3. Translational and Rotational Dynamics MAE 345 2017.Pptx

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Translational and Rotational Dynamics! Robert Stengel! Robotics and Intelligent Systems MAE 345, Princeton University, 2017

Reference Frame

•! Newtonian (Inertial) Frame of Reference –! Unaccelerated Cartesian frame •! Origin referenced to inertial (non-moving) frame –! Right-hand rule –! Origin can translate at constant linear velocity –! Frame cannot rotate with respect to inertial origin ! x $ # & •! Position: 3 dimensions r = # y & –! What is a non-moving frame? "# z %&

•! Translation = Linear motion 2 Velocity and Momentum of a Particle •! Velocity of a particle ! $ ! x! $ vx dr # & # & v = = r! = y! = # vy & dt # & # ! & # & " z % vz "# %& •! Linear momentum of a particle

! v $ # x & p = mv = m # vy & # & "# vz %& 3

Newtons Laws of Motion: ! Dynamics of a Particle

First Law . If no force acts on a particle, it remains at rest or continues to move in straight line at constant velocity, . Inertial reference frame . Momentum is conserved d ()mv = 0 ; mv = mv dt t1 t2

4 Newtons Laws of Motion: ! Dynamics of a Particle Second Law •! Particle acted upon by force •! Acceleration proportional to and in direction of force •! Inertial reference frame •! Ratio of force to acceleration is particle mass d dv dv 1 1 ()mv = m = ma = Force ! = Force = I Force dt dt dt m m 3 ! $ " % fx " % fx # & 1/ m 0 0 $ ' = $ ' f Force = # fy & = force vector $ 0 1/ m 0 '$ y ' $ ' # & #$ 0 0 1/ m &' f fz $ z ' "# %& # & 5

Newtons Laws of Motion: ! Dynamics of a Particle Third Law For every action, there is an equal and opposite reaction

Force on rocket motor = –Force on exhaust gas

FR = !FE

6 One-Degree-of-Freedom Example of Newtons Second Law 2nd-order, linear, time-invariant ordinary differential equation d 2 x(t) f (t) x ! "Defined as" 2 ! ""x(t) = v"x (t) = dt m Corresponding set of 1st-order equations (State-Space Model) dx ()t 1 ! x" (t) ! x (t) ! v (t) x (t) ! x(t), Displacement dt 1 2 x 1 dx()t dx2 ()t fx (t) x2 (t) ! , Rate ! ""x1(t) = x"2 (t) = v"x (t) = dt dt m 7

State-Space Model is a Set of 1st- Order Ordinary Differential Equations State, control, and output vectors for the example

! $ ! $ x1(t) x1(t) x(t) = # & ; u(t) = u() t = fx (t); y(t) = # & "# x2 (t) %& "# x2 (t) %& Stability and control-effect matrices

! 0 1 $ ! 0 $ F= # & ; G= # & " 0 0 % " 1 / m % Dynamic equation x!(t) = Fx(t) + Gu(t) 8 State-Space Model of the 1-DOF Example Output equation

y(t) = Hxx(t) + Huu(t) Output coefﬁcient matrices

! 1 0 $ ! 0 $ Hx = # & ; Hu = # & " 0 1 % " 0 %

Dynamic System

Dynamic Process: Current state Observation Process: Measurement may depend on prior state may contain error or be incomplete x : state dim = (n x 1) y : output (error-free) u : input dim = (m x 1) dim = (r x 1) w : disturbance dim = (s x 1) n : measurement error p : parameter dim = (l x 1) dim = (r x 1) z : measurement t : time (independent variable, 1 x 1) dim = (r x 1) 10 State-Space Model of Three- Degree-of-Freedom Dynamics ! $ ! x! $ vx # & # & ! x $ ! # & # y & # vy & y # & # & ! $ # z! & ! $ # & r!(t) # vz & r(t) z x! ()t " # & = # & = = Fx()t + Gu()t x()t ! # & = # & v!x # & vx # v!(t) & # & fx / m "# v(t) %& # & " % # & # & # ! & vy vy f m # & # y / & # & "# vz %& # v!z & # & " % fz / m " %

! x $ ! 0 0 0 1 0 0 $# & ! 0 0 0 $ # & y # & 0 0 0 0 1 0 # & 0 0 0 ! $ # &# & # & fx # 0 0 0 0 0 1 & z # 0 0 0 &# & = # & + # fy & # 0 0 0 0 0 0 &# vx & # 1/ m 0 0 & # & # &# & # & fz 0 0 0 0 0 0 vy 0 1/ m 0 "# %& # &# & # & "# 0 0 0 0 0 0 %&# & "# 0 0 1/ m %& " vz % 11

What Use are the Dynamic Equations? Compute time response Compute trajectories

Determine stability Identify modes of motion

12 Forces!

External Forces: Aerodynamic/Hydrodynamic

! $ ! C $ X # X & # & 1 2 faero = Y = # CY & 'V A # & 2 # Z & # & " % " CZ % ! = air density, functionof height ! C $ "#h # X & = !sealevele dimensionless # CY & = V = airspeed # & aerodynamic coefficients CZ 1/2 1/2 " % $ 2 2 2 & $ T & =%vx + vy + vz ' = %v v' Inertial frame or body frame? A = referencearea 14 External Forces: Friction

External Forces: Gravity

•! Flat-earth approximation –! g is gravitational acceleration –! mg is gravitational force –! Independent of position –! z measured up

! 0 $ # & = = 2 mg flat m # 0 &; go 9.807 m / s "# –go %&

16 External Forces: 1/2 Gravity r = ()x2 + y2 + z2 L = Latitude ! = Longitude

•! Spherical earth, inertial ! g $ # x & frame ground = # gy & = ggravity # & –! ”Inverse-square gravitation "# gz %& –! Non-linear function of position ! x $ ! x / r $ ! cos L cos( $ µ # & µ # & µ 14 # & µ = ' 3 y = ' 2 y / r = ' 2 cos Lsin( –! = 3.986 x 10 m/s r # & r # & r # & "# z %& "# z / r %& "# sin L %& •! Spherical earth, rotating

frame gr = ggravity + grotation –! ”Inverse-square gravitation " x % " x % µ $ ' $ ' –! Centripetal acceleration = ! + (2 3 $ y ' $ y ' –5 r ! ! = 7.29 x 10 rad/s #$ z &' #$ 0 &'

State-Space Model with Round-Earth Gravity Model (Non-Rotating Frame) ! x! $ ! x $ ! 0 0 0 $ # & ! 0 0 0 1 0 0 $# & # & y! # & y 0 0 0 # & 0 0 0 0 1 0 # & # &! $ # ! & # &# & # 0 0 0 & x z # 0 0 0 0 0 1 & z # & # & = # & ' 3 ! # µ / r 0 0 &# y & # vx & # 0 0 0 0 0 0 &# vx & # & # & 3 "# z %& # v!y & 0 0 0 0 0 0 # vy & # 0 µ / r 0 & # & # &# & # & # 0 0 0 0 0 0 & 3 # v!z & " %# vz & # 0 0 µ / r & " % " % " %

" 0 0 0 1 0 0 %" x % $ '$ ' $ 0 0 0 0 1 0 '$ y ' $ 0 0 0 0 0 1 '$ z ' = 3 $ ' $ !µ / r 0 0 0 0 0 ' $ '$ vx ' 3 $ 0 !µ / r 0 0 0 0 '$ vy ' $ '$ ' !µ 3 #$ 0 0 / r 0 0 0 &'#$ vz &' Inverse-square gravity model introduces nonlinearity 18 Vector-Matrix Form of Round-Earth Dynamic Model ! $ 0 I3 ! r! $ # & ! r $ # & = # 'µ & # & v! v " % # 3 I3 0 & " % " r %

What other forces might be considered, and where would they appear in the model?

Point-Mass Motions of Spacecraft

•! For short distance and low speed, ﬂat-Earth frame of reference and gravity are sufﬁcient •! For long distance and high speed, round-Earth frame and inverse-square gravity are needed 20 Mass of an Object

xmax ymax zmax m = ! dm = ! ! ! "(x, y,z)dx dydz Body xmin ymin zmin "(x, y,z) = density of the body

Density of object may vary with (x,y,z) 21

More Forces!

22 External Forces: Linear Springs

Scalar, linear spring

f = !k() x ! xo = !k"x; k = springconstant

Uncoupled, linear vector spring " % " % kx ()x ! xo " % $ ' kx 0 0 (x $ ' $ ' $ ' $ ' fS = ! ky ()y ! yo = ! 0 ky 0 $ (y ' $ ' $ ' $ ( ' $ k z ! z ' $ 0 0 kz ' # z & # z ()o & # &

()xo, yo,zo : Equilibrium (reference) position 23

External Forces: Viscous Dampers

Scalar, linear damper f = !d() v ! vo = !d"v; d = dampingconstant Uncoupled, linear vector damper " % dx vx ! vx " % " % $ ()o ' dx 0 0 vx $ ' $ ' $ ' = ! ! (! $ 0 dy 0 ' $ vy ' fD $ dy ()vy vyo ' $ ' $ ' $ ' 0 0 dz vz $ dz vz ! vz ' #$ &' #$ &' # ()o &

()vxo ,vyo ,vzo : Equilibrium (reference) velocity [usually = 0] 24 State-Space Model with Linear Spring and Damping

! 0 0 0 0 0 0 $ # &! x ' x $ ! x! $ ! x $ # 0 0 0 0 0 0 &# ()o & # & ! 0 0 0 1 0 0 $# & ! # 0 0 0 0 0 0 &# ' & # y & # &# y & ()y yo # 0 0 0 0 1 0 & # 'k 'd &# & # z! & # z & # x 0 0 x 0 0 &# ' & # & # 0 0 0 0 0 1 &# & ()z zo ! = + # m m &# & # vx & # 0 0 0 0 0 0 &# vx & # & # 'k 'd &# vx & # ! & # & y y vy # 0 0 0 0 0 0 & vy # 0 0 0 0 &# & # & # & m m vy # 0 0 0 0 0 0 & # &# & # v!z & " %# vz & " % " % # 'kz 'dz &# v & # 0 0 0 0 &" z % " m m %

Spring Damping Effects Effects

State-Space Model with Linear Spring and Damping

Stability Effects Reference Effects

! 0 0 0 1 0 0 $ ! 0 0 0 1 0 0 $ # & # & ! x! $ # 0 0 0 0 1 0 &! x $ # 0 0 0 0 1 0 &! $ # & # & xo y! # 0 0 0 0 0 1 & y # 0 0 0 0 0 1 &# & # & # & # y & # 'k 'd & # k d & o # z! & # x 0 0 x 0 0 &# z & # x 0 0 x 0 0 &# & # & # & zo ! = # m m & + # m m &# & # vx & # vx & # 'k 'd & # k d &# 0 & # ! & y y # & y y # & vy # 0 0 0 0 & vy # 0 0 0 0 & 0 # & # m m &# & # m m &# & # v!z & # vz & # 0 & " % # 'kz 'dz &" % # kz dz &" % # 0 0 0 0 & # 0 0 0 0 & " m m % " m m %

26 Vector-Matrix Form

! $ ! $ ! $ r! 0 I3 ()r ' ro # & = # & # & " v! % # 'K / m 'D / m & # v & " % " %

! $ ! $ ! $ ! $ ! r! $ 0 I3 r 0 'I3 ro # & = # & # & + # & # & v! # 'K / m 'D / m & v # K / m D / m & # 0 & " % " % " % " % " %

Rotational Motion!

28 Assignment # 2

Document the physical characteristics and ﬂight behavior of a Syma X11 quadcopter!

https://www.youtube.com/watch?v=EwO6U7DbqSo

https://www.youtube.com/watch?v=kyiuy2CHzj0 29

Center of Mass

! $ xcm # & 1 ! = rcm # ycm & ' r dm # & m Body " zcm %

x y z ! x $ max max max # & = ' ' ' # y & ((x, y,z)dx dydz x y z min min min "# z %&

Reference point for rotational motion 30 Particle in Inverse- Square Field

Angular Momentum of a Particle

•! Moment of linear momentum of differential particles that make up the body –! Differential mass of a particle times –! Component of velocity perpendicular to moment arm from center of rotation to particle dh = ()r ! dmv = ()r ! v dm 31

Cross Product of Two Vectors

i j k r ! v = x y z = ()yvz " zvy i + ()zvx " xvz j + ()xvy " yvx k

vx vy vz ()i, j, k : Unit vectors along ()x, y,z

This is equivalent to " % yv ! zv " % $ ()z y ' " 0 !z y % vx $ ' $ ' = $ ! ' = ! v = ! $ ()zvx xvz ' $ z 0 x ' $ y ' rv $ ! ' $ ' $ ' y x 0 vz xvy ! yvx # & #$ &' #$ () &' 32 Cross-Product-Equivalent Matrix

r ! v = rv!

Cross-product equivalent of radius vector Velocity vector # & ! $ 0 "z y vx % ( # & r ! ! r" = % z 0 "x ( v = # vy & # & % "y x 0 ( v $ ' "# z %& 33

Angular Momentum of an Object

Integrate moment of linear momentum of differential particles over the body

xmax ymax zmax h ! " ()r ! v dm = " " " ()r ! v #(x, y,z)dx dydz Body xmin ymin zmin # & hx xmax ymax zmax % ( = ! ! " " " ()rv (x, y,z)dx dydz " % hy ( xmin ymin zmin % ( % hz ( $ ' 34 Angular Velocity and Corresponding Velocity Increment " % Angular velocity of object ! x with respect to inertial $ ' frame of reference ! = $ ! y ' $ ' ! z #$ &'I

" !v % Linear velocity $ x ' increment at a point, ! = ! = ( ) = * )(( (x,y,z), due to v()x, y,z $ vy ' r ()r $ ' angular rotation ! #$ vz &'

Angular Momentum of an Object with Respect to Its Center of Mass •! Choose center of mass as origin about which angular momentum is calculated (= center of rotation) –! i.e., r is measured from the center of mass = # ! + " % = ! + !" h ' $r ()vcm v &dm ' []r vcm dm ' []r v dm Body Body Body

By symmetry, = []rdm ! vcm # &%r ! ()r ! $ ('dm " " ! = ! = Body Body " []rdm vcm rcm vcm 0 Body $ ' = ! # r! r! " dm = &! # r! r! dm) " " I" Body & Body ) I ! Inertia Matrix % ( 36 Angular Momentum

h = I!

Three components of angular momentum ! $ ! $ ! $ h I xx 'I xy 'I xz ) # x & ! $ # & # x & = ' ! ! ) # ' ' & ) # hy & # ( r r dm& " I xy I yy I yz # y & # & # Body & # & # & " % ' ' ) # hz & # I xz I yz Izz & # z & " % " % " %

Inertia Matrix, I! Angular rate has equal effect on all particles

# 0 !z y & # 0 !z y & % ( % ( I = ! " r! r! dm = ! " % z 0 !x ( % z 0 !x ( dm Body Body % !y x 0 ( % !y x 0 ( $ ' $ '

" (y2 + z2 ) !xy !xz % $ ' = $ !xy (x2 + z2 ) !yz 'dm ( $ ' Body ! ! 2 + 2 #$ xz yz (x y ) &'

38 Inertia Matrix

2 2 " % " (y + z ) !xy !xz % I !I !I $ ' $ xx xy xz ' I = $ !xy (x2 + z2 ) !yz 'dm = $ !I I !I ' ( $ xy yy yz ' Body $ 2 2 ' $ !xz !yz (x + y ) ' $ !I xz !I yz Izz ' # & # & –! Moments of inertia on the diagonal –! Products of inertia off the diagonal –! If products of inertia are zero, (x, y, z) are principal axes, and

! I 0 0 $ # xx & I = # 0 I yy 0 & # & "# 0 0 Izz %& 39

Angular Momentum and Rate are Vectors

•! Can be expressed in either an inertial or a body frame •! Vectors are transformed by the rotation matrix and its inverse

B B I I hB = IB! B = HI hI = HI II ! I hI = II ! I = HBhB = HBIB! B ! = HB! ! = HI ! B I I I B B

40 Similarity Transformations for Matrices Alternative expressions for angular momentum

B B hB = IB! B = HI hI = HI II ! I = HB HI ! I II B B Inertia matrices are transformed from one frame to the other by similarity transformations = B I = I B IB HI II HB II HBIBHI 41

Newtons 2nd Law Applied to Rotational Motion in Inertial Frame Rate of change of angular momentum = applied moment (or torque), m Inertia Matrix Expressed in Inertial Frame is Not Constant if Body is Rotating

dhI d()II ! I dII d! I = = ! I + II = mI [moment vector]I dt dt dt dt In a body reference frame, inertia matrix is constant

dh d()IB! B dI d! d! B = = B ! + I B = ()0 ! + I B dt dt dt B B dt B B dt

d! B = IB = mB [moment vector]B dt 42 How do We get Rid of d(II )/dt in the Angular Momentum Rate Equation?

•! Write the dynamic equation in body- referenced frame • ! With constant mass, I inertial properties are unchanging in body reference frame •! ... but the frame is non-Newtonian or non-inertial

Vector Derivative Expressed in a Rotating Frame Chain Rule

I dhI I dhB dHB I dhB I I dhB I = H + h = HB + ()! I " HB hB = HB + ! I " ()HBhB dt B dt dt B dt dt

dhI I dhB I dhB = HB + ! I " hI = HB + !! I hI dt dt dt

Cross-product-equivalent Similarity matrix of angular rate: transformation

# 0 "! ! & % z y ( ! I B ! = % ! z 0 "! x ( !! = H !! H % ( I B B I "! ! % y x 0 ( $ ' 44 Rate of Change of Body-Axis Angular Momentum Substitute

dhI I dhB I B I = HB + HB!! BHI hI = mI = HBmB dt dt Eliminate

I dhB I B I HB + HB!! BHI hI = HBmB dt Rearrange and Substitute

dhB = mB ! "! BhB dt 45

Rate of Change of Body-Axis Angular Velocity

d! I B = m " !! I ! B dt B B B B

d! B "1 = IB (mB " !! BIB! B ) dt

46 Rate of Change of Body-Axis Translational Velocity

I Inertial Velocity v I = HBv B

I dv dv d()HB dv I = HI B + v = HI B + !! v dt B dt dt B B dt I I dv dv Rate of change = HI B + HI !! HBv = HI B + HI !! v B dt B B I I B dt B B B

1 1 I = fI = HBfB m m

I dv B I 1 I Substitution HB + HB!! Bv B = HBfB dt m

dv B 1 Result = fB ! "! Bv B dt m 47

Rate of Change of Translational Position

Express derivative in inertial frame ! = = I rI v I HBv B

48 Rate of Change of Angular Orientation (Euler Angles)

" % ! " p % •! Body-axis angular rate $ x ' $ ' vector components ! B = $ ! y ' ! $ q ' $ ' are orthogonal $ r ' $ ! z ' # & # &B % " ( •! Euler angles form a ' * non-orthogonal ! ! ' # * vector ' $ * & ) % ( % "! ( , x •! Therefore, Euler- d! ' * ' * = ! + , + , angle rate vector is ' # * ' y * I dt ' * ' $! * , not orthogonal & ) ' z * & )I 49

Transformation From Euler-Angle Rates to Body-Axis Rates

!˙ is measured in the Inertial Frame !˙ is measured in Intermediate Frame #1 !˙ is measured in Intermediate Frame #2

50 Sequential Transformations from Euler-Angle Rates to Body-Axis Rates !˙ is measured in the Inertial Frame !˙ is measured in Intermediate Frame #1 ! ˙ is measured in Intermediate Frame #2 ! $ ! ! $ p ' ! 0 $ ! 0 $ # & # & # & B ! B 2 # & # q & = I3 # 0 & + H2 ( + H2 H1 0 # & # & # & # & # & # )! & r 0 " 0 % " % " % "# %& ... which is ! $ ! ! $ p ! 1 0 'sin( $ ) # & # & # & = ) ) ( # (! & B+! # q & # 0 cos sin cos & " LI # & # *! & " r % # 0 'sin) cos) cos( & " % " % 51

Inversion to Transform Body-Axis Rates to Euler-Angle Rates Transformation is not orthonormal $ 1 0 !sin" ' & ) B # # " LI = & 0 cos sin cos ) & ! # # " ) % 0 sin cos cos ( Inverse transformation is not the transpose !1 T B ! I " B ()LI LB ()LI !1 $ ' 1 0 !sin" B !1 & ) Adj()LI B = # # " = ()LI & 0 cos sin cos ) B det L & 0 !sin# cos# cos" ) ()I % ( 52 Euler-Angle Rates !1 $ ' $ ! " ! " ' B 1 0 !sin" 1 sin tan cos tan !1 Adj()LI & ) & ) B = = # # " = 0 cos! #sin! ()LI B & 0 cos sin cos ) & ) L det()I & ! # # " ) & ! " ! " ) % 0 sin cos cos ( % 0 sin sec cos sec (

Euler-angle rates from body-axis rates $ ' !! $ 1 sin! tan" cos! tan" ' $ p ' & ) & ) & ) ! I & " ) = & 0 cos! *sin! ) & q ) = LB+ B & #! ) & ) & ) % ( 0 sin! sec" cos! sec" % r ( % (

Can the inversion become singular? What does this mean?

Summary of Six-Degree-of-Freedom (Rigid Body) Equations of Motion Rigid-body dynamic equations are nonlinear Translational position I and velocity r!I = HBv B ! x $ ! u $ # & # & = = 1 rI # y &; v B # v & v! = f ! "" v # z & "# w %& B m B B B " % Rotational position ! I # = LB" B and velocity !1 % " ( % p ( ! " ' * ' * " B = IB (mB ! " BIB" B ) ! = # + = ' *; B ' q * ' $ * ' * & ) & r ) 54 Next Time:! Flying and Swimming Robots!

!"##$%&%'()$** +)(%,-)$*

56 Moments and Products of Inertia for Common Constant-Density Objects are Tabulated

Moments and Products of Inertia (Bedford & Fowler)

58 " ! ! % $ I xx I xy I xz ' I = $ !I I !I ' Construction of $ xy yy yz ' $ !I xz !I yz Izz ' Inertia Matrix # & Build up moments and products of inertia from components using parallel-axis theorem, e.g., = + + + + Ixxairplane Ixxwings Ixx fuselage Ixxhorizontal tail Ixxvertical tail ...

… or use software, e.g., Autodesk, Creo Parametric, SimMechanics, …