CarND-Controls-MPC

Self-Driving Car Engineer Nanodegree Program

Rubrik Points

The Model

The simulator passes several information into the program:

x and y position of waypoints in global coordinates
x and y position of the vehicle in global coordinates
orientation angle of the vehicle
velocity of the vehicle
steering angle of the vehicle
throttle of the vehicle

Model Predictive Control (MPC) is used to control the throttle a and the steering angle delta of the car. The MPC is a kinematic model and uses the following equations:

x[t+1] = x[t] + v[t] * cos(psi[t]) * dt
y[t+1] = y[t] + v[t] * sin(psi[t]) * dt
psi[t+1] = psi[t] + v[t] / Lf * delta[t] * dt
v[t+1] = v[t] + a[t] * dt
cte[t+1] = f(x[t]) - y[t] + v[t] * sin(espi[t]) * dt
epsi[t+1] = psi[t] - psides[t] + v[t] * delta[t] / Lf * dt

Where:

x: x position of the vehicle
y: y position of the vehicle
psi: orientation of the vehicle
v: velocity of the vehicle
cte: cross-track error, which is the deviation in y-direction from the desired path
epsi: orientation error, which is the deviation from the desired orientation
f(x): the polynomial which is mapping the path
psides: the desired orientation which is atan(f'(x))

The state of the vehicle is consists of the first six elemements of the list above.

The MPC reframes the problem of following a trajectory as an optimization problem. This involves simulating different actuator inputs, predicting the resulting trajectory and selecting the trajectory with the minimum cost. After the lowest cost trajectory is found, the first set of actuation commands is implemented. The rest of the calculated trajectory is thrown away. The reason for that is that we cannot be sure of a perfect execution of the commands which could result in a deviation from the desired trajectory. Instead, we take the new state and use that to perform the same steps again.

The described optimization problem tries to find the optimal acceleration a and steering angle delta according to a cost function. For this implementation the cost function considered the following factors:

squared cte
squared epsi
squared difference of the v to a desired velocity which avoids that the vehicle stops driving when it is on the desired trajectory but not at the desired location yet
squared delta and squared a to avoid erratic control inputs
squared delta[t-1] - delta[t] and a[t-1] - a[t] to smooth sequential actuations

All those terms are added together for the cost function. The weights of those cost function terms will influence the solver in handling those terms. The higher the weights the closer together sequential values will be and the smoother the output will be.

Timestep Length and Elapsed Duration

The elapsed duration dt together with the number of steps N determine the horizon of the prediction. Having N = 10 and dt = 0.1 s would result in a prediction horizon of 1 second. N also impacts the performance of the controller as it influences the length of the solution vector. As we try to control a car at higher speeds the environment might change very fast which is why I decided to use a prediction horizon of 1 second. Because of the speed and some strong curves on the track it is beneficial to have a short duration between control inputs. Because we have to deal with a latency of 100 ms I decided that it would not make sense to calculate a new set of control parameters before the previous ones have been executed. This is why a dt of 100 ms was chosen. In order to get to a prediction horizon of 1 second this resulted in 10 steps N.

Polynomial Fitting and MPC Preprocessing

The waypoints and the vehicle's coordinates are passed to the controller in global coordinates. As a first step the waypoints are converted into vehicle coordinates. This means the position of the vehicle is the origin of the coordinate system and the x-axis pointing in the direction of the heading of the car and the y-axis pointing to the left. This can be done using homogenous transformation. First we shift the origin of the global coordinate system to the origin of the vehicle's coordinate system by subtracting x and y of the vehicle from x and y of the waypoints. Then we rotate the coordinate system to match the vehicle's orientation:

x_transformed = x * cos(-psi) - y * sin(-psi)
y_transformed = x * sin(-psi) + y * cos(-psi)

After that we fit a third-degree polyonomial to the transformed waypoints. This is the desired trajectory the vehicle should follow (drawn in yellow in the simulator). Because we are now acting in the coordinate system of the vehicle x, y, and psi of the initial state are all zero. The cte and epsi can be calculated using the polynomial.

Model Predictive Control with Latency

A problem a lot of controller struggle with is latency. Latency describes the delay between calculation and actual implementation of the control parameters. This is a problem because when the parameters are applied the state is already different and the control parameters react to a state that is no longer valid.

In my implementation, the model accounts for latency by predicting the state for the time the control parameters are applied. This means the current state is taken and based on that the state of 100 ms (latency) in the future is predicted. This predicted state is then passed to the controller to calculate the delta and a that should be applied.

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.
Ipopt and CppAD: Please refer to this document for installation 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

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

Tips

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.
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.
For visualization this C++ matplotlib wrapper could be helpful.)
Tips for setting up your environment are available here
VM Latency: Some students have reported differences in behavior using VM's ostensibly a result of latency. Please let us know if issues arise as a result of a VM environment.

Editor Settings

We've purposefully kept editor configuration files out of this repo in order to keep it as simple and environment agnostic as possible. However, we recommend using the following settings:

indent using spaces
set tab width to 2 spaces (keeps the matrices in source code aligned)

Code Style

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

Project Instructions and Rubric

Note: regardless of the changes you make, your project must be buildable using cmake and make!

More information is only accessible by people who are already enrolled in Term 2 of CarND. If you are enrolled, see the project page for instructions and the project rubric.

Hints!

You don't have to follow this directory structure, but if you do, your work will span all of the .cpp files here. Keep an eye out for TODOs.

Call for IDE Profiles Pull Requests

Help your fellow students!

We decided to create Makefiles with cmake to keep this project as platform agnostic as possible. Similarly, we omitted IDE profiles in order to we ensure that students don't feel pressured to use one IDE or another.

However! I'd love to help people get up and running with their IDEs of choice. If you've created a profile for an IDE that you think other students would appreciate, we'd love to have you add the requisite profile files and instructions to ide_profiles/. For example if you wanted to add a VS Code profile, you'd add:

/ide_profiles/vscode/.vscode
/ide_profiles/vscode/README.md

The README should explain what the profile does, how to take advantage of it, and how to install it.

Frankly, I've never been involved in a project with multiple IDE profiles before. I believe the best way to handle this would be to keep them out of the repo root to avoid clutter. My expectation is that most profiles will include instructions to copy files to a new location to get picked up by the IDE, but that's just a guess.

One last note here: regardless of the IDE used, every submitted project must still be compilable with cmake and make./

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bentocin / sdcnd_model_predictive_control