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Floquet Multipliers Do Not Correlate with Gross Mechanical Stability in a

Neuromusculoskeletal Model

Mukul Talaty1

Antonie J. van den Bogert2

1 Gait & Motion Analysis Laboratory; MossRehab; Elkins Park, PA 19027

2 Department of Biomedical Engineering; Lerner Research Institute; 9500 Euclid Avenue;

Cleveland, Ohio 44195

Introduction

Bipedal walking in inherently unstable. Nevertheless, humans can walk without much

apparent control effort. This is generally thought to be a consequence of sensory feedback and

the organization of the nervous system. Neuromusculoskeletal models have been developed to

study the basic mechanisms of locomotion and its stability (Hase, 1999, Hase and Yamazaki,

1998, Taga, 1994, Taga, 1995a, Taga, 1995b, Taga, et al., 1991). These show that stable

locomotion can be achieved with specific neural control architectures. These and other models


can capture many of the salient observable features of human gait dynamics. Those with explicit

and physiologically inspired neural control schemes may be ideally useful for exploring clinical

scenarios. Such models must, however, be able to respond in ways similar humans to realize this

potential.

Having commensurate stability to the real (i.e. human) system is one aspect of the validity of

model response that has not been well addressed. By and large, model stability – if evaluated at

all – has been demonstrated qualitatively and anecdotally. Some of the notable exceptions

include the following : Taga quantitatively evaluated stability by applying backward directed

impulsive forces of varying magnitudes and at varying times of gait cycle, applying masses to

various segments and changing the ground slope (Taga, 1995b). For the first approach, he

reported maximum loads that could be tolerated while walking maintained. Gerritsen et al.

demonstrated the contribution of muscle properties to stability by similarly applying loads until

models fell over (Gerritsen, et al., 1998). They developed a family of models with varying

inclusion of F-l and F-v muscle properties.

Assessing global (mechanical) stability may involve finding the maximum perturbation(s)

that a system could tolerate without falling over (Dingwell and Kang, 2007). However there is

no efficient or elegant way to find this solution. Floquet analysis may be a step in the direction

towards a concise conceptual framework and efficient means to apply it. Floquet analysis has

been used with some success as a means to objectively quantify walking stability in humans and

simple models (Dingwell and Kang, 2007, Garcia, et al., 1998, Hurmuzlu, et al., 1996,

Poulakakis, et al., 2006, Su and Dingwell, 2007) but the relationship between the computed

Floquet results and the more tangible aspects of stability has not been well described.


Some efforts have shown sensitivity and/or association of Floquet calculations to

experimental conditions (walking velocity, albeit without consensus (Bruijn, et al., 2009,

Dingwell, et al., 2007, Granata and Lockhart, 2008), mass distribution and walking slope of

passive walker (Norris, et al., 2008), mental distractions (Dingwell, et al., 2008) and to

calculation conditions such as number of cycles used (Bruijn, et al., 2009, Scheiner, et al., 1995),

experimental noise (Su and Dingwell, 2007)). Others have demonstrated relevance of Floquet to

more gross stability but by inference For example, Hurmuzlu showed that Polio subjects had

higher overall range of max than healthy normal subjects. Similarly Granata et al. showed

statistical differences in max for healthy and fall prone adults (Granata and Lockhart, 2008). Su

et al. measured max on a passive dynamic walker over varying “bumpiness” slopes. However,

they found as they expected that the max did not vary as the model itself (and thus its basin of

attraction) was not changed across conditions. Thus a rather direct question remains relatively

unanswered : to what aspect(s) of stability does Floquet relate?

The purpose of the current work is to clarify the interpretation of Floquet multiplier analysis

for estimating stability in a neuromusculoskeletal model by comparing Floquet results to those

from empirical stability testing. We define empirical stability as how much applied external force

can the model tolerate before falling over. We hypothesize that Floquet multipliers would not

reflect relative differences in the gross stability of a family of models. Floquet analysis measures

the linear response to small perturbations and our measure of gross stability is predicated upon

applying relatively large forces to cause the model to fall. The latter combined with the

considerable nonlinearities involved in such a model – particularly in the neural control – was

the basis for expecting that Floquet and the gross empirical stability would measure different

characteristics of the models and would not correlate. Clarifying this characteristic of Floquet


multipliers will help to establish its role as part of an overall assessment of stability as well as in

validation for neuromusculoskeletal models.

Methods

The neuromusculoskeletal model.

The model contains 8 segments and is constrained to move in the sagittal plane under the

control of a central pattern generator (CPG) coupled to 16 Hill-type muscle models with first

order activation and fixed muscle moment arms. The muscle models have been used extensively

in simulation studies for over a decade (Gerritsen, et al., 1998). The neural aspects are

fundamentally those as described by Taga (Taga, 1995a). Seven neural oscillators generate

control signals to the muscles. Each oscillator consists of a coupled pair of flexor and extensor

neurons. Each neuron is self-inhibiting as well as reciprocally inhibiting to its antagonist neuron

and are known to have stable limit cycles (Matsuoka, 1985). The joint between the pelvis and

trunk was heavily damped resulting in only approximately 2.2 degrees of motion over each gait

cycle. The ground contact model consists of 10 visco-elastic elements per foot and was of the

form described previously (Gerritsen, et al., 1995). A family of 3 models was created by shifting

the center of mass of the trunk up and down 75mm from the base model. The models were

simulated using Matlab (The Mathworks, Natick MA) ode23 variable step integrator using a 1e-6

absolute and 1e-3 relative error tolerances.

The model performance captures many of the salient features of walking (Figure 1) and can

walk indefinitely. It does not appear to have a fixed point limit cycle based on observation of a

large number of cycles (hundreds) – even though the gross performance appears to have settled

into a steady state pattern. This cycle to cycle variability was the basis for using Floquet


techniques to estimate orbital stability.

[Figure 1 here]

Floquet theory.

Well established methods for estimating Floquet multipliers were used (Dingwell and Kang,

2007, Granata and Lockhart, 2008, Hurmuzlu, et al., 1996).In short, multiple cycles of walking

kinematics are used to estimate the relationship between one stride and the next for each state.

Each state is referenced to a steady state value which is estimated as the average of all steady

state cycles. Because the neuromusculoskeletal model demonstrates step to step variability –

similar to that of a human, we are able to apply the methods similarly to how they have been

applied to date. Figures 2 and 3 below illustrate the variability present in the model

performance. Shown are phase plots of one of the more variable and one of the less variable

states. The blue traces show the transient variability in the first approximately 5 cycles (first 15

are shown). The red traces demonstrate the model nearly, but not quite, achieving a fixed point

limit cycle during the next 695 cycles. In compiling state data from the point of maximum knee

flexion during each cycle, we have estimated an equivalent system to the one we have actually

simulated; stability assessments were performed on this equivalent system.

[Figures 2,3]

Floquet calculations.

Each model was simulated continuously for 900 seconds of continuous walking or about 700 gait


cycles. Eight pairs of angular orientations and angular velocities (pelvis and bilateral hip, knee

and ankle) were used to estimate the Jacobian. Data were compiled at the time of maximum knee

flexion in each gait cycle. The estimation of Floquet multipliers was as described previously

(Hurmuzlu, et al., 1996), except that all data were obtained from a single long simulation. In

brief, step to step change matrices were compiled, dA, dB :

dA = A(k) – Xss; dB = A(k+1) – Xss;

where k was cycle number and Xss was the steady state value :

Xss = (1/n)*( x(k)); k=1,n (i.e. cycle #)

The Jacobian J was the function relating step k and the next step k+1:

dB = J · dA

so was computed by the least squares solution of the over-determined system defined by :

J = [dB]-1

/ dA

The maximum Floquet multiplier, max, was the largest eigenvalue of this Jacobian.

Empirical Stability Testing


100ms horizontal or vertical impulses were applied to pelvis or foot of the models during

midswing and during double support of steady state walking. The exact time of the gait cycle at

which the perturbation was applied was selected by the similarity of the joint angles and ground

reaction forces – as shown below in Figure 4a,b.

[Figure 4]

Results

[Table 1]

The horizontal and vertical perturbation forces and result (fall or walk) are presented in Table 2

as the empirical measure of stability.

[Table 2]

Discussion

Floquet multiplier and empirical testing results were not consistent with each other suggesting

orbital stability and falls risk are not related for this model. For the purposes of this study, we

assumed that being able to tolerate greater forces without falling over was equivalent to having a

lower falls risk. Statistical analysis of the maximum Floquet multipliers ( max) indicated that all

three models were equally stable, though the high CM model trended towards being less stable


(p=0.10). The empirical results suggested that the base and the high models were similarly

stable and the low model was less stable.

This is the first such assessment using Floquet techniques on data from a complex model with

human-like muscle properties under closed loop neural control. These results are consistent with

two other recent findings that suggest orbital stability does not correlate with falls risk (Su and

Dingwell, 2007, Verdaasdonk, et al., 2007). Both of these studies were also on simulated data

from a model. In the Su work, a passive dynamic walker model was perturbed at varying levels

and the max calculated. The max did not vary with perturbation levels despite the model

performing closer to the edges of the basin of attraction with larger perturbations. Verdaasdonk

et al. used a very simple (2 rigid legs) but actively controlled model and applied varying levels of

a moment to block swing leg progression and reported that max did not correlate with the

varying levels of risk.

Floquet estimation of orbital stability in the conditions tested may not correlate to empirical

(gross) stability because Floquet analysis is based on measuring the system response to small

perturbations whereas the empirical stability involved large perturbations. The perturbations

used to calculate max were the inherently small cycle to cycle variations during steady state

(Figure2,3) However the model responses to perturbations near the levels that cause it to fall are

considerably greater (c.f. perturbation magnitudes in Table 2) than the cycle to cycle variability

exhibited by the model when unperturbed. Because the perturbations are large, the model

response is likely outside of the linear response region estimated by the least-squares best-fit

Jacobian. It has been suggested but not rigorously demonstrated that the asymptotic stability of a

nonlinear system can be estimated from the linearized map (Granata and Lockhart, 2008,

Mombaur, et al., 2001). It is certainly conceivable that there is a neurophysiological or systems


level relationship between the small and large perturbation response, but that relationship – if it

exists – is not known to the authors at this time. Recent studies on human walking suggest max

may be related to gross stability.

The current study results and the other model-based ones have a clearer basis of „falls risk‟

and oppose the notion that stride to stride fluctuations of relatively steady state walking have

predictive value for assessing falls potential (Granata and Lockhart, 2008, Hurmuzlu, 1994,

Hurmuzlu, et al., 1996).Granata et al. assumed reliability in differences in stability between

groups based on a self-reported history of falls during overground community walking and then

calculated Floquet estimates from controlled treadmill walking. They provided a strong

statistical basis for concluding max could identify fall prone individuals. However, the self-

report falls risk and discrepancy between venue of reported falls and laboratory testing are

sources of potential error. The accuracy of self-reported falls and falls risk is not without debate

(Hauer, et al., 2009, Kanten, et al., 1993, Srygley, et al., 2009). Differences along many

dimensions between treadmill and overground walking have been noted to exist for several

decades and continue to be reported (Alton, et al., 1998, Brouwer, et al., 2009, Lemke, et al.,

1995, Phadke, et al., 2007, Powell, et al., 2009, Puh and Baer, 2009, Riley, et al., 2007, van

Ingen Schenau, 1980, Vogt, et al., 2002, White, et al., 1998). Other possible sources of the

discrepancy with the human studies may stem from Floquet methodology used which is not yet

fully mature and varies across researchers and even across studies by the same group. Granata et

al. used only ~35 cycles in estimating max, while our own work (unpublished) as well as that of

Bruijn (Bruijn, et al., 2009) suggests at least 150 cycles are needed for a reliable estimate of max.

Still others have suggested 85 cycles may be enough (Scheiner, et al., 1995)


One other potential source of discrepancy between human and model results is that the source

of cycle to cycle variability is likely fundamentally different in the model and the human studies

It has been argued that “…Natural variance is attributed to mechanical disturbances or

neuromotor control errors.” (Granata and Lockhart, 2008) And yet it is the neural controllers and

the musculoskeletal system that attenuate these disturbances. So there should be a relationship

between this variability and the system‟s ability to attenuate disturbances. The cause of the

variability in our neuromusculoskeletal model is not clear. That a deterministic model shows

human-like variability in steady state is in and of itself interesting. We have ruled out that

technical issues such as appropriateness of computation time step, event selection, and number of

cycles may be contributing to this variability. The similarity of our Floquet results to those from

human studies perhaps suggests that some fundamental characteristics of the model that

contribute to this variability is in accord with actual human characteristics. This is an unexpected

and could be a potentially novel approach to establishing model validity.

Finally, how well the empirical testing adequately sampled all the relevant aspects of the

model‟s stability responses is not known. It is always a question with empirical testing whether

all modes by which the model may fall and thus all aspects of the models relevant performance

space have been evaluated. This is central to the motivation to discover a formal stability

analysis method, such as Floquet, that can be practically implemented. It is possible, but not

likely, that applying other load configurations (timing, location, magnitude, etc.) would produce

results more consistent with the Floquet analysis. Applying perturbation forces during both

single and double support, to proximal as well as distal segments, and in vertical as well as

horizontal directions sampled many aspects of the performance space and minimized this

possibility. Our basis for selecting the variations was not guided by a theoretical knowledge of


the overall performance space. We are unaware of any practical means to estimate this. A 16

dimensional space has 2^16 = 65536 corners and 150 cycles of a 700 cycle simulation would not

seem to adequately cover this entire space. What part of this space is relevant to perturbation

response is not clear. Despite our own and Bruijn‟s sensitivity analysis, perhaps more cycles are

needed to estimate Floquet Bruijn used a robust bootstrapping technique to demonstrate that

approximately 150 cycles was sufficient to produce statistically precise estimates of max.

Conclusions

We present a case of bipedal, powered, human-like gait in which the maximum Floquet

multiplier does not correlate with falls risk. This is similar to what a few other studies using

models have suggested, but contrary to what others have reported using the same methods

applied to human walking. It is unclear whether differences in the model and human gait may

account for this discrepancy in interpretation of Floquet results. Alternatively, perhaps the notion

of falls risk in humans has not been thoroughly or rigorously defined. This study extends work

done on very basic models and is the first to demonstrate Floquet results in a neurally-controlled,

muscle-actuated human-like walking simulation.

To what practical aspect of stability does max relate is still unclear. Further clarification is

needed for both this particular neuromusculoskeletal model as well as for human walking. It is

unclear also whether subtler methodological choices could affect results. For example, the effect

of which and how many states to include has not been sufficiently explored. The influence of

source of the variability on the results also seems to deserve further evaluation. Floquet


multipliers do seem to have a role in understanding walking stability but are not the holy grail of

stability and need to be interpreted with caution and in the context of not yet well-understood

limitations.


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Figures

0.8 1 1.2 1.4 1.6 1.8 2 2.2 2.4 2.6

0

0.5

1

1.5

Figure 1. The base model in early steady state for approximately one cycle


-1.8 -1.75 -1.7 -1.65 -1.6 -1.55-1.5

-1

-0.5

0

0.5

1

1.5

2

2.5

Pelvis angle (rad)

Pelv

is a

ngula

r velo

city (

rad/s

)

-1.4 -1.2 -1 -0.8 -0.6 -0.4 -0.2 0 0.2-10

-5

0

5

10

159

Knee angle (rad)

Knee a

ngula

r velo

city (

rad/s

)

Figures 2,3 Phase plots of a more variable (left) and a less variable state (right) in the NMS


305.2 305.4 305.6 305.8 306 306.2 306.4 306.6 306.8 307-0.2

0

0.2

0.4

0.6

0.8

1

1.2

1.4

Time: 0.250(sample=501) | dt: 5.000000E-004; | Speed=15x | <any key> to continue

305.6 305.8 306 306.2 306.4 306.6 306.8 307 307.2 307.4-0.2

0

0.2

0.4

0.6

0.8

1

1.2

1.4

Time: 0.600(sample=1201) | dt: 5.000000E-004; | Speed=15x | <any key> to continue

Figure 4. Illustration of the times during the gait cycle at which perturbations were applied.

Left Panel : MidSwing, Right Panel : Double Support


Tables

Table 1. Maximum Floquet Multipliers ( max) for the Models.

Cycles used for max

Model

Base High Low

1-150 0.55 0.51 0.52

151-300 0.55 0.63 0.50

301-450 0.54 0.66 0.45

451-600 0.53 0.64 0.70

Mean 0.54 0.61* 0.54**

Standard deviation 0.01 0.07 0.11

1-600 0.49 0.62 0.52

Notes. * p value was 0.10. ** p value was 1.00. Data row for cycles 1-600 were

not used for any comparisons but are presented for reference only.


Table 2. Perturbation Testing Results

Segment

Direction

Time

Magni-

tude (N)

Model

Base High Low

Pelvis

Vert.

MS 5250 Walk Walk Fall

5400 Fall Fall Fall

DS 700 Walk Walk Fall

800 Fall Fall Fall

Horiz.

MS 150 Walk Walk Fall

250 Walk Fall Fall


250 Walk Fall Fall

Foot

Vert.

MS 50 Walk Walk Walk

75 Walk Fall Walk

DS 900 Walk Walk Walk

1500 Walk Walk Walk

Horiz.

MS 350 Fall Fall Fall

150 Fall Fall Fall


250 Fall Fall Fall

Thigh

Vert.

MS 900 Fall Fall Fall

1500 Fall Fall Fall

DS 1200 Fall Fall Fall

900 Fall Walk Fall

Horiz.

MS 150 Fall Walk Fall

250 Fall Fall Fall

DS 250 Walk Walk Walk

350 Walk Fall Walk

Total # of Falls 11 13 18

*MS=MidSwing; DS=Double Support; Bold indicate differences relative to the base

model; Italic indicate changes contrary to the Floquet results.

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