Introduction to
Robotics
Alfred Bruckstein
Yaniv Altshuler
Course
• 4 Home assignments (10% each)
• Final exam (60%)
Plan
• Introduction and mathematical tools
• Forward kinematics
• Inverse kinematics
• Navigation
• Multi robotics
Bibliography
• Physics-Based Animation,
Kenny
Erleben , Jon Sporring , Knud
Henriksen , Henrik Dohlmann, Charles
River Media 2005
Introduction
Czech playwright Karel Capek in 1921 described a robot (from robota =
work, labour) - a machine resembling humans and which can work
without effort.
Robots are artificial physical or virtual/software agents that can sense
and interact with environment using their sensors and effectors.
Introduction:
type 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.
Introduction:
types 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).
Introduction:
Articulated figures
Introduction:
Articulated figures
•
•
Link : a solid rod, cannot change shape nor lenght
Joint : connection between two links (can rotate/translate
with several degrees of freedom)
Introduction:
effectors 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).
Introduction:
effectors part 2
Katana 6M180 has only 5 DOF, therefore in general its
end-effector cannot be aligned with arbitrary 6 DOF
pose of a manipulated object.
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 endeffector with arbitrary orientations is called the
dextrous workspace.
Introduction:
effectors part 2
Manipulators , r
1 r  4.5
0   50o

x = r cos 
y = r sin 
workspace
Introduction:
effectors 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.
Control of robotic manipulators:
joints
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:
Revolute joints
Prismatic joints
1 controllable DOF
1 controllable DOF
Spherical joints
3 controllable DOFs
Kinematics
• How can a robot move ?
• Kinematics : “study of the motion of
parts, without considering mass or
forces”
Kinematics
• Kinematics are subdivided in two
groups :
• forward kinematics
• inverse kinematics
Kinematics
• Forward kinematics
•
•
•
Knowing the starting point, what’s the final
destination ?
Forward kinematics = computing final
destination
Easy, and unique.
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
Kinematics
Mathematical tools
• A three dimensional point, in the system
{A} :
Mathematical tools
• The point P is located along the three
axes of the coordinate system
z
y
x
Transformations
• A rotation matrix :
Transformations
• The product : transformation matrix R
by vector point P in the system {B}
gives us the vector point P in the
system {A}
Transformations
• Example
Transformations
• With homogeneous coordinates
• Pure translation matrix, of vector v :
Point P
x
x
Translation vector v
Point P+v
Transformations
• Combining rotation and translation
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, 0]T?
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 > j
Very difficult to describe the motion in a system common to all
links and joints !
Solution : the Paired Joint Coordinates method
Paired Joint Coordinates
• Each link has three predefined
i
orthogonal coordinates systems :
• The Body Frame (BF )
• The Inner Frame (IF )
• The Outer Frame (OF )
i
i
i
Paired Joint Coordinates
• The Body Frame (BF )
• Associated with link
• Origin generally at its center of mass
i
i
Paired Joint Coordinates
• The Inner Frame (IF )
• Associated with link and joint
• Origin on the axis of joint
• One axe parallel to the direction of the
i
i
i
i
motion of the joint
Paired Joint Coordinates
• The Outer Frame (OF )
• Associated with link and joint
• Origin on the axis of joint
• One axe parallel to the direction of the
i
i
i+1
i+1
motion of the joint
Paired Joint Coordinates
• With this method we can derive
transformations between different
frames
• But it is a general approach, not easy
too use