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A method to combine numerical optimization and EMG data for the estimation of joint moments under dynamic conditions David Amarantini*, Luc Martin Laboratoire Sport et Performance Motrice EA 597, Universite! Joseph Fourier, Grenoble cedex 9 38041, France Accepted 16 December 2003
Abstract To solve the problem of muscle redundancy at the level of opposing muscle groups, an alternative method to inverse dynamics must be employed. Considering the advantages of existing alternatives, the present study was aimed to compute knee joint moments under dynamic conditions using electromyographic (EMG) signals combined with non-linear constrained optimization in a single routine. The associated mathematical problems accounted for muscle behavior in an attempt to obtain accurate predictions of the resultant moment as well as physiologically realistic estimates of agonist and antagonist moments. The experiment protocol comprised (1) isometric trials to determine the most effective EMG processing for the prediction of the resultant moment and (2) stepping-in-place trials for the calculation of joint moments from processed EMG under dynamic conditions. Quantitative comparisons of the model predictions with the output of a biological-based model, showed that the proposed method (1) produced the most accurate estimates of the resultant moment and (2) avoided possible inconsistencies by enforcing appropriate constraints. As a possible solution for solving the redundancy problem under dynamic conditions, the proposed optimization formulation also led to realistic predictions of agonist and antagonist moments. r 2004 Elsevier Ltd. All rights reserved. Keywords: Optimization; Electromyography; Co-contraction; Inverse dynamics; Stepping-in-place
1. Introduction The assessment of joint moments for the quantification of co-contraction (Falconer and Winter, 1985; Winter, 1990) remains a significant challenge. Such data represent expressive information for rehabilitation applications (Kellis, 1998), and could provide a powerful means for exploring the control of joint stability (De Serres and Milner, 1991; Milner, 2002), body posture (Winter et al., 1998, 2001) and movement (Latash, 1992). From a mechanical point of view, the estimation of the moments produced by opposing muscle groups requires implementation of an alternative method to the inverse dynamics approach. Indeed, the problem of redundancy inherent in the musculoskeletal system (i.e., more muscles than degrees of freedom) yields not
*Corresponding author. Tel.: +33-3-76-63-50-88; fax: +33-3-76-5144-69. E-mail address:
[email protected] (D. Amarantini). 0021-9290/$ - see front matter r 2004 Elsevier Ltd. All rights reserved. doi:10.1016/j.jbiomech.2003.12.020
enough equilibrium equations to uniquely calculate individual moments (An et al., 1995; Happee, 1994). As a solution for either opposing muscle groups or individual muscles, numerical optimization methods have been used under static and dynamic conditions (Crowninshield and Brand, 1981; Happee, 1994; Pedotti et al., 1978; Seireg and Arvikar, 1973; Stokes and Gardner-Morse, 2001). Using these approaches, forces or moments are obtained by minimizing an appropriate criterion (e.g., sum of normalized forces squared: Pedotti et al., 1978; sum of muscle stresses cubed: Crowninshield and Brand, 1981) subject to constraints (e.g., equilibrium equations: Pedotti et al., 1978; physiological bounds: Happee, 1994). Nevertheless, Buchanan and Shreeve (1996) and Challis and Kerwin (1993) and Challis (1997) highlighted that the solutions from optimization methods could be unsuitable (physiologically unrealistic), specially because they fail to consider an indicator of muscle activity (Cholewicki et al., 1995; Gagnon et al., 2001). A convincing alternative approach should associate numerical optimization with the use of
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electromyographic (EMG) signals, which provide information on the level of muscle activity. Under isometric contractions the development of powerful algorithms for computing efforts from EMG (Buchanan et al., 1993; Laursen et al., 1998; van Dieen and Visser, 1999) is more conveniently performed because the relationship between force and EMG can be considered as linear (Hof, 1997; Onishi et al., 2000). Conversely, the use of EMG as an input variable under dynamic conditions involves major limitations because of muscle dynamics and possible signal artefacts (Kellis, 1998; Rainoldi et al., 2000). To solve this problem, Olney and Winter (1985) introduced a powerful solution, accounting for the moment–angle and the moment–velocity effects. Their influence was incorporated into the model with two biological-based constants depending, respectively, upon angular changes and angular velocity (for details see Olney and Winter, 1985, p.10). The associated experiment protocol comprised two sessions to (1) calculate coefficients to establish the relationship between EMG and the resultant moment under isometric contractions and (2) compute the joint moments during gait according to the coefficients calculated during the first session. Nevertheless, the solutions provided by this method should be adjusted by enforcing sub-weights, in order to perfectly balance the resultant joint moment (Cholewicki and McGill, 1994). Whether including this optimization algorithm as a separate routine can fully satisfy the equilibrium equations associated with the resultant moment (Cholewicki et al., 1995; Gagnon et al., 2001), the agonist and antagonist moments may remain physiologically unrealistic. Moreover with such methods, the associated criterion does not exploit the possibility of reflecting ‘‘a mechanism via which the human body recruits muscles to produce a joint moment’’ (Challis, 1997, p. 254). So, the purpose of the present study was to develop a procedure associating the two alternatives presented above in a single routine for solving the redundancy problem at the level of knee muscle groups under dynamic conditions. To address this issue, we used an approach inspired by Olney and Winter (1985) during stepping-in-place. This task was used for studies on the control of body posture and locomotion (Breniere and Ribreau, 1998; Ivanenko et al., 2000) and presents spatio-temporal characteristics similar to gait (Garcia et al., 2001). From a mathematical point of view: (1) non-linear constrained optimization was implemented for the calculations and (2) the expressions were developed to account for the influence of the biarticularity of the muscles crossing the knee joint (Basmajian and De Luca, 1985, pp. 232–239). The results obtained from our approach are compared with those given by using other algorithms (Olney and Winter, 1985).
2. Methods The proposed method is decomposed in two successive steps (Fig. 1): the first one is aimed to provide the best EMG processing in order to match the knee resultant joint moment under step isometric contractions determined using kinematic and kinetic data. An attempt was made to improve the performance of the model through EMG exponentiation since either linear, quasi-linear or quadratic relationships have been reported to exist between EMG and moment (Marras and Granata, 1997; Metral and Cassar, 1981). In the second step, the agonist and antagonist knee muscle moments are estimated with processed EMG signals (i.e., computed using the coefficients assessed from isometric calibration) and the resultant moment at the knee as input variables in dynamic conditions (stepping-inplace). The problem is formulated as a general optimization problem (Boggs and Tolle, 1996) with a criterion function that combined several components, as previously done by others (Seireg and Arvikar, 1973; Stokes and Gardner-Morse, 2001). Given the significance of muscle function and the possible distortions caused in the EMG, each muscle moment was balanced using a coefficient method inspired by Cholewicki and McGill (1994). Nine healthy male volunteers (age, 23.274.3 years; height, 1.8070.05 m; mass, 74.276.1 kg, means7SD) participated in this study. The anthropometric data used for calculations were taken from tables (Winter, 1990). EMG data were acquired using MYODATAs. Both kinematic and kinetic data were recorded with DATACtm . All data were synchronized by using a single electrical pulse acquired with DATACtm and MYODATAs simultaneously. All computations were done using MATLAB (Math Works, Natick, MA).
2.1. Electromyography For all procedures, surface EMG was recorded through bipolar electrodes (Meditrace, diameter 2 cm, Graphic Controls, Canada) with a 2 cm inter-electrode spacing. After appropriate skin preparation, the electrodes were positioned on the right leg, over the bellies of representative extensor and flexor muscles that function at the knee. Following the recommendations of Olney and Winter (1985) and Manal and Buchanan (2000), the biceps femoris (BF) and the gastrocnemius (GA) were chosen to adequately represent the muscular activity of the knee flexor group while the rectus femoris (RF) and the vastus medialis (VM) represent that of the knee extensor group. Under this hypothesis, the force production capacity of the selected muscles is assumed to be equivalent to that of their corresponding muscle group. The EMG signals were sampled at 1024 Hz, pre-
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Fig. 1. Schematic representation of the processing stages and the experiment protocols used to estimate knee joint moments under dynamic conditions. The equations refer to the expressions detailed in method and appendix sections.
amplified near the recording site (gain 1–600) and then full-wave rectified. 2.2. Methods for isometric EMG to moment processing The isometric calibration procedure was performed while the subjects stood upright with their left foot flat on the floor, arms akimbo. The right leg of each subject was attached to a rigid experimental apparatus designed in an attempt to minimize movement of the individual body segments (Fig. 1). Velcrotm straps were placed around the thigh, the shank and the ankle joint in order to fix the right leg firmly to the frames of the apparatus. Each joint of the right leg of the subjects was stabilized in the middle of the angular range covered during stepping-in-place to yield accurate estimates of the resultant moment (van Dieen and Visser, 1999). Numeric photographs were taken to plot the positions of three markers placed on the lateral malleolus, the lateral epicondyle, and the greater trochanter of the right leg. For all subjects, the hip was flexed to 45 ðyHc Þ
with ankle at 90 ðyAc Þ: The angle at the knee joint was calculated according to the convention in Fig. 2. Thus, knee flexion angle ranged from 61 to 75 between subjects (69.1274.37 , mean7SD). The components of the force generated by the subjects in the anterior ðRx Þ and the vertical ðRy Þ directions during isometric contractions were measured by a load cell (Schlumberger, model CD7501, Ve! lizy-Villacoubay, France; sampling rate: 100 Hz) attached by a harness to the right ankle. In this posture, the EMG–moment relationship was estimated by having each subject performing isometric step efforts. The calibration procedure comprised two experimental sessions that consisted of knee flexion (S1) and knee extension (S2) contractions. For each session, prior to the test, the subjects performed three maximal isometric efforts of 2 s separated by a 5 s rest. The true maximum voluntary contraction (MVC) level used for each session was defined as the maximum of these three attempts. To perform the test the subjects exerted isometric contractions varying from 20% to 80% of
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fc ¼ ffcGA ; fcBF ; fcRF ; fcVM g and p ¼ fpGA ; pBF ; pRF ; pVM g that minimize Ciso ¼ a;fc;p
1X # Kiso ðtÞÞ2 ; ðMKiso ðtÞ M 2 t
ð1Þ
# Kiso are, respectively, the resultant where MKiso and M moment computed from classical mechanical laws and the resultant moment estimated from EMG data, expressed by 3 2 3 2 0 cosðyHc þ yKc Þ 7 6 7 6 0 4 5 ¼ 4 sinðyHc þ yKc Þ 5 MKiso ðtÞ 0 2 3 lK Rx ðtÞ 6 7 #4 lK ððmH þ mK þ mA Þ g Ry ðtÞÞ 5; 0 ð2Þ # Kiso ðtÞ ¼ M # Kiso Extensors ðtÞ M # Kiso Flexors ðtÞ; M
ð3Þ
where # Kiso Extensors ðtÞ ¼ aRF rEMGRF ðtÞpRF M
fcRF
þ aVM rEMGVM ðtÞpfcVM ; VM
ð4Þ
# Kiso Flexors ðtÞ ¼ aGA rEMGGA ðtÞpGA M fcGA þ aBF rEMGBF ðtÞpfcBFBF
ð5Þ
Fig. 2. Configuration of the two-dimensional mechanical model of the lower limb. The segments are modeled as rigid links; the joints are modeled as frictionless hinges. H, K, A are, respectively, the center of the hip, knee and ankle joint. The indexes H, K and A refer to the segments of the thigh, shank and foot, respectively. For each segment i ¼ fH; K; Ag : li is the length, mi the mass, Gi the center of mass, ri the distance between Gi and joint i and Ii ,the moment of inertia. Cp is the position of the center of pressure under the right foot.
subject to 8 > < a > 0:0; 3:0 > fc > 1:0; > : 2:0 > p > 1:0;
their previously measured MVC in steps of 20%. They were asked to do five trials of voluntary efforts up to the current level of MVC for 5 s with 5 s of relaxation between each contraction. The subjects were provided with visual feedback of the required level of MVC on an oscilloscope and were instructed to maintain this level of force for the duration of contraction. The resting period between S1 and S2 was 5 min. The coefficients establishing the EMG signals to resultant moment relation were optimally estimated using a non-linear least-squares curve-fitting procedure with upper and lower bounds on the design variables (a, fc, p). The associated optimization problem can be mathematically formulated as follows:
where a, fc and p are, respectively, the optimal gains, cut-off frequencies and exponents applied to the rectified EMG (rEMG) for each individual muscle. lK is the length of the shank and mH, mK and mA are the masses of the thigh, shank and foot, respectively. The optimal Butterworth filter cut-off frequencies (fc) were subject to constraint boundaries selected from previously reported EMG-assisted models (Olney and Winter, 1985; Potvin et al., 1996; van Dieen and Visser, 1999). The order of the Butterworth low-pass digital filter was set to 4 for all EMG signals. The values of the exponents (p) were optimally estimated in the range of 1.0–2.0 to produce accurate estimates of the resultant moment from EMG data (Clancy and Hogan, 1997; Cholewicki et al., 1995). The minimization of Ciso was carried out with a initially set to 10.0. fc and p were initially set to the mean of their lower and upper bounds.
find a ¼ faGA ; aBF ; aRF ; aVM g;
ð6Þ
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2.3. Methods to estimate knee moments during dynamic exercise To provide data for the estimation of knee moments during dynamic contractions, EMG, kinematic and kinetic data were collected from the subjects during stepping-in-place. The positions of four infrared light emitting diode markers fixed on the fifth metatarsal head, the lateral malleolus, the lateral epicondyle, and the greater trochanter of the right leg were recorded at 400 Hz using a two-dimensional motion measurement system . (Selspot I, Molndal, Sweden). The angular positions, angular velocities and angular accelerations of the body segments were obtained with twice differentiation and smoothed with a second-order Butterworth filter (10 Hz low-pass cut-off frequency). The ground reaction force components necessary to calculate the resultant moment at the knee joint during stepping-in-place were acquired at 200 Hz using an AMTI force plate (AMTI, Newton, MA). The ground reaction force data were computed using the calibration matrix of the force plate and smoothed with a low-pass digital filter (25 Hz cut-off frequency). In the starting position the subjects stood upright, with their feet parallel to each other and spaced about 20 cm, with arms akimbo. The subjects were instructed to perform smooth sequences of stepping-in-place at their preferred speed on the force plate. Each subject was given about 10 cycles training steps before the start of the experiment. The experimental session included five trials of 20 s for each subject. Data sampling started 5 s after the onset of stepping, and lasted for 10 s for each trial to avoid sampling of anticipatory and joint movements induced by the initiation of the movement (Breniere and Dietrich, 1992). For each subject, the optimization was performed using the averaged data from 10 stepping cycles. Thus, the number of cycles considered for the calculations was sufficient to produce reliable data for EMG results interpretation (Arsenault et al., 1986; Shiavi et al., 1998). The resultant knee joint moment during stepping-inplace (MKdyn) was calculated using an inverse dynamics method with external forces and the data from the markers as inputs. The right leg was modeled as a threebar linkage in plane motion (Fig. 2); the joints were modeled as frictionless hinges. Considering the sensitivity of joint moments to uncertainties in input data (Hatze, 2000, 2002), this method yields valid estimates of the resultant joint moments for movements of large amplitude (Cahou.et et al., 2002). So, the resultant moment vector ðM ¼ /MHdyn ; MKdyn ; MAdyn ST Þ was computed by solving the inverse dynamics problem using the Lagrangian formalism, represented by the following expression: M ¼ AðhÞ h. þ BðhÞ h’ h’ þ CðhÞ h’ þ GðhÞ; 2
ð7Þ
1397
where h ¼ / yH ; yK ; yA ST is the angular position vector, h. ¼ /y. H ; y. K ; y. A ST is the angular acceleration vector, 2 h’ ¼ /y’ 2H ; y’ 2K ; y’ 2A ST and h’ h’ ¼ /y’ H y’ K ; y’ H y’ A ; y’ K y’ A ST are velocity vectors. A, B and C are, respectively, inertial, Coriolis’ and centrifugal coefficient matrices. G is the vector sum of external forces (see Appendix A). Using the model reported by Olney and Winter (1985) as a basis, a minimization problem was formulated for the estimation of knee joint moments during dynamic exercise. The objective function to be minimized (Cdyn) was designed in order to (1) find the best fit between MKdyn and the resultant moment estimated from EMG # Kdyn Þ; and (2) produce physiologically realistic data ðM results by avoiding high muscle moments. Thus, Cdyn was formed as the sum of two criteria: the sum of # Kdyn Þ; and squared differences between MKdyn and ðM the sum of individual muscle moments squared. The minimization problem was expressed by find
w; b; d
that minimize Cdyn ¼ w;b;d
1 X MKdyn ðtÞ 2 t # Kdyn ðtÞÞ2 þ M
2 i¼4 # X MKi ðtÞ t;i¼1
wi ðtÞ
! ;
ð8Þ # Kdyn are, respectively, the resultant where MKdyn and M moment computed from inverse dynamics and the resultant moment estimated from EMG data, expressed by # Kdyn ðtÞ ¼ M # Kdyn Extensors þ M # Kdyn Flexors M ð9Þ ¼ ðwðtÞ SðtÞÞT ½1 þ E ðb DhÞ ’ E ðd hÞ subject to 8 # Kdyn Extensors > 0 and M # Kdyn Flexors o0; >M < b > 0 and d > 0; > : 1 > w > 0;
ð10Þ
ð11Þ
# Ki represents the moments of individual where M muscles. S is the vector of EMG signals processed according to the coefficients assessed from isometric calibration and 1 is a four-component vector of length unity. b and d are the coefficient matrices included to account for changes in angular positions and velocities. w and E are, respectively, the matrices of individual muscle gains and biarticularity (see Appendix B). The moments at the knee joint were subject to non-linear inequality constraints so that a positive sign was assigned to each extensor muscle moment whilst each flexor muscle moment was defined as negative. The sequential quadratic programming (SQP; Boggs and Tolle, 1996) approach was used to estimate the
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design variables. Cdyn was minimized with the values of w, b and d initially set to 0.5, 0.1 and 0.1, respectively. 2.4. Data analysis
Table 1 Optimal values (means7SD) of the gains ðaÞ; cut-off frequencies (fc) and exponents (p) used on individual muscle EMG signals to establish the EMG–moment relationship in step isometric contractions Muscle
The ability of the proposed method to estimate MKiso and MKdyn was compared with that obtained from leastsquares estimation (Olney and Winter, 1985) by using the Relative RMS error (Cahou.et et al., 2002) between the knee moment calculated by mechanical laws and the knee moment estimated from the EMG. The cocontraction (CI) of antagonist muscle groups acting across the knee joint was computed at each time t using the expression given by Falconer and Winter (1985) and Winter (1990). 2.5. Statistics For statistical purposes, paired Student’s t-tests were conducted to compare Relative RMS errors under either isometric or dynamic conditions. The variables (a, fc, p) obtained from incremental and optimization procedures under isometric conditions were compared using a 2 Models 4 Muscles analysis of variance (ANOVA) with repeated measures. We proceeded in a similar way to compare w, b and d obtained from non-linear optimization and least-squares estimation under dynamic conditions. Descriptive statistics are means 7 standard deviations throughout. An alpha level of 0.05 was used for all statistical tests.
3. Results 3.1. Isometric calibration results The mean values of the coefficients computed by nonlinear least-squares curve-fitting are presented in Table 1. Apart from the exponents, there was no main effect of Models neither for fc (t8=0.40, p>0.05) nor for a (t8=3.74, p>0.05). Considering the results obtained using our model, the mean value of the optimal cut-off frequency was 2.0370.47 Hz and the contrast analysis revealed no Muscle effect (F3,24=0.98, p>0.05). The results concerning a showed a significant Muscle effect (F3,24=12.37, po0.05) but no significant interaction of Models Muscle (F3,24=1.47, p>0.05). This reveals that (1) the contributions of BF, GA, RF and VM to the resultant moment were similar in the two models and (2) the weights were different between the muscles for each model. It should be noted that inter-subjects variability of a was high because of the high variability of EMG data, inherent in the use of such information. The exponents obtained from curve-fitting differed between the four studied muscles (F3,24=4.40, po0.05) and ranged from 1.0270.03% for VM to 1.2070.20% for
BF GA RF VM
Isometric EMG–Moment coefficients a
fc
13.1775.59 (14.41) 16.08711.80 (21.47) 27.95722.24 (34.31) 8.3878.01 (7.80)
2.1870.76 1.9470.87 1.8770.75 2.1470.80
p (2.34) (2.04) (1.80) (2.18)
1.0470.06 1.1670.15 1.2070.20 1.0270.03
(1.00) (1.00) (1.00) (1.00)
Values between brackets are the solutions determined from the incremental method. BF, biceps femoris; GA, gastrocnemius; RF, rectus femoris; VM, vastus medialis.
RF. The relative RMS error associated with the # Kiso was not improved by predictions of MKiso and M using optimization and EMG exponentiation (t8=0.42, p>0.05). The two methods produced very satisfactory estimates of the resultant moment. Specifically, the # Kiso was mean relative RMS error between MKiso and M 4.1670.64% (mean coefficient of determination (r2)=0.9870.01) for the results obtained using nonlinear curve-fitting. Additionally, the two models produced similar estimates of the agonist and antagonist moments, despite the results showed a main Model effect for the exponents (t8=3.74, po0.05). The EMG signals recorded from BF, GA, RF and VM and the corresponding moments computed for a typical trial with non-linear curve fitting are presented in Fig. 3. Nevertheless, we found that the incremental approach presented by Olney and Winter (1985) could produce inconsistent estimates of the design variables in several calibration trials. This problem did not dramatically affect the mean results but constituted a major limitation because the sign convention (i.e., polarity of muscle moments) was not satisfied. 3.2. Stepping-in-place results (dynamic conditions) # Kdyn and the corresponding The patterns of MKdyn, M agonist and antagonist moments calculated using leastsquares estimation (Olney and Winter, 1985) and those obtained from non-linear constrained optimization are presented in Fig. 4. Contrary to isometric conditions, the optimization method significantly improved the performance of the model during stepping-in-place (t8=8.73, po0.05). Without optimization the resultant moment was wrongly estimated (see Fig. 4): the mean relative RMS deviation associated with MKdyn and # Kdyn M from least-squares estimation was 44.62712.27% (r2=0.2270.21). Consequently, the muscle moments were apparently calculated with major discrepancies: the predictions were frequently overestimated (i.e., exceeded the moment reached at 100% MVC under isometric conditions) and/or failed to
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Fig. 3. (a) Raw-rectified EMG signals recorded during step isometric efforts at 20%, 40%, 60% and 80% MVC under knee extension and flexion contractions and (b) the corresponding knee joint moments estimated using non-linear constrained curve-fitting (BF, biceps femoris; GA, gastrocnemius; RF, rectus femoris; VM, vastus medialis).
satisfy the sign convention. Conversely, the development of the mathematical expressions associated with the optimization problem (Eqs. (8)–(11)) produced very accurate estimates of MKdyn: the mean relative RMS error decreased up to 1.0170.48% (r2E1.0). Also, the solution from the proposed optimization appeared to provide physiologically realistic estimations of the agonist and antagonist moments (Fig. 4) and produced a mean CI of 55.41711.13% in good agreement with Falconer and Winter (1985). The mean values of b; d and w computed using optimization are presented in Table 2. The results showed no significant Models effect on either bK
(t8=0.56, p>0.05) or dK (t8=1.55, p>0.05) but revealed sign discrepancies when using least-squares estimation. The mean of the added weights designed to balance each muscle moment was 0.4870.02 (see Table 2), indicating that no muscle remained inactive over the duration of the cycle.
4. Discussion The present study was aimed to predict the agonist and antagonist muscle moments at the knee joint during dynamic conditions. The associated method used EMG
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Fig. 4. (a) Schematic representation of limb postures during one step (S.S., single-stance phase; D.S., double-stance phase) and (b) the corresponding dynamic raw-rectified EMG signals (BF, biceps femoris; GA, gastrocnemius; RF, rectus femoris; VM, vastus medialis). The resultant, agonist and antagonist moments about the knee computed using least-squares estimation are represented in (c); those obtained from the proposed non-linear constrained optimization method are represented in (d).
ARTICLE IN PRESS D. Amarantini, L. Martin / Journal of Biomechanics 37 (2004) 1393–1404 Table 2 Optimal values (means7SD) of the variables used on processed dynamic EMG signals to predict knee joint moments from non-linear optimization during stepping-in-place BF
GA
Mean individual muscle gains w% 0.5770.08 0.5670.07 Least squares
Hip 1
RF
VM
0.3870.12
0.4270.08
Knee
Ankle
1
Optimal values for b (rad ) and d (s rad ) b 0.1470.75 10370.00a 10370.00a d 0.0370.06 0.1770.10 0.0270.04
2.0171.06 0.0470.08
a Indicates that the solution equals to the lower bound enforced for the resolution (103) for all subjects. BF, biceps femoris; GA, gastrocnemius; RF, rectus femoris; VM, vastus medialis.
data and the estimates of the resultant moment as inputs, and implements non-linear optimization. We used an experimental protocol inspired by Olney and Winter (1985) that comprised an isometric calibration and stepping-in-place trials. For isometric calibration, the incremental (Olney and Winter, 1985) and the optimization algorithms (Eqs. (1)–(6)) produced very accurate estimates of the resultant moment according to obtained relative RMS errors. Although the optimization procedure yielded finer estimates of the design variables, the relationship linking EMG signals and the resultant moment was not improved when implementing non-linear curve-fitting. On the other hand, the optimization method avoided emerging of discrepancies because constraints were enforced. This advantage has been expressed by Challis (1997) for the calculation of realistic muscle forces. Here, the results proved that imposing constraints was necessary to assure that the sign convention was satisfied. Whether evidence was given for the use of optimization, the results concerning the exponents (p) and the cut-off frequencies (fc) suggested that the expressions associated with the optimization problem could be simplified. By using EMG exponentiation, the optimal relationships linking each individual muscle moment and EMG data were not closely linear, in agreement with the results found by others (Onishi et al., 2000). However, from a mathematical point of view, the lack of improvement in the performance of the model indicated that a linear relationship between moment and EMG could be sufficiently adequate under isometric conditions (Olney and Winter, 1985; van Dieen and Visser, 1999). Similarly, the results suggested that fc could be separately assigned a constant value for all muscles. Considering the previous estimates of the optimal cut-off frequency (Olney and Winter, 1985; Potvin et al., 1996) and those from the current method, the use of 2.5 Hz may be relevant for the prediction of the efforts.
1401
Under dynamic conditions, the optimal coefficients obtained from isometric calibration were used to process EMG signals. Large deviations were produced between # Kdyn using a similar routine to that MKdyn and M proposed by Olney and Winter (1985). The predictions of the resultant moment were not sufficiently accurate for our purpose and thus, yielded unsuitable estimates of the agonist and antagonist moments. These discrepancies may be attributed to distortions in EMG signals, differences in the protocol and the algorithms used for isometric calibration, as well as the use of a different task for dynamic conditions. Nevertheless, criticisms could be leveled at the mathematical formulation of the problem. Indeed, inconsistencies regarding the polarity of either design coefficients or muscle moments remained under dynamic conditions when using least-squares estimation. With the developments of the mathematical expressions and the use of optimization algorithms in the proposed method, the accuracy of the fit was dramatically improved since the relative RMS error decreased from 44.62712.27% to 1.0170.48%. The solutions for bK and dK resulting from the two approaches were identical, that is, the improvement of ability to predict MKdyn from EMG data could be explained by (1) the coefficients (w) incorporated to adjust each muscle moment and (2) the terms included to account for the biarticularity of the muscles. Here, the adjustments inspired by the method proposed by Cholewicki and McGill (1994) in a way to assure mechanical equilibrium contributed to accurately estimate the resultant moment. The results indicated the relevancy of considering how angular positions and velocities both affect the production of muscle force (Olney and Winter, 1985) as well as the biarticularity of the muscles crossing the knee joint. Specifically, the high value of bA (associated to the ankle joint) suggested that it was justified to take the biarticularity of GA in account to produce accurate estimates of knee joint moments. Nevertheless, the patterns of b and d were not obvious because of the high inter-subject variability. By using the optimization method, the discrepancies observed from least-squares estimation did not emerge because constraints (boundaries and polarity) were enforced. In addition to producing an excellent degree # Kdyn ; the use of a multiof fit between MKdyn and M component objective function for the non-linear optimization problem fulfilled two major objectives. First, the non-linearity of Cdyn predicted all muscles to be active (Challis and Kerwin, 1993). Secondly, the solution produced realistic estimates of the agonist and antagonist moments since they did not overcome the moments reached at 100% MVC under isometric conditions. In conclusion, our method associates the use of EMG and optimization in a single routine and provides accurate estimates of the muscle moments under
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dynamic conditions. Despite the impossibility of comparing our solutions with the muscle moments measured in vivo, efforts were made to obtain physiologically realistic results (according to the definition given by Challis and Kerwin, 1993, p. 139). Indeed, the model accounts for an indicator of muscle activity, incorporates the isometric moment–EMG relationships, accounts for the moment–angle and the moment–velocity effects and integrates the influence of muscle biarticularity.
C11 ¼ C22 ¼ C33 ¼ 0; C12 ¼ C21 ¼ mK rK lH sin yK mA lH lK sin yK C13
mA rA lH sinðyK þ yA Þ; ¼ C31 ¼ mA rA lK sin yA mA rA lH sinðyK þ yA Þ;
C23 ¼ C32 ¼ mA rA lK sin yA : The vector sum of external forces ðGÞ was expressed by G ¼ QðyÞ þ DðyÞ R; where R represents the reaction force vector and Q1 ¼ g½mH rH cos yH þ mK ðlH cos yH þ rK cosðyH þ yK ÞÞ þ mA ðlH cos yH þ lK cosðyH þ yK Þ
Acknowledgements The authors thank Prof. Alain Barraud (Laboratoire d’Automatique de Grenoble, INPG, France) for his expert suggestions concerning the mathematical intricacies of optimization.
According to Eq. (7) and the notations reported in Fig. 2, the resultant moments at the ankle (MAdyn), knee (MKdyn) and hip (MHdyn) joints were computed from the components of inertial (A), Coriolis’ (B) and centrifugal (C) matrices expressed by 2 A11 ¼ IH þ mH r2H þ IK þ mK r2K þ IA þ mA r2A þ mK lH 2 2 þ mA lH þ mA lK þ 2ðmA rA lH cosðyK þ yA Þ
þ mA lH lK cos yK þ mA rA lK cos yA þ mK rK lH cos yK Þ; mK r2K
mA r2A
2 mA lK
þ IA þ þ A12 ¼ A21 ¼ IK þ þ mK rK lH cos yK þ mA lH lK cos yK
þ mA rA lH cosðyK þ yA Þ þ 2mA rA lK cos yA ; A13 ¼ A31 ¼ IA þ mA r2A þ mA rA lK cos yA þ mA rA lH cosðyK þ yA Þ; 2 A22 ¼ IK þ mK r2K þ IA þ mA r2A þ mA lK
þ 2mA rA lK cos yA ; A23 ¼ A32 ¼ IA þ mA r2A þ mA rA lK cos yA ; mA r2A :
B21 ¼ B32 ¼ B33 ¼ 0; B11 ¼ 2mK rK lH sin yK 2mA lH lK sin yK B12
þ rA cosðyH þ yK þ yA ÞÞ ; Q3 ¼ g½mA rA cosðyH þ yK þ yA Þ : D13 ¼ D23 ¼ D33 ¼ 0; D11 ¼ lH sin yH þ lK sinðyH þ yK Þ þ ðyCp yA Þ; D12 ¼ lH cos yH lK cosðyH þ yK Þ ðxCp xA Þ;
Appendix A
A33 ¼ IA þ
þ rA cosðyH þ yK þ yA ÞÞ ; Q2 ¼ g½mK rK cosðyH þ yK Þ þ mA ðlK cosðyH þ yK Þ
2mA rA lH sinðyK þ yA Þ; ¼ 2mA rA lK sin yA 2mA rA lH sinðyK þ yA Þ;
B13 ¼ 2mA rA lH sinðyK þ yA Þ 2mA rA lK sin yA ; B22 ¼ 2mA rA lK sin yA ; B23 ¼ 2mA rA lK sin yA ; B31 ¼ 2mA rA lK sin yA :
D21 ¼ lK sinðyH þ yK Þ þ ðyCp yA Þ; D22 ¼ lK cosðyH þ yK Þ ðxCp xA Þ; D31 ¼ yCp yA ; D32 ¼ xA xCp :
Appendix B ’ and matrices The components of the vectors ðS; Dh; hÞ ðb; d; w; EÞ designed for the mathematical computation of knee moments (Eq. (10)) are expressed at each time t as follows: 2 3 aGA rEMGGA ðtÞpfcGA GA 6 7 p 6 aBF rEMGBF ðtÞfcBFBF 7 7 SðtÞ ¼ 6 6 a rEMG ðtÞpRF 7; 4 RF RF fcRF 5 aVM rEMGVM ðtÞpfcVM VM
where rEMG represents the raw-rectified EMG signal recorded under dynamic conditions. DhðtÞ ¼ /yH ðtÞ yHc ; yK ðtÞ yKc ; yA ðtÞ yAc ST and ’ ¼ /y’ H ; y’ K ; y’ A ST : hðtÞ b and d are positive diagonal 3 3 matrices such that: b11 ¼ bH ; b22 ¼ bK ; b33 ¼ bA and d11 ¼ dH ; d22 ¼ dK ; d33 ¼ dA : w(t) is a positive diagonal 4 4 matrix such that w11 ðtÞ ¼ wGA ðtÞ; w22 ðtÞ ¼ wBF ðtÞ; w33 ðtÞ ¼ wRF ðtÞ and w44 ðtÞ ¼ wVM ðtÞ: In a attempt to account for the biarticularity of the muscles crossing the knee joint, the matrix E was
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introduced. So, the angular and velocity changes were expressed as a linear combination of DyH ; DyK and DyA ; according to individual physiological muscle functions: 2
0
1
6 1 1 6 E¼6 4 1 1 0
1
1
3
07 7 7: 05 0
References An, K.-N., Kaufman, K.R., Chao, E.Y.-S., 1995. Estimation of muscle and joint forces. In: Allard, P., Stokes, I.A.F., Blanchi, J.P. (Eds.), Three-Dimensional Analysis of Human Movement. Human Kinetics, Champaign, pp. 201–214. Arsenault, A.B., Winter, D.A., Marteniuk, R.G., Hayes, K.C., 1986. How many strides are required for the analysis of electromyographic data in gait? Scandinavian Journal of Rehabilitation Medicine 18, 133–135. Basmajian, J.V., De Luca, C.J., 1985. Muscles Alive, Their Functions Revealed by Electromyography, 5th Edition. Williams & Wilkins, Baltimore. Boggs, P.T., Tolle, J.W., 1996. Sequential quadratic programming. Acta Numerica 4, 1–52. Breniere, Y., Dietrich, G., 1992. Heel-off perturbation during gait initiation: biomechanical analysis using triaxial accelerometry and a force plate. Journal of Biomechanics 25, 121–127. Breniere, Y., Ribreau, C., 1998. A double-inverted pendulum model for studying the adaptability of postural control to frequency during human stepping-in-place. Biological Cybernetics 79, 337–345. Buchanan, T.S., Shreeve, D.A., 1996. An evaluation of optimization techniques for the prediction of muscle activation patterns during isometric tasks. Journal of Biomechanical Engineering 118, 565–574. Buchanan, T.S., Moniz, M.J., Dewald, J.P., Zev Rymer, W., 1993. Estimation of muscle forces about the wrist joint during isometric tasks using an EMG coefficient method. Journal of Biomechanics 26, 547–560. Cahou.et, V., Martin, L., Amarantini, D., 2002. Static optimal estimation of joint accelerations for inverse dynamics problem solution. Journal of Biomechanics 35, 1507–1513. Challis, J.H., 1997. Producing physiologically realistic individual muscle force estimations by imposing constraints when using optimization techniques. Medical Engineering and Physics 19, 253–261. Challis, J.H., Kerwin, D.G., 1993. An analytical examination of muscle force estimations using optimization techniques. Proceedings of the Institution of Mechanical Engineers. Part H, Journal of Engineering in Medicine 207, 139–148. Cholewicki, J., McGill, S.M., 1994. EMG assisted optimization: a hybrid approach for estimating muscle forces in an indeterminate biomechanical model. Journal of Biomechanics 27, 1287–1289. Cholewicki, J., McGill, S.M., Norman, R.W., 1995. Comparison of muscle forces and joint load from an optimization and EMG assisted lumbar spine model: towards development of a hybrid approach. Journal of Biomechanics 28, 321–331. Clancy, E.A., Hogan, N., 1997. Relating agonist–antagonist electromyograms to joint torque during isometric, quasi-isotonic, nonfatiguing contractions. IEEE Transactions on Biomedical Engineering 44, 1024–1028.
1403
Crowninshield, R.D., Brand, R.A., 1981. A physiologically based criterion of muscle force prediction in locomotion. Journal of Biomechanics 14, 793–801. De Serres, S.J., Milner, T.E., 1991. Wrist muscle activation patterns and stiffness associated with stable and unstable mechanical loads. Experimental Brain Research 86, 451–458. Falconer, K., Winter, D.A., 1985. Quantitative assessment of cocontraction at the ankle joint in walking. Electromyography and Clinical Neurophysiology 25, 135–149. Gagnon, D., Lariviere, C., Loisel, P., 2001. Comparative ability of EMG, optimization, and hybrid modelling approaches to predict trunk muscle forces and lumbar spine loading during dynamic sagittal plane lifting. Clinical Biomechanics (Bristol, Avon) 16, 359–372. Garcia, R.K., Nelson, A.J., Ling, W., van Olden, C., 2001. Comparing stepping-in-place and gait ability in adults with and without hemiplegia. Archives of Physical Medicine and Rehabilitation 82, 36–42. Happee, R., 1994. Inverse dynamic optimization including muscular dynamics, a new simulation method applied to goal directed movements. Journal of Biomechanics 27, 953–960. Hatze, H., 2000. The inverse dynamics problem of neuromuscular control. Biological Cybernetics 82, 133–141. Hatze, H., 2002. The fundamental problem of myoskeletal inverse dynamics and its implications. Journal of Biomechanics 35, 109–115. Hof, A.L., 1997. The relationship between electromyogram and muscle force. Sportverletz Sportschaden 11, 79–86. Ivanenko, Y.P., Grasso, R., Lacquaniti, F., 2000. Influence of leg muscle vibration on human walking. Journal of Neurophysiology 84, 1737–1747. Kellis, E., 1998. Quantification of quadriceps and hamstring antagonist activity. Sports Medicine 25, 37–62. Latash, M.L., 1992. Independent control of joint stiffness in the framework of the equilibrium-point hypothesis. Biological Cybernetics 67, 377–384. Laursen, B., Jensen, B.R., Nemeth, G., Sjogaard, G., 1998. A model predicting individual shoulder muscle forces based on relationship between electromyographic and 3D external forces in static position. Journal of Biomechanics 31, 731–739. Manal, K., Buchanan, T.S., 2000. Limitations of inverse dynamics in identifying quadriceps avoidance. Proceedings of the American Society of Biomechanics 24, 143. Marras, W.S., Granata, K.P., 1997. The development of an EMGassisted model to assess spine loading during whole-body freedynamic lifting. Journal of Electromyography and Kinesiology 7, 259–268. Metral, S., Cassar, G., 1981. Relationship between force and integrated EMG activity during voluntary isometric anisotonic contraction. European Journal of Applied Physiology and Occupational Physiology 46, 185–198. Milner, T.E., 2002. Adaptation to destabilizing dynamics by means of muscle cocontraction. Experimental Brain Research 143, 406–416. Olney, S.J., Winter, D.A., 1985. Predictions of knee and ankle moments of force in walking from EMG and kinematic data. Journal of Biomechanics 18, 9–20. Onishi, H., Yagi, R., Akasaka, K., Momose, K., Ihashi, K., Handa, Y., 2000. Relationship between EMG signals and force in human vastus lateralis muscle using multiple bipolar wire electrodes. Journal of Electromyography and Kinesiology 10, 59–67. Pedotti, A., Krishnan, V.V., Stark, L., 1978. Optimization of muscleforce sequencing in human locomotion. Mathematical Biosciences 38, 57–76. Potvin, J.R., Norman, R.W., McGill, S.M., 1996. Mechanically corrected EMG for the continuous estimation of erector spinae muscle loading during repetitive lifting. European Journal of Applied Physiology and Occupational Physiology 74, 119–132.
ARTICLE IN PRESS 1404
D. Amarantini, L. Martin / Journal of Biomechanics 37 (2004) 1393–1404
Rainoldi, A., Nazzaro, M., Merletti, R., Farina, D., Caruso, I., Gaudenti, S., 2000. Geometrical factors in surface EMG of the vastus medialis and lateralis muscles. Journal of Electromyography and Kinesiology 10, 327–336. Seireg, A., Arvikar, R.J., 1973. A mathematical model for evaluation of forces in lower extremeties of the musculo–skeletal system. Journal of Biomechanics 6, 313–326. Shiavi, R., Frigo, C., Pedotti, A., 1998. Electromyographic signals during gait: criteria for envelope filtering and number of strides. Medical and Biological Engineering and Computing 36, 171–178. Stokes, I.A., Gardner-Morse, M., 2001. Lumbar spinal muscle activation synergies predicted by multi-criteria cost function. Journal of Biomechanics 34, 733–740.
van Dieen, J.H., Visser, B., 1999. Estimating net lumbar sagittal plane moments from EMG data. The validity of calibration procedures. Journal of Electromyography and Kinesiology 9, 309–315. Winter, D.A., 1990. Biomechanics and Motor Control of Human Movement. Wiley, New York. Winter, D.A., Patla, A.E., Prince, F., Ishac, M., Gielo-Perczak, K., 1998. Stiffness control of balance in quiet standing. Journal of Neurophysiology 80, 1211–1221. Winter, D.A., Patla, A.E., Rietdyk, S., Ishac, M.G., 2001. Ankle muscle stiffness in the control of balance during quiet standing. Journal of Neurophysiology 85, 2630–2633.
