The equations of motion for a standing human reveal three mechanisms for balance

ARTICLE IN PRESS

Journal of Biomechanics 40 (2007) 451–457

Short communication

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The equations of motion for a standing human reveal three mechanisms for balance At L. Hof Laboratory of Human Movement Analysis, Center for Rehabilitation, University Medical Center Groningen, Center for Human Movement Sciences, University of Groningen, PO Box 196, 9700 AD Groningen, The Netherlands Accepted 20 December 2005

Abstract The equations of motion for a standing multi-segment human model are derived. Output quantity of these equations is the horizontal acceleration of the whole-body centre of mass (CoM). There are three input terms and they can be identified as the three mechanisms by which balance can be maintained: (1) by moving the centre of pressure with respect to the vertical projection of the CoM, (2) by counter-rotating segments around the CoM, and (3) by applying an external force, other than the ground reaction force. For the first two mechanisms the respective contributions to CoM acceleration can be obtained from force plate recordings. This is illustrated by some example data from experiments, which show that the contribution from mechanism 2 can be considerable, e.g. in one-legged standing. r 2006 Elsevier Ltd. All rights reserved. Keywords: Balance; Balance strategy; Equations of movement; Centre of mass; Centre of pressure; Balance mechanisms

1. Introduction A current and widely used model for human balance is the ‘inverted pendulum’ (IP) model (Winter, 1995a, b). The basic idea of this model is schematically drawn in Fig. 1a. The whole body has a centre of mass (CoM), the position and velocity of which are crucially important for balance (Hof et al., 2005). A ground reaction force (GRF) FG acts on the body, the point of attack of which is called the centre of pressure (CoP). In the IP model the body is modelled as a stick put on the ground at the CoP, with a mass at the CoM, which falls to the right as soon as the CoM is to the right of the CoP, and vice versa. The mechanism by which balance is maintained in this unstable situation, is that the CoP can be moved to some extent by means of muscle action, in such a way Tel.: +31 50 3632645; fax: +31 50 3633150.

E-mail address: [email protected]. URL: http://www/ihms.nl. 0021-9290/$ - see front matter r 2006 Elsevier Ltd. All rights reserved. doi:10.1016/j.jbiomech.2005.12.016

that the CoM is driven to remain within safe boundaries. In the following we will call this action: moving the CoP. To execute this mechanism, several synergies or ‘strategies’ are available, the ankle strategy, the load– unload strategy and the stepping strategy (Horak and Nashner, 1986; Winter, 1995a). In spite of its simplicity, the IP model has been remarkably successful in many applications (Winter et al., 1996, 1997; Gage et al., 2004). Nevertheless, a number of situations remain in which the IP model is obviously not applicable, or only applicable with ad-hoc modifications. Examples are standing or walking with a support (Maki et al., 2003) or situations in which arm or trunk motions are involved (Allum et al., 2002; Otten, 1999). The purpose of this paper is to give a formulation of the equations of motion for balance in a very general form. This formulation will show that next to the ‘moving the CoP’-mechanism, at least two more mechanisms can be discriminated. It will also be shown,

ARTICLE IN PRESS A.L. Hof / Journal of Biomechanics 40 (2007) 451–457

452

mi ME rE ri rCoM

Nomenclature The co-ordinate system is according to the ISB recommendations: X-axis forward, Y-axis vertically upward, Z-axis to the right. aCoM acceleration whole-body CoM horizontal components aCoMx and aCoMz ai acceleration of body segment i CoM whole-body CoM CoM0 vertical projection of CoM on the ground CoP CoP, point of attack of GRF FE external force FG GRF g acceleration of gravity (0 –9.81 0)T h leg length, trochanteric height H_ time derivative of angular momentum w.r.t. CoM, see (3) H_ x component of H_ in YZ-plane (medio-lateral) _ of in XY-plane (sagittal) H_ z component H Ibody moment of inertia of whole body w.r.t. CoM Ii moment of inertia of segment i w.r.t. segment CoM l effective pendulum length m body mass

rCoM0 r0 CoM rCoP

mass of segment i (rErCoM0 )  FE ¼ moment of FE w.r.t. CoM0 point of attack of external force position CoM of segment i position whole-body CoM ¼ ðxCoM ; yCoM ; zCoM ÞT position COM0 ¼ (xCoM, 0, zCoM)T position of equivalent CoM, corrected for pendulum length l position CoP ¼ ðxCoP ; 0; zCoP ÞT

xa

Ex Þ l ðF GxFþF Gy

xe

Ez M F Gy

xh

þ FHGyz

_

xCoM, xCoP, etc. See rCoM, rCoP Ez Þ za l ðF GzFþF Gy ze

Ex þM F Gy

_

zh  FHGyx zCoM, zCoP, etc. See rCoM, rCoP aCoM angular acceleration of pendulum ai angular acceleration of segment i

• H

FGy

(a)

CoP

maxz

max

(b)

CoM′

CoM′

ME

CoM

(c)

FE

CoM•

Fig. 1. Diagrams illustrating the three mechanisms for balance. (a) mechanism 1: ‘moving the CoP’, (b) mechanism 2: ‘counter-rotation’, (c) mechanism 3: ‘applying an external force’.

ARTICLE IN PRESS A.L. Hof / Journal of Biomechanics 40 (2007) 451–457

that the contribution from each of these mechanisms can be quantified on the basis of force plate recordings. This will be illustrated by some experiments.

2. Theory

The IP model describes the movement of a single segment. In this paper we will extend the balance model to cover a number of rigid segments, connected at joints. The number of segments can be left unspecified, which makes the model very general. For the present purpose we will consider the complete human body as the system of interest and analyse the effect of external forces, of which three categories can be discerned: gravity forces which act on the CoM of each segment, the GRF FG and possibly some other external force FE. Using d’Alembert’s principle the moment equation can be given for such a model (Hof, 1992). The moments are calculated with respect to the CoM0 , the vertical projection of the CoM on the ground.1 ðrCoP  rCoM0 Þ  F G þ ðrE  rCoM0 Þ  F E X  þ ðri  rCoM0 Þ  mi g i

¼

X

 X ðri  rCoM0 Þ  mi ai þ I i ai

i

ð1Þ

i

(For meaning of symbols see nomenclature.) On the lefthand side of this equation are the moments of the external forces, on the right-hand side the outcome: the acceleration of segments. We will now decompose the positions and accelerations of all segments, as used in (1) into positions/accelerations of- and positions/ accelerations with respect to the CoM. The result is ðrCoP  rCoM0 Þ  F G þ ðrE  rCoM0 Þ  F E ¼ ðrCoM  rCoM0 Þ  m aCoM þ I body aCoM þ H_

ð2Þ

in which X  H_ ¼ ðri  rCoM Þ  mi ðai  aCoM Þ i

þ

X

I i ðai  aCoM Þ

the point around which to calculate the moments, the gravity term has disappeared from (2), although it is still implicitly contained in the GRF. It is now _ instructive to take the H-term in (2) to the left-hand side, so that only terms related to CoM acceleration remain on the right: ðrCoP  rCoM0 Þ  F G  H_ þ ðre  rCoM0 Þ  F E ¼ ðrCoM  rCoM0 Þ  m aCoM þ I body aCoM .

2.1. Equations of motion—moment equation

ð3Þ

i

i.e. the derivative of the total angular momentum H with respect to the CoM is equal to the moment due to the accelerations of all segments with respect to the CoM. In _ principle, this H-term is quite complicated, as it consists of the sum of the linear and angular accelerations of all segments, multiplied by the perpendicular distance to the CoM. Thanks to the convenient choice of CoM0 as 1 It seems more logical to use the CoP as the point with respect to which the moments are determined. As it should, the result is identical, but it turns out that the derivation is more complicated.

453

ð4Þ

According to theory (Prentis, 1979) the inertia of a compound pendulum, with mass m, length lc and moment of inertia I w.r.t. the CoM, is mechanically identical to a mathematical pendulum with the same mass, but a slightly longer length l ¼ l c þ I=ml c . In this way, the terms rCoM  maCoM þ I body aCoM can be taken together, to form r0 CoM  maCoM, in which r0 CoM is a point just a little (about 10%) above the actual whole-body CoM (Kodde et al., 1979; Massen and Kodde, 1979). In the following we will use this equivalent length and no longer include the rotation term IbodyaCoM. Next to this it can be noted that the vector (rCoPrCoM0 ) is horizontal and (r0 CoMrCoM0 ) vertical. When the moment related to the external force ðrE  rCoM0 Þ  F E is denoted as ME, the scalar versions of (4) become ðxCoP  xCoM ÞF Gy  H_ z þ M Ez ¼ lmaCoMx ,  ðzCoP  zCoM ÞF Gy  H_ x þ M Ex ¼ þlmaCoMz .

ð5Þ

2.2. Three mechanisms for balance control The Eqs. (4) and (5) have the form of three ‘input’ terms on the left-hand side and one ‘output’ term, containing CoM acceleration, on the right-hand side. The three terms on the left-hand side can be identified with three mechanisms by which balance can be maintained. In the first term, mechanism 1, we recognise the ‘move the CoP’-mechanism well known from the standard IP model, Fig. 1a. Next to this mechanism, two more can be discerned. One can identified with _ the H-term. It is especially seen in situations where the base of support available to the CoP’ is insufficiently wide to accommodate sufficient displacement of the CoP. In such cases it is seen that parts of the trunk are rotated with respect to the CoM (Otten, 1999). Due to the conservation of angular momentum, this means that the whole body rotates in the opposite direction, Fig. 1b. We may call this mechanism 2: counter-rotation. The ‘hip strategy’ (Horak and Nashner, 1986) belongs to this mechanism, as well as the arm and leg motions that are seen when balancing on narrow supports. Finally, mechanism 3 can be defined: applying an external force, as when the subject leans against a wall or holds on to a handrail, Fig. 1c.

ARTICLE IN PRESS A.L. Hof / Journal of Biomechanics 40 (2007) 451–457

454

2.3. Equations of motion—force equation

xa

The force equation for the same set of connected segments is X X FG þ FE þ mi g ¼ m i ai . (6) i

i

CoM

xh+xe

. H

y l

FGy

FG

FE

Applying the definition of the whole-body CoM, this reduces to: F G þ F E ¼ mðaCoM  gÞ.

FGx

(7)

x

It is seen that (4) and (7) give two different expressions for the acceleration of the whole-body CoM. When only the GRF and gravity are present as external forces, these expressions can be used to calculate CoM0 position from forceplate data only, CoP position and horizontal forces (Zatsiorsky and King, 1998; Hof, 2005).

CoP

CoM′

Fig. 2. Explanation of Eq. (10). When only mechanism 1 is active (the classical inverted pendulum model) the GRF FG is directed toward the whole-body CoM. When mechanisms 2 or 3 are present, this is no longer the case; xa is not equal to (xCoMxCoP). The difference in horizontal distance of the CoM to the line of action of FG equals _ and external force FE corresponding to xh+xe. The directions of H positive xh and xe are shown. For the mediolateral YZ plane the figure is identical.

2.4. Calculation of H_ _ can be calculated by combining (5) and (7) H ðxCoP  xCoM ÞF Gy  H_ z þ M Ez ¼ l ðF Gx þ F Ex Þ,  ðzCoP  zCoM ÞF Gy  H_ x þ M Ex ¼ þl ðF Gz þ F Ez Þ. ð8Þ In this form all terms in the equations are moments. To arrive at distances we divide both sides of (8) by the vertical component of the GRF, to give H_ z M Ez ðF Gx þ F Ex Þ  ¼l , F Gy F Gy F Gy H_ x M Ex ðF Gz þ F Ez Þ zCoM  zCoP  þ ¼l . F Gy F Gy F Gy

xCoM  xCoP þ

ð9Þ

All terms in both equations are in length units. For sake of clarity they can be expressed as xCoM  xCoP þ xh þ xe ¼ xa , zCoM  zCoP þ zh þ ze ¼ za

ð10Þ

in which H_ z ; F Gy H_ x zh ¼  ; F Gy

xh ¼ þ

xe ¼ 

M Ez ; F Gy

xa ¼ l

ðF Gx þ F Ex Þ , F Gy

ze ¼ þ

M Ex ; F Gy

za ¼ l

ðF Gz þ F Ez Þ . ð11Þ F Gy

The different plus and minus signs for x and z are due to the fact that rotations have different signs in the XY and the ZY plane. A geometrical interpretation of (10) can be seen in Fig. 2. When balance is only maintained by mechanism 1, the line of action of the GRF always runs through the CoM. When mechanism 2 or 3 is working in addition, this is no longer the case. The distance between CoM

and GRF line of action, measured at the level of the CoM, equals xh þ xe . As a by-product of this analysis, Eq. (10) gives a simple method to find xh or zh from force plate data alone, for the case when there are no external forces apart from GRF and gravity: xh ¼ ðxCoP  xCoM Þ þ xa , zh ¼ ðzCoP  zCoM Þ þ za .

ð12Þ

Note that in this procedure CoM position has to be calculated beforehand, but this can be done with forceplate data as well. If an external force is present, this force and its position relative to the CoM0 are to be measured, in order to obtain ME and xe and then a similar procedure based on (10) can be followed.

3. Experiment In the following some experimental data will be presented to show that in several common balance situations, mechanism 2 is very relevant. Next to this, the forceplate method (10) can so be demonstrated. Experiments will be presented of a 21 year male subject, body mass 88.5 kg, stature 1.93 m and leg length 1.05 m. CoP position and GRF are measured with a Bertec type 4060 forceplate, converted by a 16-bit A/D converter and processed on a PC with Matlab. The conditions are: (1) standing quietly on two feet, (2) standing on one foot, (3) tilting from left to right, in which weight is shifted alternately from the left to the right foot, while the body is kept as stiff as possible, (4) rotating the trunk from left to right, the feet remaining in place, (5) the arms extended sideways

ARTICLE IN PRESS A.L. Hof / Journal of Biomechanics 40 (2007) 451–457 one-legged standing

2

2

1.5

1.5

1

1

0.5

0.5

(cm)

(cm)

two-legged standing

455

0 -0.5

-0.5

zCoP zh

-1

-1

zCoM

-1.5

0

-1.5

za

-2

-2 10

12

(a)

14 16 time (s)

18

20

20

22

(b)

tilting L/R

24 26 time (s)

28

30

28

30

28

30

moving trunk L/R 10

30

8 20

6 4

10 (cm)

(cm)

2 0

0 -2

-10

-4 -6

-20

-8 -30

20

22

(c)

24 26 time (s)

28

-10 20

30

22

(d) 5

4

4

3

3

2

2

1

1

0

-1

-2

-2

-3

-3

-4

-4 20

(e)

0

-1

-5 22

24 26 time (s)

28

-5

30

(f)

bar

standing on bar

5

(cm)

(cm)

moving arms

24 26 time (s)

20

22

24 26 time (s)

Fig. 3. Data on a number of balance situations. CoM position zCoM thick drawn line, CoP position zCoP thin drawn line, za thin dashed line, zh thick dashed line. (a) Quiet standing on two feet, eyes open, (b) Standing on one foot, (c) Tilting, weight shifting from left to right foot, which were 40 cm apart, (d) Subject stood with feet wide apart and rotated his trunk from left to right, (e) Subject stood erect, arms were sideways and moved up and down in counter phase, and (f) Subject stood with one foot lengthwise on a wooden bar of 4  4 cm. In Figs. e and f the lines for za and zh, and for zCoP and zCoM, are closely together, indicating that mechanism 2 preponderated.

swaying up and down, in counterphase, and (6) standing with one foot lengthwise on a 4 cm wide wooden bar. Experiments 1–5 lasted 60 s, experiment 6 for 30 s. All data refer to medio-lateral movements. CoM position is calculated from both the CoP and the horizontal force data (Hof, 2005) and (12) has been used for the

calculation of zh. Fig. 3a–f shows 10-s sections of the four signals, Table 1 gives peak and root-mean square (r.m.s.) values of za, (zCoP– zCoM) and zh. Peak and r.m.s. errors in CoP position and the horizontal component of the GRF are calculated on the basis of a recording of the unloaded forceplate.

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456 Table 1 Results Test

max. (cm) za

Standing 2 legs Standing 1 leg Tilting Trunk sway Arms sway Standing on bar Error

0.4421 1.7023 14.3000 10.5839 2.4733 3.7450 0.0650

zCopzCom 0.2295 1.0845 12.2860 5.7025 1.1765 0.8884 0.0350

rms zh

za

0.2732 1.3542 3.9887 6.2202 1.9535 3.1789 0.1000

0.0831 0.4186 8.2661 4.5015 0.9022 0.9956 0.0200

zCopzCom 0.0739 0.3173 7.3442 2.0954 0.3558 0.2989 0.0070

zh

%c

%h

0.0647 0.2057 1.0768 2.6006 0.8944 0.8981 0.0300

(60)* 62 89 43 9 12

(41)* 34 15 57 91 87

Results for one subject, male, 21 year, stature 1.93 m, body mass 88.5 kg. Medio-lateral movement. For each test the maximum and r.m.s. values of za, (zCoP– zCoM) and zh are given. The last two colums give the percentual contribution of mechanisms 1 and 2, respectively, to the total horizontal CoM acceleration, calculated by Eq. (13). Error levels in za and zCoP are calculated on the basis of the noise with unloaded force plate. Error of zh is the sum of these two. Possible errors in CoM have not been included. Entries with (*) are inaccurate, because the signal amplitudes were too close to the error levels.

4. Discussion The simple relation (10) is based on a very general multi-segment human body model, in which even the number of segments is left unspecified. In fact the only assumption refers to the relation between ‘real’ CoM height and the effective length of the compound pendulum. From model studies it turns out that length l depends on the mode of movement. For foreaft movements (XY plane) Geursen et al. (1976) found (a) l ¼ 1:20 and (b) 1.24 h, for the cases (a) in which the trunk is kept vertical and (b) rotates in line with the legs, respectively. For left-right movements Kodde et al. (1979) found l ¼ 1:34 h. With the same mass distribution the real CoM is at a height of 1.10 h. The data in Table 1 suggest that in many postures and actions balance mechanism 2 cannot be neglected. In standing on one leg, in our subject the proportion mechanism 1: mechanism 2 was about 2:1. In standing on a narrow base mechanism 2 is dominant; remarkably high and fast fluctuations in zh are shown in Fig. 3f. These correspond to the violent trunk, arm and free leg movements that can be observed in this situation. The ‘tilting’ is a kind of left–right stepping movement, in which mechanism 1 is dominant, Fig. 3c. Although not small in an absolute sense, zh is much less than ðzCoP  zCoM Þ in this case. The trunk and arm motion, Figs. 3d and e, give an indication of the possible range of zh:74 cm (peak) for trunk motion and 71 cm for the arm swing. The proportion of zh with respect to the total acceleration term za was expressed in Table 1 as the linear regression coefficient   mean yh ðtÞya ðtÞ   %h ¼ 100% (13) mean y2a ðtÞ

and similarly for (zCoPzCoM). Although care was taken to use a high amplification for the horizontal forces, the accuracy in CoP position and horizontal force (related to za) was hardly sufficient for the two-legged quiet standing situation, the more so as zh is the difference between these two inaccurate quantities. In this case the percentage values cannot be relied upon. In principle all relevant quantities, positions and accelerations, can be obtained from kinematic data as well. When comparing the present, very general, equations of movement (5) with those of the standard IP model, the correspondence is evident. This may explain why such a simple model performs so well (Gage et al., 2004). Two differences emerge. The major difference is that in the standard IP model CoP position xCoP is the only input (mechanism 1 only). In the present model the input term has been replaced by (xCoPxhxe) to accommodate mechanisms 2 and 3 in addition. The minor difference is that the eigen (angular) frequency of the IP is now rffiffiffiffiffiffiffiffiffi F Gy o0 ¼ (14) ml which may be time-varying, but is on the average equal to the ‘classical’ value rffiffiffiffiffiffiffi rffiffiffi mg g ¼ . (15a) o0 ¼ ml l In a previous analysis of stability (Hof et al., 2005) the usual IP model was used. To accommodate the present results, this means that around the usual base of support, which is determined by the possible positions of the CoP, a wider area can be delimited in which balance can be maintained when mechanism 2 is applied. The width of this region will be determined by the achievable range of xh. The results in Table 1 and Fig. 3d suggest that this additional area is over 4 cm

ARTICLE IN PRESS A.L. Hof / Journal of Biomechanics 40 (2007) 451–457

wide. In a more challenging situation, balancing on a ridge of only 4 mm, some subjects of Otten (1999) even reached 6 cm. In the same paper it can also be seen how H_ is generated by various joint moments.

Acknowledgement Mireille Bolster, Hedwig van der Sluis and Michiel Nagel are thanked for conducting the reported experiments. References Allum, J.H.J., Carpenter, M.G., Honegger, F., Adkin, A.L., Bloem, B.R., 2002. Age-dependent variations in the directional sensitivity of balance corrections and compensatory arm movements in man. Journal of Physiology 542, 643–663. Gage, W., Winter, D., Frank, J., Adkin, A., 2004. Kinematic and kinetic validity of the inverted pendulum model in quiet standing. Gait and Posture 19, 124–132. Geursen, J.B., Altena, D., Massen, C.-H., Verduin, M., 1976. A model for the description of the standing man and his dynamic behaviour. Agressologie 17, 63–69. Hof, A.L., 1992. An explicit expression for the moment in multi-body systems. Journal of Biomechanics 25, 1209–1211. Hof, A.L., 2005. Comparison of three methods to estimate the center of mass during balance assessment (Letter to the Editor). Journal of Biomechanics 38, 2134–2135.

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