Definition of an Industrial Robot A robot is

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

Modeling and Control of Manipulators Ø Modeling • Kinematics • Differential kinematics • Dynamics

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

Mechanical Components

“Degrees of Freedom”

Mechanical Configurations Industrial robots are categorized by the first three joint types Ø Five different robot configurations: Ø • • • Cartesian (or Rectangular), Cylindrical, Spherical (or Polar), Jointed (or Revolute), and SCARA

Position and Orientation of a Rigid Body

3 -D Homogeneous Transformations Ø Coordinate transformation (translation+rotation)

3 -D Homogeneous Transformations Ø Homogeneous vector Ø Homogeneous transformation matrix

3 -D Homogeneous Transformations Ø Composition of coordinate transformations

Euler Angles Ø Minimal representation of orientation Ø Three parameters are sufficient Ø Euler Angles Ø Two successive rotations are not made about parallel axes Ø How many kinds of Euler angles are there?

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

Direct Kinematics Ø The direct kinematics function is expressed by the homogeneous transformation matrix

Open Chain

Ø Ø Description of end-effector task Ø position: coordinates (easy) w. r. t base frame Ø orientation: (n s a) (difficult) Function of time Operational space Independent variables Ø Joint space Prismatic: d Revolute: theta

Ø 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.

Differential Kinematics and Statics

Differential Kinematics Ø 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

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 Ø 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 Ø Ø Ø 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 Ø Ø Ø relationship between the joint actuator torques and the motion of the structure Derivation of dynamic model of a manipulator Ø Simulation of motion Ø Design of control algorithms Ø Analysis of manipulator structures Method based on Lagrange formulation

Lagrange Formulation Ø Generalized coordinates Ø Ø n variables which describe the link positions of an n-degree-of-mobility manipulator The Lagrange of the mechanical system

Lagrange Formulation Ø The Lagrange’s equations Ø Generalized force Ø Ø Given by the nonconservative force Joint actuator torques, joint friction torques, joint torques induced by interaction with environment

Computation of Kinetic Energy Ø Consider a manipulator with n rigid links Kinetic energy of link i Kinetic energy of the motor actuating link i

Kinetic Energy of Link Ø Express the kinetic energy as a function of the generalized coordinates of the system, that are the joint variables

Computation of Potential Energy Ø 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 Ø Direct dynamics: Given joint torques and initial joint position and velocity, determine joint acceleration Ø Useful for simulation Ø Ø Inverse dynamics: Ø Given joint position, velocity and acceleration, determine joint torques Ø Useful for trajectory planning and control algorithm implementation

Trajectory Planning

Trajectory Planning Ø 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 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 Ø Polynomial interpolation Ø Trapezoidal velocity profile

Motion Control

Motion Control Ø 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 Ø Joint space control problem Open loop

Independent Joint Control Ø Regard the manipulator as formed by n independent systems (n joints) Ø control Ø treat each joint as a SISO system coupling effects as disturbance

Independent Joint Control Ø Assuming that the actuator is a rotary dc motor

Position and Velocity Feedback