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FWIW, in the context of robotics, where there are real inputs you set, the motion model ends up being extremely useful and behaves much differently than just a

FWIW, in the context of robotics, where there are real inputs you set, the motion model ends up being extremely useful and behaves much differently than just a frequency-domain filter. But the overall subject of comparison to just a low pass filter is very interesting

jbay808 4y ago
Absolutely! And thank you for saying so. (To be clear, the answer to the question posed in the title was never intended to be yes. The answer was intended to be sometimes.) In one of my other articles where I applied the Kalman filter to a more complex system, a frequency-domain filter would have failed. But in this article I tried to stick to the simplest possible system, so that the math wouldn't get in the way of illustrating the way in which Kalman and Wiener filters relate to each other, since it's an interesting connection that doesn't get much attention.
kcexn 4y ago
Not a robotics expert. But my understanding of the effect of input/motion models on the state estimation of Kalman filters is that it conditions the error covariance of the state estimate. So, when the quality of measurements is very high (almost no noise), then the motion model provides very little benefit. And when the measurement quality is very low, the motion model reduces the error ellipse around the real state of the system. So, even when there are inputs and a motion model you could imagine the Kalman Filter as a frequency-domain filter (E.G. IIR/FIR filter), where the tap weights are being updated based on some known function. If the quality of the measurements are high enough, and the sampling frequency is high enough. Then the Kalman filter should be able to converge to a very good estimate of state, even with no motion model right? In which case it would have the exact same characteristics as an adaptive, slowly time-varying frequency-domain filter.
- blt 4y ago
  > If the quality of the measurements are high enough, and the sampling frequency is high enough. Then the Kalman filter should be able to converge to a very good estimate of state, even with no motion model right? No. Consider this discrete-time linear system with 2-dimensional state [x1, x2] and one output y: x1' = 2 * x1 x2' = x2 y = x1 + x2 Suppose we see the output sequence 1, 2, 4, 8, ... If we know the motion model, then we know that the initial state must have been [1, 0]. (In technical terms, the pair (A, C) is observable.) But if we don't have the motion model -- all we know is that y = x1 + x2 -- there's no hope of ever getting a state estimate, even if our y measurement has zero noise.

jbay808 4y ago

Absolutely! And thank you for saying so. (To be clear, the answer to the question posed in the title was never intended to be yes. The answer was intended to be sometimes.) In one of my other articles where I applied the Kalman filter to a more complex system, a frequency-domain filter would have failed. But in this article I tried to stick to the simplest possible system, so that the math wouldn't get in the way of illustrating the way in which Kalman and Wiener filters relate to each other, since it's an interesting connection that doesn't get much attention.

kcexn 4y ago

Not a robotics expert. But my understanding of the effect of input/motion models on the state estimation of Kalman filters is that it conditions the error covariance of the state estimate. So, when the quality of measurements is very high (almost no noise), then the motion model provides very little benefit. And when the measurement quality is very low, the motion model reduces the error ellipse around the real state of the system. So, even when there are inputs and a motion model you could imagine the Kalman Filter as a frequency-domain filter (E.G. IIR/FIR filter), where the tap weights are being updated based on some known function. If the quality of the measurements are high enough, and the sampling frequency is high enough. Then the Kalman filter should be able to converge to a very good estimate of state, even with no motion model right? In which case it would have the exact same characteristics as an adaptive, slowly time-varying frequency-domain filter.

blt 4y ago
> If the quality of the measurements are high enough, and the sampling frequency is high enough. Then the Kalman filter should be able to converge to a very good estimate of state, even with no motion model right? No. Consider this discrete-time linear system with 2-dimensional state [x1, x2] and one output y: x1' = 2 * x1 x2' = x2 y = x1 + x2 Suppose we see the output sequence 1, 2, 4, 8, ... If we know the motion model, then we know that the initial state must have been [1, 0]. (In technical terms, the pair (A, C) is observable.) But if we don't have the motion model -- all we know is that y = x1 + x2 -- there's no hope of ever getting a state estimate, even if our y measurement has zero noise.