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The best way to solve this is to position your IMU onto the Car's COG, so you can associate directly the IMU information to the COG. In case that's not possible (design choices, space, etc), two things need to be done: 1) Find the transformation from the IMU to the COG. You need to find the translation and rotation from the IMU to the COG so you can ...

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I actually spent a while trying to figure this out, and I realized that they're talking about Final value theorem. Here's a paper (warning: PDF) that gives the equation you've provided in your paper with a correction: $$\left(sI - \left(A-BK\right)\right)^{-1}B \\$$ Note the Laplace operator $s$! The final value theorem says that you can get $\lim_{t\... 5 A coordinate transformation of a point P from Frame 1 to Frame 0 is given by: $$\mathbf{p}^0=\mathbf{o}^0_1+\mathbf{R}^0_1\mathbf{p}^1.$$ Differentiating with respect to time gives: $$\dot{\mathbf{p}}^0=\dot{\mathbf{o}}^0_1+\mathbf{R}^0_1\dot{\mathbf{p}}^1+\dot{\mathbf{R}}^0_1\mathbf{p}^1.$$ Considering that$\dot{\mathbf{p}}^1=0$as$\mathbf{p}^1$is ... 2 Lynch and Park's Modern Robotics book uses the product of exponentials formula and screw axes to describe manipulators, and they have a well-documented library available in Python, MATLAB, and Mathematica. Plus there is a community-released C++ port using CMake/Eigen. Book is available on this site (for free): http://modernrobotics.org/ Original library is ... 2 Quite easily, by applying the definition of the Null Space straight away, you have to solve for: $$\mathbf{J} \left( q_1,q_2=0\right) \cdot \left[\dot{q_1}, \dot{q_2} \right]^T = 0.$$ You'll come up with the following relation: $$\frac{\dot{q_1}}{\dot{q_2}} = - \frac{l_2}{l_1+l_2},$$ which in turn can be summarized by$\mathcal{N} \left( \mathbf{J} \...

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"A car with n trailers" is known to be a differentially flat system. Flatness implies that the Lie algebra generated by the system's vector fields (f and g_i's) is full. Therefore, the example you give is controllable. In the example, f is zero, but you have two vector fields g1 and g2 that are associated with u1, u2, respectively. You need to work on the ...

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While Screw theory has only been explored and deployed in robotics in research and academic capacity, I found this library on github that could help you. Screw-Theory-Toolbox-for-Robotics-ST24R I hope you find this useful.

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