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When you have one fast-moving source and want to fuse it with a slow-moving source, a complimentary filter should be sufficient. Hopefully, it's a lot easier to understand than Kalman filters. There are plenty of examples where they use a complimentary filter to combine accelerometer and Gyroscope. When you say the Z-Axis, I assume you mean vertical axis ...


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After some work I got it working. You can find the C++ source code on github. Don't use the python code in the question. It has some flaws. In the end I just abandoned the gyro drift (I would argue that this is unobservable with just one gyro). Furthermore I used the classic covariance propagation $P_{k+1} = FP_kF^T + GQG^T$.


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@fibonatic was absolutely correct in the comments about the NASA paper talking about the continuous version of it. By just one look of the equation, I could tell since all of the variables were a function of time and it is a differential equation.


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