1. Team Members: Robert Muntean 1832066
ELG 4152 Modern Control PID Control for a Path Following Car Professor: Riadh Habash TA: Wei Yang
Team Members:
Robert Muntean
Yanhui Gao
Muntaber Al Timini
Yirong Wang

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Introduction
The main goal of our project is to control a car and make it able to follow a desired path based on a PID controller. The car must sense its position with respect to the desired path and adjust itself automatically if it is deviated from the desired path.
We have created a Matlab simulation environment to test our kinematics model. The simulation shows the car’s behavior. We have implemented the sensors and controller as closely as possible to the actual system.
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Introduction
References:
Rezaei, S.; Guivant, J.; Nebot, E.M. “Car-like robot path following in large unstructured environments” Intelligent Robots and Systems, (IROS 2003). Proceedings IEEE/RSJ International Conference on, Vol.3, Iss., Oct. 2003
L. Consolini􀀀, A. Piazzi , M.Tosques, “Motion Planning for steering car-like vehicles”, Proceeding of the European Control Conference 2001
Naranjo, J.E.; Gonzalez, C.; Garcia, R.; de Pedro, T.; Haber, R.E, “Power-steering control architecture for automatic driving”, IEEE Transaction on Intelligent Transportation Systems, Volume 6,  Issue 4,  Dec Page(s):406 – 415
Eric N Moret, “Dynamic Modeling and Control of a Car-Like Robot”, IEEE Conference on Decision and Control. Feb 5, 2003.
C. Hatipoglu, U. Ozguner and Keith A. Redmil, “Automated Lane change Controller Design”, IEEE Transaction on Intelligent Transportation Systems, Volume4, Issue 1, March 2003
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4. Mathematical Modeling
Derivation of the kinematics Model
* v1 is the linear velocity of the rear wheels
* v2 is the angular velocity of steering wheels.
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5. Mathematical Modeling
Simulink representation of the cars kinematics model
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6. Mathematical Modeling
Derivation of the Dynamic Model
c(s) is the path’s curvature and is defined as: c(s)=dθt/ds.
The magnitude of c(s) is 1/R. If the path is turning left, c(s) is positive and if the path is turning right, c(s) is negative.
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Systems Block Diagram
Gc(s) = k1+k2/s +k3s
The values of the parameters used in the simulation were:
Lambda = 8.
k1= lambda^3 = 512
k2=3*lambda^2 = 192
k3=3*lambda = 24
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Control Algorithms
With the system in chained form, the controller to perform path following can be developed. For the car model with four states, the chain form system is:
Finally, the control law is defined as
Where k1, k2, and k3 are arbitrary gains. This control law takes the dynamics of the servo into account. As will be shown in the simulation part, this small addition makes for much less error.
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Simulation Program
Flowchart of the program
The initialization involves creating the car and path.
The car’s position on the path is determined and the values needed by the controller are calculated.
The controller then calculates the necessary velocity and steering inputs to make the car follow the path.
These inputs are used in the kinematic model to update the car’s position.
These steps are repeated until the end of the simulation is reached
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Simulation Program
The simulation was designed to match the car’s actual environment as closely as possible, and in an effort to keep true to this, the simulation used both sampling times as well as digitized data.
In the simulation, an ideal path is created for the car to follow. The car’s movement is given by the dynamic model.
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11. Initialization of Simulation
There are three files involved with the initialization of the simulation.
One of the files is simply to create the car.
Another file is to placed the car in the proper initial position.
The third file is to create the path for the vehicle to follow, a path that contains a straight segment, followed by a curve, followed by another straight segment. This file also contains all the constants involved with the vehicle such as mass, wheelbase, number of sensors, and motor and servo constants. It is this file that the user chooses the desired velocity of the vehicle.
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12. Determination of the car’s position
There are three files involved with the determination of the position of the car or the error
One of the files deals with finding the error between the position of the car with respect to the path that needs to be followed.
The second file is to simulate an array of sensors that detects the position of the line that the car needs to follow with respect to the center of the car. For instance, if the car was slightly to the left of the center of the path, the resultant sensor reading would be where the zeros show the position of the line beneath the vehicle.
The third file is to simulate the heading angle, θp, which is the angle between the car and the path.
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Simulation Program
Control Calculation
There are two parts involved with the control calculation part of the simulation.
The first part in the program is to determine the steering and velocity inputs to move the car along the path.
In the second part, the main program, creates the control equations and then starts the dynamic simulation. The velocity and the heading angle are calculated then these are used to update the position of the car in the kinematics model.
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Simulation Results
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Simulation Results
Comparison of path and car traverse at different speed
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Simulation Results
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Simulation Results
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Methodology
Task Distribution
Finding references and Research was done by all members of the group
Robert and Muntader - research of the mathematical model
Yanhui and Yirong - Simulation in Matlab and Simulink
Tools used
Matlab
Simulink
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19. Conclusion/Limitations
We have successfully simulate the behavior of a car that follows a path at different speeds using a PID controller
Limitations
The improvement of the steering response at higher speed
Check the behavior of the car that is using a different type of controller
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Questions?
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