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5 Mobile Robot Manueverability

This document discusses the maneuverability of mobile robots. It defines maneuverability as the combination of mobility and steerability. Mobility is based on the sliding constraints of the robot platform, while steerability provides additional degrees of freedom from steering mechanisms. The degree of maneuverability is calculated as the degree of mobility plus the degree of steerability. The document also examines instantaneous centers of rotation, degrees of freedom, and how holonomic and non-holonomic constraints impact a robot's ability to maneuver.

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Aisha Qamar
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620 views12 pages

5 Mobile Robot Manueverability

This document discusses the maneuverability of mobile robots. It defines maneuverability as the combination of mobility and steerability. Mobility is based on the sliding constraints of the robot platform, while steerability provides additional degrees of freedom from steering mechanisms. The degree of maneuverability is calculated as the degree of mobility plus the degree of steerability. The document also examines instantaneous centers of rotation, degrees of freedom, and how holonomic and non-holonomic constraints impact a robot's ability to maneuver.

Uploaded by

Aisha Qamar
Copyright
© © All Rights Reserved
We take content rights seriously. If you suspect this is your content, claim it here.
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Download as PDF, TXT or read online on Scribd
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EE-877 Mobile Robotics

Mobile Robot Kinematics

Dr. Latif Anjum


Assist. Prof. NUST – SEECS
PG EE (Electronics, Power and Control)
Mobile Robot Maneuverability
 The maneuverability of a mobile robot is the combination
 of the mobility available based on the sliding constraints
 plus additional freedom contributed by the steering

 It can be derived using the equation seen before


 Degree of mobility m
 Degree of steerability  s
 Robots maneuverability  M   m   s

Introduction to Autonomous Mobile Robots (2nd Ed), Seigwart


Instructor: Dr. Latif Anjum
Mobile Robot Maneuverability: Degree of Mobility
 To avoid any lateral slip the motion vector R( )I has to satisfy the
following constraints:
C1 f R ( )I  0  C1 f 
C1 (  s )   

C1s (  s ) R( ) I  0 
 1s s 
C ( )

 Mathematically:
 R( )I must belong to the null space of the projection matrix C1 (  s )

 Null space of C1 (  s ) is the space N such that for any vector n in N

C1 (  s )  n  0
 Geometrically this can be shown by the Instantaneous Center of Rotation
(ICR)

Introduction to Autonomous Mobile Robots (2nd Ed), Seigwart


Instructor: Dr. Latif Anjum
Mobile Robot Maneuverability: Instantaneous Center of Rotation

 ICR is always a single point for a robot irrespective of no. of wheels

 Ackermann Steering . Bicycle

Zero Motion Line

Introduction to Autonomous Mobile Robots (2nd Ed), Seigwart


Instructor: Dr. Latif Anjum
Mobile Robot Maneuverability: Instantaneous Center of Rotation

 ICR is always a single point for a robot irrespective of no. of wheels

Introduction to Autonomous Mobile Robots (2nd Ed), Seigwart


Instructor: Dr. Latif Anjum
Mobile Robot Maneuverability: More on Degree of Mobility

 Robot chassis kinematics is a function of the set of independent


constraints   C  C1 f R ( ) I  0
rank C1 ( s ) C1 (  s )   1 f 
C1s (  s ) C1s (  s ) R( )I  0

 the greater the rank of , C1 (  s ) the more constrained is the mobility

 Mathematically
 m  dim N C1 ( s )  3  rank C1 ( s ) 0  rank C1 ( s )  3
 no standard wheels rank C1 ( s )  0
 all direction constrained rank C1 ( s )  3

 Examples:
 Unicycle: One single fixed standard wheel
 Differential drive: Two fixed standard wheels
 wheels on same axle

 wheels on different axle

Introduction to Autonomous Mobile Robots (2nd Ed), Seigwart


Instructor: Dr. Latif Anjum
Mobile Robot Maneuverability: Degree of Steerability

 Indirect degree of motion


 s  rank C1s ( s )
 The particular orientation at any instant imposes a kinematic constraint
 However, the ability to change that orientation can lead additional degree of
maneuverability
 Range of  s : 0 s  2

 Examples:
 Steerability 0: No steering wheel
 Steerability 1: one steered wheel: Tricycle
car (Ackermann steering): Nf = 2, Ns=2 -> common axle
 Steerability 2: two steered wheels: No fixed standard wheel

Introduction to Autonomous Mobile Robots (2nd Ed), Seigwart


Instructor: Dr. Latif Anjum
Mobile Robot Maneuverability: Robot Maneuverability

 Degree of Maneuverability
M  m s

 Two robots with same  M are not necessary equal


 Example: Differential drive and Tricycle (last slide)

 For any robot with  M  2 the ICR is always constrained


to lie on a line
 For any robot with  M  3 the ICR is not constrained and
can be set to any point on the plane

 The Synchro Drive example:  M   m   s  1  1  2

Introduction to Autonomous Mobile Robots (2nd Ed), Seigwart


Instructor: Dr. Latif Anjum
Synchro Drive
Nf = 0
Ns = 3
- No common axle
- Wheels are not independently steerable

Video: J. Borenstein

 M   m   s 11  2

Introduction to Autonomous Mobile Robots (2nd Ed), Seigwart Synchro drive can only move in x – y plane
Instructor: Dr. Latif Anjum
Five Basic Types of Three-Wheel Configurations

Introduction to Autonomous Mobile Robots (2nd Ed), Seigwart


Instructor: Dr. Latif Anjum
Mobile Robot Workspace: Degrees of Freedom
 The Degree of Freedom (DOF) is the robot’s ability to achieve various
poses.

 But what is the degree of vehicle’s freedom in its environment?


 Car example

 Workspace
 how the vehicle is able to move between different configuration in its
workspace?
 The robot’s independently achievable velocities
 = differentiable degrees of freedom (DDOF) = m

 Bicycle:  M   m   s  1 1 DDOF = 1; DOF=3

 Omni Drive:  M   m   s  3 0 DDOF=3; DOF=3

Introduction to Autonomous Mobile Robots (2nd Ed), Seigwart


Instructor: Dr. Latif Anjum
Mobile Robot Workspace: Degrees of Freedom, Holonomy

 DOF degrees of freedom:


 Robots ability to achieve various poses
 DDOF differentiable degrees of freedom:
 Robots ability to achieve various path

DDOF   M  DOF
 Holonomic Robots
 A holonomic kinematic constraint can be expressed as an explicit function of
position variables only.
 A non-holonomic constraint requires a different relationship, such as the
derivative of a position variable
 Fixed and steered standard wheels impose non-holonomic constraints
A robot with no non-holonomic constraints is a holonomic robot and vice versa

Steering locked bicycle and omni drive are both holonomic


Introduction to Autonomous Mobile Robots (2nd Ed), Seigwart
Instructor: Dr. Latif Anjum

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