Implement a Quintic Polynomial Solver

In this exercise you will implement a quintic polynomial solver. This will let you take boundary conditions as input and generate a polynomial trajectory which matches those conditions with minimal jerk.

Inputs

Your solver will take three inputs.

start - [s_i, \dot{s_i}, \ddot{s_i}]
end - [s_f, \dot{s_f}, \ddot{s_f}]
T - the duration of maneuver in seconds.

Instructions

Implement the JMT(start, end, T) function in main.cpp
Hit Test Run and see if you're correct!

Tips

Remember, you are solving a system of equations: matrices will be helpful! The Eigen library used from Sensor Fusion is included.

The equations for position, velocity, and acceleration are given by:

s(t) = s_i + \dot{s_i}t + \frac{\ddot{s\_i}}{2}t^2 + \alpha_3t^3 + \alpha_4t^4 + \alpha\_5t^5

\dot{s}(t) = \dot{s_i} + \ddot{s\_i}t + 3 \alpha_3t^2 + 4\alpha_4t^3 + 5\alpha_5t^4

\ddot{s}(t) = \ddot{s\_i} + 6 \alpha\_3t + 12\alpha\_4t^2 + 20\alpha\_5t^3

and if you evaluate these at t=0 you find the first three coeffecients of your JMT are:

[\alpha_0, \alpha_1, \alpha_2] = [s_i, \dot{s_i}, \frac{1}{2}\ddot{s_i}]

and you can get the last three coefficients by evaluating these equations at t = T. When you carry out the math and write the problem in matrix form you get the following:

All these quantities are known except for \alpha_3, \alpha_4, \alpha_5

Start Quiz:

main.cpp

#include <iostream>
#include <fstream>
#include <cmath>
#include <vector>

#include "Dense"

using namespace std;
using Eigen::MatrixXd;
using Eigen::VectorXd;

// TODO - complete this function
vector<double> JMT(vector< double> start, vector <double> end, double T)
{
    /*
    Calculate the Jerk Minimizing Trajectory that connects the initial state
    to the final state in time T.

    INPUTS

    start - the vehicles start location given as a length three array
        corresponding to initial values of [s, s_dot, s_double_dot]

    end   - the desired end state for vehicle. Like "start" this is a
        length three array.

    T     - The duration, in seconds, over which this maneuver should occur.

    OUTPUT 
    an array of length 6, each value corresponding to a coefficent in the polynomial 
    s(t) = a_0 + a_1 * t + a_2 * t**2 + a_3 * t**3 + a_4 * t**4 + a_5 * t**5

    EXAMPLE

    > JMT( [0, 10, 0], [10, 10, 0], 1)
    [0.0, 10.0, 0.0, 0.0, 0.0, 0.0]
    */
    return {1,2,3,4,5,6};
    
}

bool close_enough(vector< double > poly, vector<double> target_poly, double eps=0.01) {


	if(poly.size() != target_poly.size())
	{
		cout << "your solution didn't have the correct number of terms" << endl;
		return false;
	}
	for(int i = 0; i < poly.size(); i++)
	{
		double diff = poly[i]-target_poly[i];
		if(abs(diff) > eps)
		{
			cout << "at least one of your terms differed from target by more than " << eps << endl;
			return false;
		}

	}
	return true;
}
	
struct test_case {
	
		vector<double> start;
		vector<double> end;
		double T;
};

vector< vector<double> > answers = {{0.0, 10.0, 0.0, 0.0, 0.0, 0.0},{0.0,10.0,0.0,0.0,-0.625,0.3125},{5.0,10.0,1.0,-3.0,0.64,-0.0432}};

int main() {

	//create test cases

	vector< test_case > tc;

	test_case tc1;
	tc1.start = {0,10,0};
	tc1.end = {10,10,0};
	tc1.T = 1;
	tc.push_back(tc1);

	test_case tc2;
	tc2.start = {0,10,0};
	tc2.end = {20,15,20};
	tc2.T = 2;
	tc.push_back(tc2);

	test_case tc3;
	tc3.start = {5,10,2};
	tc3.end = {-30,-20,-4};
	tc3.T = 5;
	tc.push_back(tc3);

	bool total_correct = true;
	for(int i = 0; i < tc.size(); i++)
	{
		vector< double > jmt = JMT(tc[i].start, tc[i].end, tc[i].T);
		bool correct = close_enough(jmt,answers[i]);
		total_correct &= correct;

	}
	if(!total_correct)
	{
		cout << "Try again!" << endl;
	}
	else
	{
		cout << "Nice work!" << endl;
	}

	return 0;
}