Introduction to Robotics

1
Introduction to Robotics

• Alfred Bruckstein
• Yaniv Altshuler

2
Course

• 4 Home assignments (10 each)
• Final exam (60)

3
Plan

• Introduction and mathematical tools
• Forward kinematics
• Inverse kinematics
• Navigation
• Multi robotics

4
Bibliography

• Physics-Based Animation, Kenny Erleben , Jon
  Sporring , Knud Henriksen , Henrik Dohlmann,
  Charles River Media 2005

5
Introduction
Czech playwright Karel Capek in 1921 described a
robot (from robota work, labour) - a machine
resembling humans and which can work without
effort.
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Introductiontype of robots
Most of physical robots fall into one of the
three categories

• Manipulators/robotic arms which are anchored to
  their workplace and built usually from sets of
  rigid links connected by joints.
• Mobile robots which can move in their environment
  using wheels, legs, etc.
• Hybrid robots which include humanoid robots are
  mobile robots equipped with manipulators.

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Introductiontypes of sensors
Traditionally robot sensors can be split into two
categories

• Proprioceptive sensors which provide information
  about robot internal state configuration of
  joints (shaft encoders), force and torque
  measured at robots wrist, battery charge, etc.
• Exteroceptive sensors which enables a robot to
  sense its environment. The group involves imaging
  sensors (cameras), tactile sensors, range
  finders, GPS, and many others.

Alternatively, sensors can be

• Active if they involve direct interaction
  with environment to be able to sense it (sonars,
  range finders).
• Passive if they do not require such interaction
  (cameras).

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IntroductionArticulated figures
9
IntroductionArticulated figures

• Link a solid rod, cannot change shape nor
  lenght
• Joint connection between two links (can
  rotate/translate with several degrees of freedom)

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Introductioneffectors part 1
Effectors enabe robots to interact with the
environment i.e. to change their physical
configuration.
The kinematic state of a robot effector
(constructed of rigid bodies) can be uniquely
specified by a constant number of parameters
called number of degrees of freedom (DOF).
The dynamic state involves additionally the rate
of change of each parameters.
Rigid bodies can have up to 6 DOF which define
pose of the body, specified by e.g. Cartesian
position (3 DOF) and angular orientation (3 DOF).
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Introductioneffectors part 2
Set of all end-effector positions which can be
reached for some configuration of joint angles is
called the reachable workspace.
Set of all positions which can be reached by the
end-effector with arbitrary orientations is
called the dextrous workspace.
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Introductioneffectors part 2

• Manipulators ?, r
• 1? r ? 4.5
• 0 ? ?? 50o

x r cos ? y r sin ?
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Introductioneffectors part 3
Mobile robots can have more DOF than the number
of actuators. For instance, a car has 3 effective
degree of freedom, but only 2 controllable degree
of freedom.
A robot is nonholonomic when it has more
effective DOF than controllable DOF, and
holonomic if these numbers are the same.
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Control of robotic manipulatorsjoints
Joints provide a consistent way of connecting arm
links.
The configuration of each joint is determined by
a specific number of DOF, where each DOF is
usually driven by attached motor.
The most common types of joints are
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Kinematics

• How can a robot move ?
• Kinematics study of the motion of parts,
  without considering mass or forces

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Kinematics

• Kinematics are subdivided in two groups
• forward kinematics
• inverse kinematics

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Kinematics

• Forward kinematics
• Knowing the starting point, whats the final
  destination ?
• Forward kinematics computing final destination
• Easy, and unique.

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Kinematics

• Inverse kinematics
• I have to get there, how do I do it ?
• Inverse kinematics computing how to arrive to a
  precise final destination
• Not easy, and not always unique !
• Additional constraints may be added
• Smoothness
• Dynamic limitations
• Obstacles

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Kinematics
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Mathematical tools

• A three dimensional point, in the system A

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Mathematical tools

• The point P is located along the three axes of
  the coordinate system

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Transformations

• A rotation matrix

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Transformations

• The product transformation matrix R by vector
  point P in the system B gives us the vector
  point P in the system A

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Transformations

• Example

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Transformations

• With homogeneous coordinates
• Pure translation matrix, of vector v

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Transformations

• Combining rotation and translation

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Transformations

• Example rotating a frame B relative to a frame
  A about Z axis by 30 and moving it 10 units in
  direction of X and 5 units in the direction of Y.
  What will be the coordinates of a point in frame
  A if in frame B the point is 3, 7, 0T?

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Paired Joint Coordinates

• Articulated figure many links and joints
• Each joint can move
• The motion of linkj and jointj affects the motion
  of linki and jointi for i gt j
• Very difficult to describe the motion in a system
  common to all links and joints !
• Solution the Paired Joint Coordinates method

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Paired Joint Coordinates

• Each linki has three predefined orthogonal
  coordinates systems
• The Body Frame (BFi)
• The Inner Frame (IFi)
• The Outer Frame (OFi)

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Paired Joint Coordinates

• The Body Frame (BFi)
• Associated with linki
• Origin generally at its center of mass

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Paired Joint Coordinates

• The Inner Frame (IFi)
• Associated with linki and jointi
• Origin on the axis of jointi
• One axe parallel to the direction of the motion
  of the joint

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Paired Joint Coordinates

• The Outer Frame (OFi)
• Associated with linki and jointi1
• Origin on the axis of jointi1
• One axe parallel to the direction of the motion
  of the joint

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Paired Joint Coordinates

• With this method we can derive transformations
  between different frames
• But it is a general approach, not easy too use