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Motion Predictive Control Project

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Introduction

  • This is Udacity's Self-Driving Car Nanodegree Term2 MPC Project.
  • In this project you'll implement Model Predictive Control (MPC) to drive the car around the track.
  • The basic idea of MPC is to predict the states of the car by considering the equations of motions of the car and minimizing a cost.

Model

  • The model of the MPC is a basic kinematic model. Not dynamic model. For more details, refer to this paper.
  • The state of the vehicle consists of x-position, y-position, psi (orientation(heading direction)), v (velocity), CTE (cross-track-eror), epsi (orientation error)
  • Actuator consists of steering angle and acceleration.(positive = throttle, negative = brake)
  • In simulator, MPC predicted trajectory is represented in green, and reference to the vehicle's coordinate is represented in yellow.

Environment

  • ubuntu 16.04 (64bit)
    • memory : 62.8 GB
    • CPU : Intel Core i7-6850K CPU @ 3.60GHz x 12 ea
    • GPU : Titan X (Pascal) x 4 ea
  • Ipopt-3.12.1
  • Tested Simulator
    • Term_2_Simulator_V1.3_Updated_MPC_ubuntu
    • Term_2_Simulator_v1.4_Localization_ubuntu
    • Term_2_Simulator_v1.45_PID_ubuntu

Dependencies

  • cmake >= 3.5
  • All OSes: click here for installation instructions
  • make >= 4.1
  • gcc/g++ >= 5.4
  • uWebSockets
    • Run either install-mac.sh or install-ubuntu.sh.
    • If you install from source, checkout to commit e94b6e1, i.e. 
      git clone https://github.com/uWebSockets/uWebSockets 
cd uWebSockets
git checkout e94b6e1
      Some function signatures have changed in v0.14.x. See this PR for more details.
  • Fortran Compiler
    • Mac: brew install gcc (might not be required)
    • Linux: sudo apt-get install gfortran. Additionall you have also have to install gcc and g++, sudo apt-get install gcc g++. Look in this Dockerfile for more info.
  • Ipopt
    • Mac: brew install ipopt
    • Linux
      • You will need a version of Ipopt 3.12.1 or higher. The version available through apt-get is 3.11.x. If you can get that version to work great but if not there's a script install_ipopt.sh that will install Ipopt. You just need to download the source from the Ipopt releases page or the Github releases page.
      • Then call install_ipopt.sh with the source directory as the first argument, ex: bash install_ipopt.sh Ipopt-3.12.1.
    • Windows: TODO. If you can use the Linux subsystem and follow the Linux instructions.
  • CppAD
    • Mac: brew install cppad
    • Linux sudo apt-get install cppad or equivalent.
    • Windows: TODO. If you can use the Linux subsystem and follow the Linux instructions.
  • Eigen. This is already part of the repo so you shouldn't have to worry about it.
  • Simulator. You can download these from the releases tab.
  • Not a dependency but read the DATA.md for a description of the data sent back from the simulator.

Basic Build Instructions

  1. Clone this repo.
  2. Make a build directory: mkdir build && cd build
  3. Compile: cmake .. && make
  4. Run it: ./mpc.

Details

Model

  • The vehicle model is a kinematic model and these are the equations for the model :
x_[t] = x[t-1] + v[t-1] * cos(psi[t-1]) * dt
y_[t] = y[t-1] + v[t-1] * sin(psi[t-1]) * dt
psi_[t] = psi[t-1] + v[t-1] / Lf * delta[t-1] * dt
v_[t] = v[t-1] + a[t-1] * dt
cte[t] = f(x[t-1]) - y[t-1] + v[t-1] * sin(epsi[t-1]) * dt
epsi[t] = psi[t] - psides[t-1] + v[t-1] * delta[t-1] / Lf * dt

Timestep and Elapsed Duration( N, dt)

  • N is the number of timesteps in the horizon. dt is how much time elapses between actuations.
  • They are hyperparameters and T(N*dt) should be as large as possible, while dt should be as small as possible.
  • To find the appropriate value for N and dt, I tested several values(e.g. N = 7, 10, 15, 20 and corresponding dt value).
  • In my model, N = 10, dt = 0.1 were best values.

fitting waypoints & latency

  • The waypoints of the road are given in the global coordinate.
  • So I need to convert it to the vehicle coordinate.
  • After that, the waypoints are fitted with a 3rd order polynomial.
for (int i = 0; i < ptsx.size(); ++i) {

	// move to origin and rotate around origin to align with axis
	double shift_x = ptsx[i] - px;
	double shift_y = ptsy[i] - py;

	ptsx[i] = shift_x * cos(0 - psi) - shift_y * sin(0 - psi);
	ptsy[i] = shift_x * sin(0 - psi) + shift_y * cos(0 - psi);
}
  • In the simulator, there is a latency between actuations commands.(100 ms)
  • I considered this latency and the equation is as below.
double latency_x = 0 * v * (100 / 1000);
double latency_y = 0;
double latency_psi = 0;
double cte = polyeval(coeffs, 0) - 0;
double epsi = latency_psi - atan(coeffs[1] + 2 * latency_x * coeffs[2] + 3 * pow(latency_x, 2) * coeffs[3]);