International Conference on Electromechanical Engineering ICEE Plenary II. Ahmed CHEMORI

Transcription

1 International Conference on Electromechanical Engineering ICEE 2012 Plenary II Ahmed CHEMORI Laboratory of Informatics, Robotics and Microelectronics of Montpellier LIRMM, CNRS/University of Montpellier 2 161, rue Ada Montpellier, France Skikda, November, 20 th, 2012

2 Montpellier Montpellier is a city in the south of France It is the capital of Languedoc Roussillon region as well as Hérault department It is the 8 th city in the country Montpellier 2

3 LIRMM Laboratory LIRMM Laboratory of Informatics, Robotics and Microelectronics of Montpellier (LIRMM) is a research laboratory supervised by both University Montpellier 2 and the French National Center for Scientific Research (CNRS) 359 permanent and 80 temporary employees, working together in 3 research units : Department of Computer science Department of Robotics Department of Microelectronics 3

4 LIRMM Laboratory Robotics department The robotics department constitutes one of the vital forces of robotics in France It comprised of 30 researchers and lecturer-researchers It contains 5 main research teams IDH DEXTER ICAR Robotics Department Of LIRMM EXPLORE DEMAR 4

5 LIRMM Laboratory Robotics department DEXTER Research team E. DOMBRE (DR CNRS) P. POIGNET (PR UM2) Medical robotics N. ZEMITI (MC UM2) C. LIU (CR CNRS) F. PIERROT (DR CNRS) O. COMPANY (MC UM2) S. KRUT (CR CNRS) A. CHEMORI (CR CNRS) M. GOUTTEFARDE (CR CNRS) Parallel robotics 5

6 LIRMM Laboratory Robotics department Facilities 6

7 LIRMM Laboratory LIRMM in the robotics platforms network in France (Equipex ROBOTEX) 7

8 Outline of the presentation Part I Part II Part III Part IV Part V o Context of the study Mechanical systems Under-actuated mechanical systems Dynamic modeling o Description of our demonstrator Description of the experimental testbed Description of the mechanical part Dynamic modeling Linearized and descritized dynamics o Control for stabilization Formulation of the stabilization problem Proposed solution for stabilization Obtained results o Control for stable limit cycle generation Formulation of the limit cycle generation problem Proposed solution 1 : Optimal partial feedback linearization Proposed solution 2 : Dual model-free controller o 8

9 Context Part I Mechanical systems Under-actuated mechanical systems Dynamic modeling 9

10 Context Mechanical systems Adept Viper S650 PendCon Acrobot Barrett Wam 6 degrees of freedom 6 actuators 2 degrees of freedom 1 actuator 6 degrees of freedom 7 actuators Nbr Act = DOF Nbr Act < DOF Nbr Act > DOF Fully actuated Under-actuated Redundant 10

11 Context Under-actuated mechanical systems Systems with less actuators than degrees of freedom Under-actuation is either : Chosen in the design stage Minimize the cost, the weight, consumption, etc or because of the deficiency one/more actuators Nonlinear coupling between actuated and unactuated coordinates Internal dynamics often unstable non minimum phase Systems Some Examples : Acrobot Pendubot Grippers Flexible arms PVTOL Walking robots Underwater vehicles 11

12 Context Under-actuated mechanical systems Dynamics Consider the case of 2-dof and 1 control input under-actuated systems Nonlinear dynamics : M(q)Äq + H(q; _q) + G(q) = Ru M(q) 2 R 2 2 is the inertia matrix, H(q; _q) 2 R 2 1 is the Coriolis and centrifugal vector, G(q) 2 R 2 1 is the gravity vector, u 2 R is the control input, R 2 R 2 1 a matrix distributing the e ect of u on generalized coordinates, q; _q; Äq 2 R 2 1 are the vectors of positions, velocities and accelerations. The vector of generalized coordinates can be spit-up as : 12

13 Context Under-actuated mechanical systems Dynamics The dynamics can be rewritten as with: M(q) = m 11 Äq a + m 12 Äq na + h 1 + g 1 = u m 21 Äq a + m 22 Äq na + h 2 + g 2 = 0 h1 g1 ; H(q; _q) = ; G(q) = m 21 m 22 h 2 g 2 m11 m 12 From the equation of actuated coordinate : Äq a = m 1 11 ( m 12Äq na h 1 g 1 + u) Replaced in the dynamics of unactuated coordinate gives : where: m 2 Äq na + h 2 + g 2 = m 21 m 1 11 u m 2 = m 22 m 21 m 1 11 m 12 h 2 = h 2 m 21 m 1 11 h 1 g 2 = g 2 m 21 m 1 11 g 1 13

14 Context Stabilization L.C. generation Demonstrator Part II Basic principle of the gyrostabilizer Description of experimental testbed Description of the mechanical part Dynamic modeling 14

15 Context Stabilization L.C. generation Demonstrator Principle of the gyrostabilizer The effect of a torque (e.g. gravity / excitation moments) Causes a variation in the spin axis Reaction of a spinning wheel Output torque orthogonal to the input torque and spin axes The phenomenon provides an effective means of motion control and balance Gyro s flywheel must be in motion to resist gravity Two early examples of application : The Schilovski Gyrocar (1914) Brennan monorail (1924) Single gimbal active stabilizer unit With 40 inch diameter & 4.5 inch thick Flywheel operated at rpm Twin type active stabilizer system (3000 rpm) 40 feet long and weighted 22 tons Developed primary for military applications 15

16 Context Stabilization L.C. generation Demonstrator Examples of some applications [Townsend et al 2007] Marine systems Robotics Aerospace Underwater vehicles Gyrostabilizer Applications Monorails Academic Cars IKURA AUV ECP 750 Two-wheel gyro car 16

17 Context Stabilization L.C. generation Demonstrator Description of the experimental testbed encoder Micro strain FAS-G Inclinometer Maxon EC-Powermax 30 (DC Brushless) Control PC Pendulum body Power supply (12V) Speed variator Inertia wheel Input/output card Mechanical part Electric/electronic part 17

18 Context Stabilization L.C. generation Demonstrator Description of the mechanical part Inclinometer Pendulum body Inertia wheel Active articulation Passive articulation Frame Mechanical part of the system : inertia wheel inverted pendulum 18

19 Context Stabilization L.C. generation Demonstrator Schematic view of principle θ 2 2 G 2 z x O y G 1 θ O Equilibrium points of the system A. Stable equilibrium point B. Intermediate state C. Unstable equilibrium point 19

20 Context Stabilization L.C. generation Demonstrator Functioning principle The actuator is controlled to produce a torque on the inertia wheel Torque can produce an acceleration of the rotating wheel Thanks to the dynamic coupling, a torque acting on the passive joint is generated This passive joint can be controlled through the acceleration of the inertia wheel A G 2 G 2 Rotation G 1 B G 1 O O Initial mechanical model Equivalent mechanical model 20

21 Context Stabilization L.C. generation Demonstrator Functioning principle A G 2 B G 1 O Three moments are acting on the passive joint : Moment relative to the force : Moment relative to the force : Moment du to the gravity force : Rotation 1 Rotation 2 O One moment (torque of actuator) is acting on the active joint (inertia wheel) 21

22 Context Stabilization L.C. generation Demonstrator Generalized coordinates : Dynamic modeling q 1 = µ 1 ; q 2 = µ 2 Propose to use the formalism of Lagrange : The application of Lagrange principle gives : d i = Q i ; i = 1::2 T = 1 2 (m 1l m 2 l i 1 ) _ µ i 2 ( _ µ 1 + _ µ 2 ) 2 V = (m 1 l 1 + m 2 l 2 )g cos(µ 1 ) L = T V = 1 2 I _ µ i 2 ( _ µ 1 + _ µ 2 ) 2 mlg cos(µ 1 ) ĵ 1 = 1 I 2 + mlg sin µ 1 ml = m 1 l 1 + m 2 l 2 I = m 1 l m 2l i 1 I + i2 i 2 i 2 i 2 ĵ1 Ä µ 2 ĵ 2 = 1 Ii 2 (i2 + I) 2 i 2 mlg sin µ 1 mlg sin µ1 + 0 = 0 2 M(q)Äq + H(q; _q) + G(q) = Ru 22

23 Context Part III Control problem formulation Proposed solution Real-time experimental results 23

24 Context Control problem formulation Assume the system in some initial condition Find a control input u to bring to and maintain it around this point O O Proposed solutions : State feedback control Optimal control Generalized Predictive Control (GPC) Linear discrete dynamics 24

25 Context Recall the nonlinear dynamics : Linearization of the dynamics Consider the state vector : and The nonlinear dynamics can be written in nonlinear state space as : The unstable equilibrium of the system : The linearization of the dynamics around the unstable equilibrium gives : With : 25

26 Context Recall the linearized dynamics : Discritization of the dynamics The discretization of the dynamics gives : Summary of geometric and dynamic parameters of the system 26

27 Context Proposed control approach : GPC Consider the case of the GPC approach with penalty on the end-state Recall the discrete dynamics of the system : Consider the extended state vector : The variation on the control input : The dynamics can be written as : with : This last dynamics will be used in the controller design 27

28 Context Proposed control approach : GPC Consider the following optimization function : From the system model one can have the state predictions : The prediction of future outputs be : 28

29 Context In a matrix form we have : Proposed control approach : GPC The control input is computed as the minimum of : with The optimal solution is : 29

30 Context Real-time experiments Four proposed experimental scenarios Scenario 1 : Stabilization in the nominal case Scenario 2 : Case with persistent disturbance Scenario 3 : Case with punctual disturbance Scenario 4 : Combination of the two disturbances 30

31 Context Scenario 1 : Stabilization in the nominal case 31

32 Context Scenario 2 : Case with persistent disturbance 32

33 Context Scenario 3 : Case with punctual disturbance 33

34 d 2 [rad/s] U[N m] 1 [rad] d 1 [rad/s] Context Scenario 4 : Combination of the two disturbances 10 La position angulaire du pendule inversé 6 La vitesse angulaire du pendule inversé Temps[s] temps[s] 2000 La vitesse angulaire du volant d'inertie 10 Le couple du volant d'inertie Temps[s] Temps[s] 34

35 Context Part IV Control problem formulation Reference trajectories generation Proposed solution 1 : Optimal partial feedback linearization Proposed solution 2 : Dual model-free control 35

36 Context Control problem formulation Assume that the system in some initial condition Find a control input u to bring and maintain it around an oscillating trajectory While keeping the internal dynamics stable Stable O O Proposed solutions : Solution 1: Partial feedback linearization with optimization Solution 2 : Dual model-free control Both solutions need reference trajectories generation 36

37 Context 37

38 Context Reference trajectories generation The first step is parameterized reference trajectories generation To be tracked on unactuated coordinate Consider parameterized trajectories that should be : Continuous and derivable Periodic in order to generate limit cycles Boundary conditions Split up in half period Use symmetry to generate the whole cycle The parameterization of these trajectories allows the controller to update p corresponds to time at which trajectory crosses zero 38

39 Context Reference trajectories generation For a given period and amplitude the boundary conditions : A 6-degree polynomial function Normalized reference trajectories For : Plot on a half of period To keep oscillatory shape Symmetry property 39

40 Context Solution 1 40

41 Context First proposed solution for limit cycle generation 41

42 Context First proposed solution for limit cycle generation 2 = 42

43 Context First proposed solution for limit cycle generation 43

44 Context Simulation results 44

45 Context Simulation results 45

46 Context Simulation results 46

47 Context Real-time experiments Two proposed experimental scenarios Scenario 1 : Without external disturbances Scenario 2 : With external disturbances 47

48 Context Scenario 1 : Without external disturbances _ µ 1[rad=s] µ1[rad] _ µ 1[rad=s] Time [s] Time [s] Optimization parameter p µ 1 [rad] Time [s] 48

49 Context Scenario 1 : Without external disturbances _ µ 2[rad=s] U [V] Time [s] Absolute motor velocity [rpm] Time [s] Absolute motor torque [Nm] 49

50 Context Scenario 2 : With external disturbances _ µ 1[rad=s] µ1[rad] _ µ 1[rad=s] Time [s] External disturbance Time [s] Optimization parameter p µ 1 [rad] Time [s] 50

51 Context Scenario 2 : With external disturbances _ µ 2[rad=s] U [V] Time [s] Absolute motor velocity [rpm] Time [s] Absolute motor torque [Nm] 51

52 Context Solution 2 52

53 Context Second proposed solution for limit cycle generation Objective : generation stable limit cycles on both coordinates Proposed control architecture: + - The reference trajectories are generated given a parameter p These trajectories are tracked (on unactuated coordinate) by a first model-free controller Parameter p is updated by the second model-free controller (stabilize actuated coordinate) 53

54 Context Second proposed solution for limit cycle generation Model-free control : Background Model-free control (called also intelligent PID) : recently developed Its design relies on a local modeling valid for a short time interval This local modeling is updated based on input-output behavior Controlled system dynamics can be linear/nonlinear and/or time-varying Assume that the input-output behavior of the system can be expressed as : E(y; _y; : : : ; y (a) ; u; _u; : : : ; u (b) ) = 0 Consider then the following approximation (valid only on a short time interval) y (n) = F + u The parameters and are chosen by the designer 54

55 Context Second proposed solution for limit cycle generation Model-free control : Background The numerical value of is computed at each sample time : F(k) = [y (n) (k)] e u(k 1) [y (n) (k)] e is the estimation of the derivative of the output The desired behavior is obtained trough the following control u = F Z + y(n) + K pe + K i e + K d _e is the output reference trajectory, and is its derivative y e = y y K p ; K i ; K d is the tracking error y (n) are the controller gains (chosen by the designer) 55

56 Context Second proposed solution for limit cycle generation The first controller : Tracking of reference trajectories q na(p; ; t) Recall the dynamics of the unactuated coordinate : m 2 Äq na + h 2 + g 2 = m 21 m 1 11 u on unactuated coordinate Consider on a short time interval the following approximation : Äq na = F u According to the principle of model-free control : u = F 1 + Äq na (p; ; t) + K p1 e + K i1 1 1 Z e + K d1 _e with: e = q d na q na At each sample time this control input is computed and applied to the system 56

57 Context Second proposed solution for limit cycle generation The second controller : The parameter p used in the previous controller is constant over half a period 8t 2 [k 2 (k + 1) 2 [ k 2 N Updated at the end of each half a period at time (k + 1) 2 The idea is to use the second model-free controller to update p Consider the actuated coordinate velocity v a = _q a Its variation over a half period : v a = _q a (k 2 ) _q a((k 1) 2 ) It is replaced by the discrete local approximation : v a = F p According to the principle of model-free control : p = F Z 2 + K p2 e a + K i2 e a + K d2 _e a with 2 At each half period this control input is computed e a = q d a q a 57

58 Context 58

59 Context µ 1 Simulation results _µ 2 _µ 1 External disturbance u _µ 1 j _ µ m j p µ 1 juj 59

60 Context 60

61 Pendulum velocity [rad/s] Phase portrait Pendulum position [rad] Inertia-wheel velocity [rad/s] Context µ 1 Experimental results _µ 2 _µ 1 External disturbances _µ 1 µ 1 61

62 Control input [N. m] Power admissibility Context u Experimental results j _ µ m j p juj 62

63 Context Experimental results : movie 63

64 Context Application 1 : Classical inverted pendulum Application 2 : Pendubot 64

65 Context Application 1 : inverted pendulum (cart-pole system) A classical under-actuated system It consists of a pendulum beam Attached to a cart through a passive joint 2 dof versus one actuator Measurement noise : System parameters : Reference trajectories parameters : Controller parameters : 65

66 unactuated coordinate Actuated coordinate Context Application 1 : inverted pendulum (cart-pole system) Disturbance 0.2N.m 66

67 Control signals Actuated coordinate unactuated coordinate Context Application 1 : inverted pendulum (cart-pole system) 67

68 Context Application 2 : a 2-dof planar under-actuated manipulator (Pendubot) A classical system in robotics It consists of two-link manipulator Two joints (one active & one passive) 2 dof versus one actuator Control absolute angles with : 68

69 Context Application 2 : a 2-dof planar under-actuated manipulator (Pendubot) Measurement noise : System parameters : Reference trajectories parameters : Controller parameters : 69

70 unactuated coordinate Actuated coordinate Context Application 2 : a 2-dof planar under-actuated manipulator (Pendubot) Disturbance 8 N.m 70

71 Control signals Actuated coordinate unactuated coordinate Context Application 2 : a 2-dof planar under-actuated manipulator (Pendubot) 71

72 Context Part V 72

73 Context Problem : Control of under-actuated mechanical systems for Stabilization and limit cycle generation Our demonstrator : Inertia wheel inverted pendulum Proposed solution for stabilization : GPC approach Proposed solutions for limit cycle generation : Solution 1 : Partial feedback linearization with optimization Solution 2 : A dual model-free control scheme Implementation & validation : all the controllers validated on our demonstrator 73