Instant center of rotation estimation using the Reuleaux technique and a Lateral Extrapolation technique
Introduction
Numerous reports have described how the Reuleaux technique can be used to calculate the planar instant center of rotation (ICR) of human joints (Frankel et al., 1971; Grant, 1973; Tamea and Henning, 1981; Gerber and Matter, 1983; Haher et al., 1992; Maganaris, 2000). Most reports describe how joint movement is associated with ICR movement along a pathway. The majority also describe how ICR pathways can be affected by injury and disease. ICR pathway measurement can therefore provide valuable information about a joint's kinematic characteristics. This information can aid both the detection of injury and the assessment of treatment outcome.
The main drawback with the Reuleaux technique is that it does not accurately deduce a joint's ICR. Reuleaux's original article describes his technique as yielding a temporary center of rotation (TCR) that only approximates to an ICR for small rotation angles (Reuleaux, 1876). While this observation is theoretically correct, investigations have shown that in practice the technique is susceptible to error magnification with small rotation angles (Panjabi, 1979; Bryant et al., 1984). Despite these observations the technique remains interesting due to its simplicity and ease of application.
The remainder of this section examines the Reuleaux technique and the errors encountered when it is used to approximate Instant Center data. The following section then describes how Instant Center data can be more accurately estimated by applying a Lateral Extrapolation technique to the Reuleaux TCR data. Finally, the article quantifies the accuracy improvement of the extrapolation technique using data collected from a mathematical model of a rolling wheel.
Vector analysis of a rolling wheel shows that at any instant its ICR is its contact point with the ground (Barton, 1984), as shown in Fig. 1a. Further analysis shows that slippage causes the ICR to displace vertically, as shown in Fig. 1b. An ICR overlay on an object's image therefore indicates its kinematic action at an instant in time.
Fig. 2 shows the Reuleaux technique applied to a radius mark on a model of a wheel rolling with 10% slippage (slip ratio s=0.1). The technique derives a TCR, that can also be referred to as a pole (Hartenberg and Denivit, 1964). The TCR (Pole) indicates a point about which the wheel can rotate to achieve the same displacement as the roll, by flying through the air in an arc. In the X direction the Pole lies midway between the roll start and end points at X=rθ(1−s)/2. In the Y direction the offset is described by Eq. (1), derived as (A.9) in the appendix:If rotation angle θ is small, then the pole will lie close to the roll's start and end ICRs, and it can be used to estimate both of these locations. As the rotation angle increases, the pole can be used to estimate an ICR midway through the roll, at angle θ/2.
In this model situation the position of the wheel can be calculated at angle θ/2. However, in practical kinematic measurement situations such an extrapolation may not be possible. For instance, if a pole is deduced from knee radiographs at 10° and 20° flexion, there may be no 15° image to overlay the result on. It is therefore desirable that pole data are extrapolated into ICR data that matches the available images. The ICRs can then be overlaid on the images so that an assessment of a joint's kinematic action can be made. The next section describes a technique for achieving this.
Section snippets
Method
The Lateral Extrapolation technique is a development of the Parallelogram Conversion technique, briefly described in an earlier report (Montgomery et al., 1998). Both techniques use the same X co-ordinate algorithm, but the updated version has an improved Y co-ordinate estimation. The new technique is outlined in Fig. 3.
Fig. 3a shows the calculated positions of a radius mark on a rolling wheel at 0°, 30° and 90° rotation, along with its associated poles and ICRs. Fig. 3b shows the ICRs
Results
In Fig. 4 the 10 cm radius wheel experienced a pure roll of 45°, followed by a 45° roll with 50% slip. The pure roll moved the wheel forward by 1/8 of its circumference producing a horizontal displacement of (2πr/8=)7.8 cm. The 50% slipping roll moved the wheel forward by half this amount. In the X direction, pole P12 lies midway between ICR1 and ICR2a, and pole P23 lies midway between ICR2b and ICR3. In the Y direction, both poles have small offset errors from the true ICRs.
If P12 is taken as an
Discussion
Examination of the offset errors outlined in Fig. 4, shows that P12 is a poor estimation of both ICR1 and ICR2a. The 45° roll between these points produces a pole with an error radius of 3.96 cm from each ICR. If this error is taken as a percentage of the wheel's radius, then the Reuleaux pole has a 39.6% offset error from the ICRs. By comparison the 0.52 cm lateral extrapolation error is only 5.2%. This error is smaller because the extrapolation removes the X error from the estimation.
Further
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