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Design Changes in Ankle-Foot Orthosis Intended to Alter Stiffness Also Alter Orthosis Kinematics

Robert Singerman, PhD
David J. Hoy, BS, CPO
Joseph M. Mansour, PhD

ABSTRACT

The specific aim of the present research was to determine whether kinematics of an ankle-foot orthosis (AFO) are dependent on design changes made to alter stiffness. Four AFOs of different design were loaded in plantarflexion/dorsiflexion. The applied moment, coupled moments, and screw axis were determined. Increased trim resulted in decreased stiffness and a posterior shift in the location of the screw axis. The flexible-hinge design exhibited a substantial increase in the variability of the screw axis location compared to the locked-hinge design. Results indicate that design changes intended to alter stiffness also alter the kinematics of the orthoses.

Key Words: ankle-foot orthoses, dorsiflexion/plantarflexion, adduction/abduction, inversion/eversion, moment, screw axis

Introduction

It is generally accepted that wearing an ankle-foot orthosis (AFO) can have a positive effect on gait in patients with hemiplegia.^1–3 These orthoses aid several functions in ambulation, including control of dorsiflexion and plantarflexion in both the stance and swing phases of gait. AFOs are also thought to stabilize the ankle in the frontal and transverse planes during balance and gait activities. Historically, conventional metal (BiCaal) AFOs have been studied to further evaluate clinically observed joint stability in hemiplegic patients wearing these devices.^4,5 These traditional metal orthoses, replete with problems of excessive weight, bulk, and poor patient acceptance, were commonly prescribed because of the inherent strength and rigidity of their structural framework design, as well as the availability of adjustable ankle-locking mechanisms for controlling the sagittal-plane positioning of the foot and ankle.⁶

To meet the apparent needs of a varied patient population, numerous designs have evolved. The proliferation of designs^7–10 has been particularly pronounced because of the introduction of thermoplastic-bracing concepts, which have increased patient acceptance and compliance.⁷ Hinged or articulated thermoplastic designs are now commonly accepted in pediatric orthotic practice, and the clinical effectiveness of devices using this method is widely known.¹¹ To increase the stiffness and ankle adjustability of articulated thermoplastic AFOs, a hybrid design of metal joints and plastic was introduced to blend the mechanical characteristics of both materials into an external support system.¹² The dynamic AFO, an extremely thin and flexible supramalleolar-level orthosis, is reported to allow small increments of stable graded movements of the foot and ankle to occur within the orthosis.^10,13 Specific contouring used in certain AFO designs is thought to provide better foot alignment and possibly influence spasticity.^8–10

The concept of "biotechnical matching," as cited by Sarno,¹⁴ was introduced by Lehneis1 approximately 25 years ago. This provided a general approach to selecting the best orthosis through a systematic study of gait abnormalities, the function of the available orthoses, and the expected changes in gait that would result from using a particular orthosis. The implementation of these ideas generally relies on qualitative assessment rather than on quantitative data.^14–16

More recently, the prescription of AFOs has moved toward a more quantitative process.^17–20 These attempts have shown clearly measured differences in stiffness among orthoses,^18,21 and that the dorsiflexion/plantar flexion stiffness of an AFO had a measurable effect on certain gait characteristics.²¹ Yamamoto et al.¹⁷ measured the resistance to deformation and defined an overall average stiffness for 11 AFO designs. Klasson et al.²⁰ measured in three planes the motion of the orthosis resulting from an applied moment in a single plane. Sumiya et al.¹⁹ used the method of Rouleaux to determine the instant centers of rotation for posterior-type plastic AFOs. Sumiya et al.,¹⁸ comparing the amount of trim to the effectiveness of orthoses in gait, showed a relatively narrow range of acceptable trim for a particular individual. As the amount of trim is associated with stiffness, other design variables that alter stiffness may have equally narrow ranges of acceptability. The two somewhat dominant quantitative design considerations used to describe the mechanical behavior of AFOs, stiffness and the axis of rotation, are usually not independent. When a change in orthosis stiffness is made, by altering trim line or ankle-hinge design for example, the kinematics of the orthosis may be inadvertently changed.

The long-term goal of the present research is to develop orthosis-design techniques that allow for the independent control of orthosis stiffness and orthosis kinematics. The specific aim of the present investigation was to determine the extent that the kinematics of an orthosis are dependent on original design or design changes made to alter stiffness for four specific AFOs.

Methods

Four orthoses were tested. Two hingeless all-plastic AFOs that differed in amount of trim were tested, the solid-ankle design, and the posterior-spring design (Figure 1 ). The locked-hinge design (Camber Axis Hinge?, Analog Orthotics, Mansfield, OH; Figure 2 ) uses an ankle hinge comprised of two 40 mm-diameter coaxial circular disks. One disk has an annular slot that spans 115°. When a cam is placed in the slot, the hinge is free to rotate relatively unconstrained until the cam comes in contact with the end of the slot, at which point the hinge is essentially locked. A cam that completely filled the slot was used for the present study, thus there was no unconstrained rotation in the orthosis ankle joint. The fourth orthosis tested has a flexible-hinge design (Tamarack Flexure Joint, Tamarack Habilitation Technologies, St. Paul, MN; Figure 3 ). This design uses a flexible silastic hinge connecting the shank and foot sections of the orthosis. A posterior silastic stop is connected to the foot section of the orthosis and acts to limit plantarflexion when it comes into contact with the shank section. A posterior strap connects the shank section and foot section of the orthosis and acts to limit dorsiflexion.

The four orthoses tested were fabricated by placing a cast on the leg and foot of one of the investigators. The cast was bivalved and removed. The halves were realigned to form a mold for casting a foam impression of the shank and foot. Orthoses were formed over this positive impression of the shank and foot strictly according to manufacturers' specifications for the locked-hinge and flexible-hinge orthotic designs,²² as well as published orthotic fabrication techniques for the solid and posterior leaf-spring orthoses.^23,24 A second foam impression of the shank was made, into which a T-shaped metal frame had been cast. The T-shaped frame was used to mount the surrogate shank to a rigid frame (Figure 4 ). Each orthosis was attached to the surrogate shank with a proximal strap similar to clinical usage. To prevent rotation of the orthosis with respect to the surrogate, a bolt was placed through the orthosis and surrogate just proximal to the strap. The bolt location was far enough from the ankle so as not to alter the stiffness or resistance to deformation of the orthosis in the ankle region. A fixed "stop" was used on the posterior aspect of the orthosis when testing in dorsiflexion/plantar flexion to prevent the orthosis from pulling away from the shank when the foot was moved into plantarflexion (Figure 4 ). This constraint is similar to the limitation that would be applied by wearing a shoe. In this configuration, the shank section of the orthosis was fixed on the surrogate shank, and the foot section of the orthosis was loaded and deformed relative to the shank section.

Kinematics

The motion of the foot section of the orthosis with respect to the shank section was described in terms of finite screw axis displacements. The screw axis is a line located in space about which the foot section can be thought to rotate and also along which the foot section can be thought to translate when moving from one position to a second position (see Appendix at the end of this document). Together this translation and rotation are referred to as a screw displacement because they are described with respect to the same axis. This terminology is derived from the motion of a screw advancing through a threaded hole; the screw's translation is in the direction of the axis of the screw and the screw's rotation is about the axis of the screw. The orientation of the screw axis was characterized by the angle it made with respect to a line perpendicular to the plane of dorsiflexion/plantarflexion in which the orthoses were deformed. The location of the screw axis with respect to the orthoses was found by computing the coordinates of the point where the screw axis intersected a sagittal plane tangent to the most medial point of the orthoses covering the medial malleolus.

The screw axis orientation and location, and the translation along the screw axis, were determined from the measured position of four markers on both the foot and shank section of the orthosis (see Appendix at the end of this document). The locations of these points were determined with the Optotrak? (Northern Digital, Waterloo, Canada). This system tracked the three-dimensional coordinates of markers (infrared light-emitting diodes) as the foot section was rotated in dorsiflexion/plantar flexion. Markers were arranged into two rigid arrays of four markers each with one marker of the plane determined by the other three. One of these arrays was attached to the shank and one to the foot (Figure 4 ). In addition, a single marker was attached to the anterior region of the dorsal surface of the orthoses.

The kinematic data collected during cyclic loading of the orthoses were divided into subsets, each subset consisting of one half of a load cycle during which the foot angle changed monotonically from maximum plantarflexion (dorsiflexion) to maximum dorsiflexion (plantarflexion). The screw displacements were calculated in each subset for the two positions represented by the first data frame and the first subsequent data frame for which the angular rotation of the foot section was 6°. This process was repeated beginning with the second, third, and remaining data frames. The location of the screw axis was associated with the mean angle of plantarflexion or dorsiflexion of the first and last frame of each subset of the total motion. The angular rotation of the foot section was known to within ± 0.05° (see Appendix at the end of this document). The uncertainty of the location of the screw axis was ± 2.2 mm for rotations of 6° (see Appendix at the end of this document).

Kinetics

An instrumented lever arm was used to apply moments to the foot section of the orthoses (Figure 4 ). The lever arm was attached to the plantar surface of an orthosis by means of a rigid mounting bracket. Dorsiflexion/plantarflexion moments were generated by applying forces to the (posterior) free end of the lever arm. The lever arm was instrumented with 12 strain gages arranged in three circuits to measure three orthogonal moments (dorsiflexion/plantarflexion, inversion/eversion, and adduction/abduction). The gages were mounted between the free end of the lever arm where loads were applied and the mounting bracket. The uncertainties of the moment measurements were estimated to be 1% of full scale. Full scale was 75 N·m in dorsiflexion/plantarflexion, 25 N·m in adduction/abduction, and 18 N·m in inversion/eversion. According to our sign convention, dorsiflexion, inversion, and adduction moments and rotations were positive.

The weight of the lever arm and mounting bracket induced a parasitic load on the orthosis, creating an arbitrary rotation of the device. Prior to attaching the mounting bracket and lever arm to the orthosis, the three-dimensional coordinates of the single marker attached to the toe of the foot section were recorded. The mounting bracket and the loading arm were then attached. The lever arm was then positioned so that this marker was returned to its location when the orthosis was unloaded. Typically, this point was returned to within 2 mm of its original position. The three moment channels were zeroed with the lever arm in this position. To generate dorsiflexion/plantarflexion moments, we moved the lever arm from this starting position along a vertical guide located at the point of load application. Adduction/abduction and inversion/eversion rotations of the foot section of the orthosis were substantially constrained by the restrictions placed on the lever arm used to load the orthosis.

For each of the four orthoses tested, a varying dorsiflexion/plantarflexion moment was applied. Moments were measured about the dorsiflexion/plantarflexion axis, the primary axis, and the two orthogonal secondary axes. The marker position measurements were synchronized with moment measurements, and both were digitized at 60 samples per second. Resistance to deformation was represented by plotting moment as a function of angular deformation. Stiffness is defined as the slope of this curve. In the most general case, stiffness will vary from point to point on this curve and, because of the presence of hysteresis in the load-deformation curve, will have two values at each value of angular deformation. The stiffness at three points (the neutral, unloaded position and two extreme positions of angular deformation) was used to characterize each orthosis. The stiffness at these points was defined as the average of the two slopes of the two branches of the hysteresis loop of the moment-angular rotation curve at each point.

Results

Increased trim resulted in both reduced stiffness and reduced resistance to dorsiflexion/plantarflexion (Figure 5 ). The stiffness of the solid-ankle design decreased continuously from a maximum at 6° of plantarflexion to a minimum at 8° of dorsiflexion. The stiffness of the posterior-spring design, although relatively uniform over the range tested, was maximum at the neutral position, and decreased slightly with increasing plantarflexion and dorsiflexion. Both the stiffness and the resistance to dorsiflexion/plantarflexion were reduced when the flexible-hinge design was used compared to when the locked-hinge design was used (Figure 6 ). The stiffness of the locked-hinge orthosis was maximum at the extremes of plantarflexion and dorsiflexion and minimum at the neutral position. The stiffness of the flexible-hinge orthosis was a minimum in plantarflexion and increased with increasing dorsiflexion.

The location of the screw axis for the solid-ankle design moved along a path orientated predominantly in the inferosuperior direction (Figure 7A ). This path was approximately 40 mm in length and was located 25 mm posterior to the medial malleolus (Figure 7B ). Increased trim resulted in a shift of this path approximately 30 mm posterior relative to the path for the solid-ankle orthosis (Figure 8A ). In addition, the path was no longer in a relatively straight line, but rather included a loop-shaped region (Figure 8B ) seen in all load cycles.

In contrast, the location of the screw axis for the locked-hinge design moved in both the inferosuperior and anteroposterior directions, but always remained within 7 mm of the center of the hinge, located at the medial malleolus (Figure 9A and 9B ). Replacing the locked-hinge design with the flexible-hinge design resulted in a substantial increase in the movement of the location of the screw axis (Figure 10A and 10B ). The path was displaced approximately 15 mm posteriorly with respect to the medial malleolus and moved inferosuperiorly almost 50 mm.

The screw axis was oriented predominantly in the mediolateral direction for all four orthoses tested. The angle that the screw axis made with respect to a line perpendicular to a sagittal plane ranged between 5.2° to 9.2° for the solid-ankle design, 3.6° to 6.0° for the locked-hinge design, 3.4° to 18.1° for the posterior-spring design, and 0.0° to 4.9° for the flexible-hinge design. The orientation of the screw axis was nearly constant over the ranges of dorsiflexion/plantarflexion tested for the locked-hinge orthosis (Figure 11 ), whereas that of the posterior-spring design varied the most (Figure 12 ). The average displacement along the screw axis associated with a 6° rotation of the foot section ranged from 0.2 mm for the locked-hinge design to 0.4 mm for the flexible-hinge design.

The stiffness of the solid-ankle design was 7.2 N·m/deg at maximum plantarflexion and decreased monotonically to 5.9 N·m/deg at the neutral position and 3.6 N·m/deg at maximum dorsiflexion (Table 1 ). In contrast, the locked-hinge design was stiffest at the extremes of angular deformation (9.0 N·m/deg in plantarflexion and 7.0 N·m/deg in dorsiflexion) and least stiff in the neutral position (2.8 N·m/deg). The stiffness of the posterior-spring design was relatively constant over the range of angular deformations tested (1.3 N·m/deg at maximum plantarflexion, 1.6 N·m /deg at the neutral position, and 1.2 N·m/deg at maximum dorsiflexion). The flexible-hinge design was the least stiff of the four designs tested (0.9 N·m /deg at maximum plantarflexion and the neutral position), stiffening in dorsiflexion (2.0 N·m/deg at maximum dorsiflexion).

The secondary coupled moments were one to two orders of magnitude smaller than the primary applied moments (Table 2 ). Although the posterior-spring design was substantially less stiff than the solid-ankle design, the adduction/abduction coupled moments were approximately twice as high. The coupled inversion/eversion moments tended to be low in magnitude and the sign of these moments was not clearly related to the sign of the primary dorsiflexion/plantarflexion moment. For the solid-ankle, posterior-spring, and flexible-hinge designs an applied primary dorsiflexion moment produced a secondary abduction moment resisting abduction of the foot section of the orthosis. For the locked-hinge design, an applied dorsiflexion moment produced an adduction moment.

Discussion

Kinematic and kinetic properties of four AFOs were evaluated quantitatively. Results showed clear differences in the mechanical behavior among the orthoses tested. These differences have potentially important implications for the design and performance of ankle-foot orthoses.

Changes in orthosis design that are made ostensibly to alter stiffness or resistance to deformation can also alter the kinematics of the orthosis. For the solid-ankle design, increasing the amount of trim reduced the stiffness and resistance to deformation during dorsiflexion/plantarflexion, but the screw axis was moved 30 mm posteriorly and the coupled adduction moments were almost doubled. This result is similar to that reported by Sumiya et al.,¹⁹ who determined the instant centers of rotation for all-plastic orthoses of varying trim. Their results show that the instant centers of rotation tend to be centered in the region bounded by the anterior and posterior edges of the orthosis at approximately ankle height. A similar posterior shift in the location of the screw axis is seen as the anterior margin of the plastic is moved posteriorly because of increased trim (Figure 7 and 8 ). If a locked-hinge orthosis is replaced by a flexible-hinge orthosis to reduce stiffness and resistance to deformation, a substantial alteration in kinematics also takes place. The location of the screw axis for the locked-hinge design remains within 7 mm of the medial malleolus, indicating hinge-like rotations of the foot section. Changing to the flexible-hinge design shifts the location of the screw axis 15 mm posteriorly and increases the inferosuperior range of the screw axis to 40 mm. Differences between the location of the ankle's screw axis and the location of the screw axis for different designs and for different dorsiflexion/plantarflexion angles for a single design may result in a mechanical mismatch between the orthosis and the ankle, resulting in pistoning.

The four designs tested illustrated the rather wide range of kinematic and kinetic characteristics of AFOs and how these characteristics are related to design. Measurements among three orthoses with substantially different designs (solid-ankle, locked-hinge, and flexible-hinge) showed that the locked-hinge design was stiffest at the extremes of plantarflexion and dorsiflexion but that the solid-ankle design was stiffest at extreme plantarflexion. It appears that as the solid-ankle orthosis is deformed in plantarflexion, the structural response is strongly determined by tension developed along its anterior medial and lateral edges, resulting in a slight increase in stiffness with increasing plantarflexion. In contrast, the stiffness decreased with increasing dorsiflexion. This appeared to be because of buckling of the medial and lateral edges of the orthosis as they were compressed during dorsiflexion. This was the only orthosis that showed this substantial (50%) decrease in stiffness with increased dorsiflexion. This might be important for patient acceptance as decreasing stiffness in dorsiflexion may result in a feeling that the orthosis was collapsing or "giving way" and may result in altered gait to avoid putting the foot into dorsiflexion.

The two least-stiff orthoses, the flexible-hinge design and the posterior-spring design, exhibited the most uniform absolute stiffness over the range of dorsiflexion/plantarflexion tested. When the flexible-hinge orthosis is deformed in plantarflexion, the posterior stop makes contact with the posterior aspect of the shank section and resistance to deformation is supplied by compression of this column and extension of the flexible silastic hinges. In dorsiflexion, the posterior strap resists motion of the most posterior aspect of the foot section of the orthosis and continued dorsiflexion is accompanied by deformation of the silastic hinge in compression. The relatively low stiffness of the silastic results in low stiffness for the orthosis in general.

The sagittal-plane symmetry of the four designs was reflected in the fact that the screw axis tended to be parallel to a line perpendicular to the sagittal plane. The orientation of the screw axis for the flexible-hinge design was most nearly perpendicular to a sagittal plane. It appears that the foot section of the orthosis tends to rotate about the center of the relatively flexible silastic hinges. The orientation of the axis was qualitatively similar for the locked-hinge design and the design. However, the locked-hinge design showed the least variation in orientation of the designs tested. The orientation of the screw axis for these designs is likely a result in slight ásymmetries in the location of the center of the locked hinges or the amount of medial and lateral trim. The posterior-spring design exhibited the widest variation in the orientation of the screw axis, up to 18° in extreme dorsiflexion. The deviation of the screw axis from the mediolateral ankle axis is reflected in the presence of non-zero coupled moments. The magnitude of the coupled moments is a function of the stiffness characteristics of each orthosis. The presence of the coupled moments may be because of asymmetries in trim line, hinge construction and location, material properties, or applied loads.

Whereas the moments about the axis of primary loading can be thought of as characteristics of the orthoses, the coupled moments and rotations that are induced by the primary loading depend on the particular constraints that are applied during the loading. In our study, the coupled rotations were limited by the restraint applied to the free end of the lever arm used to load the orthosis in dorsiflexion/plantarflexion. If the foot section is constrained from rotating about the axes orthogonal to the axis of primary load, then reaction moments will be developed in the orthosis. In the present study, constraint on the free end of the loading bar substantially constrained the foot section of the orthoses from adduction/abduction rotations. In contrast, the loading method used by Klasson et al.²⁰ provided little constraint of motion under a primary dorsiflexion/plantarflexion moment. They reported coupling in terms of rotations rather than moments.

One long-standing criticism of thermoplastic AFOs is the inadvertent skin breakdown associated with excessive pressure or shear forces caused by poor fit or poor orthotic design. In light of the results of the present study, these frictional skin problems may be related to a mechanical mismatch of the axes of rotation of the orthosis and the human ankle, as demonstrated by the performance results of the solid-ankle, posterior-leaf, and flexible orthotic designs. The results of the present investigation show clearly the coupling between the kinetics and kinematics of the orthoses tested. Changes made to an orthosis with satisfactory fit that are intended to improve the stiffness may result in concomitant changes in the kinematics of the brace that may reduce the quality of the fit.

AFOs have been prescribed historically to improve the gait patterns of patients. Clinical theories and technical products promoting increased material stiffness with enhanced performance have been put forward with no direct methods for measuring clinical outcomes or gait function. Within the limitations of the present study, we have identified several important factors relating clinical orthotic practice to the quantification of AFO mechanical characteristic in the prescription process of patient management with various types of AFOs. Resistance to collapsing AFO dorsiflexion in the terminal stance phase of gait can be essential to the patient requiring stabilization of the impaired limb through ground-reaction forces. The stiffness characteristics exhibited by the orthoses tested indicate that the solid-ankle and locked-hinge AFOs may be best suited for this clinical tasks. The locked-hinge orthosis clearly demonstrates increased resistance to increased dorsiflexion in comparison with the other designs tested. Plantarflexion control with an orthosis in the presence of a flail ankle and foot during the swing phase of gait places no great demand upon an AFO and all the orthoses tested responded accordingly in static testing.

The secondary coupled moments recorded during quasistatic loading may provide clinical insight into actual dynamic wearing conditions where random loads are imposed upon AFOs during gait activities. Where stiffness control of the unstable hind foot is required, more flexible orthotic systems may not be suitable for stabilization of the deformity.

The clinical implications of the present study derive from the recognition that orthosis stiffness and orthosis kinematics are both important mechanical characteristics of AFOs. Similarly, there is subject-to-subject variation in ankle stiffness and ankle kinematics. Thus, two subjects who prefer similar AFO stiffness may have inherently different ankle kinematics and require their preferred AFO to have a unique screw axis pattern.

To met these needs, AFO design can be improved by having the ability to design independently the orthosis stiffness and orthosis kinematics. Judicious selection of AFO design continues to require thoughtful matching of patient needs with available treatment modalities.

Appendix

The finite motion of the foot section with respect to the shank section of each orthosis from an initial position to a final position was characterized by the finite screw displacement. Finite motion refers to the fact that position was determined at discrete times, separated by a finite rotation, in this case 6°. Orthoses' kinematics are then determined at the initial and final positions. In the present study, the kinematic data were divided into subsets, each subset representing a 6° rotation of the foot section of the orthosis in dorsiflexion/plantarflexion. The initial and final positions mentioned previously refer to the initial and final position for each of these subsets. In principle, the path of the motion between the initial and final positions is not known. The finite screw axis is one of several ways of describing the relation between the initial and final positions.

The screw axis is a line about which the rigid body can be thought to rotate, and along which the rigid body can be thought to translate, in a manner that takes the rigid body from its initial to its final position. The screw axis was determined from measurements of the three-dimensional coordinates of four points on the foot section of an orthosis in its initial and final positions.²⁵ The relationship between the final value of the three-dimensional coordinates and the initial value of the three-dimensional coordinates of these four points is given by

where (A_x1,A_y1,A_z1), (B_x1,B_y1,B_z1), (C_x1,C_y1,C_z1), and (D_x1,D_y1,D_z1) are the three-dimensional coordinates of the initial location of the four points; (A_x2,A_y2,A_z2),(B_x2,B_y2,B_z2), (C_x2,C_y2,C_z2), and (D_x2,D_y2,D_z2) are the three-dimensional coordinates of the final location of the four points; u_x, u_y, and u_z are the direction cosines of the screw axis; D is the translation along the screw axis; and

is the rigid-body rotation matrix where

theta; is the rotation about the screw axis. The following equation can be used to solve for the screw matrix:

For values of theta less than or equal to 45°

The intersection of the screw axis with a plane is calculated by first defining the plane. In the present investigation, the plane is a sagittal plane that is tangent to the medial aspect of the medial malleolus. The coordinate system used in this investigation was oriented such that this

plane was defined as where y_o is determined by digitizing (with the Optotrak?) the coordinates of the medial aspect of the medial malleolus. Once this point is determined, the following three equations can be solved for the translation D along the screw axis and the coordinates of the intersection of the screw axis, with the defined sagittal plane, x_o, and z_o:

In the present study, the coordinates of the individual markers were determined to within ± 0.1 mm in the coronal plane and ± 0.15 mm in the sagittal plane of the orthosis. To determine the accuracy of rotation measurements, a marker array was mounted on a rotary head (accurate to 0.003°) in a configuration similar to that used in the orthosis experiments. Rotational accuracy measured with the Optotrak was determined to be ± 0.05°. The accuracy of the location of the intersection of the screw axis with a predefined plane was estimated by inserting a three-jaw chuck in the rotary head with its long axis aligned with the axis of rotation of the rotary head. The Optotrak was used to digitize a point at the tip of the three-jaw chuck on the axis of rotation. The screw displacement axis of the rotary head was determined, and the coordinates of the point where this axis intersected a plane passing through the point digitized at the tip of the three-jaw chuck were computed. The accuracy of the determination of this point increased as the magnitude of the angular difference between the initial and final positions increased to 6°, increasing only marginally for larger angles. The uncertainty of this point was defined as the magnitude of the radial distance between the digitized and calculated points (mean plus two standard deviations). This uncertainty was determined to be 2.2 mm.

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