Robots, Robots, Everywhere!

Recursive Bayesian Filtering ¶
Gaussian Distribution ¶

"On the ground, in the air, robots, robots, everywhere! Up in space, beneath the seas, robots make discoveries."

Recursive Bayesian Filtering ¶

$$Bel(x_t) = p(x_t | z_{1:t})$$$$Bel(x_t) = \frac{1}{\eta} p(z_t|x_t)\int{p(x_t|x_{t-1})Bel(x_{t-1})}dx_{t-1}$$$$\eta=p(z_t|z_{1:t-1})=\int{p(z_t|x_t)p(x_t|z_{1:t-1})}dx_t$$

Gaussian Distribution ¶

$$X \sim \mathcal{N}(\mu,\,\Sigma)$$$$P(x) = \frac{e^{-\frac{1}{2}(x-\mu)^T\Sigma^{-1}(x-\mu)}}{\sqrt{((2\pi)^k|\Sigma|}}$$

In [5]:

# Plot Gaussian Distribution

import matplotlib.pyplot as plt
import numpy as np
import scipy.stats as stats
import math

mu = 0
sigma = 1
x = np.linspace(mu - 3*sigma, mu + 3*sigma, 100)
plt.plot(x, stats.norm.pdf(x, mu, sigma), color='b', label='Gaussian pdf')

samples = np.random.normal(loc=mu, scale=sigma, size=100)
plt.plot(samples, stats.norm.pdf(samples, mu, sigma), '*r', label='Gaussian samples')
plt.legend()
plt.show()

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This blog is converted from hello_robot.ipynb

Written on April 29, 2021