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
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