LUMPED MASS MODELLING WITH APPLICATIONS

Transcription

1 LUMPED MASS MODELLING WITH APPLICATIONS July, 8, 07 Lumped Mass Modelling with Applications

2 . Motivation. Some examples 3. Basic control scheme 4. Single joint single link - single mass INDEX 5. Extensions of single joint single link - single mass 6. Single joint single link several masses 7. Two joints single link single mass 8. Two joints two links two masses 9. Three joints two link single mass 0. Interaction with the environment. Some conclusions Lumped Mass Modelling with Applications

3 . MOTIVATION: definition FLEXIBLE ROBOT: Any robot that has at least one mechanical component which is flexible, i.e., that is elastically deformed by a force or a torque applied to it Lumped Mass Modelling with Applications 3

4 . MOTIVATION: definition z i KINDS OF FLEXIBILITY y i- y y i x i y i x i z x z i z i- xi- x i z i x i y i Flexible link Flexible joint Lumped Mass Modelling with Applications 4

5 . MOTIVATION: why flexible robots? PRECEDENTS SPACE STATION REMOTE MANIPULATOR SYSTEM (SSRMS) ANTENNA PRISMATIC CRANE WHEELCHAIR WITH A ROBOTIZED ARM ELASTIC WRIST (REMOTE COMPLIANCE CENTER) FLEXIBLE ROBOT OF THREE DEGREES OF FREEDOM Lumped Mass Modelling with Applications 5

6 . MOTIVATION: why flexible robots? SPACE STATION REMOTE MANIPULATOR SYSTEM (SSRMS) Lumped Mass Modelling with Applications 6

7 . MOTIVATION: why flexible robots? PRISMATIC CRANE Lumped Mass Modelling with Applications 7

8 . MOTIVATION: why flexible robots? ELASTIC WRIST (REMOTE COMPLIANCE CENTER) Lumped Mass Modelling with Applications 8

9 . MOTIVATION: why flexible robots? FLEXIBLE ROBOT OF THREE DEGREES OF FREEDOM Lumped Mass Modelling with Applications 9

10 . MOTIVATION: why flexible robots? WHEELCHAIR WITH A ROBOTIZED ARM Lumped Mass Modelling with Applications 0

11 . MOTIVATION: why flexible robots? ANTENNA Lumped Mass Modelling with Applications

12 . MOTIVATION: why flexible robots? INCREASE THE ROBOT MOBILITY - MORE LIGHTWEIGHT ARMS INCREASE THE ROBOT SPEED - COMPLIANCE IN CONTACT TASKS - IMPACT DAMAGE REDUCTION - SENSING SYSTEMS Lumped Mass Modelling with Applications

13 . MOTIVATION: why lumped mass models? FLEXIBLE ROBOTS EXHIBIT VERY COMPLEX DYNAMICS: DISTRIBUTED DYNAMICS (INFINITE ORDER), MULTIVARIABLE, NONLINEAR, NONMINIMUM PHASE, UNDAMPED. THEN WE MUST ACCEPT EITHER [] - Models of high order. - Models of low accuracy. - Models not appropriate for the full range of operation. In particular, decisions made on the basis of these models should recognize their approximate and specialized nature. Lumped Mass Modelling with Applications 3

14 . MOTIVATION: why lumped mass models? MODELLING METHODS [] - THE ASSUMED-MODES METHOD: only the first several modes are retained after truncation while the higher modes are neglected. - THE FINITE ELEMENT METHOD: is the most versatile among the three and can deal with complicated boundary conditions, but it is not computationally efficient and difficult to deal with, and the boundary conditions are often uncertain for flexible manipulators. - THE LUMPED MASSES METHOD: is the simplest one, but because the manipulator is modelled as spring and mass systems, it often does not yield sufficiently accurate models. Lumped Mass Modelling with Applications 4

15 . MOTIVATION: why lumped mass models? ADVANTAGES OF THE LUMPED MASS MODELS: - THEY ARE EASY TO OBTAIN: specialized knowledge of mechanics is not needed. - THEY ARE RELATIVELY SIMPLE MODELS: at least, simpler than other methods. - THEY CAN BE USED IN MULTI-BODY SYSTEMS: assumed-modes method is difficult to apply [3] and the finite elements method does not provide an analytical model. - THEY CAN BE EASILY MATCHED TO EXPERIMENTAL DATA: e.g., time or frequency responses [4]. - ANALYSIS AND DESIGN OF CONTROL SYSTEMS IS SIMPLER: the analysis is more intuitive and controllers obtained from model based methodologies are simpler. Lumped Mass Modelling with Applications 5

16 . MOTIVATION: why lumped mass models? DISADVANTAGES OF THE LUMPED MASS MODELS: - THEY ARE SIMPLIFIED MODELS THAT MAY BE INACCURATE: they may not be adequate to model: ) Links with a mass of the order of the robot payload. ) Links with variable sections. 3) Shearing and rotary inertia effects. - SPILLOVER EFFECTS CAN NOT BE ANALYZED USING THESE MODELS - EXTERNAL DISTURBANCES ON THE LINK CAN NOT BE EASILY INCLUDED: tasks that include interaction with the environment. Lumped Mass Modelling with Applications 6

17 HOWEVER:. MOTIVATION: why lumped mass models? THESE MODELS YIELD REASONABLE MODELS THAT ALLOW TO DESIGN HIGH PERFORMANCE CONTROL SYSTEMS IN MANY PRACTICAL CASES, I.E. These models may not be enough accurate for the ANALYSIS and DESIGN of the MECHANISM, but are often good for the ANALYSIS AND DESIGN of the CONTROL SYSTEM. THIS TALK WILL BE ABOUT THIS Lumped Mass Modelling with Applications 7

18 . SOME EXAMPLES: preamble LUMPED PARAMETERS MODELS: the infinite dimension nature of these distributed systems is approximated by finite order models in order to make the problem easy to handle. THE ASSUMED-MODES METHOD THE LUMPED MASSES METHOD THE FINITE ELEMENT METHOD LUMPED PARAMETERS MODELS IDENTIFICATION BASED METHOD Lumped Mass Modelling with Applications 8

19 Lumped Mass Modelling with Applications 9 Massless elastic chains with servo controlled joints [5] 0 0 T N A N E N E A E A X XYi Yi i XYi ZYi i Yi Zi i i s s Z s s Y s s X E. SOME EXAMPLES

20 . SOME EXAMPLES E i XF XF i i F F Y Z ii ii C i F X ii XM XM i i M M Z Y ii ii F i F i F F Y Z ii ii 0 M i M i M M Z Y ii ii F i F Y ii 0 M T i X M ii i M Z ii F i F Z ii 0 M T i M X ii i M Y ii Fx ij, F Yij, F Zij = forces at the end of beam i, in terms of coordinate system j. F ijt = [F Xij F Yij F Zij ] M Xij, M Yij, M Zij = moments at the end of beam i, in terms of coordinate system j. M ijt = [M Xij M Yij M Zij ] α Ci = coefficient in compression, displacement/unit force α FXi = coefficient in bending, displacement/unit force α MXi = coefficient in bending, displacement/unit moment α θfi = coefficient in bending, angle/unit force ( = α MXi ) α θmi = coefficient in bending, angle/unit moment α T = coefficient in torsion, angle/unit moment Lumped Mass Modelling with Applications 0

21 Lumped Mass Modelling with Applications C K M M M F F F C P P P NO NO NO NO NO NO NO NO NO NO NO NO Z Y X Z Y X Z Y X Z Y X C J of are the eigenvalues P P P J M M M F F F P P P P P P C i e i i i i i Z Y X Z Y X Z Y X Z Y X e Z Z e Y Y e X X e Z Z e Y Y e X X NO NO NO NO NO NO NO NO NO NO NO NO NO NO NO NO NO NO NO NO NO NO NO NO. SOME EXAMPLES

22 . SOME EXAMPLES A lumped spring-mass model [6] Lumped Mass Modelling with Applications

23 Lumped Mass Modelling with Applications 3 Deflections given by the constrained assumed modes: ) ( ) ( 0 L F dx x F M L e ) ( ) ( 0 " L F dx x F EI K L Me gives equivalent Kinetic energy and K gives equivalent elastic potential energy:. SOME EXAMPLES

24 . SOME EXAMPLES Dynamic model: M M p Kp L I h t K e p L Lumped Mass Modelling with Applications 4

25 . SOME EXAMPLES Distributed elasticity modelled as a set of elastic joints [7] The system considered is composed of rigid bodies connected together by rigid joints (revolute or prismatic) or elastic joints (torsional or translational spring). A body with n degrees of freedom with respect to its antecedent is modelled as n- virtual bodies with zero mass and inertia plus the real body. Link elasticity Multiple joint elasticity Lumped Mass Modelling with Applications 5

26 . SOME EXAMPLES A spring damper joint model of the actuator-flexible arm [8] Lumped Mass Modelling with Applications 6

27 . SOME EXAMPLES A beam with very large deflections [9] Lumped Mass Modelling with Applications 7

28 Lumped Mass Modelling with Applications 8 Model: a compliance matrix whose elements vary with the tip force and are calculated numerically F X F X F X F X F X F X F X C. SOME EXAMPLES

29 . SOME EXAMPLES Extensible robot: prismatic and revolute joints [0]. n successive equal rods with equal mass and inertia, connected with torsional springs. Lumped Mass Modelling with Applications 9

30 3. BASIC CONTROL SCHEME Single link flexible arm []. Lumped Mass Modelling with Applications 30

31 3. BASIC CONTROL SCHEME Lumped Mass Modelling with Applications 3

32 Lumped Mass Modelling with Applications 3 3. BASIC CONTROL SCHEME Equivalent transfer function of the inner loop (motor loop): ), ( ˆ coul coup n J K Controllers: h s s a s a s a s C 0 ) ( h s b s b s C 0 ) ( Motor dynamics (): * ; ) ( ) ( ) ( p s s s s M

33 4. SINGLE JOINT SINGLE LINK - SINGLE MASS: assumptions Links with constant circular cross sections. Links with constant elasticity. Links geometrically linear: relative deflection lower than 0% sin( x), tan( x) Shearing and rotary inertia effects. x Lumped Mass Modelling with Applications 33

34 4. SINGLE JOINT SINGLE LINK - SINGLE MASS: basics coup M M L q e coup K q ; c K Lumped Mass Modelling with Applications 34

35 Lumped Mass Modelling with Applications 35 Taking Laplace transforms: ) ( ) ( s Q s L M M s e coup 4. SINGLE JOINT SINGLE LINK - SINGLE MASS: basics ) ( ) ( ) ( s Q s K s coup ) ( ) ( ) ( L M M K s s s Q s G e q ) ( ) ( ) ( s s K s s s G

36 4. SINGLE JOINT SINGLE LINK - SINGLE MASS: basics M e yields the same momentum as the distributed mass of the beam considered as rigid: M e M e yields the same rotational inertia as the distributed mass of the beam considered as rigid: L M e 3 L Lumped Mass Modelling with Applications 36

37 4. SINGLE JOINT SINGLE LINK - SINGLE MASS: basics M e yields the same kinetic energy as the distributed mass of the beam assuming that its deflection is given by the third order polynomial obtained assuming a massles flexible beam: M e M e yields the same kinetic energy as the first assumed flexible mode of the distributed mass beam and K the e same elastic potential energy as the first assumed mode [6] Lumped Mass Modelling with Applications 37 L

38 4. SINGLE JOINT SINGLE LINK - SINGLE MASS: basics M e obtained from the Rayleigh s method: in an harmonic movement, the maximum strain energy is equal to the maximum kinetic energy. Equating them, the first vibration mode is obtained and, hence, the equivalent mass. M e obtained from the Ritz s method: it is a refinement of the Rayleigh s method in which deflections are assumed function of several parameters tuned which are obtained by making the computed frequency a minimum. M e obtained from fitting the model to an experimental response, e.g, obtaining from the frequency response and subsequently calculating Lumped Mass Modelling with Applications 38 M e

39 4. SINGLE JOINT SINGLE LINK - SINGLE MASS: basics Procedure useful in many problems for finding approximations to both the natural frequencies and the mode shapes []. Lumped Mass Modelling with Applications 39

40 4. SINGLE JOINT SINGLE LINK - SINGLE MASS: method Modeling a massless Euler-Bernouilli beam [3] F e and T e are exerted by the beam end E I d 4 y( x) 4 dx 0 Lumped Mass Modelling with Applications 40

41 4. SINGLE JOINT SINGLE LINK - SINGLE MASS: method Boundary conditions: Lumped Mass Modelling with Applications 4

42 4. SINGLE JOINT SINGLE LINK - SINGLE MASS: method Dynamic equations: Make. Then Combining the last two boundary conditions: and the solution of the Euler Bernouilli is Lumped Mass Modelling with Applications 4

43 Lumped Mass Modelling with Applications 43 Particularizing this equation to : substituting the dynamic equations and operating: being x L 3 3 ) ( ) ( ) ( L I E L F L I E L T L y 4. SINGLE JOINT SINGLE LINK - SINGLE MASS: method e e F L T q K q L m 3 L I E K 3 L m K

44 4. SINGLE JOINT SINGLE LINK - SINGLE MASS: method Moreover: Substituting the dynamic equations yields coup T e m L q L F e and substituting the previous solution: coup K q Te Lumped Mass Modelling with Applications 44

45 4. SINGLE JOINT SINGLE LINK - SINGLE MASS: example of adaptive control CONTROL SCHEME [4] Lumped Mass Modelling with Applications 45

46 4. SINGLE JOINT SINGLE LINK - SINGLE MASS: example of adaptive control Dynamic model: q coup q K q q K coup GPI controller: Feedforward term Feedback term Lumped Mass Modelling with Applications 46

47 Lumped Mass Modelling with Applications SINGLE JOINT SINGLE LINK - SINGLE MASS: example of adaptive control Algebraic identification technique: Tuning laws: Feedforward term Feedback term * * * ˆ ˆ ˆ p s a a s a s a s q q

48 4. SINGLE JOINT SINGLE LINK - SINGLE MASS: example of adaptive control EXPERIMENTAL RESULTS Industrial robot control Adaptive flexible arm control Lumped Mass Modelling with Applications 48

49 4. SINGLE JOINT SINGLE LINK - SINGLE MASS: example of adaptive control Tip angle q Tip angle (zoom at the beginning). Assumed: ω = 9 rad/s; real one: ω = 5. rad/s Lumped Mass Modelling with Applications 49

50 4. SINGLE JOINT SINGLE LINK - SINGLE MASS: example of adaptive control Frequency estimation Lumped Mass Modelling with Applications 50

51 Lumped Mass Modelling with Applications 5 Rayleigh damping: Adding to F e, the model becomes: 5. EXTENSIONS OF SINGLE JOINT SINGLE LINK SINGLE MASS: damping m K, q K m F e e F L q K m L T q K q L m 3 e F e K L T K q q q 3

52 5. EXTENSIONS OF SINGLE JOINT SINGLE LINK SINGLE MASS: rotational inertia at the tip Adding to T e, the model becomes: m L q K 3 3 q Te J t q g L Fe Moreover Since and taking into account that Lumped Mass Modelling with Applications 5

53 5. EXTENSIONS OF SINGLE JOINT SINGLE LINK SINGLE MASS: rotational inertia at the tip it yields that and combining both dynamic equations: Lumped Mass Modelling with Applications 53

54 5. EXTENSIONS OF SINGLE JOINT SINGLE LINK SINGLE MASS: rotational inertia at the tip The coupling torque becomes coup K K 6 4 q q g L Fe Lumped Mass Modelling with Applications 54

55 5. EXTENSIONS OF SINGLE JOINT SINGLE LINK SINGLE MASS: geometrically nonlinear beam General equation of the massless Euler-Bernouilli beam E I d dl T (l) Lumped Mass Modelling with Applications 55

56 5. EXTENSIONS OF SINGLE JOINT SINGLE LINK SINGLE MASS: geometrically nonlinear beam Linear hypothesis: Nonlinear hypothesis: Lumped Mass Modelling with Applications 56

57 5. EXTENSIONS OF SINGLE JOINT SINGLE LINK SINGLE MASS: geometrically nonlinear beam Static (kinematic) equation of the beam [5] : Elliptic integral F ( L) ( L), ( L), K, L numerical solution d p m Fe ( L), q, K, dt L Lumped Mass Modelling with Applications 57

58 Lumped Mass Modelling with Applications 58 Geometrically linear Geometrically non linear Tip radius Tip angle Friction term Linear hypothesis: Nonlinear: 5. EXTENSIONS OF SINGLE JOINT SINGLE LINK SINGLE MASS: geometrically nonlinear beam q L K 3 3 q L K q L K q K m L q K q L m q K m L q K q K q L m 3 3 ) (

59 5. EXTENSIONS OF SINGLE JOINT SINGLE LINK SINGLE MASS: example of nonlinear beam Lumped Mass Modelling with Applications 59

60 5. EXTENSIONS OF SINGLE JOINT SINGLE LINK SINGLE MASS: example of nonlinear beam Sensorial system - Camera-based 3D position measurement Actuator - Harmonic Drive mini servo DC motor - Embedded encoder Flexible beam data Material Density Diameter Fiberglass 76.7 kg/m 3 3 mm Young s Modulus Cross section inertia Length Pa m 4 m Tip mass data Material Mass Diameter Height Nylon g 9.4 mm 5.3 mm Lumped Mass Modelling with Applications 60

61 Trajectory 5. EXTENSIONS OF SINGLE JOINT SINGLE LINK SINGLE MASS: example of nonlinear beam - Constant velocity: - Total manoeuver rotation: Polar coordinates Big deflections - In angle 0.85 radians (48.7º) - In radius 7 cm (7% of the total length) Lumped Mass Modelling with Applications 6

62 5. EXTENSIONS OF SINGLE JOINT SINGLE LINK SINGLE MASS: example of nonlinear beam Model fitting Least-squares method MATLAB TM Levenberg-Marquardt algorithm Linear model: Nonlinear model: , K , K 3 K 0.385, 0.05, Lumped Mass Modelling with Applications 6

63 5. EXTENSIONS OF SINGLE JOINT SINGLE LINK SINGLE MASS: example of nonlinear beam Mean error Linear Non linear angle Linear rad Non linear rad error (rad) t (s) Lumped Mass Modelling with Applications 63

64 5. EXTENSIONS OF SINGLE JOINT SINGLE LINK SINGLE MASS: example of nonlinear beam Mean error Linear Non linear radius Linear mm Non linear -.8 mm error (mm) t (s) Lumped Mass Modelling with Applications 64

65 5. EXTENSIONS OF SINGLE JOINT SINGLE LINK SINGLE MASS: gravity effect Adding to F e, the model becomes: and m L q K q m g L cosq L F e Te 3 Lumped Mass Modelling with Applications 65

66 5. EXTENSIONS OF SINGLE JOINT SINGLE LINK SINGLE MASS: gravity effect If the link mass m link is not negligible with respect to the payload mass, the link mass is included in the model by adding two separated terms to ponderate inertial and gravitational effects of the link mass. m m link acc L q K q m mlink grv g Lcos q L Fe Te 3 coup K q Te Lumped Mass Modelling with Applications 66

67 5. EXTENSIONS OF SINGLE JOINT SINGLE LINK SINGLE MASS: example of link with gravity effect [6] Parameters Symbol Value Link Length l.0 m Equivalent Link Mass inertial contribution Equivalent Link Mass gravitational contribution m link acc m link grv 0.035Kg 0.38 Kg Flexural Rigidity E I 746 N m Payload Mass m tip.46 Kg Experimental setup Gravitational Constant Identified parameters g 9.8 m s Lumped Mass Modelling with Applications 67

68 5. EXTENSIONS OF SINGLE JOINT SINGLE LINK SINGLE MASS: example of link with gravity effect Experimental validation of the dynamic model Lumped Mass Modelling with Applications 68

69 5. EXTENSIONS OF SINGLE JOINT SINGLE LINK SINGLE MASS: example of link with gravity effect Input-State Feedback Linearization Controller Diagram Demonstration Video Note: The blocks indicated in red have been developed considering the dynamic model previously presented Lumped Mass Modelling with Applications 69

70 6. SINGLE JOINT SINGLE LINK - SEVERAL MASSES: basics Lumped Mass Modelling with Applications 70

71 Lumped Mass Modelling with Applications 7 6. SINGLE JOINT SINGLE LINK - SEVERAL MASSES: basics 0,, 0 n i L i j j i i i i i i i i i i i x n i x u x u x u u x y x y 3,3,,,0, ) ( ) (

72 6. SINGLE JOINT SINGLE LINK - SEVERAL MASSES: basics T i x i Lumped Mass Modelling with Applications 7

73 Lumped Mass Modelling with Applications SINGLE JOINT SINGLE LINK - SEVERAL MASSES: basics i i i i i i L x i i i e e i i i e e dx dy dt d J T T q m F F ˆ ˆ i i e i i e i i T L T F L F ˆ ) ( ˆ ) (

74 6. SINGLE JOINT SINGLE LINK - SEVERAL MASSES: basics Lumped Mass Modelling with Applications 74

75 6. SINGLE JOINT SINGLE LINK - SEVERAL MASSES: example two masses plus rotational inertia [7] Lumped Mass Modelling with Applications 75

76 Lumped Mass Modelling with Applications SINGLE JOINT SINGLE LINK - SEVERAL MASSES: example two masses plus rotational inertia ) ( ) ( s U A I E A I E s M s Q

77 Lumped Mass Modelling with Applications SINGLE JOINT SINGLE LINK - SEVERAL MASSES: example two masses plus rotational inertia ) ( ) ( s s s s s s L I E s s coup

78 6. SINGLE JOINT SINGLE LINK - SEVERAL MASSES: sensing antenna: example three masses Whisker and Antenna Sensors - in nature- Touch is not only an alerting stimulus, it is used to solve complex perceptual tasks. The bending moment, or torque, at the whisker base is used to generate the 3D environment. Lumped Mass Modelling with Applications 78

79 6. SINGLE JOINT SINGLE LINK - SEVERAL MASSES: sensing antenna: example three masses [7] Lumped Mass Modelling with Applications 79

80 6. SINGLE JOINT SINGLE LINK - SEVERAL MASSES: sensing antenna: example three masses K m. 47g L Lumped Mass Modelling with Applications 80

81 6. SINGLE JOINT SINGLE LINK - SEVERAL MASSES: sensing antenna: example three masses Lumped Mass Modelling with Applications 8

82 7. TWO JOINTS SINGLE LINK - SINGLE MASS: basics Q,,, Lumped Mass Modelling with Applications 8

83 7. TWO JOINTS SINGLE LINK - SINGLE MASS: basics E e K K p L Lumped Mass Modelling with Applications 83

84 Lumped Mass Modelling with Applications TWO JOINTS SINGLE LINK - SINGLE MASS: basics,,, cos J J L M L M diag I n cos sin L M C e

85 7. TWO JOINTS SINGLE LINK - SINGLE MASS: basics K a K, M L K g g L Lumped Mass Modelling with Applications 85

86 7. TWO JOINTS SINGLE LINK - SINGLE MASS: example: sensing antenna [7] Single mass model to obtain a controller that linearizes and decouples the antenna dynamics. Three mass model to control the residual vibrations (three modes) in azimuthal and attitude angles. Lumped Mass Modelling with Applications 86

87 7. TWO JOINTS SINGLE LINK - SINGLE MASS: example: sensing antenna [7] Sensing antenna Lumped Mass Modelling with Applications 87

88 7. TWO JOINTS SINGLE LINK - SINGLE MASS: example: sensing antenna [7] with motor control with tip control Single mass model to obtain a controller that linearizes and decouples the antenna dynamics. Three mass model to control the residual vibrations (three modes) in azimuthal and attitude angles. Lumped Mass Modelling with Applications 88

89 8. TWO JOINTS TWO LINKS - TWO MASSES: basics Lumped Mass Modelling with Applications 89

90 8. TWO JOINTS TWO LINKS - TWO MASSES: basics E e K q Lumped Mass Modelling with Applications 90

91 yields 8. TWO JOINTS TWO LINKS - TWO MASSES: basics Substituting and taking into account boundary conditions: u i u, 0 i, 0 Lumped Mass Modelling with Applications 9

92 8. TWO JOINTS TWO LINKS - TWO MASSES: basics Γ = I q q + C q, q + G q + F(q q = q q g q θ θ Lumped Mass Modelling with Applications 9

93 8. TWO JOINTS TWO LINKS - TWO MASSES: basics (m + m l m l l cos (q g q + q ) m l l cos (q g q + q ) 0 0 I(q = m l l cos (q g q + q ) J + m l m l 0 J m l l cos (q g q + q ) m l m l J 0 0 J 0 0 J F(q = 6 EI l (q g q + θ ) EI l ( q g 3 q + θ ) 3 EI l (θ q EI l q g 3 q + θ 3 EI l (θ q G q = C(q, q = m l l sin q g q + q m + m g l co s( q m g l co s( q g + q m g l co s( q g + q 0 0 (q g) + (q g)(q ) + (q ) (q ) (q ) 0 0 Lumped Mass Modelling with Applications 93

94 9. THREE JOINTS TWO LINKS TIP PAYLOAD: example: tip lumped mass [8] Robot Puma 560 Flexible robot Workspace 0,9 m,00 m Maximum load,5 kg,9 kg Tip acceleration g g Weight 6,0 kg 0,8 kg / 0 kg Lumped Mass Modelling with Applications 94

95 9. THREE JOINTS TWO LINKS TIP PAYLOAD: example: tip lumped mass [8] Lumped Mass Modelling with Applications 95

96 Lumped Mass Modelling with Applications THREE JOINTS TWO LINKS TIP PAYLOAD: example: tip lumped mass e F L q K q L m e F q L L L K q L m g r F p p K p m C K

97 9. THREE JOINTS TWO LINKS TIP PAYLOAD: example: tip lumped mass Lumped Mass Modelling with Applications 97

98 9. THREE JOINTS TWO LINKS TIP PAYLOAD: example: tip lumped mass m Response to a joint pulse s t Standard control Vibration control Lumped Mass Modelling with Applications 98

99 9. THREE JOINTS TWO LINKS TIP PAYLOAD: example: tip lumped mass Strain gauge signals. Vibration control is connected after s. Lumped Mass Modelling with Applications 99

100 9. THREE JOINTS TWO LINKS TIP PAYLOAD: example: tip mass with rotational inertia Lumped Mass Modelling with Applications 00

101 Lumped Mass Modelling with Applications 0 9. THREE JOINTS TWO LINKS TIP PAYLOAD: example: tip mass with rotational inertia e e g g t F L T q q K q q J L m Angular momentum. Time rate of change about a center of rotation:

102 Lumped Mass Modelling with Applications 0 9. THREE JOINTS TWO LINKS TIP PAYLOAD: example: tip mass with rotational inertia e e g g t g t F L T q q L L L L K q dt dj q q L J m w r t t P g m K J J I m f r

103 9. THREE JOINTS TWO LINKS TIP PAYLOAD: example: tip mass with rotational inertia Lumped Mass Modelling with Applications 03

104 9. THREE JOINTS TWO LINKS TIP PAYLOAD: example: tip mass with rotational inertia Linear speed about 0.8 m/s Maximum acceleration about 3 m/s Max. position error mm Max. orientation error 3 mrad Lumped Mass Modelling with Applications 04

105 Lumped Mass Modelling with Applications INTERACTION WITH THE ENVIRONMENT: example: three degrees of freedom-two links-tip lumped mass [9] g r F p p K p m * * * p p K r coup

106 0. INTERACTION WITH THE ENVIRONMENT: example: three degrees of freedom-two links-tip lumped mass Impact detection Estimation of time (known position) Strain gauge measurement ς is experimentally determined. η is the máximum level of noise of the torque sensor Lumped Mass Modelling with Applications 06

107 Amplitude 0. INTERACTION WITH THE ENVIRONMENT: example: three degrees of freedom-two links-tip lumped mass Estimation of time (known position) Strain gauge measurement Gauge signal F.F.T. Filtered signal Frequency (Hz) Filter Time (s) Time (s) Frequency (Hz) Lumped Mass Modelling with Applications 07

108 0. INTERACTION WITH THE ENVIRONMENT: example: three degrees of freedom-two links-tip lumped mass Lumped Mass Modelling with Applications 08

109 0. INTERACTION WITH THE ENVIRONMENT: example: three degrees of freedom-two links-tip lumped mass Force control (Scheme) Lumped Mass Modelling with Applications 09

110 0. INTERACTION WITH THE ENVIRONMENT: example: three degrees of freedom-two links-tip lumped mass Force control (Experiments) Lumped Mass Modelling with Applications 0

111 Force (Nw) 0. INTERACTION WITH THE ENVIRONMENT: example: three degrees of freedom-two links-tip lumped mass Force control (Force measurements) Tip force during dynamometer experiment Time (seg) Lumped Mass Modelling with Applications

112 0. INTERACTION WITH THE ENVIRONMENT: single link with contact at an intermediate point Defining incremental values: Mechanical impedance of the contacted object Lumped Mass Modelling with Applications

113 0. INTERACTION WITH THE ENVIRONMENT: single link with contact at an intermediate point L L Lumped Mass Modelling with Applications 3

114 Lumped Mass Modelling with Applications 4 0. INTERACTION WITH THE ENVIRONMENT: single link with contact at an intermediate point F e I E L q I E L m u ,0

115 Lumped Mass Modelling with Applications 5 0. INTERACTION WITH THE ENVIRONMENT: single link with contact at an intermediate point L I E K 3 4 q q c coup

116 0. INTERACTION WITH THE ENVIRONMENT: example: force control with a robotic gripper Lumped Mass Modelling with Applications 6

117 0. INTERACTION WITH THE ENVIRONMENT: example: force control with a robotic gripper Lumped Mass Modelling with Applications 7

118 0. INTERACTION WITH THE ENVIRONMENT: example: force control with a robotic gripper Grasping a fragile object Lumped Mass Modelling with Applications 8

119 . SOME CONCLUSIONS Simple to use, either with Newton-Euler or Lagrange equations. Not adequate to study complex mechanical systems. But yield simple dynamical models that give a deep insight into the control problem Facilitate control system design and control based applications Lumped Mass Modelling with Applications 9

120 THANK YOU FOR YOUR ATTENTION Prof. Vicente Feliu Batlle Lumped Mass Modelling with Applications 0

121 REFERENCES [] W.J. Book, Modeling, Design, and Control of Flexible Manipulator Arms: a Tutorial Review, Proceedings 9th IEEE Conference on Decision and Control, Honolulu, USA, December 990: [] C.T. Kiang, A. Spowage and C.K. Yoong, Review of Control and Sensor System of Flexible Manipulator, J Intell Robot Syst (05) 77:87 3. [3] S.K. Dwivedy and P. Eberhard, Dynamic Analysis of Flexible Manipulators, a Literature Review, Mechanism and Machine Theory (006) 4(7): [4] G.G. Hastings and W.J. Book, Verification of a Linear Dynamic Model for Flexible Robotic Manipulators, Proceedings of IEEE Conference on Robotics and Automation, San Francisco (USA), April 986: Lumped Mass Modelling with Applications

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