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probabilistic_robotics_._book [2016/10/24 21:11] nikolaj |
probabilistic_robotics_._book [2016/10/31 20:03] nikolaj |
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| Sections: | Sections: | ||
| - Introduction: "Probabilistic state estimation algorithms compute //believe distributions// over possible world states" | - Introduction: "Probabilistic state estimation algorithms compute //believe distributions// over possible world states" | ||
| - | - Probability theory basics | + | - Probability theory basics (see also [[Introduction To Modern Bayesian Econometrics]]) |
| - Mathematical world representation | - Mathematical world representation | ||
| - Bayes filters | - Bayes filters | ||
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| This all has more connections to path integrals and stochastic integrals than I previously thought, so, to me, that's great and fun. | This all has more connections to path integrals and stochastic integrals than I previously thought, so, to me, that's great and fun. | ||
| - | ----- | + | == Exercises == |
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| - | So I read into the Wikipedia page an think we now understand the notorious **Kalman filter**: | + | |
| - | It's when the true state evolution is given by a linear relation | + | $bel_0(\neg faulty)=\frac{9}{10}$ |
| - | $ x_k = {F}_k x_{k-1} $. | + | $bel_0(faulty)=\frac{1}{10}$ |
| - | (+possibly by a noise term) | + | $ p(z\in [0,1]\,|\, faulty) = 1 $ |
| - | and when the sensor is set to measure $ {H}_{k} x_k $. | + | $ p(z\notin [0,1]\,|\, faulty) = 0 $ |
| - | (the H-matrix can be a projection, thus taking into account that you only measure particular features of the truth, and you can't catch em all.) | + | $ p(z\in [0,1]\,|\, \neg faulty) = \frac{1}{3} $ |
| - | and you apply the Bayes Filter with | + | $ p(z\notin [0,1]\,|\, \neg faulty) = \frac{2}{3} $ |
| - | $ p( x_k \mid x_{k-1}) = \mathcal{N} ( {F}_k x_{k-1}, {Q}_k) $ | + | $ p(faulty \,|\, z\in [0,1]) \propto p(z\in [0,1]\,|\, faulty)\cdot bel_0(faulty)$ |
| - | $ p( {z}_k\mid x_k) = \mathcal{N}( {H}_{k} x_k, {R}_k) $ | + | $ N = \sum_{x = faulty, \neg faulty} p(z\in [0,1]\,|\, x)\cdot bel_0(x)$ |
| - | where $ \mathcal{N}( x, \sigma^2) $ is the normal distribution, except of course with multivariate arguments. | ||
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