1. 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)

2. Components of Industrial Robot
Mechanical structure or manipulator
Actuator
Sensors
Control system

3. Modeling and Control of Manipulators
Kinematics
Differential kinematics
Dynamics

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

5. 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

6. “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

7. 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

8. 3-D Kinematics

9. Position and Orientation of a Rigid Body

10. 3-D Homogeneous Transformations
Coordinate transformation (translation+rotation)

11. 3-D Homogeneous Transformations
Homogeneous vector
Homogeneous transformation matrix

12. 3-D Homogeneous Transformations
Composition of coordinate transformations

13. 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?

14. Direct Kinematics

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

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

17. Open Chain

18. Denavit-Hartenberg Convention

19. 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

20. 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.

21. Inverse Kinematics

22. 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!

23. 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)

24. Differential Kinematics and Statics

25. 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

26. 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

27. Jacobian Computation

28. 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.

29. 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

30. Dynamics

31. 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

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

33. 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

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

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

36. Computation of Potential Energy
Consider a manipulator with n rigid links

37. 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

38. 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

39. Trajectory Planning

40. 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

41. 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

42. Point-to-point Motion
Polynomial interpolation
Trapezoidal velocity profile

43. Motion Control

44. 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

45. The Control Problem
Joint space control problem
Open loop

46. 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

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

48. Position and Velocity Feedback