Accelerometers and gyroscopes (Inertial Sensors) · 2018. 12. 9. · Problems: 1°) An accelerometer is sensing ax = 2m.s-2 and az = -9.8m.s-2. The corresponding gyroscope is sensing

Accelerometers and gyroscopes (Inertial Sensors)

I) Introduction

2

What did you need with LEGO robot to solve your challenge ?

Why acceleration ? Easy to imagine a mechanical system

https://en.wikipedia.org/wiki/Centrifugal_governor

Examples

Why angular velocity ? Easy to imagine a mechanical system

Inverted Pendulum explanations (duration 6:24)https://www.youtube.com/watch?v=OB3ufWYpj-IClassic Inverted Pendulum – Equations of Motionhttps://www.youtube.com/watch?v=5qJY-ZaKSicSelf Balancing Stick - Dual Axis Reaction Wheel Inverted Pendulumhttps://www.youtube.com/watch?v=woCdjbsjbPg

And also pedometers and shock detectors

II) Accelerometers

Youtube : MPU-6050 data with complementary filterhttps://www.youtube.com/watch?v=qmd6CVrlHOM

Questions : is the accelerometer sensitive to gravitation ?How many directions for this accelerometer (DOF) ?What happen to this accelerometer in a free fall ?

g

z

x

Dans cette situation :Gz = -gGx = 0

g

z

x

Dans cette situation :Gz = -g.cos()Gx = -g.sin()

g

z

x

Dans cette situation :Gz = -g.cos(-)=-g.cos()Gx = -g.sin(-)=g.sin()

First application : Tilt sensors (or Inclinometers)Needs a 0Hz low cutoff frequency accelerometer

0-g measurments1-g measurments

AN-1057

3

(AN3397)

4

Second application : 1D-Trajectory computing

(AN3397)

Whouh : where is the mistake ?

This works properly. Is it always the case ? No !

What happen during a 2D circular motion ? Answer during the lecture.

Accéléromètres en français :* F. Ferrero : users.polytech.unice.fr/~ferrero/TPelec2/arduino2.pdf

g

x

z

ygz

gy

gx

The attitude problem with 3D accelerometer

III) GyroscopesGyroscope is invented by Léon Foucault in 1852

https://www.youtube.com/watch?v=ho2P3KHhEMQ

Physics - Mechanics: The Gyroscope (2 of 5) The Torque of a Non-Spinning Gyroscope

Gyroscopes physics (Lecture by Walter Lewin)https://www.youtube.com/watch?v=XPUuF_dECVI14:30 Moment cinétique22:00 Expérience surprenante avec roue de vélo39:20 La même avec petit gyroscope44:00 Gros gyroscope

Physics - Mechanics: The Gyroscope (3 of 5) The Torque of a Spinning Gyroscope

https://www.youtube.com/watch?v=qS_dcNqs3d4

Thinking with cross product : the right-hand rule

τ= r×F L= r×m⋅v=I⋅ω τ=d Ldt

(Linear) Momentum : quantité de mouvementAngular momentum : moment cinétiqueTorque : coupleMoment of inertia : moment d'inertieAngular velocity : vitesse de rotationAngular acceleration is denoted in English books lectures...Precession : précession

https://www.youtube.com/watch?v=1KmhTfhaWG8

Gyrocar #1 (gyroscope stabilized 2-wheeled toy)

If the gyroscope stabilizes the 2-whelled toy above, how would you install it on the self balancing bot below.

First gyroscope application : stabilization

MEMS

MEMS :https://www.youtube.com/watch?v=eqZgxR6eRjoAnimation Gyroscope 1:26

IV Inertial sensors properties

Why accalerometer data are not enough for 2D and 3D problem ?

http://www.geekmomprojects.com/gyroscopes-and-accelerometers-on-a-chip/

What are the other difficulties with 3D rotations ?3D rotations are not commutative !

V Inertial navigation

Problems :1°) An accelerometer is sensing ax = 2m.s-2 and az = -9.8m.s-2.

The corresponding gyroscope is sensing z = 0.2 radian.s-1.What can you say about the trajectory, the velocity and so on ?

2°) A robot is following the below sine trajectory. The velocity vx is perfectly constant vx = 1m.s-1.Can you calculate what is sensing the accelerometer in function of time when fixed on a stable platform ?Can you calculate what is sensing the gyroscope in function of time ?Can you deduce what is sensing the accelerometer when fixed directly on the robot ?

x0

y

Lx=10m

Ly=6m

VI) Application : Self Balancing Robot

xi

j

Cart Free Body Diagram

FF

T P=−mc⋅g⋅ j

N

T=−T sin (θ)⋅i +T cos(θ)⋅jF=F⋅i

FF

then

a p= ac+ ap / c= x i +L θ [−cos (θ) i −sin(θ) j ]−L θ ²[−sin(θ) i+cos (θ) j ]

[1 ] ⇒ T⋅sin (θ)=m p x−mp L θ⋅cos (θ)+mp L θ ² sin(θ) [3]

[2] ⇒ −T⋅cos (θ)−m p⋅g=−mp Lθ⋅sin (θ)−mp L θ ² cos (θ) [4 ]

Cancel out T with [3]⋅cos (θ)+[4 ]⋅sin(θ)

−m p⋅g⋅sin (θ)=m p x⋅cos (θ)−mp L θ [5 ]

Cancel out T cos() avec [0] + [3]

F+m p Lθ⋅cos(θ)−mpL θ ² sin(θ)=(m p+m c) x [6]

Pendulum Free Body Diagram

−TP=−mp⋅g⋅j

−T =T sin (θ)⋅ i−T cos (θ)⋅ j

Newton 2nd LawCarting

i : F−T sin (θ)=mc x [0]

j : not interesting :no move

pendulumi : T sin(θ)=m papx [1]

j : −T cos (θ)−m p⋅g=m papy [2]

FFF

er

eθ

a p= ac+ ap / c= x i +[L θ eθ−L θ ² er ]

Mouvement circulaire

Mathematical Modeling