Model Predictive Control Project

View the running version here http://recordit.co/pezFcQgEdP

Description

This project is implementation of Model Predictive Control to drvie the car around the track.

In MPC, at each time step, we compute control by solving an openloop optimization problem for the prediction horizon. The we apply the first value of the computed control sequence.At the next time step, get we get the model state and re-compute.

The Model

The model used in this project is Kinematic Bicycle Model which consists of following states:

• x: The x position of the vehicle.
• y: The y position of the vehicle.
• psi: The orientation of the vehicle.
• v: The current velocity.
• cte: The Cross-Track-Error
• epsi: The orientation error

The update equations are as following:

fg[1 + x_start + t] = x1 - (x0 + v0 * CppAD::cos(psi0) * dt);
fg[1 + y_start + t] = y1 - (y0 + v0 * CppAD::sin(psi0) * dt);
fg[1 + psi_start + t] = psi1 - (psi0 + v0 * delta0 / Lf * dt);
fg[1 + v_start + t] = v1 - (v0 + a0 * dt);
fg[1 + cte_start + t] = cte1 - ((f0 - y0) + (v0 * CppAD::sin(epsi0) * dt));
fg[1 + epsi_start + t] = epsi1 - ((psi0 - psides0) + v0 * delta0 / Lf * dt);

Polynomial Fitting and MPC Preprocessing

The waypoints from simulator are global coordinate system, but we have to transform them because all computations are performed in the vehicle coordinate system.

double x = (ptsx[i] - px) * cos(psi) + (ptsy[i] - py) * sin(psi);
double y = -(ptsx[i] - px) * sin(psi) + (ptsy[i] - py) * cos(psi);

Timestep Length and dt

At first, I started with N=25 and dt=0.05. But I changed the values to N=10 and dt=0.2 because:

• Smaller dt is better because it gives finer resolution.
• But we have the 100ms latency, so I chose the larger value to deal with the latency.
• Larger value than N=10 is easy to miscalculate.
• Smaller value thant N=10 is not enough to caculate the trajectory.

Latency

As noted above, I chose dt=0.2 to deal with the latency. Also, I incorporated the latency into the model.

Eigen::VectorXd state_vector(6);

double latency = 0.1;
double Lf = 2.67;

double x = v * cos(psi) * latency;
psi = v / Lf * delta * latency;
v = v + a * latency;

state_vector << x, 0, psi, v, cte, epsi;

auto results = mpc.Solve(state_vector, coeffs);

Basic Build Instructions

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

Dependencies

• cmake >= 3.5
• All OSes: click here for installation instructions
• make >= 4.1(mac, linux), 3.81(Windows)
  • Linux: make is installed by default on most Linux distros
  • Mac: install Xcode command line tools to get make
  • Windows: Click here for installation instructions
• gcc/g++ >= 5.4
  • Linux: gcc / g++ is installed by default on most Linux distros
  • Mac: same deal as make - [install Xcode command line tools]((https://developer.apple.com/xcode/features/)
  • Windows: recommend using MinGW
• 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
  • If challenges to installation are encountered (install script fails). Please review this thread for tips on installing Ipopt.
  • Mac: brew install ipopt
    • Some Mac users have experienced the following error:

    Listening to port 4567
    Connected!!!
    mpc(4561,0x7ffff1eed3c0) malloc: *** error for object 0x7f911e007600: incorrect checksum for freed object
    - object was probably modified after being freed.
    *** set a breakpoint in malloc_error_break to debug
    

    This error has been resolved by updrading ipopt with brew upgrade ipopt --with-openblas per this forum post.
  • 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.
    • Then call install_ipopt.sh with the source directory as the first argument, ex: sudo 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.

Tips

1. It's recommended to test the MPC on basic examples to see if your implementation behaves as desired. One possible example is the vehicle starting offset of a straight line (reference). If the MPC implementation is correct, after some number of timesteps (not too many) it should find and track the reference line.
2. The lake_track_waypoints.csv file has the waypoints of the lake track. You could use this to fit polynomials and points and see of how well your model tracks curve. NOTE: This file might be not completely in sync with the simulator so your solution should NOT depend on it.
3. For visualization this C++ matplotlib wrapper could be helpful.

Code Style

Please (do your best to) stick to Google's C++ style guide.