Análisis de la Silla de Ruedas “Triesférica” y Dinámica de ...font/downloads/IMAC_MBS.pdf · q=Pq Pq=v v c a ca++ Decomposition of the Impulsive Motion. Decomposition of the

Josep Maria Font [email protected]

Mecánica de Sistemas Multicuerpo:Análisis de la Silla de Ruedas “Triesférica” y Dinámica de la Marcha de Sistemas Bípedos

Universidad Pública de Navarra

12 de Noviembre de 2008

Departamento de Ingeniería Mecánica,

Energética y de Materiales

Departamentode Ingeniería Mecánica

McGill University

Presentation Contents

Wheelchair Kinematics

Wheelchairs with Conventional Wheels

Wheelchair with Omnidirectional Wheels

Introduction to Wheelchair Dynamics

Introduction to Dynamic Walking

Dynamic Model of the Walking System

Decomposition of the Impulsive Motion

Numerical Results and Discussion

Mechanics of Wheelchairs

Biomechanics of Bipedal Systems


Degrees of Freedom of a Wheelchair


Control of a Wheelchair with Differential Steering




Control of a Wheelchair with Direct Steering


Kinematics in Wheelchair Control













Wheelchair with Differential Steering










Wheelchair with Tricycle Steering


Tricycle Wheelchair with Steering-Driving Wheel


Tricycle Wheelchair with Steering-Driving Wheel


Kinematics of a Tricycle Wheelchair


Control of a Tricycle Wheelchair












Wheelchairs with Omnidirectional Wheels

Mobility of the Centre of the Wheel


Omnidirectional Wheel with Rollers at 45º


Omnidirectional Wheel with Rollers at 90º


3-DOF Platform with 3 Omnidirectional Wheels


Omnidirectional Wheel with Spherical Rollers


Omnidirectional Wheel with Spherical Rollers



LONGITUDINAL DOF



TRANSVERSAL DOF



ROTATIONAL DOF




Wheelchair with 3 Omnidirectional Wheels


Wheelchair with 3 Omnidirectional Wheels


Wheelchair Motion Modes


Control of the Motion Modes


Longitudinal motion

Rotation

Transverse motion

General motion














Equations of Motion: Method of Virtual Work



Concluding Remarks

Use of Omnidirectional Wheels. Conclusions

Josep Maria Font1,2 and József Kövecses1

Effects of Mass Distribution and Configuration on the Energetic Losses at Impacts of Bipedal Walking Systems

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
















Dynamic Walking models are used to increase the understanding of the

principles underlying bipedal locomotion.

Starting point: Passive Dynamic Walking [McGeer 1990]

Dynamic Walking or ‘Limit Cycle Walking’


Passive walker with knees [Nagoya Institute of Technology]



Passive Walking resembles Human Walking [Nagoya Institute of Technology]

Dynamic Walking models are used to increase the understanding of the

principles underlying bipedal locomotion.

Starting point: Passive Dynamic Walking [McGeer 1990]

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]












Phases of the Walking Motion


Single-support phase (Finite Motion) Heel Strike (Impulsive Motion)


( ) ( ) ( ), TA S S+ + = +M q q c q q u q f A λ

S =A q 0 Bilateral constraints






( ) T

I IT

++ −

−

∂= − = ∂

M q q Aq

λ

I+ =A q 0 Impulsive constraints

0n+S Sv += ≥B q






• Main cause of energy loss.

• Topology transition (some constraints are added and other become passive).



Compass-Gait Biped with Upper Body


l = 0.8 m

lT = 0.4 m

a = b = 0.4 m

mB = 30 kg

• Generalized coordinates:

• Kinetic energy: ( ) ( )1,2

TT =q q q M q q

[ ]1 2 3 4 5, , , , Tq q q q q=q

2H

mm

µ = Lower bodymass distribution

Upper bodymass distribution

TT

H

mm

µ =













• Impulse-momentum level dynamic equations:

( ) T

I IT

++ −

−

∂= − = ∂

M q q Aq

λ

• Impulsive constraints:

I+ =A q 0

AI : constraint Jacobian matrix. This matrix has different representations depending on which foot collides the ground.

(defines post-impact kinematic condition)

Heel Strike Dynamics

1 0 0 0 00 1 0 0 0R

=

A

( ) ( )( ) ( )

3 4 3 4 3

3 4 3 4 3

1 0 cos cos cos 00 1 sin sin sin 0L

l q l q q l q ql q l q q l q q

− − − = − − − −

A

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

( ) 11 1T Tc I I I I

−− −=P M A A M A A

( ) 11 1T Ta I I I I

−− −= −P I M A A M A A

Space of Constrained Motion (SCM)

Space of Admissible Motion (SAM)

The generalized velocities and impulses can be decoupled as

T Tc a c a+ +f = P f P f = f f

c a c a+ +q = P q P q = v v


Decomposition of the Dynamic Equations

This gives a complete decoupling of the dynamic equations

Space of Constrained Motion (SCM)

Space of Admissible Motion (SAM)

( )

( )

c Tc c I I

c

aa a

a

T

T

+−

−

+−

−

∂ = − = ∂

∂ = − = ∂

+

+

M v v Av

M v v 0v

λ

1 12 2

T Tc a c c a aT T T= + = v Mv + v Mv

and the kinetic energy of the system

Solution: c =+v 0 and a a−=+v v a a

−= =+ +q v P q


( )3

12

2 sin

T Tc c

cL

S

TL l q

− −−

−ξ = =q P MP q

Kinetic Energy Decomposition at the Pre-Impact Time

Kinetic Energy of Admissible Motion

( ) ( )1 12 2

T Tc a c c a aT T T− − − − − − −= + = +v M v v M v

Kinetic Energy of Constrained Motion

LOSTat Heel Strike

STAYSin 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:













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. (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.

nSv+

• Concentrating the mass of the lower body at the legs reduces the energy loss at impact.

• A low impact angle q4 reduces the kinetic energy loss (for a given mass distribution).

Kinetic Energy Tc Kinetic Energy Ta

Effects of the Lower Body on the Kinetic Energy Decomposition


Cost of transport ξL (J/m)

• Concentrating the mass of the lower body at the legs reduces the energy loss per unit distance.

• A low impact angle q4 (small steps) reduces the energy loss per unit distance.

Effects of the Lower Body on the Cost of Transport


Effects of the Upper Body on the Foot Separation


• Concentrating the mass of the upper body at the hip increases the post-impact normal velocity of the trailing foot.

Post-impact vel. (m/s) nSv+


• Concentrating the mass of the upper body at the top reduces the kinetic energy loss.

• A torso leaning forward (q5=0) improves the efficiency of the impact (for a given mass distribution).

Kinetic Energy Tc Kinetic Energy Ta

Effects of the Upper Body on the Kinetic Energy Decomposition

Conclusions

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.

Josep Maria Font [email protected]

Mecánica de Sistemas Multicuerpo:Análisis de la Silla de Ruedas “Triesférica” y Dinámica de la Marcha de Sistemas Bípedos





Departamentode Ingeniería Mecánica

McGill University

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Análisis de la Silla de Ruedas “Triesférica” y Dinámica de ...font/downloads/IMAC_MBS.pdf · q=Pq Pq=v v c a ca++ Decomposition of the Impulsive Motion. Decomposition of the