Introduction to Robotics

Transcription

1 Jianwei Zhang Universität Hamburg Fakultät für Mathematik, Informatik und Naturwissenschaften Technische Aspekte Multimodaler Systeme 04. April 2014 J. Zhang 1

2 General Information Outline General Information Introduction Architectures of Sensor-based Intelligent Systems Conclusions and Outlook J. Zhang 2

3 General Information General Information (1) Lecture: Room: Web: Friday 10:15 s.t - 11:45 s.t. F334 Name: Prof. Dr. Jianwei Zhang Office: F308 zhang@informatik.uni-hamburg.de Consultation hour: (Thursday 15:00-16:00) Secretary: Tatjana Tetsis Office: F311 Tel.: tetsis@informatik.uni-hamburg.de J. Zhang 3

4 General Information General Information (2) Exercises: Room: Web: Friday 9:15 s.t - 10:00 s.t. F334 Name: Hannes Bistry Office: F313 Tel.: bistry@informatik.uni-hamburg.de Consultation hour: by arrangement J. Zhang 4

5 General Information Exercises: Criteria for Course Certificate: 60 % of points in the exercises regular presence in exercises presentation of two tasks everyone of a group should be able to present the tasks J. Zhang 5

6 General Information Previous knowledge Basics in physics (Basics of electrical engineering) Linear algebra Elementary algebra of matrices Programming knowledge J. Zhang 6

7 General Information Content Mathematic concepts (description of space and coordinate transformations, kinematics, dynamics) Control concepts (movement execution) Programming aspects(ros, RCCL) Task-oriented movement J. Zhang 7

8 Introduction Universität Hamburg Outline General Information Introduction Basic terms Robot classification Coordinate systems Concatenation of rotation matrices Inverse transformation Transformation equation Summary of homogenous transformations Architectures of Sensor-based Intelligent Systems Conclusions and Outlook J. Zhang 8

9 Introduction - Basic terms Introduction Basic terms Components of a robot Robotics: intelligent combination of computers, sensors and actuators. J. Zhang 9

10 Introduction - Basic terms An interdisciplinary field J. Zhang 10

11 Introduction - Basic terms 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 or specialized systems, with varied and programmed movements, with the aim of carrying out varied tasks. J. Zhang 11

12 Introduction - Basic terms Background of some terms Robot became popular through a stage play by Karel Capek in 1923, being a capable servant. Robotics was invented by Isaac Asimov in Autonomous : (literally) (gr.) living by one s own laws (Auto: Self; nomos: Law) Personal Robot : a small, mobile robot system with simple skills regarding vision system, speech, movement, etc. (from 1980). Service Robot : a mobile handling system featuring sensors for sophisticated operations in service areas (from 1989). J. Zhang 12

13 Introduction - Basic terms A robot s degree of freedom Degrees of freedom (DOF): The number of independent coordinate planes or orientations on which a joint or end-point of a robot can move. The DOF are determined by the number of independent variables of the control system. On a plane: translational / rotational movement In a space: translational / rotational movement - location + orientation (the maximum DOF of a solid object?) The DOF of a manipulator: Number of joints which can be controlled independently. A Robot should have at least two degrees of freedom. J. Zhang 13

14 Introduction - Basic terms 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) J. Zhang 14

15 Introduction - Robot classification Robot classification by engine type electrical hydraulic pneumatic J. Zhang 15

16 Introduction - Robot classification Robot classification by field of work stationary arms with 2 DOF arms with 3 DOF... arms with 6 DOF redundant arms (> 6 DOF) multi-finger hand mobile automated guided vehicles portal robot mobile platform running machines and flying robots anthropomorphic robots (humanoids) J. Zhang 16

17 Introduction - Robot classification Robot classification by type of joint translatory ( linear joint, translational, cartesian, prismatic ) rotatory combinations J. Zhang 17

18 Introduction - Robot classification Robot classification by robot coordinate system cartesian cylindrical spherical J. Zhang 18

19 Introduction - Robot classification Robot classification by usage object manipulation object modification object processing transport assembly quality testing deployment in non-accessible areas agriculture and forestry unterwater building industry service robot in medicine, housework,... J. Zhang 19

20 Introduction - Robot classification Robot classification by intelligence manuel control programmable for repeated movements featuring cognitive ability and responsiveness adaptive on task level J. Zhang 20

21 Introduction - Robot classification Robotics is fun! robots move - computers don t interdisciplinarity: soft- and hardware technology sensor technology mechatronics control engineering multimedia,... A dream of mankind: "Computers are the most ingenious product of human laziness to date." computers robots J. Zhang 21

22 Introduction - Robot classification Literature The official slides (including more literature references) are available through the TAMS website under lectures Important secondary literature: K. S. Fu, R. C. Gonzales and C. S. G. Lee, Robotics: Control, Sensing, Vision and Intelligence, McGraw-Hill, 1987 R. P. Paul, Robot Manipulators: Mathematics, Programming and Control, MIT Press, 1981 J. J. Craig., Addison-Wesley, J. Zhang 22

23 Introduction - Coordinate systems Coordinate systems The position of objects, in other words their location and orientation in Euclidian space can be described through specification of a cartesian coordinate system (CS) in relation to a base coordinate system (B). e zk CS e xk e yk p P p e zb B e xb e yb e unit vectors p, p position vectors J. Zhang 23

24 Introduction - Coordinate systems Specification of location and orientation position (object-coordinates): translation along the axes of the base coordinate system (here B) e zk CS P p e yk e xk p given by p = [p x, p y, p z ] T R 3 e zb B e xb e yb e unit vectors p, p position vectors J. Zhang 24

25 Introduction - Coordinate systems Specification of location and orientation (cont.) orientation (in space): Euler-angles φ, θ, ψ rotations are performed successively around the axes of the new coordinate systems, e. g. ZY X or ZX Z (12 possibilities) Gimbal-angles (Roll-Pitch-Yaw) relative to object coordinates (used in aviation and maritime) rotation with respect to fixed axes (X - Roll, Y - Pitch, Z - Yaw) given by Rotationmatrix R R 3 3 redundant; 9 parameters for 3 DOF R = Y Y α β α Y X β α β X Hinweis: α Winkel nach Euler β Winkel nach RPY r 11 r 12 r 13 r 21 r 22 r 23 r 31 r 32 r 33 X J. Zhang 25

26 Introduction - Coordinate systems Specification of location and orientation - summary Position: given through p R 3 Orientation: given through projection n, o, a R 3 of the axes of the CS to the origin system [ ] summarized to rotation matrix R = n o a R 3 3 redundant, since there are 9 parameters for 3 degrees of freedom other kinds of representation possible, e.g. roll, pitch, yaw angle J. Zhang 26

27 Introduction - Coordinate systems Coordinate-Transformations Transform of Coordinate systems: frame: a reference CS typical frames: robot base end-effector table (world) object camera screen... T 6 Frame-transformations transform one frame into another. J. Zhang 27

28 Introduction - Coordinate systems Homogenous transformation [ ] R p Combination of p and R to T = R Concatenation of several T through matrix multiplication not commutative, in other words A B B A J. Zhang 28

29 Introduction - Coordinate systems Homogeneous transformation Homogeneous transformation matrices: [ ] R T H = P S whereas P depicts the perspective transformation and S the scaling. J. Zhang 29

30 Introduction - Coordinate systems Translatory transformation A translation with a vector [p x, p y, p z ] T is expressed through a transformation H: p x H = T (px,p y,p z ) = p y p z J. Zhang 30

31 Introduction - Coordinate systems Rotatory transformation (shortened representation: S : sin, C : cos) The transformation corresponding to a rotation around the x-axis with angle ψ: R x,ψ = 0 Cψ Sψ 0 0 Sψ Cψ J. Zhang 31

32 Introduction - Coordinate systems Rotatory transformation The transformation corresponding to a rotation around the y-axis with angle θ: Cθ 0 Sθ 0 R y,θ = Sθ 0 Cθ J. Zhang 32

33 Introduction - Coordinate systems Rotatory transformation The transformation corresponding to a rotation around the y-axis with angle φ: Cφ Sφ 0 0 R z,φ = Sφ Cφ J. Zhang 33

34 Introduction - Coordinate systems Multiple rotations Sequential left-multiplication of the transformation matrices by order of rotation. An example: 1. A rotation ψ around the x-axis R x,ψ - yaw 2. A rotation θ around the y-axis R y,θ - pitch 3. A rotation φ around the z-axis R z,φ - roll J. Zhang 34

35 Introduction - Concatenation of rotation matrices Concatenation of rotation matrices R φ,θ,ψ = R z,φ R y,θ R x,ψ Cφ Sφ 0 0 Cθ 0 Sθ = Sφ Cφ Cψ Sψ Sθ 0 Cθ 0 0 Sψ Cψ CφCθ CφSθSψ SφCψ CφSθCψ + SφSψ 0 = SφCθ SφSθSψ + CφCψ SφSθCψ CφSψ 0 Sθ CθSψ CθCψ Remark: Matrice multiplication is not commutative: AB BA J. Zhang 35

36 Introduction - Concatenation of rotation matrices Coordinate frames They are represented as four vectors using the elements of homogenous transformation. H = [ r1 r 2 r 3 ] p r 11 r 12 r 13 p x = r 21 r 22 r 23 p y r 31 r 32 r 33 p z (1) J. Zhang 36

37 Introduction - Inverse transformation Inverse transformation The inverse of a rotation matrix is simply its transpose: R 1 = R T and RR T = I whereas I is the identity matrix. The inverse of (1) is: r 11 r 21 r 31 p r 1 H 1 = r 12 r 22 r 32 p r 2 r 13 r 23 r 33 p r whereas r 1, r 2, r 3 and p are the four column vectors of (1) and represents the scalar product of vectors. J. Zhang 37

38 Introduction - Inverse transformation Relative transformations One has the following transformations: Z: World Manipulator base T 6 : Manipulator base Manipulator end E: Manipulator end Endeffector B: World Object G: Object Endeffector J. Zhang 38

39 Introduction - Transformation equation Transformation equation There are two descriptions of the endeffector position, one in relation to the object and the other in relation to the manipulator. Both descriptions are equal to eachother: ZT 6 E = BG In order to find the manipulator transformation: T 6 = Z 1 BGE 1 In order to determine the position of the object: B = ZT 6 EG 1 This is also called kinematic chain. J. Zhang 39

40 Introduction - Transformation equation Example: coordinate transformation J. Zhang 40

41 Introduction - Summary of homogenous transformations Summary of homogenous transformations A homogenous transformation depicts the position and orientation of a coordinate frame in space. If the coordinate frame is defined in relation to a solid object, the position and orientation of the solid object is unambiguously specified. The depiction of an object A can be derived from a homogenous transformation relating to object B. This is also possible the other way around using inverse transformation. J. Zhang 41

42 Introduction - Summary of homogenous transformations Summary of homogenous transformations Several translations and rotations can be multiplied. The following applies: If the rotations / translations are performed in relation to the current, newly defined (or changed) coordinate system, the newly added transformation matrices need to be multiplicatively appended on the right-hand side. If all of them are performed in relation to the fixed reference coodinate system, the transformation matrices need to be multiplicatively appended on the left-hand side. A homogenous transformation can be segemented into a rotation and a translation part. J. Zhang 42

43 Introduction - Summary of homogenous transformations Robot kinematics Quite often, only position and orientation of the robot gripper is of interest. In that case, a robot is treated just like a regular object, depicted through a transformation like all others. J. Zhang 43

44 Introduction - Summary of homogenous transformations Coordinates of a manipulator Joint coordinates: A vector q(t) = (q 1 (t), q 2 (t),..., q n (t)) T (a robot configuration) Endeffector coordinates (Object coordinates): A Vector p = [p x, p y, p z ] T Description of orientations: Euler angle φ, θ, ψ Rotation matrix: r 11 r 12 r 13 R = r 21 r 22 r 23 r 31 r 32 r 33 J. Zhang 44

45 Introduction - Summary of homogenous transformations Denavit Hartenberg Convention (outlook) Definition of one coordinate system per segment i = 1..n Definition of 4 parameters per segment i = 1..n Definition of one transformation A i per segment i = 1..n T 6 = n i=1 A i Later Denavit Hartenberg Convention will be presented more detailed! J. Zhang 45

46 Introduction - Summary of homogenous transformations Kinematics The direct kinematic problem: Given the joint values and geometrical parameters of all joints of a manipulator, how is it possible to determine the position and orientation of the manipulator-endeffector? The inverse kinematic problem: Given a desired position and orientation of the manipulator-endeffector and the geometrical parameters of all joints, is it possible for the manipulator to reach this position / orientation? If it is, how many manipulator configurations are capable of matching these conditions? (An example: A two-joint-manipulator moving on a plane) J. Zhang 46

47 Introduction - Summary of homogenous transformations Position T 6 defines, how the n joint angles are supposed to be consolidated to 12 non-linear formulas in order to descriebe 6 cartesian degrees of freedom. Forward kinematics K defined as: K : θ R n x R 6 Joint angle Position + Orientation Inverse kinematics K 1 defined as: K 1 : x R 6 θ R n Position + Orientation Joint angle non-trivial, since K is usually not unambiguously invertible J. Zhang 47

48 Introduction - Summary of homogenous transformations Differential movement Non-linear kinematics K can be linearized through the Taylor series f (x) = f (n) (x 0 ) n=0 n! (x x 0 ) n. The Jacobi matrix J as factor for n = 1 of the multi-dimensional Taylor series is defined as: J( θ) : θ R n x R 6 Joint speed kartesian speed Inverse Jacobi matrix J 1 defined as: J 1 ( θ) : x R 6 θ R n kartesian speed Joint speed non-trivial, since J not necessarily invertible (e.g. not quadratic) J. Zhang 48

49 Introduction - Summary of homogenous transformations Motion planning Since T 6 describes only the target position, explicit generation of a trajectory is necessary - depending on constraints differently for: joint angle space cartesiaan space Interpolation through: piecewise straight lines piecewise polynoms B-Splines... J. Zhang 49

50 Introduction - Summary of homogenous transformations Suggestions 1.1 Read: J. F. Engelberger. Robotics in Service, The MIT Press, (available in key texts) 1.3 Repeat your linear algebra knowledge, especially regarding elementary algebra of matrices. J. Zhang 50