1. Introduction to Robotics
Alfred Bruckstein
Yaniv Altshuler

2. Course
4 Home assignments (10% each)
Final exam (60%)

3. Plan Introduction and mathematical tools Forward kinematics
Inverse kinematics
Navigation
Multi robotics

4. Bibliography
Physics-Based Animation, Kenny Erleben , Jon Sporring , Knud Henriksen , Henrik Dohlmann, Charles River Media 2005

5. Introduction
Czech playwright Karel Capek in 1921 described a robot (from robota = work, labour) - a machine resembling humans and which can work without effort.
Robots are artificial physical or virtual/software agents that can sense and interact with environment using their sensors and effectors.

6. Introduction: type of robots
Most of physical robots fall into one of the three categories:
Manipulators/robotic arms which are anchored to their workplace and built usually from sets of rigid links connected by joints.
Mobile robots which can move in their environment using wheels, legs, etc.
Hybrid robots which include humanoid robots are mobile robots equipped with manipulators.

7. Introduction: types of sensors
Traditionally robot sensors can be split into two categories:
Proprioceptive sensors which provide information about robot internal state: configuration of joints (shaft encoders), force and torque measured at robots wrist, battery charge, etc.
Exteroceptive sensors which enables a robot to sense its environment. The group involves imaging sensors (cameras), tactile sensors, range finders, GPS, and many others.
Alternatively, sensors can be:
Active – if they involve direct „interaction” with environment to be able to sense it (sonars, range finders).
Passive – if they do not require such interaction (cameras).

8. Introduction: Articulated figures

9. Introduction: Articulated figures
Link : a solid rod, cannot change shape nor lenght
Joint : connection between two links (can rotate/translate with several degrees of freedom)

10. Introduction: effectors part 1
Effectors enabe robots to interact with the environment – i.e. to change their physical configuration.
The kinematic state of a robot effector (constructed of rigid bodies) can be uniquely specified by a constant number of parameters called number of degrees of freedom (DOF).
The dynamic state involves additionally the rate of change of each parameters.
Rigid bodies can have up to 6 DOF which define pose of the body, specified by e.g. Cartesian position (3 DOF) and angular orientation (3 DOF).

11. Introduction: effectors part 2
Katana 6M180 has only 5 DOF, therefore in general its end-effector cannot be aligned with arbitrary 6 DOF pose of a manipulated object.
Set of all end-effector positions which can be reached for some configuration of joint angles is called the reachable workspace.
Set of all positions which can be reached by the end-effector with arbitrary orientations is called the dextrous workspace.

12. Introduction: effectors part 2
Manipulators , r
1 r  4.5
0   50o
x = r cos 
y = r sin 
workspace

13. Introduction: effectors part 3
Mobile robots can have more DOF than the number of actuators. For instance, a car has 3 effective degree of freedom, but only 2 controllable degree of freedom.
A robot is nonholonomic when it has more effective DOF than controllable DOF, and holonomic if these numbers are the same.

14. Control of robotic manipulators: joints
Joints provide a consistent way of connecting arm links.
The configuration of each joint is determined by a specific number of DOF, where each DOF is usually driven by attached motor.
The most common types of joints are:
Revolute joints
1 controllable DOF
Prismatic joints
1 controllable DOF
Spherical joints
3 controllable DOFs

15. Kinematics How can a robot move ?
Kinematics : “study of the motion of parts, without considering mass or forces”

16. Kinematics Kinematics are subdivided in two groups :
forward kinematics
inverse kinematics

17. Kinematics Forward kinematics
Knowing the starting point, what’s the final destination ?
Forward kinematics = computing final destination
Easy, and unique.

18. Kinematics Inverse kinematics I have to get there, how do I do it ?
Inverse kinematics = computing how to arrive to a precise final destination
Not easy, and not always unique !
Additional constraints may be added :
Smoothness
Dynamic limitations
Obstacles

19. Kinematics

20. Mathematical tools
A three dimensional point, in the system {A} :

21. Mathematical tools x y z
The point P is located along the three axes of the coordinate system

22. Transformations
A rotation matrix :

23. Transformations
The product : transformation matrix R by vector point P in the system {B} gives us the vector point P in the system {A}

24. Transformations
Example

25. Transformations With homogeneous coordinates
Pure translation matrix, of vector v :
Translation vector v x
Point P
Point P+v

26. Transformations
Combining rotation and translation

27. Transformations
Example : rotating a frame B relative to a frame A about Z axis by 30° and moving it 10 units in direction of X and 5 units in the direction of Y. What will be the coordinates of a point in frame A if in frame B the point is : [3, 7, 0]T?

28. Paired Joint Coordinates
Articulated figure = many links and joints
Each joint can move
The motion of linkj and jointj affects the motion of linki and jointi for i > j
Very difficult to describe the motion in a system common to all links and joints !
Solution : the Paired Joint Coordinates method

29. Paired Joint Coordinates
Each linki has three predefined orthogonal coordinates systems :
The Body Frame (BFi)
The Inner Frame (IFi)
The Outer Frame (OFi)

30. Paired Joint Coordinates
The Body Frame (BFi)
Associated with linki
Origin generally at its center of mass

31. Paired Joint Coordinates
The Inner Frame (IFi)
Associated with linki and jointi
Origin on the axis of jointi
One axe parallel to the direction of the motion of the joint

32. Paired Joint Coordinates
The Outer Frame (OFi)
Associated with linki and jointi+1
Origin on the axis of jointi+1
One axe parallel to the direction of the motion of the joint

33. Paired Joint Coordinates
With this method we can derive transformations between different frames
But it is a general approach, not easy too use