 Definition of an Industrial Robot

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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
Kinematics Differential kinematics Dynamics

Modeling and Control of Manipulators
Trajectory planning Motion control Hardware/software architecture

Mechanical Components
Robots are serial “chain” mechanisms made up of “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 independent movements the robot is capable of Ideally, each joint has exactly one degree of freedom degrees of freedom = number of joints Industrial robots typically have 6 DoF, but 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 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?

Direct 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

Denavit-Hartenberg Convention

Joint 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 Joint space Prismatic: d Revolute: theta Independent variables

Kinematic 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.

Inverse Kinematics

Inverse 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 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 Kinematics
Find the relationship between the joint velocities and the end-effector linear and angular velocities. Linear velocity Angular velocity for a revolute joint for a prismatic joint

The contribution of single joint i to the end-effector linear velocity
Jacobian Computation The contribution of single joint i to the end-effector linear velocity 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 the motor actuating link i Kinetic energy of 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 Actuation torques Coulomb friction torques Force and moment exerted on the environment Multi-input-multi-output; Strong coupling; Nonlinearity

Direct Dynamics and Inverse 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 system
Goal: to generate the reference inputs to the motion control system which ensures that the manipulator executes the planned trajectory Motion control system Robot Trajectory planning system torques Position, velocity, acceleration

Joint Space Trajectory
Trajectory parameters in operation space Trajectory parameters in joint space Inverse kinematics algorithm Trajectory planning algorithm Initial and final end-effector location, traveling time, etc. Joint (end-effector) trajectories in terms of position, velocity and acceleration

Point-to-point Motion
Polynomial interpolation 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 steady-state 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

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