Mecánica de Sistemas Multicuerpo:

Transcription

1 Universidad Pública de Navarra 12 de Noviembre de 2008 Departamento de Ingeniería Mecánica, Energética y de Materiales Mecánica de Sistemas Multicuerpo: Análisis de la Silla de Ruedas Triesférica y Dinámica de la Marcha de Sistemas Bípedos Josep Maria Font Llagunes josep.m.font@upc.edu Departamento de Ingeniería Mecánica McGill University

2 Presentation Contents Wheelchair Kinematics Wheelchairs with Conventional Wheels Wheelchair with Omnidirectional Wheels Mechanics of Wheelchairs Introduction to Wheelchair Dynamics Introduction to Dynamic Walking Dynamic Model of the Walking System Decomposition of the Impulsive Motion Biomechanics of Bipedal Systems Numerical Results and Discussion

3 Degrees of Freedom of a Wheelchair Wheelchair Kinematics

4 Control of a Wheelchair with Differential Steering Wheelchair Kinematics

5 Control of a Wheelchair with Differential Steering Wheelchair Kinematics

6 Control of a Wheelchair with Direct Steering Wheelchair Kinematics

7 Kinematics in Wheelchair Control Wheelchair Kinematics

8 Presentation Contents Wheelchair Kinematics Wheelchairs with Conventional Wheels Wheelchair with Omnidirectional Wheels Mechanics of Wheelchairs Introduction to Wheelchair Dynamics Introduction to Dynamic Walking Dynamic Model of the Walking System Decomposition of the Impulsive Motion Biomechanics of Bipedal Systems Numerical Results and Discussion

9 Wheelchair with Differential Steering Wheelchairs with Conventional Wheels

10 Control of a Wheelchair with Differential Steering Wheelchairs with Conventional Wheels

11 Wheelchairs with Conventional Wheels

12 Wheelchairs with Conventional Wheels

13 Wheelchairs with Conventional Wheels

14 Wheelchairs with Conventional Wheels

15 Wheelchairs with Conventional Wheels

16 Wheelchairs with Conventional Wheels

17 Wheelchair with Tricycle Steering Wheelchairs with Conventional Wheels

18 Tricycle Wheelchair with Steering-Driving Wheel Wheelchairs with Conventional Wheels

19 Tricycle Wheelchair with Steering-Driving Wheel Wheelchairs with Conventional Wheels

20 Kinematics of a Tricycle Wheelchair Wheelchairs with Conventional Wheels

21 Control of a Tricycle Wheelchair Wheelchairs with Conventional Wheels

22 Presentation Contents Wheelchair Kinematics Wheelchairs with Conventional Wheels Wheelchair with Omnidirectional Wheels Mechanics of Wheelchairs Introduction to Wheelchair Dynamics Introduction to Dynamic Walking Dynamic Model of the Walking System Decomposition of the Impulsive Motion Biomechanics of Bipedal Systems Numerical Results and Discussion

23 Mobility of the Centre of the Wheel Wheelchairs with Omnidirectional Wheels

24 Omnidirectional Wheel with Rollers at 45º Wheelchairs with Omnidirectional Wheels

25 Omnidirectional Wheel with Rollers at 90º Wheelchairs with Omnidirectional Wheels

26 3-DOF Platform with 3 Omnidirectional Wheels Wheelchairs with Omnidirectional Wheels

27 Omnidirectional Wheel with Spherical Rollers Wheelchairs with Omnidirectional Wheels

28 Omnidirectional Wheel with Spherical Rollers Wheelchairs with Omnidirectional Wheels

29 Wheelchairs with Omnidirectional Wheels

30 Wheelchairs with Omnidirectional Wheels

31 LONGITUDINAL DOF Wheelchairs with Omnidirectional Wheels

32 Wheelchairs with Omnidirectional Wheels

33 TRANSVERSAL DOF Wheelchairs with Omnidirectional Wheels

34 Wheelchairs with Omnidirectional Wheels

35 ROTATIONAL DOF Wheelchairs with Omnidirectional Wheels

36 Wheelchairs with Omnidirectional Wheels

37 Wheelchair with 3 Omnidirectional Wheels Wheelchairs with Omnidirectional Wheels

38 Wheelchair with 3 Omnidirectional Wheels Wheelchairs with Omnidirectional Wheels

39 Wheelchair Motion Modes Wheelchairs with Omnidirectional Wheels

40 Control of the Motion Modes Wheelchairs with Omnidirectional Wheels

41 Wheelchairs with Omnidirectional Wheels

42 Longitudinal motion Transverse motion Rotation General motion Wheelchairs with Omnidirectional Wheels

43 Presentation Contents Wheelchair Kinematics Wheelchairs with Conventional Wheels Wheelchair with Omnidirectional Wheels Mechanics of Wheelchairs Introduction to Wheelchair Dynamics Introduction to Dynamic Walking Dynamic Model of the Walking System Decomposition of the Impulsive Motion Biomechanics of Bipedal Systems Numerical Results and Discussion

44 Equations of Motion: Method of Virtual Work Introduction to Wheelchair Dynamics

45 Introduction to Wheelchair Dynamics

46 Introduction to Wheelchair Dynamics

47 Use of Omnidirectional Wheels. Conclusions Concluding Remarks

48 Universidad Pública de Navarra 12 de Noviembre de 2008 Departamento de Ingeniería Mecánica, Energética y de Materiales Effects of Mass Distribution and Configuration on the Energetic Losses at Impacts of Bipedal Walking Systems Josep Maria Font 1,2 and József Kövecses 1 1: Department of Mechanical Engineering and Centre for Intelligent Machines McGill University, Montréal, Canada 2: Department of Mechanical Engineering Universitat Politècnica de Catalunya, Barcelona, Spain

49 Presentation Contents Wheelchair Kinematics Wheelchairs with Conventional Wheels Wheelchair with Omnidirectional Wheels Mechanics of Wheelchairs Introduction to Wheelchair Dynamics Introduction to Dynamic Walking Dynamic Model of the Walking System Decomposition of the Impulsive Motion Biomechanics of Bipedal Systems Numerical Results and Discussion

50 Dynamic Walking or Limit Cycle Walking Dynamic Walking models are used to increase the understanding of the principles underlying bipedal locomotion. Starting point: Passive Dynamic Walking [McGeer 1990] Passive walker with knees [Nagoya Institute of Technology] Introduction to Dynamic Walking

51 Dynamic Walking or Limit Cycle Walking Dynamic Walking models are used to increase the understanding of the principles underlying bipedal locomotion. Starting point: Passive Dynamic Walking [McGeer 1990] Passive Walking resembles Human Walking [Nagoya Institute of Technology] Introduction to Dynamic Walking

52 Dynamic Walking or Limit Cycle Walking Actuated Dynamic Walkers have been recently developed (e.g., robot Flame developed at TU Delft). Walk on level ground, Orbitally stable (limit cycle), Human-like motion, Energetically efficient. Robot Flame [TU Delft] Introduction to Dynamic Walking

53 Presentation Contents Wheelchair Kinematics Wheelchairs with Conventional Wheels Wheelchair with Omnidirectional Wheels Mechanics of Wheelchairs Introduction to Wheelchair Dynamics Introduction to Dynamic Walking Dynamic Model of the Walking System Decomposition of the Impulsive Motion Biomechanics of Bipedal Systems Numerical Results and Discussion

54 Phases of the Walking Motion Single-support phase (Finite Motion) Heel Strike (Impulsive Motion) Dynamic Model of the Walking System

55 Phases of the Walking Motion Single-support phase (Finite Motion) Heel Strike (Impulsive Motion) ( ) ( ) ( ) Mqq + cqq + uq= f + Aλ AS q = 0 T, A S S Bilateral constraints Dynamic Model of the Walking System

56 Phases of the Walking Motion Single-support phase (Finite Motion) Heel Strike (Impulsive Motion) ( ) ( ) ( ) Mqq + cqq + uq= f + Aλ AS q = 0 T, A S S Bilateral constraints q T + + AI q + = v S S n T ( ) I λ I = M q q = A = Bq 0 Impulsive constraints Dynamic Model of the Walking System

57 Phases of the Walking Motion Single-support phase (Finite Motion) Heel Strike (Impulsive Motion) ( ) ( ) ( ) Mqq + cqq + uq= f + Aλ AS q = 0 T, A S S Bilateral constraints Main cause of energy loss. Topology transition (some constraints are added and other become passive). Dynamic Model of the Walking System

58 Compass-Gait Biped with Upper Body l = 0.8 m l T = 0.4 m a = b = 0.4 m m B = 30 kg µ = 2m m H Lower body mass distribution m µ T = m T H Upper body mass distribution Generalized coordinates: q = T Kinetic energy: T ( qq, ) = ( ) Dynamic Model of the Walking System 1 q M q q 2 [ q1, q2, q3, q4, q5] T

59 Presentation Contents Wheelchair Kinematics Wheelchairs with Conventional Wheels Wheelchair with Omnidirectional Wheels Mechanics of Wheelchairs Introduction to Wheelchair Dynamics Introduction to Dynamic Walking Dynamic Model of the Walking System Decomposition of the Impulsive Motion Biomechanics of Bipedal Systems Numerical Results and Discussion

60 Heel Strike Dynamics Impulse-momentum level dynamic equations: q T + + T ( ) I λ I = M q q = A Impulsive constraints: AI q + = 0 (defines post-impact kinematic condition) A I : constraint Jacobian matrix. This matrix has different representations depending on which foot collides the ground. A A R L = ( ) ( ) ( ) ( ) 1 0 lcos q3 lcos q4 q3 lcos q4 q3 0 = 0 1 lsin q3 lsin q4 q3 lsin q4 q3 0 Decomposition of the Impulsive Motion

61 Decomposition of the Dynamic Equations The tangent space of the walking system can be decomposed to two subspaces mutually orthogonal with respect to the mass metric of the system [Kövecses 2003] This is achieved based on the following projection operators T ( ) 1 P = M A A M A A 1 T 1 c I I I I T ( ) 1 P = I M A A M A A 1 T 1 a I I I I Space of Constrained Motion (SCM) Space of Admissible Motion (SAM) The generalized velocities and impulses can be decoupled as q=pq + Pq=v + v c a c a f=pf+ Pf=f+ f T T c a c a Decomposition of the Impulsive Motion

62 Decomposition of the Dynamic Equations This gives a complete decoupling of the dynamic equations + Tc = + T M( vc vc ) = AI λ I vc + Ta = = v a + ( a a) M v v 0 Space of Constrained Motion (SCM) Space of Admissible Motion (SAM) + Solution: c = v 0 and v a = v a q = va = Pq a and the kinetic energy of the system 1 T 1 T T = T + T = v Mv + v Mv 2 2 c a c c a a Decomposition of the Impulsive Motion

63 Kinetic Energy Decomposition at the Pre-Impact Time 1 ( ) T 1 ( ) T T = T + T = v Mv + v Mv 2 2 c a c c a a Kinetic Energy of Constrained Motion LOST at Heel Strike Kinetic Energy of Admissible Motion STAYS in the system Useful tool to analyze energetic losses at heel strike and gain insight into the behaviour of dynamic walkers at impact. Energy loss per unit distance: 1 T T ( q ) Pc MPcq Tc ξ 2 L = = L 2lsin q S 3 Decomposition of the Impulsive Motion

64 Presentation Contents Wheelchair Kinematics Wheelchairs with Conventional Wheels Wheelchair with Omnidirectional Wheels Mechanics of Wheelchairs Introduction to Wheelchair Dynamics Introduction to Dynamic Walking Dynamic Model of the Walking System Decomposition of the Impulsive Motion Biomechanics of Bipedal Systems Numerical Results and Discussion

65 Simulation Results Goal: Analyze the effect of the body configuration and mass distribution on the dynamics of heel strike. Results and Discussion

66 Effects of the Lower Body on the Foot Separation Post-impact vel. v + S n (m/s) Concentrating the mass of the lower body at the hip increases the range of angles for which the trailing foot passively lifts up. Results and Discussion

67 Effects of the Lower Body on the Kinetic Energy Decomposition Kinetic Energy T c Kinetic Energy T a Concentrating the mass of the lower body at the legs reduces the energy loss at impact. A low impact angle q 4 reduces the kinetic energy loss (for a given mass distribution). Results and Discussion

68 Effects of the Lower Body on the Cost of Transport Cost of transport ξ L (J/m) Concentrating the mass of the lower body at the legs reduces the energy loss per unit distance. A low impact angle q 4 (small steps) reduces the energy loss per unit distance. Results and Discussion

69 Effects of the Upper Body on the Foot Separation Post-impact vel. v + S n (m/s) Concentrating the mass of the upper body at the hip increases the post-impact normal velocity of the trailing foot. Results and Discussion

70 Effects of the Upper Body on the Kinetic Energy Decomposition Kinetic Energy T c Kinetic Energy T a Concentrating the mass of the upper body at the top reduces the kinetic energy loss. A torso leaning forward (q 5 =0) improves the efficiency of the impact (for a given mass distribution). Results and Discussion

71 Conclusions We presented a Lagrangian formulation applicable to the study of the impulsive dynamics of heel strike. We introduced a decomposition of the dynamic equations and the kinetic energy to the spaces of constrained and admissible motions. This is useful to analyze the kinetic energy redistribution and the velocity change at heel strike. A low inter-leg angle at heel strike and a torso leaning forward reduce the energetic consumption per unit distance due to impacts. Conclusions

72 Universidad Pública de Navarra 12 de Noviembre de 2008 Departamento de Ingeniería Mecánica, Energética y de Materiales Mecánica de Sistemas Multicuerpo: Análisis de la Silla de Ruedas Triesférica y Dinámica de la Marcha de Sistemas Bípedos Josep Maria Font Llagunes josep.m.font@upc.edu Departamento de Ingeniería Mecánica McGill University