The ‘extrapolated center of mass’ concept suggests a simple control of balance in walking

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Human Movement Science 27 (2008) 112–125 www.elsevier.com/locate/humov

The ‘extrapolated center of mass’ concept suggests a simple control of balance in walking At L. Hof

*

Laboratory of Human Movement Analysis, Center for Rehabilitation, University Medical Center Groningen, The Netherlands Available online 1 November 2007

Abstract Next to position x and velocity v of the whole body center of mass (CoM) the ‘extrapolated center of mass’ (XcoM) can be introduced: n = x + v/x0, where x0 is a constant related to stature. Based on the inverted pendulum model of balance, the XcoM enables to formulate the requirements for stable walking in a relatively simple form. In a very simple walking model, with the effects of foot roll-over neglected, the trajectory of the XcoM is a succession of straight lines, directed in the line from center of pressure (CoP) to the XcoM at the time of foot contact. The CoM follows the XcoM in a more sinusoidal trajectory. A simple rule is sufficient for stable walking: at foot placement the CoP should be placed at a certain distance behind and outward of the XcoM at the time of foot contact. In practice this means that a disturbance which results in a CoM velocity change Dv can be compensated by a change in foot position (CoP) equal to Dv/x0 in the same direction. Similar simple rules could be formulated for starting and stopping and for making a turn.  2007 Elsevier B.V. All rights reserved. PsycINFO classification: 2330; 4140 Keywords: Stability; Control; Gait; Locomotion; Robotics

*

Present Address: Center for Human Movement Sciences, University of Groningen, P.O. Box 196, 9700 AD Groningen, The Netherlands. Tel.: +31 50 363 2645; fax: +31 50 363 3150. E-mail address: [email protected] URL: http://www/ihms.nl 0167-9457/$ - see front matter  2007 Elsevier B.V. All rights reserved. doi:10.1016/j.humov.2007.08.003

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Nomenclature bx bz CoM CoP g k1 l sc sn t Tc Tn ux uxn uz, uzn vx, vz vxn, vzn w wc x, z xn, zn XcoM DT Dvx r(Æ) n, f nn, f n x0

offset in forward direction offset in lateral direction center of mass, coordinates (x, y, z) center of pressure, coordinates (ux, 0, uz) acceleration of gravity = 9.81 m s1 gain of proportional control lateral movement effective pendulum length desired step length actual step length of nth step time desired step duration duration of nth step forward CoP position constant forward CoP position during step n lateral CoP position CoM velocity in forward/lateral direction CoM velocity at the time of foot placement of step n step width desired step width forward/lateral position of CoM position of CoM at beginning of step n extrapolated center of mass, coordinates (n, 0, f) difference between actual step time and Tc difference between actual velocity and steady-state value standard deviation of noise in (Æ) forward/lateral position of XcoM forward/lateral position of XcoM at beginning of step n pffiffiffiffiffiffiffi eigenfrequency of pendulum ¼ g=l

1. Introduction Two central concepts in the study of balance are the center of mass (CoM) and the center of pressure (CoP) (Winter, 1995a, 1995b). The force of gravity attacks at the CoM and the ground-reaction force at the CoP. The small horizontal distance between CoP and the CoM projection on the ground produces a destabilizing moment that has to be controlled by a timely displacement of the CoP. These concepts originate from the ‘inverted pendulum’ model of balance (Fig. 1), which gives a good description of balance mechanics, simple as it is (Hof, 2007). In a previous paper (Hof, Gazendam, & Sinke, 2005), we proposed to add a third concept to CoM and CoP, for which we proposed the name ‘extrapolated center of mass’ (XcoM). When the projection of the CoM position on the ground is denoted as x and CoM velocity as vx, XcoM position n is defined as

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CoM mg l

-Fy y

x x ux CoM’ CoP

Fig. 1. Schematic diagram of the inverted pendulum model. The inertia of the body is represented by a single mass m, balancing at the CoM on a stick of length l. CoM 0 is the vertical projection of the CoM on the ground. Because CoM 0 and CoP are not at the same horizontal position, a moment (x  ux)Fy is exerted with respect to CoM 0 . This leads to a horizontal acceleration of the CoM, with lmax = (x  ux)Fy.

vx ; ð1Þ x0 pffiffiffiffiffiffiffi in which x0 ¼ g=l is the eigenfrequency of the (inverted) pendulum (see Nomenclature). It has been shown that the relative positions of CoP and XcoM can predict balance. This n¼xþ

v/ ω

0

0.5

0

-0.5 -0.5

0

0.5

x - ux Fig. 2. Phase diagram of the inverted pendulum model (2) for the case that ux is constant. Trajectories are 2 hyperbolas ðx  ux Þ2  xvx0 ¼ constant (Bottaro et al., 2005). At initial condition (x  ux) = 0.3 two very different trajectories can be followed, depending on the initial velocity. With vx/x0 = 0.28 (dashed line) velocity decreases and becomes negative, while position does not reach x = u, but returns to the initial value. When vx/ x0 = 0.32 (solid line) velocity decreases also, but increases afterward. Position crosses x = ux and reaches the symmetrical position (x  ux) = +0.3 at the initial velocity.

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can nicely be illustrated by the phase diagram for the inverted pendulum model (Bottaro, Casadio, Morasso, & Sanguineti, 2005), Fig. 2. This diagram is valid for the case that CoP position ux is constant. When the initial value of jnj < ux, i.e., when jvx/x0j < j(ux  x)j, the CoM will not reach the CoP, but slow down and eventually return with opposite velocity, left and right branches of the phase diagram. As soon as jnj > ux the upper or lower branches in the phase diagram will be followed, velocity will decrease, but not reach zero and increase again. In this case, the CoM will continue its course in the same direction. When exactly n = ±ux (dotted lines in Fig. 2), the CoM would in principle come to a standstill right above the CoP. This is a hypothetical case, however, because the fixed point (x  ux) = 0, vx = 0 is unstable. The special role of the XcoM in the inverted pendulum model suggests the application of this concept to investigate stability in walking. The first problem is how to define stable walking. In this paper, we will confine ourselves to the following requirements of walking stability. (1) The walker should not fall down, i.e., the CoP should not get beyond the reach of foot placement (Hof et al., 2005). (2) The walker should be able to follow a well-defined course (straight or curved). (3) Forward speed should not increase or fluctuate too much, e.g., stay within 20% of the reference value. (4) Step length should not become too large, e.g., not greater than leg length. In the following, we will present some ideas to this effect. It will be shown that the XcoM concept suggests some simple control strategies to achieve a stable walking gait. 2. Theory In the inverted pendulum (IP) model, the body is represented by a single mass m supported by a massless leg with length l (Fig. 1). For the horizontal acceleration of the CoM position x, the following holds (Hof, 2007; Hof et al., 2005) d2 x ¼ x20 ðx  ux Þ; ð2Þ dt2 in which ux is the CoP position. By differentiating the XcoM definition (1) with respect to time we find dn dx 1 d2 x ¼ þ : dt dt x0 dt2

ð3Þ

Eq. (1) can be rewritten as dx ¼ x0 ðx  nÞ: dt

ð4Þ

Inserting (2) and (4) in (3) then gives dn ¼ x0 ðn  ux Þ: dt

ð5Þ

In this way, the second-order differential equation (2) has been replaced by a set of two first order differential equations (4) and (5). In terms of state-space, the IP model can thus be represented by two states: CoM position x and XcoM position n. The two differential equations (4) and (5) look alike, but there is an important difference. Eq. (4) has a minus sign and is therefore stable (Khoo, 2000) (the pole of the transfer function is x0).

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Eq. (5), in contrast, is unstable with pole +x0. This is apparent in the solution for the case the input is a step function ux(t) = uxn for t > 0: nðtÞ ¼ ðnn  uxn Þ ex0 t þ uxn :

ð6Þ

Here, the exponential term is positive, which implies that the difference between n and ux grows exponentially with time. The above equations were in the forward X-coordinate. The same equations apply to the lateral Z-coordinates z, f and uz, respectively (in the ISB coordinate convention (Wu & Cavanagh, 1995)). From the stability of the x–n relation (4) and the instability of the n–ux relation (5), we obtain the following: Property 1. A sufficient condition for stability of the CoM trajectory, is that the XcoM trajectory is stable. What is meant here by ‘stable’ may depend on the context: in a constant direction, with a constant speed, without excessive step lengths or widths, etc. (Hasan, 2005). In Section 1, we proposed some requirements for walking. A second property follows directly from the differential equation (5) and its counterpart in the Z-coordinates. For the direction of the XcoM trajectory it holds: on dn=dt x0 ðn  ux Þ n  ux : ¼ ¼ ¼ of df=dt x0 ðf  uz Þ f  uz

ð7Þ

Property 2. At any point the tangent to the trajectory of the XcoM (n(t), f(t)) runs through the CoP(ux(t), uz(t)). 3. Possibilities for control The properties derived in the preceding section suggest that the control of walking as represented in an IP model can be simplified if the XcoM is controlled, instead of the CoM. We present here a few possibilities. Only the idealized case is considered, in which the CoP (i.e., foot placement) is constant during a step and changes instantaneously between steps, so there is no double contact phase modeled and no roll over of the foot during stance. Some consequences of these approximations will be considered in Section 4. In this case of constant CoP, it follows from Property 2 that the trajectory of the XcoM consists of a sequence of straight lines, Fig. 3a. Furthermore, the solution (6) of the XcoM position applies. The problem is now to find a control law which gives for each step n the step duration Tn and the CoP position (uxn, uzn) of a new step as a function of the XcoM position (nn, fn) at the time of foot contact, in such a way that a stable trajectory of the XcoM is obtained. From Property 1, it then follows that the CoM trajectory is stable as well. Results are presented of simulations on the basis of the equations in the preceding section. Each simulation consisted of 50 steps, starting and ending in standstill, i.e., with velocity zero. It was started with the left leg, therefore steps with n odd are left and with n even are right. The parameter x0 was set at 3.0 s1, corresponding to l = 1.09 m, or a leg length of 0.91 m (Hof et al., 2005). In part of the simulations CoP position and timing were exact, but a disturbance was introduced at step 20. The disturbance could be in

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117

CoP XcoM CoM

4.5 4 3.5 3

x(m)

2.5 2 1.5 1 0.5 0

1

-0.5 -0.05

0

0.05

0.1

z(m) 15

CoM CoP XcoM avg. CoP

14.5 14

x(m)

13.5 13 12.5 12 11.5 11 10.5 10 0.26

0.28

0.3

0.32

0.34

0.36

0.38

0.4

z(m) Fig. 3. (a) Theoretical example of the trajectories of XcoM and CoM, according to Eqs. (4)–(7). The CoP (·) is assumed constant during a step. The XcoM follows a course of broken straight line segments. The directions of the line segments in determined by the line from CoP to the XcoM at foot contact (dotted). The walk is initiated by placing the CoP behind and to the left of the initial CoM/XcoM position (·). CoM trajectory follows XcoM according to (4). (b) Measured trajectories of CoP, XcoM and CoM from a human subject walking at 1.25 m s1. CoP recorded on a treadmill with built-in force transducers (Verkerke et al., 2005). CoP position is not constant during stance, because of foot roll-over, and crosses over during bipedal stance (dashed line with dots every 0.1 s). Average CoP position in stance has been indicated with (·). Straight-line extensions of XcoM trajectory are given with dotted lines.

XcoM, which simulated a push given to the walker, or in the CoP, which simulated the situation that foot placement was not free, e.g., due to an obstacle.

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3.1. Forward control Parameters of forward locomotion are step length s, step time T, and forward speed s/T. For forward control, a stable gait requires that these factors will not deviate too much from their average values (see Section 1). Control can in principle be achieved either by adapting T or by adapting s. In the simulations, the setpoints were Tc = 0.5 s, sc = 0.75 m, thus an average speed of 1.5 m s1. To begin with, the case without feedback, thus keeping both T and s constant, results in an unstable gait. As soon as the slightest disturbance is encountered, the distance between CoM and CoP increases exponentially, clearly an undesirable situation. Control of step time T was achieved by putting Tn ¼

  1 s ln þ1 ; x0 nn  uxn

ð8Þ

which follows from (6) and keeping step length s = sc = uxn  ux(n1) constant. This indeed results in a continuous gait. There are two problems, however. Locomotion does not start spontaneously from standstill and a disturbance in XcoM or CoP results in a persistent change of step time, and thus of speed, see Fig. 4. The simplest stable control of CoP position could be made by positioning the CoP at a constant distance behind the XcoM ‘constant offset control’: uxn ¼ nn  bx ;

ð9Þ

T controlled, push +20 cm 3 steplength (m) step time (s) speed (m/s)

2.5

2

1.5

1

0.5

0 0

10

20

30

40

50

step # Fig. 4. Step length (s) in m, step time (þ) in s and speed (·) in m s1 for a simulation in which step length is kept constant at 0.75 m and step time is controlled. At step 20 a forward push of 0.2 m was added to the XcoM. This results in a permanently lower step time and increased speed.

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119

in which the offset bx was a constant distance, calculated from (6) as bx ¼

sc : ex0 T c  1

ð10Þ

In the numerical example given bx was 21.5 cm. This simple control law results in a completely stable gait. A disturbance in nn is immediately reacted upon by an equal change in uxn, so step length sn also increases by this amount. The subsequent step lengths are constant again, Fig. 5a. A disturbance in CoP, e.g., a shorter step than according to (9), is compensated by one longer step to follow, Fig. 5b. Similarly, a long step is compensated by one short step. Again, step length and speed are constant after the disturbed steps. The extra long or short steps result in a forward or backward displacement with respect to the ideal constant speed and this extra displacement is not compensated in subsequent steps. It is questionable, however, if such a compensation would occur in real life. The effect of a disturbance Dux in ux can directly be calculated from (6) and (9). If the disturbance is applied at step n, the resulting step lengths are sn ¼ sc þ Dux

and

snþ1 ¼ sc  ðex0 T c  1ÞDux :

ð11Þ

With the adopted values for x0 and Tc the factor ðex0 T c  1Þ amounts to 3.48. The correction in step n + 1 is thus considerably larger than the disturbance itself. For the case that every step is somewhat disturbed by a random and uncorrelated error with SD r(u), the calculation in (11) can serve to show that the standard deviation r(s) of sn equals qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 ð12Þ rðsÞ ¼ rðuÞ 12 þ ðex0 T c  1Þ : In our example the multiplication factor is 3.62. This was in good agreement with simulations in which a random error was added to the ‘correct’ value of uxn according to Eq. (9). In a similar way the effect of a deviation DT from the correct step time Tc can be calculated:   x0 DT ; ð13Þ sn ¼ sc 1 þ ð1  ex0 T c Þ and the result of noise r(T) in T: sc x0 rðsÞ ¼ rðT Þ: 1  ex0 T c

ð14Þ

3.2. Lateral control In Section 1, we required for a stable lateral control in walking that the subject could walk a straight course and would not fall. The latter requirement implies that the difference between CoP and CoM (or XcoM) should not exceed anatomical constraints. A stable gait could in this case also be achieved with a constant offset control bypositioning the left foot to the left, and the right foot the same distance bz to the right of the XcoM: n

uzn ¼ fn þ ð1Þ bz ;

ð15Þ

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A.L. Hof / Human Movement Science 27 (2008) 112–125

a

3 steplength (m) step time (s) speed (m/s)

2.5

2

1.5

1

0.5

0 0

10

20

30

40

50

step #

b

short step 65 cm 3 steplength (m) step time (s) speed (m/s)

2.5

2

1.5

1

0.5

0 0

10

20

30

40

50

step # Fig. 5. Step length (s) in m, step time (+) in s and speed (·) in m s1 for a simulation with ‘constant offset control’: CoP is positioned always a constant distance behind the XcoM. Step time is constant at 0.5 s. (a) At step 20 a forward disturbance of 0.2 m in the XcoM is given. The reaction is one 0.2 m longer step. (b) In a different simulation run, step 20 is made 0.65 instead of 0.75 m. The reaction is that step 21 is 0.75 + 0.35 = 1.10 m, see Eq. (11).

in which bz is related to the desired stride width wc as wc : bz ¼ x T c 0 e þ1

ð16Þ

With the example data and a stride width of 10 cm, bz is 1.82 cm. This control law results in a stable gait with a stride width close to wc. A disadvantage is that after a disturbance

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the path is still in the right direction, but is shifted sideways, Fig. 6a. While the similar effect in forward control is probably not serious, in lateral control it may be undesirable, especially on treadmills or narrow pathways. It can be remedied by including a proportional control, with the ‘middle of the road’ xc as the setpoint:

a

lateral simulation with CoM 0.2 0.15

CoP XcoM CoM

+ 3 cm

0.1

z(t)

0.05 0

-0.05 -0.1 -0.15 -0.2 5

b

10 t(s)

15

lateral simulation with CoM 0.2 CoP XcoM CoM

0.15 + 3 cm 0.1

z(t)

0.05 0

-0.05 -0.1 -0.15 -0.2 5

10 t(s)

15

Fig. 6. CoP, XcoM and CoM as a function of time in simulations of lateral control. wc = 0.1 m. (a) Constant offset control. The CoP is positioned a constant distance left/right of the XcoM at the time of foot placement. In step 5 the CoP is put 3 cm too much to the right. This is corrected in the next step by a greater leftward CoP placement. The leftward deviation from the straight course is not corrected. (b) Offset plus proportional control, with k1 = 0.1 and wc = 0.1 m. The resulting stridewidth is now 15.6 cm. Same disturbance as in (a). The deviation from the ‘middle of the road’ z = 0 is now corrected in a few steps.

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A.L. Hof / Human Movement Science 27 (2008) 112–125 n

uzn ¼ fn þ ð1Þ bz þ k 1 ðfn  xc Þ:

ð17Þ

Such an ‘offset plus proportional control’ results in a stable control in which the CoP tends to alternate left and right of xc. The steady-state stride width now depends also on k1: w ¼ bz

ex0 T c þ 1 : 1  k21 ðex0 T c  1Þ

ð18Þ

2-D simulation with CoM 17 16.5 16

x(m)

15.5 15 14.5 14

CoP XcoM CoM

13.5 13 -1

0

1

2

3

4

z(m) 16 15.5 15

x(m)

14.5 14 13.5 13

CoP XcoM CoM

12.5 12 -1

0

1

2

3

4

z(m) Fig. 7. Simulations of walking turns. (a) Three turns of 30 each. (b) A 90 turn in a single step. Without the turn, the CoP would have been positioned at the square (h) at the first step of the turn.

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It should hold that k1 P 0, otherwise the deviation of the straight path steadily increases. It should also hold that k 1 < 2=ðex0 T c  1Þ, otherwise step width increases exponentially. An example, with k1 = 0.1 is shown in Fig. 6b. 3.3. Start and stop walking In constant offset control the start of walking is provided for. In standing CoM, XcoM, and CoP are all closely together. Initiation of walking is achieved when the CoP is moved backward. As a result the XcoM moves forward, followed by the CoM, Fig. 3a, first step. To stop walking, the CoP should be placed just at the XcoM. This therefore requires a step longer than average, or a shorter step time. 3.4. Walking in turns Walking in turns can simply be achieved by rotating the x- and z-offsets over an angle u: 

ux uz



¼

cosðuÞ

sinðuÞ

 sinðuÞ cosðuÞ



bx n

ð1Þ bz

 :

ð19Þ

Fig. 7a shows a simulation with a 90 turn in three steps, Fig. 7b shows a military parade style 90 turn in a single step. 4. Discussion The theory as given here can be divided in two parts with different status. The first part, Eqs. (1)–(5) and (7), is based on the inverted pendulum model of walking and on the definition of the XcoM, but within these bounds it can be considered very generally valid. It has been shown (Hof, 2007) that the IP model is widely applicable for human balance, the more so when effects of segment movements with respect to the body CoM, e.g., arm movements and hip strategies, are taken into account. Section 3, on the other hand, just serves to show that simple ‘constant-offset’ or ‘offset plus proportional’ control laws are sufficient to achieve stable gait. This may be useful information for the construction of walking robots (Collins, Ruina, Tedrake, & Wisse, 2005), but whether these laws are also followed in human locomotion, can only be investigated experimentally. It has been shown, to give a counterexample, that after stumbling most subjects do not take one (very) long step, keeping step time constant, but make a number of quick short steps (Forner Cordero, Koopman, & Van der Helm, 2003). Control of forward walking stability in humans by a constant step length/variable timing control law, however, has the shortcomings that gait is not spontaneously initiated and that walking speed is not constant after a disturbance. In this respect, it is equivalent to the ‘rimless wheel’ model of walking (Coleman, Chatterjee, & Ruina, 1997). The model for foot placement in this section is very elementary: the double-contact time is zero and the CoP remains constant during the step. It is in fact more a model of stiltwalking. In human biped walking there is a double contact period during which the CoM can be accelerated or decelerated. Next, the human foot has a considerable base

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of support (Hof et al., 2005) and CoP position follows a path of some 8–10 cm forward and 5 cm medio-laterally during stance, as the foot rolls over. In Fig. 3b, an XcoM trajectory has been drawn as calculated from measured CoP data. It suggests that the model is a sensible approximation if for the ‘constant’ CoP position the average value during single stance is chosen. It has been shown that regulation of the roll-over plays a significant role in lateral balance because it can correct inaccurate or incorrect foot placements (Hof, Van Bockel, Schoppen, & Postema, 2007). It may be expected that a similar mechanism is also effective in controlling forward velocity. The formulas as presented may cast doubt on the possibility whether all relevant calculations can be processed on-line by a walking human. Every step the next foot placement should according to (9) be at vxn uxn ¼ nn  bx ¼ xn þ  bx : ð20Þ x0 First it should be noted that for a subject foot placement u is always relative to his own body CoM x. Moreover, for a steady-state gait (20) reduces to sc ð21Þ uxn ¼ xn þ : 2 When steady-state gait is disturbed and the horizontal velocity at the time of foot placement is changed by an amount Dvxn, the foot should be placed at uxn ¼ xn þ

sc Dvxn þ ; 2 x0

ð22Þ

instead, in the case of constant step time. The foot placement rule is in fact thus very simple: when forward velocity is increased/decreased by an amount Dvxn, the next step should 0.03

CoP - XcoM(m)

0.02 0.01 0 -0.01 -0.02 -0.03 -0.04 0.2

0.25

0.3

0.35

0.4

0.45

XcoM position(m) Fig. 8. Distance between average lateral CoP and XcoM at footfall against XcoM position. Experimental data from a subject walking at 1.25 m s1 (Hof et al., 2007). By linear regression from these data (dashed line) it was found that k1 was 0.12 and 0.14, bz = 1.40 cm and 1.28 cm for the right and left leg, respectively. Correlation coefficients were 0.49 and 0.43.

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be made a distance of Dvxn/x0 longer/shorter. A corresponding rule works for lateral foot placement. Expressed in vector form this means that a disturbance which results in a CoM velocity change Dv can be compensated by a change in foot position (CoP) in the next step equal to Dv/x0 in the same direction. Velocity changes of the CoM can obtained by integrating horizontal CoM acceleration, which may be sensed by the otolith organs or by shear forces on the foot sole (Ting & Macpherson, 2004). A remaining problem is to predict these velocity changes in advance, as foot placement takes a considerable time. A previous study on lateral balance (Hof et al., 2007) gives data that support the offsetplus-proportional control of lateral balance, although the correlations are not really convincing, Fig. 8. The initiation of walking as described by the constant-offset model closely resembles experimental findings, in which the CoP is also moved backward at the start of walking (Winter, 1995a). Published research on straight or curved walking (Gregoire, Papaxanthis, & Schieppati, 2006) unfortunately does not give data on both foot placement and timing. It should be admitted, however, that reliable data on a feedback control system can only be obtained from an experimental set-up in which the control loop is opened in some way, e.g., by applying a known disturbance (Van der Kooij, Van Asseldonk, & Van der Helm, 2005). References Bottaro, A., Casadio, M., Morasso, P., & Sanguineti, V. (2005). Body sway during quiet standing: Is it the residual chattering of an intermittent stabilization process? Human Movement Science, 24, 588–615. Coleman, M., Chatterjee, A., & Ruina, A. (1997). Motions of a rimless spoked wheel: A simple 3D system with impacts. Dynamics and Stability of Systems, 12, 139–160. Collins, S., Ruina, A., Tedrake, R., & Wisse, M. (2005). Efficient bipedal robots based on passive-dynamic walkers. Science, 307, 1082–1085. Forner Cordero, A., Koopman, H. F. J. M., & Van der Helm, F. C. T. (2003). Multiple-step strategies to recover from stumbling perturbations. Gait & Posture, 18, 47–59. Gregoire, G., Papaxanthis, C., & Schieppati, M. (2006). Coordinated modulation of locomotor synergies constructs straight-ahead and curvilinear walking in humans. Experimental Brain Research, 170, 320–335. Hasan, Z. (2005). The human motor control system’s response to mechanical perturbation: Should it, can it, and does it ensure stability? Journal of Motor Behavior, 37, 484–493. Hof, A. L. (2007). The equations of motion for a standing human reveal three mechanisms for balance. Journal of Biomechanics, 40, 451–457. Hof, A. L., Gazendam, M., & Sinke, W. E. (2005). The condition for dynamic stability. Journal of Biomechanics, 38, 1–8. Hof, A. L., Van Bockel, R., Schoppen, T., & Postema, K. (2007). Control of lateral balance in walking. Experimental findings in normal subjects and above-knee amputees. Gait & Posture, 25, 250–258. Khoo, M. C. K. (2000). Physiological control systems. Analysis, simulation, estimation. New York: Wiley. Ting, L., & Macpherson, J. (2004). Ratio of shear to load ground-reaction force may underlie the directional tuning of the automatic postural response to rotation and translation. Journal of Neurophysiology, 92, 808–823. Van der Kooij, H., Van Asseldonk, E., & Van der Helm, F. C. T. (2005). Comparison of different methods to identify and quantify balance control. Journal of Neuroscience Methods, 145, 175–203. Verkerke, G. J., Hof, A. L., Zijlstra, W., Ament, W., & Rakhorst, G. (2005). Determining the centre of pressure during walking and running using an instrumented treadmill. Journal of Biomechanics, 38, 1881–1885. Winter, D. A. (1995a). Human balance and posture control during standing and walking. Gait & Posture, 3, 193–214. Winter, D. A. (1995b). ABC of balance during standing and walking. Waterloo, CA: Waterloo Biomechanics. Wu, G., & Cavanagh, P. R. (1995). ISB recommendations for standardization in the reporting of kinematic data. Journal of Biomechanics, 28, 1257–1260.