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Journal of Biomechanics 44 (2011) 984–987 Contents lists available at ScienceDirect Journal of Biomechanics journal homepage: www.elsevier.com/locate/jbiomech www.JBiomech.com Short communication An improved cost function for modeling of muscle activity during running Ali Asadi Nikooyan 1, Amir Abbas Zadpoor n,1 Department of Biomechanical Engineering, Faculty of Mechanical, Maritime, and Materials Engineering, Delft University of Technology, Mekelweg 2, Delft 2628 CD, The Netherlands a r t i c l e i n f o a b s t r a c t Article history: Accepted 23 November 2010 This paper tries to improve a recently developed mass-spring-damper model of the human body during running. The previous model took the muscle activity into account using a nonlinear controller that tuned the mechanical properties of the soft-tissue package based on two physiological hypotheses, namely ‘‘constant-force’’ and ‘‘constant-vibration’’. Three cost functions were used, out of which one was based on the constant-force hypothesis and two others were based on the constant-vibration hypothesis. The results of the study showed that the proposed cost functions are only partially successful in capturing the experimentally observed trends of the ground reaction force and vibration. The current paper proposes an improved cost function that combines both above-mentioned hypotheses. It is shown that the improved cost function can capture all the trends that were observed in the measurements of the ground reaction force and vibration level. It is therefore advised to use the new cost function in place of the previous ones. & 2010 Elsevier Ltd. All rights reserved. Keywords: Mass-spring-damper model Muscle tuning Running Ground reaction force Vibrations 1. Introduction According to the muscle-tuning paradigm, the human body tries to regulate the muscle activity to minimize the vibrations of the lower body’s soft-tissue package (Boyer and Nigg, 2006, 2007; Wakeling et al., 2003, 2002, 2001; Wakeling and Nigg, 2001) or to keep the ground reaction force (GRF) constant (Clarke et al., 1983; Nigg et al., 1987, 1983; Snel et al., 1985) during running. In a recent study (Zadpoor and Nikooyan, 2010), a previously developed mass-spring-damper model of the human body (Liu and Nigg, 2000; Zadpoor and Nikooyan, 2006; Zadpoor et al., 2007) was improved by taking the pre-landing muscle activity into account. For a full description of the model and the physiological questions it tries to answer see Zadpoor and Nikooyan (2010) and the electronic supplement of the current paper. In the so-called improved model, the stiffness and damping properties of the lower body wobbling mass (LBWM) were controlled by a mechanism that mimicked the functionality of the central nervous system (CNS) in determining and issuing the motor signals that are required for muscle tuning. Based on experimental observations, three cost functions, namely force, vibration amplitude, and vibration acceleration, were defined to study the effects of shoe hardness on the loading of the human body during running. The force criterion was based on the constant-force hypothesis whereas the other two criteria were based on the constant-vibration muscle-tuning hypothesis. The ‘‘vibration level’’ n Corresponding author. Tel.: + 31 15 2786794; fax: + 31 15 2784717. E-mail addresses: [email protected], [email protected], [email protected] (A.A. Zadpoor). 1 Both authors have equally contributed to this article and should therefore be considered as joint first authors. 0021-9290/$ - see front matter & 2010 Elsevier Ltd. All rights reserved. doi:10.1016/j.jbiomech.2010.11.032 was defined based on either the vibration amplitude or the half-sine peak acceleration following impact. The results of the previous study (Zadpoor and Nikooyan, 2010) showed that by applying the force cost function, the controller can keep the GRF constant in a specific range of shoe hardness parameters (i.e. safe region). By applying the vibration cost functions, the controller was able to minimize the vibrations for the whole range of shoe parameters. Although each control strategy could capture a part of the experimentally observed trends, none could cover all aspects. Therefore, an improved cost function capable of simultaneously satisfying force and vibration hypotheses is needed. In this study, two cost functions (force and vibration amplitude) are combined into a new criterion. The model is simulated using the new cost function and the simulation results are compared with experimental observations. 2. Materials and methods 2.1. Basic and improved models The ‘‘basic model’’ presented in Zadpoor and Nikooyan (2006) and Zadpoor et al. (2007) was later improved by considering the LBWM as an active element (Fig. 1; Zadpoor and Nikooyan, 2010). The system is connected to the ground via an element (Fg), whose force is a function of the displacement (x1) and velocity (v1) of the LBWM (Liu and Nigg, 2000): ( Ac ½axb1 þ cxd1 ve1 ðx1 4 0Þ Fg ¼ ð1Þ 0 ðx1 r 0Þ The hardness of the shoe is determined by two parameters b and d. The governing equations of the motion can be written as follows: m1 x€ 1 ¼ m1 gFg k1 ðx1 x3 Þk2 /wSðx1 x2 Þc1 ðx_ 1 x_ 3 Þc2 /wSðx_ 1 x_ 2 Þ m2 x€ 2 ¼ m2 g þ k2 /wSðx1 x2 Þk3 ðx2 x3 Þ þ c2 /wSðx_ 1 x_ 2 Þ A.A. Nikooyan, A.A. Zadpoor / Journal of Biomechanics 44 (2011) 984–987 985 Method for bound-constraint minimization (Lewis and Torczon, 1999)), and the same optimization bound limits are used. The limits within which k2 and c2 could change are as follows (Zadpoor and Nikooyan, 2010): bound limits 1:14 k2,0 r k2,i r 4k2,0 , 1 4c2,0 r c2,i r 4c2,0 2 :13 k2,0 r k2,i r 3k2,0 , 3:12 k2,0 r k2,i r 2k2,0 , 1 3c2,0 r c2,i r 3c2,0 1 2c2,0 r c2,i r 2c2,0 bound limits bound limits ð6Þ For each pair of b and d, the k2 and c2 values that minimized the objective functions were calculated using the controller. Simulations were carried out for each bound limit, resulting in three sets of simulations. 3. Results Fig. 1. Schematic drawing of the ‘improved model’. m3 x€ 3 ¼ m3 g þ k1 ðx1 x3 Þ þ k3 ðx2 x3 Þðk4 þ k5 Þðx3 x4 Þ þ c1 ðx_ 1 x_ 3 Þc4 ðx_ 3 x_ 4 Þ m4 x€ 4 ¼ m4 g þ ðk4 þ k5 Þðx3 x4 Þ þ c4 ðx_ 3 x_ 4 Þ ð2Þ The model and its parameters (Fig. 1) are fully described elsewhere (Liu and Nigg, 2000; Zadpoor and Nikooyan, 2010; Zadpoor et al., 2007). In the improved model, for each pair of the shoe hardness parameters (b and d), the controller minimizes the cost function by adjusting the stiffness (k2) and damping (c2) coefficients of the LBWM (Zadpoor and Nikooyan, 2010). The stiffness coefficient k2 and damping coefficient c2 represent the muscle-tendon properties and are in the range of experimentally measured values (Liu and Nigg, 2000). The bound limits used for optimization of k2 and c2 are based on experimental observations and guarantee that the optimized values are within the physiological range (Zadpoor and Nikooyan, 2010). The parameters k2 and c2 are considered time-independent. This assumption is thoroughly discussed in Zadpoor and Nikooyan (2010). Similar to the simulations that used the constant-force hypothesis, there is a particular area of the b–d plane within which the normalized optimization error (optimization residual) is very close to zero (Fig. 2a–c). This region is called the ‘‘safe region’’ and is defined as a part of the b–d plane within which the normalized error is less than 0.05 (the region lies between two dashed lines in each subfigure of Fig. 2). However, comparing the current simulation results with those of the previous study shows that the new safe region has reduced in size by about 50%. This is because satisfaction of the combined cost functions is stricter that satisfying either force or vibration cost functions. The maximum and minimum values of k2 and c2, when the error is brought to zero by the controller, are listed in Table 1. One can see in this table that the selected bound limits for the pattern search optimization considerably affect the minimum value of c2, while they have no sizable impact on the minimum value of k2 and maximum value of c2. The maximum values of k2 are only slightly (2–6%) affected. The GRF plots for different pairs of shoe hardness parameters (Fig. 3) confirm that only inside the safe region the peak forces remain unchanged (compared to the default values). For soft shoes, the pattern of the time response of the GRF (Fig. 3) when using the improved cost function is similar to that of the force criterion. For hard shoes, the pattern resembles the simulation results of the improved model, when vibration amplitude cost function was used. 4. Discussions 2.2. Combined force–vibration criterion Two cost functions, namely force (Jf) and vibration amplitude (Jv1), were defined as follows (Zadpoor and Nikooyan, 2010): Jf ¼ 9p1 /bi ,di Sp1,0 /b0 ,d0 S9þ 9p2 /bi ,di Sp2,0 /b0 ,d0 S9 ð3Þ Jv1 ¼ 9Li /bi ,di SL0 /b0 ,d0 S9 ð4Þ where p1 and p2 are the first and second ground reaction force peaks. The default force values (p1,0 ¼ 1436.8 N and p2,0 ¼ 2026.4 N) are the force peaks calculated using the default values of the shoe hardness parameters b0 (¼ 1.38) and d0 (¼0.75) (Zadpoor et al., 2007). L is the vibration amplitude (the vertical displacement of the soft tissues) and L0 (¼ 10.1 cm) is the default value of the vibration amplitude that were calculated using the default values of b0 and d0 (Nikooyan and Zadpoor, 2007). The new cost function is defined as sum of the two above-mentioned cost functions. The objective functions are first nondimensionalized by normalizing each term with respect to its default value. The new criterion (Jfv) can be therefore written as p1 /bi ,di Sp1,0 p2 /bi ,di Sp2,0 Li /bi ,di SL0 ð5Þ Jfv ¼ þ þ p1,0 p2,0 L0 2.3. Simulations To make comparison of simulation results between the current and the previous study (Zadpoor and Nikooyan, 2010) as easy as possible, the same range of the shoe hardness parameters (b and d), the same optimization technique (Pattern Search In order to assess the usefulness of the combined cost function, the simulation results are compared with three sets of experimental observations. The first comparison is between the GRF values calculated here and the ones measured during actual running exercises (Clarke et al., 1983) as well as the findings of the study by Ferris and Farley (1997). Both in the simulation results and in experimental observations, there is a range of hardness values (safe region) for which the GRF remains constant. Outside the safe region, the GRF deviates from default values. Many other researchers have observed similar trends in their experiments. Lieberman et al. (2010) observed that the mean impact transient force is not dependent on the hardness of the platform nor is it dependent on whether the runners are barefoot or shod. Other researchers (Dixon et al., 2000; Kerdok et al., 2002) have observed that by varying the stiffness of the surface and/or footwear, there is a range of stiffness parameters for which the ground reaction force does not significantly change. Comparing the force cost function (Jf) with the improved cost function (Jfv), one can see that the latter is less sensitive to the selected bound limits (Table 1): when using Jfv, the minimum values of the stiffness show no dependency on the bound limit, while this is not the case for Jf. 986 A.A. Nikooyan, A.A. Zadpoor / Journal of Biomechanics 44 (2011) 984–987 Fig. 2. Simulation results: the normalized error (a)–(c), the absolute error of the force part (d)–(f), the absolute error of the amplitude part (g)–(i), stiffness coefficients (j)–(l), damping coefficients (m)–(o), the first force peak (p)–(r), the second force peak (s)–(u), and vibration amplitude (v)–(x). Columns 1–3 of this figure relate to bound limits 1–3 defined by Eq. (6). The shoe hardness changes from ‘‘the softest’’ at the left top to ‘‘the hardest’’ at the right down. A.A. Nikooyan, A.A. Zadpoor / Journal of Biomechanics 44 (2011) 984–987 Table 1 Values of stiffness (k2) and damping (c2) coefficients when the optimization error (optimization residual) is close to zero for three different bound limits (UB, upper bound; LB, lower bound). These values show that the involved muscles do not need to be extremely stiff to keep the GRF and the level of vibrations constant. k2 (N/m) c2 (kg/s) Bound limit 1 Bound limit 2 Bound limit 3 UB Max. 24,000 8048 18,000 7536 12,000 7350 LB Min. 1500 4976 2000 4976 3000 4976 UB Max. 2600 907 1950 907 1300 907 LB Min. 162.5 174 216.67 279 325 327 987 latest stages of contact with the ground, when the post-landing muscle activity plays an important role. Conflict of interest statement We hereby state that we have not had any financial or personal relationships with other people or organizations that could inappropriately influence (bias) our work. Appendix A. Supplementary Materials Supplementary data associated with this article can be found in the online version at doi:10.1016/j.jbiomech.2010.11.032 References Vertical Ground Reaction Force (N) 2500 2000 default model Jf safe region 1500 Jv1 safe region Jfv safe region Jf soft shoe 1000 Jv1 soft shoe Jfv soft shoe Jf hard shoe 500 Jv1 hard shoe Jfv hard shoe 0 0 0.05 0.1 0.15 0.2 0.25 Stance Time (s) Fig. 3. Ground reaction force vs. time applying some selected pairs of (b, d) parameters; default values b¼ 1.38, d¼ 0.75, relatively soft shoe: b¼ 1.24, d¼ 0.81, relatively hard shoe: b¼ 1.71, d¼ 0.74, and inside the safe region: b¼1.33, d¼0.69. For all cost functions, the results related to bound limit 1 are presented. The second comparison is between the experimental observations about vibration level (Nigg and Wakeling, 2001) and the vibration level predicted by the model (Fig. 2g–i). One can see that, in agreement with experimental observations, there is only negligible change in the predicted vibration level even when the shoe hardness varies substantially. The third comparison is between the stiffness and damping coefficients calculated using the combined cost function and the previous experimental measurements that suggested the role of damping in muscle tuning is more than that of the stiffness (Wakeling et al., 2002; Wilson et al., 2001). According to these studies, stiffness coefficients change far less than the damping coefficients. The same pattern is observed in the simulation results (Fig. 3m–o). Damping coefficient increases as the shoe hardness increases. However, the stiffness coefficient remains almost unchanged for a wide range of the shoe hardness parameters (Fig. 3j–l). 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