Rico贾若童的博客

Electronics - Communication Protocols

Encoding / Decoding, Error Checking (CRC), Base64, EtherCAT

Error Checking CRC GF(2) Math GF(2) is pronounced as “Galois Field 2”. Often used in cryptography, it’s a binary field that contains only 0 and 1 for error correction. Some properties of GF(2):...

Posted by Rico's Nerd Cluster on June 1, 2017

Electronics - EMF and Eddy Current

Electromotive Force (EMF，电动势) In short, EMF is: 1 changing magnetic field -> induced voltage It’s characterized by the Faraday’s law: \[\begin{gather*} \begin{aligned} & \epsilon = -N \...

Posted by Rico's Nerd Cluster on June 1, 2017

Math - Sherman-Morrison-Woodbury Equation

How Kalman Filter Reduces Computation For Inverse

The Sherman-Morrison-Woodbury Equation is: \[\boxed{ AB(D+CAB)^{-1} = \left(A^{-1}+BD^{-1}C\right)^{-1}BD^{-1} }\] Assume all displayed inverses exist. Direct proof Define \[X=AB(D+CAB)^...

Posted by Rico's Nerd Cluster on March 5, 2017

Math - Differential Entropy

Differential Entropy Differential entropy is the entropy version for a continuous random variable. For a discrete variable: 1 2 3 4 H(x) = -sum(p(x) * log(p(x))) // continuous version h(X) = -∫ ...

Posted by Rico's Nerd Cluster on March 5, 2017

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 (...