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This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution license (http://creativecommons.org/licenses/by/3.0/).

We propose a simple and direct method for measuring the first-order mass moment of a human body segment. With the proposed method, the first-order mass moment of the body segment can be directly measured by using only one precision scale and one digital camera. In the dummy mass experiment, the relative standard uncertainty of a single set of measurements of the first-order mass moment is estimated to be 1.7%. The measured value will be useful as a reference for evaluating the uncertainty of the body segment inertial parameters (BSPs) estimated using an indirect method.

The knowledge of body segment inertial parameters (BSPs) is required in the field of biomechanics. These parameters can be accurately estimated using dual x-ray absorptiometry (DXA) [

In this paper, a simple and direct method for measuring the first-order mass moment, M^{(1)} = ML, is proposed, where M is the mass of a body segment and L is the length from the axis of rotation to the position of the center of mass (CoM) of the segment.

In the proposed method, the first-order mass moment of a human body segment M^{(1)} is obtained from the two positions of the CoM of the entire body, in which the body segment under test has two different rotation angles. Only one precision scale and one digital camera are used in the experiment.

The first-order mass moment obtained with the proposed method can be used to evaluate the validity of the values of the BSPs obtained from other non-direct measurement methods. The moment value is also used for the posture error analysis in measuring human body masses in spacecraft with Space Scale [

_{1} and that of the rest of the body S_{2} as follows:
_{1}, and CoM_{2} are the center of mass of the entire body, of the body segment S_{1}, and of the rest of the body S_{2}, respectively. M, M_{1}, and M_{2} are the mass of the entire body, of the body segment S_{1}, and of the rest of the body S_{2}, respectively.

_{1} has two different rotation angles:
_{1} has two different rotation angles. When the rotation occurs in the horizontal plane, CoM_{1,β} can be expressed using the rotation matrix R(θ) and CoM_{1,α} as follows:
_{1} from the initial state (expressed using suffix α) to the final state (expressed using suffix β). On the basis of _{2} M_{2} can be eliminated as follows:
^{(1)}_{1} of the segment under test S_{1} can be calculated using the rotation matrix R(θ)^{−1}; the center of mass of the entire body before and after the rotation of segment S_{1}, _{α} and CoM_{β}; and the mass of the entire body M as follows:

The values of CoM at the two different states are measured using a common technique [

The center of rotation of segment S_{1}, CoR, is set as the origin of the coordinate system, O. The entire body is placed on a horizontal board, which is supported by the following three points: Support Point-1, Support Point-2, and Support Point-3. The weights applied at these three points are measured using a precision scale (capacity 3100 g, resolution: 0.1 g, precision: 1.5 g, model: GX-30KR, manufacturer: A and D Co., Ltd. (Japan)) one by one, while the attitude of the body is fixed. During the measurement, the three support points are supported by the scale or the height adjustment blocks whose total height is equal to that of the scale. CoM is calculated as follows:
_{P1}, P_{P2}, and P_{P3} are the horizontal positions of Support Point-1, Support Point-2, and Support Point-3, respectively. M_{P1}, M_{P2}, and M_{P3} are the readings of the scale at Support Point-1, Support Point-2, and Support Point-3, respectively. The horizontal positions P_{P1}, P_{P2}, and P_{P3}; the center of rotation CoR (O); and the angle of rotation θ are manually measured using the pictures taken by the digital network cameras [image capture resolution: 640 × 480, model: BL-C1, manufacturer: Panasonic Co., Ltd. (Japan)] placed at a distance of approximately 2.4 meters above the board. The rotation angle is manually measured using the printed overlapped images that are taken by the fixed digital camera with the help of a straightedge and protractor. Here, we did not compensate the distortion of the image due to the perspective projection of the camera when we estimate the center of rotation. The distortion is thought to be negligible because the camera is placed sufficiently far from the board (approximately 2.4 m, compared to the body thickness of less than 0.2 m and lens center line to margin length of about 0.5 m).

The first-order mass moment M^{(1)} of a dummy mass is evaluated for demonstrating the performance of the proposed method.

An aluminum bar (mass: 3.013 kg, length: 77.2 cm) is used as the segment under test S_{1}, which simulates an arm or a leg under the test. The other aluminum block with a mass of 52.066 kg is also used as segment S_{2}, which simulates the rest of the human body. Both S1 and S2 are placed on the horizontal board with a mass of 11.627 kg. The center of rotation CoR and the origin of the coordinate system O are defined at the end of segment S_{1}. The first-order mass moment M^{(1)} of segment S_{1} around the origin O (the center of rotation CoR) is calculated to be 116.3 kg·cm. In the experiment, 30 sets of measurements are conducted with a rotation angle between 30° and 180°.

^{(1)}, their mean value M^{(1)}_{mean}, and the calibrated value M^{(1)}_{cal} ^{(1)}_{mean} and the calibrated value M^{(1)}_{cal} are 117.6 kg·cm and 116.3 kg·cm, respectively. The standard deviation of the measured values of M^{(1)} is 1.619 kg·cm, which corresponds to 1.4% of the calibrated value M^{(1)}_{cal}. The root mean square value (RMS value) of the difference between the measured values of M^{(1)} and the calibrated value M^{(1)}_{cal} is 2.0 kg·cm, which corresponds to 1.7% of the calibrated value M^{(1)}_{cal}. This value is considered to be the relative standard uncertainty of a single set of measurements of M^{(1)}.

The first-order mass moment M^{(1)} of a leg of a human body is measured using the proposed method.

^{(1)} and their mean value M^{(1)}_{mean} ^{(1)} is 388.8 kg·cm. The standard deviation of the 15 measured values of M^{(1)} is 5.3 kg·cm, which corresponds to 1.4% of the mean value.

The relative standard uncertainty of a single set of measurements of M^{(1)} in the dummy mass experiment is estimated to be 1.7%. The standard deviations of the values of M^{(1)} in the experiments using the dummy mass for 30 measurements with various rotation angles and the human body for 15 measurements with a constant rotation angle are both 1.4% of the mean value. Although these error values are not directly comparable, the random errors of both the experiments are thought to be at the same level. This indicates that the change of posture of the other part of the human body during one set of measurements can be ignored even when the leg is rotated. With regard to the bias error, the bias error in the human body experiment might be slightly larger than that in the dummy mass experiment since the accuracy of the estimation of the rotation center CoR in the human body experiment might be lesser than that in the dummy mass experiment. In the experiments, the rotation canter CoR and the rotation angle θ are manually measured using the printed overlapped images that are taken by the fixed digital camera with the help of a straightedge and protractor. The accuracy of estimating the rotation center will be improved with more careful consideration [

The measured value will be useful as a reference for evaluating the uncertainty of the body segment inertial parameters (BSPs) estimated using an indirect method, in which the mass and the center of mass of a segment are estimated [^{(1)} of a body segment, which is directly measured using the proposed method, is compared with the value, which is calculated from the mass, the center of mass and the center of rotation estimated using the indirect method.

In the experiment, only one scale is used for measuring the weights applied at the three supporting points of the board. Therefore, during the measurements of the three points, the subject should not be moved; this requires a certain amount of effort in the case of the human subject. If three scales are used, then the weights at the three points can be obtained immediately and the measurement time can be reduced.

A simple and direct method for measuring the first-order mass moment M^{(1)} of the human body segment has been proposed. By using the proposed method, the first-order mass moment of the body segment can be directly measured by using only one precision scale and one digital camera. The relative standard uncertainty of a single set of measurements of M^{(1)} in the dummy mass experiment is estimated to be 1.7%. The measured value will be useful as a reference for evaluating the uncertainty of the body segment inertial parameters (BSPs) estimated using an indirect method.

This work was supported in part by a research-aid fund of the Asahi Glass Foundation, and Grant-in-Aid for Scientific Research (B) 21360196(KAKENHI 21360196).

Principle of the method (top-down view).

Setup for measuring the center of mass CoM of a subject body.

Overlapped pictures used in the analysis (subject: dummy mass, M_{1}: 3.013 kg, M^{(1)}_{1}: 116.3 kg·cm).

Relationship between the 30 measured values of M^{(1)}, their mean value M^{(1)}_{mean}, and the calibrated value M^{(1)}_{cal}

Overlapped pictures used in the analysis.

Relationship between the 15 measured values of M^{(1)} and their mean value M^{(1)}_{mean}