vault backup: 2025-06-12 23:41:18

Affected files:
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99 Work/0 OneSec/OneSecNotes/30 Engineering Skills/Robotics/Kalman Filter.md
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.obsidian/workspace.json

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99 Work/0 OneSec/OneSecNotes/30 Engineering Skills/Robotics/Kalman Filter.md

@@ -65,6 +65,18 @@ $$ x_k \sim \mathcal{N}(\hat{x}_k, P_k)$$
 - Q: is the process noise: it's just there you cannot really get rid of it
 $$ \hat{x}^-_k = A \hat{x}_{k-1}$$
 $$ P_k^- = AP_{k-1} A^T + Q$$
+
+$$Q=\begin{pmatrix}
+
+\sigma_{1}^{2} & 0              & \cdots & 0\\
+
+0              & \sigma_{2}^{2} & \ddots & \vdots\\
+
+\vdots         & \ddots         & \ddots & 0\\
+
+0              & \cdots         & 0      & \sigma_{n}^{2}
+
+\end{pmatrix}$$, where $\sigma_i^2$ is the variance of the process noise
 ### Estimation Step
 #### II. Compute Kalman Gain
 This is the dynamic update of the Kalman gain.
@@ -86,7 +98,27 @@ Note:
 - $H$: is the measurement noise: you can control it with better sensors
   
 
-### Simple Example
+## Dynamic Attitude Determination
+Sensors:
+- Gyroscope and accelerometer 
+
+Lets integrate the gyro data to visualize what we could do to get the attitude.
+![[Pasted image 20250612230104 1.png]]
+As we can see below, there is a lot of drift meaning the numerical integration is no good.
+![[Pasted image 20250612230414.png]]
+
+System Model
+
+## Key Takeaways
+
+Use the Kalman filter to estimate velocity from position samples. Its much more precise than a simple derivative.
+![[Pasted image 20250612225625.png]]
+
+
+Why do IMUs have gyro and accelerometers? Because they work really well together. The gyro is fast and gives quite good data on a short time horizon, but has terrible drift. On contrary, the accelerometer is quite noisy, but gives a true value about attitude, because it can measure gravity. Thus with accelerometer measurmenets every now and then, you can fix the gyro drift and estimate it to fix it.   
+Accelerometer data can give you pitch and roll. --> this is used in the kalman filter to update the estimates --> sensor fusion
+Magnetometer data can give you yaw.
+
 
 ---
 ## Sources