Mobile Robot
Simulation and Control 

? Maplesoft, a division of Waterloo Maple Inc., 2008 

Given the model of a two wheel differential drive mobile robot, we would like to simulate its trajectory response to various inputs in both open and closed loop settings. The robot model has been derived in a separate application example worksheet titled Mobile Robot Modeling and Simulation . 

System Definition 

Define the Variables and the Parameters that describe the system. 

Parameters 

Name	Value	Units
Robot Parameters
Robot Chassis Radius ()
Robot Chassis Mass ()
Robot Chassis Moment of Inertia ()
Wheel Parameters
Wheel Radius ()
Wheel Mass ()
Wheel Moment of Inertia ()
Motor Parameters
DC Motor Armature Resistance ()
DC Motor Armature Inductance ()
DC Motor Torque Constants ()
DC Motor EMF Constants ()

# Retrieve Parameters Defined

Variables 

 

Name	Description
Robot State
	Robot Position (x and y coordinates)
	Robot Heading (counter clockwise from the x-axis when viewed from top)
	Robot Linear Velocity (w.r.t. Body Fixed Frame, positive going forward)
	Robot Angular Velocity (w.r.t. Body Fixed Frame, about vertical axis)
Wheel Variables
	Wheel Angular Speed (left and right)
	Wheel Applied Torque (left and right)
	Wheel Traction Force (left and right)
Motor Variables
	Motor Current (left and right)
	Motor Input Voltage (left and right)

Robot System Model 

The model of the robot has been derived in the MobileRobotModel worksheet.  The overall system of equations is given as follows: 

# Overall System Model 

 

Equations for the linear acceleration in the x and y direction and the angular acceleration: 

 

(2.1)

 

Equations for the linear and angular velocities: 

 

(2.2)

 

Equations for the angular velocities of the wheels: 

 

(2.3)

 

 

(2.4)

 

Equations for the wheel torques: 

 

(2.5)

 

Equations for the motor currents: 

 

(2.6)

 

 

Simulation 

Open Loop Case 

Here, we specify the input voltage to each of the two motors and observe the system response (uncontrolled). 

System Input 

# Open Loop System Input 

The inputs to the combined system are the applied voltage to the motor. For the simulation, they are defined as piecewise continuous functions.
(3.1.1.1)
(3.1.1.2)

System Initial Conditions 

# Open Loop Initial Conditions 

The robot is assumed to be at the origin at rest. The following initial conditions describe this state.
Robot initial conditions    (3.1.2.1)	Wheel initial conditions    (3.1.2.2)	Motor initial conditions    (3.1.2.3)

System Response 

# Open Loop System Response 

 

Robot X and Y Position	Robot Heading
Robot Chassis Linear and Angular Velocities	Trajectory Animation

 

 

Speed Control 

We want to implement a proportional-integral PI controllers to control the linear and angular speeds. The linear speed determines how fast the robot is moving forward, while the angular speed determines how fast the robot is turning. 

System Input 

# Speed Control System Input 

The following inputs are applied to the system. The robot will first turn (rotate), then start moving forward while rotating, and finally move forward without turning before coming to a stop.

Controller Definition 

We will use two independent PI controllers to control the forward () and turning () speed of the robot. 

The output of the PI controllers are the forward control input and the turning control input  

 

(3.2.2.1)

 

 

 

(3.2.2.2)

 

 

 

where the error (and its integral) in and are defined as: 

 

 

(3.2.2.3)

 

 

 

(3.2.2.4)

 

 

(3.2.2.5)

 

 

(3.2.2.6)

 

 

Parameter values for the Speed PI Controllers  Gain  Value  Forward Speed Control          Turning Speed Control          Initial Conditions:      (3.2.2.7)	(3.2.2.10)     (3.2.2.11)     Substituting the forward and turning controls in and into the following, we get:       (3.2.2.12)     (3.2.2.13)
The forward speed controller drives both wheels forward, while the turning controller drives each wheel in opposite direction. As such, the forward and turning control inputs relate to the left and right motor inputs as follows:      (3.2.2.8)       (3.2.2.9)     Writing the motor inputs in terms of the forward and turning controls:    >

Substituting the numeric values of the controller gains, we get: 

 

 

(3.2.2.14)

 

 

System Response 

# Response with Speed Control 

Forward Speed	Angular Speed
Motor Control Signals	Trajectory Animation

Heading Control 

Here, we specify the desired heading for the mobile robot to follow. The heading controller is implemented as an outer loop controller to the speed controlled robot system.  Again, we will be using a PI controller in this example. 

System Input 

# Heading Control System Input 

The following inputs are applied. The robot is asked to start moving forward and turning at the same time. Then it will keep moving forward while maintaining a constant heading. After that, it will turn again and eventually come to a stop while pointing at the desired heading.

 

Controller Definition 

Similar to the speed controller, a PI controller is used for the heading control.  In this case, the output of the controller is the desired turning rate :  

 

(3.3.2.1)

 

 

 

where the heading error (and its integral) is defined as: 

 

(3.3.2.2)

 

 

(3.3.2.3)

 

 

Parameters for the Heading PI Controller:    Gain  Value  Heading Control            Initial Conditions:      (3.3.2.4)

Substituting the numeric values of the controller gains, we get:    >   (3.3.2.5)

System Response 

# System Response with Heading 

Forward Speed	Heading
Angular Speed	Trajectory Animation