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Research Papers

Optimized Design of an Instrumented Spatial Linkage that Minimizes Errors in Locating the Rotational Axes of the Tibiofemoral Joint: A Computational Analysis

[+] Author and Article Information
Daniel P. Bonny

Biomedical Engineering Graduate Group,
University of California, Davis,
One Shields Ave,
Davis, CA 95616-5270

M. L. Hull

Department of Mechanical Engineering,
Department of Biomedical Engineering,
University of California, Davis,
One Shields Ave,
Davis, CA 95616-5270
e-mail: mlhull@ucdavis.edu

S. M. Howell

Biomedical Engineering Graduate Group,
Department of Mechanical Engineering,
University of California, Davis,
One Shields Ave,
Davis, CA 95616-5270

1Corresponding author.

Contributed by the Bioengineering Division of ASME for publication in the Journal of Biomechanical Engineering. Manuscript received May 7, 2012; final manuscript received October 22, 2012; accepted manuscript posted December 8, 2012; published online February 11, 2013. Assoc. Editor: Sean S. Kohles.

J Biomech Eng 135(3), 031003 (Feb 11, 2013) (11 pages) Paper No: BIO-12-1175; doi: 10.1115/1.4023135 History: Received May 07, 2012; Revised October 22, 2012; Accepted December 08, 2012

An accurate method to locate of the flexion-extension (F-E) axis and longitudinal rotation (LR) axis of the tibiofemoral joint is required to accurately characterize tibiofemoral kinematics. A method was recently developed to locate these axes using an instrumented spatial linkage (ISL) (2012, “On the Estimate of the Two Dominant Axes of the Knee Using an Instrumented Spatial Linkage,” J. Appl. Biomech., 28(2), pp. 200–209). However, a more comprehensive error analysis is needed to optimize the design and characterize the limitations of the device before using it experimentally. To better understand the errors in the use of an ISL in finding the F-E and LR axes, our objectives were to (1) develop a method to computationally determine the orientation and position errors in locating the F-E and LR axes due to transducer nonlinearity and hysteresis, ISL size and attachment position, and the pattern of applied tibiofemoral motion, (2) determine the optimal size and attachment position of an ISL to minimize these errors, (3) determine the best pattern of pattern of applied motion to minimize these errors, and (4) examine the sensitivity of the errors to range of flexion and internal-external (I-E) rotation. A mathematical model was created that consisted of a virtual “elbow-type” ISL that measured motion across a virtual tibiofemoral joint. Two orientation and two position errors were computed for each axis by simulating the axis-finding method for 200 iterations while adding transducer errors to the revolute joints of the virtual ISL. The ISL size and position that minimized these errors were determined from 1080 different combinations. The errors in locating the axes using the optimal ISL were calculated for each of three patterns of motion applied to the tibiofemoral joint, consisting of a sequential pattern of discrete tibiofemoral positions, a random pattern of discrete tibiofemoral positions, and a sequential pattern of continuous tibiofemoral positions. Finally, errors as a function of range of flexion and I-E rotation were determined using the optimal pattern of applied motion. An ISL that was attached to the anterior aspect of the knee with 300-mm link lengths had the lowest maximum error without colliding with the anatomy of the joint. A sequential pattern of discrete tibiofemoral positions limited the largest orientation or position error without displaying large bias error. Finally, the minimum range of applied motion that ensured all errors were below 1 deg or 1 mm was 30 deg flexion with ±15 deg I-E rotation. Thus a method for comprehensive analysis of error when using this axis-finding method has been established, and was used to determine the optimal ISL and range of applied motion; this method of analysis could be used to determine the errors for any ISL size and position, any applied motion, and potentially any anatomical joint.

FIGURES IN THIS ARTICLE
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Copyright © 2013 by ASME
Topics: Rotation , Errors , Linkages , Design
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References

Figures

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Fig. 1

The virtual tibiofemoral joint. The F-E axis was perpendicular to the i∧Fk∧F-plane and passed through the origin of the femoral anatomic coordinate system. The LR axis was perpendicular to the i∧Tj∧T-plane and passed 2.5 mm anterior to the origin tibial anatomic coordinate system.

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Fig. 2

The variables describing the size of the virtual ISL, shown with the coordinate systems of the virtual tibiofemoral joint at full extension, were the link length l and the “elbow” angle φ. If the ISL “wrist” was attached to tibia, the “elbow” was the origin of link 4 (e=4). If the ISL “wrist” was attached to femur, the “elbow” was the origin of link 3 (e=3).

Grahic Jump Location
Fig. 3

The variables describing attachment position of the virtual ISL, shown with the femoral anatomic coordinate system of the virtual tibiofemoral joint, transverse view. The variable β defined the angular position of the ISL attachment about the tibiofemoral joint, the variable d defined the distance of the ISL “wrist” and “shoulder” from the k∧F axis, and the variable γ defined the orientation of the axes of the revolute joints of ISL links 1 and 6 at full extension.

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Fig. 4

Flow chart of error calculations

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Fig. 10

Maximum bias, precision, and RMSE in locating the (a) F-E axis and (b) LR axis, using the optimal ISL and the “sequential discrete” pattern of applied motion for seven I-E rotation cycles (±20 deg) across 30 deg flexion at any starting angle

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Fig. 9

P-D RMSE and A-P RMSE in locating the F-E axis as a function of initial flexion angle for seven I-E rotation cycles (±20 deg) across 30 deg flexion using the “sequential discrete” pattern of applied motion. While the error in locating the F-E axis was largest in the P-D direction at full extension, the error becomes larger in the A-P direction when starting from 40 deg flexion or greater.

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Fig. 8

(a) P-D RMSE in locating the F-E axis and (b) M-L RMSE in locating the LR axis as a function of range of flexion and range of I-E rotation using “sequential discrete” applied motion. Results for ±5 deg are only partially shown in (b) for clarity. The error in locating the F-E axis in the P-D direction was only slightly affected by range of I-E rotation, while the error in locating the LR axis in the M-L direction was only slightly affected by range of flexion.

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Fig. 7

Bias, precision, and RMSE in locating the (a) F-E axis and (b) LR Axis, using the optimal ISL and the “sequential discrete” pattern of applied motion for thirteen I-E rotation cycles (±20 deg) across 120 deg flexion. The error in locating the LR axis in the M-L direction was an order of magnitude larger than the errors in locating the F-E axis.

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Fig. 6

Bias, precision, and RMSE in locating the LR axis in the M-L direction, for (a) “random discrete,” (b) “sequential discrete,” and (c) “sequential continuous” patterns of applied motion. “Random discrete” motion had the largest precision error, wile “sequential continuous” motion had the largest bias error.

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Fig. 5

The optimal ISL had link lengths of 300 mm and was attached to the anterior aspect of the tibiofemoral joint. The axes of the revolute joints of ISL links 1 and 6 were parallel to the F-E axis at full extension, and the “wrist” and “shoulder” were offset 200 mm from the k∧F axis.

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