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Analytical Formulations in Lagrangian Dynamics: Theoretical Aspects and Applications to Interactions with Virtual and Physical

Environments

József Kövecses

Department of Mechanical Engineering and Centre for Intelligent Machines

McGill University Montreal, Quebec, Canada

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McGill University

• Close to 200 years old• ~25-30,000 students• No. 1 research university in Canada (2006 Research

Infosource, highest proportion of graduate students in Canada

• No. 18 worldwide (2009 Times Higher Education Supplement); No. 20 in Engineering

• No. 1 in Maclean’s magazin rankings (ranking of Canadian universities)

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Faculty of Engineering

• ~150 faculty members (professors), approximately, within the following academic units

• Departments of – Chemical Engineering– Civil Engineering– Electrical and Computer Engineering – Mechanical Engineering (30 professors) – Mining and Materials Engineering – Architecture– Urban Planning

• Mechanical Engineering – 30 professors (assistant, associate, full) – Support staff (technicians, secretaries, etc.)

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Mechanical Systems• Simple systems can exhibit very complex behaviours

– Impact: different scales in space and time – Friction: different scales in space

• Example: Spinning and Sliding CoinsZ. Farkas, G. Bartels, T. Unger, D.E. Wolf: Frictional Coupling between Sliding and Spinning Motion, Physical Review Letters, 90, 248-302, (2003). T.C. Halsey: Friction in a Spin, Nature, 424, 1005, (2003).

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• Basic concepts in mechanical system dynamics

• Proposed concept: based on principle of relaxation of constraints

• Usual approaches and further possibilities

• Applications – Variable-topology multibody systems: topology transitions– Force feedback devices and interactions with virtual environments– Robotic vehicles for unstructured terrain

Outline

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• Biomechanical systems and humanoid robots• Force feedback devices and interactions with virtual environments

Two Examples

Anybody Technologies

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global representation

Representation of Geometric Vectors

local representation

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• Newton’s law

• Specification: force vs. motion

• Lagrangian approach

Some Basic Relations for Mechanical Systems

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• n coordinates and n velocities

• m constraints (holonomic and/or nonholonomic)

• Dynamics equations (in global frame)

Usual Approaches in Multibody System Dynamics

v

constraint Jacobian

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Possibility 1:

Possibility 2:

Meaning of the Lagrange Multipliers

m

L

gy

xO

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• Augmented set of equations using Lagrange multipliers

• Reduction to a minimum set of coordinates/velocities (n-m independent variables)

Usual Approaches in Multibody System Dynamics

v

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• Mostly developed for ideal bilateral constraints• Problematic situations can arise

– For example: non-ideal realization of constraints – Constrained and admissible dynamics are coupled– Constrained motion dynamics needs to be analyzed first

Usual Approaches

vr

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• Problematic situations can arise – Constrained motion dynamics needs to be analyzed first

Usual Approaches

• bilateral and unilateral constraints • nonideal contacts/constraints • imperfect constraints: e.g. contacts

with finite stiffness• nonideal servo constraints• also impulse/momentum level

description• etc.

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• Full development of the Principle of Relaxation of Constraints

• Moving the force-motion specification to a later phase: two-step process• Interpret distinguished directions/transformations in “space”

Proposed Approach

v

– constraints relaxed --- transformation

– possibility to generalize the idea of the free-body diagram

– mathematical formulation• variational• direct geometric

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• Freeing the system: – body or system: “free-body diagram”– representation of forces (Newtonian) – definition of a specific subspace with

transformation (Lagrangian): the space of constrained motion

Proposed Approach

v“free” body or system

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• Augmented set of equations using Lagrange multipliers

• Extension via relaxation

• Conditions for and – motion or force specification

(bilateral, unilateral, constitutive, etc.)

Extension of Usual Approaches

v

constrained

admissible

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• Reduction to a minimum set of coordinates/velocities (n-m) independent variables)

• Extension (local parameterization)

– Constrained motion dynamics

– Admissible motion dynamics

• Conditions: motion or force specification

Extension of Usual Approaches

v

constrained

admissible

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• Transformations and “Constraints”– Constrained motion (r dimensional subspace) – Admissible motion (n-r dimensional subspace)

space of admissible motion

– Local and global parameterizations of physicalquantities

Possible Unified Description

v

uconstrained

admissible

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• Possibility of redundant constraints/transformation

• Physical coordinates\components: homogeneous in units

• Definition of subspaces– space of constrained motion – given – space of admissible motion – defined by analyst

– non-orthogonal and orthogonal definition

Parameterization and Definition of Subspaces

v

uconstrained

admissible

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• Definition – Non-orthogonal: traditional way

• independent coordinates, Lagrange multipliers, etc.

– Orthogonal definition (mapping is determined)• decoupling – generally requires nonholonomic quasi-velocities

• Parameterization– local– global

– combination of the two – physical components– coupling between subspaces

Subspaces

v

uconstrained

admissible

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• Expressed with geometric vectors

– Calculus of base vectors

• Expressed with variations

– Variations (coordinates, velocities, accelerations)

Physical Principle

v

uconstrained

admissible

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• Balance of geometric vectors

– base vectors – M – mass matrix associated with v

– Expressed in global parameterization, v

– Expressed in local parameterization, u

Fundamental Equations

v

uconstrained

admissible

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• Definition – Non-orthogonal: traditional way

• independent coordinates, Lagrange multipliers, etc.

– Orthogonal definition (mapping is determined)• decoupling – generally requires nonholonomic quasi-velocities

• Parameterization– local– global

– combination of the two – physical components– coupling between subspaces

Subspaces

v

uconstrained

admissible

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• Dynamic equations for constrained motion

• Dynamic equations for admissible motion

Global Parameterization with Orthogonal Definition of Subspaces

v

uconstrained

admissible

– Motion vs. force specification– Transformations can be specified

redundantly – Physically consistent

manipulations/computations– Constrained motion space projector

of physical components - important

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• Transformations

• Orthogonality condition

• Dynamic equations for constrained motion

• Dynamic equations for admissible motion

Local Parameterization with Orthogonal Definition of Subspaces

v

uconstrained

admissible– Motion vs. force specification– Nonideal force terms

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• Variable-topology multibody systems: topology transitions

• Force feedback devices and interactions with virtual environments

• Robotic vehicles for unstructured terrain

Application Examples

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Applications to Multibody Systems with Time-Varying Topology

• Nature of forces developed during topology transitions• Dynamics of contact transitions: impulse-momentum

– unification of concepts: kinematic, kinetic, energetic • Qualitative investigation for impulsive motion

– Performance measures

Dynamics of constrained directions can be decoupled (constrained and admissible subspaces)

Can be done in any set of coordinates (measurability)

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• Decoupling of the kinetic energy

• Analysis of incremental work (energy loss for ideal case: space of constrained motion)

• Generalized definition for energetic coefficient of restitution:

• Original definition given by Stronge:

Application of Global Parameterization with Orthogonal Subspaces

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Applications to Contact Dynamics: Example of the Falling Rod

• Impact dynamics highly depends on the variation of Tc

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Variable Topology Systems: Transitions

Experimental Analysis McGill / UPC

d = 0 m

d = 0.2 m

γ = 90º

γ = 60º

γ = 90º

γ = 60º

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• Energy redistribution – “Rayleigh’s quotient”

• Eigen-value problem

• Constrained motion space projector for physical components

Characterization of Topology Change

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Example: Generalized Particle

Impulsive constraint:

SCM projector for physical components:

α

ρ

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Example: Generalized Particle• Eigenvalue

– Eigenvector

• Eigenvalue– Eigenvector

α

ρ

∗1v

∗2v

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Experimental Setup

• Experimental test-bed based on two 6-DoF dual-pantograph devices (equipped with high resolution force sensors and optical encoders).

• The end-effector of the active device impacts the passive device.

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Dynamics of Heel Strike in Bipedal Locomotion

• Sudden change of topology

• Constraints are added on the leading foot

• The trailing foot also undergoes a transition (must leave the ground)

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• Interaction between sensing/instrumentation and dynamic parameterization

• Dynamics of physical and virtual system assemblies – modelling, simulation, and control

Force Feedback Devices and Virtual Environments

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Maple Environment

Simulation Framework

Modelling the Multibody System

Generating Essential Functions

Translating to S-Functions

Visualization Dynamics Analysis

Integration & Filtering

Matlab Simulink Environment

Via MuT interface•Mass Matrix•Force Vector•Constraints Jacobian•Constraints at Position level •etc.

MuT interface

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Simulation with no corrections and stabilization

Simulation with two configuration corrections

Simulation with velocity filtering algorithm

Simulation with no corrections and stabilization

Constraint violations at the velocity level

Constraint violations at the configuration level

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Comparison with SimMechanics Model

Simulation with our frameworkSimulation with SimMechanics

Simulation with SimMechanics Simulation with our framework

Trajectory along the Y direction

Trajectory along the X direction

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Simulation modelQuanser 2-DoF Pantograph

F

K D

Emulation of a Bouncing Ball

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Physical DeviceHuman User Virtual Model

2DoF Pantograph

Controller

Virtual ModelDerivative

X

Fvirtual

TorquesT

X,X,X. .. Human User

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• Nonholonomic parameterization important

• Operational space: extension of traditional concept

Robotic Vehicles for Unstructured Terrain

Rocky 7 – NASA JPL

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• Modelling and analysis based on the principle of relaxation of constraints

• Opens up new possibilities for describing mechanical systems

• Parameterizations are important – Physical components – Nonholonomic generalized velocities

Kövecses, J, ASME J. Applied Mechanics, Vol. 75, 2008, 061012 and 061013.

Conclusion

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• Ali Azimi• Josep Maria Font Llagunes • Kamran Ghaffari • Farnood Gholami• Bahareh Ghotbi• Martin Hirschkorn • Arash Mohtat• Ali Modarres Najafabadi • Javier Ros• Bilal Ruzzeh• Sara Shayan Amin• Majid Sheikholeslami

Acknowledgement

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• Financial support of sponsors is gratefully acknowledged:

– Natural Sciences and Engineering Research Council of Canada – Canada Foundation for Innovation – Fonds Québecois de Recherche sur la Nature et les Technologies – Canadian Space Agency– Quanser, Inc. – CMLabs Simulations, Inc.

Acknowledgement