**6. Model results and model sensitivity analysis**

160 Injury and Skeletal Biomechanics

different subject.

**-1**

**A**

**c**

**c**

**ele**

**r**

**atio**

**n [g]**

**0**

**1**

**2**

**-4**

**A**

**c**

**c**

**ele**

**r**

**atio**

**n [g]**

**0**

**4**

**8**

**12**

Due to nonlinearity of the stiffness/damping properties of the joints of the leg [e.g. 20,37], we were not generally able to estimate the model parameters while assuming that they remain constant over the heel-strike period. Thus, the heel-strike period was divided into two equal periods (22 ms each) and the parameters were estimated separately for each of these periods, using the Gauss-Marquardt [38-39] method of non-linear estimation. For the first period the initial conditions were as prescribed in equation (2), and for the second period

Figure 6 shows the model prediction of the shank mass (*m2* in Figure 1) acceleration (continuous line) versus the experimentally measured tibial tuberosity acceleration (denoted as +). The results of two different subjects are demonstrated (vertical partition of the Figure) for two time points of running (horizontal partition): 1st min, fatigue-free, and 15th min of the running test. The subtle, brief, discontinuity in the traces at mid-cycle represents the

**0**

**5 15 25 35 45**

**5 15 25 35 45**

**<sup>12</sup> Sub: ME, 15th min.**

**Sub: AV, 1st min.**

**4**

**8**

**Figure 6.** The model prediction of the *m2* acceleration (continuous line) versus the experimentally measured tibial tuberosity acceleration (denoted as +). The upper panels represent the solution for the 1st min of running and the lower panels represent the solution for the 15th min of running. The subtle, short, discontinuity in the traces at mid-cycle represents the transition between the two parameter estimation periods, with piece-wise constant stiffness each. Each of the left and right panels represents a

**time [ms] time [ms]**

**-4**

**0**

**4**

**8**

the initial conditions used were the values reached at the end of the first period.

transition between the two parameter estimation periods.

**Sub: CH, 1st min.**

**5 15 25 35 45**

**5 15 25 35 45**

**Sub: PL, 15th min.**

Table 1 shows the sample results of the stiffness and damping parameters for the two time zones. A sensitivity analysis can provide an indication to the quality of the estimation of parameters. Thus, sensitivity of the *m2* and *m4* model results to each of the estimated parameters was performed by two-fold multiplying and/or dividing each of the estimated parameters separately and for each time zone. The two-fold variation demonstrated a strong sensitivity of *m2* to *k1* and *k2* in the first time zone and to parameters *k1*, *k2*, *k3* and *c3* in the second time zone. Small to medium sensitivity was obtained in the second time zone to parameters *c1* and *k4*, repectively. The *m4* acceleration was not sensitive in the first time zone to none of the parameters, while in the second time zone it demonstrated strong sensitivity to parameters *k2* and *k3* and small sensitivity to parameters *k4* and *c3*.


**Table 1.** Summary of the sensitivity of the *m2* and *m4* accelerations to the model parameters. A twofold change up and down was checked in the two time solution-zones, separately for each parameter. Parameters for which the accelerations were not sensitive were checked also for a change in one order of magnitude (up and down). The results of this latter test are shown in parentheses (\* = negligibly low effect; \*\* = low to medium effect; \*\*\* = high effect, i.e. more than 10% change in peak acceleration of *m2* or *m4*).

In cases of low- or no-sensitivity the parameters were varied, up and down, by one order of magnitude with the results shown in parentheses. The one order of magnitude variation in the parameter values did not evoke sensitivity beyond that of the twofold variation, except for *c1* on *m2* and *k1* on *m4*.

### 162 Injury and Skeletal Biomechanics

Due to the fact that the parameter estimation of the model coefficients was performed in short time intervals, fractions of the heel strike event, the damping coefficients disclosed a high variability. Better estimations would have probably resulted if the period of estimation was higher. Accordingly, the stiffness parameters *k1* and *k2*, in the first time zone, and *k1*, *k2*  and *k3* in the second time zone, considered more reliably estimated, were reported in what follows.

Modeling the Foot-Strike Event in Running Fatigue via Mechanical Impedances 163

**Stiffness [kN/m] 1 15 30**

19.3 (28.2) 16.4 (16.1) 13.6 (4.7) *k1*

11.4 (4.3) 14.7 (6.0) 19.8 (18.0) *k2*

140.2**\*** (90.1) 190.3**\*** (53.8) 171.0**\*** (73.2) *k1*

17.1**\*** (4.5) 17.8 (4.4) 23.1 (15.6) *k2*

14.0 (9.3) 17.8 (7.3) 15.8 (9.4) *k3*

**Time in running [min]** 

**Table 2.** Parameter means (S.D) of 10 subjects: *k1*, *k2* in ZONE 1 (upper part of Table), and *k1*, *k2*, *k3* in ZONE 2 (lower part of Table). Parameter-values are from 3 time points, i.e. at the 1st, 15th and 30th min of running. Parameter-means change with fatigue was significant with p>0.148. The asterisks in the lower

The *m1* displacement obtained from our results revealed that the foot sinks about one cm in the first time zone of heel-strike, followed by a bouncing back phase. Previous reports also indicated a 1 cm deformation of the heel [e.g. 48]. The elastic properties of the heel-pad of various mammals were studied and it has been found that full up and down oscillations might result before actual settling down of the foot [53]. At cases, the foot may even temporarily lose contact with the ground during these oscillations. In our case, the

The obtained average knee stiffness *k3* of 15.8 kN/m was within the range of 6.89 – 112.0

Further exploration of the effect of fatigue in the course of running was performed by correlating, using linear regression, the tibial tuberosity peak acceleration to the parameter values obtained from the model results. Figure 7 shows the correlation for which the Pearson's coefficient was statistically different from zero (with 95% significance). The parameters are *k1* in the first and second zones, upper and middle rows, respectively, and *k3* in the second time zone of stance. The 1st, 15th and 30th min of running are shown in the left, middle and right columns, respectively. Table 3 summarizes the regression results for the cases displayed in Figure 7. The results show the following: (a) For *k1* in time zone 1, the Pearson correlation *rp* starts off by a low value of 0.26 in the 1st min of running but increases in the 15th (0.49) and the 30th (0.44) min of running, indicating that a higher peak in the tibial tuberosity acceleration is associated with a lower *k2* in time zone 1; (b) there is a high correlation (0.86) between the peak tibial tuberosity acceleration and the *k1* parameter in time zone 2 at the initial stage of running (1st min).

part of Table signify significant changes (p<0.05) from ZONE 1 to ZONE 2.

oscillation phase would succeed the first time zone.

*ZONE 1*

*ZONE 2* 

kN/m, previously reported [12-14,16,19,22,29,45,54-55].

Difficulties in estimating the damping coefficients are not unusual due to their expected low values. It has been reported that in repetitive physical activity, such as in running, the subject bounces on the ground in a spring-like manner [17, 40-45]. Depending on the range of joint flexion and on the frequency of motion, a considerable amount of elastic energy can be stored and re-used. It has been shown that the dissipated energy in muscles increase when the amplitudes of joint movement are increased [46]. It has also been also commented that the utilization of stored elastic energy depended on the shortness in latency between the stretch and shortening phases of the muscles [47]. Accordingly, during the groundcontact period of running, the leg was modeled as a one-dimensional four-degree-offreedom piece-wise linear spring, with no damping. During heel-strike, the joints did not have a damping effect, to contribute to energy dissipation.

Summary of the values of these parameters for the two solution time zones is given in the rightmost column of Table 2. The values, averages for 10 subjects (SD), are calculated at each of the 1st, 15th and 30th min of running to evaluate the effect of fatigue. The asterisks indicate a significant change *p* < 0.05 between the values of zone 1 and zone 2. The differences between *k1* in zone 1 and *k1* in zone 2 were significant with *p* < 0.0007, for each of the 1st, 15th and 30th min of running. For the *k2* parameter a statistically significant difference was obtained for the 1st min of running only, with *p* < 0.009. The differences in parameter values due to fatigue were not statistically different, despite the notable differences in the averages. The reason was the big variability among the tested subjects. On the individual level, though, the differences due to fatigue were statistically significant in most of the cases.

It should be noted that the stiffness *k1*, relating to the heel-pad may be alternatively obtained from the ratio between the foot-ground reaction force and the heel-pad deformation. Past reports using this method have reported an increased stiffness with the heel-pad deformation, as occurs during heel-strike [27,48-52]. These studies have reported an approximately tenfold increase in heel-pad stiffness, i.e., from 6.6 – 135 kN/m to 77 – 1430 kN/m, using different methodologies including actual running in conjunction with measurements of the heel pad by means of X-rays; heel-strike simulation by in vivo pendulum impact, or Instron measurements made on cadavers. Despite the differences with the method applied in this study, the stiffness *k1* indicates similar values with those obtained by other methods. Our model provides further the fatigue effect on *k1*, which is found opposite in the first time zone, where the stiffness decreases with fatigue, to the second time zone where the stiffness is found to increase.


162 Injury and Skeletal Biomechanics

in most of the cases.

zone where the stiffness is found to increase.

follows.

Due to the fact that the parameter estimation of the model coefficients was performed in short time intervals, fractions of the heel strike event, the damping coefficients disclosed a high variability. Better estimations would have probably resulted if the period of estimation was higher. Accordingly, the stiffness parameters *k1* and *k2*, in the first time zone, and *k1*, *k2*  and *k3* in the second time zone, considered more reliably estimated, were reported in what

Difficulties in estimating the damping coefficients are not unusual due to their expected low values. It has been reported that in repetitive physical activity, such as in running, the subject bounces on the ground in a spring-like manner [17, 40-45]. Depending on the range of joint flexion and on the frequency of motion, a considerable amount of elastic energy can be stored and re-used. It has been shown that the dissipated energy in muscles increase when the amplitudes of joint movement are increased [46]. It has also been also commented that the utilization of stored elastic energy depended on the shortness in latency between the stretch and shortening phases of the muscles [47]. Accordingly, during the groundcontact period of running, the leg was modeled as a one-dimensional four-degree-offreedom piece-wise linear spring, with no damping. During heel-strike, the joints did not

Summary of the values of these parameters for the two solution time zones is given in the rightmost column of Table 2. The values, averages for 10 subjects (SD), are calculated at each of the 1st, 15th and 30th min of running to evaluate the effect of fatigue. The asterisks indicate a significant change *p* < 0.05 between the values of zone 1 and zone 2. The differences between *k1* in zone 1 and *k1* in zone 2 were significant with *p* < 0.0007, for each of the 1st, 15th and 30th min of running. For the *k2* parameter a statistically significant difference was obtained for the 1st min of running only, with *p* < 0.009. The differences in parameter values due to fatigue were not statistically different, despite the notable differences in the averages. The reason was the big variability among the tested subjects. On the individual level, though, the differences due to fatigue were statistically significant

It should be noted that the stiffness *k1*, relating to the heel-pad may be alternatively obtained from the ratio between the foot-ground reaction force and the heel-pad deformation. Past reports using this method have reported an increased stiffness with the heel-pad deformation, as occurs during heel-strike [27,48-52]. These studies have reported an approximately tenfold increase in heel-pad stiffness, i.e., from 6.6 – 135 kN/m to 77 – 1430 kN/m, using different methodologies including actual running in conjunction with measurements of the heel pad by means of X-rays; heel-strike simulation by in vivo pendulum impact, or Instron measurements made on cadavers. Despite the differences with the method applied in this study, the stiffness *k1* indicates similar values with those obtained by other methods. Our model provides further the fatigue effect on *k1*, which is found opposite in the first time zone, where the stiffness decreases with fatigue, to the second time

have a damping effect, to contribute to energy dissipation.

**Table 2.** Parameter means (S.D) of 10 subjects: *k1*, *k2* in ZONE 1 (upper part of Table), and *k1*, *k2*, *k3* in ZONE 2 (lower part of Table). Parameter-values are from 3 time points, i.e. at the 1st, 15th and 30th min of running. Parameter-means change with fatigue was significant with p>0.148. The asterisks in the lower part of Table signify significant changes (p<0.05) from ZONE 1 to ZONE 2.

The *m1* displacement obtained from our results revealed that the foot sinks about one cm in the first time zone of heel-strike, followed by a bouncing back phase. Previous reports also indicated a 1 cm deformation of the heel [e.g. 48]. The elastic properties of the heel-pad of various mammals were studied and it has been found that full up and down oscillations might result before actual settling down of the foot [53]. At cases, the foot may even temporarily lose contact with the ground during these oscillations. In our case, the oscillation phase would succeed the first time zone.

The obtained average knee stiffness *k3* of 15.8 kN/m was within the range of 6.89 – 112.0 kN/m, previously reported [12-14,16,19,22,29,45,54-55].

Further exploration of the effect of fatigue in the course of running was performed by correlating, using linear regression, the tibial tuberosity peak acceleration to the parameter values obtained from the model results. Figure 7 shows the correlation for which the Pearson's coefficient was statistically different from zero (with 95% significance). The parameters are *k1* in the first and second zones, upper and middle rows, respectively, and *k3* in the second time zone of stance. The 1st, 15th and 30th min of running are shown in the left, middle and right columns, respectively. Table 3 summarizes the regression results for the cases displayed in Figure 7. The results show the following: (a) For *k1* in time zone 1, the Pearson correlation *rp* starts off by a low value of 0.26 in the 1st min of running but increases in the 15th (0.49) and the 30th (0.44) min of running, indicating that a higher peak in the tibial tuberosity acceleration is associated with a lower *k2* in time zone 1; (b) there is a high correlation (0.86) between the peak tibial tuberosity acceleration and the *k1* parameter in time zone 2 at the initial stage of running (1st min). This correlation, however, was lower in the 15th min (0.27) and in the 30th min (0.49) of running, thus suggesting that high peak acceleration at the tibial tuberosity is associated with a higher *k1* value in time zone 2; (c) a medium correlation (0.73) was found between *k3* and peak acceleration at the tibial tuberosity in the 1st mi of running. But here too this correlation was decreased with the development of fatigue, suggesting that higher peak acceleration at the tibial tuberosity is associated with a high *k3* parameter value in time zone 2.

Modeling the Foot-Strike Event in Running Fatigue via Mechanical Impedances 165

**t1 t15 t30**


3.261.70 153.1019.48 0.27 0.07

0.660.20 11.452.34 0.42 0.18


7.942.02 89.9523.05 0.49 0.24

1.190.24 4.262.74 0.57 0.33

**Table 3.** Linear regression results (Y= aX+b) of Figure 7, at 95% confidence intervals for coefficients a


23.562.20 40.3617.92 0.86 0.74

> 1.760.26 1.522.08 0.73 0.54

The correlation found between low stiffness in the first time zone to the high stiffness in the second time zone is obvious from the anatomy of the heel pad, which consists of nearly closed collagen cells filled with fatty cells [27,48]. The vertical orientations of these cells, together with the high viscosity of the fat tissue are the major factors responsible for the absorption of the impact energy at heel strike. Initially, the fat flows sideways and small loads result in high deformation (low stiffness). In the second time zone, after the heel pad has already considerably deformed, further increase in deformation provokes a high load, thus high stiffness. The effect of fatigue could be explained by means of the heating effect during the course of running. With nearly 80 heel-strikes per min, the whole running duration of 30 min results in some 2500 heel-strikes, each of which causing a rapid deformation of the heel pad and during which the fatty tissue frictions while squeezed out of the collagen cells. The accumulated heat due to friction reduces the viscosity and the vertical displacement is accelerated, causing a reduction in stiffness during the first zone of heel strike. In the second zone, however, the thinner remaining tissue together with the

Stress fractures in long bones of the lower limbs are believed to originate from repetitive and/or excessive loading, such as may take place in long-distance running at a speed exceeding the anaerobic threshold. In the present study the average running distance per test was 6.30 km (30 min of running at the average speed of 12.6 km/h) in agreement with the definition of 'long distance' [57]. We have measured and analyzed the following: respiratory data to monitor global fatigue; and accelerometry, to provide quantitative information on loading of the major segments of the lower limb. While providing accelerometer boundary values for the model system, accelerometry is an advantageous

and b ; *rp* = Pearson correlation coefficient ; and *r2* = coefficient of determination.

**Regression coefficients**

> a b *rp r 2*

> a b *rp r 2*

> a b *rp r 2*

**Tested parameter**

Z1\_*k*<sup>1</sup>

Z2\_ *k*<sup>1</sup>

Z2\_ *k*3

underlying bone evokes an increase in stiffness.

method due to its being non-invasive.

**7. Conclusion** 

**Figure 7.** Simple linear-regression made to express the relationship between the tibial tuberosity peak acceleration and parameter values (results of 10 subjects). The regression was performed for the 1st min (leftmost panels), the 15th min (middle panels) and for the 30th min (rightmost panels). The correlation coefficient of the presented regression lines was significantly different from zero, at 95% significance level. The 3 upper panels - for *k1* in ZONE 1 (Z1); 3 middle panels for *k1* in ZONE 2 (Z2); and 3 lower panels - for *k3* in ZONE 2.

It has been shown that in running with shoes, the foot is restricted from bulging sideways, thus limiting the vertical deformation to an average of 5.5 mm, as opposed to 9 mm when running barefoot [48,56]. This explains the higher stiffness during the first time zone compared to bare foot running. It also explains the lower stiffness during the second time zone compared to bare foot running. It has also been shown by that better energy absorption and impact shock attenuation is associated with lower stiffness [51].


**Table 3.** Linear regression results (Y= aX+b) of Figure 7, at 95% confidence intervals for coefficients a and b ; *rp* = Pearson correlation coefficient ; and *r2* = coefficient of determination.

The correlation found between low stiffness in the first time zone to the high stiffness in the second time zone is obvious from the anatomy of the heel pad, which consists of nearly closed collagen cells filled with fatty cells [27,48]. The vertical orientations of these cells, together with the high viscosity of the fat tissue are the major factors responsible for the absorption of the impact energy at heel strike. Initially, the fat flows sideways and small loads result in high deformation (low stiffness). In the second time zone, after the heel pad has already considerably deformed, further increase in deformation provokes a high load, thus high stiffness. The effect of fatigue could be explained by means of the heating effect during the course of running. With nearly 80 heel-strikes per min, the whole running duration of 30 min results in some 2500 heel-strikes, each of which causing a rapid deformation of the heel pad and during which the fatty tissue frictions while squeezed out of the collagen cells. The accumulated heat due to friction reduces the viscosity and the vertical displacement is accelerated, causing a reduction in stiffness during the first zone of heel strike. In the second zone, however, the thinner remaining tissue together with the underlying bone evokes an increase in stiffness.

### **7. Conclusion**

164 Injury and Skeletal Biomechanics

**t1**

**40**

**Z1\_K1**

**Z2\_K1**

**Z2\_K3**

**[kN/m]**

**[kN/m]**

**[kN/m]**

**t1**

**t1**

**0 4 8 12 16 Peak T.T. Acceleration [g]**

panels - for *k3* in ZONE 2.

zone 2.

**Z1\_K1**

**Z2\_K3**

**[kN/m]**

**Z2\_K1**

**[kN/m]**

**[kN/m]**

This correlation, however, was lower in the 15th min (0.27) and in the 30th min (0.49) of running, thus suggesting that high peak acceleration at the tibial tuberosity is associated with a higher *k1* value in time zone 2; (c) a medium correlation (0.73) was found between *k3* and peak acceleration at the tibial tuberosity in the 1st mi of running. But here too this correlation was decreased with the development of fatigue, suggesting that higher peak acceleration at the tibial tuberosity is associated with a high *k3* parameter value in time

**0**

**2**

**Z2\_K3**

**[kN/m]**

**t30**

**t30**

**t30**

**6 10 14 18 22 Peak T.T. Acceleration [g]**

**Z1\_K1**

**Z2\_K1**

**[kN/m]**

**[kN/m]**

**t15**

**t15**

**t15**

**Figure 7.** Simple linear-regression made to express the relationship between the tibial tuberosity peak acceleration and parameter values (results of 10 subjects). The regression was performed for the 1st min (leftmost panels), the 15th min (middle panels) and for the 30th min (rightmost panels). The correlation coefficient of the presented regression lines was significantly different from zero, at 95% significance level. The 3 upper panels - for *k1* in ZONE 1 (Z1); 3 middle panels for *k1* in ZONE 2 (Z2); and 3 lower

**0 5 10 15 20 25**

**Peak T.T. Acceleration [g]**

It has been shown that in running with shoes, the foot is restricted from bulging sideways, thus limiting the vertical deformation to an average of 5.5 mm, as opposed to 9 mm when running barefoot [48,56]. This explains the higher stiffness during the first time zone compared to bare foot running. It also explains the lower stiffness during the second time zone compared to bare foot running. It has also been shown by that better energy absorption

and impact shock attenuation is associated with lower stiffness [51].

Stress fractures in long bones of the lower limbs are believed to originate from repetitive and/or excessive loading, such as may take place in long-distance running at a speed exceeding the anaerobic threshold. In the present study the average running distance per test was 6.30 km (30 min of running at the average speed of 12.6 km/h) in agreement with the definition of 'long distance' [57]. We have measured and analyzed the following: respiratory data to monitor global fatigue; and accelerometry, to provide quantitative information on loading of the major segments of the lower limb. While providing accelerometer boundary values for the model system, accelerometry is an advantageous method due to its being non-invasive.

We have addressed a major fatigue-related factor taking part in exposing the shank to stress fractures risk: the decline in end tidal carbon dioxide pressure, the latter expressing metabolic fatigue [31,58]. The mechanical consequence of fatigue in long-distance running is two-fold: enhanced impact acceleration due to global fatigue and muscle activity imbalance due to local fatigue before and during foot contact, resulting in the development of excessive shank-bone bending stresses and higher risk of stress injury [11].

Modeling the Foot-Strike Event in Running Fatigue via Mechanical Impedances 167

[5] Mizrahi J, Voloshin A, Russek D, Verbitsky O, Isakov E. The Influence of Fatigue on EMG and Impact Acceleration in Running. Basic Appl. Myol. 1997;7: 111-118. [6] Mizrahi J, Verbitsky O, Isakov E. Shock Accelerations and Attenuation in Downhill and

[7] Voloshin A, Mizrahi J, Verbitsky O, Isakov E. Dynamic Loading on the Human Musculoskeletal System- Effect of Fatigue. *Clin. Biomechanics* 1998;13: 515-520. [8] Verbitsky O, Mizrahi J, Voloshin A, Treiger J, Isakov E. Shock Absorption and Fatigue in

[9] Baker J, Frankel VH, Burstein A. Fatigue Fractures: Biomechanical Considerations. The

[10] Nordin M, Frankel V. Biomechanics of Bone. In: Nordin M., Frankel V. (eds). Basic Biomechanics of the Musculoskeletal System. Philadelphia (PA): Lea and Febiger, 1989,

[11] Mizrahi J, Verbitsky O, Isakov E. Fatigue-Related Loading Imbalance on the Shank in Running: A Possible Factor in Stress Fractures. *Annals Biomed. Eng*., 2000;28: 463-469. [12] Greene PR, McMahon TA. Reflex Stiffness of Man's Anti-Gravity Muscles During

[13] Mizrahi J, Susak Z. In-Vivo Elastic and Damping Response of the Human Leg to Impact

[14] Ozguven HN, Berme N. An Experimental and Analytical Study of Impact Forces

[15] Kim W, Voloshin AS, Johnson SH. Modeling of Heel Strike Transients during Running.

[16] Farley CT, Gonzalez O. Leg Stiffness and Stride Frequency in Human Running. *J.* 

[17] Farley C T, Morgenroth D C. Leg Stiffness Primarily Depends on Ankle Stiffness during

[18] Spagele T, Kistner A, Gollhofer A. Modeling, Simulation and Optimization of a Human Vertical Jump. ASME Journal of Biomechanical Engineering 1999;32: 521-530. [19] McMahon TA, Green PR. The Influence of Track Compliance on Running. *J* 

[20] Rapoport S, Mizrahi J, Kimmel E, Verbitsky O, Isakov E. Constant and Variable Impedance of the Leg Joints in Human Hopping. *ASME Journal of Biomechanical* 

[21] Cavanagh PR, Valiant GA, Misevich KW. Biological Aspects of Modeling Shoe/Foot Interaction during Running. In: Frederick EC. (ed.) Sport Shoes and Playing Surfaces.

*[22]* McMahon TA, Valiant G, Frederick EC. Groucho Running. *J Appl. Physiol.* 1987;62: 2326-

[23] Nigg BM. Experimental Techniques Used in Running Shoe Research. In: Nigg BM. (ed.). Biomechanics of Running Shoes. Champaign: Human Kinetics Publishers; 1986; p27-62.

Kneebends while Carrying Extra Weights. *J. Biomechanics,* 1979;12*:* 881-891.

Level Running. Clinical Biomech 2000;15: 15-20.

Human Running. *J. Appl. Biomechanics*, 1998;14: 300-311.

Journal of Bone and Joint Surgery 1972;54A: 1345-1346.

during Human Jumping. *J Biomechanics* 1988;21: 1061-1066.

Human Hopping. Journal of Biomechanics 1999;32: 267-273.

Forces. *J . Biomech. Engng.* 1982;104: 63-66.

*Human Movement Science,* 1994;13: 221-244.

*Biomechanics,* 1996;29: 181-186.

*Biomechanics,* 1979;12: 893-904.

*Engineering* 2003;125: 507-514.

2337.

Champaign: Human Kinetics; 1984. p24-46.

p. 3-29.

While departing from the stiffness constancy concept, the model revealed that a correct and sufficient variability of the joint stiffness is a two-region piece-wise constant stiffness indicating that a higher order of non-linearity is not necessary. This result should be considered meaningful in those problems where the constant stiffness representation is not sufficient and in cases where the system's representation has to be improved. Joint stiffness is dominated by muscular activation [59-60] and as the joints stiffen, they undergo smaller angular displacements during the ground-contact phase, also resulting in smaller excursion of the hip and higher leg stiffness. Thus, since stiffness is related to muscle activation, the piece-wise constant stiffness obtained solution also provides, through the obtained stiffness profiles, an insight into the patterns of the muscular activation in the legs' joints.

The fact that the simple model of a piece-wise constant stiffness can predict major features of the running exercise makes it an effective tool for future designing of artificial legs and robots and also for the development of more accurate control strategies.
