introduction to robotics - uni-hamburg.detams- · i j. j. craig. introduction to robotics,...

Universitat Hamburg

MIN-FakultatDepartment Informatik

Introduction to Robotics

Introduction to Robotics

Jianwei [email protected]

Universitat HamburgFakultat fur Mathematik, Informatik und NaturwissenschaftenDepartment Informatik

Technische Aspekte Multimodaler Systeme

04. April 2014

J. Zhang 1

Universitat Hamburg


General Information Introduction to Robotics

Outline

General InformationIntroductionArchitectures of Sensor-based Intelligent SystemsConclusions and Outlook

J. Zhang 2

Universitat Hamburg



General Information (1)

Lecture: Friday 10:15 s.t - 11:45 s.t.Room: F334Web: http://tams-www.informatik.uni-hamburg.de/lehre/

Name: Prof. Dr. Jianwei ZhangOffice: F308E-mail: [email protected] hour: (Thursday 15:00 - 16:00)

Secretary: Tatjana TetsisOffice: F311Tel.: +49 40 - 42883-2430E-mail: [email protected]

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Universitat Hamburg



General Information (2)

Exercises: Friday 9:15 s.t - 10:00 s.t.Room: F334Web: http://tams-www.informatik.uni-hamburg.de/lehre/

Name: Hannes BistryOffice: F313Tel.: +49 40 - 42883-2398E-mail: [email protected] hour: by arrangement

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Universitat Hamburg



Exercises:

Criteria for Course Certificate:

I 60 % of points in the exercises

I regular presence in exercises

I presentation of two tasks

I everyone of a group should be able to present the tasks

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Universitat Hamburg



Previous knowledge

I Basics in physics

I (Basics of electrical engineering)

I Linear algebra

I Elementary algebra of matrices

I Programming knowledge

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Universitat Hamburg



Content

I Mathematic concepts (description of space and coordinatetransformations, kinematics, dynamics)

I Control concepts (movement execution)

I Programming aspects(ROS, RCCL)

I Task-oriented movement

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Universitat Hamburg


Introduction Introduction to Robotics

Outline

General InformationIntroduction

Basic termsRobot classificationCoordinate systemsConcatenation of rotation matricesInverse transformationTransformation equationSummary of homogenous transformations

Architectures of Sensor-based Intelligent SystemsConclusions and Outlook

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Universitat Hamburg


Introduction - Basic terms Introduction to Robotics

IntroductionBasic terms

Components of a robot

Robotics: intelligent combination of computers, sensors andactuators.

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Universitat Hamburg



An interdisciplinary field

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Universitat Hamburg



Definition of Industry robots

According to RIA (Robot Institute of America), a robot is:

...a reprogrammable and multifunctional manipulator,devised for the transport of materials, parts, tools orspecialized systems, with varied and programmedmovements, with the aim of carrying out varied tasks.

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Universitat Hamburg



Background of some terms

“Robot” became popular through a stage play by Karel Capek in1923, being a capable servant.

“Robotics” was invented by Isaac Asimov in 1942.

“Autonomous”: (literally) (gr.) “living by one’s own laws” (Auto:Self; nomos: Law)

“Personal Robot”: a small, mobile robot system with simple skillsregarding vision system, speech, movement, etc. (from1980).

“Service Robot”: a mobile handling system featuring sensors forsophisticated operations in service areas (from 1989).

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Universitat Hamburg



A robot’s degree of freedom

Degrees of freedom (DOF):The number of independent coordinate planes or orientations onwhich a joint or end-point of a robot can move.The DOF are determined by the number of independent variablesof the control system.

I On a plane: translational / rotational movement

I In a space: translational / rotational movement - location +orientation (the maximum DOF of a solid object?)

I The DOF of a manipulator: Number of joints which can becontrolled independently. A “Robot” should have at least twodegrees of freedom.

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Universitat Hamburg



A robot’s degree of freedom - Examples

• Kuka LBR 4+ robot arm: 7 (without gripper)• Shadow Air Muscle Robot Hand: 20 (+4 unactuated joints)• 80’s Toy Robot (Quickshot): 4 (without gripper)

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Universitat Hamburg


Introduction - Robot classification Introduction to Robotics

Robot classification

by engine type

I electrical

I hydraulic

I pneumatic

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Universitat Hamburg




by field of workI stationary

I arms with 2 DOFI arms with 3 DOFI ...I arms with 6 DOFI redundant arms (> 6 DOF)I multi-finger hand

I mobileI automated guided vehiclesI portal robotI mobile platformI running machines and flying robotsI anthropomorphic robots (humanoids)

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Universitat Hamburg




by type of joint

I translatory (“linear joint”, “translational”, “cartesian”,“prismatic”)

I rotatory

I combinations

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Universitat Hamburg




by robot coordinate system

I cartesian

I cylindrical

I spherical

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Universitat Hamburg




by usage

I object manipulation

I object modification

I object processing

I transport

I assembly

I quality testing

I deployment in non-accessible areas

I agriculture and forestry

I unterwater

I building industry

I service robot in medicine, housework, ...J. Zhang 19

Universitat Hamburg




by intelligence

I manuel control

I programmable for repeated movements

I featuring cognitive ability and responsiveness

I adaptive on task level

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Universitat Hamburg



Robotics is fun!

I robots move - computers don’tI interdisciplinarity:

I soft- and hardware technologyI sensor technologyI mechatronicsI control engineeringI multimedia, ...

I A dream of mankind:

"Computers are the most ingenious product of human

laziness to date."

computers ⇔ robots

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Universitat Hamburg



Literature

The official slides (including more literature references) areavailable through the TAMS website under ”lectures”

Important secondary literature:

I K. S. Fu, R. C. Gonzales and C. S. G. Lee, Robotics:Control, Sensing, Vision and Intelligence, McGraw-Hill, 1987

I R. P. Paul, Robot Manipulators: Mathematics, Programmingand Control, MIT Press, 1981

I J. J. Craig. Introduction to Robotics, Addison-Wesley, 1989.

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Universitat Hamburg


Introduction - Coordinate systems Introduction to Robotics

Coordinate systems

The position of objects, in other words their location andorientation in Euclidian space can be described throughspecification of a cartesian coordinate system (CS) in relation to abase coordinate system (B).

ezK

exB

ezB

exK

eyK

p’

p

P

B

eyB

e −− unit vectors

p, p’ −− position vectors

CS

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Universitat Hamburg



Specification of location and orientation

position (object-coordinates):

I translation along the axes of the base coordinate system (hereB)

ezK

exB

ezB

exK

eyK

p’

p

P

B

eyB

e −− unit vectors

p, p’ −− position vectors

CS

I given by p = [px , py , pz ]T ∈ R3

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Universitat Hamburg



Specification of location and orientation (cont.)

orientation (in space):

I Euler-angles φ, θ, ψI rotations are performed successively around

the axes of the new coordinate systems, e. g.ZY ′X ′′ or ZX ′Z ′′ (12 possibilities)

Y

X

α’

’β

’’β

α’’

Hinweis:

−− Winkel nach Euler

β

α

−− Winkel nach RPY

X’’

Y’’

X’

Y’

α β

I Gimbal-angles (Roll-Pitch-Yaw)I relative to object coordinates

(used in aviation and maritime)I rotation with respect to fixed axes (X - Roll,

Y - Pitch, Z - Yaw)

I given by Rotation-matrix R ∈ R3×3

I redundant; 9 parameters for 3 DOFR =

r11 r12 r13

r21 r22 r23

r31 r32 r33

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Universitat Hamburg



Specification of location and orientation - summary

I Position:I given through ~p ∈ R3

I Orientation:I given through projection ~n, ~o, ~a ∈ R3 of the axes of the CS to the

origin systemI summarized to rotation matrix R =

[~n ~o ~a

]∈ R3×3

I redundant, since there are 9 parameters for 3 degrees of freedomI other kinds of representation possible, e.g. roll, pitch, yaw angle

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Universitat Hamburg



Coordinate-Transformations

I Transform of Coordinate systems:frame: a reference CStypical frames:I robot base

T6

I end-effectorI table (world)I objectI cameraI screenI ...

Frame-transformations transform one frame into another.

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Universitat Hamburg



Homogenous transformation

I Combination of ~p and R to T =

[R ~p

0 0 0 1

]∈ R4×4

I Concatenation of several T through matrix multiplication

I not commutative, in other words A · B 6= B · A

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Universitat Hamburg



Homogeneous transformation

I Homogeneous transformation matrices:

H =

[R TP S

]whereas P depicts the perspective transformation and S thescaling.

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Universitat Hamburg



Translatory transformation

A translation with a vector [px , py , pz ]T is expressed through atransformation H:

H = T(px ,py ,pz ) =

1 0 0 px0 1 0 py0 0 1 pz0 0 0 1

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Universitat Hamburg



Rotatory transformation

(shortened representation: S : sin, C : cos)The transformation corresponding to a rotation around the x-axiswith angle ψ:

Rx ,ψ =

1 0 0 00 Cψ −Sψ 00 Sψ Cψ 00 0 0 1

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Universitat Hamburg




The transformation corresponding to a rotation around the y -axiswith angle θ:

Ry ,θ =

Cθ 0 Sθ 00 1 0 0−Sθ 0 Cθ 0

0 0 0 1

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Universitat Hamburg




The transformation corresponding to a rotation around the y -axiswith angle φ:

Rz,φ =

Cφ −Sφ 0 0Sφ Cφ 0 00 0 1 00 0 0 1

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Universitat Hamburg



Multiple rotations

Sequential left-multiplication of the transformation matrices byorder of rotation.

An example:1. A rotation ψ around the x-axis Rx,ψ - ”yaw”2. A rotation θ around the y -axis Ry ,θ - ”pitch”3. A rotation φ around the z-axis Rz,φ - ”roll”

J. Zhang 34

Universitat Hamburg


Introduction - Concatenation of rotation matrices Introduction to Robotics

Concatenation of rotation matrices

Rφ,θ,ψ = Rz,φRy ,θRx ,ψ

=

Cφ −Sφ 0 0Sφ Cφ 0 00 0 1 00 0 0 1

Cθ 0 Sθ 00 1 0 0−Sθ 0 Cθ 0

0 0 0 1

1 0 0 00 Cψ −Sψ 00 Sψ Cψ 00 0 0 1

=

CφCθ CφSθSψ − SφCψ CφSθCψ + SφSψ 0SφCθ SφSθSψ + CφCψ SφSθCψ − CφSψ 0−Sθ CθSψ CθCψ 0

0 0 0 1

Remark: Matrice multiplication is not commutative:

AB 6= BA

J. Zhang 35

Universitat Hamburg


Introduction - Concatenation of rotation matrices Introduction to Robotics

Coordinate frames

They are represented as four vectors using the elements ofhomogenous transformation.

H =

[r1 r2 r3 p0 0 0 1

]=

r11 r12 r13 pxr21 r22 r23 pyr31 r32 r33 pz0 0 0 1

(1)

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Universitat Hamburg


Introduction - Inverse transformation Introduction to Robotics

Inverse transformation

The inverse of a rotation matrix is simply its transpose:R−1 = RT and RRT = I

whereas I is the identity matrix.The inverse of (1) is:

H−1 =

r11 r21 r31 −p · r1

r12 r22 r32 −p · r2

r13 r23 r33 −p · r3

0 0 0 1

whereas r1, r2, r3 and p are the four column vectors of (1)and · represents the scalar product of vectors.

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Universitat Hamburg


Introduction - Inverse transformation Introduction to Robotics

Relative transformations

One has the following transformations:

I Z: World → Manipulator base

I T6: Manipulator base → Manipulator end

I E: Manipulator end → Endeffector

I B: World → Object

I G: Object → Endeffector

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Universitat Hamburg


Introduction - Transformation equation Introduction to Robotics

Transformation equation

There are two descriptions of the endeffector position, one in relation tothe object and the other in relation to the manipulator. Both descriptionsare equal to eachother:

ZT6E = BG

In order to find the manipulator transformation:

T6 = Z−1BGE−1

In order to determine the position of the object:

B = ZT6EG−1

This is also called kinematic chain.J. Zhang 39

Universitat Hamburg


Introduction - Transformation equation Introduction to Robotics

Example: coordinate transformation

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Universitat Hamburg


Introduction - Summary of homogenous transformations Introduction to Robotics

Summary of homogenous transformations

I A homogenous transformation depicts the position andorientation of a coordinate frame in space.

I If the coordinate frame is defined in relation to a solidobject, the position and orientation of the solid object isunambiguously specified.

I The depiction of an object A can be derived from ahomogenous transformation relating to object B. This is alsopossible the other way around using inverse transformation.

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Universitat Hamburg



Summary of homogenous transformations

I Several translations and rotations can be multiplied. Thefollowing applies:

I If the rotations / translations are performed in relation to thecurrent, newly defined (or changed) coordinate system, the newlyadded transformation matrices need to be multiplicativelyappended on the right-hand side.

I If all of them are performed in relation to the fixed referencecoodinate system, the transformation matrices need to bemultiplicatively appended on the left-hand side.

I A homogenous transformation can be segemented into arotation and a translation part.

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Universitat Hamburg



Robot kinematics

Quite often, only position and orientation of the robot gripper isof interest. In that case, a robot is treated just like a regularobject, depicted through a transformation like all others.

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Universitat Hamburg



Coordinates of a manipulator

I Joint coordinates:

A vector q(t) = (q1(t), q2(t), ..., qn(t))T

(a robot configuration)

I Endeffector coordinates

(Object coordinates):A Vector p = [px , py , pz ]T

I Description of orientations:I Euler angle φ, θ, ψI Rotation matrix:

R =

r11 r12 r13

r21 r22 r23

r31 r32 r33

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Universitat Hamburg



Denavit Hartenberg Convention (outlook)

I Definition of one coordinate system per segment i = 1..n

I Definition of 4 parameters per segment i = 1..n

I Definition of one transformation Ai per segment i = 1..n

I T6 =∏n

i=1 Ai

Later Denavit Hartenberg Convention will be presented moredetailed!

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Universitat Hamburg



Kinematics

I The direct kinematic problem:Given the joint values and geometrical parameters of all jointsof a manipulator, how is it possible to determine the positionand orientation of the manipulator-endeffector?

I The inverse kinematic problem:Given a desired position and orientation of themanipulator-endeffector and the geometrical parameters of alljoints, is it possible for the manipulator to reach this position /orientation? If it is, how many manipulator configurations arecapable of matching these conditions?(An example: A two-joint-manipulator moving on a plane)

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Universitat Hamburg



Position

T6 defines, how the n joint angles are supposed to be consolidatedto 12 non-linear formulas in order to descriebe 6 cartesian degreesof freedom.I Forward kinematics K defined as:

I K : ~θ ∈ Rn → ~x ∈ R6

I Joint angle → Position + Orientation

I Inverse kinematics K−1 defined as:I K−1 : ~x ∈ R6 → ~θ ∈ Rn

I Position + Orientation → Joint angleI non-trivial, since K is usually not unambiguously invertible

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Universitat Hamburg



Differential movement

Non-linear kinematics K can be linearized through the Taylor series

f (x) =∑∞

n=0f (n)(x0)

n! (x − x0)n.

I The Jacobi matrix J as factor for n = 1 of themulti-dimensional Taylor series is defined as:

I J(~θ) : ~θ ∈ Rn → ~x ∈ R6

I Joint speed → kartesian speed

I Inverse Jacobi matrix J−1 defined as:I J−1(~θ) : ~x ∈ R6 → ~θ ∈ Rn

I kartesian speed → Joint speedI non-trivial, since J not necessarily invertible (e.g. not quadratic)

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Universitat Hamburg



Motion planning

Since T6 describes only the target position, explicit generation ofa trajectory is necessary - depending on constraints differently for:

I joint angle space

I cartesiaan space

Interpolation through:

I piecewise straight lines

I piecewise polynoms

I B-Splines

I ...

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Universitat Hamburg



Suggestions

1.1 Read: J. F. Engelberger. Robotics in Service, The MIT Press,1989. (available in key texts)

1.3 Repeat your linear algebra knowledge, especially regardingelementary algebra of matrices.

J. Zhang 50

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