Definition of an Industrial Robot

A robot is a A robot is a re-programmable multifunctionalre-programmable multifunctional manipulator designed to move material, manipulator designed to move material, parts, tools, or specialized devices through parts, tools, or specialized devices through variable programmed motions for the variable programmed motions for the performance of a variety of tasks.performance of a variety of tasks.

Robot Institute of America

(Group within Society of Manufacturing Engineers)

Components of Industrial Robot

Mechanical structure or manipulatorMechanical structure or manipulator

ActuatorActuator

SensorsSensors

Control systemControl system

Modeling and Control of Manipulators

ModelingModeling

• KinematicsKinematics

• Differential kinematicsDifferential kinematics

• DynamicsDynamics

Modeling and Control of Manipulators

ControlControl

• Trajectory planning

• Motion control

• Hardware/software architecture

Mechanical Components

Robots are serial “chain” mechanisms Robots are serial “chain” mechanisms made up of made up of • ““links” (generally considered to be rigid), and links” (generally considered to be rigid), and • ““joints” (where relative motion takes place) joints” (where relative motion takes place)

Joints connect two links Joints connect two links • PrismaticPrismatic• revoluterevolute

“Degrees of Freedom”

Degrees of freedom (DoF) is the number Degrees of freedom (DoF) is the number of independent movements the robot is cof independent movements the robot is capable of apable of

Ideally, each joint has exactly one degree Ideally, each joint has exactly one degree of freedom of freedom • degrees of freedom = number of joints degrees of freedom = number of joints

Industrial robots typically have 6 DoF, but Industrial robots typically have 6 DoF, but 3, 4, 5, and 7 are also common 3, 4, 5, and 7 are also common

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

3-D Kinematics

Position and Orientation of a Rigid BodyPosition and Orientation of a Rigid Body

Coordinate transformation (translation+rotation)

3-D Homogeneous Transformations

Homogeneous vector

Homogeneous transformation matrix

3-D Homogeneous Transformations

Composition of coordinate transformations

3-D Homogeneous Transformations

Euler AnglesEuler 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?

][

Direct Kinematics

Compute the position and orientation of the end effector as a function of the joint variables

Aim of Direct Kinematics

The direct kinematics function is expressed by the homogeneous transformation matrix

Direct Kinematics

Open Chain

Denavit-Hartenberg ConventionDenavit-Hartenberg Convention

Joint Space and Operational SpaceJoint Space and Operational Space

Description of end-effector task

position: coordinates (easy)

orientation: (n s a) (difficult)

w.r.t base frame

Function of time

Operational space

Independent variables

Joint space

Prismatic: d

Revolute: theta

Kinematic RedundancyKinematic Redundancy

Definition

A manipulator is termed kinematically redundant

when it has a number of degrees of mobility whic

h is greater than the number of variables that are

necessary to describe a given task.

Inverse Kinematics

Inverse KinematicsInverse Kinematics

we know the desired “world” or “base” coordinates for the end-effector or tool

we need to compute the set of joint coordinates that will give us this desired position (and orientation in the 6-link case).

the inverse kinematics problem is much more difficult than the forward problem!

Inverse KinematicsInverse KinematicsInverse KinematicsInverse Kinematics

there is no general purpose technique that will guarantee a closed-form solution to the inverse problem!

Multiple solutions may exist Infinite solutions may exist, e.g., in the case

of redundancy There might be no admissible solutions

(condition: x in (dexterous) workspace)

Differential Kinematics and Statics

Differential KinematicsDifferential Kinematics

Find the relationship between the joint velocities and the end-effector linear and angular velocities.

Linear velocity

Angular velocity

i

ii dq

for a revolute joint

for a prismatic joint

Jacobian ComputationJacobian Computation

nOn

Pni

Oi

Pi

O

P qJ

Jq

J

Jq

J

Jv

1

1

1

The contribution of single joint i to the end-effector angular velocity

The contribution of single joint i to the end-effector linear velocity

Jacobian ComputationJacobian Computation

Kinematic SingularitiesKinematic 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 SingularitiesReasons 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

DynamicsDynamics

relationship between the joint actuator torques relationship between the joint actuator torques and the motion of the structureand the motion of the structure

Derivation of dynamic model of a manipulatorDerivation of dynamic model of a manipulator

Simulation of motion

Design of control algorithms

Analysis of manipulator structures

Method based on Lagrange formulationMethod based on Lagrange formulation

Lagrange FormulationLagrange Formulation

Generalized coordinatesGeneralized coordinates n variables which describe the link positions of an

n-degree-of-mobility manipulator

The Lagrange of the mechanical system

Lagrange FormulationLagrange Formulation

The Lagrange’s equations

Generalized force Given by the nonconservative force Joint actuator torques, joint friction torques, joint to

rques induced by interaction with environment

Computation of Kinetic EnergyComputation 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 LinkKinetic 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 EnergyComputation of Potential Energy

Consider a manipulator with n rigid links

Joint Space Dynamic ModelJoint Space Dynamic Model

Viscous friction torques Coulomb

friction torques

Actuation torques

Force and moment exerted on the environment

Multi-input-multi-output; Strong coupling; NonlinearityMulti-input-multi-output; Strong coupling; Nonlinearity

Direct Dynamics and Inverse Dynamics

Direct Dynamics and Inverse Dynamics

Direct dynamics:Direct dynamics: Given joint torques and initial joint position and Given joint torques and initial joint position and

velocity, determine joint accelerationvelocity, determine joint acceleration Useful for simulationUseful for simulation

Inverse dynamics:Inverse dynamics:

Given joint position, velocity and acceleration, determine joint torques

Useful for trajectory planning and control algorithm implementation

Trajectory Planning

Trajectory PlanningTrajectory Planning

Goal: to generate the reference inputs to the Goal: to generate the reference inputs to the motion control system which ensures that the motion control system which ensures that the manipulator executes the planned trajectorymanipulator executes the planned trajectory

Motion control system

Robot Trajectory planning system

torques

Position, velocity, acceleration

Joint Space TrajectoryJoint Space Trajectory

Inverse kinematics algorithm

Trajectory parameters in

operation space

Joint (end-effector) trajectories in terms of position, velocity and acce

leration

Trajectory parameters in joint space Trajectory

planning algorithmInitial and final en

d-effector location, traveling time, etc.

Polynomial interpolationPolynomial interpolation

Trapezoidal velocity profileTrapezoidal velocity profile

Point-to-point MotionPoint-to-point Motion

011

1)( atatatatq nnnn

Motion Control

Motion ControlMotion 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 steady-

state requirements

The Control ProblemThe Control Problem

Joint space control problem

Open loop

Independent Joint ControlIndependent Joint Control

Regard the manipulator as formed by n Regard the manipulator as formed by n

independent systems (n joints)independent systems (n joints)

control each joint as a SISO systemcontrol each joint as a SISO system

treat coupling effects as disturbancetreat coupling effects as disturbance

Independent Joint ControlIndependent Joint Control

Assuming that the actuator is a rotary dc motor Assuming that the actuator is a rotary dc motor

Position and Velocity FeedbackPosition and Velocity Feedback