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A kinematic approach to calculating ground reaction forces in dance.

The high prevalence of musculoskeletal injuries and pain in dancers has been frequently acknowledged. (1) Microtraumatic (overuse) injuries may be due to demands placed on dancers' bodies through endless repetitions of dance movements that are beyond their capabilities. Using biomechanical techniques, loads within the body's joints and muscles, which may indicate the injury potential of specific dance movements, can be calculated. (2) Similar load assumptions have been made concerning the contribution of impact forces to injury in running and gymnastics during take-off or landing phases. (3,4) The information required to perform this calculation includes positional data, typically obtained from a 3-dimensional optical tracking system. (5) The forces within the body are also a function of external forces acting upon the body. During dance, the primary such factor is the ground reaction force generated between the floor and the foot. This force is typically greater than during gait due to the large accelerations experienced by the body while dancing.

Ground reaction forces are conventionally measured in a gait laboratory using floor-mounted force plates. These are high precision instruments that measure forces in three orthogonal directions, together with center of pressure and torque about a vertical axis. However, there are disadvantages associated with the use of force plates, especially in the context of measuring dance movement. A force plate has a small surface area (e.g., 0.6 m x 0.6 m). This is generally not an issue in a clinical gait study, as only a single step is considered. To analyze a dance sequence, on the other hand, a number of steps need to be analyzed that do not physically fit on a force plate. Furthermore, force plates are very stiff. Again, this is not a problem in the study of gait, but dancing on a stiff structure may cause injuries, especially when juxtaposed with a sprung dance floor. (6)

Ren and colleagues (7) investigated the calculation of ground reaction forces from gait kinematic data and mass distribution. This approach was found to provide reasonable results in the sagittal plane (which accounts for the greatest amount of movement during gait), but large errors were associated with movements in the frontal and transverse planes. Shippen (8) calculated the mass distribution from kinematic data but again within a gait context. In contrast, dancers perform complex movements in three orthogonal planes. While a kinematic approach to calculating ground reaction forces would be highly suitable for analyzing this kind of movement, none has yet been validated. However, the redundancy introduced by the use of whole body kinematic data and force plate data can improve the accuracy of the estimated joint moments, (9) joint accelerations, (10) or body segment parameters. (11)

The standard anthropometric tables for calculation of segmental masses used in a kinematic approach may not accurately represent the anthropometrics of dancers, however, as the aesthetics of dance dictate an atypically slim body type. (12-17) Furthermore, since the soft tissue mass of a body segment may move relative to the bony structures, (18) the segmental mass center does not have a fixed anatomical location. Its acceleration will lie somewhere between the acceleration of the bony structures and that of the soft tissue mass. To further complicate matters, both the bony structures and the soft tissue mass are covered by the skin, which may have a different acceleration altogether, especially in impact situations.

This study presents an application of mechanics theory that can be used for calculating a dancer's ground reaction forces from movement data in the absence of force plates. The method is then extended to investigate a technique for tuning the calculated mass distribution of an individual dancer to improve measurement of the predetermined ground reaction forces.


Theoretical Background

For this validation study, a single dancer was used. The kinematic approach to calculating ground reaction force with force plates has been demonstrated previously in runners and gymnasts using single human subjects, as well as single animal subjects such as horses and elephants. (3,4,19,20)

In the calculation of ground reaction force, the body is considered to be comprised of a finite number of rigid segments connected via joints. The 15 segments identified for this study were the head, thoracic abdomen, pelvic abdomen, upper arm, lower arm, hand, thigh, shin, and foot.

Using this segmentation, ground reaction force acting between the dancer and the floor can be calculated using Newton's Second Law (Foruma 1), where GRF is the calculated ground reaction force vector and

[m.sub.i] is the mass of the ith segment, [x.sub.i] is the position matrix of the center of mass of the ith segment for all times in all directions,

GRF = [n.summation over (i=1)] [m.sub.i] ([[??].sub.i] - g)

Formula 1

n is the number of segments, and g is the acceleration due to gravity vector.

To calculate ground reaction force, it is also necessary to know the mass associated with each segment of the body and the acceleration of each segmental center of mass. Winter (16) provides a decomposition of the total body mass into segmental proportions as shown in Table 1. Winter also locates the center of mass of the segments within the segmental coordinate system as illustrated in Figure 1 and annotated in Table 1. (16)


For this study, movement of the segmental coordinate systems was measured using a 12-camera MX40 Vicon optical tracking system, captured at 50 Hz. Markers were placed in accordance with the Vicon full body Plug-in-Gait marker set. (21) Each segment was defined by at least three retro-reflective markers attached to the segment or two real markers and a point defined in a more proximal segment. Figure 2 illustrates the retro-reflective markers attached to a dancer. The biomechanical model was written in Matlab 7.0 (The MathWorks, Inc.,



Formula 2

Natick, MA, USA) specifically for this study and was based on the formulation presented previously. (8)

Having defined the movement of the coordinate systems from the movement of the markers defining them, the movement of the center of mass of each segment was calculated (Table 1). The position of the center of mass was filtered with a low pass 4th order Butterworth filter (a mathematical technique with a 20 Hz cut-off frequency). The velocity and acceleration time histories were then calculated using the first and second derivatives of the interpolating cubic spline between the positional data. Interpolating cubic splines were calculated between the positional data, which ensured continuity of position, velocity, and acceleration. Free spline conditions were applied at the beginning and end of the trial.

The mass distribution from Winter (16) is generic in that it is averaged across a large sample of subjects. Consequently, some subjects may vary considerably from the population mean, especially when considering dancers, whose mass distribution may differ significantly from that of the general populace. (22) Therefore, an approach was investigated that could calculate a mass distribution specific to an individual subject that could subsequently be used to calculate the ground reaction force for that particular subject.

This approach to fine-tuning the generic mass distribution required a force plate to measure the ground reaction force produced when the study subject made unstructured whole body movements while standing on the plate. The movement of the body segments was simultaneously measured by a 3-dimensional optical tracking system, and hence the movement of the segmental centers of mass could be calculated as described above.

From equation 1, the error between the calculated and measured ground reaction force can be defined and calculated with the equation shown in Formula 2, where [GRF.sub.m] is the ground reaction force vector measured by the force plate.

Therefore, the mass distribution, mi, which minimizes this error, is assumed to be the mass distribution for the individual when subsequently calculating ground reaction forces during dance movements.

An optimization routine was written in Matlab based on the Trust Region method (23) to calculate mi, which minimizes the error in equation 2, subject to the inequality constraints that all segmental masses are positive ([m.sub.i] > 0) and the equality constraints of anatomical symmetry (although this constraint can be removed if appropriate: e.g., amputees).

Experimental Protocol

The above technique was applied to a 23-year-old female amateur dancer. The participant gave informed consent, and the study was approved by the University Research Ethics Committee. Prior to dynamic testing, an anthropometric survey of the subject was undertaken that included height, mass, segmental lengths, and joint width measurements in order to create the subject-specific biomechanical model.

The subject was asked to move her whole body in an unstructured manner while standing on a force plate. This type of movement was selected in order to obtain accelerations of all segments of the body in all orthogonal planes and also to measure the ground reaction force components, collected at 1500 Hz, that occur during propulsive phases of movement. An example of the movement of the subject and measured ground reaction forces during a trial can be seen at: No leaps or jumps were included in this analysis.

As previously mentioned, the motion of the subject was recorded using a 3-dimensional optical tracking system, and the method described above was used to calculate the ground reaction forces. During the trial, simultaneous measurements of the ground reaction forces were recorded using the force plate in order to assess the accuracy of the kinematic-approach calculations.

To investigate the tuning of a generic mass distribution to the specific distribution for this dancer, the ground reaction forces and subject's movement were recorded using the force plate and optical tracking system respectively. The mass distribution of the subject, which minimized the ground reaction force error, was calculated using the above method. This mass distribution for the subject was then used to calculate the ground reaction forces for the unstructured movements.


Figure 3 shows an example of the vertical ground reaction force as measured by the force plate during large movements in the sagittal, frontal, and transverse planes. Overlaid on this measurement is the kinematic-approach calculation of the vertical ground reaction force for the same movements. The mean calculated ground reaction force was 678 N over the 15 second duration of the trial. The mean absolute error between measured and calculated ground reaction forces in this example was 18.7 N, or 2.76% of the mean of measured ground reaction force.

Figure 4 shows an example of a plot of the calculated vertical ground reaction force using the generic mass distribution from the literature. (16) Overlaid on this measurement is the kinematic-approach calculated vertical ground reaction force using the mass distribution calculated specifically for this subject. The mean absolute error between the vertical ground reaction force calculated using generic and specific mass distribution was 8.9 N. The mean absolute error between vertical ground reaction force measured by the force plate and vertical ground reaction force calculated using the specific mass distribution was 16.5 N, or 2.43%.



Models based on kinematics are increasingly of interest as options for data capture with optical systems, inertial sensors, or video become more flexible and permit capture outside the laboratory for activities as varied as dancing, skiing, ice skating, or outdoor running. (7,19,24-26)

Despite the elementary nature of the analysis procedure based on Newton's Second Law applied on a segmental basis, the technique described here produced calculated ground reaction forces with a mean error less than 3% of the equivalent measured forces. Maximum error between measured and calculated ground reaction forces occurred during impact between the foot and the floor, when the assumption of the body consisting of rigid jointed segments was violated due to soft tissue vibration.

The results of the study have focused on vertical ground reaction force, although the analysis is independent of direction and, therefore, is also suitable for calculation of the propulsive and breaking ground shear forces. Future studies are required to investigate the accuracy of the method during those phases of dance.


There was no observed correlation between error in the calculated ground reaction forces and movement in the sagittal plane, as has been observed in previous studies applying similar methods in a gait analysis context. There was a slight improvement in the accuracy of the kinematic-approach calculated ground reaction forces when the generic mass distribution was modified to reduce ground reaction force error. The reduction in error using the tuning method is a function of the subject's variance from the mass distribution norm, with subjects demonstrating greater variance presumably benefiting more from mass tuning.

The technique relies on the calculation of acceleration from positional data; hence, there is the probability of noise being introduced during double differentiation. For this reason the method is dependent on the implementation of appropriate filtering.

The technique also requires that the markers attached to the subject remain firmly in place. For this reason, a tight fitting lycra suit is normally worn. Experience at the authors' and other laboratories indicates that subjects quickly accommodate to the presence of the suit and markers so that they do not affect performance. Of more significance is the effect of the suit sliding over the skin of the subject causing displacement of the markers from their original anatomical positions. This is a problem in all motion capture studies, and it can be minimized by visually checking for correct alignment prior to each trial and, if practical, during the trial.

If calculated ground reaction forces using the kinematic approach are subsequently used to calculate the loads within muscles, it should be recognized that errors associated with the approximate mass distribution inherent in the kinematic data measurement will adversely affect muscle force results.


An exploratory technique has been described that can calculate ground reaction force for a dancer in the absence of force plates, and hence is applicable for movement across a large performance area and on a sprung floor. Subject to the approximations and assumptions inherent in the described technique, this kinematic approach to calculation of ground reaction force enables subsequent calculation of loads within muscles and joints during dance movements that may be informative in identifying steps with a high risk of injury. The accuracy of the method may be further improved by calculating a specific mass distribution that minimizes ground reaction force error for the specific dancer under consideration. This approach is suitable for the analysis of dancers whose mass distribution is atypical of dancers in general.

Caption: Figure 1 The segmental coordinate systems.

Caption: Figure 2 Retro-reflective markers attached to a subject.

Caption: Figure 3 A plot of the measured ground reaction force and the calculated ground reaction force.

Caption: Figure 4 A plot of the calculated ground reaction force for the generic mass distribution and the specific mass distribution.


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James Shippen, B.Sc., Ph.D., C.Eng. M.I.Mech.E., and Barbara May, M.B.A., Ph.D.

James Shippen, B.Sc., Ph.D., C.Eng. M.I.Mech.E., and Barbara May, M.B.A., Ph.D., are at Coventry University, United Kingdom.

Correspondence: James Shippen, B.Sc., Ph.D., C.Eng. M.I.Mech.E., Department of Industrial Design, Coventry University, Coventry CV1 5FB, United Kingdom;
Table 1 Mass Distribution and Location of Center
of Mass, after Winter (16)

Segment          Mass Fraction   CoM Position

Skull                0.081       [0.05 0 0.05]
Thorax               0.286       [0 0 0.1214]
Pelvis               0.212        [-0.25 0 0]
Humerus              0.028        [0 0 0.436]
Radius / Ulna        0.016        [0 0 0.43]
Hand                 0.006        [0 0 0.506]
Femur                0.1          [0 0 0.433]
Tibia / Fibula       0.0465       [0 0 0.433]
Foot                 0.014         [0 0 0.5]

CoM = center of mass.
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Title Annotation:Technical Report
Author:Shippen, James; May, Barbara
Publication:Journal of Dance Medicine & Science
Date:Jan 1, 2012
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