剛體轉動牛頓尤拉方程 講義

OMG!

(ex.)

(ex.)

A:

B: Lagrange()

C:

Python

DEMO

nutationprecession

1. xyzbodyspace?xyzbodyBody:bodyxyz?

?

=

+

L

L=I

?non-inertialb (body)inertials (space)spacessspacebody??

sspace framexyzbodysspace?

body framebody?()?

(strap-down angular rate sensor)? body frame???! ???

bodyspace?

b?

b???

bs?

?

- ?

=

A Rodrigues rotation formulaCayley-Klein parameters

body framespace framesbody framespace frame

sdt=dsspacebody

sb frames frame

sdt=dssb

s

body xyzLIdiagonals frame b frameti -> ti+1

ti

ti+1

tibzSzti+1bz

Body xyz=

=

+

ti

ti+1

sb frameti+1 , ti+2s frame

b(t0)xyz(t0)

(1)=CK(b(t0)dt) (0)

ti+1

ti+2

ti+1

ti+2

lab frame

sspaces frameti!!bt+dtst

Sz (t1)b(t1)

ii+1

lab frame

(direction cosine matrix)

(ps. )

Lagrange

? pythonLagrange!

B Lagrange Equation

A Recursive Newton-Euler Equation

C

Python3D

B method

LagraneEuler Anglesreduce

Demo Lagranges method

BProf. HasbunClassical Mechanics with Matlab Applications matlab code.

Show top.py file.

!! 0!!!

Matlab

ODE solver

Demo Lagranges method

Run file Gyroscope-TeachDemo-1.py

xyzomegaL

ARecursive NE equation

ARecursive NE equation

EulerDCMiter()

ODE

Lagrange

ARecursive NE equation

Demo ring motion (5,8,0)wave motion (5,0,-4)

Demo space cone body cone, file Gyro_spaceBodyCone.py

body coneconezwwL

Demo file Gyroscope-TeachDemo-2-circular.py

ADEMO-1

A?

(!)

090sin cos!!!!!

?

AB

ABNEvs Lagrange

Run file Gyroscope-TeachDemo-3.pyDrawOption?

580,50-4,400

numerical drift

bodylab

A

BLagrange(t)(t)(t)

Bb(t)cBc

B

EulerW2bodyW()

B

A? C!!

directDCMiter()

? AC-AC

DEMO-3

BCLagrangevs

Run file Gyroscope-TeachDemo-4.py

A

PCICgyro sensorPortable device

C

Lagrange

GPSfusion

(Kalman filter)

?

IncludeNoiseInOmega(1)

Show and run Gyroscope-TeachDemo-5.py

1 ~ -1 rad/sec 1 rad/sec = 57 deg / sec

2000Hz-> (~MHz)

J-cycle

2014/1 6/MPU6050

0.01

:

http://blog.sina.com.cn/zhaowenwin

bodybody

(tsb frame)body

=

body xyzbody xyz

NE

2

gyro sensor

http://whymranderson.blogspot.tw/2014/03/4th-runge-kutta.html

!

?