Mechanical Work Rate Minimization and Freely Chosen Stride Frequency of Human Walking: A Mathematical Model

Optimization principles seem to control many biological functions, particularly repetitive processes. The hypothesized ‘controller’ has to choose from an appropriate number of extensive variables to satisfy some optimization criteria: e.g. performance maximization, energy saving, comfort or safety maintenance (Hämäläinen, 1978). In this respect, simple mathematical models can be used in parallel with experimental work to design or verify hypotheses about the optimization mechanisms.

In the past such modelling has been applied in studies of the respiratory (Otis et al. 1950; Mead, 1960; Hämäläinen and Viljanen, 1978) and cardiovascular systems (Yamashiro et al. 1979; Hämäläinen and Hämäläinen, 1985), whereas investigators of human locomotion (Margaria, 1938; Cavagna et al. 1977; Cavagna and Franzetti, 1986; Alexander, 1989) have shown experimentally that optimization phenomena also occur in gait. There is a striking similarity between breathing and walking, as pointed out by Cavagna and Franzetti (1986). In order to cope with particular performance requirements (alveolar ventilation and progression speed) the system has to select the appropriate combination of extensive parameters (tidal volume/breathing frequency, step length/step frequency) following an optimization criterion, for instance the minimization of mechanical work rate or average force. In both systems, for a fixed ventilation rate or speed, the total work curves attain high values at the extremities of the frequency range, whereas somewhere in the middle they have a point which corresponds to the minimum work rate. In breathing, such behaviour is produced by a balance between the viscous and elastic components of the total work done, each of which increases or decreases monotonically with an increase in breathing frequency. During walking, mutatis mutandis, the total work components displaying the same trend are the internal work rate (Ẇ _int), due to the movements of the limbs with respect to the body centre of mass, and the external work rate (Ẇ_ext), due to the displacement of body centre of mass with respect to the environment. The rationale underlying the choice of a certain stride frequency might be the following: when walking at a constant speed using low-frequency longer steps, most of the energy expended is used to raise and lower the body centre of mass, resulting in a greater external component, while at the highest stepping frequencies the body centre of mass moves along a relatively straight line and most of the energy is devoted to accelerating and decelerating limbs with respect to it, resulting in a greater internal component. Cavagna and Franzetti (1986) showed experimentally that the total mechanical work rate (Ẇ_tot), obtained by adding Ẇ_int and Ẇ_ext, at each constant progression speed displays a minimum for a step frequency roughly corresponding (−25%) to the one freely chosen by the subjects. Their predictions have been further improved (−12%) by using more recent (Dempster et al. 1959) average anthropometric data (Minetti et al. 1990a).

Despite the multiplicity of methods available for calculating the mechanical work of locomotion (which yield results varying over three orders of magnitude; Williams and Cavanagh, 1983), the technique adopted by Cavagna and Franzetti (1986) seems to be the only one to face the optimization phenomena related to the chosen stride frequency. There is thus a need to incorporate such a method into a comprehensive and simple mathematical model capable of verifying the assumptions about minimization of the mechanical work rate during walking..

The aim of the present study is to provide simple equations for Ẇ _int and Ẇ_ext and to compare the model predictions of natural stride frequency with experimental data.

Fig. 1 shows the stick diagram adopted in the formalization of the Ẇ_int model. The body is composed of five stiff segments, four of which (two lower and two upper limbs) are involved in the model computation. The values of segment length, mass, position of the centre of mass (circles in Fig. 1) and radius of gyration have been taken from the literature (Dempster et al. 1959).

In this paper the calculation of Ẇ_ext is based on the following rationale: the only fluctuations in the energy curve of the body centre of mass (concentrated in the hip joint) to be taken into account are those produced by the loss of energy occurring between the end of a step and the beginning of the next one due to the angle formed by the trajectory vectors in that transition. When the maximum angle between the front and the hind limbs (2φ_max in Fig. 2, where φ_max is the maximum angle between the vertical and the limbs) is zero the body centre of mass energy could ideally be completely recovered, but when it is π/2 the trajectory vector will be in line with the front limb, precluding any energy transfer to the subsequent step. On the basis of an elastic collision, the energy loss in the forward direction is proportional to cos²(2φ_max), but we opted, for a number of reasons (see Results and Discussion), to approximate the function with a simple cosine.

The adequacy of the model has been tested by comparing our graphs of Ẇ_int, Ẇ_ext and Ẇ_totversus stride frequency at constant progression speed (Fig. 3) with the ones provided by Cavagna and Franzetti (1986; Fig. 2B) at the same speeds (1.278, 1.5 and 1.8ms⁻¹). The most crucial difference in the work rate curves occurs for Ẇ_int, which is smaller and linear in the present study, while the experimental data show a curved relationship with stride frequency. This discrepancy is due to the different methodology adopted in the estimation of the internal work rate. Cavagna and Franzetti (1986) took the displacement curves of four subjects investigated in a previous experiment (Cavagna and Kaneko, 1977), who walked at speeds ranging from about 1 to 5.5ms⁻¹ at their natural stride frequency, and forced those curves, by re-scaling the time axis, to correspond to three given speeds [4.6, 5.5 and 6.5 kmh⁻¹ using their units (1.3,1.5 and 1.8 ms⁻¹, respectively)], maintaining the original step length range at each new speed. In so doing, they clustered estimates of Ẇ_int calculated from very different motion patterns of the limbs at every constant speed.

In our model only the stride frequency was allowed to change, while the motion pattern was invariant (a sinusoidal oscillation). This treatment implies, for a given progression speed, a constant maximum speed of the limbs with respect to the body centre of mass, regardless of the stride frequency adopted, causing Ẇ_int to be a linear function of f (within the limits indicated by equation A18). Another reason for the observed discrepancy in Ẇ_int is the fact that Cavagna and Franzetti (1986) adopted in their computations the anthropometric data of Braune and Fischer (1892), which report the mass of four limbs equal to 50.22% of the entire body mass, while more recent measurements (Dempster et al. 1959) set this at 42.20%. This difference has been shown to be responsible for a 20% overestimation of the internal work rate (Minetti et al. 1990a). Recently, Minetti and Saibene (1992) found good agreement between the preliminary predictions of the Ẇ_int model and experimental data for freely chosen and forced stride frequencies at different walking speeds (range 0.69 to 2.08ms⁻¹).

Conversely, modelling the external work according to cos²(2φ_max) (the energy loss due to the impact during the step transition) leads to unrealistic high work values when compared with the experimental results of Cavagna and Franzetti (1986). We adopted a simple cos(2φ_max) function, which behaves similarly but attenuates the energy losses, because it is supported by three circumstances occurring during walking. First, at low step frequencies (and longer step lengths) the horizontal trunk rotation separates the hips, corresponding to an increase in the lower limb length. This increase in r results in a decrease in the external work rate while having no effect on the internal one, whose equation is unaffected by changes in the lower limb length. Second, the spread of the four limbs moves the body centre of mass vertically with respect to the hip, reducing the angle between the lines of impact. When modelling this aspect with the walking geometry assumed in this study and standard anthropometric data, it turns out that the vertical range of the movement of the body centre of mass is about 60% of the hip movement range. Finally, foot extension acts to smooth the step transition, by increasing the radius before the collision. All three occurrences imply a decrease in 2φ_max and, concurrently, in the energy losses related to the step transition, resulting in a lower external work rate.

The Ẇ_ext values obtained using this approach, whose rationale is based on observations about the energy loss of walking made by others (Cavagna et al. 1977), nevertheless resemble both quantitatively and qualitatively those obtained by Cavagna and Franzetti (1986), who used a force platform. Only at high speed is there a tendency to overestimate the experimental data because of the fourth power in equation A24.

The mathematical model proposed in the present paper predicts the relationship between average walking speed and the natural stride frequency according to the criterion of minimum work rate (Fig. 4) (curve Ẇ_min, ‘standard’ subject). These predictions seem to be in agreement with our experimental data at low speeds, but deviate considerably when the speed increases. Although Strathy et al. (1983) found that treadmill walking overestimates the stride frequency by about 7%, the accuracy of our measurements is confirmed by values reported by others (Cavagna and Franzetti, 1986). It is important to keep in mind that the curve reports the stride frequencies at which the rate of positive internal work is minimal during the stride (only the positive differences in kinetic energy are included in the Ẇ_int computation) and does not take into account the work necessary for decelerating the limbs. From another viewpoint, the same curve can be regarded as the one that minimizes the positive and negative internal work rates pertaining only to one stride phase, for example the stance, while the other phase (the swing) is considered as a ballistic (i.e. passive) movement, a hypothesis also suggested by others (Mochon and McMahon, 1980). In Fig. 4 a horizontal line marks the frequency at which a frictionless pendulum (with length equal to the mean of the lower and upper limb proximal distance of the centre of mass) oscillates naturally. Above this frequency the limbs have to be continuously accelerated and decelerated, with increasing positive and negative work (see below).

The calculation of Ẇ_tot partly follows the algorithm proposed by Cavagna and Kaneko (1977) but, as anticipated in the Introduction, there does not seem to be agreement about the computational technique to adopt when estimating the mechanical work of locomotion (Williams and Cavanagh, 1983; Minetti et al. 1990b). The major uncertainties reside in the energy transfer among segments, the incorporation of negative work and the recovery of the stored elastic energy. For this reason we incorporate in Fig. 4 a shaded area whose right-hand boundary, with equation ⁠, represents the minimization of the total work rate whose internal component has been doubled, thus also taking into account the negative work rate of both stride phases. Such high weighting for the negative work (100% of positive Ẇ _int) has been used in another study on the estimation of the mechanical work of locomotion (Winter, 1979), although other investigators (Williams and Cavanagh, 1983) adopted a value of about 33% (that is to say, they considered the efficiency of negative work as being three times greater than the efficiency of positive work, provided that, during level walking, positive work equalled negative work). The shaded area within the two lines includes the curves which minimize total mechanical work with different negative work efficiencies. Within this area, a departure from Ẇ_min towards the right-hand limit could be interpreted as the work minimization (positive plus negative) when one stride phase (the swing) becomes progressively less passive. The last assumption about two active phases (no ballistic return) is mirrored by the behaviour of the respiratory system during increased alveolar ventilation, when expiration can no longer be considered to be a passive return to the initial operating condition (Otis et al. 1950).

All frequency predictions made by the present model are linear functions of the progression speed, resulting in a constancy of stride length (step length 0.73 and 1.03 m for Ẇ_min and the right-hand limit, respectively). These findings are not reflected by the experimental points which, although almost linear, seem to intersect several iso-length lines when the speed increases. As stated above, those lines pertain to different weightings of negative internal work (or different proportions in the activity of the swing phase). However, when a subject is asked to walk at speeds normally achieved by running (>2.1ms⁻¹), a tendency to maintain a constant step length is found.

Another phenomenon only partially taken into account by the model is the foot extension between successive steps that occurs at high walking speeds. This results both in a reduction of energy loss due to the angle between the trajectory vectors (see Fig. 2), as suggested by Cavagna et al. (1977) (Fig. 3), and in an increase in the system energy. To simulate the first effect we included a greater segment length (lower limb plus foot) in the Ẇ_ext computation and obtained the prediction labelled in Fig. 4 ⁠, which corresponds to a lower optimal stride frequency.

A stability analysis of our model predictions reveals a marked insensitivity to changes in body mass (only a different mass distribution within the body plays a role in the Ẇ_int/Ẇ_ext balance). However, an increase in subject height results in lower optimal stride frequencies. Fig. 5 shows the Ẇ _min and curves given in Fig. 4 (thick solid and dashed lines) together with the limits, ceteris paribus, for the tallest and the shortest subjects of our group (right-hand and left-hand thin lines, respectively). The narrow bands between the corresponding thin lines reflect the variability of the experimental data in Fig. 4, which do not show height-dependence in the scatterplot.

The reasons for the discrepancies lie partly in the assumptions within the model (stiff limbs, no double support, body centre of mass located in the hip joint) and partly in the possibility that the minimum work rate criterion is not the only one involved in performance optimization.

A recent paper about the energy expenditure of locomotion in mammals argues that a force minimization criterion is probably operating (Kram and Taylor, 1990) in species of very different sizes. Such a hypothesis, supported by measurements of foot-substratum contact time during running, could not apply to walking because of the different gait mechanism (Cavagna et al. 1977). As pointed out by Alexander (1991), who reviewed several optimization criteria that could operate during locomotion, future models will have to consider both energetic aspects: the energy spent to generate work and the energy necessary to generate (isometric) force. In addition, our model of Ẇ_ext is very similar to the one proposed by Alexander (1991), which applies the principle of conservation of angular momentum, as also suggested by McGeer (1990). We tried to incorporate Alexander’s equation in our model and found that it overestimates the experimental data of Cavagna and Franzetti (1986). However, because of the additive procedure for total work rate computation, the incorporation of that model results in comparable predictions about the optimal stride frequency.

The present model could be further improved by taking into account a frictional component and allowing for a penalty factor related to the departure of the stride frequency from the one for the ideal pendulum at increasing speeds. In addition, accurate measurements could quantify the energy-saving effects of trunk rotation, foot extension and segmental mass displacements as contributors to the elevation of the centre of rotation, with a consequent decrease in 2 φ_max (see Fig. 2). By increasing the values and curvature of Ẇ_int and decreasing Ẇ_ext, these considerations are expected to improve the model predictions about the optimal stride frequency.

The values in parentheses have been used in this paper to predict the natural stride frequency of walking of a ‘standard man’ (height=1.75 m).

 
• a

  proximal distance of lower limb centre of mass as a fraction of r (44.7%)

 
• b

  upper limb length as a fraction of r (83.2%)

 
• f

  stride frequency (Hz)

 
• f_opt

  stride frequency for minimum rate of total work (Hz)

 
• g_L,g_U

  lower and upper limb radii of gyration (limb length fraction) (32.6, 36.8%)

 
• go

  mean radius of gyration (lower limb length fraction)

 
• I_L, I_U

  lower and upper limb moment of inertia

 
• m

  body mass (70 kg)

 
• m_L, m_U

  mass of lower and upper limbs (11.27 and 3.50 kg)

 
• r

  lower limb length (0.928 m)

 
• RKE_L, RKE_U

  rotational kinetic energy of lower and upper limbs

 
• 

  average progression speed (ms⁻¹)

 
• s_L(t),s_U(t)

  movement of lower and upper extremities

 
• s_Lcom(t),s_Ucom(t)

  movement of lower and upper limb centre of mass

 
• ṡ_Lcom(t), ṡ_Ucom(t)

  speed of lower and upper limb centre of mass

 
• TE_max

  total energy of the body centre of mass: maximum level

 
• TE_min

  total energy of the body centre of mass: minimum level

 
• TKG_Lcom(t),TKE_Ucom(t)

  translational kinetic energy of lower and upper limbs

 
• Ẇ_int

  rate of internal mechanical work (Wkg⁻¹)

 
• Ẇ_ext

  rate of external mechanical work (Wkg⁻¹)

 
• Ẇ_tot

  rate of total mechanical work (Wkg⁻¹)

 
• Δ RKE_t

  total range of rotational kinetic energy

 
• Δ TE

  total energy range of the body centre of mass

 
• Δ TKE_Lcom, Δ TKE_Ucom

  TKE range of lower and upper limbs

 
• Δ TKE_t

  total range of translational kinetic energy

 
• ρ _L, ρ _U

  lower and upper limb radii of gyration (about the centre of mass)

 
• φ

  angle between the vertical line and limbs (rad)

 
• φ(t)

  φat time t (rad)

 
• φ_max

  maximum φduring the stride (rad)

 
• 

  angular speed

 
• ω

  swing frequency

The authors wish to thank Giovanni Cavagna, Institute of Physiology, University of Milan, and Giancesare Belli, Institute of Mathematics, Politecnico of Milan, for their helpful suggestions and criticisms. Part of this work was presented at the Xlllth International Congress on Biomechanics, Perth (Australia), 9-13 December 1991.