1. Inertial Measurement Unit (IMU) Basics

2. IMU (Inertial Measurement Unit)
Accelerometer
Gyroscope
Magnetometer (Compass)
Acceleration along 3 axes
Rotation speed around 3 axes
Direction of magnetic north
{ 𝑑 2 𝑥 𝑑 𝑡 2 ,
𝑑 2 𝑦 𝑑 𝑡 2 ,
𝑑 2 𝑧 𝑑 𝑡 2 }
{ 𝑑 𝜃 𝑥 𝑑𝑡 , 𝑑 𝜃 𝑦 𝑑𝑡 , 𝑑 𝜃 𝑧 𝑑𝑡 }
{ 𝑚 𝑥 , 𝑚 𝑦 , 𝑚 𝑧 }
Popular since they are inexpensive, small, and power efficient
Can be embedded inside any object to enable intelligence

3. Accelerometer

4. Basic model (1D)
Accelerometer measures 𝐹 𝑘 𝑚
𝑚𝑎 𝑎𝑐𝑐𝑙
𝐹 𝑘 =𝑚𝑔
𝐴 𝑚𝑒𝑎𝑠𝑢𝑟𝑒𝑑 = 𝐹 𝑘 𝑚 = 𝑔
Accelerometer output under rest is non-zero. It measures g
𝑚𝑎 𝑎𝑐𝑐𝑙 = 𝐹 𝑘 −𝑚𝑔
𝐴 𝑚𝑒𝑎𝑠𝑢𝑟𝑒𝑑 = 𝐹 𝑘 𝑚 =𝑔+ 𝑎 𝑎𝑐𝑐𝑙
Accelerometer output under motion
Accelerometers measure sum of acceleration due to motion and gravity
Smartphone accelerometers are tri-axial – can measure 3D acceleration

5. Accelerometer under free fall

6. Accelerometer measures ZERO in free fall
G = 9.8 m/s2

7. Measuring linear motion (1D)
𝐴 𝑚𝑒𝑎𝑠𝑢𝑟𝑒𝑑 =𝑔+ 𝑎 𝑎𝑐𝑐𝑙
𝑎 𝑎𝑐𝑐𝑙 = 𝐴 𝑚𝑒𝑎𝑠 − 𝑔
Need to subtract gravity to obtain acceleration due to motion
𝑑 2 𝑥 𝑑 𝑡 2 = A meas −g
𝑑 2 𝑥 𝑑 𝑡 2 = A meas −g + n
Hardware noise
𝑑𝑥 𝑑𝑡 = (𝐴 −𝑔+ 𝑛 ) 𝑑𝑡= 𝐴 −𝑔 𝑑𝑡+ 𝑛 𝑑𝑡
𝑥= 0 𝑡 𝐴−𝑔 𝑑𝑡+ 0 𝑡 𝑛 𝑑𝑡
Error accumulates dramatically with time

8. Subtraction of gravity non-trivial in 2D or 3D

9. Subtraction of gravity in 2D
𝑎 𝑥
𝑎 𝑦 y
1D equation → 𝐴 𝑚𝑒𝑎𝑠 = 𝑎 𝑎𝑐𝑐𝑙 + 𝑔
2D equations (y axis pointing upwards) 𝜃 x
𝐴 𝑥 = 𝑎 𝑥 g
𝐴 𝑦 = 𝑎 𝑦 + 𝑔

10. Subtraction of gravity in 2D
𝑎 𝑥
𝑎 𝑦
1D equation → 𝐴 𝑚𝑒𝑎𝑠 = 𝑎 𝑎𝑐𝑐𝑙 + 𝑔 y x
2D equations (y axis pointing upwards) 𝜃
𝐴 𝑥 = 𝑎 𝑥 g
𝐴 𝑦 = 𝑎 𝑦 + 𝑔
𝐴 𝑥 = 𝑎 𝑥 + 𝑔𝑠𝑖𝑛𝜃
𝐴 𝑦 = 𝑎 𝑦 + 𝑔𝑐𝑜𝑠𝜃
2D equations (arbitrary rotation of phone)
We need to know the the orientation to subtract gravity 𝜃
Inaccurate orientation will not eliminate gravity, resulting in errors (which accumulate over time)

11. Accelerometer is an orientation sensor!!
An interesting idea
𝐴 𝑥 = 𝑎 𝑥 + 𝑔𝑠𝑖𝑛𝜃
𝐴 𝑦 = 𝑎 𝑦 + 𝑔𝑐𝑜𝑠𝜃
2D equations (arbitrary rotation of phone)
Suppose the phone is at rest, then
𝑎 𝑥 =0, 𝑎 𝑦 =0
𝐴 𝑥 = 𝑔𝑠𝑖𝑛𝜃
𝑠𝑖𝑛𝜃 = 𝑔 𝐴 𝑥
Accelerometer is an orientation sensor!!
Need the accelerometer to be at rest to estimate rotation
However accelerometer alone is not sufficient if we need 3D orientation, more later ..

12. Accelerometer to measure distance
𝑥= 0 𝑡 𝐴−𝑔 𝑑𝑡+ 0 𝑡 𝑛 𝑑𝑡
Double integration fails dramatically –
However, accelerometer is good in tracking steps
Steps
Distance = step_count * step_size
Combining distance estimates with compass directions, we can dead reckon

13. Accelerometer Measures gravity + linear acceleration
When static, gravity measurement can be used to sense orientation
Double integrating accelerometer will accumulate error dramatically, step counting is reasonable

14. Magnetometer

15. Measures magnetic north (2D example)y x
Consider a 2D example
𝑀 𝑦 = 𝑀
𝑀 𝑥 =0

16. Measures magnetic north (2D example)y
Suppose, phone rotates in 2D by an angle
𝑀 𝑦 = 𝑀𝑐𝑜𝑠𝜃
𝑀 𝑥 =𝑀 𝑠𝑖𝑛𝜃 𝜃 x
The same concept generalizes to 3D,
3D magnetometer output { 𝑀 𝑥 , 𝑀 𝑦 , 𝑀 𝑧 } depends on phone orientation
Magnetometer can be used to sense the orientation
𝑠𝑖𝑛𝜃= 𝑀 𝑥 𝑀
However, in 3D magnetometer alone is insufficient to determine the orientation

17. 3D orientation

18. Foundations of 3D Orientation
East
North Up X Y Z
Local Frame
Global Frame

19. Consider a phone in random orientation
East
North Up X Y Z
3D Orientation captures the misalignment between global and local frames

20. 3x3 Rotation matrix captures the full 3D orientation
In global frame
In local frame 𝑉
East
North Up X Y Z
𝑥 𝑙 𝑦 𝑙 𝑧 𝑙 = ⋯ ⋮ ⋱ ⋮ 𝑥 𝑔 𝑦 𝑔 𝑧 𝑔
Rotation Matrix R
R is the 3x3 Rotation Matrix
3x3 Rotation matrix captures the full 3D orientation

21. How can we estimate rotation matrix?
Key idea  use globally known reference vectors
which can also be measured in the local frame of reference
Gravity
Magnetic North

22. Gravity equation
Gravity globally known, measurable in local frame with accelerometer
𝑎 𝑥 𝑎 𝑦 𝑎 𝑧 = ⋯ ⋮ ⋱ ⋮ −𝑔
Rotation Matrix R
𝑚 𝑥 𝑚 𝑦 𝑚 𝑧 = ⋯ ⋮ ⋱ ⋮ 𝑀 0
Rotation Matrix R
Magnetic north, globally known, measurable in local frame with magnetometer
6 equations and 9 unknowns (3x3 rotation matrix) can we solve ?
Yes, these 9 unknowns are all not independent (rotation matrix satisfies special properties)
It does not change length of a vector
Columns are orthogonal unit vectors
The above 6 equations are sufficient to solve the rotation matrix
Accelerometer and Magnetometer can be used to determine the rotation matrix (3D orientation)

23. Decomposing the rotation matrix
yaw
pitch X Y Z
roll
3x3 Rotation Matrix R
cos⁡(𝑝𝑖𝑡𝑐ℎ) 0 −sin⁡(𝑝𝑖𝑡𝑐ℎ) sin⁡(𝑝𝑖𝑡𝑐ℎ) 0 cos⁡(𝑝𝑖𝑡𝑐ℎ)
cos⁡(𝑟𝑜𝑙𝑙) sin⁡(𝑟𝑜𝑙𝑙) 0 −sin⁡(𝑟𝑜𝑙𝑙) 𝑐𝑜𝑠⁡(𝑟𝑜𝑙𝑙)
cos⁡(𝑦𝑎𝑤) −sin⁡(𝑦𝑎𝑤) 0 sin⁡(𝑦𝑎𝑤) cos⁡(𝑦𝑎𝑤) =
Orientation can be represented as 3D yaw, pitch, roll
Estimating yaw, pitch, roll will determine the orientation

24. Accelerometer equation
𝑎 𝑥 𝑎 𝑦 𝑎 𝑧 = ⋯ ⋮ ⋱ ⋮ 𝑔
Rotation Matrix R
𝑎 𝑥 𝑎 𝑦 𝑎 𝑧
cos⁡(𝑟𝑜𝑙𝑙) sin⁡(𝑟𝑜𝑙𝑙) 0 −sin⁡(𝑟𝑜𝑙𝑙) 𝑐𝑜𝑠⁡(𝑟𝑜𝑙𝑙)
cos⁡(𝑦𝑎𝑤) −sin⁡(𝑦𝑎𝑤) 0 sin⁡(𝑦𝑎𝑤) cos⁡(𝑦𝑎𝑤)
0 0 −𝑔
cos⁡(𝑝𝑖𝑡𝑐ℎ) 0 −sin⁡(𝑝𝑖𝑡𝑐ℎ) sin⁡(𝑝𝑖𝑡𝑐ℎ) 0 cos⁡(𝑝𝑖𝑡𝑐ℎ) =

25. Accelerometer equation
𝑎 𝑥 𝑎 𝑦 𝑎 𝑧 = ⋯ ⋮ ⋱ ⋮ 𝑔
Rotation Matrix R
𝑎 𝑥 𝑎 𝑦 𝑎 𝑧
cos⁡(𝑟𝑜𝑙𝑙) sin⁡(𝑟𝑜𝑙𝑙) 0 −sin⁡(𝑟𝑜𝑙𝑙) 𝑐𝑜𝑠⁡(𝑟𝑜𝑙𝑙)
0 0 −𝑔
cos⁡(𝑝𝑖𝑡𝑐ℎ) 0 −sin⁡(𝑝𝑖𝑡𝑐ℎ) sin⁡(𝑝𝑖𝑡𝑐ℎ) 0 cos⁡(𝑝𝑖𝑡𝑐ℎ) =
Accelerometer output does not depend on yaw!
Hence, yaw cannot be estimated using accelerometer

26. Accelerometer equation
𝑎 𝑥 𝑎 𝑦 𝑎 𝑧 = ⋯ ⋮ ⋱ ⋮ −𝑔
Rotation Matrix R
𝑎 𝑥 𝑎 𝑦 𝑎 𝑧
−sin 𝑝𝑖𝑡𝑐ℎ . cos 𝑟𝑜𝑙𝑙 .𝑔 − sin 𝑟𝑜𝑙𝑙 .𝑔 − cos 𝑝𝑖𝑡𝑐ℎ . cos 𝑟𝑜𝑙𝑙 .𝑔 =
The above equations estimate pitch and roll

27. Magnetometer equation
𝑚 𝑥 𝑚 𝑦 𝑚 𝑧 = ⋯ ⋮ ⋱ ⋮ 𝑀 0
Rotation Matrix R
𝑚 𝑥 𝑚 𝑦 𝑚 𝑧
cos⁡(𝑟𝑜𝑙𝑙) sin⁡(𝑟𝑜𝑙𝑙) 0 −sin⁡(𝑟𝑜𝑙𝑙) 𝑐𝑜𝑠⁡(𝑟𝑜𝑙𝑙)
cos⁡(𝑦𝑎𝑤) −sin⁡(𝑦𝑎𝑤) 0 sin⁡(𝑦𝑎𝑤) cos⁡(𝑦𝑎𝑤)
0 𝑀 0
cos⁡(𝑝𝑖𝑡𝑐ℎ) 0 −sin⁡(𝑝𝑖𝑡𝑐ℎ) sin⁡(𝑝𝑖𝑡𝑐ℎ) 0 cos⁡(𝑝𝑖𝑡𝑐ℎ) =

28. Magnetometer equation
𝑚 𝑥 𝑚 𝑦 𝑚 𝑧 = ⋯ ⋮ ⋱ ⋮ 𝑀 0
Rotation Matrix R
𝑚 𝑥 𝑚 𝑦 𝑚 𝑧
cos⁡(𝑟𝑜𝑙𝑙) sin⁡(𝑟𝑜𝑙𝑙) 0 −sin⁡(𝑟𝑜𝑙𝑙) 𝑐𝑜𝑠⁡(𝑟𝑜𝑙𝑙)
cos⁡(𝑦𝑎𝑤) −sin⁡(𝑦𝑎𝑤) 0 sin⁡(𝑦𝑎𝑤) cos⁡(𝑦𝑎𝑤)
0 𝑀 0
cos⁡(𝑝𝑖𝑡𝑐ℎ) 0 −sin⁡(𝑝𝑖𝑡𝑐ℎ) sin⁡(𝑝𝑖𝑡𝑐ℎ) 0 cos⁡(𝑝𝑖𝑡𝑐ℎ) =
Pitch, roll known from accelerometer
Unknown yaw can be determined from above equations
yaw, pitch, roll together determine the rotation matrix (3D orientation) of a system

29. Gyroscope

30. Error accumulates over time
1D example
Measures angular velocity
𝑔𝑦𝑟𝑜=𝜔
𝑑𝜃 𝑑𝑡 = 𝜔
𝑑𝜃 𝑑𝑡 =𝜔+𝑛𝑜𝑖𝑠𝑒
Error
Time
𝜃= 0 𝑡 𝜔 .𝑑𝑡+ 0 𝑡 𝑛𝑜𝑖𝑠𝑒 . 𝑑𝑡
Error accumulates over time

31. 3D rotation estimation with gyroscope
𝜔 𝑥 𝜔 𝑦 𝜔 𝑧
3D angular velocity
Rotation Matrix R(t+1) =
dR: 3x3 Matrix
(from Gyroscope)
Rotation Matrix R(t)
Captures relative rotation between two times

32. 3D rotation estimation with gyroscope
Gyroscope Measurements
Rotation Matrix R(t+N-1)
dR(t+N-1)
3x3
dR(t+1)
3x3
dR(t)
3x3
Rotation Matrix R(t) …… =
dRs have errors
Error accumulates with time with gyroscope integration

33. Summary: two ways to measure orientation
Using gyroscope
Error accumulates
Using accelerometer and compass
Big advantage: No error accumulation ( since there is no integration involved)
Accelerometer’s gravity measurement can be corrupted with linear motion
Accelerometer can measure orientation only when the phone is static
Magnetometer is susceptible to electromagnetic interference
Error
Time
𝐴 𝑚𝑒𝑎𝑠𝑢𝑟𝑒𝑑 =𝑔𝑟𝑎𝑣𝑖𝑡𝑦 + 𝑎 𝑚𝑜𝑡𝑖𝑜𝑛

34. Recall UnLoc

35. Can we correct gyro drift using accelerometer/magnetometer like UnLoc

36. High level idea to combine the two
Use gyroscope to track orientation in general
Errors will accumulate (drift)
Reset errors with accelerometer/magnetometer (When the phone is static and no magnetic interference)
Error
Time
Gyro
Gyro + Magn + Accl