Rico's Nerd Cluster

「离开世界之前 一切都是过程」

Math - Integration Approximations

Trapezoidal Rule

Trapezoidal Rule If the robot drives forward while turning left, using yaw_mid, moves the displacement along the middle of the arc rather than entirely along the old heading. Integrate pose using...

Posted by Rico's Nerd Cluster on March 2, 2017

Math - Trace, Determinant, Frobenius Norm

Von-Neumann's Trace Inequality

Determinant $\det(AB) = \det(A)\,\det(B)$ For an orthogonal matrix $Q$ (i.e., $Q^\top Q = I$), we have $\det(Q) = \pm 1$: \[\begin{aligned} Q^\top Q &= I \\ \Rightarrow\; \det(Q^\top Q)...

Posted by Rico's Nerd Cluster on February 28, 2017

Math - Linear Fitting

Plane Fitting, Line Fitting, Isotropic Matrix, Condition Number of Covariance Matrix

Given a group of points x_i = [x, y, z], how do we find a plane and a line that best fit them? Plane Fitting Points on a plane satisfy: \[\begin{gather*} \begin{aligned} & n^Tx + d = 0 \end{...

Posted by Rico's Nerd Cluster on February 25, 2017

Math - SO(3) Perturbations

Left Perturbation, Right Perturbation, BCH Formula, Left Jacobian, Right Jacobian

Left and Right Perturbations Imagine we have a world frame, a car frame, and a pedestrian. Now the car has rotated from the car1 pose to the car2 pose. We assume the pedestrain has not moved: In ...

Posted by Rico's Nerd Cluster on February 22, 2017

Math - Lie Group

Group Definition, Lie Group and Lie Algebras, Skew Matrices, Manifold, SO(3) Derivatives

Group Definition Group (Group) is an algebraic structure consisting of a set and an operation. Denote the set as A, and the operation as ⋅. Then, the group can be written as G=(A,⋅). Closure (...

Posted by Rico's Nerd Cluster on February 22, 2017

Math - Multivariate Normal Distribution

PDF, Linear Transformation, Covariance Matrix

PDF “Variate” is basically a synonym to “random variable”. If we have a vector of random variables and we want to find their joint distributution, then we call this “multivariate” distribution. I...

Posted by Rico's Nerd Cluster on February 13, 2017

Math - MAP(Maximum A-Posteriori) and MLE (Maximum Likelihood Estimate)

Likelihood vs Probabilities

Likelihood VS Probabilities Probabilities describe the chances of discrete, mutually-exclusive possible states. These chances should sum up to 1. Likelihoods describe the chances or the plausibili...

Posted by Rico's Nerd Cluster on February 13, 2017

Math - t-SNE

Introduction van der Maaten, Geoff Hinton and Yoshua Bengio came up with “t-SNE” (pronounced as “tee-snee”) to reduce high dimensional data down to low dimensional data, so clusters between data i...

Posted by Rico's Nerd Cluster on February 7, 2017

Math - Approximation to SVD

How to find Null Space? How to implement that using Singular Value Decomposition (SVD)?

Introduction For a square full-rank matrix $G$, the usual SVD is \[G = U\Sigma V^\top\] and the closest orthogonal matrix to $G$ in Frobenius norm is the polar factor \[Q = UV^\top.\] But comp...

Posted by Rico's Nerd Cluster on February 7, 2017

Math - Null Space and Singular Value Decomposition (SVD)

How to find Null Space? How to implement that using Singular Value Decomposition (SVD)?

Introduction The three most basic & important matrix factorizations in pure & applied linear algebra are: QR Decomposition (Gram-Schmidt Orthogonalization) LU Decomposition (Gaussian...

Posted by Rico's Nerd Cluster on February 7, 2017

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