Description

24-677 Special Topics: Modern Control – Theory and Design
Prof. D. Zhao
• Your online version and its timestamp will be used for assessment.
• We will use Gradescope to grade. The link is on the panel of CANVAS. If you are confused about the tool, post your questions on Campuswire.
• Submit your controller.py to Gradescope under P4-code and your solutions in .pdf format to P4-writeup. Insert the performance plot image in the .pdf. We will test your controller.py and manually check all answers.
• We will make extensive use of Webots, an open-source robotics simulation software, for this project. Webots is available here for Windows, Mac, and Linux.
• Please familiarize yourself with Webots documentation, specifically their User Guide and their Webots for Automobiles section, if you encounter difficulties in setup or use. It will help to have a good understanding of the underlying tools that will be used in this assignment. To that end, completing at least Tutorial 1 in the user guide is highly recommended. • If you have issues with Webots that are beyond the scope of the documentation (e.g. the software runs too slow, crashes, or has other odd behavior), please let the TAs know via Campuswire. We will do our best to help.
• We advise you to start with the assignment early. All the submissions are to be done before the respective deadlines of each assignment. For information about the late days and scale of your Final Grade, refer to the Syllabus in Canvas.
1 Introduction
In this project, you will complete the following goals:
1. Design a EKF SLAM to estimate the position and heading of the vehicle.
[Remember to submit the write-up, plots, and codes on Gradescope.]
2 P4: Problems
Exercise 1. In this final part of the project, you will design and implement an Extended Kalman Filter Simultaneous Localization and Mapping (EKF SLAM). In previous parts, we assume that all state measurements are available. However, it is not always true in the real world. Localization information from GPS could be missing or inaccurate in the tunnel, or closed to tall infrastructures. In this case, we do not have direct access to the global position X, Y and heading ψ and have to estimates them from ˙x, ˙y, ψ˙ on the vehicle frame and range and bearing measurements of map features. Consider the discrete-time dynamics of the system:
Xt+1 = Xt + δtX˙ t + ωtx
Yt+1 = Yt + δtY˙t + ωty (1) ψt+1 = ψt + δtψ˙t + ωtψ
Substitute X˙t = x˙t cosψt − y˙t sinψt and Y˙t = x˙t sinψt + y˙t cosψt in to (1),
Xt+1 = Xt + δt(x˙t cosψt − y˙t sinψt) + ωtx
Yt+1 = Yt + δt(x˙t sinψt + y˙t cosψt) + ωty (2) ψt+1 = ψt + δtψ˙t + ωtψ
δt is the discrete time step. The input ut is [˙xt y˙t ψ˙t]T . Let pt = [Xt Yt]T . Suppose you have n map features at global position for j = 1,…,n. The ground truth of these map feature positions are static but unknown, meaning they will not move but we do not know where they are exactly. However, the vehicle has both range and bearing measurements relative to these features. The range measurement is defined as the distance to each feature with the measurement equations for j = 1,…,n. The bearing measure is defined as the angle between the vehicle’s heading (yaw angle) and ray from the vehicle to the feature with the measurement equations yt,bearingj =
, where the and are
the measurement noises. Let the state vector be
 Xt 
 Yt 
 
 ψt 
 1 
 mx 
 1 
 my 
xt =  m2x  (3)

 2 
 my 
 … 


 mnx  
mny
The measurement system be
 km1 − ptk   vt,distance1 
 …   … 
 n − ptk− Xt) t  +  vvt,distancent,bearing1  km
yt =  1 − Yt,m1x − ψ  
 atan2(my

 …   …  atan2(mny − Yt,mnx − Xt) − ψt vt,bearingn (4)
Derive Ft and Ht for EKF SLAM to estimate the vehicles state X, Y , ψ and feature
positions simultaneously.
Hints:
• write out the complete dynamical system with the state in (3) and then follow the standard procedure of deriving EKF. Think what is the dynamic for static landmarks?
Exercise 2. For this exercise, you will implement EKF SLAM in the ekf slam.py file by completing four functions (marked with TODO in the script). You can use the same controller from your previous parts or write a new one. Before integrating this module with Webots, you can run the script by $ python ekf slam.py to test your implementation. Feel free to write your own unit-testing scripts. You should not use any existing Python package that implements EKF. [Hints: remember to wrap heading angles to [−π,π]]
Submit your controller ekf slam.py, ekf slam.py, and the final completion plot as described on the title page. You do not need to submit the plot generated by running the test script (by running $ python ekf slam.py). Your controller is required to achieve the following performance criteria to receive full points:
1. Time to complete the loop = 250 s
2. Maximum deviation from the reference trajectory = 10.0 m
3. Average deviation from the reference trajectory = 5 m
Debugging tips:
• Do not hardcode the number of map features. Instead, use n in your code.
• Check all the signs carefully.
• Check all the numpy array indexing.
• Use wrap to pi function smartly. Only wrap the angle terms but not the distance terms.
• When you run $ python ekf slam.py, it is normal if the estimation diverge gradually, but you should have some reasonable tracking performance.
3 Appendix
(Already covered in P1)

Figure 1: Bicycle model[2]

Figure 2: Tire slip-angle[2]
We will make use of a bicycle model for the vehicle, which is a popular model in the study of vehicle dynamics. Shown in Figure 1, the car is modeled as a two-wheel vehicle with two degrees of freedom, described separately in longitudinal and lateral dynamics. The model parameters are defined in Table 2.
3.1 Lateral dynamics
Ignoring road bank angle and applying Newton’s second law of motion along the y-axis:
Combining the two equations, the equation for the lateral translational motion of the vehicle is obtained as:

Moment balance about the axis yields the equation for the yaw dynamics as
ψI¨ z = lfFyf − lrFyr
The next step is to model the lateral tire forces Fyf and Fyr. Experimental results show that the lateral tire force of a tire is proportional to the “slip-angle” for small slip-angles when vehicle’s speed is large enough – i.e. when ˙x ≥ 0.5 m/s. The slip angle of a tire is defined as the angle between the orientation of the tire and the orientation of the velocity vector of the vehicle. The slip angle of the front and rear wheel is
αf = δ − θV f
αr = −θV r
where θV p is the angle between the velocity vector and the longitudinal axis of the vehicle, for p ∈ {f,r}. A linear approximation of the tire forces are given by
!
where Cα is called the cornering stiffness of the tires. If ˙x < 0.5 m/s, we just set Fyf and Fyr both to zeros.
3.2 Longitudinal dynamics
Similarly, a force balance along the vehicle longitudinal axis yields:
x¨ = ψ˙y˙ + ax
max = F − Ff Ff = fmg
3.3 Global coordinates
In the global frame we have:
X˙ = x˙ cosψ − y˙ sinψ
Y˙ = x˙ sinψ + y˙ cosψ
3.4 System equation
Gathering all of the equations, if ˙x ≥ 0.5 m/s, we have:

Y˙ = x˙ sinψ + y˙ cosψ
otherwise, since the lateral tire forces are zeros, we only consider the longitudinal model.
3.5 Measurements
The observable states are:
x˙ 
y˙ 
ψ˙  y =  
X
 
Y  ψ
3.6 Physical constraints
The system satisfies the constraints that:

F > 0 and F 6 15736 N
x˙ > 10−5 m/s
Table 1: Model parameters.
Name Description Unit Value
(x,˙ y˙) Vehicle’s velocity along the direction of vehicle frame m/s State
(X,Y ) Vehicle’s coordinates in the world frame m State
ψ, ψ˙ Body yaw angle, angular speed rad, rad/s State
δ or δf Front wheel angle rad State
F Total input force N Input
m
lr
mass
lf Length from front tire to the center of mass m 1.55
Cα
Iz 25854
Fpq p ∈ {x,y},q ∈ {f,r}
f sec
3.7 Simulation

Figure 3: Simulation code flow
Several files are provided to you within the controllers/main folder. The main.py script initializes and instantiates necessary objects, and also contains the controller loop. This loop runs once each simulation timestep. main.py calls your controller.py’s update method on each loop to get new control commands (the desired steering angle, δ, and longitudinal force, F). The longitudinal force is converted to a throttle input, and then both control commands are set by Webots internal functions. The additional script util.py contains functions to help you design and execute the controller. The full codeflow is pictured in Figure 3.
Please design your controller in the your controller.py file provided for the project part you’re working on. Specifically, you should be writing code in the update method. Please do not attempt to change code in other functions or files, as we will only grade the relevant your controller.py for the programming portion. However, you are free to add to the CustomController class’s init method (which is executed once when the CustomController object is instantiated).
3.8 BaseController Background
The CustomController class within each your controller.py file derives from the BaseController class in the base controller.py file. The vehicle itself is equipped with a Webots-generated GPS, gyroscope, and compass that have no noise or error. These sensors are started in the BaseController class, and are used to derive the various states of the vehicle. An explanation on the derivation of each can be found in the table below.
Table 2: State Derivation.
Name Explanation
(X,Y ) From GPS readings

3.9 Trajectory Data
The trajectory is given in buggyTrace.csv. It contains the coordinates of the trajectory as (x,y). The satellite map of the track is shown in Figure 4.

Figure 4: Buggy track[3]
4 Reference
1. Rajamani Rajesh. Vehicle Dynamics and Control. Springer Science & Business Media, 2011.
2. Kong Jason, et al. “Kinematic and dynamic vehicle models for autonomous driving control design.” Intelligent Vehicles Symposium, 2015.
3. cmubuggy.org, https://cmubuggy.org/reference/File:Course_hill1.png