(Computational Methods in Applied Sciences) Carlo L. Bottasso - Multibody Dynamics_ Computational Methods and Applications (Computational Methods in Applied Sciences)-Springer (2008)

The conventional approach to solve this problem, the static opti-mization, is computationally efficient but neglects the dynamics involved in muscleforce generation and requires the use of

Multibody Dynamics

Computational Methods in Applied Sciences

Computational Methods and Applications

123

Multibody Dynamics

Carlo L Bottasso

All Rights Reserved

c

2009 Springer Science + Business Media B.V.

No part of this work may be reproduced, stored in a retrieval system, or transmitted in any form or by any means, electronic, mechanical, photocopying, microfilming, recording or otherwise, without written permission from the Publisher, with the exception of any material supplied specifically for the purpose

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springer.com

ISBN 978-1-4020-8828-5 e-ISBN 978-1-4020-8829-2

Printed on acid-free paper

Library of Congress Control Number: 2008

Multibody Dynamics is an area of Computational Mechanics which blendstogether various disciplines such as structural dynamics, multi-physics me-chanics, computational mathematics, control theory and computer science,

in order to deliver methods and tools for the virtual prototyping of complexmechanical systems Multibody dynamics plays today a central role in themodeling, analysis, simulation and optimization of mechanical systems in avariety of fields and for a wide range of industrial applications

The ECCOMAS Thematic Conference on Multibody Dynamics was ated in Lisbon in 2003, and then continued in Madrid in 2005 with the goal

initi-of providing researchers in Multibody Dynamics with appropriate venues forexchanging ideas and results The third edition of the Conference was held

at the Politecnico di Milano, Milano, Italy, from June 25 to June 28, 2007.The Conference saw the participation of over 250 researchers from 32 differ-ent countries, presenting 209 technical papers, and proved to be an excellentforum for discussion and technical exchange on the most recent advances inthis rapidly growing field

This book is a collection of revised and expanded versions of papers sented at the Conference Goal of this collection of works is to offer an up-to-date view on some of the most recent cutting edge research developments

pre-in Multibody Dynamics Contributions have been selected from all sessions

of the Conference, and cover the areas of biomechanics (Ackermann andSchiehlen, Millard et al.), contact dynamics (Tasora and Anitescu), control,mechatronics and robotics (Bottasso), flexible multibody dynamics (Cugnon

et al., Lunk and Simeon, Betsch and S¨anger), formulations and numericalmethods (Jay and Negrut), optimization (Collard et al.), real-time simulation(Binami et al.), software development, validation, education (Pennestr`ı andValentini), and vehicle systems (Ambr´osio et al.)

I hope you will find the reading of this collection enjoyable and stimulating,

as we anxiously wait for the 2009 edition of this excellent Conference Series

V

Preface V

Physiological Methods to Solve the Force-Sharing Problem

in Biomechanics

Marko Ackermann and Werner Schiehlen 1

Multi-Step Forward Dynamic Gait Simulation

Matthew Millard, John McPhee, and Eric Kubica 25

A Fast NCP Solver for Large Rigid-Body Problems

with Contacts, Friction, and Joints

Alessandro Tasora and Mihai Anitescu 45

Solution Procedures for Maneuvering Multibody Dynamics

Problems for Vehicle Models of Varying Complexity

Carlo L Bottasso 57

Synthesis and Optimization of Flexible Mechanisms

Frederic Cugnon, Alberto Cardona, Anna Selvi, Christian Paleczny,

and Martin Pucheta 81

The Reverse Method of Lines in Flexible Multibody Dynamics

Christoph Lunk and Bernd Simeon 95

A Nonlinear Finite Element Framework for Flexible

Multibody Dynamics: Rotationless Formulation

and Energy-Momentum Conserving Discretization

Peter Betsch and Nicolas S¨ anger 119

A Second Order Extension of the Generalized–α Method

for Constrained Systems in Mechanics

Laurent O Jay and Dan Negrut 143

VII

Kinematical Optimization of Closed-Loop Multibody Systems

Jean-Fran¸ cois Collard, Pierre Duysinx, and Paul Fisette 159

A Comparison of Three Different Linear Order Multibody

Dynamics Algorithms in Limited Parallel Computing

Environments

Adarsh Binani, James H Critchley, and Kurt S Anderson 181

Linear Dual Algebra Algorithms and their Application

to Kinematics

Ettore Pennestr`ı and Pier Paolo Valentini 207

A Memory Based Communication in the Co-simulation

of Multibody and Finite Element Codes

for Pantograph-Catenary Interaction Simulation

Jorge Ambr´ osio, Jo˜ ao Pombo, Frederico Rauter, and Manuel Pereira 231

the Force-Sharing Problem in Biomechanics

Marko Ackermann and Werner Schiehlen

Summary The determination of individual muscle forces has many applications

including the assessment of muscle coordination and internal loads on joints andbones, useful, for instance, for the design of endoprostheses Because muscle forcescannot be directly measured without invasive techniques, they are often estimatedfrom joint moments by means of optimization procedures that search for a uniquesolution among the infinite solutions for the muscle forces that generate the samejoint moments The conventional approach to solve this problem, the static opti-mization, is computationally efficient but neglects the dynamics involved in muscleforce generation and requires the use of an instantaneous cost function, leadingoften to unrealistic estimations of muscle forces An alternative is using dynamicoptimization associated with a motion tracking, which is, however, computation-ally very costly Other alternative approaches recently proposed in the literature arebriefly reviewed and two new approaches are proposed to overcome the limitations

of static optimization delivering more realistic estimations of muscle forces whilebeing computationally less expensive than dynamic optimization

1 Introduction

Inverse dynamics is used to compute the net joint moments required to erate a measured motion Although giving a clue about the intensity of theactuation required to accomplish the observed motion, net joint moments fail

gen-in delivergen-ing gen-information on the forces applied by the gen-individual muscles andother structures spanning the joints Because the skeletal system is redun-dantly actuated by muscles, i.e there are many more muscles than actuateddegrees of freedom, and many muscles are multi-articular, spanning more thatone joint, the direct translation of net moments into muscle forces is not pos-sible Therefore, conclusions about muscle activity from net joint momentsare not very reliable (Zajac et al [33]) Furthermore, the energy consumptionC.L Bottasso (ed.),Multibody Dynamics: Computational Methods and Applications, 1 c

Springer Science+Business Media B.V 2009

involved, represented by the metabolic cost during human motion cannot beaccurately assessed.

In order to solve the mathematically indeterminate problem and assessmuscle forces, optimization approaches are employed The classical static op-timization approach is characterized by the search for muscle forces that mini-mize a cost function and fulfill constraints, given basically by bounded muscleforces and by the equations of motion or joint moments computed by inversedynamics, respectively The cost functions are mathematical expressions as-sumed to model some physiological criteria optimized by the central nervoussystem during a particular activity

In spite of being computationally efficient, the static optimization proach assumes an instantaneous optimal distribution of muscle forces suffer-ing from two important limitations Firstly, it neglects the muscle contractionand activation dynamics, what might lead to unphysiological estimations ofmuscle forces Secondly, the cost functions must be an instantaneous measure

ap-of performance, what excludes the possibility ap-of using time-integral criteria asfor example total metabolic cost expended The latter limitation is speciallyimportant for the analysis of human walking, since metabolic cost is accepted

to play an important role during locomotion

The muscle activation and contraction dynamics can be taken into account

by using dynamic optimization associated with the tracking of the prescribedkinematics This approach is based on the search for optimal controls, inthis case the neural excitations, that drive a forward-dynamics model of themusculoskeletal system to track the prescribed motion Due to the severalnumerical integrations of the differential equations necessary, a prohibitivecomputational effort is required to achieve a solution This drawback preventsthis approach from being widely used and stimulated recent efforts to reducethe computational burden Some strategies based on dynamic optimizationare presented in more details in Section 3.2

An approach to solve the distribution problem in biomechanics is proposed

in Section 4, see also Ackermann and Schiehlen [3] It considers the musclecontraction and activation dynamics and permits the use of time-integral costfunctions as the total metabolic cost This approach is called extended inversedynamics (EID) because it requires, in addition to the inversion of the skele-tal system dynamics, the inversion of the muscle contraction and activationdynamics Since no numerical integration of the differential equations is re-quired, the extended inverse dynamics is computationally less costly than thedynamic optimization A second, simplified approach, called modified staticoptimization (MSO), to permit computation of muscle forces that fulfill theconstraints given by the activation and contraction dynamics is also proposedand presented in Section 5 The latter approach maintains computational ef-fort similar to the ones for static optimization, while considering the dynamicsinvolved in the muscle force generation process

2 Musculoskeletal System Dynamics and Energetics

The skeletal system is often modeled by a multibody system composed of rigidbodies whose dynamics is described by its equations of motion as

M (y) ¨ y + k(y, ˙y) = q(y, ˙y, f gr , h) = q r (y, ˙y, f gr ) + R(y) f m , (1)

where M is the symmetric, positive definite f ×f-mass matrix, k is the f vector of generalized Coriolis forces, q r is the f ×1-vector of generalized forces

×1-other than the ones caused by the muscles, f m is the m × 1-vector of muscle

forces, R is the f ×m-matrix that transforms the muscle forces into generalized

forces, and h = A f m , where A is a k × m-matrix that contains the muscles

moment arms The vector q r includes the vector of ground reaction forces f gr

for both feet since the contact forces between feet and ground are modeled asexternal applied forces For more details see Schiehlen [23]

When large-scale musculoskeletal models are considered Hill-type cle models, see e.g Zajac [32], are almost exclusively used Hill-type musclemodels are composed by a contractile element CE that generates force andrepresents the muscle fibers, and by passive elements in parallel and series

mus-to the CE modeling the tissue in parallel and in series mus-to the muscle fibers.Figure 1 illustrates the three-element Hill-type muscle model, showing the

CE, a series elastic element SE and a parallel elastic element PE

Figure 2 shows a scheme of the dynamics of the complete musculoskeletalsystem having the neural excitations as controls The activation dynamics

describing the process leading to a muscle activation state a from the neural excitation u is modeled by a first order differential equation, see e.g He et al.

[13], as

The first order muscle contraction dynamics arising from the presence of theseries elastic element (SE) and from the muscle force-length-velocity relation,refer e.g to Ackermann [1], reads as

MuscleContractionDynamics

SkeletalSystemDynamics

˙

l m (y), v m (y, y)

metabolic cost rate E˙

Fig 2 Scheme of the musculoskeletal system dynamics

where f mmeans the muscle force Note in Fig 2 that the muscle contractiondynamics is coupled to the skeletal system dynamics because it depends on

the muscle length l m and on the muscle shortening velocity v m The vectorsresulting from these values are defined by the geometry of the musculoskeletal

system and are modeled as functions of the generalized coordinates y and their derivatives ˙y as l m (y) and v m (y, ˙y).

In order to estimate the metabolic cost rate ˙E consumed by the muscles,

phenomenological muscle energy expenditure expressions recently proposed inthe literature, e.g by Umberger et al [31], can be used as

˙

E = ˙ E(u, a, v ce , l ce , f ce , p m ) , (4)

where the muscle parameters are summarized in the vector p m, and the

quan-tities of the contraction elements are found in the vectors v ce , l ce and f ce

3 Muscle Force-Sharing Problem in Biomechanics

The knowledge of loads in individual joint structures is important in fields likemedicine, sport science or prosthesis design For instance, the determination

of muscle and ligament forces is required for an analysis of the risk of damage

of the ACL (anterior cruciate ligament) in specific sport activities, for thedesign of hip and knee endoprostheses or for the planning and evaluation oforthopedic surgeries, Delp [10] However, muscle forces cannot be measureddirectly in-vivo without invasive techniques This stresses the importance oftechniques that permit the estimations of tissue loads from the skeletal systemmotion and the external applied forces, quantities that can be measured non-invasively

Since many muscles span each joint of the skeletal system, a redundantsystem arises because muscle forces may contribute differently to the samejoint moments, i.e muscle forces cannot be uniquely determined from jointmoments Under a mathematical point of view, there are more unknown mus-cle forces than equilibrium equations available, i.e there is an infinite number

of solutions for the muscle forces that fulfill the equilibrium equations and cangenerate the same observed motions of the skeletal system.

The dynamics of the skeletal system is described by the correspondingequations of motions of its multibody system model, (1) If the kinematics

of the movement y(t), ˙y(t), ¨ y(t) and the ground forces applied f gr (t) are

known, (1) can be solved for h, since the number of degrees of freedom of

the mechanical system f is greater or equal to the number k of unknown

joint moments in the vector h On the other hand, the number of muscles

m is always greater than the number of degrees of freedom of the skeletal system f , leading to the mentioned underdetermination It is reasonable to

assume that the central nervous system distributes the muscle forces f m insuch a way as to optimize some physiological criteria, for instance, energy,fatigue or pain This assumption is the basis for the optimization procedures

to determine muscle forces presented further on

A typical formulation of the optimization is as follows: find f m (t) that imizes a cost function J , subject to equality constraints given by the equa-

min-tions of motion in the form R f m = b(y, ˙y, ¨ y, f gr), which represent linear

constraints on the muscle forces, where b = M ¨ y + k − q r Additionally, equality constraints are given either by bounded muscle forces or by boundedneural excitations as explained further on

in-3.1 Static Optimization Approach

The static optimization is a computationally efficient approach to solve themuscle force-sharing problem presented in the previous section Static opti-mization is based on the assumption of an instantaneous cost function This

allows the solution of the force-sharing problem for each time instants t j

in-dependently The vector of muscle forces f m

j at the instant t j is searched that

minimizes a cost function J s (f j) The optimization is subject to physiologicallower and upper bounds for the muscle forces and constraints given by theequations of motion in (1)

Some variations of this approach have been proposed in the literaturemainly to better account for muscle physiology, refer to Tsirakos et al [30],but the basic strategy remains the same Several instantaneous cost functions

J s have been proposed to solve the muscle force-sharing problem, refer, e.g

to Tsirakos et al [30] and da Silva [24] for extensive reviews and applications.The cost function proposed by Crowninshield and Brand [8] is one of the mostfrequently employed due to its physiological background related to musclefatigue and reads as

P CSA i

3

where f m

ij is the force applied by muscle i at time instant t j , P CSA i is the

cross-sectional area of muscle i, and m is the number of muscles considered.

3.2 Dynamic Optimization Approach and Alternative Methods

The investigation of human motion coordination and muscle recruitment bysolving the optimal control problem using neural excitations as controls wasused in the past, e.g., in Hatze [11], Hatze and Buys [12] and in Davy andAudu [9] Pandy et al [20] proposed the use of an alternative computationalmethod for these problems consisting in the conversion of the optimal controlproblem into a parameter optimization problem, where the neural excita-tion histories are parameterized using a set of nodal points This approach isclaimed to circumvent the numerical difficulties that arise to solve the two-point boundary-value problem derived from the necessary conditions of opti-mal control theory This approach has been successfully implemented to studynormal walking using metabolic energy cost per unit of distance traveled ascost function, e.g in Anderson and Pandy [5], Bhargava et al [6], Umberger

et al [31] These studies could mimic human normal walking patterns such askinematics, optimal walking velocity and metabolic energy cost reasonably,using forward-dynamics models of the musculoskeletal system Other applica-tion fields include simulation of vertical jumping (Sp¨agele [25], Anderson andPandy [4], Nagano and Gerritsen [17]), and cycling (Neptune and van denBogert [18]) This approach is denoted dynamic optimization in opposition tostatic optimization

The advantage of using dynamic optimization over static optimizationresides in the consideration of the muscle contraction and activation dynamics,and in the possibility of using a time-integral cost function such as totalmetabolic cost However, performing dynamic optimization with large-scalemusculoskeletal models is extremely costly in terms of computational effort,requiring as much as weeks for a 2-D musculoskeletal model with a reducednumber of degrees of freedom, Menegaldo et al [15], or months for a 3-Dcomplex musculoskeletal model for walking using parallel super-computingfacilities, Anderson and Pandy [5] The high CPU times result mainly fromthe several numerical integrations of the differential equations required.The typical applications of dynamic optimization look simultaneously foroptimal controls, muscle forces and optimal motion patterns that minimize acost function For the cases in which the motion and the external applied forcesare completely or partially prescribed or measured, the mechanical model mustadditionally track the known kinematics and apply the prescribed forces onthe environment This is achieved by augmenting the cost function with aterm that quantifies the deviation from the prescribed kinematics and appliedforces, see e.g Neptune and van den Bogert [18], Neptune and Hull [19],Strobach et al [27], Davy and Audu [9] The introduction of the tracking term

to the cost function transforms the problem in a multi-criteria optimization,which compromises the interpretation of the results, since the solution of theproblem depend on the weighting factors chosen Because the objective criteriaare usually competing, the use of different weighting factors leads to differentsolutions

In order to reduce the prohibitive computational effort required to solvethe muscle force distribution problem by dynamic optimization new methodsare being proposed For instance, Menegaldo et al [16] propose recently adynamic optimization approach based on the tracking of the moments at thejoints, which are computed for example from measured kinematics by con-ventional inverse dynamics By avoiding the necessity of forward integration

of the skeletal system dynamics, it considerably reduces the computationaltime required while considering the muscle activation and contraction dy-namics and allowing for the use of time-integral cost functions Although thismethod seems very promising with respect to computational time, it involvesthe solution of a multi-criteria optimization

Thelen et al [29] and Thelen and Anderson [28] propose an algorithmcalled Computed Muscle Control (CMC) to solve the problem of muscle forcedistribution for known movement kinematics, based on a control algorithmthat tracks the kinematics of a measured movement and uses measured exter-nal forces as input This method is much faster than dynamic optimizationapproaches, because it requires only one forward integration of the state equa-tions It efficiently enforces the musculoskeletal system dynamics, but, in order

to solve the muscle redundancy, it still requires the use of an instantaneouscost function Therefore, in opposition to dynamic optimization, the use of atime-integral cost function such as total metabolic cost is not possible.Two new methods are proposed here, which present some advantages overthe other methods described to solve the muscle redundancy problem Bothmethods depend on the inversion of the contraction and activation dynamics.The first one named extended inverse dynamics (EID) and described in detail

in Section 4 permits the computation of muscle forces by using time-integralcost functions as total metabolic cost and is computationally more efficientthan dynamic optimization The second method is described in Section 5 and

is called modified static optimization (MSO) It is based on the tion of an instantaneous cost function and characterized by constraints onthe muscle forces at the current time step derived from the muscle forces atthe previous time step that arise due to the activation and the contractiondynamics A more detailed comparison of the approaches proposed here withothers from the literature is presented in Ackermann [1]

minimiza-4 Extended Inverse Dynamics

In this section a novel optimization procedure is proposed that considers thecontraction and activation dynamics, and permits the use of time-integral costfunctions as the total metabolic cost, while reducing the computation timescompared to dynamic optimization The proposed approach consists in formu-lating the problem as a large-scale optimization problem, whose optimizationvariables are the muscle forces at all time steps considered The optimiza-tion is subject to equality constraints given by the equations of motion at

Forces

Joint Moments

Muscle Moment Arms

Measured Kinematics

& GRF

Metabolic Cost

of Transport

Cost Function Constraints

Figure 3 shows the general optimization scheme of the EID The mization variables are parameterized muscle force histories The activationand neural excitations for each one of the muscles considered are computed

opti-by inverting the contraction and activation dynamics The optimization issubject to two sets of constraints: (1) the constraints represented by neuralexcitations bounded by 0 and 1; (2) the constraints given by the equations ofmotion at all time steps, ensuring the compatibility between the muscle forcesand the measured skeletal system motion and ground reaction forces Thus,optimal (parameterized) muscle force histories are searched for that minimizetotal metabolic cost, fulfill the constraints given by the equations of motion forthe given kinematics and measured ground reaction forces (GRF), and ensurephysiological neural excitations bounded by 0 and 1 In the next sections, theelements of the optimization scheme proposed are explained in details

4.1 Parameterization of Muscle Forces

The parameterization of the force f m

i (t) applied by the ith muscle is performed using a set of n nodes uniformly distributed along the duration of the motion of

interest resulting in a vector of time steps t = [t1· · · t j · · · t n ], where t j −t j−1=

∆t The optimization variables are the muscle forces f m

ij for all muscles i,

i = 1 m, at all nodes j, j = 1 n, summarized in a mn × 1-vector of

global muscle forces

F m = [f m1T f m2 T · · · f m

j

T · · · f m

n T

where f m

j = [f 1j · · · f ij · · · f mj]T is the vector of muscle forces at the time

step t j

4.2 Inversion of the Contraction and Activation Dynamics

The inversion of the contraction and activation dynamics are important stepsfor the implementation of the approach proposed The first step to invert thecontraction dynamics (3) and find the time history of the muscle activation

a(t), is to compute the total muscle length l m(including the tendon) and the

total muscle shortening velocity v m from the generalized coordinates in y and their time derivatives in ˙y as

The second step consists in numerically differentiating the time history

of the muscle force f m (t) obtaining ˙ f m (t) Since the value of f m (t) is only available at discrete time steps t j, ˙f m (t) is estimated only at these time in-

stants by using the centered finite-difference formula, Chapra and Canale [7],for internal nodes or by using forward and backward finite-difference formulas

for the initial (j = 1) and final nodes (j = n) In the case of perfectly periodic

motions, centered finite-divided-difference formulas for the extreme nodes arepreferred, since they deliver more accurate estimations of the derivatives, seee.g Chapra and Canale [7]

The muscle serial elastic element (SE) length l se and shortening velocity

v se can be then computed, respectively, by

l se = l se (f m) and v se = v se( ˙f m ) , (8)

where l se (f m) models the force-length relation of the SE The contractile

ele-ment (CE) shortening velocity, length and force, v ce , l ce and f ce, are computed

according to Fig 1 and with α p ≈ constant, respectively, as

Finally, the muscle activation a is obtained by solving the muscle length-velocity relation for a as

For some models of the muscle force-length-velocity relation a can be written explicitly as function of v ce , l ce and f ce For other models, this is not possi-

ble and a has to be computed numerically through (10) using a zero-finder

algorithm, which considerably increases the computation time required The

procedure explained is repeated for all n nodes and m muscles considered.

The neural excitations are assessed by inverting the activation dynamics

described by (2) The first step is to find the time derivative ˙a(t) of the muscle excitation a(t) This is achieved by numerical differentiation of a(t)

using finite-divided-difference formulas as the ones used to compute the time

derivatives of muscle forces The values for the muscle activation a(t) and its first time derivative ˙a(t) are then inserted into (2) resulting in an algebraic equation, which is either linear or quadratic in u If a linear equation in u arises, e.g for the activation dynamics model of Zajac [32], solving (2) for u

is trivial If a quadratic equation in u arises, e.g for the activation dynamics

model of He et al [13], the two roots of the polynomial are computed and one

of them is chosen as solution The choice of the appropriate root is in mostcases straightforward Checking which root is bounded by 0 and 1 and is most

proximal to the muscle activation a proved to be efficient rules.

4.3 Constraints

The optimization is subject to two kinds of constraints as depicted in Fig 3,the constraints that ensure the fulfillment of the equations of motion and thelower and upper bounds for the required neural excitations The fulfillment ofthe constraints is checked at the nodes considered so that small infringements

in the region between nodes might occur The magnitude of these

infringe-ments is dictated by the number of nodes used A properly chosen ∆t leads

to negligible inter-node infringements of the constraints

Fulfillment of Equations of Motion

The vectors of muscle forces f m

j contained in F m, (6), have to satisfy the

equations of motion (1) at all time instants t j considered as

M (y j) ¨y j + k(y j , ˙y j ) = q r (y j , ˙y j , f gr j ) + R(y j ) f m j , j = 1 n (11)

Since the kinematics of the movement in y j , ˙y j, ¨y j and the ground reaction

forces in f gr j are measured or specified, the only unknowns are the muscle

forces f m

j Rearranging (11) yields a set of linear equations in the elements

of f m,

Rj f m j = b j , (12)

where b j = M (y j) ¨y j + k(y j , ˙y j)− q r (y j , ˙y j , f gr j ) Writing all the constraint

equations given by the equations of motion at all time steps j in a single

matrix equation yields

Bounds for Neural Excitations

The second group of constraints is represented by neural excitations bounded

by 0 and 1 as

0≤ u ij ≤ 1 , i = 1 m , j = 1 n (15)The fulfillment of these constraints for the whole period considered guaranteesthe fulfillment of the lower and upper bounds for the activations and muscleforces However, although not strictly necessary, constraints on muscle acti-vations and muscle forces can be additionally formulated This measure, inspecial the explicit bounds on the optimization variables, reduces the searchspace of the optimization variables, which can, depending on the optimiza-tion algorithm employed, reduce the number of iterations required to achieveconvergence to an optimum

4.4 Cost Function

One of the advantages of using the extended inverse dynamics approach overstatic optimization is the use of time-integral cost functions as the total energyexpenditure, which is accepted to be the primary performance criteria duringwalking, Ralston [21] Hence, for the simulation results discussed further on, alldealing with walking, the total metabolic cost is adopted as cost function Thetotal metabolic cost can be estimated by recently proposed phenomenologicalmuscle energy expenditure models, e.g Umberger et al [31], as a function of

the neural excitation u, muscle activation a and muscle CE force f ce, length

l ce and shortening velocities v ce, and from a set of muscle specific parameters

p m for all muscles m considered as

forces F m

opt that minimizes the total metabolic cost (16), subject to linearequality constraints given by the equations of motion at all time steps (13), tothe inequality constraints for the neural excitations (15), and, if advantageous,

to additional lower and upper bounds for the activations and muscle forces,respectively There are many different numerical methods that can be used

to solve a nonlinear optimization problem with nonlinear constraints Here,

the Sequential Quadratic Programming (SQP) implemented in the fmincon

function available in the Optimization Toolbox of Matlab
R is used.

5 Modified Static Optimization

The extended inverse dynamics approach proposed in the previous sectionaccounts for the activation and contraction dynamics and uses a time-integralcost function The price for these desirable features is a computational effortsome orders of magnitude higher than the one required in static optimization,although being lower than the one required for dynamic optimization, seeAckermann [1] The high computational effort is a limiting factor for the use

of the more elaborate approaches dynamic optimization and extended inversedynamics

Specially in applications in which a rather gross estimation of muscle forces

is sufficient and instantaneous cost functions are assumed to be reasonablemodels of the underlying muscle force distribution laws adopted by the centralnervous system (CNS) static optimization will still be the first choice How-ever, neglecting completely the activation and contraction dynamics, assum-ing the muscle as a perfect force generator, capable of delivering the requiredamount of force instantaneously, can lead to unphysiological solutions In thissection an alternative approach is proposed, which modifies the static opti-mization approach in such a way as to consider the muscle activation andcontraction dynamics, while requiring a reduced computational effort.The approach, named modified static optimization, formulates the opti-mization problem in the same way as in the static optimizations for eachtime step considered with the difference of defining additional nonlinear con-straints that ensure neural excitations bounded by 0 and 1 This measureguarantees the compatibility of the current muscle forces with the activation

max (f m j−1, a j−1)

∆f − max (f m j−1, a j−1)

Fig 4 Schematic

repre-sentation of the implicitadditional lower and up-per bounds on the muscle

force f j m as a function ofthe states at the previous

time instant t j −1 in themodified static optimizationapproach

and contraction dynamics These additional constraints can be interpreted

as additional upper and lower bounds on the current muscle forces obtained

by the maximal allowed variations of muscle forces that are still compatiblewith the activation and contraction dynamics The upper and lower bounds,

f m

j,max and f m

j,min, respectively, are implicitly formulated depending on the

states, muscle forces f m

j −1 and activations a j−1, at the previous time instant

t j−1 as depicted in Fig 4

Therefore, the formulation of the optimization problem for the time step j

is identical to the one for the static optimization, Section 3.1, with additionalupper and lower bounds for the neural excitations of the muscles considered as

0≤ u ij ≤ 1 , i = 1 m , j = 2 n (17)The computation of the neural excitations requires the inversion of the con-traction and activation dynamics as done for the extended inverse dynamicsapproach, Section 4.2, with the difference of using only the information fromthe previous time steps The procedure is briefly explained in the following

focusing on the differences from the extended inverse dynamics The index i

referring to a specific muscle is omitted

The first step consists in computing the first derivatives of the muscleforces ˙f m

j from the values of the muscle forces at the previous and current

time steps f m

j −1 and f j m, respectively, using a backward finite-divided formula

The total muscle length l m

j and shortening velocity v m

j are computed from

the generalized coordinates y j and their derivatives With this informationthe contraction dynamics (3) is inverted as explained in Section 4.2 in such a

way a j is computed by

a j = a j( ˙f j m , v j m , l j m , f j m ) , j = 2 n (18)The next step consists in the inversion of the activation dynamics which

requires the estimation of the first time derivatives of the activations ˙a by

using also a backward finite-divided formula It follows the computation of

the neural excitations, as explained in Section 4.2, by inserting a j and ˙a j into

(2) and solving it for u j as

˙a j − ˙a j (a j , u j ) = 0 , j = 2 n (19)

The explained procedure to estimate the neural excitation at t j shows that

u j is a function of the muscle force f j−1 m and activation a j−1 at the previous

time step t j−1 , of the time step size ∆t, of the muscle total length l j m and

0≤ u j (f j m , f j−1 m , a j −1 , y j , ˙y j , ∆t) ≤ 1 , j = 2 n , (20)

for all muscles considered Therefore, muscle forces at t jare searched that imize the instantaneous cost functions of Section 3.1, and fulfill the nonlinearconstraints given by (20) and the linear constraints given by the equations ofmotion (11) The first time step in the modified static optimization receives a

min-special treatment The muscle forces f m

1at the first time step t1are computed

by conventional static optimization without the additional constraints on the

neural excitations The activations a i1 , i = 1 m, at t1are approximately timated by inversion of the contraction dynamics, as explained in Section 4.2,with the difference that the required estimations of the derivatives of the mus-cle forces ˙f1are obtained by a forward finite-divided-difference formula using

es-the muscle forces computed at es-the first and second time steps, f m1 and f m2,respectively, computed by conventional static optimization

The constraints on the rate of change of the muscle forces can be so strictive, that infeasibilities may occur at a time instant for which no solutionfor the muscle forces can be found that fulfills all the constraints This occurs

re-to a great extent due re-to the fixed values of the computed muscle forces atthe previous time steps The incidence of such infeasibilities for the extendedinverse dynamics approach is much lower, because there the complete timehistories of the muscle forces can be accommodated in such a way as to guar-antee fulfillment of the constraints at all time steps One drawback of themodified static optimization is the necessity of using backward finite-divided-difference formulas to estimate derivatives numerically, which causes greatertruncation errors in comparison to centered finite-divided-difference formulas.Furthermore, the activations at the initial time step have to be determinedapproximately, because no values for the muscle forces at the previous timestep are available

6 Application to Normal and Disturbed Gaits

In this section both approaches to solve the muscle force-sharing problem

in biomechanics proposed are applied to the normal and to mechanically

disturbed gaits measured in a gait analysis laboratory, Ackermann andGros [2] The extended inverse dynamics and the modified static optimiza-tion are compared to the static optimization.

6.1 Model of the Musculoskeletal System

A 2-D mechanical model of the skeletal system of the right lower limb isadopted here, composed by three rigid bodies, the thigh, the shank and thefoot The motion is performed in the sagittal plane and is described by threegeneralized coordinates and two rheonomic constraints, refer to Fig 5b The

generalized coordinates are the angle α describing the rotation of the thigh, the angle β describing the knee flexion, and the angle γ for the ankle plantar

flexion The two rheonomic constraints are the horizontal and vertical

posi-tions of the hip joint, x hip and z hip, respectively The pelvis and trunk areassumed to remain in the vertical position throughout the gait cycle, what

is reasonable for normal walking The masses, center of mass locations, andthe mass moment of inertia of the three segments in the sagittal plane arecomputed using the tables in de Leva [14] as functions of the subject’s bodymass, stature, thigh length and shank length The motion and the groundreaction forces were measured in a gait analysis laboratory

The eight muscle groups considered in this analysis are shown in Fig 5a.The Hill-type muscle model is composed by a contractile element CE and a

1

4 5

8 6

Fig 5 Musculoskeletal model of the lower limb: (a) muscle units; (b) mechanical

model with generalized coordinates

series elastic element SE, while the force of the parallel elastic element PE isset to zero, Fig 1 In this model all the structures in parallel to the CE andthe SE are represented by total passive moments at the joints, which includethe moments generated by all other passive structures crossing the joints, likeligaments, too The formula for the passive moments at the hip, knee and

ankle are functions of α, β and γ as proposed by Riener and Edrich [22] A

linear damping is added to the knee and hip joints and their values are theapproximate average values obtained by pendulum experiments performed byStein et al [26] The models adopted for the muscle activation and muscleforce-length-velocity relation are based to a great extent on the models inNagano and Gerritsen [17] with some few modifications For details on themodels used and for the corresponding parameters refer to Ackermann [1]

6.2 Application to the Normal Walking

The results obtained by using extended inverse dynamics (EID), modifiedstatic optimization (MSO) and the static optimization (SO) for the normalgait of one subject are presented in Table 1 For the SO and for the MSOthe cost function proposed by Crowninshield and Brand [8] is used, refer toSection 3.1 The cost function for the EID adopted is the total metabolic cost(16) The metabolic cost for the MSO is computed by inverting the contractionand activation dynamics after the computation of optimal muscle forces for

each time instant j The initial guesses for the muscle forces in the static

optimization are zero The initial guesses used in the many low-dimensionoptimizations involved in the MSO and in the unique large-scale optimization

in the EID are the optimal muscle forces obtained as solutions of the SO.The analysis of the results in Table 1 shows that the computation timerequired for the EID approach is four orders of magnitude higher than thecomputation times required for the SO and MSO This difference can be ex-plained by the much higher dimension of the optimization problem in the EIDwith respect to the optimizations in the SO and MSO While in the EID aunique large-scale optimization problem with several optimization variables

is solved, the SO and the MSO require the solution of many low-dimensionoptimization problems This is the cost that has to paid for the use of a

Table 1 Solutions for the force-sharing problem for the measured normal walking

using: Static Opt – static optimization; Mod Static Opt – modified static mization; Ext Inverse Dyn – extended inverse dynamics

opti-Initial Computation Opt Metabolicguess time (s) variables cost (J)Static Opt (SO) f m j,0= 0 5.3 8 (67×) –

Mod Static Opt (MSO) Solution SO 7.6 8 (67×) 254.8

Ext Inverse Dyn (EID) Solution SO 4.5 × 104 400 (8× 50) 201.0

time-integral cost function as metabolic cost Although the computational

ef-fort for the EID (33.6 h) can be considered high with respect to the SO and

MSO, it is probably much lower than the required for a dynamic optimizationwith tracking of the kinematics as discussed in Section 3.2

The results show the computational efficiency of the MSO The MSO quired a computational effort comparable to the one required by the SO, whileconsidering the activation and the contraction dynamics This shows the po-tential of the MSO for applications that need fast and realistic estimations

re-of muscle forces, muscle activation and neural excitations when instantaneouscost function are adopted The MSO can, however, lead to infeasibilities due

to the restrictive constraints imposed by the limitation on the amount of lowable changes in the muscle forces Nevertheless, the possible infeasibilitiescannot be attributed to a fail of the method, but rather to inconsistenciesarising from errors in the measurements, oversimplification of the models orassumption of an incorrect cost function

al-The metabolic cost for the SO cannot be estimated using (16), because themuscle forces computed lead to muscle activations and neural excitations thatinfringe their lower and upper bounds, see Fig 6, being out of the range towhich these expressions are valid This occurs because the SO does not con-sider the activation and contraction dynamics On the contrary, the solutionfor the muscle forces delivered by the MSO can be used to compute muscleactivations and neural excitations that fulfill the constraints by inverting thecontraction and the activation dynamics These values can then be insertedinto (16), which delivers estimations of metabolic cost as shown in Table 1

Fig 6 Results for the static optimization (SO); TOr – toe off of the right foot;

HSr – heel strike of the right foot

TOr HSr TOr TOr HSr TOr

Fig 7 Results for modified static optimization (MSO)

The metabolic cost estimated from the muscle forces computed with the MSO

is 27% greater than the metabolic cost estimated with the EID A difference

is expected, since the EID minimizes the metabolic cost, while in the MSOthe instantaneous cost function of Crowninshield and Brand [8] is minimized,which is related to muscle fatigue The relatively big differences show thatthe cost function of Crowninshield and Brand [8] is not well related to themetabolic energy consumption The absolute values of the total metaboliccost of transport obtained with the EID, assuming the total metabolic costexpended during walking is approximately two times the expended for one ofthe legs, is 311 J/m, which agrees well with values found in the literature.The optimal muscle forces for the normal gait computed using SO, MSOand EID and the corresponding muscle activations and neural excitations arepresented in Figs 6, 7 and 8, respectively Figure 6 shows clearly the infringe-ment of the lower and upper bounds for the neural excitations indicating non-physiological muscle force histories Figure 7 shows that the MSO smoothesthe muscle force curves avoiding unphysiological fast variations, maintainingthe neural excitations bounded by 0 and 1 The smoothing effect is speciallyvisible for the muscle groups rectus femoris (RF) and vasti (Vas) Also theresults of the EID in Fig 8 fulfill the bounds on the activations and neuralexcitations

The constraints on the neural excitations cause cocontraction, even insingle joint antagonist muscles, although cocontraction of antagonists is clearlynoneconomical This occurs due to the fact that muscles are not ideal actuatorsand cannot be switched on and off instantaneously Indeed, the single jointantagonistic muscle pairs iliopsoas (Ilio) and glutei (Glu), and tibialis anterior

01

TOr HSr TOr TOr HSr TOr

Fig 8 Results for the extended inverse dynamics (EID)

(TA) and soleus (Sol) present practically no cocontraction, i.e almost nosimultaneous activation, when SO is used, see Fig 6 On the contrary, ifMSO or EID are used cocontraction is observed for these muscles, refer toFigs 7 and 8 In order to reduce energy consumption, the results of EIDhave less cocontraction than the results of MSO, but a considerable amount

of cocontraction in the results of EID can still be observed for the mentionedmuscles, specially at the regions of activation and de-activation

Application to the Walking with an Ankle Weight

In this section the effect of adding a 1.7 kg ankle weight during the swingphase of the gait is investigated using the extended inverse dynamics with thetotal metabolic cost as cost function The bars in Fig 9 show the metaboliccost estimated for three different scenarios The first bar from the left showsthe metabolic cost, 48.3 J, computed for the measured kinematics of the lowerlimb of the subject during the swing phase without weight The second barshows the metabolic cost, 49.9 J, for the measured kinematics during theswing phase with a 1.7 kg ankle weight The third bar from the left showsthe metabolic cost using the kinematics measured for the normal walking andadding a 1.7 kg ankle weight to the skeletal system model, 59.6 J

As expected, the swing phase with an ankle weight requires a highermetabolic cost than the normal, although the observed difference of 3% isslight An interesting result is shown by the third bar from the left It shows

an increase of about 23% in the metabolic cost if the kinematics is tained the same as for the normal walking when the ankle weight is added

Fig 9 The bars in the diagram in the left show the metabolic cost for the swing

phase obtained with the EID, from left to right, during normal walking, during ing with a 1.7 kg ankle weight, and during the walking with the normal kinematicsand an added 1.7 kg ankle weight to the model In the right a schematic represen-tation of the musculoskeletal model of the lower limb with the ankle weight added

walk-is depicted In the middle a picture of the lower limb of the subject with an ankleweight attached to his ankle is shown

This means that, after the addition of the ankle weight, the subject naturallyadapts the motion of the lower limb in such a way as to reduce the metaboliccost required, leading to a slight increase of energy expenditure with respect

to the undisturbed gait If the adaptations in the kinematics were not formed, the increase in the metabolic cost would be much higher (third bar inFig 9) This observation evidence the importance of the motion adaptations

per-in the reduction of energy expenditure durper-ing gait

7 Conclusion

The conventional method to solve the muscle force-sharing problem is calledstatic optimization and, although being computationally efficient, suffers fromtwo important limitations: (1) it neglects the dynamics involved in the muscleforce generation process, what can lead to unphysiological muscle force histo-ries, and (2) it requires the use of instantaneous cost functions, excluding thepossibility of using time-integral cost functions as the total metabolic cost,which was shown to play a key role during walking

In order to overcome these limitations dynamic optimization associatedwith a tracking of the measured kinematics is an alternative This approachrequires, however, extremely high computational costs due to the several nu-merical integrations of the high-dimensional system equations necessary Twoalternative approaches are proposed to overcome the limitations of static

optimization delivering more realistic muscle force estimations while beingcomputationally less expensive than dynamic optimization One approachnamed extended inverse dynamics delivers physiological estimations of mus-cle forces by considering the muscle activation and contraction dynamics and

by permitting the use of time-integral cost functions as total metabolic cost.Although the improvements provided by this approach makes it computation-ally much more expensive than static optimization, it is less expensive thandynamic optimization, because it does not require any numerical integration

of the state equations The second proposed approach, named modified staticoptimization, offers a viable alternative to static optimization by consideringthe muscle activation and contraction dynamics while requiring a low compu-tational effort

The two proposed approaches are applied to estimate muscle force historiesfor the normal gait of a subject measured in a gait analysis laboratory using

a musculoskeletal model of the lower limbs The approaches are compared tothe static optimization with respect to computational effort and fulfillment

of constraints that guarantee the consideration of the dynamics involved inthe process of muscle force generation The extended inverse dynamics is used

to investigate the walking with an ankle weight It is shown that the ject naturally adapts the kinematics of the swinging leg to reduce energyconsumption as a response to the addition of an ankle weight The choice

sub-of a proper approach depends on the accuracy required, the computationalfacilities available, and the particularities of the problem

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of submaximal steady-state cycling using a forward dynamic model J BiomechEng 120:334–341

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Esti-27 Strobach D, Kecskemethy A, Steinwender G, Zwick B (2005) A simplified proach for rough identification of muscle activation profiles via optimization andsmooth profile patches In: Proceedings of MULTIBODY DYNAMICS 2005,ECCOMAS Thematic Conference, Madrid, Spain

ap-28 Thelen DG, Anderson FC (2006) Using computed muscle control to generateforward dynamic simulations of human walking from experimental data J Bio-mech 39:1107–1115

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31 Umberger BR, Gerritsen KGM, Martin PE (2003) A model of human muscleenergy expenditure Comput Meth Biomech Biomed Eng 6:99–111

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to biomechanics and motor control Crit Rev Biomed Eng 19:359–411

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coordina-Matthew Millard, John McPhee, and Eric Kubica

Systems Design Engineering, University of Waterloo, 200 University Ave West,Waterloo ON, Canada

E-mail: mjhmilla@engmail.uwaterloo.ca

Summary A predictive forward-dynamic simulation of human gait would be

ex-tremely useful to many different researchers, and professionals Metabolic efficiency

is one of the defining characteristics of human gait Forward-dynamic simulations

of human gait can be used to calculate the muscle load profiles for a given ing pattern, which in turn can be used to estimate metabolic energy consumption.One approach to predict human gait is to search for, and converge on metabolicallyefficient gaits This approach demands a high-fidelity model; errors in the kineticresponse of the model will affect the predicted muscle loads and thus the calculatedmetabolic cost If the kinetic response of the model is not realistic, the simulatedgait will not be reflective of how a human would walk The foot forms an importantkinetic and kinematic boundary condition between the model and the ground: jointtorque profiles, muscle loads, and thus metabolic cost will be adversely affected by

walk-a poorly performing foot contwalk-act model A recent walk-approwalk-ach to predict humwalk-an gwalk-ait

is reviewed, and new foot contact modelling results are presented

1 Introduction

Human and animal gait has been studied by using experiments to tease outthe neural, muscular and mechanical mechanisms that are employed to walk.Inverse dynamic simulation is the most common simulation technique used

to study human gait Inverse dynamics works backwards from an observedmotion in an effort to find the forces that caused the motion – inverse dynamics

is not predictive In contrast, forward dynamics can be used to determinehow a mechanism will move when it is subjected to forces – making forwarddynamics predictive

Forward dynamic human gait simulations usually only simulate a singlestep [4,11] in an effort to avoid modelling foot contact and balance control sys-tems The few multi-step forward-dynamic simulations in the literature haveused a relatively fixed gait [24, 27] In contrast, Peasgood et al.’s [23] forwarddynamic simulation is predictive: the simulated gait is altered in an effort toC.L Bottasso (ed.),Multibody Dynamics: Computational Methods and Applications, 25 c

Springer Science+Business Media B.V 2009

find metabolically efficient or ‘human-like’ gaits, allowing it to estimate how aperson would walk in a new situation – e.g with a new lower-limb prosthetic,

or more flexible muscles

A computer simulation that is able to reliably predict how a person wouldwalk in a new situation would be extremely useful to many health care pro-fessionals and researchers studying human gait Peasgood et al.’s system finds

‘human-like’ or metabolically minimal gaits by searching for joint trajectoriesfor the hip, knee and ankle that minimize metabolic cost per distance trav-eled The model is not supported or balanced by any artificial means, and so,poorly chosen trajectories can overwhelm the balance controller, causing themodel to fall This study was undertaken to evaluate and extend Peasgood

et al.’s work, and to identify the shortcomings of current multi-step forwarddynamic gait simulations

2 Methods

Peasgood et al.’s system represents the first attempt at developing a tive, multi-step gait simulation that searches for metabolically efficient gaits.Nearly 1,000, ten-step simulations were required to find a metabolically effi-cient, ‘human-like’ gait Originally the 1,000 gait simulations took 10 days toperform on a single computer using the popular mechanical modeling pack-age MSC.Adams [21] DynaFlexPro [9], another modeling package, developedsince Peasgood et al.’s work, offers substantial performance advantages overAdams: the updated version of Peasgood et al.’s predictive system now takesonly 8 hours to run Peasgood et al.’s work was taken, carefully examined,analyzed, improved and implemented in DynaFlexPro

predic-2.1 Dynamic Model

Peasgood et al developed a predictive gait simulation using a 2D, seven ment, nine degree of freedom (dof), anthropomorphic model shown in Fig 1with a continuous foot contact model This is a fairly standard model topologyfor gait studies The upper body is simplified into a single body representingthe head, arms and trunk (HAT); the thigh and shank are each one segment,

seg-as is the foot [1, 3, 13] An additional simplification hseg-as been made in thismodel by fusing the HAT to the pelvis There was an unintended error inPeasgood et al.’s original model: there was an extra body attached to the footthat had a moment of inertia of 1.5 kg m2, which is comparable to the HATsegment

A convergence study was performed on both the DynaFlexPro and the rected Adams gait models by dropping both unactuated models onto the floorfrom the same initial conditions The convergence of each model was checkedindividually The results from the DynaFlexPro model converged for everysimulation, whereas the Adams model failed to converge with an integrator

cor-Fig 1 Peasgood et al.’s seven segment, nine degree of freedom, planar gait model

with a 2-point continuous foot contact model

Table 1 Performance comparison between the Adams and DynaFlexPro 2D seven

segment gait models for a 10 second simulation The Adams simulation with anintegrator error tolerance of 10−5 failed to converge The relative error increasesfrom the hip position to the foot angle: the large mass of the HAT attenuatesposition error of the hip, while foot position is more sensitive to errors due to itslight mass The stiffness of the heel contact makes the simulated contact forces verysensitive to errors

Adams DynaFlexPro Maximum relative error (%)Integrator GSTIFF (I3) ode15s (NDF) Left hip Right ankle Right heelerror tol Simulation time disp (x) angle contact force

as shown in Table 1

2.2 Foot Contact

Foot contact forces were calculated using a two-point foot contact model,with a point contact located at the heel and metatarsal Normal forces werecalculated using the Adams implementation [22] of the continuous Hunt-Crossley [18] point contact model:

The Hunt-Crossley contact model calculates normal force (f n) as a

func-tion of penetrafunc-tion depth (y), penetrafunc-tion rate ( ˙ y), material stiffness (k, p), and material damping (c(y)) The implementation of the model ramps up damping (c(y)) as a function of penetration depth, to prevent an instanta-

neous normal force that would be created using a simple damping term such

as (c max y) A dry Coulomb model was used to calculate the force of friction˙between the points and the plane:

This friction model has stiction (µ s ) and dynamic friction (µ d) values thatare interpolated using a cubic step function [22] between the stiction velocity

(v s ) and the sliding velocity (v d ) using the tangential contact velocity ( ˙x) as

an input The particular contact and friction parameters used for the gaitsimulation were chosen by the pattern search routine (described later) tomatch the ground reaction forces created during healthy gait [26]

2.3 Joint Trajectory Control

Pre-computed joint trajectories are used to define the gait of the model atthe position level Each joint is actuated using a proportional-derivative (PD)controller that modifies and regulates the predefined joint trajectories Theinitial joint trajectories were taken from an existing experimental data set of

a healthy gait of an average-sized male [26] and interpolated using a five-termFourier series:



+ B k cos



2πkt period



Some adjustments were made to the trajectories in order to apply them

to a sagittal plane gait model: the swing phase of the ankle trajectory had

to be altered to prevent the foot from dragging on the ground This makessense because the 2D sagittal plane model cannot use hip roll and body sway

in the frontal plane to adjust the floor clearance of the swing limb, unlike thesubject used in the experiment data set The interpolated joint trajectorieswere applied to the PD joint controllers to achieve an initial simulated gait.The optimization routine adjusts the values of the Fourier series coefficientsfor each limb to search for new gaits The same Fourier coefficients are used

for each limb, offset in phase by π radians, restricting the model to walk with

a symmetric gait

2.4 Balance and Velocity Control

A balanced gait and a desired forward velocity is achieved by manipulating thepitch of the HAT The pitch controller works by monitoring the orientation

of the HAT relative to a desired set angle and speeding up or slowing downthe progression of the legs through the joint trajectories to keep the HAT

at a desired angle When the HAT pitches forward (backward) beyond thedesired set angle, the legs are driven faster (slower) to walk ahead (behind) ofthe HAT The velocity controller is very similar to the pitch controller: whenthe model is moving too slowly (quickly), the reference angle for the pitchcontroller is increased (decreased), causing the model to lean forward (back-ward), making the balance controller force the model to walk faster (slower)

A detailed account of the pitch and velocity controllers can be found in good et al.’s original paper [23] The pitch and velocity controllers balancedthe model, but only over a very narrow range: the model could not initiategait from a stand still, but had to begin the simulation with carefully selectedinitial conditions These initial conditions were used for every simulation

Peas-2.5 Pattern Search Optimization Routine

Peasgood et al tuned the control system parameters and the joint trajectoriesusing a pattern search optimization routine The algorithm is conceptuallydescribed below A more formal treatment of the material can be found inLewis et al [20]

1 Repeat for all parameters:

(a) Add amounts +∆ and −∆ (called the grid size) to one parameter.

(b) Evaluate the objective function Save parameter changes that improvethe objective function for later use

2 Update all parameters with the improved values from Step 1

3 Evaluate the objective function If it improves, accept the new parameterset from Step 2; else use the original parameter set

4 Decrease ∆ by half, return to Step 1 Continue until ∆ is below a

prede-fined tolerance

The performance of this algorithm relies on the assumption that a set ofindividual changes to the joint trajectories will collectively result in an im-provement This assumption is valid if the set of parameters are independent.Peasgood et al.’s assumption of independence does not hold when applied tojoint trajectories: a beneficial change to the hip joint trajectory may causethe model to fall when combined with a beneficial change to the knee jointtrajectory Thus this search routine only ever improved the objective functionwhen a set of individual parameter changes was found that just happened tocollectively improve the simulated gait

The pattern search optimization routine was used to find joint trajectoriesthat minimized metabolic cost In an optimization run that had 717 simula-tions only once did all of the individual improvements found by the patternsearch routine result in a more efficient gait when used collectively This onesingle improvement was able to decrease the metabolic cost of the simulatedgait by 21.5% An examination of the optimization log file revealed that therewere many individual parameter changes that improved the objective func-tion but were ignored Further investigation showed that a set of individuallybeneficial parameter changes caused the model to fall when applied simul-taneously The pattern search algorithm was adjusted to take advantage ofgood individual parameter changes immediately, resulting in a greedy pat-tern search routine A further adjustment was made by allowing the patternsearch to continue making adjustments to a single parameter that improvedthe objective function until the improvements ceased.

3 Results

The joint angles for the final simulated gait and a healthy human gait [26] areshown in Fig 2 The standard deviation of the joint angles, torques and groundreaction forces for the current results are negligible, indicating that the gait isvery consistent The joint trajectories of the knee and hip are similar betweenall three data sets, but the ankle joint trajectories, and torques are quitedissimilar The log file of the optimization routine revealed that increasing theankle extension led to a significant reduction in metabolic cost The adjustedpattern search routine was able to find a gait that resulted in 47.6% lessmetabolic cost, a 26.1% improvement over Peasgood et al.’s original approach.The foot contact model produced ground reaction forces that differ sub-stantially from those observed during normal human gait [26], as shown inFig 3 The poor performance of the foot contact model is partly responsi-ble for the joint torque differences seen between healthy human gait and thesimulated results in Fig 2 The kinematics of the foot contact model alsoexhibited heel and metatarsal compressions exceeding 40.0 mm, far greaterthan compression levels of real human heel [10] and metatarsal pads [7] Thekinematics and kinetics of this gait differ from healthy human gait [26], andare highly influenced by differences between the simulated foot contact modeland a human foot

4 Discussion

One of the biggest shortcomings of the current system is that the balance troller is so sensitive to changes in gait parameters, that very little of the gaitspace can be searched without making the model fall The latest optimiza-tion run consisted of 721 simulations; 543 of these simulations resulted in the

−1 Std,

−0.05 0 0.05 0.1 0.15 0.2 0.25 0.3

Percent of Stride (%)

Ankle Joint Torque (Plantar Flexion)

Winter Millard et al + 1 Std

Percent of Stride (%)

Hip Joint Torque (Extension)

Fig 2 Joint trajectory and torque comparison between Winter’s recordings of

human gait [26], and the current results

model falling As well, the current system is not well suited to making changes

to single parameters without having potentially disastrous effects: changingany one of the Fourier coefficients will alter the entire gait cycle A parameterchange that improves the efficiency of the stance phase, may cause the model

to fall during the swing phase A more advanced balance control system thatallows the swing and stance phases to be tuned separately would be a greatimprovement to the current system

Percent of Stride (%)

Normal Ground Reaction Force

Winter Millard et al + 1 Std

−1 Std

Fig 3 Normal and friction force comparison between Winter’s recordings [26] and

the current two-point foot contact model

The computationally efficient, but low-fidelity foot contact model duced ground reaction forces and foot pad compressions that were drasticallydifferent than those observed in healthy human gait, and negatively affectedthe simulated joint kinetics A high-fidelity foot contact model is especiallyimportant for a predictive gait simulation: contact forces at the foot will af-fect the loads at the joints of the legs, and thus the metabolic cost of theleg muscles If the model does not have a realistic foot contact model, it will

pro-be impossible to produce metabolic cost estimates that correspond to whatone would expect from a human [28] A predictive gait simulation without ahigh-fidelity foot contact model could not converge to a ‘human-like’ gait

5 Foot Contact Modeling

Foot contact models are typically not validated separately from the gait ulation [23, 24, 27] This approach is problematic: if the ground reaction forcerepresentation is poor, it is impossible to know if its due to an error in the footcontact model or due to the way the foot is being used by the assumed con-trol system The only foot contact model that was validated separately fromthe gait simulation [12] was validated in a naive way: ankle joint torques andforces estimated from an inverse dynamics analysis were applied to a forwarddynamic simulation of the foot model; the fidelity of the foot model was eval-uated by comparing the kinematics of the simulated foot to the experimentaldata This approach is naive because the quantization and measurement errorthat is inherent in an experimental inverse dynamics analysis will cause theforward dynamic simulation to diverge from the experimental observations,even if the model is perfect None of the lumped-parameter foot contact mod-els published to date [12, 23, 24, 27], provide convincing results of emulating areal human foot

sim-The approach taken in the current work to assess candidate foot contactmodels is different from previous attempts [12]: a contact model that was suit-able for modeling heel tissue was first identified, then candidate foot contactmodels were created using this contact model Ground reaction force profileswere used assess the fidelity each model: a realistic foot contact model shoulddevelop the same ground reaction forces as a human foot when driven throughthe same kinematic path A simple experiment was undertaken to gather thedata required to test the candidate foot contact models: a subject’s ankleposition and ground reaction force profiles during normal gait were recordedusing Optotrak infrared diodes (IREDs) and a force plate The subject walked

at three different subjective paces (slow, normal and quickly) in two differentload conditions: bodyweight (BW) and 113% bodyweight The different ve-locity and loading conditions were used to assess the sensitivity of the model

to cadence and load The heavier loading condition was achieved by havingthe subject carry a cinder block The following sections will detail recent work

to create and validate a new foot contact model

5.1 Foot Pad Contact Properties

Studies to determine the stiffness and damping properties of human foot pads

have failed to produce consistent results Traditionally in vivo experimental results disagree by orders of magnitude from in vitro experiments In the past, in vivo experiments have measured the tissue compression and load

by impacting an instrumented mass into a subject’s heel [19, 25] As long asthe skeletal system of the body acts like a perfect ground, the deceleration

of the mass will be entirely due to the compression of the heel pad Aerts

et al [2] was able to experimentally demonstrate that this assumption isinvalid: significant amounts of energy is lost through the body, skewing the

stiffness values reported from in vivo pendular experiments to be nearly sixth the published in vitro values In vitro stiffness and damping estimates

one-obtained using an Instron material testing machine are also suspect becausethe tissue may not be representative of living foot pad tissue from the general

population An in vivo experimental procedure was developed to estimate foot

stiffness and damping:

1 The compression of the heel pad was inferred by tracking the position ofthe fibular trochlea of the calcaneus using an Optotrak IRED The fibulartrochlea of the calcaneus is a bony protrusion on the lateral side (outside)

of the heel bone A marker was also placed on the medial (inside) side ofthe calcaneus

2 The force acting on the heel pad was measured using a force plate Onlythe heel was placed on the force plate

3 The subject voluntarily lowered their heel on the force plate at threesubjective speeds: slow, medium and fast The heel was slowly raised Thefast trials had to be discarded due to undersampling, despite sampling thedata at 200 Hz