Navigation Mathematics: Angular and Linear Velocity EE 570: Location and Navigation

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1 Lecture Nvigtion Mthemtics: Angulr n Liner Velocity EE 57: Loction n Nvigtion Lecture Notes Upte on Februry, 26 Kevin Weewr n Aly El-Osery, Electricl Engineering Dept., New Mexico Tech In collbortion with Stephen Bruer, Electricl & Computer Engineering, Embry-Rile Aeronuticl University. Lecture Topics Contents Review 2 Intro to Vel 2 3 t C n ω 2 3. Approch : Structure n mechnics Approch 2: Angle-xis Properties of SS Mtrices 6 5 A Angulr Velocity 7 6 Pos, Vel & Accel 7.2 Review Review z b y b r bc z c y c trnsltion between frmes {} n {c}: x b r b r c x c r c = r b + r bc written wrt/frme {}.3 z y r c = r b + r bc = r b + Cb r b bc x

2 2 Introuction to Velocity Introuction to Velocity Given reltionship for trnsltion between moving rotting n trnslting frmes wht is liner velocity between frmes? r c = r b + C b r b bc r c t r c = r t b + Cb r b bc = r b + Ċ b r b bc + Cb r bc b Why is Ċ b in generl? Recoorintiztion of r b bc is time-epenent. Ċ b is irectly relte to ngulr velocity between frmes {} n {b}..4 3 Derivtive of Rottion Mtrix n Angulr Velocity 3. Approch : Structure n mechnics First pproch to tc n ngulr velocity Given rottion mtrix C, one of its properties is [C b ] T C b = C b [C b ] T = I Tking the time-erivtive of the right-inverse property C t b [Cb ] T = t I Ċ b [Cb ] T + Cb [Ċ b ] T = Ω b Ċ b [Cb ] T T [Ω b ]T Ω b + [Ω b] T = Ω b is skew-symmetric!.5 First pproch to tc n ngulr velocity Define this skew-symmetric mtrix Ω b ω z ω y Ω b = [ ω b ] = ω z ω x ω y ω x Note Ω b = Ċ b [C b ]T Ċ b = Ω bc b is mens of fining erivtive of rottion mtrix provie we cn further unerstn Ω b..6 2

3 First pproch to tc n ngulr velocity Now for some insight into physicl mening of Ω b. Consier point p on rigi boy rotting with ngulr velocity ω = [ω x, ω y, ω z ] T = θ k = θ[k x, k y, k z ] T with k unit vector. ω θ k r p p.7 First pproch to tc n ngulr velocity ω θ k r p v p p From mechnics, liner velocity v p of point is v p = ω r p = ω x ω y r x r y = ω yr z ω z r y ω z r x ω x r z = ω z ω y ω z ω x r x r y ω z r z ω x r y ω y r x ω y } ω x {{ } r z?.8 First pproch to tc n ngulr velocity ω θ k r p v p p From mechnics, liner velocity v p of point is ω x r x ω y r z ω z r y 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 ω x r y ω z r z ω x r y ω y r x ω y } ω x {{ } r z Ω=[ ω ] Ω represents ngulr velocity n performs cross prouct.9 3

4 First pproch to tc n ngulr velocity Now let s fixe frme {} n rotting frme {b} ttche to moving boy such tht there is ngulr velocity ω b between them. ω b {} {b} r p r p n tke erivtive wrt time r p = Ċb Ω b C b r b bp + C b r b bp Strt with position p r p = r b +Cb r b bp = Ω bcb r b bp = Ω b r bp = [ ω b ] r bp from which it is observe compre to v p = ω r p tht Ω b represents cross prouct with ngulr velocity ω b Approch 2: Angle-xis Secon pproch to tc n ngulr velocity Another pproch to eveloping erivtive of rottion mtrix n ngulr velocity is bse upon ngle-xis representtion of orienttion n rottion mtrix s exponentil. This pproch is inclue in notes.. Secon pproch to tc n ngulr velocity Since the reltive n fixe xis rottions must be performe in prticulr orer, their erivtives re somewht chllenging The ngle-xis formt, however, is reily ifferentible s we cn encoe the 3 prmeters by K K t ktθt = K 2 t K 3 t where θ = K Hence, Kt t = K t K 2 t K 3 t.2 Secon pproch to tc n ngulr velocity For sufficiently smll time intervl we cn often consier the xis of rottion to be constnt i.e., kt = k Kt t kθt t = k θt This is referre to s the ngulr velocity ωt or the so clle boy reference ngulr velocity Angulr Velocity ωt k θt.3 4

5 Secon pproch to tc n ngulr velocity This efinition of the ngulr velocity cn lso be relte bck to the rottion mtrix. Reclling tht C b t = R k b,θt = eκ b θt Hence, t C b t = t eκ b θt = eκ b θt θ θ t = κ be κ b θt θt = κ θt b Cb t Ċ b t [C b t] T = κ b θt.4 Secon pproch to tc n ngulr velocity Notice tht κ θt b = Skew [kb] θt [ = Skew kb θt ] = Skew [ ω b] = Ω b Therefore, or Ċ b t [C b t] T = Ω b Ċ b = Ω bc b.5 Secon pproch to tc n ngulr velocity Note κ = k 3 k 2 k 3 k 2 = k 2 3 k 3 2 k 3 k 3 = k k 2 k 3 k 2 k 2 Hence, we cn think of the skew-symmetric mtrix s κ = [ k ] or, in the cse of ngulr velocity Ω = [ ω ].6 5

6 4 Properties of Skew-symmetric Mtrices Properties of Skew-symmetric Mtrices ] CΩC T b = C [ ω C T b = C ω CC T b = C ω b = [C ω ] b Therefore from bove, n vi istributive property CΩC T = C[ ω ]C T = [C ω ] C[ ω ] = [C ω ]C.7 Properties of Skew-symmetric Mtrices Ċb = Ω bcb = [ ω b ]Cb = [Cb ω b b ]Cb = Cb [ ω b b ] = Cb Ω b b Ċ b = Ω bc b = C b Ω b b.8 Summry of Angulr Velocity n Nottion Angulr velocity cn be escribe s vector the ngulr velocity of the b-frme wrt the -frme resolve in the c-frme, ω c b ω b = ω b escribe s skew-symmetric mtrix Ω c b = [ ω c b ] the skew-symmetric mtrix is equivlent to the vector cross prouct when premultiplying nother vector relte to the erivtive of the rottion mtrix Ċ b = Ω bc b = C b Ω b b Ċ b = Ω bc b = C b Ω b b.9 6

7 5 Propgtion/Aition of Angulr Velocity Propgtion/Aition of Angulr Velocity Consier the erivtive of the composition of rottions C 2 = C C 2. t C 2 = t C C 2 Ċ 2 = Ċ C 2 + C Ċ 2 Ω 2C2 = Ω CC 2 + CC 2Ω 2 2 Ω 2 = Ω C2 [ ] C T 2 + C 2 Ω 2 [ ] 2 C T 2 [ ω 2 ] = [ ω ] + [C 2 ω 2 2 ] ω 2 = ω + ω 2 ngulr velocities s vectors so long s resolve common coorinte system.2 6 Liner Position, Velocity n Accelertion Liner Position Consier the motion of fixe point origin of frme {2} in rotting frme frme {} s seen from n inertil frme {} frmes {} n {} hve the sme origin frme {} rottes bout unit vector k wrt frme {} origin of frme {2} is fixe wrt frme {} Position: r 2t = r t + r 2t = C t r 2.2 Liner Velocity Liner velocity: r 2t = t C t r 2 = Ċ t r 2 = [ ω ]C t r 2 = ω r 2t.22 Liner Accelertion Liner ccelertion: r 2 = t ω C t r 2 = ω Ct r 2 + ω Ċ t r 2 = ω r 2t + ω ω r 2t Trnsverse ccel Centripetl ccel ω 2 r.23 7

8 Liner Position We cn get bck to where we strte... frmes n their erivtives. z motion trnsltion n rottion between x y r r 2 z 2 r 2 z x 2 y 2 Trnsltion position between frmes {} n {}: r 2 = r + r 2 = r + C r 2.24 y x Liner Velocity Liner velocity: r 2t = t r + C r 2 = r + Ċ r 2 + C r 2 = r + [ ω ]C r 2 + C r 2 = r + ω C r 2 + C r 2.25 Liner Accelertion Liner ccelertion: r 2 = t r + ω C r 2 + C r 2 = r + ω C r 2 + ω Ċ r 2 + ω C r 2 + Ċ r 2 + C r 2 = r + ω r 2t + ω ω r 2t + 2 ω C r 2 + C r 2 ccel of {} s origin from {} in {} The En Trnsverse ccel Centripetl ccel ω 2 r Coriolis ccel 2ω v ccel of {2} s origin from {} in {}