In this paper, we study dynamic stability during running, focusing on the effects of speed, and the use of a leg prosthesis. We compute and compare the maximal Lyapunov exponents of kinematic time-series data from subjects with and without unilateral transtibial amputations running at a wide range of speeds. We find that the dynamics of the affected leg with the running-specific prosthesis are less stable than the dynamics of the unaffected leg and also less stable than the biological legs of the non-amputee runners. Surprisingly, we find that the center-of-mass dynamics of runners with two intact biological legs are slightly less stable than those of runners with amputations. Our results suggest that while leg asymmetries may be associated with instability, runners may compensate for this effect by increased control of their center-of-mass dynamics.

1.
See www.mpipks-dresden.mpg.de/~tisean for TISEAN, version 3.0.1.
2.
H.
Abarbanel
,
R.
Brown
, and
M. B.
Kennel
, “
Variation of Lyapunov exponents on a strange attractor
,”
J. Nonlinear Sci.
1
,
175
(
1991
).
3.
S.
Bruijn
,
O.
Meijer
,
P.
Beek
, and
J.
van Dieen
, “
Assessing the stability of human locomotion: A review of current measures
,”
J. R. Soc. Interface
10
,
20120999
(
2013
).
4.
S.
Bruijn
,
J.
van Dieen
,
O.
Meijer
, and
P.
Beek
, “
Is slow walking more stable?
J. Biomech.
42
,
1506
1512
(
2009
).
5.
S.
Bruijn
,
J.
van Dieen
,
O.
Meijer
, and
P.
Beek
, “
Statistical precision and sensitivity of measures of dynamic gait stability
,”
J. Neurosci. Methods
178
,
327
333
(
2009
).
6.
U.
Buzzi
,
N.
Stergiou
,
M.
Kurz
,
P.
Hagman
, and
J.
Heidel
, “
Nonlinear dynamics indicates aging affects variability during gait
,”
Clin. Biomech.
18
,
435
443
(
2003
).
7.
S.
Carver
,
N.
Cowan
, and
J.
Guckenheimer
, “
Lateral stability of the spring-mass hopper suggests a two step control strategy for running
,”
Chaos
19
,
026106
(
2009
).
8.
R.
Clewley
,
J.
Guckenheimer
, and
F.
Valero-Cuevas
, “
Estimating effective degrees of freedom in motor systems
,”
IEEE Trans. Biomed. Eng.
55
,
430
442
(
2008
).
9.
J.
Dingwell
and
J.
Cusumano
, “
Nonlinear time series analysis of normal and pathological human walking
,”
Chaos
10
,
848
863
(
2000
).
10.
J.
Dingwell
and
L.
Marin
, “
Kinematic variability and local dynamic stability of upper body motions when walking at different speeds
,”
J. Biomech.
39
,
444
452
(
2006
).
11.
S.
England
and
K.
Granata
, “
The influence of gait speed on local dynamic stability of walking
,”
Gait Posture
25
,
172
178
(
2007
).
12.
R.
Enoka
,
D.
Miller
, and
E.
Burgess
, “
Below-knee amputee running gait
,”
Am. J. Phys. Med.
61
,
66
84
(
1982
).
13.
J.
Frank
,
S.
Mannor
, and
D.
Precup
, “
Activity and gait recognition with time-delay embeddings
,” in
Proceedings of the Twenty-Fourth AAAI Conference on Artificial Intelligence
,
2010
, pp.
1581
1586
.
14.
A.
Fraser
and
H.
Swinney
, “
Independent coordinates for strange attractors from mutual information
,”
Phys. Rev. A
33
(
2
),
1134
1140
(
1986
).
15.
R.
Full
,
T.
Kubow
,
J.
Schmitt
,
P.
Holmes
, and
D.
Koditschek
, “
Quantifying dynamic stability and maneuverability in legged locomotion
,”
Integr. Comp. Biol.
42
,
149
157
(
2002
).
16.
A.
Grabowski
,
C.
McGowan
,
W.
McDermott
,
M.
Beale
,
R.
Kram
, and
H.
Herr
, “
Running-specific prostheses limit ground-force during sprinting
,”
Biol. Lett.
6
,
201
204
(
2010
).
17.
L.
Gundersen
,
D.
Valle
,
A.
Barr
,
J.
Danoff
,
S.
Stanhope
, and
L.
Snyder-Mackler
, “
Bilateral analysis of the knee and ankle during gait: An examination of the relationship between lateral dominance and symmetry
,”
Phys. Ther.
69
,
640
650
(
1989
).
18.
M.
Hausdorff
, “
Gait dynamics in Parkinson's disease: Common and distinct behavior among stride length, gait variability, and fractal-like scaling
,”
Chaos
19
,
026113
(
2009
).
19.
P.
Holmes
,
R.
Full
,
D.
Koditschek
, and
J.
Guckenheimer
, “
The dynamics of legged locomotion: Models, analyses, and challenges
,”
SIAM Rev.
48
,
207
304
(
2006
).
20.
K.
Jordan
,
J.
Challis
,
J.
Cusumano
, and
K.
Newell
, “
Stability and the time-dependent structure of gait variability in walking and running
,”
Hum. Mov. Sci.
28
,
113
128
(
2009
).
21.
H.
Kantz
, “
A robust method to estimate the maximal Lyapunov exponent of a time series
,”
Phys. Lett. A
185
,
77
(
1994
).
22.
H.
Kantz
and
T.
Schreiber
,
Nonlinear Time Series Analysis
(
Cambridge University Press
,
Cambridge
,
1997
).
23.
M.
Kennel
,
R.
Brown
, and
H.
Abarbanel
, “
Determining minimum embedding dimension using a geometrical construction
,”
Phys. Rev. A
45
,
3403
3411
(
1992
).
24.
T.
Lockhart
and
J.
Liu
, “
Differentiating fall-prone and healthy adults using local dynamic stability
,”
Ergonomics
51
,
1860
1872
(
2008
).
25.
P.
McAndrew
,
J.
Wilken
, and
J.
Dingwell
, “
Dynamic stability of human walking in visually and mechanically destabilizing environments
,”
J. Biomech.
44
,
644
649
(
2011
).
26.
S.
McGregor
,
M.
Busa
,
J.
Skufca
,
J.
Yaggie
, and
E.
Bollt
, “
Control entropy identifies differential changes in complexity of walking and running gait patterns with increasing speed in highly trained runners
,”
Chaos
19
,
026109
(
2009
).
27.
J.
Milton
, “
Introduction to the focus issue on ‘Bipedal locomotion: From robots to humans’
,”
Chaos
19
,
026101
(
2009
).
28.
J.
Nessler
,
C.
De Leone
, and
S.
Gilliland
, “
Nonlinear time series analysis of knee and ankle kinematics during side by side treadmill walking
,”
Chaos
19
,
026104
(
2009
).
29.
N.
Scafetta
,
D.
Marchi
, and
B.
West
, “
Understanding the complexity of human gait dynamics
,”
Chaos
19
,
026108
(
2009
).
30.
M.
Seeley
,
B.
Umberger
, and
R.
Shapiro
, “
A test of the functional asymmetry hypothesis in walking
,”
Gait Posture
28
,
24
28
(
2008
).
31.
H.
Skinner
and
D.
Effeney
, “
Gait analysis in amputees
,”
Am. J. Phys. Med.
64
,
82
89
(
1985
).
32.
F.
Takens
, “
Detecting strange attractors in fluid turbulence
,” in
Dynamical Systems and Turbulence
, edited by
D.
Rand
and
L.-S.
Young
(
Springer
,
Berlin
,
1981
), pp.
366
381
.
33.
P.
Weyand
,
D.
Sternlight
,
M.
Bellizzi
, and
S.
Wright
, “
Faster top running speeds are achieved with greater ground forces not more rapid leg movements
,”
J. Appl. Physiol.
89
,
1991
1999
(
2000
).
34.
S.
Wurdeman
,
S.
Myers
, and
N.
Stergiou
, “
Transtibial amputee joint motion has increased attractor divergence during walking compared to non-amputee gait
,”
Ann. Biomed. Eng.
41
,
806
813
(
2013
).
35.

See the penultimate paragraph of Sec. III for a detailed explanation of this point.

36.
This parameter specifies the size of the neighborhood whose points are tracked for the calculation of the spreading factor S. Too-small values of ϵ cause numerical problems because the neighborhood contains only a few points; too-large values cause the calculation to sample the dynamics too broadly.
37.
The second hypothesis is not at issue in this section, since the sacrum position data do not effectively isolate the dynamics of the individual lower limbs.
38.

Indeed, Kantz and Schreiber22 quote Salman Rushdie to make this point.

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