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Article

Interpolating Three-Dimensional Kinematic Data Using Quaternion Splines and Hermite Curves

[+] Author and Article Information
James Coburn

 Bioengineering Laboratory, Dept. of Orthopaedics, Brown Medical School/Rhode Island Hospital 1 Hoppin Street, Coro West, Suite 404, Providence, RI 02903

Joseph J. Crisco1

 Bioengineering Laboratory, Dept. of Orthopaedics, Brown Medical School/Rhode Island Hospital 1 Hoppin Street, Coro West, Suite 404, Providence, RI 02903

1

Corresponding author

J Biomech Eng 127(2), 311-317 (Dec 06, 2004) (7 pages) doi:10.1115/1.1865195 History: Received December 05, 2003; Revised November 03, 2004; Accepted December 06, 2004

Kinematic interpolation is an important tool in biomechanics. The purpose of this work is to describe a method for interpolating three-dimensional kinematic data, minimizing error while maintaining ease of calculation. This method uses cubic quaternion and hermite interpolation to fill gaps between kinematic data points. Data sets with a small number of samples were extracted from a larger data set and used to validate the technique. Two additional types of interpolation were applied and then compared to the cubic quaternion interpolation. Displacement errors below 2% using the cubic quaternion method were achieved using 4% of the total samples, representing a decrease in error over the other algorithms.

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Copyright © 2005 by American Society of Mechanical Engineers
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Figures

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Figure 1

Example quaternion spline curve. An example curve showing the location of control points ai,bi,qi−1, and qi+3. Each interpolated segment is calculated using two tangent points and two key frames. This is a dimensionally reduced example of the four-dimensional quaternion for visualization purposes.

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Figure 2

Quaternion variable versus Hermite variable. Equally spaced quaternion intervals at top with linear time scale at the bottom for an arbitrary 10 steps. Scalar multiples of quaternion parameter q at top define increments of Hermite parameter ti at bottom.

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Figure 3

Components of QLand thamover the test motion. The left vertical axis shows the HAM location QL broken into its component at each sample. The right vertical axis shows the HAM translation tham at each sample. Note the large changes in all components of QL toward the end of the data set, showing a possible correspondence to the increase in interpolation error shown in Figs.  56.

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Figure 4

Components of ϕand nover the test motion. The left vertical axis shows the HAM orientation n broken into its components. The right vertical axis shows the HAM rotation ϕ. The rotation tracks a steady pattern through the motion while the orientation changes dramatically at the beginning and end of the samples.

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Figure 5

Displacement error in data set 7s. Distance between calculated sensor location and original sensor location for three interpolation methods. The cubic quaternion spline mitigates the errors of the cubic spline while maintaining continuity, unlike the linear interpolation. Key frames are located where the rms error is zero. Sensor with the largest error is displayed.

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Figure 6

Displacement error in data set 5s. Distance between calculated sensor location and acquired sensor location for three interpolation methods. With so few frames, errors keeping up with changes in HAM location dominate, reducing the advantage of the cubic quaternion spline. Key frames are located where the rms error is zero. The sensor with the largest error is displayed.

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Figure 7

Example showing interpolated capitate. Scanned capitate positions or key frames (colors) and interpolated positions (gray) moving from extension (left) to flexion (right). This space is the motion envelope of this wrist bone and shows a full range of flexion extension using only these five frames.

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