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Muscle coordination of mediolateral balance in normal walking
Journal of Biomechanics 43 (2010) 2055–2064
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Muscle coordination of mediolateral balance in normal walking Marcus G. Pandy n, Yi-Chung Lin, Hyung Joo Kim Department of Mechanical Engineering, University of Melbourne, Parkville, Victoria 3010, Australia
a r t i c l e in fo
abstract
Article history: Accepted 13 April 2010
The aim of this study was to describe and explain how individual muscles control mediolateral balance during normal walking. Biomechanical modeling and experimental gait data were used to quantify individual muscle contributions to the mediolateral acceleration of the center of mass during the stance phase. We tested the hypothesis that the hip, knee, and ankle extensors, which act primarily in the sagittal plane and contribute significantly to vertical support and forward progression, also accelerate the center of mass in the mediolateral direction. Kinematic, force plate, and muscle EMG data were recorded simultaneously for five healthy subjects who walked at their preferred speeds. The body was modeled as a 10-segment, 23 degree-of-freedom skeleton, actuated by 54 muscles. Joint moments obtained from inverse dynamics were decomposed into muscle forces by solving an optimization problem that minimized the sum of the squares of the muscle activations. Muscles contributed significantly to the mediolateral acceleration of the center of mass throughout stance. Muscles that generated both support and forward progression (vasti, soleus, and gastrocnemius) also accelerated the center of mass laterally, in concert with the hip adductors and the plantarflexor everters. Gravity accelerated the center of mass laterally for most of the stance phase. The hip abductors, anterior and posterior gluteus medius, and, to a much lesser extent, the plantarflexor inverters, actively controlled balance by accelerating the center of mass medially. & 2010 Elsevier Ltd. All rights reserved.
Keywords: Musculoskeletal Gait Center-of-mass acceleration Stability Model Simulation
1. Introduction The vertical and fore-aft components of the ground reaction force for normal walking are stereotypic and well known. The vertical ground reaction force is marked by the appearance of two peaks, one in early stance and the other in late stance, whereas the shape of the fore-aft component resembles a sine wave. The mediolateral ground reaction force is also stereotypic, but has received less attention in the gait literature, presumably because it is much smaller than the peak forces applied in the vertical and fore-aft directions. The mediolateral ground reaction is directed medially for most of the stance phase, and its magnitude remains less than 5% of body weight (BW) (Giakas and Baltzopoulos, 1997). The net acceleration of the center of mass in the mediolateral direction can be deduced directly from force plate measurements of the mediolateral ground reaction force. Irregularities in the mediolateral acceleration have been used to diagnose and treat elderly patients who are prone to falling (Lord and Sturnieks, 2005), as well as patients with balance problems arising from neuromuscular disorders such as Parkinson’s disease (Chou et al., 2003; Latt et al., 2009). The next step in understanding the biomechanical causes of gait abnormalities in
n
Corresponding author. Tel.: + 61 3 8344 4054; fax: + 61 3 8344 4290. E-mail address: [email protected] (M.G. Pandy).
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patients with balance problems is establishing a causal link between muscle action and the mediolateral acceleration of the center of mass. In walking, the leg muscles fulfil three distinct functions during stance: (1) they generate support by opposing the downward pull of gravity; (2) they generate progression by accelerating the body forward; and (3) they control sideways (mediolateral) balance during each step (Perry, 1967; Winter, 1995). The contributions of individual muscles to vertical support and forward progression have been evaluated for walking over a wide range of speeds (Anderson and Pandy, 2003; Neptune et al., 2004; Liu et al., 2006, 2008; Neptune et al., 2008). Although different models were used in these studies, the results are consistent in their predictions of leg–muscle function. Support and progression are generated mainly by the actions of five muscle groups: gluteus maximus, gluteus medius, vasti, soleus, and gastrocnemius (Anderson and Pandy, 2003; Neptune et al., 2004; Liu et al., 2006). Gluteus maximus, gluteus medius, and vasti provide most of the vertical acceleration of the center of mass and also decrease the forward speed of the body during the first half of stance, whereas soleus and gastrocnemius support the body and propel it forward during the second half of stance. Little is known about how muscles accelerate the center of mass in the mediolateral direction. Many studies have measured the mediolateral acceleration of the center of mass for level
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walking (Helbostad and Moe-Nilssen, 2003; Kavanagh et al., 2004), but these studies are descriptive and do not consider how leg–muscle action is coordinated to produce the measured accelerations. MacKinnon and Winter (1993) and Kuo (1999) used biomechanical modeling to study mediolateral balance during gait, but their models did not include muscles. On the basis of an inverted pendulum model, MacKinnon and Winter (1993) found that gravity accelerated the center of mass medially, and that an abductor moment was needed about the stance-leg hip to maintain balance. Kuo (1999) used a more complex 3D model of walking and showed that mediolateral balance cannot be maintained without the addition of active control at the stance-leg hip. Although these studies have identified the importance of a hip abductor moment in controlling mediolateral balance, no quantitative data are available to show how the individual leg muscles coordinate motion of the center of mass in the mediolateral direction. In the present study, a three-dimensional, muscle-actuated model of the body was used to better understand muscle coordination of mediolateral balance during normal walking. Because the dynamical equations of motion of a multi-joint system are coupled, each muscle contributes to the accelerations of all the joints, and therefore to the acceleration of the center of mass of the body, at each instant (Zajac and Gordon, 1989; Pandy, 2001). We hypothesized, therefore, that muscles such as gluteus maximus, vasti, soleus, and gastrocnemius, which act primarily in the sagittal plane and contribute significantly to vertical support and forward progression, also accelerate the center of mass in the mediolateral direction. Our specific aims were firstly, to calculate the contributions of the individual leg muscles to the net joint moments and joint angular accelerations developed about the hip, knee, and ankle during stance; secondly, to assess the net contributions of all muscle forces, gravity, and other external forces to the mediolateral acceleration of the center of mass; and finally, to determine which muscles contribute most significantly to the acceleration of the center of mass in the mediolateral direction.
2. Methods Gait experiments were performed on five healthy males (age: 2674 yrs; weight: 707 5 kg; height: 17874 cm). Joint motion, ground reaction forces, and muscle EMG data were recorded simultaneously as each subject walked at his preferred speed on level ground. Three-dimensional locations of retro-reflective markers attached to each subject’s body were measured using a 9-camera, videobased, motion capture system (Vicon, Oxford Metrics Ltd., Oxford). Foot-ground forces were measured using three strain-gauged force plates (Advanced Mechanical Technology Inc., Watertown, MA). Pairs of pre-amplified EMG surface electrodes (Motion Laboratory Systems, Baton Rouge, LA) were attached to both legs to record activity from eight muscles in each leg: gluteus medius, gluteus maximus, lateral hamstrings, rectus femoris, vastus medialis, medial gastrocnemius, soleus, and tibialis anterior. Details of retro-reflective marker and EMG electrode placement are given by Jancic (2009). Video data were sampled at 120 Hz, whereas analog force plate and EMG data were recorded at 1080 Hz. Marker trajectories were lowpass filtered using a fourth-order Butterworth filter with a cut-off frequency of 4 Hz. All experiments were conducted in the Human Motion Laboratory at the University of Melbourne, after approval was obtained from the University’s Human Research Ethics Committee. Each subject gave written informed consent prior to participating in the study. A 3D muscle-actuated model was used to calculate lower-limb muscle forces for one gait cycle. The model was identical in structure to that described by Anderson and Pandy (1999). The skeleton was represented as a 10-segment, 23 degree-of-freedom linkage. The head, arms, and torso were modeled as a single rigid body, which articulated with the pelvis via a ball-and-socket back joint. Each hip was modeled as a ball-and-socket joint, each knee as a hinge joint, each anklesubtalar complex as a universal joint, and each metatarsal joint as a hinge. The locations of the joint centers and the orientations of the joint axes in the model were found by minimizing differences between the positions of surface markers located on the subject and virtual markers defined in the model (Reinbolt et al., 2005; Kim et al., 2009). Subject-specific models of the skeleton were created by scaling the segmental inertial properties of the model to each subject’s height and weight. The model was actuated by 54 muscle–tendon units, with each actuator
represented as a Hill-type muscle in series with tendon. The force-generating properties, attachment sites, and paths of all the muscles in the model were the same as those identified by Anderson and Pandy (1999). Muscle forces were found using inverse dynamics and static optimization. Measurements of the subject’s joint motion and ground reaction forces were input into the skeletal model, and inverse dynamics was used to calculate the net moments exerted about the back, hip, knee, and ankle joints for one gait cycle. The net joint moments were decomposed into individual muscle forces by solving an optimization problem that minimized the sum of the squares of the muscle activations. The optimization problem was solved subject to the physiological bounds on muscle force imposed by each muscle’s force–length–velocity property (Anderson and Pandy, 2001). A pseudo-inverse force decomposition method was used to quantify the contributions of all muscle forces, gravity, and other external forces to the ground reaction force generated during walking (Lin et al., 2010). Five foot–ground contact points were identified on the sole of the foot in the model: four points located at the perimeters of the hindfoot segment and one at the distal end of the toes segment. At each instant of the gait cycle, the dynamical equations of motion were used to compute the contribution of each muscle force to the joint angular accelerations and to the forces acting at each of the five foot–ground contact points. Previous attempts to decompose the ground reaction force (e.g., Anderson and Pandy, 2003) rigidly fixed the foot–ground contact points by prescribing zero acceleration at each of these points. The present ‘pseudo-inverse method’ defined transition stages between free and fixed foot–ground contact points, based on the position of the center of pressure; specifically, the accelerations of the foot–ground contact points were permitted to vary linearly from free to fixed (i.e., zero acceleration) as the foot transitioned from heel-contact to foot-flat and from fixed to free as the foot transitioned from foot-flat to heel-off (see Lin et al. (2010) for details). Once the contribution of a muscle force to the ground reaction force was known, Newton’s 2nd law of motion (i.e., force¼ mass acceleration) was used to determine the muscle’s contribution to the acceleration of the center of mass of the body. The contributions from gravity and other external forces were found similarly. The contributions of all muscle forces, gravity, and other external forces to the joint angular accelerations and to the acceleration of the center of mass were calculated for one representative trial for each subject, and the data were then averaged across all subjects. Data were obtained for one complete gait cycle; however, only the results for the stance phase are presented below.
3. Results Gluteus maximus and hamstrings contributed significantly to the hip extensor moment during the first half of stance, whereas iliopsoas dominated the hip flexor moment in the second half of stance (Fig. 1). Anterior and posterior gluteus medius contributed nearly all of the abductor moment applied about the hip. These muscles also dominated the hip axial rotation moment, with gluteus maximus contributing significantly as well. Soleus and gastrocnemius generated nearly all of the plantarflexor moment at the ankle (Fig. 2). The plantarflexor inverters and everters contributed most significantly to the net inversion–eversion moment applied about the subtalar joint. Gluteus maximus, hamstrings, and vasti accelerated the hip into extension during the first half of stance, whereas iliopsoas accelerated the hip into flexion during the second half of stance (Fig. 3). Soleus accelerated the hip into extension during the second half of stance. Muscles accelerated the hip into adduction for most of the stance phase (Fig. 3, shaded region). Iliopsoas, vasti, soleus, gastrocnemius, the hip adductors, and erector spinae all accelerated the hip into adduction, while only anterior and posterior gluteus medius accelerated the hip into abduction (Fig. 3 and Table 1). Whereas anterior and posterior gluteus medius acted in unison to accelerate the hip into abduction, these muscles induced opposing joint angular accelerations in the transverse plane to control hip axial rotation. Iliopsoas, gluteus maximus, and soleus also contributed significantly to the angular acceleration of the hip in rotation. Gravity accelerated the hip into adduction for most of the stance phase (Fig. 3, GRAV). Gluteus maximus and soleus, two uniarticular muscles crossing the hip and ankle, respectively, accelerated the knee into
M.G. Pandy et al. / Journal of Biomechanics 43 (2010) 2055–2064
extension with as much vigour as did vasti during stance (Fig. 4). Iliopsoas accelerated the knee into flexion throughout stance. Soleus accelerated the ankle into plantarflexion during late stance, whereas iliopsoas induced a dorsiflexor acceleration (Fig. 4). Soleus, gastrocnemius, posterior gluteus medius, and the plantarflexor inverters accelerated the subtalar joint into inversion during late stance, while iliopsoas, anterior gluteus medius, and the plantarflexor everters accelerated the joint into eversion (Fig. 4 and Table 1). Gravity accelerated the subtalar joint into eversion. Muscles and gravity contributed significantly to the mediolateral acceleration of the center of mass during stance (Fig. 5). Gravity accelerated the center of mass laterally for most of the stance phase; in late stance, gravity accelerated the center of mass
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medially. The summed effect of all muscle forces was to accelerate the center of mass medially in early and midstance; in late stance, all muscles combined to accelerate the center of mass laterally (Fig. 5, ALL MUSCLES). Other external forces, such as the inertial forces, contributed significantly to the mediolateral acceleration of the center of mass only during double-leg stance. The summed contributions of all muscle forces, gravity, and other external forces to the mediolateral acceleration of the center of mass in the model was closely similar to the mean mediolateral acceleration of the center of mass measured for the subjects (Fig. 5, compare EXPT and MODEL). The root-mean-square (RMS) value of the difference between the measured and calculated mediolateral acceleration of the center of mass over the entire stance phase was 0.12 m/s2.
Hip Extension RHS
LTO
LHS
RTO
MS
Moment (Nm)
HA
50
GM AX
0 RF
-50
I L PS
O
HAMS
Hip Abduction Moment (Nm)
60 40 20
DA GME
P ED GM GMAX
0 -20
AD
D
GMED
Hip Internal Rotation GM ED A
Moment (Nm)
20 10
ILPSO
0 -10
GMAX
DP
E GM
GM AX
20
40
60
Gait Cycle (%) Fig. 1. Contributions of individual muscles to the net moments exerted about the hip. The shaded regions represent the net moments exerted by all the muscles spanning the hip. Positive moments generated about the hip represent extension, abduction, and internal rotation; negative moments represent flexion, adduction, and external rotation. Results are the mean of all subjects and are shown for the stance phase of walking. Symbols defining the major gait events are as follows: RHS, heel-strike of the ipsilateral (right) leg; LTO, toe-off of the contralateral (left) leg; LHS, left heel-strike; and RTO, right toe-off. Muscle symbols appearing in the graphs are: GMAX, medial, and lateral portions of gluteus maximus combined; GMEDA and GMEDP, anterior and posterior portions of gluteus medius/minimus, respectively; ILPSO, iliacus and psoas major combined; RF, rectus femoris; HAMS, semimembranosus, semitendinosus and biceps femoris long head combined; ADD, adductor magnus, adductor longus, and adductor brevis combined. The horizontal bars indicate the periods of EMG activity recorded for the muscles shown. Note that GMAX represents activity recorded from medial gluteus maximus; GMED represents activity recorded from anterior gluteus medius; HAMS represents activity recorded from lateral hamstrings (biceps femoris long head). No EMG data were recorded for ILPSO and ADD.
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Knee Extension RHS
LTO
LHS
RTO
Moment (Nm)
40 VAS
RF
20 0 S
M
HA
-20
GAS
-40 VAS RF
Ankle Plantarflexion Moment (Nm)
100 L SO
50
GAS
0
GAS SOL
Subtalar Inversion
Moment (Nm)
10
PF I N
5
GAS
SO L
0 PFEV
-5 -10 0
20
40
60
Fig. 2. Contributions of individual muscles to the net moments exerted about the knee, ankle, and subtalar joints. The shaded regions represent the net moments exerted by all the muscles spanning each joint. Positive moments represent knee extension, ankle plantarflexion, and subtalar inversion; negative moments represent knee flexion, ankle dorsiflexion, and subtalar eversion. Results are the mean of all subjects and are shown for the stance phase of walking. Muscle symbols appearing in the graphs are: VAS, vastus medialis, vastus intermedius, and vastus lateralis combined; GAS, medial and lateral compartments of gastrocnemius combined; SOL, soleus, RF, rectus femoris; HAMS, semimembranosus, semitendinosus, and biceps femoris long head combined; PFIN, tibialis posterior, flexor digitorum longus, and flexor hallucis longus combined; PFEV, peroneus brevis and peroneus longus combined. The horizontal bars indicate the periods of EMG activity recorded for the muscles shown. Note that VAS represents activity recorded from vastus medialis; GAS represents activity recorded from medial gastrocnemius. No EMG data were recorded for PFIN and PFEV.
Gluteus maximus, gluteus medius, vasti, soleus, and gastrocnemius contributed most significantly to the vertical and fore-aft accelerations of the center of mass during stance (Fig. 6, TOTAL). Vasti, soleus, gastrocnemius, the hip adductors, and the plantarflexor everters accelerated the center of mass laterally, whereas anterior and posterior gluteus medius and the plantarflexor inverters accelerated the center of mass medially.
4. Discussion Muscles that lie primarily in the sagittal plane and contribute significantly to vertical support and forward progression (i.e., vasti, soleus, and gastrocnemius) also regulate the
mediolateral acceleration of the center of mass. The reason is that the skeleton is a system of jointed segments which are dynamically coupled; specifically, each muscle force is transmitted by all the joints, and thus, each muscle accelerates all the body segments simultaneously (Zajac and Gordon, 1989; Pandy, 2001). Vasti generated support and also decreased the forward speed of the center of mass during the first half of stance. In the mediolateral direction, vasti (together with the hip adductors and gravity) accelerated the body laterally during the first half of stance (Fig. 6). If balance is to be maintained, these actions must be resisted by the actions of other muscles. Anterior and posterior gluteus medius acted synergistically to oppose the actions of vasti and gravity by accelerating the center of mass medially during the first half of stance. Similarly, soleus and gastrocnemius supported
M.G. Pandy et al. / Journal of Biomechanics 43 (2010) 2055–2064
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Hip Extension
Angular Acceleration (rad/s2)
RHS
LTO
LHS
RTO
H
AM
500
G MA X
S
SOL
VAS
0
ILP S
-500
O
-1000
Angular Acceleration (rad/s2)
Hip Abduction GM E D A
100
GMEDP 0
ADD
VA S
GAS
GRAV
SO
ILPS O
-100
L
-200
Hip Internal Rotation Angular Acceleration (rad/s2)
GM E DA SO ILP
100
0 P ED L GM SO
GM AX
-100
0
20
40
60
Gait Cycle (%) Fig. 3. Contributions of individual muscles to the joint angular accelerations of the hip. The shaded regions represent the contributions from all muscles to the joint angular accelerations of the hip in each plane. Positive joint angular accelerations represent extension, abduction, and internal rotation; negative joint angular accelerations represent flexion, adduction, and external rotation. Results are the mean of all subjects. Muscle symbols as defined in the captions for Figs. 1 and 2. GRAV is the acceleration induced by gravity. See also Table 1.
the body and propelled it forward during the second half of stance, and these muscles also accelerated the body laterally at this time (Fig. 6). Anterior and posterior gluteus medius actively controlled balance by accelerating the center of mass medially during the second half of stance. As a general principle, muscles which accelerated the hip into abduction also accelerated the center of mass medially (i.e., anterior and posterior gluteus medius), whereas muscles which accelerated the hip into adduction also accelerated the center of mass laterally (i.e., vasti, soleus, gastrocnemius, and the hip adductors) (Table 1). This principle is illustrated in Fig. 7, which shows a double pendulum model of the body in the frontal plane. Anterior and posterior gluteus medius applied abductor moments about the stance hip and accelerated the hip into abduction (Figs. 1 and 3). A hip abductor moment applies a counter-clockwise moment to the stance limb that transmits a laterally directed force to the ground. The ground must therefore apply an equal and opposite force in the medial direction, which
explains why anterior and posterior gluteus medius accelerated the center of mass medially. Similarly, vasti and the hip adductors accelerated the stance hip into adduction in the first half of stance, whereas iliopsoas, soleus, and gastrocnemius accelerated the stance hip into adduction during the second half (Fig. 3 and Table 1). These muscles induce forces that tend to turn the stance limb clockwise in the frontal plane, thereby transmitting a medially directed force to the ground. The ground must therefore apply an equal and opposite force in the lateral direction, which explains why vasti, soleus, gastrocnemius, iliopsoas, and the hip adductors accelerated the center of mass laterally. In contrast to MacKinnon and Winter (1993), we found that gravity accelerated the center of mass laterally for most of the single-leg stance phase (Figs. 5 and 6). MacKinnon and Winter used a single pendulum model of balance to conclude that gravity accelerates the center of mass medially during single-leg stance. In single-leg stance, the position of the center of mass lies
2060 Table 1 Contributions of muscles and gravity to the net joint moments at the hip, knee and ankle; the joint angular accelerations of the hip, knee, and ankle; the vertical, fore-aft, and mediolateral accelerations of the center of mass (COM); and the vertical, fore-aft, and mediolateral ground reaction forces. Data presented are the peak contributions calculated for the stance phase, averaged across all subjects. Muscle symbols are defined in the captions of Figs. 1 and 2. Hip extension, hip abduction, hip internal rotation, knee extension, ankle plantarflexion, and subtalar inversion are positive for moments and joint angular accelerations. Forward, upward, and lateral are positive for COM accelerations and ground reaction forces. Joint acceleration (rad/s2)
Joint moment (Nm) Hip
GMEDA GMEDP ADD ILPSO HAMS RF VAS SOL GAS PFIN PFEV ERCSPN Gravity
Mean STD Mean STD Mean STD Mean STD Mean STD Mean STD Mean STD Mean STD Mean STD Mean STD Mean STD Mean STD Mean STD Mean STD
Ankle
Hip
Knee
Ankle
Exten
Abd
IntRot
Exten
Plantar
Inver
Exten
Abd
IntRot
Exten
Plantar
Inver
37.8 10.4 3.2 8.4 14.1 1.2 1.1 7.2 77.6 4.0 77.5 17.8 24.5 5.2 – – – – – – – – – – – – 49.0 86.5
7.6 1.9 33.1 5.5 26.5 5.2 19.9 6.8 3.1 0.2 0.2 0.8 6.3 1.9 – – – – – – – – – – – – 52.2 32.2
14.3 5.0 23.3 7.5 12.3 2.5 2.2 0.5 3.6 1.4 1.1 0.3 0.4 0.2 – – – – – – – – – – – – 5.2 11.3
– – – – – – – – – – 40.2 9.3 22.4 4.6 33.2 6.0 0.0 0.0 29.4 5.9 – – – – – – 41.7 23.7
– – – – – – – – – – – – – – – – 67.7 11.9 47.2 9.2 3.5 2.0 6.3 1.5 – – 87.4 51.4
– – – – – – – – – – – – – – – – 5.2 1.0 3.5 0.5 8.2 4.6 8.5 2.2 – – 7.9 11.6
402.5 129.0 60.2 93.9 161.3 16.1 26.8 88.1 777.0 117.4 586.3 170.3 160.7 80.2 183.2 20.4 239.2 33.9 40.9 6.3 17.5 11.9 20.8 6.4 288.0 128.4 20.9 46.3
1.6 40.6 128.8 28.4 74.9 17.3 63.4 15.2 121.5 47.3 65.8 32.5 19.6 7.5 58.2 18.9 108.0 19.9 31.7 65.8 1.7 12.0 7.5 12.9 101.0 44.8 52.8 8.1
125.4 30.9 156.6 50.2 65.6 12.8 30.3 11.1 143.4 31.3 76.7 43.9 36.0 18.1 71.9 12.1 90.3 24.6 3.4 30.8 14.4 7.5 0.6 6.3 29.1 14.5 35.1 4.2
237.1 83.5 113.8 52.0 99.8 9.4 18.0 50.7 488.2 27.6 146.6 24.6 68.2 17.0 246.1 41.7 257.1 46.5 13.8 84.8 19.6 11.0 35.5 13.8 67.3 24.6 12.0 69.3
139.4 90.5 45.4 14.3 55.5 17.8 44.1 29.3 289.3 20.9 101.4 82.9 33.4 9.5 235.2 108.3 229.2 34.5 219.1 169.7 38.3 13.9 60.5 38.3 34.6 9.2 123.9 110.1
1.1 17.5 20.2 8.8 8.5 11.6 26.3 7.3 35.3 43.6 25.6 14.8 18.8 7.2 17.9 20.0 29.7 28.5 41.1 13.0 16.0 6.3 18.5 11.6 11.4 3.9 35.2 4.1
Ground reaction force (BW)
Fore-aft
Vertical
Mediolateral
Fore-aft
Vertical
Mediolateral
0.3 0.4 0.1 0.4 0.0 0.5 0.2 0.3 0.6 0.3 0.7 0.3 0.8 0.2 1.5 0.2 1.9 0.2 1.3 0.3 0.3 0.1 0.2 0.2 0.1 0.2 0.7 0.7
2.8 0.8 1.6 0.4 2.3 0.6 1.0 0.5 0.5 2.8 1.8 0.7 1.0 0.2 2.3 2.2 7.1 0.7 3.8 0.8 0.5 0.6 0.0 0.7 1.1 0.6 9.6 0.0
0.1 0.3 0.7 0.2 0.7 0.1 0.5 0.1 0.1 0.5 0.2 0.2 0.2 0.0 0.4 0.2 0.8 0.3 0.6 0.2 0.1 0.1 0.2 0.1 0.1 0.1 0.0 0.5
0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.1 0.1 0.0 0.1 0.0 0.2 0.0 0.2 0.0 0.1 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.1 0.1
0.3 0.1 0.2 0.0 0.2 0.0 0.1 0.1 0.1 0.3 0.2 0.1 0.1 0.0 0.2 0.2 0.7 0.1 0.4 0.1 0.1 0.1 0.0 0.1 0.1 0.0 0.3 0.1
0.0 0.0 0.1 0.0 0.1 0.0 0.0 0.0 0.0 0.1 0.0 0.0 0.0 0.0 0.0 0.0 0.1 0.0 0.1 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0
M.G. Pandy et al. / Journal of Biomechanics 43 (2010) 2055–2064
GMAX
Knee
COM acceleration (m/s2)
M.G. Pandy et al. / Journal of Biomechanics 43 (2010) 2055–2064
2061
Angular Acceleration (rad/s2)
Knee Extension RHS
LTO
LHS
RTO
400 VA S
200
S OL
GM A X
0 -200 ILP S
O
-400
L SO
V
200
AS
Angular Acceleration (rad/s2)
Ankle Plantarflexion
GMAX
GAS
0 ILP S
-200
O
Angular Acceleration (rad/s2)
Subtalar Inversion 50 GAS
SO L
GMEDP
0 GMEDA GR A V ILPSO
-50 0
20
40
60
Gait Cycle (%) Fig. 4. Contributions of individual muscles to the joint angular accelerations of the knee, ankle, and subtalar joints. Each shaded region represents the contributions from all muscles to the angular acceleration of the joint. Positive joint angular accelerations represent knee extension, ankle plantarflexion, and subtalar inversion; negative joint angular accelerations represent knee flexion, ankle dorsiflexion, and subtalar eversion. Results are the mean of all subjects. Muscle symbols as defined in the captions for Figs. 1 and 2. GRAV is the acceleration induced by gravity. The contributions of the plantarflexor inverters and everters to subtalar joint angular acceleration were smaller than those of the other muscles shown in the graph (see Table 1).
medial to the base of support. If the body is modeled as a single pendulum in the frontal plane, gravity will act to accelerate the center of mass medially (i.e., to the left in Fig. 7). In the present study, gravity accelerated the center of mass in the opposite direction, laterally. This result may seem counterintuitive, but it is readily understood by considering, again, the double pendulum model in Fig. 7. In both the single- and doublependulum models, gravity’s effect on the acceleration of the center of mass is determined by the orientation of the stance leg in the frontal plane. In the double pendulum, whenever the angle between the leg and the ground is less than 901 (as shown in Fig. 7), gravity will tend to tip the body to the right, and the center of mass will be accelerated laterally. Conversely, if the angle between the stance leg and the ground is greater than 901, gravity will tend to tip the body in the other direction, to the left, and the center of mass then will be accelerated medially. As shown in Fig. 3, gravity acted to accelerate the hip into adduction during
single-leg stance, and, as explained above, this action causes the center of mass to be accelerated laterally. Nonetheless, the angle between the stance leg and the ground remains the critical factor in determining the direction in which gravity accelerates the center of mass. In the present study, the angle between the stance leg and the ground in the frontal plane was slightly less than 901 for most of single-leg stance, which explains why gravity accelerated the body laterally. We note here that the angle between the stance leg and the ground is a function of step width; increasing one’s step width will increase this angle (y for the double pendulum model shown in Fig. 7) and could thereby cause gravity to accelerate the center of mass medially during single-leg stance. We may predict, therefore, based on the results of Fig. 6, that an increase in step width would require less active control from anterior and posterior gluteus medius, as gravity would then assist these muscles in accelerating the center of mass medially. This may be
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Gait Cycle (%) Fig. 5. Contributions of all muscle forces (ALL MUSCLES), gravity (GRAV), residual forces (RESIDUAL), centrifugal and Coriolis forces (CENTRIFUGAL), and inertial forces (INERTIAL) to the mediolateral acceleration of the center of mass of the body. Residual forces were the forces acting on the pelvis segment and were calculated from an inverse dynamics analysis applied to the model skeleton. Inertial forces were comprised of fictitious forces that arose because of the assumption of rigid contact imposed at each of the foot–ground contact points in the model. The inertial forces were computed using Eq. (7) given by Anderson and Pandy (2003). The experimental result (EXPT) was obtained directly from the mean mediolateral ground reaction force measured for the subjects. MODEL represents the summed contributions of all muscle forces, gravity, centrifugal forces, residual forces, and inertial forces at each instant of the gait cycle. The shaded regions represent 7 1 standard deviation from the mean for both the model and experimental results.
one reason why young children and older adults walk with relatively wide steps compared with healthy young adults (Whittle, 1996; Murray et al., 1969). Based on the predictions of the single pendulum model, Winter (1995) concluded that the plantarflexor inverters and everters have negligible involvement in balance control. Our results indicate that although these muscles do not contribute as significantly to the mediolateral acceleration of the center of mass as the other plantarflexors, soleus and gastrocnemius (see Fig. 6), their involvement is not negligible. The plantarflexor inverters applied an internal rotation moment about the subtalar joint, accelerating this joint internally and the center of mass medially. Conversely, the plantarflexor everters applied an external rotation moment about the subtalar joint, accelerating it externally and the center of mass laterally (Table 1). We conclude that the plantarflexor inverters assist anterior and posterior gluteus medius in controlling mediolateral balance during stance. Although the present analysis is based on gait data obtained from only five subjects, we have confidence in the conclusions
derived from the model. The time history of the mean mediolateral acceleration of the center of mass measured for the five subjects is consistent with measurements reported in the literature, with a peak medial acceleration of approximately 0.75 m/s2 occurring in double-leg stance (compare EXPT in Fig. 5 with results presented by MacKinnon and Winter (1993) in their Fig. 4). Muscle contributions to joint angular accelerations in the frontal and transverse planes have not previously been reported for walking, but our results are consistent with those published for the sagittal plane. Arnold et al. (2005) analysed a dynamic optimization solution of normal walking and found that gluteus maximus, vasti, and hamstrings contributed significantly to hip extensor acceleration during the first half of stance, whereas posterior gluteus medius and soleus accelerated the hip into extension during the second half of stance. At the knee, vasti and gluteus maximus contributed significantly, and in nearly equal proportions, to accelerating the joint into extension in the first half of stance, whereas posterior gluteus medius and soleus accelerated the knee into extension in the second half. Results given in Figs. 3 and 4 and Table 1 agree with these findings. The patterns of muscle
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Fig. 6. Contributions of individual muscles and gravity (GRAV) to the vertical, fore-aft, and mediolateral accelerations of the center of mass in normal walking. The shaded regions represent the accelerations of the center of mass in the vertical, fore-aft, and mediolateral directions contributed by all the stance leg muscles. TOTAL represents the summed contribution of all muscles shown for each direction; specifically, for vertical acceleration (top row), TOTAL is the sum of GMAX, GMEDA, GMEDP, VAS, SOL, and GAS; for fore-aft acceleration (middle row), TOTAL is the sum of GMAX, VAS, GMEDP, SOL, and GAS; for mediolateral acceleration (bottom row), TOTAL is the sum of GMEDA, GMEDP, ADD, VAS, SOL, GAS, PFIN, and PFEV. Positive accelerations are directed upward, forward, and laterally; negative accelerations are directed downward, backward, and medially. Results are the mean of all subjects. Muscle symbols as defined in the captions for Figs. 1 and 2.
contributions to vertical support and forward progression obtained in this study (Fig. 6) are also consistent with those reported previously by Anderson and Pandy (2003), Liu et al. (2006), Neptune et al. (2008), and Liu et al. (2008). Another potential limitation of the present study is that static optimization was used to solve the muscle-force distribution problem. The static solution was constrained by the force–length– velocity property of muscle, but activation dynamics was neglected. Anderson and Pandy (2001) compared the lower-limb muscle forces obtained from static and dynamic optimization and showed that muscle activation dynamics has little influence on the solution derived from static optimization (see also Supplementary Material). Fig. 5 shows that muscles and gravity contributed significantly to the mediolateral acceleration of the center of mass throughout stance, and that the inertial forces also contributed significantly, but only during double-leg stance. The inertial forces were comprised of fictitious forces that arose because of the assumption of rigid contact imposed at each of the foot–ground contact points
in the model. Had the assumption of rigid foot–ground contact been sound, the magnitude of the contribution of the inertial forces would have been small throughout stance, and the solution of the force decomposition problem then would have been more accurate. Results of Fig. 5 suggest, therefore, that interpretations of leg–muscle function in double-leg stance ought to be viewed more cautiously than those derived for single-leg stance. In conclusion, it is tempting to view mediolateral balance during normal walking as a dynamic equilibrium created by the interaction of gravity and the hip abductor muscles alone (MacKinnon and Winter, 1993). In the framework of the single pendulum model of balance, gravity creates a moment that accelerates the center of mass medially, while the hip abductors apply a moment that accelerates the center of mass in the opposite direction to counter the effect of gravity. The present analysis shows that this is too simplistic a view to take. Muscles that generate both vertical support and forward progression (vasti, soleus, and gastrocnemius) also accelerate the center of mass laterally, in concert with the hip adductors, the plantarflexor
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Ackland, Tom Correa, and Anthony Schache for reviewing an earlier draft of this paper. This work was supported by a VESKI Innovation Fellowship and Australian Research Council Discovery Grants DP0772838 and DP0878705.
Mabductor mg
Appendix A. Supplementary Material
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Supplementary data associated with this article can be found in the online version at doi:10.1016/j.jbiomech.2010.04.010.
Y Fground Z Fground reaction
α mg
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Fig. 7. Top: double-pendulum model of the body in the frontal plane used to illustrate how a medially directed ground reaction force can be induced by an abductor moment applied at the hip. The hip abductor muscles apply a net abductor moment, Mabductor, at the hip, which in turn applies a counter-clockwise moment on the stance leg as shown. This moment generates a laterally-directed force (pointing to the right in the diagram) on the ground (Fground). The ground applies an equal and opposite medially-directed force to the leg (Fground reaction). Bottom: single-pendulum and double-pendulum models of the body in the frontal plane. Because the center of mass remains medial to the base of support (i.e., the stance foot) during single-leg stance, the angle between the stance leg and the ground, y, must be greater than 901 for the single pendulum. Gravity therefore acts to tilt the pendulum medially in this model. The double pendulum lumps the mass of the body at the tip of the pelvis segment, and so the direction in which gravity acts to tip the model is determined by the angle between the stance leg and the ground. This angle could be smaller or greater than 901. If the angle between the stance leg and the ground is smaller than 901, gravity will act to tip the model laterally (to the right); otherwise, gravity will act to tip the model medially (to the left). a represents the angle between the stance leg and the pelvis.
everters, and gravity. The hip abductors, anterior and posterior gluteus medius, and, to a much lesser extent, the plantarflexor inverters, actively control balance by accelerating the center of mass medially.
Conflict of interest The authors do not have any financial or personal relationships with other people or organizations that could inappropriately influence their work.
Acknowledgements We thank Mirjana Jancic and Anthony Schache for their help with data collection. We are also grateful to Richard Baker, David
References Anderson, F.C., Pandy, M.G., 1999. A dynamic optimization solution for vertical jumping in three dimensions. Computer Methods in Biomechanics and Biomedical Engineering 2, 201–231. Anderson, F.C., Pandy, M.G., 2001. Static and dynamic optimization solutions for gait are practically equivalent. Journal of Biomechanics 34, 153–161. Anderson, F.C., Pandy, M.G., 2003. Individual muscle contributions to support in normal walking. Gait & Posture 17, 159–169. Arnold, A.S., Anderson, F.C., Pandy, M.G., Delp, S.L., 2005. Muscular contributions to hip and knee extension during the single stance phase of normal gait: a framework for investigating the causes of crouch gait. Journal of Biomechanics 38, 2181–2189. Chou, L.S., Kaufman, K.R., Hahn, M.E., Brey, R.H., 2003. Medio-lateral motion of the center of mass during obstacle crossing distinguishes elderly individuals with imbalance. Gait & Posture 18, 125–133. Giakas, G., Baltzopoulos, V., 1997. Time and frequency domain analysis of ground reaction forces during walking: an investigation of variability and symmetry. Gait & Posture 5, 189–197. Helbostad, J.L., Moe-Nilssen, R., 2003. The effect of gait speed on lateral balance control during walking in healthy elderly. Gait & Posture 18, 27–36. Jancic, M., 2009. Muscle function during walking and running. Master’s thesis. Department of Mechanical Engineering, University of Melbourne. Kavanagh, J.J., Barrett, R.S., Morrison, S., 2004. Upper body accelerations during walking in healthy young and elderly men. Gait & Posture 20, 291–298. Kim, H.J., Fernandez, J.W., Akbarshahi, M., Walter, J., Fregly, B.J., Pandy, M.G., 2009. Evaluation of predicted knee-joint muscle forces during gait using an instrumented knee implant. Journal of Orthopaedic Research 27, 1326–1331. Kuo, A.D., 1999. Stabilization of lateral motion in passive dynamic walking. International Journal of Robotics Research 18, 917–930. Latt, M.D., Menz, H.B., Fung, V.S., Lord, S.R., 2009. Acceleration patterns of the head and pelvis during gait in older people with Parkinson’s disease: a comparison of fallers and nonfallers. Journal of Gerontology, Series A: Biological Sciences Medical Sciences. 64, 700–706. Lin Y.C., Kim H.J., Pandy M.G. A computationally efficient method for assessing muscle function during human locomotion. International Journal for Numerical Methods in Biomedical Engineering. in press, 2010. Liu, M.Q., Anderson, F.C., Pandy, M.G., Delp, S.L., 2006. Muscles that support the body also modulate forward progression during walking. Journal of Biomechanics 39, 2623–2630. Liu, M.Q., Anderson, F.C., Schwartz, M.H., Delp, S.L., 2008. Muscle contributions to support and progression over a range of walking speeds. Journal of Biomechanics 41, 3243–3252. Lord, S.R., Sturnieks, D.L., 2005. The physiology of falling: assessment and prevention strategies for older people. Journal of Science and Medicine in Sport 8, 35–42. MacKinnon, C.D., Winter, D.A., 1993. Control of whole body balance in the frontal plane during human walking. Journal of Biomechanics 26, 633–644. Murray, M.P., Kory, R.C., Clarkson, B.H., 1969. Walking patterns in healthy old men. Journal of Gerontology 24, 169–178. Neptune, R.R., Sasaki, K., Kautz, S.A., 2008. The effect of walking speed on muscle function and mechanical energetics. Gait & Posture 28, 135–143. Neptune, R.R., Zajac, F.E., Kautz, S.A., 2004. Muscle force redistributes segmental power for body progression during walking. Gait & Posture 19, 194–205. Pandy, M.G., 2001. Computer modeling and simulation of human movement. Annual Review of Biomedical Engineering 3, 245–273. Perry, J., 1967. The mechanics of walking. Physical Therapy 47, 778–801. Reinbolt, J.A., Schutte, J.F., Fregly, B.J., Koh, B.I., Haftka, R.T., George, A.D., Mitchell, K.H., 2005. Determination of patient-specific multi-joint kinematic models through two-level optimization. Journal of Biomechanics 38, 621–626. Whittle, M.W., 1996. Gait Analysis: An Introduction. Butterworth-Heinemann, Oxford. Winter, D.A., 1995. Human balance and posture control during standing and walking. Gait & Posture 3, 193–214. Zajac, F.E., Gordon, M.E., 1989. Determining muscle’s force and action in multiarticular movement. Exercise and Sport Sciences Reviews 17, 187–230.
Contents lists available at ScienceDirect
Journal of Biomechanics journal homepage: www.elsevier.com/locate/jbiomech www.JBiomech.com
Muscle coordination of mediolateral balance in normal walking Marcus G. Pandy n, Yi-Chung Lin, Hyung Joo Kim Department of Mechanical Engineering, University of Melbourne, Parkville, Victoria 3010, Australia
a r t i c l e in fo
abstract
Article history: Accepted 13 April 2010
The aim of this study was to describe and explain how individual muscles control mediolateral balance during normal walking. Biomechanical modeling and experimental gait data were used to quantify individual muscle contributions to the mediolateral acceleration of the center of mass during the stance phase. We tested the hypothesis that the hip, knee, and ankle extensors, which act primarily in the sagittal plane and contribute significantly to vertical support and forward progression, also accelerate the center of mass in the mediolateral direction. Kinematic, force plate, and muscle EMG data were recorded simultaneously for five healthy subjects who walked at their preferred speeds. The body was modeled as a 10-segment, 23 degree-of-freedom skeleton, actuated by 54 muscles. Joint moments obtained from inverse dynamics were decomposed into muscle forces by solving an optimization problem that minimized the sum of the squares of the muscle activations. Muscles contributed significantly to the mediolateral acceleration of the center of mass throughout stance. Muscles that generated both support and forward progression (vasti, soleus, and gastrocnemius) also accelerated the center of mass laterally, in concert with the hip adductors and the plantarflexor everters. Gravity accelerated the center of mass laterally for most of the stance phase. The hip abductors, anterior and posterior gluteus medius, and, to a much lesser extent, the plantarflexor inverters, actively controlled balance by accelerating the center of mass medially. & 2010 Elsevier Ltd. All rights reserved.
Keywords: Musculoskeletal Gait Center-of-mass acceleration Stability Model Simulation
1. Introduction The vertical and fore-aft components of the ground reaction force for normal walking are stereotypic and well known. The vertical ground reaction force is marked by the appearance of two peaks, one in early stance and the other in late stance, whereas the shape of the fore-aft component resembles a sine wave. The mediolateral ground reaction force is also stereotypic, but has received less attention in the gait literature, presumably because it is much smaller than the peak forces applied in the vertical and fore-aft directions. The mediolateral ground reaction is directed medially for most of the stance phase, and its magnitude remains less than 5% of body weight (BW) (Giakas and Baltzopoulos, 1997). The net acceleration of the center of mass in the mediolateral direction can be deduced directly from force plate measurements of the mediolateral ground reaction force. Irregularities in the mediolateral acceleration have been used to diagnose and treat elderly patients who are prone to falling (Lord and Sturnieks, 2005), as well as patients with balance problems arising from neuromuscular disorders such as Parkinson’s disease (Chou et al., 2003; Latt et al., 2009). The next step in understanding the biomechanical causes of gait abnormalities in
n
Corresponding author. Tel.: + 61 3 8344 4054; fax: + 61 3 8344 4290. E-mail address: [email protected] (M.G. Pandy).
0021-9290/$ - see front matter & 2010 Elsevier Ltd. All rights reserved. doi:10.1016/j.jbiomech.2010.04.010
patients with balance problems is establishing a causal link between muscle action and the mediolateral acceleration of the center of mass. In walking, the leg muscles fulfil three distinct functions during stance: (1) they generate support by opposing the downward pull of gravity; (2) they generate progression by accelerating the body forward; and (3) they control sideways (mediolateral) balance during each step (Perry, 1967; Winter, 1995). The contributions of individual muscles to vertical support and forward progression have been evaluated for walking over a wide range of speeds (Anderson and Pandy, 2003; Neptune et al., 2004; Liu et al., 2006, 2008; Neptune et al., 2008). Although different models were used in these studies, the results are consistent in their predictions of leg–muscle function. Support and progression are generated mainly by the actions of five muscle groups: gluteus maximus, gluteus medius, vasti, soleus, and gastrocnemius (Anderson and Pandy, 2003; Neptune et al., 2004; Liu et al., 2006). Gluteus maximus, gluteus medius, and vasti provide most of the vertical acceleration of the center of mass and also decrease the forward speed of the body during the first half of stance, whereas soleus and gastrocnemius support the body and propel it forward during the second half of stance. Little is known about how muscles accelerate the center of mass in the mediolateral direction. Many studies have measured the mediolateral acceleration of the center of mass for level
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walking (Helbostad and Moe-Nilssen, 2003; Kavanagh et al., 2004), but these studies are descriptive and do not consider how leg–muscle action is coordinated to produce the measured accelerations. MacKinnon and Winter (1993) and Kuo (1999) used biomechanical modeling to study mediolateral balance during gait, but their models did not include muscles. On the basis of an inverted pendulum model, MacKinnon and Winter (1993) found that gravity accelerated the center of mass medially, and that an abductor moment was needed about the stance-leg hip to maintain balance. Kuo (1999) used a more complex 3D model of walking and showed that mediolateral balance cannot be maintained without the addition of active control at the stance-leg hip. Although these studies have identified the importance of a hip abductor moment in controlling mediolateral balance, no quantitative data are available to show how the individual leg muscles coordinate motion of the center of mass in the mediolateral direction. In the present study, a three-dimensional, muscle-actuated model of the body was used to better understand muscle coordination of mediolateral balance during normal walking. Because the dynamical equations of motion of a multi-joint system are coupled, each muscle contributes to the accelerations of all the joints, and therefore to the acceleration of the center of mass of the body, at each instant (Zajac and Gordon, 1989; Pandy, 2001). We hypothesized, therefore, that muscles such as gluteus maximus, vasti, soleus, and gastrocnemius, which act primarily in the sagittal plane and contribute significantly to vertical support and forward progression, also accelerate the center of mass in the mediolateral direction. Our specific aims were firstly, to calculate the contributions of the individual leg muscles to the net joint moments and joint angular accelerations developed about the hip, knee, and ankle during stance; secondly, to assess the net contributions of all muscle forces, gravity, and other external forces to the mediolateral acceleration of the center of mass; and finally, to determine which muscles contribute most significantly to the acceleration of the center of mass in the mediolateral direction.
2. Methods Gait experiments were performed on five healthy males (age: 2674 yrs; weight: 707 5 kg; height: 17874 cm). Joint motion, ground reaction forces, and muscle EMG data were recorded simultaneously as each subject walked at his preferred speed on level ground. Three-dimensional locations of retro-reflective markers attached to each subject’s body were measured using a 9-camera, videobased, motion capture system (Vicon, Oxford Metrics Ltd., Oxford). Foot-ground forces were measured using three strain-gauged force plates (Advanced Mechanical Technology Inc., Watertown, MA). Pairs of pre-amplified EMG surface electrodes (Motion Laboratory Systems, Baton Rouge, LA) were attached to both legs to record activity from eight muscles in each leg: gluteus medius, gluteus maximus, lateral hamstrings, rectus femoris, vastus medialis, medial gastrocnemius, soleus, and tibialis anterior. Details of retro-reflective marker and EMG electrode placement are given by Jancic (2009). Video data were sampled at 120 Hz, whereas analog force plate and EMG data were recorded at 1080 Hz. Marker trajectories were lowpass filtered using a fourth-order Butterworth filter with a cut-off frequency of 4 Hz. All experiments were conducted in the Human Motion Laboratory at the University of Melbourne, after approval was obtained from the University’s Human Research Ethics Committee. Each subject gave written informed consent prior to participating in the study. A 3D muscle-actuated model was used to calculate lower-limb muscle forces for one gait cycle. The model was identical in structure to that described by Anderson and Pandy (1999). The skeleton was represented as a 10-segment, 23 degree-of-freedom linkage. The head, arms, and torso were modeled as a single rigid body, which articulated with the pelvis via a ball-and-socket back joint. Each hip was modeled as a ball-and-socket joint, each knee as a hinge joint, each anklesubtalar complex as a universal joint, and each metatarsal joint as a hinge. The locations of the joint centers and the orientations of the joint axes in the model were found by minimizing differences between the positions of surface markers located on the subject and virtual markers defined in the model (Reinbolt et al., 2005; Kim et al., 2009). Subject-specific models of the skeleton were created by scaling the segmental inertial properties of the model to each subject’s height and weight. The model was actuated by 54 muscle–tendon units, with each actuator
represented as a Hill-type muscle in series with tendon. The force-generating properties, attachment sites, and paths of all the muscles in the model were the same as those identified by Anderson and Pandy (1999). Muscle forces were found using inverse dynamics and static optimization. Measurements of the subject’s joint motion and ground reaction forces were input into the skeletal model, and inverse dynamics was used to calculate the net moments exerted about the back, hip, knee, and ankle joints for one gait cycle. The net joint moments were decomposed into individual muscle forces by solving an optimization problem that minimized the sum of the squares of the muscle activations. The optimization problem was solved subject to the physiological bounds on muscle force imposed by each muscle’s force–length–velocity property (Anderson and Pandy, 2001). A pseudo-inverse force decomposition method was used to quantify the contributions of all muscle forces, gravity, and other external forces to the ground reaction force generated during walking (Lin et al., 2010). Five foot–ground contact points were identified on the sole of the foot in the model: four points located at the perimeters of the hindfoot segment and one at the distal end of the toes segment. At each instant of the gait cycle, the dynamical equations of motion were used to compute the contribution of each muscle force to the joint angular accelerations and to the forces acting at each of the five foot–ground contact points. Previous attempts to decompose the ground reaction force (e.g., Anderson and Pandy, 2003) rigidly fixed the foot–ground contact points by prescribing zero acceleration at each of these points. The present ‘pseudo-inverse method’ defined transition stages between free and fixed foot–ground contact points, based on the position of the center of pressure; specifically, the accelerations of the foot–ground contact points were permitted to vary linearly from free to fixed (i.e., zero acceleration) as the foot transitioned from heel-contact to foot-flat and from fixed to free as the foot transitioned from foot-flat to heel-off (see Lin et al. (2010) for details). Once the contribution of a muscle force to the ground reaction force was known, Newton’s 2nd law of motion (i.e., force¼ mass acceleration) was used to determine the muscle’s contribution to the acceleration of the center of mass of the body. The contributions from gravity and other external forces were found similarly. The contributions of all muscle forces, gravity, and other external forces to the joint angular accelerations and to the acceleration of the center of mass were calculated for one representative trial for each subject, and the data were then averaged across all subjects. Data were obtained for one complete gait cycle; however, only the results for the stance phase are presented below.
3. Results Gluteus maximus and hamstrings contributed significantly to the hip extensor moment during the first half of stance, whereas iliopsoas dominated the hip flexor moment in the second half of stance (Fig. 1). Anterior and posterior gluteus medius contributed nearly all of the abductor moment applied about the hip. These muscles also dominated the hip axial rotation moment, with gluteus maximus contributing significantly as well. Soleus and gastrocnemius generated nearly all of the plantarflexor moment at the ankle (Fig. 2). The plantarflexor inverters and everters contributed most significantly to the net inversion–eversion moment applied about the subtalar joint. Gluteus maximus, hamstrings, and vasti accelerated the hip into extension during the first half of stance, whereas iliopsoas accelerated the hip into flexion during the second half of stance (Fig. 3). Soleus accelerated the hip into extension during the second half of stance. Muscles accelerated the hip into adduction for most of the stance phase (Fig. 3, shaded region). Iliopsoas, vasti, soleus, gastrocnemius, the hip adductors, and erector spinae all accelerated the hip into adduction, while only anterior and posterior gluteus medius accelerated the hip into abduction (Fig. 3 and Table 1). Whereas anterior and posterior gluteus medius acted in unison to accelerate the hip into abduction, these muscles induced opposing joint angular accelerations in the transverse plane to control hip axial rotation. Iliopsoas, gluteus maximus, and soleus also contributed significantly to the angular acceleration of the hip in rotation. Gravity accelerated the hip into adduction for most of the stance phase (Fig. 3, GRAV). Gluteus maximus and soleus, two uniarticular muscles crossing the hip and ankle, respectively, accelerated the knee into
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extension with as much vigour as did vasti during stance (Fig. 4). Iliopsoas accelerated the knee into flexion throughout stance. Soleus accelerated the ankle into plantarflexion during late stance, whereas iliopsoas induced a dorsiflexor acceleration (Fig. 4). Soleus, gastrocnemius, posterior gluteus medius, and the plantarflexor inverters accelerated the subtalar joint into inversion during late stance, while iliopsoas, anterior gluteus medius, and the plantarflexor everters accelerated the joint into eversion (Fig. 4 and Table 1). Gravity accelerated the subtalar joint into eversion. Muscles and gravity contributed significantly to the mediolateral acceleration of the center of mass during stance (Fig. 5). Gravity accelerated the center of mass laterally for most of the stance phase; in late stance, gravity accelerated the center of mass
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medially. The summed effect of all muscle forces was to accelerate the center of mass medially in early and midstance; in late stance, all muscles combined to accelerate the center of mass laterally (Fig. 5, ALL MUSCLES). Other external forces, such as the inertial forces, contributed significantly to the mediolateral acceleration of the center of mass only during double-leg stance. The summed contributions of all muscle forces, gravity, and other external forces to the mediolateral acceleration of the center of mass in the model was closely similar to the mean mediolateral acceleration of the center of mass measured for the subjects (Fig. 5, compare EXPT and MODEL). The root-mean-square (RMS) value of the difference between the measured and calculated mediolateral acceleration of the center of mass over the entire stance phase was 0.12 m/s2.
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40
60
Gait Cycle (%) Fig. 1. Contributions of individual muscles to the net moments exerted about the hip. The shaded regions represent the net moments exerted by all the muscles spanning the hip. Positive moments generated about the hip represent extension, abduction, and internal rotation; negative moments represent flexion, adduction, and external rotation. Results are the mean of all subjects and are shown for the stance phase of walking. Symbols defining the major gait events are as follows: RHS, heel-strike of the ipsilateral (right) leg; LTO, toe-off of the contralateral (left) leg; LHS, left heel-strike; and RTO, right toe-off. Muscle symbols appearing in the graphs are: GMAX, medial, and lateral portions of gluteus maximus combined; GMEDA and GMEDP, anterior and posterior portions of gluteus medius/minimus, respectively; ILPSO, iliacus and psoas major combined; RF, rectus femoris; HAMS, semimembranosus, semitendinosus and biceps femoris long head combined; ADD, adductor magnus, adductor longus, and adductor brevis combined. The horizontal bars indicate the periods of EMG activity recorded for the muscles shown. Note that GMAX represents activity recorded from medial gluteus maximus; GMED represents activity recorded from anterior gluteus medius; HAMS represents activity recorded from lateral hamstrings (biceps femoris long head). No EMG data were recorded for ILPSO and ADD.
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Knee Extension RHS
LTO
LHS
RTO
Moment (Nm)
40 VAS
RF
20 0 S
M
HA
-20
GAS
-40 VAS RF
Ankle Plantarflexion Moment (Nm)
100 L SO
50
GAS
0
GAS SOL
Subtalar Inversion
Moment (Nm)
10
PF I N
5
GAS
SO L
0 PFEV
-5 -10 0
20
40
60
Fig. 2. Contributions of individual muscles to the net moments exerted about the knee, ankle, and subtalar joints. The shaded regions represent the net moments exerted by all the muscles spanning each joint. Positive moments represent knee extension, ankle plantarflexion, and subtalar inversion; negative moments represent knee flexion, ankle dorsiflexion, and subtalar eversion. Results are the mean of all subjects and are shown for the stance phase of walking. Muscle symbols appearing in the graphs are: VAS, vastus medialis, vastus intermedius, and vastus lateralis combined; GAS, medial and lateral compartments of gastrocnemius combined; SOL, soleus, RF, rectus femoris; HAMS, semimembranosus, semitendinosus, and biceps femoris long head combined; PFIN, tibialis posterior, flexor digitorum longus, and flexor hallucis longus combined; PFEV, peroneus brevis and peroneus longus combined. The horizontal bars indicate the periods of EMG activity recorded for the muscles shown. Note that VAS represents activity recorded from vastus medialis; GAS represents activity recorded from medial gastrocnemius. No EMG data were recorded for PFIN and PFEV.
Gluteus maximus, gluteus medius, vasti, soleus, and gastrocnemius contributed most significantly to the vertical and fore-aft accelerations of the center of mass during stance (Fig. 6, TOTAL). Vasti, soleus, gastrocnemius, the hip adductors, and the plantarflexor everters accelerated the center of mass laterally, whereas anterior and posterior gluteus medius and the plantarflexor inverters accelerated the center of mass medially.
4. Discussion Muscles that lie primarily in the sagittal plane and contribute significantly to vertical support and forward progression (i.e., vasti, soleus, and gastrocnemius) also regulate the
mediolateral acceleration of the center of mass. The reason is that the skeleton is a system of jointed segments which are dynamically coupled; specifically, each muscle force is transmitted by all the joints, and thus, each muscle accelerates all the body segments simultaneously (Zajac and Gordon, 1989; Pandy, 2001). Vasti generated support and also decreased the forward speed of the center of mass during the first half of stance. In the mediolateral direction, vasti (together with the hip adductors and gravity) accelerated the body laterally during the first half of stance (Fig. 6). If balance is to be maintained, these actions must be resisted by the actions of other muscles. Anterior and posterior gluteus medius acted synergistically to oppose the actions of vasti and gravity by accelerating the center of mass medially during the first half of stance. Similarly, soleus and gastrocnemius supported
M.G. Pandy et al. / Journal of Biomechanics 43 (2010) 2055–2064
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Hip Extension
Angular Acceleration (rad/s2)
RHS
LTO
LHS
RTO
H
AM
500
G MA X
S
SOL
VAS
0
ILP S
-500
O
-1000
Angular Acceleration (rad/s2)
Hip Abduction GM E D A
100
GMEDP 0
ADD
VA S
GAS
GRAV
SO
ILPS O
-100
L
-200
Hip Internal Rotation Angular Acceleration (rad/s2)
GM E DA SO ILP
100
0 P ED L GM SO
GM AX
-100
0
20
40
60
Gait Cycle (%) Fig. 3. Contributions of individual muscles to the joint angular accelerations of the hip. The shaded regions represent the contributions from all muscles to the joint angular accelerations of the hip in each plane. Positive joint angular accelerations represent extension, abduction, and internal rotation; negative joint angular accelerations represent flexion, adduction, and external rotation. Results are the mean of all subjects. Muscle symbols as defined in the captions for Figs. 1 and 2. GRAV is the acceleration induced by gravity. See also Table 1.
the body and propelled it forward during the second half of stance, and these muscles also accelerated the body laterally at this time (Fig. 6). Anterior and posterior gluteus medius actively controlled balance by accelerating the center of mass medially during the second half of stance. As a general principle, muscles which accelerated the hip into abduction also accelerated the center of mass medially (i.e., anterior and posterior gluteus medius), whereas muscles which accelerated the hip into adduction also accelerated the center of mass laterally (i.e., vasti, soleus, gastrocnemius, and the hip adductors) (Table 1). This principle is illustrated in Fig. 7, which shows a double pendulum model of the body in the frontal plane. Anterior and posterior gluteus medius applied abductor moments about the stance hip and accelerated the hip into abduction (Figs. 1 and 3). A hip abductor moment applies a counter-clockwise moment to the stance limb that transmits a laterally directed force to the ground. The ground must therefore apply an equal and opposite force in the medial direction, which
explains why anterior and posterior gluteus medius accelerated the center of mass medially. Similarly, vasti and the hip adductors accelerated the stance hip into adduction in the first half of stance, whereas iliopsoas, soleus, and gastrocnemius accelerated the stance hip into adduction during the second half (Fig. 3 and Table 1). These muscles induce forces that tend to turn the stance limb clockwise in the frontal plane, thereby transmitting a medially directed force to the ground. The ground must therefore apply an equal and opposite force in the lateral direction, which explains why vasti, soleus, gastrocnemius, iliopsoas, and the hip adductors accelerated the center of mass laterally. In contrast to MacKinnon and Winter (1993), we found that gravity accelerated the center of mass laterally for most of the single-leg stance phase (Figs. 5 and 6). MacKinnon and Winter used a single pendulum model of balance to conclude that gravity accelerates the center of mass medially during single-leg stance. In single-leg stance, the position of the center of mass lies
2060 Table 1 Contributions of muscles and gravity to the net joint moments at the hip, knee and ankle; the joint angular accelerations of the hip, knee, and ankle; the vertical, fore-aft, and mediolateral accelerations of the center of mass (COM); and the vertical, fore-aft, and mediolateral ground reaction forces. Data presented are the peak contributions calculated for the stance phase, averaged across all subjects. Muscle symbols are defined in the captions of Figs. 1 and 2. Hip extension, hip abduction, hip internal rotation, knee extension, ankle plantarflexion, and subtalar inversion are positive for moments and joint angular accelerations. Forward, upward, and lateral are positive for COM accelerations and ground reaction forces. Joint acceleration (rad/s2)
Joint moment (Nm) Hip
GMEDA GMEDP ADD ILPSO HAMS RF VAS SOL GAS PFIN PFEV ERCSPN Gravity
Mean STD Mean STD Mean STD Mean STD Mean STD Mean STD Mean STD Mean STD Mean STD Mean STD Mean STD Mean STD Mean STD Mean STD
Ankle
Hip
Knee
Ankle
Exten
Abd
IntRot
Exten
Plantar
Inver
Exten
Abd
IntRot
Exten
Plantar
Inver
37.8 10.4 3.2 8.4 14.1 1.2 1.1 7.2 77.6 4.0 77.5 17.8 24.5 5.2 – – – – – – – – – – – – 49.0 86.5
7.6 1.9 33.1 5.5 26.5 5.2 19.9 6.8 3.1 0.2 0.2 0.8 6.3 1.9 – – – – – – – – – – – – 52.2 32.2
14.3 5.0 23.3 7.5 12.3 2.5 2.2 0.5 3.6 1.4 1.1 0.3 0.4 0.2 – – – – – – – – – – – – 5.2 11.3
– – – – – – – – – – 40.2 9.3 22.4 4.6 33.2 6.0 0.0 0.0 29.4 5.9 – – – – – – 41.7 23.7
– – – – – – – – – – – – – – – – 67.7 11.9 47.2 9.2 3.5 2.0 6.3 1.5 – – 87.4 51.4
– – – – – – – – – – – – – – – – 5.2 1.0 3.5 0.5 8.2 4.6 8.5 2.2 – – 7.9 11.6
402.5 129.0 60.2 93.9 161.3 16.1 26.8 88.1 777.0 117.4 586.3 170.3 160.7 80.2 183.2 20.4 239.2 33.9 40.9 6.3 17.5 11.9 20.8 6.4 288.0 128.4 20.9 46.3
1.6 40.6 128.8 28.4 74.9 17.3 63.4 15.2 121.5 47.3 65.8 32.5 19.6 7.5 58.2 18.9 108.0 19.9 31.7 65.8 1.7 12.0 7.5 12.9 101.0 44.8 52.8 8.1
125.4 30.9 156.6 50.2 65.6 12.8 30.3 11.1 143.4 31.3 76.7 43.9 36.0 18.1 71.9 12.1 90.3 24.6 3.4 30.8 14.4 7.5 0.6 6.3 29.1 14.5 35.1 4.2
237.1 83.5 113.8 52.0 99.8 9.4 18.0 50.7 488.2 27.6 146.6 24.6 68.2 17.0 246.1 41.7 257.1 46.5 13.8 84.8 19.6 11.0 35.5 13.8 67.3 24.6 12.0 69.3
139.4 90.5 45.4 14.3 55.5 17.8 44.1 29.3 289.3 20.9 101.4 82.9 33.4 9.5 235.2 108.3 229.2 34.5 219.1 169.7 38.3 13.9 60.5 38.3 34.6 9.2 123.9 110.1
1.1 17.5 20.2 8.8 8.5 11.6 26.3 7.3 35.3 43.6 25.6 14.8 18.8 7.2 17.9 20.0 29.7 28.5 41.1 13.0 16.0 6.3 18.5 11.6 11.4 3.9 35.2 4.1
Ground reaction force (BW)
Fore-aft
Vertical
Mediolateral
Fore-aft
Vertical
Mediolateral
0.3 0.4 0.1 0.4 0.0 0.5 0.2 0.3 0.6 0.3 0.7 0.3 0.8 0.2 1.5 0.2 1.9 0.2 1.3 0.3 0.3 0.1 0.2 0.2 0.1 0.2 0.7 0.7
2.8 0.8 1.6 0.4 2.3 0.6 1.0 0.5 0.5 2.8 1.8 0.7 1.0 0.2 2.3 2.2 7.1 0.7 3.8 0.8 0.5 0.6 0.0 0.7 1.1 0.6 9.6 0.0
0.1 0.3 0.7 0.2 0.7 0.1 0.5 0.1 0.1 0.5 0.2 0.2 0.2 0.0 0.4 0.2 0.8 0.3 0.6 0.2 0.1 0.1 0.2 0.1 0.1 0.1 0.0 0.5
0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.1 0.1 0.0 0.1 0.0 0.2 0.0 0.2 0.0 0.1 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.1 0.1
0.3 0.1 0.2 0.0 0.2 0.0 0.1 0.1 0.1 0.3 0.2 0.1 0.1 0.0 0.2 0.2 0.7 0.1 0.4 0.1 0.1 0.1 0.0 0.1 0.1 0.0 0.3 0.1
0.0 0.0 0.1 0.0 0.1 0.0 0.0 0.0 0.0 0.1 0.0 0.0 0.0 0.0 0.0 0.0 0.1 0.0 0.1 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0
M.G. Pandy et al. / Journal of Biomechanics 43 (2010) 2055–2064
GMAX
Knee
COM acceleration (m/s2)
M.G. Pandy et al. / Journal of Biomechanics 43 (2010) 2055–2064
2061
Angular Acceleration (rad/s2)
Knee Extension RHS
LTO
LHS
RTO
400 VA S
200
S OL
GM A X
0 -200 ILP S
O
-400
L SO
V
200
AS
Angular Acceleration (rad/s2)
Ankle Plantarflexion
GMAX
GAS
0 ILP S
-200
O
Angular Acceleration (rad/s2)
Subtalar Inversion 50 GAS
SO L
GMEDP
0 GMEDA GR A V ILPSO
-50 0
20
40
60
Gait Cycle (%) Fig. 4. Contributions of individual muscles to the joint angular accelerations of the knee, ankle, and subtalar joints. Each shaded region represents the contributions from all muscles to the angular acceleration of the joint. Positive joint angular accelerations represent knee extension, ankle plantarflexion, and subtalar inversion; negative joint angular accelerations represent knee flexion, ankle dorsiflexion, and subtalar eversion. Results are the mean of all subjects. Muscle symbols as defined in the captions for Figs. 1 and 2. GRAV is the acceleration induced by gravity. The contributions of the plantarflexor inverters and everters to subtalar joint angular acceleration were smaller than those of the other muscles shown in the graph (see Table 1).
medial to the base of support. If the body is modeled as a single pendulum in the frontal plane, gravity will act to accelerate the center of mass medially (i.e., to the left in Fig. 7). In the present study, gravity accelerated the center of mass in the opposite direction, laterally. This result may seem counterintuitive, but it is readily understood by considering, again, the double pendulum model in Fig. 7. In both the single- and doublependulum models, gravity’s effect on the acceleration of the center of mass is determined by the orientation of the stance leg in the frontal plane. In the double pendulum, whenever the angle between the leg and the ground is less than 901 (as shown in Fig. 7), gravity will tend to tip the body to the right, and the center of mass will be accelerated laterally. Conversely, if the angle between the stance leg and the ground is greater than 901, gravity will tend to tip the body in the other direction, to the left, and the center of mass then will be accelerated medially. As shown in Fig. 3, gravity acted to accelerate the hip into adduction during
single-leg stance, and, as explained above, this action causes the center of mass to be accelerated laterally. Nonetheless, the angle between the stance leg and the ground remains the critical factor in determining the direction in which gravity accelerates the center of mass. In the present study, the angle between the stance leg and the ground in the frontal plane was slightly less than 901 for most of single-leg stance, which explains why gravity accelerated the body laterally. We note here that the angle between the stance leg and the ground is a function of step width; increasing one’s step width will increase this angle (y for the double pendulum model shown in Fig. 7) and could thereby cause gravity to accelerate the center of mass medially during single-leg stance. We may predict, therefore, based on the results of Fig. 6, that an increase in step width would require less active control from anterior and posterior gluteus medius, as gravity would then assist these muscles in accelerating the center of mass medially. This may be
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RHS
2
LTO
LHS
RTO
Lateral
1 GRAV
0 ALL MU S C LES
-1 Medial
Mediolateral Acceleration (m/s2)
-2
2
Lateral
1
INER TIAL
0 C E N T R IF U
RESID U
GAL
AL
-1 Medial
-2
2
Lateral
1
OD E
EX P T L
M
0 -1 Medial
-2 0
20
40
60
Gait Cycle (%) Fig. 5. Contributions of all muscle forces (ALL MUSCLES), gravity (GRAV), residual forces (RESIDUAL), centrifugal and Coriolis forces (CENTRIFUGAL), and inertial forces (INERTIAL) to the mediolateral acceleration of the center of mass of the body. Residual forces were the forces acting on the pelvis segment and were calculated from an inverse dynamics analysis applied to the model skeleton. Inertial forces were comprised of fictitious forces that arose because of the assumption of rigid contact imposed at each of the foot–ground contact points in the model. The inertial forces were computed using Eq. (7) given by Anderson and Pandy (2003). The experimental result (EXPT) was obtained directly from the mean mediolateral ground reaction force measured for the subjects. MODEL represents the summed contributions of all muscle forces, gravity, centrifugal forces, residual forces, and inertial forces at each instant of the gait cycle. The shaded regions represent 7 1 standard deviation from the mean for both the model and experimental results.
one reason why young children and older adults walk with relatively wide steps compared with healthy young adults (Whittle, 1996; Murray et al., 1969). Based on the predictions of the single pendulum model, Winter (1995) concluded that the plantarflexor inverters and everters have negligible involvement in balance control. Our results indicate that although these muscles do not contribute as significantly to the mediolateral acceleration of the center of mass as the other plantarflexors, soleus and gastrocnemius (see Fig. 6), their involvement is not negligible. The plantarflexor inverters applied an internal rotation moment about the subtalar joint, accelerating this joint internally and the center of mass medially. Conversely, the plantarflexor everters applied an external rotation moment about the subtalar joint, accelerating it externally and the center of mass laterally (Table 1). We conclude that the plantarflexor inverters assist anterior and posterior gluteus medius in controlling mediolateral balance during stance. Although the present analysis is based on gait data obtained from only five subjects, we have confidence in the conclusions
derived from the model. The time history of the mean mediolateral acceleration of the center of mass measured for the five subjects is consistent with measurements reported in the literature, with a peak medial acceleration of approximately 0.75 m/s2 occurring in double-leg stance (compare EXPT in Fig. 5 with results presented by MacKinnon and Winter (1993) in their Fig. 4). Muscle contributions to joint angular accelerations in the frontal and transverse planes have not previously been reported for walking, but our results are consistent with those published for the sagittal plane. Arnold et al. (2005) analysed a dynamic optimization solution of normal walking and found that gluteus maximus, vasti, and hamstrings contributed significantly to hip extensor acceleration during the first half of stance, whereas posterior gluteus medius and soleus accelerated the hip into extension during the second half of stance. At the knee, vasti and gluteus maximus contributed significantly, and in nearly equal proportions, to accelerating the joint into extension in the first half of stance, whereas posterior gluteus medius and soleus accelerated the knee into extension in the second half. Results given in Figs. 3 and 4 and Table 1 agree with these findings. The patterns of muscle
M.G. Pandy et al. / Journal of Biomechanics 43 (2010) 2055–2064
LTO
LHS
RTO
RHS
LTO
LHS
RTO
10
10
L SO
TO TA L
Vertical Acceleration (m/s 2 )
RHS
2063
5
GA S
0 VA S
G ME
0
GMEDP
DA
GRAV
G M AX
4 Fore
4 Fore
2
2 G M ED P
0
X G MA
S VA
S GA
GRAV
0
AL T OT
SO L
-2
-2 Aft
Aft
Lateral
Lateral
1 TO T
1 VA S
AL
G
D AD
PFEV
0
0
GM ED GM E A DP
-1 Medial 0
L SO A S
Mediolateral Acceleration (m/s 2 )
Fore-aft Acceleration (m/s 2 )
-10
P F IN
-1 20 40 Gait Cycle (%)
60
G R AV
Medial
0
20 40 Gait Cycle (%)
60
Fig. 6. Contributions of individual muscles and gravity (GRAV) to the vertical, fore-aft, and mediolateral accelerations of the center of mass in normal walking. The shaded regions represent the accelerations of the center of mass in the vertical, fore-aft, and mediolateral directions contributed by all the stance leg muscles. TOTAL represents the summed contribution of all muscles shown for each direction; specifically, for vertical acceleration (top row), TOTAL is the sum of GMAX, GMEDA, GMEDP, VAS, SOL, and GAS; for fore-aft acceleration (middle row), TOTAL is the sum of GMAX, VAS, GMEDP, SOL, and GAS; for mediolateral acceleration (bottom row), TOTAL is the sum of GMEDA, GMEDP, ADD, VAS, SOL, GAS, PFIN, and PFEV. Positive accelerations are directed upward, forward, and laterally; negative accelerations are directed downward, backward, and medially. Results are the mean of all subjects. Muscle symbols as defined in the captions for Figs. 1 and 2.
contributions to vertical support and forward progression obtained in this study (Fig. 6) are also consistent with those reported previously by Anderson and Pandy (2003), Liu et al. (2006), Neptune et al. (2008), and Liu et al. (2008). Another potential limitation of the present study is that static optimization was used to solve the muscle-force distribution problem. The static solution was constrained by the force–length– velocity property of muscle, but activation dynamics was neglected. Anderson and Pandy (2001) compared the lower-limb muscle forces obtained from static and dynamic optimization and showed that muscle activation dynamics has little influence on the solution derived from static optimization (see also Supplementary Material). Fig. 5 shows that muscles and gravity contributed significantly to the mediolateral acceleration of the center of mass throughout stance, and that the inertial forces also contributed significantly, but only during double-leg stance. The inertial forces were comprised of fictitious forces that arose because of the assumption of rigid contact imposed at each of the foot–ground contact points
in the model. Had the assumption of rigid foot–ground contact been sound, the magnitude of the contribution of the inertial forces would have been small throughout stance, and the solution of the force decomposition problem then would have been more accurate. Results of Fig. 5 suggest, therefore, that interpretations of leg–muscle function in double-leg stance ought to be viewed more cautiously than those derived for single-leg stance. In conclusion, it is tempting to view mediolateral balance during normal walking as a dynamic equilibrium created by the interaction of gravity and the hip abductor muscles alone (MacKinnon and Winter, 1993). In the framework of the single pendulum model of balance, gravity creates a moment that accelerates the center of mass medially, while the hip abductors apply a moment that accelerates the center of mass in the opposite direction to counter the effect of gravity. The present analysis shows that this is too simplistic a view to take. Muscles that generate both vertical support and forward progression (vasti, soleus, and gastrocnemius) also accelerate the center of mass laterally, in concert with the hip adductors, the plantarflexor
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Ackland, Tom Correa, and Anthony Schache for reviewing an earlier draft of this paper. This work was supported by a VESKI Innovation Fellowship and Australian Research Council Discovery Grants DP0772838 and DP0878705.
Mabductor mg
Appendix A. Supplementary Material
Lateral
Mabductor
Medial
Supplementary data associated with this article can be found in the online version at doi:10.1016/j.jbiomech.2010.04.010.
Y Fground Z Fground reaction
α mg
mg Y
Y θ
θ Z
Z
Fig. 7. Top: double-pendulum model of the body in the frontal plane used to illustrate how a medially directed ground reaction force can be induced by an abductor moment applied at the hip. The hip abductor muscles apply a net abductor moment, Mabductor, at the hip, which in turn applies a counter-clockwise moment on the stance leg as shown. This moment generates a laterally-directed force (pointing to the right in the diagram) on the ground (Fground). The ground applies an equal and opposite medially-directed force to the leg (Fground reaction). Bottom: single-pendulum and double-pendulum models of the body in the frontal plane. Because the center of mass remains medial to the base of support (i.e., the stance foot) during single-leg stance, the angle between the stance leg and the ground, y, must be greater than 901 for the single pendulum. Gravity therefore acts to tilt the pendulum medially in this model. The double pendulum lumps the mass of the body at the tip of the pelvis segment, and so the direction in which gravity acts to tip the model is determined by the angle between the stance leg and the ground. This angle could be smaller or greater than 901. If the angle between the stance leg and the ground is smaller than 901, gravity will act to tip the model laterally (to the right); otherwise, gravity will act to tip the model medially (to the left). a represents the angle between the stance leg and the pelvis.
everters, and gravity. The hip abductors, anterior and posterior gluteus medius, and, to a much lesser extent, the plantarflexor inverters, actively control balance by accelerating the center of mass medially.
Conflict of interest The authors do not have any financial or personal relationships with other people or organizations that could inappropriately influence their work.
Acknowledgements We thank Mirjana Jancic and Anthony Schache for their help with data collection. We are also grateful to Richard Baker, David
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