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6

Given that an homogeneous transformation $T \in SE\left(3\right)$ can be expressed as $$T= \left( \begin{matrix} \mathbf{R} & \mathbf{p} \\ 0 & 1\end{matrix} \right),$$ where $\mathbf{R} \in \mathbb{R}^{3 \times 3}$ is symmetric and $\mathbf{p} \in \mathbb{R}^{3 \times 1}$, then we seek for the inverse $T^{-1}$, such that: $$T^{-1}=\left( \begin{... 5 A hint towards what the answer is given in the paper. Namely, one can use the generalized eigenvalue decomposition, which in this case can be formulated as finding eigenvalues \lambda \in \mathbb{R} and eigenvectors v \in \mathbb{R}^n such that$$ (\lambda\,\Lambda - K_d)\,v = 0. \tag{1} $$Consider two distinct solutions (\lambda_i,v_i) and (\... 5 Specifically, the Chi-Square Distribution(or Chi2, \chi^2, or equivalently \chi^2_1) is used to model the probability of the absolute value of the deviation of the measurement from it's expected value. This calculation is vital to tackle the measurement origin uncertainty problem. It can also be used to determine the "correctness" of a multi-hypothesis ... 4 A model of the environment in this context is an abstraction of the real world, which should be adequate for the task of the robot. For example, if you have a robot that needs to navigate an office building, you can make the abstraction that your model only needs to be in two dimensions. Further, for the task of navigation you could discretize your space in ... 4 I will try to make it as simple as possible. Imagine you have a SCREW, when you WRENCH it, it TWIST forward or backward. From your wiki link The components of the screw define the Plücker coordinates of a line in space and the magnitudes of the vector along the line and moment about this line. It means that any system can be described as those ... 3 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) 3 The way you are describing it, DOM is the number of independent dimensions in \vec q. DOF is the number of independent dimensions in \vec x. In practice, a robotics engineer will use DOF to represent the number of independent actuators of the robot, which you are calling DOM. Better notation would be to call DOM the mobility of the system, as ... 3 Complexity is not a word that's used in control theory but I think I know what you mean. The short answer would be no. Complexity in systems to be controlled is usually looked at in terms of how many state variables are needed to fully describe what's happening or, more often, how many Degrees of Freedom exist. There is a theorem that says whether a Linear ... 2 I'm not sure if you still need it, but for those who happened to google for this thread, I have made one simple version of the algorithm. Basically, it tries to build the map of the area while it cleans, and it uses the map to find the nearest unvisted node (part of the room). When it can't find any, that means the room is cleaned (or the uncleaned parts ... 2 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 ... 2 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 ... 2 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 ... 2 It's usually best to contact the author directly if you have a specific question about a specific paper. The simulations are there only to illustrate the principal of the passivity observer and controller, so it makes sense they did not elaborate on them too much. The first simulation uses velocity and position as the input and appears to have an initial ... 2 A definition of the cross product for quaternions is$$p \times q = \frac{1}{2}(pq - qp)$$We also have the identity (3.167) in Jazar$$pq = -p \cdot q + p \times q$$Apparently the vectors  \dot{e}\overset{\ast}{e}  and {}^Gr are orthogonal, so their dot product is zero and we have:$$\dot{e} \overset{\ast}{e}{}^Gr = \dot{e} \overset{\ast}{e} \times {...

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RoboEarth was the name of a European project. Knowrob is software that was developed, in part, by the RoboEarth project.

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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, ...

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If you have the path you want the robot to stay on, it sounds like you need Pure Pursuit. In this simple algorithm, you steer the robot to some look-ahead point on the path. The gist of it is illustrated by this image. The original paper is framed in terms of ackerman steering angle. But it is easy to adapt to differential drive robots. You should ...

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I haven't seen the degree symbol $^{\circ}$ to denote a percentage, but both percentages on your drawing result in the same torque: 54.7, not 57. (10.12/0.185) $\approx$ (34.12/0.6238) $\approx$ 54.7. This, in turn, is about 96 percent of the "rated torque," so I don't know if they de-rated the max torque or not. On the webpage you linked, they said, ...

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Seems that the book has an error. You are right, they should correct that. In the drawing it can be seen that those two z-axis are indeed parallel. I recommend that you considered them parallel and try to draw them to see that it fits with the image.

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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 ...

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Appending to @WalterJ's answer. Linear and nonlinear systems which form the basis for subjects like optimal control theory have rigorous math fundamentals which allow you to analyze ODEs without actually solving them and mathematically prove whether is system is stable, how fast your convergence will be or define a safe operating region. This makes ...

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The reason why ODE's are used is simply: physics. It would be great if any system could be modelled by a simple linear function like $x(t)=at$, but nature is not so simple, or linear. Even when you neglect nature, dynamical systems, like $\dot{x}(t)=f(x(t))$ still pop up everywhere, like CroCo said, it is the basis of the mathematical modelling of many ...

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It's just a phase unwrapping case in MATLAB. Anyway when real parts of complex poles/zeros are null, phase jump could be +180 or -180, it's uknown therefore both diagrams are correct

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If you are trying to implement torque control, you only need a control loop around torque. The challenge isn't in trying to integrate the torque (or acceleration) controller, and passing it to a velocity control loop, then finally passing it to a position control loop. The challenge is in finding the torque setpoints for your torque control loop so that the ...

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I would say the transferfunction becomes $\frac{1}{s^2}$ iff you assume you have a perfect drive without any dynamics. Since this system is unstable you have to be careful choosing your controllers. The best way in my opinion is to use a standard state feedback controller (LQR, Ackermann). The huge advantage using state feedback is that you can chose the ...

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You might like the algorithms run on PR-2 robot. Two papers I can think of are "A single planner for a composite task of approaching, opening and navigating through non-spring and spring-loaded doors" and "Motion planning for smooth pickup of moving objects". Both can be found on IEEE Xplore.

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I went through the Virginia Community College System (VCCS) (not saying which one specifically for myself). In Virginia, if you graduate with >= 3.0 GPA from a Virginia Community College, you are guaranteed admission to any public (Virginia) university. Community colleges here are top notch; I got the most knowledge out of community college courses and grad ...

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I disagree that implementing law 1 automatically implements the other 2. Law 1 only states that a robot may not injure a human. Nothing in that law implies that the robot must obey people or protect itself. I suggest you read the original iRobot series of short stories. They do a much better job of illustrating the different ways these 3 laws can lead ...

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According to this website http://www.cs.tufts.edu/comp/150IR/labs/wavefront.html It shows a method of converting the zero to number and calculating the fastest distance.I am assuming you mean wavefront instead of wave planning

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Something I've heard said many times (by a WHOI scientist, I think) is that The problem with models is that they work. In other words, the only things that can go wrong in a model are the things you put into the model ahead of time. Your model may give you perfect results, but in the real world any number of strange things can go wrong; the world is a ...

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