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Technical Briefs

# Marker-Based Reconstruction of the Kinematics of a Chain of Segments: A New Method That Incorporates Joint Kinematic Constraints

[+] Author and Article Information
Miriam Klous1

Department of Kinesiology, Pennsylvania State University, 20 Recreation Building, University Park, PA 16802miriam@psu.edu

Sander Klous

Nikhef, 1098 SJ, Amsterdam, The Netherlands

1

Corresponding author.

J Biomech Eng 132(7), 074501 (May 18, 2010) (7 pages) doi:10.1115/1.4001396 History: Received July 08, 2009; Revised March 05, 2010; Posted March 11, 2010; Published May 18, 2010; Online May 18, 2010

## Abstract

The aim of skin-marker-based motion analysis is to reconstruct the motion of a kinematical model from noisy measured motion of skin markers. Existing kinematic models for reconstruction of chains of segments can be divided into two categories: analytical methods that do not take joint constraints into account and numerical global optimization methods that do take joint constraints into account but require numerical optimization of a large number of degrees of freedom, especially when the number of segments increases. In this study, a new and largely analytical method for a chain of rigid bodies is presented, interconnected in spherical joints (chain-method). In this method, the number of generalized coordinates to be determined through numerical optimization is three, irrespective of the number of segments. This new method is compared with the analytical method of Veldpaus [1988, “A Least-Squares Algorithm for the Equiform Transformation From Spatial Marker Co-Ordinates,” J. Biomech., 21, pp. 45–54] (Veldpaus-method, a method of the first category) and the numerical global optimization method of Lu and O’Connor [1999, “Bone Position Estimation From Skin-Marker Co-Ordinates Using Global Optimization With Joint Constraints,” J. Biomech., 32, pp. 129–134] (Lu-method, a method of the second category) regarding the effects of continuous noise simulating skin movement artifacts and regarding systematic errors in joint constraints. The study is based on simulated data to allow a comparison of the results of the different algorithms with true (noise- and error-free) marker locations. Results indicate a clear trend that accuracy for the chain-method is higher than the Veldpaus-method and similar to the Lu-method. Because large parts of the equations in the chain-method can be solved analytically, the speed of convergence in this method is substantially higher than in the Lu-method. With only three segments, the average number of required iterations with the chain-method is $3.0±0.2$ times lower than with the Lu-method when skin movement artifacts are simulated by applying a continuous noise model. When simulating systematic errors in joint constraints, the number of iterations for the chain-method was almost a factor 5 lower than the number of iterations for the Lu-method. However, the Lu-method performs slightly better than the chain-method. The RMSD value between the reconstructed and actual marker positions is approximately 57% of the systematic error on the joint center positions for the Lu-method compared with 59% for the chain-method.

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Copyright © 2010 by American Society of Mechanical Engineers
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## Figures

Figure 1

Schematic representation of the method discussed here compared with the Veldpaus-method. The upper row shows the unmodified Veldpaus-method. In the first column of that row, the reference positions of markers 1–4 (o) on segment n are shown as well as the reference position of the JRCs that provides the connection to segment n−1 (◻) in the local coordinate system. In the second column, the measured marker positions (●) are shown with the JRCs (◼) as calculated from the adjacent segment n−1. In the third column the method from Veldpaus (1) is applied without modification. Note that the reconstructed JRC (◼) does not coincide with the JRC of n−1 (◻). The second row demonstrates the implementation of the proposed modifications in Eqs. 7,8,9. In the first column, the reference positions are reflected in the reference position of the JRCs. In the second column the measured positions are reflected in the JRCs position as calculated from the adjacent segment n−1. The result is shown in the third column: the JRCs have become the center of both the reference and the measured distribution and naturally coincide. The marker positions are optimized with the remaining degrees of freedom (the rotations around the JRCs).

Figure 2

The effects of incorporation joint kinematic constraints on the difference between reconstructed marker kinematics and true marker kinematics. The dashed line is the Veldpaus-method, the dotted line the Lu-method, and the continuous line is the chain-method. The x-axis shows the added noise simulating skin movement artifacts and the y-axis shows the RMSD between the reconstructed and the actual (noise- and error-free) marker positions normalized to the amount of noise applied.

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