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
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
Degrees of Freedom of a Wheelchair Wheelchair Kinematics
Control of a Wheelchair with Differential Steering Wheelchair Kinematics
Control of a Wheelchair with Differential Steering Wheelchair Kinematics
Control of a Wheelchair with Direct Steering Wheelchair Kinematics
Kinematics in Wheelchair Control Wheelchair Kinematics
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
Wheelchair with Differential Steering Wheelchairs with Conventional Wheels
Control of a Wheelchair with Differential Steering Wheelchairs with Conventional Wheels
Wheelchairs with Conventional Wheels
Wheelchairs with Conventional Wheels
Wheelchairs with Conventional Wheels
Wheelchairs with Conventional Wheels
Wheelchairs with Conventional Wheels
Wheelchairs with Conventional Wheels
Wheelchair with Tricycle Steering Wheelchairs with Conventional Wheels
Tricycle Wheelchair with Steering-Driving Wheel Wheelchairs with Conventional Wheels
Tricycle Wheelchair with Steering-Driving Wheel Wheelchairs with Conventional Wheels
Kinematics of a Tricycle Wheelchair Wheelchairs with Conventional Wheels
Control of a Tricycle Wheelchair Wheelchairs with Conventional Wheels
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
Mobility of the Centre of the Wheel Wheelchairs with Omnidirectional Wheels
Omnidirectional Wheel with Rollers at 45º Wheelchairs with Omnidirectional Wheels
Omnidirectional Wheel with Rollers at 90º Wheelchairs with Omnidirectional Wheels
3-DOF Platform with 3 Omnidirectional Wheels Wheelchairs with Omnidirectional Wheels
Omnidirectional Wheel with Spherical Rollers Wheelchairs with Omnidirectional Wheels
Omnidirectional Wheel with Spherical Rollers Wheelchairs with Omnidirectional Wheels
Wheelchairs with Omnidirectional Wheels
Wheelchairs with Omnidirectional Wheels
LONGITUDINAL DOF Wheelchairs with Omnidirectional Wheels
Wheelchairs with Omnidirectional Wheels
TRANSVERSAL DOF Wheelchairs with Omnidirectional Wheels
Wheelchairs with Omnidirectional Wheels
ROTATIONAL DOF Wheelchairs with Omnidirectional Wheels
Wheelchairs with Omnidirectional Wheels
Wheelchair with 3 Omnidirectional Wheels Wheelchairs with Omnidirectional Wheels
Wheelchair with 3 Omnidirectional Wheels Wheelchairs with Omnidirectional Wheels
Wheelchair Motion Modes Wheelchairs with Omnidirectional Wheels
Control of the Motion Modes Wheelchairs with Omnidirectional Wheels
Wheelchairs with Omnidirectional Wheels
Longitudinal motion Transverse motion Rotation General motion Wheelchairs with Omnidirectional Wheels
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
Equations of Motion: Method of Virtual Work Introduction to Wheelchair Dynamics
Introduction to Wheelchair Dynamics
Introduction to Wheelchair Dynamics
Use of Omnidirectional Wheels. Conclusions Concluding Remarks
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
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
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
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
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
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
Phases of the Walking Motion Single-support phase (Finite Motion) Heel Strike (Impulsive Motion) Dynamic Model of the Walking System
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
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 + = 0 + + v S S n T ( ) I λ I = M q q = A = Bq 0 Impulsive constraints Dynamic Model of the Walking System
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
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
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
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 0 0 0 = 0 1 0 0 0 ( ) ( ) ( ) ( ) 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
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
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
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
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
Simulation Results Goal: Analyze the effect of the body configuration and mass distribution on the dynamics of heel strike. Results and Discussion
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
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
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
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
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
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
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