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Research Papers

Comparative Analysis of Methods for Estimating Arm Segment Parameters and Joint Torques From Inverse Dynamics

[+] Author and Article Information
Davide Piovesan1

Robotics Laboratory, Sensory Motor Performance Program (SMPP), Rehabilitation Institute of Chicago, 345 East Superior Street, Suite 1406, Chicago, IL 60611; Ashton Graybiel Spatial Orientation Laboratory, Brandeis University, Waltham, MA 02454d-piovesan@northwestern.edu

Alberto Pierobon

Ashton Graybiel Spatial Orientation Laboratory, Brandeis University, Waltham, MA 02454pierobon@brandeis.edu

Paul DiZio

Ashton Graybiel Spatial Orientation Laboratory, Brandeis University, Waltham, MA 02454; Volen Center for Complex Studies, Brandeis University, Waltham, MA 02454dizio@brandeis.edu

James R. Lackner

Ashton Graybiel Spatial Orientation Laboratory, Brandeis University, Waltham, MA 02454; Volen Center for Complex Studies, Brandeis University, Waltham, MA 02454lackner@brandeis.edu

1

Corresponding author.

J Biomech Eng 133(3), 031003 (Feb 04, 2011) (15 pages) doi:10.1115/1.4003308 History: Received August 05, 2009; Revised December 20, 2010; Published February 04, 2011; Online February 04, 2011

A common problem in the analyses of upper limb unfettered reaching movements is the estimation of joint torques using inverse dynamics. The inaccuracy in the estimation of joint torques can be caused by the inaccuracy in the acquisition of kinematic variables, body segment parameters (BSPs), and approximation in the biomechanical models. The effect of uncertainty in the estimation of body segment parameters can be especially important in the analysis of movements with high acceleration. A sensitivity analysis was performed to assess the relevance of different sources of inaccuracy in inverse dynamics analysis of a planar arm movement. Eight regression models and one water immersion method for the estimation of BSPs were used to quantify the influence of inertial models on the calculation of joint torques during numerical analysis of unfettered forward arm reaching movements. Thirteen subjects performed 72 forward planar reaches between two targets located on the horizontal plane and aligned with the median plane. Using a planar, double link model for the arm with a floating shoulder, we calculated the normalized joint torque peak and a normalized root mean square (rms) of torque at the shoulder and elbow joints. Statistical analyses quantified the influence of different BSP models on the kinetic variable variance for given uncertainty on the estimation of joint kinematics and biomechanical modeling errors. Our analysis revealed that the choice of BSP estimation method had a particular influence on the normalized rms of joint torques. Moreover, the normalization of kinetic variables to BSPs for a comparison among subjects showed that the interaction between the BSP estimation method and the subject specific somatotype and movement kinematics was a significant source of variance in the kinetic variables. The normalized joint torque peak and the normalized root mean square of joint torque represented valuable parameters to compare the effect of BSP estimation methods on the variance in the population of kinetic variables calculated across a group of subjects with different body types. We found that the variance of the arm segment parameter estimation had more influence on the calculated joint torques than the variance of the kinematics variables. This is due to the low moments of inertia of the upper limb, especially when compared with the leg. Therefore, the results of the inverse dynamics of arm movements are influenced by the choice of BSP estimation method to a greater extent than the results of gait analysis.

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Copyright © 2011 by American Society of Mechanical Engineers
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Figures

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Figure 1

(a) Upper arm, forearm, and hand segmentation employed in our (PI) estimation method. Each section was divided in portions with length up to 0.1 m starting from the distal end. The moment of inertia IM of each portion was measured about the y-axes and combined in the matrix of inertia of the complete limb by means of the Huygens–Steiner theorem. (b) Apparatus used to measure the volume of the portions. The water was warmed to a temperature of 37°C for comfort. A graduated cylinder with a resolution of 2×10−7 m3 was used to measure the volume of water progressively displaced by each portion. The graduated cylinder was emptied after the immersion of each portion.

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Figure 2

(a) Position of Optotrak® active markers: (1) index finger tip, (2) styloid process of radius, (3) head of ulna, (4) lateral epicodyle of humerus, (5) deltoid tuberosity, (6) greater tubercle of humerus, (7) acromion, and (8) manubrium of sternum. The shape of the table was such as to restrain the movement of the subjects’ torso, minimizing the translation of the shoulder centroid (xs,ys). (b) Variables used in the floating base double-pendulum planar model. X denotes the medial-lateral direction and Y the dorso-ventral direction. An Optotrak® system was used to measure the constants l1 and l2, the coordinates of the shoulder centroid (xs,ys), and the angles θ1 and θ2 during the movement. The positions of r1 and r2 were calculated for each estimation method.

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Figure 3

(a) Distributions of the kinematic variability for the arm reaching task. In the first row, the peak and rms of joint angular accelerations are represented along with the rms of the length of reach and the displacement of the finger tip, elbow centroid, and shoulder centroid along the vertical axis. The second row depicts the dispersion ellipses of the shoulder centroid on the plane of movement across subjects. Solid line represents one standard deviation and dashed line two standard deviations, enclosing 95% probability. (b) Quartile distribution of peak and rms of acceleration for each subject.

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Figure 4

Quartile distribution of body segment parameters. The figure shows the populations of mass (M), center of mass location (CM) with respect to the proximal end, and inertial moment about the proximal end (IM) of each section as a function of the estimation method. The right hand panel indicates the pairwise compatibility of the distributions across methods: Hanavan (HV), Dempster (DE), Chandler (CH), Clauser (CL), McConville (MC), Zatsiorsky and Seluyanov (1983) (Z1), water immersion (PI), Zatsiorsky (2002) (Z2), and de Leva (DL). Boxes with gray background correspond to “hand extended” methods. (a) Hand BSPs, (b) forearm BSPs, and (c) upper-arm BSPs.

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Figure 5

(a) Quartile distribution of inertial parameters α, β, and δ. The central panel shows the populations of parameters as a function of the estimation method. The left hand panel indicates the population of parameters for each subject across methods. The right hand panel indicates the pairwise compatibility of the distributions across methods. Gray background boxes correspond to “hand extended” methods. (b) Two-way ANOVA on the inertial parameters: variance accounted for when considering the subjects’ somatotype as a factor. Subjects were divided in three groups: mesomorphs (4), endomorphs (5), and ectomorphs (4). For each of the nine methods, the influence of each factor was analyzed. The variance attributed neither to the method nor to the somatotype had to be attributed to the variability of anthropometric dimensions across subjects.

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Figure 6

Distributions of joint kinetic variables. Data relative to the shoulder and elbow kinetic variables are reported on the left and right panels, respectively. The top panels show the distribution of kinetic variables across subjects as a function of the estimation method. The central panels show the distribution of kinetic variables across subjects. The bottom bar plot illustrates the influence of different effects on the kinetic variable variance. Boxes with gray background correspond to “hand extended” methods. (a) Normalized torque peaks, (b) NRMS of torque, and (c) variance accounted for when using only hand extended methods.

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Figure 7

Normalized η2 analysis for each pairwise estimation method combination. The figure shows the contribution of method, subject, and interaction effect as well as other sources to the variance of the joint kinematic variables. The total variance of each method combination is indicated as a percentage of the total variance within the union of each pair of distributions. (a) Shoulder normalized torque peaks, (b) elbow normalized torque peaks, (c) shoulder NRMS of torque, and (d) elbow NRMS of torque.

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Figure 8

Comparison between the variance of the non-normalized elbow torque estimated with Eq. 14 and the variance of the population of non-normalized elbow torques. (a) Mean of shoulder acceleration (solid) and associated variance (dotted). (b) Mean of elbow acceleration (solid) and associated variance (dotted). (c) Components of the inertial matrices I21 (black-solid) and I22 (i.e., δ gray-solid) and their associated variance (dotted). (d) Mean of non-normalized elbow torque τ2 (black-solid) and associated variance (black-dotted). Here, we compared the different components of elbow torque τ2 as they appear in Eq. 5. τ2H is proportional to the joint angular velocities (dashed-dotted), τ2N is proportional to the shoulder linear acceleration (dashed), and τ2I is proportional to the joint angular acceleration (gray-solid). Notice how τ2H and τ2N are close to zero and tend to cancel each other, confirming the assumption used to formulate Eq. 13. (e) Components of the variance of the non-normalized torque calculated using Eq. 14. The dashed-dotted line represents the component θ̈22⋅u2(δ), while the dashed is θ̈12⋅u2(I21). The influence of the variability of the other components on the variance of the non-normalized elbow torque is negligible. (f) Comparison between the variance of τ2 estimated point by point for all subjects and all data (solid) and uc2(τ2) calculated from Eq. 14 (dashed). The maximum variance of the elbow torque aligns with the absolute torque peak, and it is mostly dependent on the variance component θ̈22⋅u2(δ). Increased dependence on θ̈12⋅u2(I21) occurs toward the end of the movement where the u2(I21) increases concurringly with I21.

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