames? r a ac = r a ab + C a b r b bc r ac a d r a ac = d ( ) r a ab + C b a r b bc = r ab a + C b a r b bc + C b a r bc b Why is C b a 0 in general? Recoordinatization of r b bc is time-dependent. C b a is directly related to angular velocity between frames {a} and {b}. Kevin Wedeward, Aly El-Osery (NMT) EE 565: Position, Navigation and Timing February 6, / 19

9 First approach to d C and angular velocity Given a rotation matrix C, one of its properties is [C a b ]T C a b = C a b [C a b ]T = I Taking the time-derivative of the right-inverse property d ( ) Cb a [C b a ]T = d I Kevin Wedeward, Aly El-Osery (NMT) EE 565: Position, Navigation and Timing February 6, / 19

10 First approach to d C and angular velocity Given a rotation matrix C, one of its properties is [C a b ]T C a b = C a b [C a b ]T = I Taking the time-derivative of the right-inverse property d ( ) Cb a [C b a ]T = d I C a b [C a b ]T } {{ } Ω a ab + Cb a [ C b a }{{ ]T } ( C b a [C b a ]T ) T }{{} [Ω a ab ]T = 0 Kevin Wedeward, Aly El-Osery (NMT) EE 565: Position, Navigation and Timing February 6, / 19

11 First approach to d C and angular velocity Given a rotation matrix C, one of its properties is [C a b ]T C a b = C a b [C a b ]T = I Taking the time-derivative of the right-inverse property d ( ) Cb a [C b a ]T = d I C a b [C a b ]T } {{ } Ω a ab + Cb a [ C b a }{{ ]T } ( C b a [C b a ]T ) T }{{} [Ω a ab ]T Ω a ab + [Ωa ab ]T = 0 = 0 Kevin Wedeward, Aly El-Osery (NMT) EE 565: Position, Navigation and Timing February 6, / 19

12 First approach to d C and angular velocity Given a rotation matrix C, one of its properties is [C a b ]T C a b = C a b [C a b ]T = I Taking the time-derivative of the right-inverse property d ( ) Cb a [C b a ]T = d I C a b [C a b ]T } {{ } Ω a ab + Cb a [ C b a }{{ ]T } ( C b a [C b a ]T ) T }{{} [Ω a ab ]T Ω a ab + [Ωa ab ]T = 0 = 0 Ω a ab is skew-symmetric! Kevin Wedeward, Aly El-Osery (NMT) EE 565: Position, Navigation and Timing February 6, / 19

13 First approach to d C and angular velocity Define this skew-symmetric matrix Ω a ab 0 ω z ω y Ω a ab = [ ω a ab ] = ω z 0 ω x ω y ω x 0 Kevin Wedeward, Aly El-Osery (NMT) EE 565: Position, Navigation and Timing February 6, / 19

14 First approach to d C and angular velocity Define this skew-symmetric matrix Ω a ab 0 ω z ω y Ω a ab = [ ω a ab ] = ω z 0 ω x ω y ω x 0 Note Ω a ab = C a b [C a b ]T C a b = Ωa ab C a b is a means of finding derivative of rotation matrix provided we can further understand Ω a ab. Kevin Wedeward, Aly El-Osery (NMT) EE 565: Position, Navigation and Timing February 6, / 19

15 First approach to d C and angular velocity Now for some insight into physical meaning of Ω a ab. Consider a point p on a rigid body rotating with angular velocity ω = [ω x, ω y, ω z ] T = θ k = θ[k x, k y, k z ] T with k a unit vector. ω θ k r p p Kevin Wedeward, Aly El-Osery (NMT) EE 565: Position, Navigation and Timing February 6, / 19

16 First approach to d C and angular velocity ω θ k r p v p p From mechanics, linear velocity v p of point is Kevin Wedeward, Aly El-Osery (NMT) EE 565: Position, Navigation and Timing February 6, / 19

17 First approach to d C and angular velocity ω θ k r p v p p From mechanics, linear velocity v p of point is ω x r x ω y r z ω z r y 0 ω z ω y r x v p = ω r p = ω y r y = ω z r x ω x r z = ω z 0 ω x r y ω z r z ω x r y ω y r x ω y } ω x {{ 0 } r z? Kevin Wedeward, Aly El-Osery (NMT) EE 565: Position, Navigation and Timing February 6, / 19

18 First approach to d C and angular velocity ω θ k r p v p p From mechanics, linear velocity v p of point is ω x r x ω y r z ω z r y 0 ω z ω y r x v p = ω r p = ω y r y = ω z r x ω x r z = ω z 0 ω x r y ω z r z ω x r y ω y r x ω y } ω x {{ 0 } r z Ω=[ ω ] Ω represents angular velocity and performs cross product Kevin Wedeward, Aly El-Osery (NMT) EE 565: Position, Navigation and Timing February 6, / 19

19 First approach to d C and angular velocity Now let s add fixed frame {a} and rotating frame {b} attached to moving body such that there is angular velocity ω ab between them. ω ab {a}{b} r ap r ap p Kevin Wedeward, Aly El-Osery (NMT) EE 565: Position, Navigation and Timing February 6, / 19

20 First approach to d C and angular velocity Now let s add fixed frame {a} and rotating frame {b} attached to moving body such that there is angular velocity ω ab between them. ω ab {a}{b} r ap r ap p Start with position r a ap = r a ab +Cb a }{{} r b bp 0 Kevin Wedeward, Aly El-Osery (NMT) EE 565: Position, Navigation and Timing February 6, / 19

21 First approach to d C and angular velocity Now let s add fixed frame {a} and rotating frame {b} attached to moving body such that there is angular velocity ω ab between them. ω ab Start with position {a}{b} r ap p r a ap = r a ab +Cb a }{{} r b bp 0 r ap and take derivative wrt time r ap a = C b a }{{} Ω a ab C b a = Ω a ab C a b r b bp r b bp + C a b r b bp }{{} 0 = Ω a ab r a bp = [ ω a ab ] r a bp Kevin Wedeward, Aly El-Osery (NMT) EE 565: Position, Navigation and Timing February 6, / 19

22 First approach to d C and angular velocity Now let s add fixed frame {a} and rotating frame {b} attached to moving body such that there is angular velocity ω ab between them. ω ab {a}{b} r ap p r ap and take derivative wrt time r ap a = C b a }{{} Ω a ab C b a = Ω a ab C a b r b bp r b bp + C a b r b bp }{{} 0 = Ω a ab r a bp = [ ω a ab ] r a bp Start with position r a ap = r a ab +Cb a }{{} r b bp 0 from which it is observed (compare to v p = ω r p ) that Ω a ab represents cross product with angular velocity ω a ab. Kevin Wedeward, Aly El-Osery (NMT) EE 565: Position, Navigation and Timing February 6, / 19

23 Second approach to d C and angular velocity Another approach to developing derivative of rotation matrix and angular velocity is based upon angle-axis representation of orientation and rotation matrix as exponential. This approach is included in notes. Kevin Wedeward, Aly El-Osery (NMT) EE 565: Position, Navigation and Timing February 6, / 19

24 Properties of Skew-symmetric Matrices Therefore (from above), and (via distributive property) )] CΩC T b = C [ ω (C T b ) = C ω (CC T b = C ω b = [C ω ] b CΩC T = C[ ω ]C T = [C ω ] C[ ω ] = [C ω ]C noting both ω and vector with which cross-product will be taken are assumed to be in the same coordinate frame and thus both need to be recoordinatized. Kevin Wedeward, Aly El-Osery (NMT) EE 565: Position, Navigation and Timing February 6, / 19

25 Properties of Skew-symmetric Matrices C a b = Ωa ab C a b = [ ω a ab ]C a b = [C a b ω b ab ]C a b = C a b [ ω b ab ] = C a b Ωb ab C a b = Ωa ab C a b = C a b Ωb ab Kevin Wedeward, Aly El-Osery (NMT) EE 565: Position, Navigation and Timing February 6, / 19

26 Summary of Angular Velocity and Notation Angular velocity can be described as a vector the angular velocity of the b-frame wrt the a-frame resolved in the c-frame, ω c ab ω ab = ω ba Kevin Wedeward, Aly El-Osery (NMT) EE 565: Position, Navigation and Timing February 6, / 19

27 Summary of Angular Velocity and Notation Angular velocity can be described as a vector the angular velocity of the b-frame wrt the a-frame resolved in the c-frame, ω c ab ω ab = ω ba described as a skew-symmetric matrix Ω c ab = [ ω c ab ] the skew-symmetric matrix is equivalent to the vector cross product when pre-multiplying another vector Kevin Wedeward, Aly El-Osery (NMT) EE 565: Position, Navigation and Timing February 6, / 19

28 Summary of Angular Velocity and Notation Angular velocity can be described as a vector the angular velocity of the b-frame wrt the a-frame resolved in the c-frame, ω c ab ω ab = ω ba described as a skew-symmetric matrix Ω c ab = [ ω c ab ] the skew-symmetric matrix is equivalent to the vector cross product when pre-multiplying another vector related to the derivative of the rotation matrix C a b = Ωa ab C a b = C a b Ωb ab C a b = Ωa ba C a b = C a b Ωb ba Kevin Wedeward, Aly El-Osery (NMT) EE 565: Position, Navigation and Timing February 6, / 19

29 Propagation/Addition of Angular Velocity Consider the derivative of the composition of rotations C 0 2 = C 0 1 C 1 2. d C 0 2 = d C 0 1 C 1 2 C 0 2 = C 0 1 C C 0 1 C 1 2 Ω 0 02 C 2 0 = Ω 0 01C1 0 C2 1 + C1 0 C2 1 Ω 2 12 Ω 0 02 = Ω 0 01C2 0 [ ] C 0 T 2 + C 0 2 Ω 2 [ ] 12 C 0 T 2 [ ω 0 02 ] = [ ω 0 01 ] + [C2 0 ω 2 12 ] ω 0 02 = ω ω 0 12 angular velocities (as vectors) add so long as resolved common coordinate system Kevin Wedeward, Aly El-Osery (NMT) EE 565: Position, Navigation and Timing February 6, / 19

30 Linear Position We can get back to where we started... motion (translation and rotation) between frames and their derivatives. z 1 x 1 y 1 r 01 r 12 r 02 z 2 y 2 Translation (position) between frames {0} and {1}: z 0 x 2 x 0 y 0 r 0 02 = r r 0 12 = r C 0 1 r 1 12 Kevin Wedeward, Aly El-Osery (NMT) EE 565: Position, Navigation and Timing February 6, / 19

31 Linear Velocity Linear velocity: r 0 02(t) = d ( r C1 0 r 1 12) = r C 1 0 r C1 0 r 1 12 = r Ω 0 01C 0 1 r C 0 1 r 1 12 = r [ ω 0 01 ]C 0 1 r C 0 1 r 1 12 = r ω 0 01 (C 0 1 r 1 12) + C 0 1 r 1 12 Kevin Wedeward, Aly El-Osery (NMT) EE 565: Position, Navigation and Timing February 6, / 19

32 Linear Acceleration Linear acceleration: r 0 02 = d 0 ( r 01 + ω 0 01 (C 0 1 r 1 ) 12 + C 0 1 r 1 ) 12 = r ω 0 01 (C 0 1 r 1 ) 12 + ω 0 ( 01 C 0 1 r 1 ) 12 + ω 0 ( 01 C 0 1 r 1 ) 12 + C 0 1 r C 0 1 r 1 12 = r ω ( ) 0 01 r 0 12 (t) + ω 0 01 ω 0 01 r 0 12 (t) + 2 ω 0 01 (C 1 0 r ) C1 0 r 1 12 accel of {1} s origin from {0} in {0} Transverse accel Centripetal accel (ω 2 r) Coriolis accel (2ω v) accel of {2} s origin from {1} in {0} Kevin Wedeward, Aly El-Osery (NMT) EE 565: Position, Navigation and Timing February 6, / 19

33 The End Kevin Wedeward, Aly El-Osery (NMT) EE 565: Position, Navigation and Timing February 6, / 19