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Definition of an Industrial
Robot
A robot is a re-programmable multifunctional
manipulator designed to move material, parts,
tools, or specialized devices through variable
programmed motions for the performance of
a variety of tasks.
Robot Institute of America
(Group within Society of Manufacturing Engineers)
Components of Industrial
Robot

Mechanical structure or manipulator

Actuator

Sensors

Control system
Modeling and Control of
Manipulators

Modeling
•
Kinematics
•
Differential kinematics
•
Dynamics
Modeling and Control of
Manipulators

Control
•
Trajectory planning
•
Motion control
•
Hardware/software architecture
Mechanical Components

Robots are serial “chain” mechanisms made
•
•

“links” (generally considered to be rigid), and
“joints” (where relative motion takes place)
Joints connect two links
•
•
Prismatic
revolute
“Degrees of Freedom”
Degrees of freedom (DoF) is the number of in
 Ideally, each joint has exactly one degree of fr

•

degrees of freedom = number of joints
Industrial robots typically have 6 DoF, but 3, 4
Mechanical Configurations
Industrial robots are categorized by the
first three joint types
 Five different robot configurations:

•
•
•
•
•
Cartesian (or Rectangular),
Cylindrical,
Spherical (or Polar),
Jointed (or Revolute), and
SCARA
Kinematics
Position and Orientation of a Rigid Body
3-D Homogeneous Transformations
 Coordinate transformation (translation+rotation)
3-D Homogeneous Transformations
 Homogeneous vector
 Homogeneous transformation matrix
3-D Homogeneous Transformations
 Composition of coordinate transformations
Euler Angles

Minimal representation of orientation
 Three parameters are sufficient

Euler Angles
  [   ]
 Two successive rotations are not made about
parallel axes
 How many kinds of Euler angles are there?
Kinematics
Aim of Direct Kinematics
Compute the position and orientation of the end
effector as a function of the joint variables
Direct Kinematics
 The direct kinematics function is expressed
by the homogeneous transformation matrix
Open Chain
rtenberg Convention
Operational Space


Description of end-effector task
 position: coordinates (easy)
w.r.t base frame
 orientation: (n s a) (difficult)
Function of time
Operational space
Independent
variables

Joint space
Prismatic: d
Revolute: theta
Redundancy
 Definition
A manipulator is termed kinematically redundant
when it has a number of degrees of mobility
which is greater than the number of variables that
are necessary to describe a given task.
e Kinematics
e Kinematics



we know the desired “world” or “base” coordinates for
we need to compute the set of joint coordinates that w
the inverse kinematics problem is much more difficult
se Kinematics




there is no general purpose technique that will guaran
Multiple solutions may exist
Infinite solutions may exist, e.g., in the case of redund
There might be no admissible solutions (condition: x i
Differential Kinematics and Statics
Differential Kinematics

Find the relationship between the joint velocities
and the end-effector linear and angular velocities.
Linear velocity
Angular velocity
i for a revolute joint
qi   
 di for a prismatic joint
Jacobian Computation
The contribution of single joint i to
the end-effector linear velocity
 J P1 
 J Pi 
 J Pn 
v    q1      q i      q n
 J O1 
 J Oi 
 J On 
The contribution of single joint i to
the end-effector angular velocity
Jacobian Computation
Kinematic Singularities

The Jacobian is, in general, a function of the
configuration q; those configurations at which J is
rank-deficient are termed Kinematic singularities.
Reasons to Find Singularities



Singularities represent configurations at which
mobility of the structure is reduced
Infinite solutions to the inverse kinematics problem
may exist
In the neighborhood of a singularity, small
velocities in the operational space may cause large
velocities in the joint space
Dynamics
Dynamics



relationship between the joint actuator torques
and the motion of the structure
Derivation of dynamic model of a manipulator

Simulation of motion

Design of control algorithms

Analysis of manipulator structures
Method based on Lagrange formulation
Lagrange Formulation

Generalized coordinates


n variables which describe the link positions of an
n-degree-of-mobility manipulator
The Lagrange of the mechanical system
Lagrange Formulation

The Lagrange’s equations

Generalized force


Given by the nonconservative force
Joint actuator torques, joint friction torques, joint
torques induced by interaction with environment
Computation of Kinetic Energy

Consider a manipulator with n rigid links
Kinetic energy
of link i
Kinetic energy
of the motor
actuating link i
Kinetic Energy of Link

Express the kinetic energy as a function of
the generalized coordinates of the system,
that are the joint variables
Computation of Potential Energy

Consider a manipulator with n rigid links
Joint Space Dynamic Model
Viscous
friction
torques
Coulomb
friction
torques
Actuation
torques
Force and moment
exerted on the
environment
Multi-input-multi-output; Strong coupling; Nonlinearity
Direct Dynamics and Inverse
Dynamics

Direct dynamics:



Given joint torques and initial joint position and
velocity, determine joint acceleration
Useful for simulation
Inverse dynamics:

Given joint position, velocity and acceleration,
determine joint torques

Useful for trajectory planning and control algorithm
implementation
Trajectory Planning
Trajectory Planning

Goal: to generate the reference inputs to the
motion control system which ensures that the
manipulator executes the planned trajectory
torques
Trajectory
planning system
Motion control
system
Robot
Position, velocity, acceleration
Joint Space Trajectory
Trajectory
parameters in
operation space
Initial and final
end-effector
location, traveling
time, etc.
Inverse
kinematics
algorithm
Trajectory
parameters in
joint space
Trajectory
planning
algorithm
Joint (end-effector)
trajectories in terms of
position, velocity and
acceleration
Point-to-point Motion

Polynomial interpolation
q(t )  a n t  a n1 t
n

n 1
   a1t  a0
Trapezoidal velocity profile
Motion Control
Motion Control

Determine the time history of the generalized
forces to be developed by the joint actuators so
as to guarantee execution of the commanded
task while satisfying given transient and steadystate requirements
The Control Problem

Joint space control problem
Open loop
Independent Joint Control

Regard the manipulator as formed by n
independent systems (n joints)

control each joint as a SISO system

treat coupling effects as disturbance
Independent Joint Control

Assuming that the actuator is a rotary dc motor
Position and Velocity Feedback