Introduction to Robotics

Alfred BrucksteinYaniv Altshuler

Course

•4 Home assignments (10% each)

•Final exam (60%)

Plan•Introduction and mathematical

tools

•Forward kinematics

•Inverse kinematics

•Navigation

•Multi robotics

Bibliography

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

IntroductionIntroduction

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.

Introduction:Introduction:type of robotstype 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.

Introduction:Introduction:types of types of sensorssensors

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).

Introduction:Introduction:Articulated figuresArticulated figures

• Link : a solid rod, cannot change shape nor lenght

• Joint : connection between two links (can rotate/translate with several degrees of freedom)

Introduction:Introduction:Articulated figuresArticulated figures

Introduction:Introduction:effectorseffectors part 1 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).

Introduction:Introduction:effectorseffectors part 2 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.

Manipulators , r

1 r 4.5

0 50o

r

x = r cos

y = r sin workspace

Introduction:Introduction:effectorseffectors part 2 part 2

Introduction:Introduction:effectorseffectors part 3 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.

Control of robotic manipulators:Control of robotic manipulators:jointsjoints

The configuration of each joint is determined by a specific number of DOF, where each DOF is usually driven by attached motor.

Revolute joints

1 controllable DOF

Prismatic joints

1 controllable DOF

Spherical joints

3 controllable DOFs

Joints provide a consistent way of connecting arm links.

The most common types of joints are:

Kinematics

•How can a robot move ?

•Kinematics : “study of the motion of parts, without considering mass or forces”

Kinematics

•Kinematics are subdivided in two groups :

•forward kinematics

•inverse kinematics

Kinematics

•Forward kinematics

• Knowing the starting point, what’s the final destination ?

• Forward kinematics = computing final destination

• Easy, and unique.

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

Kinematics

Mathematical tools

•A three dimensional point, in the system {A} :

x

y

z

Mathematical tools

•The point P is located along the three axes of the coordinate system

Transformations•A rotation matrix :

Transformations

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

Transformations•Example

Transformations

•With homogeneous coordinates

•Pure translation matrix, of vector v :

Translation vector v

x

x

Point P

Point P+v

Transformations

•Combining rotation and translation

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?

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

Paired Joint Coordinates

•Each linki has three predefined orthogonal coordinates systems :

•The Body Frame (BFi)

•The Inner Frame (IFi)

•The Outer Frame (OFi)

Paired Joint Coordinates•The Body Frame (BFi)

•Associated with linki

•Origin generally at its center of mass

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

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

Paired Joint Coordinates

•With this method we can derive transformations between different frames

•But it is a general approach, not easy too use