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Hot answers tagged probability

6

Assuming a constant update of 5Hz, your sample time is (1/5) = 0.2s. Get one position of the target, p1. Get a second position of the target, p2. Target speed is the difference in position divided by difference in time: $$v = (p_2 - p_1)/dT \\ v = (p_2 - p_1)/0.2$$ Now predict where they will be in the future, where future is $x$ seconds from now: ...

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The kalman filter that you've already been using on single robots can be broadened to apply to the swarm of robots. If you previously represented the state of a single robot with 5 variables, and you have 3 robots, then combine all 3 robot states into one state with 15 variables. That larger state representing the entire group could reasonably be called the "...

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I believe it's because you're essentially constructing an exponential distribution which has the form Because your loss function will always be >= 0, you form a valid PDF (valid in that it integrates to 1, but your loss function might not make that practically true)

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How do I have to imagine $p(m|x_t,u_t,x_{t-1}$)? In his book, he kind of just handwaves it... In SLAM, you need to build two entities, the robot's state (i.e. position and direction) and the map. In deterministic approaches, the probability of both robot's state and the map is one, meaning you know these entities with an absolute true (i.e. zero ...

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Memory of the past is required whenever failures and/or inadequacy arise in the perception layer of the robot thus affecting significantly its current representation of the world, forcing eventually to apply some sort of backtracking strategies. Quoting S.D Whitehead and Long-Ji Lin in their paper "Reinforcement learning of non-Markov decision processes&...

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As said by Jacob, sample impoverishment is inherit to the Sampling-Importance-Resampling family of particle filters. An alternative solution which does require some extra effort is to switch to a Markov Chain Monte Carlo (MCMC) particle filter. MCMC relies on constructing a Markov chain that has a stationary distribution which is equal to the distribution ...

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Your description of sample impoverishment and the way to fix it seems about right. Resampling only when the variance gets low is doing exactly what you are asking for when you say the measurements come in asynchronously. You can also improve matters by selecting the right resampling strategy. Using e.g. stratified resampling you can make sure that your ...

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I would like to mention that Fuzzy logic is still an active control system used in many industry applications. In garbage fired power plants, concrete aggregate firing, hydraulics, and the control of flow of powdered 'fluids' in foundries to name a few. However, I will admit, I've only seen them used in 'one off' difficult to model projects, such as power ...

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Short answer: Fuzzy logic (FL) isn't applicable for robotics research, The long answer is, that in the 1980s as part of the fifth computer generation fuzzy logic was researched in Japan with the attempt to build intelligent advanced parallel computers, but the Japanese researchers have failed. Fuzzy logic isn't a technical idea but a philosophical ...

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Fuzzy logic is definitely used in many of the control systems including but not limited to robotics. See this paper for an example: https://pdfs.semanticscholar.org/b9a7/332b03d46b3ee08b9d113e64714e6b668601.pdf and this: https://ieeexplore.ieee.org/document/1678143 If we consider fuzzy logic as dubious then we should do the same to probabilities. Both ...

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The algorithm can be understood by taking an example (using variables used in Probabilistic Robotics and algorithm in table 4.4 in page 110 in the same book). Algorithm: (Couldn't get math mode to work inside code mode. Hence the picture.) Consider $M = 100$ $M^{-1} = 0.01$ Let $r = 0.005$ So, $U = 0.005, 0.015,......., 0.995$ as loop progresses. If ...

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Here is what I 'believe'. Lets make the diagram more labeled. Say door 1 starting from left most is at 4 meters from the origin at left, door 2 at 7 meters, door 3 at 15 meters. When the robot senses a measurement corresponding to a door it thinks that it has high and equal probability of standing in front of any of those doors and very low probability that ...

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The robot has sensed a door, so the initial belief distribution matches the three possible door positions. i.e. the only three places that it is possible for the robot to be in that scenario. The robot moves to the right so, since the belief distribution matches the possible positions of the robot, the belief distribution must also move to the right. As the ...

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I think this is just a way of illustrating the main idea behind the probability distribution and the representation is not complete. The idea is that there is a moment when the door is detected and the prior distribution not yet considered, this is when the robot assumes this could also be door 1 and therefore the positions of the other doors are as shown. ...

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Our goal is to find a recursive expression for $P\left(\mathrm{x}_{k},\mathrm{m} | \mathrm{Z}_{0:k}, \mathrm{U}_{0:k},\mathrm{x}_{0}\right)$. This expression is called the belief for the robot's state $\mathrm{x}_{k}$ and the map $\mathrm{m}$ given all the measurements $\mathrm{Z}_{0:k} = (\mathrm{z}_{0},..., \mathrm{z}_{k})$, the control actions $\mathrm{U}... 2 To complement what Octopuscabbage correctly reported, there exists a strong theoretical foundation for using normal probability distributions in many different contexts, which builds on the Central Limit Theorem (CLT) that explains how the "exponential" distribution can work well with problems involving other types of distributions. As a result, ... 2 Multi-dimensional models In the 2D case,$x_t$is a vector with two components (e.g. position in$x$,$y$), but why stop at 2D? Often, the state vector$x_t$will have your position in two or three dimensions, in addition to velocity and acceleration. Oftentimes we also include angular state in$x_t$such as the heading and angular rate of change. Your ... 2 Use bearings-only localization to model camera informativeness, and simulate measurements with zero noise (e.g., no innovation). For a variety of reasons, this is actually theoretically sound way of estimating the informativeness of a path. There are many "measurement free" informative-ness metrics, like the Fisher Information Matrix. All you need are the ... 1 Probability and statistics. Stochastic signal processing. Estimation and Detection theory (I highly recommend that you find a class that uses Harry VanTrees's book and that offers office hours, that you enroll, and study, and that you reserve lots of time in your schedule to take it -- if you can learn that stuff by reading the book you're somewhere in the ... 1 The Probabilistic Robotics written by Thuran and his colleagues was completely helpful for me. Also, you can follow one of their colleague videos on youtube for SLAM. 1 The mean itself is precisely defined and there's no alternative definition for it. It simply is what it is. Instead you need a different summary statistic; in this case something like a mode might be useful but how do you decide which mode to use if your distribution is multi-modal? I haven't worked with this particular application so can't say what others ... 1 I have not seen any industry-grade application of fuzzy logic in space, flight, automotive control systems. Fuzzy logic came during mid-60s and it gradually faded away due to several reasons: It did not solve any control problem that cannot already be solved by the existing methods at that time. Bad news, no major advantage in terms of extending the ... 1 I think it is easy to see, when you take a look at the bayesian network: Now, we eliminate all the variables not given in your equation: Based on this baesian network, you can see that$u_t$has no effect on the other variables and thus can be omitted. This however would not work for$u_t$s that are not random. If for example$u_t$would be dependent on$...

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I'm not familiar with particle filtering, but if applying the weights between each sensor read is causing issues, why not accumulate weights to be applied between sensor calls and apply them in bulk after all sensors have been read? For example, it sounds like you're doing: [Read 1]-< apply weights >-[Read 2]-< apply weights >-[Read 3]-< apply ...

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I misread the text. r should be a random number between 0 and M^-1. Changing this should solve all your problems. The re-sampling algorithm's purpose is (roughly) to remove particles that have a low probability of representing the system that you're tracking. This is done by stacking the all particles together. Each particle's size is equal to the ...

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The referenced picture of an equation is this: and in latex it is: $$\tag{26} p(z_t|m,c_t) = \frac{1}{\sqrt{2\pi\sigma^2}} e^{-\frac{1}{2}\left\{ c_{t,*}\log{\frac{z^2_{max}}{2\pi\sigma^2}} + \sum_{k=1}^{K_t}c_{t,k}\frac{(z_t-d_{t,k})^2}{\sigma^2} + c_{t,0}\frac{(z_t-z_{max})^2}{\sigma^2} \right\}}$$ For a single ...

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When you have to multiply so many (often low) probability values or in this case probability densities, you are bound to get in trouble because of limited floating-point accuracy. It is advisable to instead sum up their logarithms, since this will give you a more accurate result. $\log p\left(z_t|x_t,m \right)=\sum_{i=1}^{n} \log p'\left(z_i|x_t,m \right)$ ...

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