John (Jizhong) Xiao

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Mobile Robot Control
G3300 Advanced Mobile Robotics

• John (Jizhong) Xiao
• Department of Electrical Engineering
• City College of New York
• jxiao_at_ccny.cuny.edu

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Content

• Mobot Kinematics
• Kinematic Motion Control
• Virtual Vehicle Approach
• Homework

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Kinematics of Mobile Robots

• Locomotion is the process of causing an
  autonomous robot to move.
• In order to produce motion, forces must be
  applied to the vehicle
• Dynamics the study of motion in which these
  forces are modeled
• Includes the energies and speeds associated with
  these motions
• Kinematics study of the mathematics of motion
  without considering the forces that affect the
  motion.
• Deals with the geometric relationships that
  govern the system
• Deals with the relationship between control
  parameters and the behavior of a system in state
  space.

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Kinematics Model

• Goal

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Differential Drive
Posture Kinematics Model Kinematics model in
world frame

• Relation between the control input and speed of
  wheels

• Kinematic equation

• Nonholonomic Constraint

Physical Meaning?
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Differential Drive
Kinematics model in robot frame ---configuration
kinematics model
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Differential drive
w(t)
V(t)
Kinematics
VL
L
VR
w ( vR - vL ) / L
ICC
R L ( vR vL ) / ( vR - vL )
R(t)
v wR ( vR vL ) / 2
robots turning radius
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Inverse Kinematics
Given a desired position or velocity, what can we
do to achieve it?
Key question
y
x
VL (t)
VR(t)
starting position
final position
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Inverse Kinematics
Given a desired position or velocity, what can we
do to achieve it?
Key question
y
x
VL (t)
VR(t)
starting position
final position
world information ? wheel information
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Inverse Kinematics
Given a desired position or velocity, what can we
do to achieve it?
Key question
y
x
VL (t)
VR(t)
starting position
final position
11
Inverse Kinematics
Given a desired position or velocity, what can we
do to achieve it?
Key question
y
x
VL (t)
VR(t)
starting position
final position
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Inverse Kinematics
Given a desired position or velocity, what can we
do to achieve it?
Key question
y
Need to solve these equations
x
VL (t)
VR(t)
starting position
final position
for VL (t) and VR(t) .
There are lots of solutions...
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Inverse Kinematics
Given a desired position or velocity, what can we
do to achieve it?
Key question
y
Finding some solution is not hard, but finding
the best solution is...
x
VL (t)
VR(t)
starting position
final position
It all depends on who gets to define best...
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Inverse Kinematics
Given a desired position or velocity, what can we
do to achieve it?
Key question
y
Finding some solution is not hard, but finding
the best solution is...

• quickest time
• most energy efficient
• smoothest velocity profiles

x
VL (t)
VR(t)
starting position
final position
VL (t)
t
VL (t)
It all depends on who gets to define best...
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Inverse Kinematics
Usual approach decompose the problem and
control only a few DOF at a time
Differential Drive
y
x
VL (t)
VR(t)
starting position
final position
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Inverse Kinematics
Usual approach decompose the problem and
control only a few DOF at a time
Differential Drive
(1) turn so that the wheels are parallel to the
line between the original and final position of
the robot origin.
y
-VL (t) VR (t) Vmax
x
VL (t)
VR(t)
starting position
final position
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Inverse Kinematics
Usual approach decompose the problem and
control only a few DOF at a time
Differential Drive
(1) turn so that the wheels are parallel to the
line between the original and final position of
the robot origin.
y
-VL (t) VR (t) Vmax
(2) drive straight until the robots origin
coincides with the destination
x
VL (t)
VL (t) VR (t) Vmax
VR(t)
starting position
final position
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Inverse Kinematics
Usual approach decompose the problem and
control only a few DOF at a time
(1) turn so that the wheels are parallel to the
line between the original and final position of
the robot origin.
Differential Drive
-VL (t) VR (t) Vmax
y
(2) drive straight until the robots origin
coincides with the destination
VL (t) VR (t) Vmax
(3) rotate again in order to achieve the desired
final orientation
x
VL (t)
VR(t)
-VL (t) VR (t) Vmax
VL (t)
starting position
final position
t
VR (t)
only 2 settings (on and off) needed
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Motion Control

• The objective of a kinematic controller is to
  follow a trajectory described by its position
  and/or velocity profiles as function of time.
• Motion control is not straight forward because
  mobile robots are nonholonomic systems.
• However, it has been studied by various research
  groups and some adequate solutions for
  (kinematic) motion control of a mobile robot
  system are available.
• Most controllers are not considering the dynamics
  of the system

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Open Loop Control

• Trajectory (path) divided in motion segments of
  clearly defined shape
• straight lines and segments of a circle.
• Control problem
• pre-compute a smooth trajectory based on line and
  circle segments
• Disadvantages
• It is not at all an easy task to pre compute a
  feasible trajectory
• limitations and constraints of the robots
  velocities and accelerations
• does not adapt or correct the trajectory if
  dynamical changes of the environment occur.
• The resulting trajectories are usually not smooth

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Feedback Control
Compute the error and change in proportion to it.
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Motion Control Methods

• Virtual Vehicle Approach
• Curvature Steering Method
• Flatness Approach
• Dynamic Path Following

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A Virtual Vehicle Approach
Read paper Control of mobile platforms using a
virtual vehicle approach Egerstedt, M. Hu, X.
Stotsky, A., IEEE Transactions on Automatic
Control, Volume 46, Issue 11, pp 1777-1782,
2001.
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A Virtual Vehicle Approach

• The robot model
• The control objective

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A Virtual Vehicle Approach

• Possible controller 1

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A Virtual Vehicle Approach

• The motion parameter

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A Virtual Vehicle Approach

• Orientation of the vehicle

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A Virtual Vehicle Approach

• The forward velocity

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Homework 3

• Consider a differential drive mobile robot and
  write a short article (56 pages) which includes
  at least the following information
• The derivation of the kinematics model of the
  mobile robot
• Include all the details, e.g., what are the
  assumptions for the kinematics model? What is the
  constraint for the kinematics model? What are the
  coordinate systems?
• The development of a motion control algorithm for
  path tracking (virtual vehicle or others)
• The simulation/experimental results for your
  algorithms with discussions
• Include the simulation results for at least two
  paths a circle and a sinusoidal wave. Please
  vary your parameters and compare your results.

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Homework 3

• The format of the short article shall follow the
  IEEE paper standard as follows
• Title
• Author with affiliation
• Abstract
• Introduction
• Kinematics Model
• Path Tracking Control
• Simulation and Discussion
• Conclusions
• Reference

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Thank you!
Homework 3 posted Next class Feb. 20, 2007