Advanced Robotics. Lecture 1

Transcription

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5 5 Advanced Robotics Lecture 1

6 6 We will build on Robotics by V. Smutny and study more advanced robot kinematics problems, e.g., 1. solving inverse kinematics of a general 6 DOF manipulator 2. identifying kinematic parameters of a manipultor 3. finding singular poses of a manipulator with more advanced mathematical tools, such as 1. space rotation and motion and 2. solving algebraic equations

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8 8 R O B O T = A G E N E R A L M A N I P U L A T O R

9 9 Industrial Robotic Applications

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13 Precision for robotic surgery 14

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15 18 Precision for industry Low (e.g. manipulation) ± 5 mm in the whole working space ± 0.5 mm locally often available High (e.g. laser welding) ± 0.5 mm in the whole working space ± 0.05 mm locally often not available

16 19 Error ± 0.5 mm

17 Modeling kinematics calibration absolute acuracy ± 0.05 mm 20 Robot-Vision calibration (courtesy Neovision s.r.o.)

18 21 Two kinds of manipulators 1. Serial manipulators 2. Parallel manipulators

19 22 Serial manipulators J KUKA manipulator

20 23 Serial manipulators Stäubli (courtesy Neovision s.r.o.) Mitsubishi (courtesy Neovision s.r.o.) 1. Direct kinematic task easy 2. Inverse kinematic task difficult

21 24 Parallel manipulators J Stewart-Gough Platform

22 25 Parallel manipulators Sliding Star (courtesy of Prof. Valášek, CTU Prague) 1. Direct kinematic task difficult 2. Inverse kinematic task easy

23 26 Kinematics in robotics Three main problems 1. Direct kinematic task (přímá kinematická úloha) 2. Inverse kinematic task (inverzní kinematická úloha) 3. Kinematic calibration (kalibrace kinematiky)

24 27 Direct kinematic task flange frame z x world frame y

25 28 Inverse kinematic task flange frame z x world frame y

26 29 Kinematic calibration z x world frame y

27 30 Kinematic Calibration

28 31 Robot Calibration

29 32 Kinematic Calibration

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33 36 Solving kinematic tasks

34 37 Solving kinematic tasks 1968 Donald L. Pieper (Ph.D. thesis) The inverse kinematics of any serial manipulator with six revolute joints, and with three consecutive joints intersecting, can be solved in closedform, i.e., analytically M. Raghavan, B. Roth. Kinematic Analysis of the 6R Manipulator of General Geometry. Int. Symp. Robotics. Research. Pp , Tokyo 1989/1990. A general technique for computing inverse kinematics for any serial manipulator with six revolute joints. leads to solving an algebraic equation of degree 16.

35 38 Solving kinematic tasks Algebraic equation of degree 16 up to 16 solutions 4 typical solutions

36 39 Solving kinematic tasks

37 40 Stäubli TX-90 Geometry

38 41 Kinematic model flange frame z y x z x world frame y

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40 43 The Standard Kinematic model in Denavit-Hartenberg Convention Stäubli TX 90 TX-90 (6 axis, RRRRRR) [Staubli] α a θ d non-trivial parameteres

41 44 The Standard Kinematic model in Denavit-Hartenberg Convention ABB IBR 140

42 45 The Standard Kinematic model in Denavit-Hartenberg Convention ABB IBR 140 IBR-140 (6 axis) [ABB] α a θ d non-trivial parameteres

43 46 The Standard Kinematic model in Denavit-Hartenberg Convention Stäubli TX 90 RV-6S (6 axis, RRRRRR) [Mitsubishi] α a θ d non-trivial parameteres

44 47 Special versus General Mechanisms Special General simple & tractable complicated & hard α a θ d α a θ d (+0.2) (+0.1) (- 0.3) (+0.1) (- 0.1) (- 0.2) non-trivial parameters 18 (+6) non-trivial parameters High precision Small misalignments important General mechanisms

45 48 Literature Linear algebra P. Pták. Introduction to Linear Algebra. Vydavatelství ČVUT, Praha, Numerical linear algebra E. Krajník. Maticový počet. Vydavatelství ČVUT, Praha, The solution M. Raghavan, B. Roth. Kinematic Analysis of the 6R Manipulator of General Geometry. Int. Symp. Robotics. Research. Pp , Tokyo 1989/1990. The numerical solution D. Manocha, J. Canny. Efficient Inverse Kinematics for General 6R Manipulators. Robotics and Automation The pedagogical solution will be developed using D. Cox, J. Little, D. O'Shea. Ideals, Varieties, and Algorithms. Springer 1998.

46 49 Software Matlab: Maple:

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48 51 S O L V I N G 1 A L G E B R A I C E Q U A T I O N 1 equation, 1 variable companion matrix eigenvalues... a simple rule It works when eig works, i.e. order 100 in Matlab is often OK.

49 52 S O L V I N G 1 A L G E B R A I C E Q U A T I O N

50 53 S O L V I N G 1 A L G E B R A I C E Q U A T I O N algebraic equation Numerical solution to f(x) is obtained by