Journal of Applied Biomechanics, 2014, 30, 737-741 http://dx.doi.org/10.1123/jab.2014-0089 © 2014 Human Kinetics, Inc.

An Official Journal of ISB www.JAB-Journal.com TECHNICAL NOTE

A New Perspective on the Walking Margin of Stability Kevin Terry,1 Christopher Stanley,2 and Diane Damiano2 1George

Mason University; 2National Institutes of Health Clinical Center

There remains a pressing need for a stability metric that can reliably identify fall susceptibility during walking, enabling more effective gait rehabilitation for reduced fall incidence. One available metric is the maximum margin of stability (MOSmax), which is calculated using the body’s center of mass (COM) position and velocity along with the location of the maximum center of pressure (COPmax). However, MOSmax has several limitations that may limit stability assessment. Specifically, the assumptions of a fixed COP and constant ground reaction force (GRF) are not applicable to gait. To address these limitations, a modified MOS equation that allows for a variable COP and is not dependent on a constant GRF is presented here. The modified MOS was ¨ significantly lower than MOSmax throughout a significant portion of single limb support for normal walking ¨x gait. This finding 2 mg theI  ml uuxxwhen mg actual MOS indicates the MOSmax metric may lack sensitivity to instability as it may still be positive 2 x indicates existing  I    ml   l ¨ or impending instability. This comparison also showed that the MOS might offer additional information about ¨ l walking stability x 2 relevant to gait assessment for fall prevention and rehabilitation. However, like other stability metrics, xIcapability this  ml must be ml 2  u  x  mg  uI xmg established with further investigations of perturbed and pathological gait. l

l

Keywords: stability, walking, balance, falls ¨ ¨

l x g Reliable and clinically meaningful identification and treatment x ¨¨x l  u  x¨ 2  u , 0  g of fall susceptibility should help reduce the personal and economic x x g  u  x  u , 0 ¨ l , (2) u  x  mg  2 Ixg ml 2¨¨ 020l  l costs associated with fall injuries and reduce the mortality rate for x g  u  x  mg I  ml ¨ l x xxl  u  gu , 0  severe fall injuries. Presently, the utility of existing stability metrics l position, 2 x  x u   u ,   where u is the COP x is the COM position, g represents g  l 0 0 2 as reliable predictors of falls is still being investigated.1–7 One of g  l distance from the 0 acceleration due to terrestrial gravity, and l is the these metrics, the margin of stability (MOS),8 incorporates center of ankle to the COM. Assuming a constant COP position, the followmass (COM) and center of pressure (COP) data that represent both ing equation (Equation 3) establishes the relationship between the kinematic and kinetic elements of movement, respectively. Another ‘extrapolated’ COM that v0 combines initial COM position (x0) and feature of the MOS is its potential to identify likely points of fall u velocity (v0) andxxCOP 0  v¨ 0position: u susceptibility within a stride, as it is measured continuously and not ¨ l 0   x g 0 ¨ l x x  u  20  ug , 0v0 averaged over multiple strides. This feature also makes the MOS an x x  u  x   ul . (3) g u , v appealing metric for assessing responses to discrete perturbations. g 02 x0  00 0 u l 0 0 0 However, in its present form, the MOS has some limitations with After substituting u with the maximal COP position, umax, this regard to walking stability. The advantage of the MOS perspective versus the traditional relationship yields the existing maximal MOS (MOSmax) equation: base of support (BOS) analysis is that it considers the trajectory  v  of the COM so that it defines stability as a condition for which the  x0  v00   MOSmax . (4)   b u max COM is within the BOS and will remain within the BOS based on v b  umax  x0  0   MOSmax x0  0  u 0   its current position and velocity. The MOS was originally developed v0  v0  x  u 0  b  inuvmax  x0 4 indicates to analyze stability during quiet stance (Figure 1) with an equation0  The absolute   MOS value symbol Equation that,max dependb  umax   x0  0    MOS 0 max 0 negative  of motion (Equation 1) developed from the fundamental Euler ing on the coordinate system, theMOS may  be for stable 0   equation for a single inverted pendulum: conditions and must be adjusted accordingly. Before proceeding ¨ further, several variables were changed to reduce ambiguity and ¨ 2 x  I  ml 2 x . (1) improve consistency. The COM position (x) and velocity (v) are  uu  xx mg   mg I  ml l changed to xCOM and vCOM, respectively. COP vCOM  Likewise,  position is 0 l v4COM   MOS x x    With these changes, Equation becomes:  max COP COM 0 Rearranging this equation to eliminate the mass (m) term and changed to xCOP.MOS max  v   xCOM 00  0  max b  umax v0   x0 max 0 xCOPMOS produce a solvable differential equation yields: max 0  v   ¨

b  umax   x0    MOS 0max vCOM    COM 0  0MOS  x0COM max  xCOPmax  0   max  xMOS  . (5) 0  COPmax   xCOM 0  0  

¨

Kevin Terry is with the Department of ¨ l x¨ Rehabilitation gScience, George x x¨VA. u  x2  Stanley u , 0 and Diane Mason University, Fairfax, l Christopher g Damino are  x g Biomechanics u  02  u ,Section, 0  National l with the Functional &xApplied Institutes g 0 l of Health Clinical Center, Bethesda, MD. Address author correspondence to Kevin Terry at [email protected].

Even though the MOS has been used to assess walking stability,9–11 there are several limitations of the MOS as currently defined. ¨ The first limitation is that the MOS inverted pendulum ¨ model was 2 x COM originally developed anterior–posterior standing  GRF xCOPfor mg I  ml2sway xCOMwith both vCOM  xxCOMmg 0 I   ml   GRF l the MOS xCOP COM ¨ feet approximated amaxsingle support. This model makes MOSmax  xas vCOM   xCOM  COP 0 0

¨ l MOSmax  xCOPmax   xCOM 0    0  xCOM xCOM  xCOP I  ml 2737 0  GRF  xCOM mg2   xCOP  GRF  xCOM mg I  ml l l

'

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738  Terry, Stanley, and Damiano

Figure 1 — Comparison of original (A) and modified (B) inverted pendulum models for MOS equation development (shown in frontal plane for M-L stability). Vectors representing gravitational (mg) and ground reaction forces (GRF) along with positions and velocities of the COM (l, xCOM, vCOM,) and COP (xCOP, vCOP) are shown. Variables used for the original MOS development8 shown in parentheses.

applicable only when the body’s supporting COP can be approximated at a single point, as in single limb support (SLS), defined as the portion of the gait cycle between final and initial contact of the contralateral foot. In addition, the MOS equation was developed with the assumption of a constant ground reaction force (GRF) equivalent to the subject’s weight. This approximation is appropriate for quiet stance, but not across a stride cycle, during which the GRF changes continuously as weight is transferred from one foot to the other and back. Because xCOP is constant in Equation 5, the typical practice is to calculate a maximum MOS (MOSmax) based on a maximal COP position (xCOPmax), which is established by the boundaries of the foot contact area. As with the constant GRF assumption, xCOPmax as defined by foot contact area is relatively stable during quiet stance. However, during walking, xCOPmax changes with each foot’s stance phase progression. For example, at initial and final contact, only part of the foot contacts the support surface and there is often considerable internal and external foot rotation throughout the stride, which will move xCOPmax medially or laterally and change the BOS. This last limitation can be addressed if xCOP is allowed to vary with time, as is xCOM, resulting in an MOS based on the actual, not maximal, COP location. Once the COP position is allowed to vary with time, it will also have a velocity and its own trajectory. The limitation of the assumed fixed COP location is cited in the original MOS development as ‘Case b’ of different stability

scenarios,8 which suggests that the foundational differential equation should be solved for ‘nonconstant u(t)’, which then leads to the inclusion of a COP velocity term (vCOP). Presented here is a solution for that scenario that eliminates the need to assume a constant GRF and a fixed maximal COP position, producing a more generalized form of the MOS equation that is applicable for both standing and walking stability assessment during SLS. Comparisons of this generalized MOS and MOSmax are then presented for individuals with normal walking gait.

Methods Generalized Margin of Stability Equation Development To eliminate the need to track a variable GRF, a modified version of the original inverted pendulum model was used to examine frontal plane stability during walking (Figure 1). Instead of treating the ankle as a planar hinge joint, the leg, ankle, and foot are treated as a single structural member with the COM at one end and the GRF applied at the other. This change places the focus on the effective moment arm created by the relative COM and COP positions and the forces at each end of the arm. Summing moments and applying

vCOM   MOSmax  xCOPmax   xCOM 0  vCOM00  MOSmax  xCOPmax   xCOM 0  0  0   vCOM 0   MOSmax  xCOPmax   xCOM 0   0   the same inertial approximation used for the original MOS equation, the following relationship is established:

t 

         vCOM 0  0 xCOM 0 

0 xCOM 0  vCOM 0 739 A New Perspective on Walking MOS      vCOM   0 xCOM 0 0     , (15)   vCOM 0  0 xCOM 0 



¨

x¨ COM xCOP  GRF  xCOM mg I  ml xCOM . (6) xCOP  GRF  xCOM mg I  ml l l By using a Galilean coordinate transformation that ¨establishes xCOM xCOP as the frame ofx reference origin, the GRFIproduces   ml 2 no effective COP  GRF  xCOM mg  moment (xCOP = 0) and no longer needs to be included in lthe equation of motion. The also produces new x' x  x (t ) ttcoordinate tttransformation x'COM  (t ) and v’COM, respectively,  xCOM    xCOP COM COM COP COM position andvelocity variables x’COM 2 2



x' ' t t xCOM t(t) xCOP (t ) , (7) v'COM  t v   COMvCOM COP v t  vCOM t   vCOP (t ) '  vCOP vv'COM  ttCOM COM COM COP v'  t  vvCOM v ttt v v ((tt)()t ) , (8) COM

COM

COP

 t  vCOM t   vCOP (t ) and Equationv6COM becomes ' '

 vCOM 0  0 xCOM 0  20

 0 is posimust result in a positive real number. Because vCOM  0 xCOM  is  xCOM tive and xCOM 0 0 negative (see Figure 1), the numerator within the brackets is positive.  0Therefore, the denominator of Equation 14 must xCOM  vCOM also be positive to produce a positive real number, or 0



' '  0 x'COM 0  0 , (16)  0vx' COM COM00  0 v  0 x COM 0'  0 0 xcondition which establishes COM 0  the COM 0  0 for instability. Therefore, a '

' v COM 0 COM 0v '

stable condition will meet the opposite condition (< 0). By dividing both sides by ω0 and reversing the coordinate transformation, the vCOM 0  vCOP0 stability condition v isv now COM 0



 (x

COP0

 x

)0

COM 0 0 COP )  (xCOM 0  xCOP vCOM 0 v v 0vCOP 0 COMCOP  0 ( xCOM xCOP ) ) 0 0, (17)  ( xCOM 0 0 xCOP 0 0 0 0

0

0

0

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 t    t   vCOP (t ) xCOM vCOM  t  vCOMxCOM t  xv COP (tt) vCOM COM  t  '  which can be rearranged to form a generalized MOS equation:    t   xvCOM  vCOM t t) tt  t  , (9) t  COM 2x v COM COP x(COM 2 COM   x t   0 COM  0 xxCOM COM 2 xCOMttt      vCOP0  v 2  02  MOS vCOP   vCOM 0  0 0 x t x0COP0 x xCOM  COM 0       COM  MOS x    with the following initial conditions:  COP   COM  0. (18) xCOM  t   x 2 t 0 0   vCOM 0  0  0 vCOP   0 0   0    COM v v       xCOM  t  MOS  x 0 COP 0 0    xCOM 0 COM 0   xt COM  t   2' MOS   xCOP0 COP   0   xCOM 0  xCOM  OM  t    2 . (10)  the0 extrapolated  ' x  t 0  x ' , x  t 0  v '     0 xCOM 0  t  COM 2 COMx ' COM Note that the COM terms that 0 comprise 0  COM xCOMCOM  t 0 0 t 0 0 v 'COM 0       COM 0 , xCOM  ' ' xx0COM '    x ' , x  t 0  v '  tt 00 x ' , x t 0  v ' are still present. Likewise, the COP terms can now be combined to       COM COM COM COM 0 COM COM COM 0 x t 0  x ' , x  t 0  v ' 0 0  COM9)0 is now  to Equation COM similar COM 0 The equation ofCOM motion (Equation 0

form an ‘extrapolated COP’, quantifying the combined COP position and velocity. Also note that the assumption of a fixed maximal COP will eliminate the vCOP term, producing Equation 5. ' x t 0  x ' , x  t 0  v ' 0    COM (v ' COM' 0 0' ' e0  COM t 0 'COM  x 'COM 0 , xCOM t COM  ('v vCOM 0tx COM 0)  COM 0  )0 x(v COM 0 )   x OM  0 0 e 0 (v 0 x COM 0' 0 Whereas COM COM 0' ' COM 0 ) the original form of the MOS quantified the maximum 00tt  x0'COM ' ' 0 ' xCOM  t  xCOM t v ' 0' '   (xt )'COM 0x, 'COM  ' 000t  COM xCOM (vv(COM 0 xCOM ee(et )(2(0vtv(COM  0x 0x COM COM COM COM 0 00)) ) (2 00))0t )available and was based solely on foot and body positions, MOS v0 'COM  x0 'COM v0e'COM  00 0 0 0x 0 COM  2 2  e 0 0  x ' ( t )  x ' ( t ) 0 0 COM COM  t  t  x 'COM (t )  ' 22 00 15 quantifies the actual MOS at any given time. In essence, 0tEquation 2   e  e 0t ' 2 ' ' 0 0 0 0 e (v COM20  00 x COM 0 ) (v COM20  0e0 xthis 0) COMmore generalized form of MOS establishes whether the effective  x ' (t ) ' e0t (v' '   x'  (v ' 0t ' ' COM  0tmoment x 'COM 0 ) arm—the distance between xCOM and xOP—is sufficient 0 COM 0 ) COM COM 2 2 e ( v   x ) ( v   x ) 0 0 e 0 0 0 COM COM 0 00t COM 'x ' 0 ' ' 0 0 . (11) 0  t ) xCOM  ) COM to  0' x 'COM ( 0t prevent instability based on current COM and COP velocities. e t '(v COM '0 COM 0 )  ( ' tv 0 COM (t ) 2' COM 0 0  2  ' 02 ' ' ) 0 x ) 0e x 'COM (t ) 2e00(v COM 0 0e00tx(vCOM e(v0xCOM 0 (v 0  x COM Including actual COP positions and velocities in the MOS provides 0 ) ) t 0 COM 0 COM 0' '  0 COM 0'2 t ' '2 '  e 0 COM 0' '  ((0v ' ' ' 0  0 0t   0 xCOM (vv  0the xCOM eee00tsame insight into some of the neuromechanical responses to varying 0x 0x COM COM COM COM 00)) ) (2 00))0t ) COM 2tv(COM e v0 'COM  x ( v   x Applying the limiting condition used for original 0 00 0 0 0 20 COM 0   0 COM 0 00t2t0 COM 0 0e dynamic conditions. This feature becomes especially useful for MOS of x’COM(t) = 00t (ie,' x22COP [t] = x [t] at the limit of stability)  t 20 20 22 eee 0x ' 0  x ' COM ) 0 e (v COM (v 'COM the study of discrete gait perturbation responses and unstable gait. 0 0 COM 0 0 0 COM 0 ) produces the equation 0 0 t ' ' '  ' ' in the x’COM term and 2, except that the COP xCOMposition 'COM 0 , xCOM t 0 is xembedded t 0 v 'COM0 therefore now has the solution: ' t ' ' ' '

e 0 (v  0)x COM 0 ) (v COM20e00tx COM 0 ) COM 20' (v 'COM 0   x ' ' 0 ) (v 'COM 0 0x COM 00t COM ' ' 0 0 0  ) 0t e (v COM 0 0 x COM ) ( v  0 x 'COM ' ' 0 0 COM  0  2  0t 2x , (12) ' ' ( v  ) 0 0e  20 2  e COM 0 0 COM 0  0 (v 20t x ) t COM e20  ' e((2vv' '0t '  ' ' e 0 0 COM 0 20xx)00'COM COM COM COM 00)) ') ' 0 v  2 2( ( v x'COM 0 COM 0x 00ttCOM  0 0 COM 0 0  t 2 0 v (  0 e e  0 COM 0 x COM 0 ) ' ' ' ' e  0 which reduces to ((vv ''  x x  '' 00)) COM COM 0 COM COM 0  (v COM 00 COM 00 ) COM 0  00 x COM e20t  (v' ' 0   x' ' ' ' COM 0  0x COM 0)) 20t (v COM  0 x COM 0 ) 0 COM 0 0t . (13) 0 (v)COM ' '  ' 020t (v' 'COM 0  e0 x 'COM 0(v   0 x COM 0 )  COM   (v COMe0  0 x COM )    vCOM 0  00xCOM   0 xCOM 0 0 vCOM  (v ' 0  ln 0x'COM   0 x0sides 0)v    of     ln v  x    12 yields     TakingCOM the natural logarithm both of COM 0 COM COM COM  0  Equation     v x  0 COM 0x 00 0    vCOM  0 COM 0  00xCOM 0  lnln ln 0  COM0vCOM 0 t    0 xCOM  t 2vv  COM 0x COM   v0 COM xCOM  00 00  COM COM COM 00  002 tt t ln 00   0     2  2     v  x  0 0  COM 0 COM   vCOM 020 0 0 xCOM 0 0  vCOM 0  0 xCOM  0   ln   . (14)   ln   0 xCOM 0 v vCOM0t   0 xCOM COM20    v  x  ln 0     0 COM t x  0  0  COM 0 v COM  0  COM 0    0  2   vCOM 0  0 xCOM will be  real 0  t The 20 time-to-contact only for a condition of 0   x  v(τ) COM 0 COM 0    2 0  0        instability, whereas a stable condition will   x  vv  x   produce an infinite τ     COM 0 COM COM 0 COM     v  x  0 xCOM vCOM 00 000 COM COMCOM  0v 0  0x  0 never  because x will reach or pass x COM 0 COM   , based on the initial    0 COP   x0COM vvCOM  0x COM 0   vCOM 00 00   positions   COM and COP  and velocities. In other words, the com x 0 COM COM  COM 00 0 COM 00   velocities are   v COM  bined COM and COP not sufficient to overcome the       x COM 0   xCOM vCOM 0  0current xCOM 0  0  0 distance  between 0 the 0COM 0 and vCOM  COP. For Equation 13 to     v  x  COM0 avreal  (unstable   0 xcondition),  0 COM 0  the bracketed quantity COM 0  COM value  0produce vCOM vCOM  xCOM 0 vCOM  0   0   v  x in Equation 13 (see Equation 15),   v v COM 0 COM  0  COM COM v0 

0t

0

0

00 COM 0

 0 vCOM  0 vCOM

Experimental Validation Validation of MOS as defined in Equation 15 was performed using data collected from 14 healthy children (7 male, 7 female; age: 12.5 ± 3.0 years; height: 152 ± 13.8 cm; weight: 48 ± 16.2 kg) at the National Institutes of Health (NIH) Clinical Center and was approved by the NIH Institutional Review Board. Parental written informed consent was obtained for each participant and written assent was obtained from each participant. Three-dimensional lower extremity kinematic and spatiotemporal data were collected at 120 Hz with a 10-camera motion capture system (Vicon, Oxford, UK) using a 34-marker model12 and analyzed using Visual3D software (C-Motion, Germantown, MD, USA). Pelvis COM data were taken from the Visual3D model and foot COP locations, and initial and final contact times were obtained using three walkway force plates (AMTI, Watertown, MA, USA) with data sampled at 1080 Hz and downsampled to 120 Hz. xCOPmax was approximated as the lateral foot marker on top of the foot near the fifth metatarsal. Equation of motion solution and data analyses were performed using MATLAB 7.12.0 (MathWorks, Natick, MA, USA). COM and COP velocities were calculated from displacements by differentiation using a central difference equation, a cubic spline interpolation, and a fourthorder Butterworth filter. Significant differences were determined

740  Terry, Stanley, and Damiano

by plotting means with the 95% confidence intervals and noting significant differences when confidence intervals did not overlap.

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Results A comparison across 14 individuals of mediolateral MOS and MOSmax during walking at self-selected speed revealed a divergence between the two metrics starting at 10–20% of SLS (Figure 2). More importantly, from about 50% of SLS to its completion, MOS is significantly less than MOSmax. This difference indicates a ‘reserve’ between the actual MOS that an individual establishes while walking and the maximum possible MOS that could be established. It also shows that the MOSmax indicates a more strongly stable condition during the latter half of SLS as compared with that indicated by the MOS. In addition, from about 70% of SLS forward, the MOSmax begins to progressively increase, whereas the MOS remains relatively stable, reflecting the medial movement of the actual COP as weight is transferred to the contralateral leg even as the stance foot remains in contact with the ground.

Discussion This expanded and more generalized MOS equation includes vCOP and maintains the original MOS features of simplicity, kinematic

and kinetic representation, and instantaneous measurement. Essentially, the MOS reflects the neuromechanical adjustments made to maintain stability versus the kinematic quantification of body trajectory and foot placement provided by MOSmax. If possible, both MOS and MOSmax should be examined simultaneously, as MOS quantifies the MOS that is maintained during SLS, whereas MOSmax quantifies the maximum MOS that could possibly be established. The comparison of MOS and MOSmax shows that past walking studies that assessed stability using MOSmax9–11,13 may have been limited because the MOSmax is significantly higher than the actual MOS. However, even this MOS equation is limited in several respects. It is still based on an inverted pendulum model, which constrains MOS stability assessment to SLS. In addition, the generalized equation does not include the effects of hip moments, but does now allow for the continuous COP adjustments accomplished primarily through ankle eversion or inversion. In addition, because kinematic data used here was originally collected for a different study, wholebody COM was not available. However, for unperturbed gait, the vertical projection of the mediolateral pelvic COM and whole-body COM will be similar and the same COM trajectory is used for both equations, so the comparative effects should be minimal. For assessing fall susceptibility, the MOS is more likely to detect instability, as its values are generally closer to zero (Figure 2) and more sensitive to sudden changes in support or body dynamics. As opposed to MOSmax, the MOS reflects COP adjustments

Figure 2 — Comparisons of left- and right-leg MOS and MOSmax means (heavier lines) and confidence intervals (shaded areas) show that the MOS was generally smaller than MOSmax and significantly smaller for the second half of single limb support (SLS).

A New Perspective on Walking MOS   741

made to accommodate constantly updated sensory input related to maintaining stability. This feature will be especially useful for examining responses during discrete perturbations and to compare stability in individuals with normal, perturbed, and pathological gait. However, the true usefulness of the MOS as a stability metric must still be established. Acknowledgement The work presented here was funded in part by the intramural research program at the NIH Clinical Center in Bethesda, MD (Protocol 10-CC0073-Physical, Functional, and Neural Effects of Two Lower Extremity Exercise Protocols in Children with Cerebral Palsy).

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