27. Motion Model Probability II
pseudo_position (x) | pre-pseudo_position | delta position | P(transition) | bel(x_{t-1}) " style="text-align:center;"> 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(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 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:
- Transition Model:
- ' i ' th Motion Model Probability: