27. Motion Model Probability II

pseudo_position (x) pre-pseudo_position delta position P(transition) bel(x_{t-1}) " style="text-align:center;"> bel(xt−1) bel(x_{t-1}) P(position)
7 1 6 1.49E-06 5.56E-02 8.27E-08
7 2 5 1.34E-04 5.56E-02 7.44E-06
7 3 4 4.43E-03 5.56E-02 2.46E-04
7 4 ? 5.40E-02 0.00E+00 0.00E+00
7 5 2 ? 0.00E+00 0.00E+00
7 6 1 3.99E-01 0.00E+00 0.00E+00
7 7 0 2.42E-01 ? 1.66E-03
7 8 -1 5.40E-02 1.79E-03 ?

Delta Position

QUESTION:

What is difference in position for an x of 7 and a pre-pseudo position of 4?

SOLUTION:

NOTE: The solutions are expressed in RegEx pattern. Udacity uses these patterns to check the given answer

Transition Probability

QUESTION:

Use normpdf (bottom of page) to determine the transition probability for x = 7 and a pre-pseudo_position of 5, and a control parameter of 1, and a standard deviation of 1. The transition probability can be determined through normpdf(delta_position, control_parameter, position_stdev) . The answer must be in scientific notation with two decimal place accuracy, for example 3.14E-15.

SOLUTION:

NOTE: The solutions are expressed in RegEx pattern. Udacity uses these patterns to check the given answer

Determine the belief state

QUESTION:

In practice we only set our initial belief state, but making the following calculation is helpful in building intuition. What is the belief state bel(xt−1) bel(x_{t-1}) for the penultimate row of our table above? Write the answer in scientific notation with an accuracy of two decimal places, for example 3.14E-15.

SOLUTION:

NOTE: The solutions are expressed in RegEx pattern. Udacity uses these patterns to check the given answer

Position Probability

QUESTION:

What is the discretized position probability for x = 7 and a pre-pseudo_position of 8, given the belief state in the table above? Write the answer in scientific notation with an accuracy of two decimal places, for example 3.14E-15.

SOLUTION:

NOTE: The solutions are expressed in RegEx pattern. Udacity uses these patterns to check the given answer

We have completed our table of discretized calculation for each i i th positon probability value. To determine the final probability returned by the motion model, we must sum the probabilities.

Aggregating Discretized P(position)

QUESTION:

Given the table above, what is the final probability returned by our motion model. Enter the answer in scientific notation with an accuracy of two decimal places, for example 3.14E-15.

SOLUTION:

NOTE: The solutions are expressed in RegEx pattern. Udacity uses these patterns to check the given answer

Recall that the transition probability can be determined through norm_pdf(delta_position, control_parameter, position_stdev)

Start Quiz: 

#include <iostream>
#include "help_functions.h"

// TODO: assign a value, the difference in distances between x_t and x_{t-1}
// for an x of 7 and a pre-pseudo position of 5

float value = ?;  // YOUR VALUE HERE


float parameter = 1.0;  // set as control parameter or observation measurement
float stdev = 1.0;  // position or observation standard deviation

int main() {

  float prob = Helpers::normpdf(value, parameter, stdev);

  std::cout << prob << std::endl;

  return 0;
}
#ifndef HELP_FUNCTIONS_H
#define HELP_FUNCTIONS_H

#include <math.h>

class Helpers {
 public:
  // definition of one over square root of 2*pi:
  constexpr static float STATIC_ONE_OVER_SQRT_2PI = 1/sqrt(2*M_PI);

  /**
   * normpdf(X,mu,sigma) computes the probability function at values x using the
   * normal distribution with mean mu and standard deviation std. x, mu and 
   * sigma must be scalar! The parameter std must be positive. 
   * The normal pdf is y=f(x,mu,std)= 1/(std*sqrt(2pi)) e[ -(x−mu)^2 / 2*std^2 ]
   */
  static float normpdf(float x, float mu, float std) {
    return (STATIC_ONE_OVER_SQRT_2PI/std)*exp(-0.5*pow((x-mu)/std,2));
  }
};

#endif  // HELP_FUNCTIONS_H

In the next concept we will implement the motion model in C++.

Reference Equations

  • Discretized Motion Model:
∑ip(xt∣xt−1(i),ut,m)bel(xt−1(i)) \sum\limits_{i} p(x_t|x_{t-1}^{(i)}, u_t, m)bel(x_{t-1}^{(i)})
  • Transition Model:
p(xt∣xt−1(i),ut,m) p(x_t|x_{t-1}^{(i)}, u_t, m)
  • ' i ' th Motion Model Probability:
p(xt∣xt−1(i)ut,m)∗bel(xt−1(i)) p(x_t|x_{t-1}^{(i)} u_t, m) *bel(x_{t-1}^{(i)})

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