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Evaluation of a long-range transmitter for use with a magnetic tracking device in motion analysis
Journal of Biomechanics 31 (1998) 957—961
Technical Note
Evaluation of a long-range transmitter for use with a magnetic tracking device in motion analysis Judd S. Day!, Genevieve A. Dumas!,*, Duncan J. Murdoch" !Department of Mechanical Engineering, Queen+s University, Kingston, Ontario, Canada K7L 3N6 "Department of Mathematics and Statistics, Queen+s University, Kingston, Ontario, Canada K7L 3N6 Received in final form 26 May 1998
Abstract The suitability of a long-range transmitter was evaluated for use with a Polhemus Fastrak' magnetic tracking device in kinesiologic studies. Performance was judged by comparing positional and rotational accuracy to a standard transmitter. Data were obtained at distances of up to 2.7 and 5.0 m for the standard and long-range transmitters, respectively. Use of the long-range transmitter improved system performance allowing reproducible measurements at a greater distance. However, it is necessary to calibrate the system in each new test environment as there can be significant distortion of the magnetic field. ( 1998 Elsevier Science Ltd. All rights reserved. Keywords: Magnetic tracking device; Accuracy; Error; Kinematic; Three-dimensional
1. Introduction
2. Methods
Magnetic tracking devices are commonly used in kinesiology. Their positional error is less than 2% of the measured distance at ranges less than 60—70 m (An et al., 1988; McKellop et al., 1993; Milne et al., 1995; Zoghi et al., 1992). They have been applied to study shoulder movement (Johnson and Anderson, 1990), and spine kinematics (Dolan and Adams 1993; Pearcy, 1992). Study of gross movements such as manual materials handling has been limited by the restricted range of the transmitter. Currently, there are two types of transmitters available for the Polhemus Fastrak' (Polhemus Incorporated, P.O. Box 560, Colchester, Vermont, 05446, http://www. polhemus.com). The standard transmitter is optimized for transmitter/sensor separations of up to 75 cm. Use of a long-range transmitter is expected to triple the effective operating range of the device. In this study, the performances of a long-range and standard transmitter were compared to evaluate their ability to record motion at distances larger than 75 cm.
2.1. Apparatus
* Corresponding author. Tel.: 001 613 545 2648; fax: 001 613 545 6489. 0021-9290/98/$19.00 ( 1998 Elsevier Science Ltd. All rights reserved. PII: S 0 0 2 1 - 9 2 9 0 ( 9 8 ) 0 0 0 8 9 - X
An apparatus consisting of an upright framework and a jig was used to evaluate positional and rotational accuracy (Fig. 1). The framework consisted of a rectangular sheet of Plexiglas' fitted vertically into a slotted wooden base. There were nine parallel slots on the base spaced 20 cm apart. A grid of holes were drilled into the Plexiglas' sheet with spacing of 20 cm, forming five columns and eight rows. The jig was composed of three mutually orthogonal Plexiglas' faces. Each face was fit with a rod that allowed it to be inserted into the upright frame. A series of levels were used to reproduce four orientations about each rod. For example, inserting Rod A into the frame caused Face A to become flush with the upright Plexiglas sheet (Fig. 1). By rotating the jig about Rod A, Levels B and C could be used to define two distinct orientations each. In total, two sets of 12 orientations could be reproduced with a constant magnetic centre, one on each side of the sheet. An Optotrak' (Northern Digital Inc., 403 Albert St., Waterloo, Ontario, Canada, N7L 3V2) was used to assess the accuracy of the above system. This active marker
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J.S. Day et al. / Journal of Biomechanics 31 (1998) 957—961
The transmitter was positioned and aligned with the frame’s reference system by positioning a sensor at a known position 60 cm from the transmitter, the minimum recommended distance for the long-range transmitter. Using the Fastrak to determine the position and orientation of the sensor with respect to the transmitter, the transmitter’s position and orientation were adjusted by moving a stand upon which the transmitter was mounted until it was aligned with the sensor. The stand was at a height of 1.4 m. Measurements were taken on both sides of the framework in five holes. Then the frame was moved 60 cm further from the source and measurements were repeated in the same five holes. After using three slots in the base, the entire frame was moved 180 cm from the source and aligned using chalk lines on the floor. This procedure was repeated until signal quality decayed noticably. For each position of the sensor, 200 samples were collected at a rate 128 Hz in each of the 12 possible orientations on each side of the framework. 2.3. Data analysis Assessment of errors required a definition for position and angular errors. Angular error was defined as the magnitude of rotation, in degrees, necessary to rotate the jig from the measured orientation to the true orientation about the screw axis. The angular error matrix was defined as the direction cosine matrix after removal of the known rotation and any small misalignment of the transmitter with respect to the calibration frame:
Fig. 1. Test setup. (a) Calibration frame. The upright frame can be positioned in any of the nine slots. The jig can be placed in any of the 40 holes on the vertical framework. The slots and holes used in the current protocol are darkened. (b) Calibration jig. The sensor was mounted such that its magnetic centre and coordinate axes are aligned with the longitudinal axes of the rods. The levels were used to accurately reproduce four orientations, about each rod axes.
opto-electronic system has an accuracy of greater than 0.45 mm at the depth of field used (system documentation). It was determined that reproducibility of position and orientation measurements were better than 0.5 cm and 0.8°, respectively. 2.2. Protocol The apparatus was used to evaluate the performance of the two transmitters in a large open room typical of an institutional setting (11.6 m]9.9 m, 3.9 m ceiling height, metal reinforced concrete floors supported by 18 in I beams).
[R ]"[R ] [R ] [E] . 3 4 or [E]"[R ]T[R ]T[R ], 4 3 .
CA
BD
1 3 + E !1 , ii 2 i/1 where R , R , R and E are the direction cosine matrices . 3 4 of the measured orientation, known rotation, source misalignment and the angular measurement error, respectively (Goldstein, 1950). Position error was defined as the vector distance between the known and measured positions after correcting the measured values for transmitter alignment. The errors were divided into three groups: (i) field distortion, (ii) random noise, and (iii) orientation dependent biases. The definitions for each type of error as well as the values calculated to quantify them are presented in Table 1. A single factor analysis of variance (ANOVA) was performed at each point to determine whether error values were dependent upon sensor orientation. error"cos~1
J.S. Day et al. / Journal of Biomechanics 31 (1998) 957—961
959
Table 1 Definition of errors and correction strategies Type of error
Definition
Calculated value
Correction strategy
Field distortion
Average error encountered over multiple orientation
Average error encountered in 12]200 measurements
Calibration procedure: Correction based on X,Y, Z position
Random noise
Deviation of measured values in a fixed position and orientation
Standard deviation of error for 200 measurements
Data smoothing of filtering
Orientation dependent biases
Deviation of average error for multiple orientations
Standard deviation of 12 average errors calculated as average of 200 measurements
Calibration procedure: Correction based on orientation at each X, Y, Z position — complex
3. Results
4. Discussion
For the standard transmitter, measurements were taken in four slots (maximum distance 2.8 m). At greater distances the signal quality deteriorated rapidly. For the long-range transmitter, measurements were taken in eight slots (maximum distance 5 m). Evaluation at greater distances was curtailed due to space constraints and large errors. Field distortion, the largest error, tended to be large near the floor and increased as the distance between the sensor and transmitter increased. Both transmitters experienced position errors of less than 3% of the total distance for measurements that were not near the floor and had distances of less than 1.2 m. At these positions angular errors were less than 2°. As the distance increased, errors increased rapidly. Measurements from both transmitters were affected by similar magnitudes of field distortion with the long-range transmitter indicating slightly better performance as the distance increased. The effects of field distortion are illustrated in Figs. 2a and 3. The amount of noise increased with distance. The performance of the standard transmitter deteriorated appreciably at distances greater than 1.5 m, while the LR transmitter operated at a much greater distance. While the standard transmitter experienced noise levels of up to 5%, the noise levels did not exceed 1% for the long-range transmitter (Fig. 2b). Orientation related effects were defined as the difference in the measurement error for different orientations at the same point. This became apparent at distances of 120 and 300 cm for the standard and long-range transmitters, respectively, increasing with distance (Fig. 2c). At smaller distances, they were negligible for the long-range transmitter and accounted for less than 1% of the position measurement for the standard transmitter. The results of the ANOVA (p(0.05) indicated high probability of a relation between orientation and the magnitude of the error.
The largest limitation of the method involved positioning the transmitter with respect to the reference frame. Measurements taken by the Fastrak' were used to position the transmitter and align it with a sensor placed in the calibration frame. At a distance of 60 cm the expected error for the standard transmitter is 1—2% ( McKellop et al., 1993; Milne et al., 1995; Zoghi et al., 1992). Day et al. (1996) used an identical test setup as in the present study. After manipulating the data to remove errors due to source misalignment, position errors for the standard source were less than 2% of the measured distance at distances up to 75 cm. It was assumed that the longrange transmitter would behave in a similar manner during alignment. Errors induced because of this limitation affect the field distortion but not the noise or orientation biases. The results of the study indicate that use of the longrange transmitter improved the performance of the system within the tested range. The increased signal strength extended the operating range by a factor of approximately 2 to 3 with respect to noise and orientation errors. However, an extremely large distortion was experienced at increased distances (Fig. 3). Field distortion, the largest source of measurement error, increased with the distance and was the largest when measurements were taken near the floor. The large amount of distortion near the floor was probably the result of the metal reinforcement of the floor. Raab et al. (1979) stated that magnetic interference due to fixed metal could be minimized by ensuring that metallic objects are at least twice as far away from the transmitter as the sensor. This may not be practical when measuring motion in a large volume. Raab et al. (1979) also stated that because field distortion is constant at any given position, compensation is possible. Fig. 3 illustrates that field distortion tended to ‘curve’ the measured space. It is recommended that a calibration procedure be used to correct this curvature.
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J.S. Day et al. / Journal of Biomechanics 31 (1998) 957—961
Fig. 2. Effect of the three types of error for the standard transmitter (ST) and long-range transmitter (LR) over the test range. (a) Field distortion errors. Error values were calculated by averaging all of the 200]12 measurements for each position. (b) Noise. The value plotted represents the standard deviation of the 200 calculated error values at each fixed position and orientation. (c) Orientation effects. The deviation of the 12 mean position and angular errors are plotted. The value plotted represents the variation in the mean measurement error associated with changes in sensor orientation at a fixed position.
J.S. Day et al. / Journal of Biomechanics 31 (1998) 957—961
961
In conclusion, it was determined that use of the longrange transmitter resulted in decreased noise and orientation errors. However, because of the large amount of distortion of the magnetic field, it is recommended to calibrate these devices when used to measure in large volumes.
Acknowledgements This research has been supported by Dupont Canada (Kingston Site), NSERC-CRD d661-001-95 and URIF dQU27-005.
References
Fig. 3. Graphical representation of average measurement error. The position and orientation at each true and measured point are represented by ‘sailboats’. The dark sailboats are located at the true position and orientation. The lighter sailboats are located at the average measured position. Position errors are indicated by the difference in position of the error boats from the true boats. Average measured angular errors are represented by the tilt of the sailboats from vertical. (a) Measurement errors in the range common to both measurement systems. (b) Top view of distortion encountered by the long-range transmitter over the entire range.
While calibration could remove the effects of field distortion, error correction is confounded by noisy measurements and orientation dependent biases. Random noise is probably introduced to the system as a result of a combination of ambient field noise and internal electronic noise. Filtering or averaging techniques may be employed to smooth noisy data. However, orientation dependent biases are very difficult to correct and it is recommended not to use the system at distances where orientation dependent biases are significant.
An, K.N., Jacobsen, M.C., Gerlund, L.J., Chao, E.Y.S., 1988. Application of a magnetic tracking device to kinesiological studies. Journal of Biomechanics 21, 613—620. Day, J.S., Dumas, G.A., Murdoch, D.J., 1996. Evaluation of the Polhemus Fastrak for the collection of biomechanical kinematic data. In: Proceedings of the Ninth Biennial Conference and Symposia of the Canadian Society for Biomechanics, Simon Fraser University, Burnaby, British Columbia, Canada, pp. 338—339. Dolan, P., Adams, M.A., 1993. Influence of lumbar and hip mobility on the bending stresses acting on the lumbar spine. Clinical Biomechanics 8, 185—192. Goldstein, H., 1950. Classical Mechanics, Addison-Wesley, Reading, MA. Johnson, G.R., Anderson, M., 1990. Measurement of three-dimensional shoulder movement by an electromagnetic sensor. Clinical Biomechanics 5, 131—136. McKellop, H., Hoffman, R., Sarmiento, A., Bin, L.U., Ebramzadeh, E., 1993. Control of motion of tibial fractures with use of a functional brace or external fixator. Journal of Bone Joint Surgery 75A, 1019—1025. Milne, A.D., Chess, D.G., Johnson, J.A., King, G.J.W., 1995. Accuracy of an electromagnetic tracking device: A study of the optimal operating range and metal interferecnce. Journal of Biomechanics 29, 79—793. Pearcy, M.J., 1992. Twisting mobility of the human back in flexed postures. Spine 18, 114—119. Raab, F.H., Blood, E.B., Steiner, T.C., Jones, H.R., 1979. Magnetic position and orientation tracking system. IEEE Transactions in Aerospace Elector Systems AES15, 709—718. Zoghi, M., Hefzy, M.S., Fu K.C., Jackson W.T., 1992. A three dimensional morphometrical study of the distal human femur. Journal of Engineering and Medicine 206, 147—157.
Technical Note
Evaluation of a long-range transmitter for use with a magnetic tracking device in motion analysis Judd S. Day!, Genevieve A. Dumas!,*, Duncan J. Murdoch" !Department of Mechanical Engineering, Queen+s University, Kingston, Ontario, Canada K7L 3N6 "Department of Mathematics and Statistics, Queen+s University, Kingston, Ontario, Canada K7L 3N6 Received in final form 26 May 1998
Abstract The suitability of a long-range transmitter was evaluated for use with a Polhemus Fastrak' magnetic tracking device in kinesiologic studies. Performance was judged by comparing positional and rotational accuracy to a standard transmitter. Data were obtained at distances of up to 2.7 and 5.0 m for the standard and long-range transmitters, respectively. Use of the long-range transmitter improved system performance allowing reproducible measurements at a greater distance. However, it is necessary to calibrate the system in each new test environment as there can be significant distortion of the magnetic field. ( 1998 Elsevier Science Ltd. All rights reserved. Keywords: Magnetic tracking device; Accuracy; Error; Kinematic; Three-dimensional
1. Introduction
2. Methods
Magnetic tracking devices are commonly used in kinesiology. Their positional error is less than 2% of the measured distance at ranges less than 60—70 m (An et al., 1988; McKellop et al., 1993; Milne et al., 1995; Zoghi et al., 1992). They have been applied to study shoulder movement (Johnson and Anderson, 1990), and spine kinematics (Dolan and Adams 1993; Pearcy, 1992). Study of gross movements such as manual materials handling has been limited by the restricted range of the transmitter. Currently, there are two types of transmitters available for the Polhemus Fastrak' (Polhemus Incorporated, P.O. Box 560, Colchester, Vermont, 05446, http://www. polhemus.com). The standard transmitter is optimized for transmitter/sensor separations of up to 75 cm. Use of a long-range transmitter is expected to triple the effective operating range of the device. In this study, the performances of a long-range and standard transmitter were compared to evaluate their ability to record motion at distances larger than 75 cm.
2.1. Apparatus
* Corresponding author. Tel.: 001 613 545 2648; fax: 001 613 545 6489. 0021-9290/98/$19.00 ( 1998 Elsevier Science Ltd. All rights reserved. PII: S 0 0 2 1 - 9 2 9 0 ( 9 8 ) 0 0 0 8 9 - X
An apparatus consisting of an upright framework and a jig was used to evaluate positional and rotational accuracy (Fig. 1). The framework consisted of a rectangular sheet of Plexiglas' fitted vertically into a slotted wooden base. There were nine parallel slots on the base spaced 20 cm apart. A grid of holes were drilled into the Plexiglas' sheet with spacing of 20 cm, forming five columns and eight rows. The jig was composed of three mutually orthogonal Plexiglas' faces. Each face was fit with a rod that allowed it to be inserted into the upright frame. A series of levels were used to reproduce four orientations about each rod. For example, inserting Rod A into the frame caused Face A to become flush with the upright Plexiglas sheet (Fig. 1). By rotating the jig about Rod A, Levels B and C could be used to define two distinct orientations each. In total, two sets of 12 orientations could be reproduced with a constant magnetic centre, one on each side of the sheet. An Optotrak' (Northern Digital Inc., 403 Albert St., Waterloo, Ontario, Canada, N7L 3V2) was used to assess the accuracy of the above system. This active marker
958
J.S. Day et al. / Journal of Biomechanics 31 (1998) 957—961
The transmitter was positioned and aligned with the frame’s reference system by positioning a sensor at a known position 60 cm from the transmitter, the minimum recommended distance for the long-range transmitter. Using the Fastrak to determine the position and orientation of the sensor with respect to the transmitter, the transmitter’s position and orientation were adjusted by moving a stand upon which the transmitter was mounted until it was aligned with the sensor. The stand was at a height of 1.4 m. Measurements were taken on both sides of the framework in five holes. Then the frame was moved 60 cm further from the source and measurements were repeated in the same five holes. After using three slots in the base, the entire frame was moved 180 cm from the source and aligned using chalk lines on the floor. This procedure was repeated until signal quality decayed noticably. For each position of the sensor, 200 samples were collected at a rate 128 Hz in each of the 12 possible orientations on each side of the framework. 2.3. Data analysis Assessment of errors required a definition for position and angular errors. Angular error was defined as the magnitude of rotation, in degrees, necessary to rotate the jig from the measured orientation to the true orientation about the screw axis. The angular error matrix was defined as the direction cosine matrix after removal of the known rotation and any small misalignment of the transmitter with respect to the calibration frame:
Fig. 1. Test setup. (a) Calibration frame. The upright frame can be positioned in any of the nine slots. The jig can be placed in any of the 40 holes on the vertical framework. The slots and holes used in the current protocol are darkened. (b) Calibration jig. The sensor was mounted such that its magnetic centre and coordinate axes are aligned with the longitudinal axes of the rods. The levels were used to accurately reproduce four orientations, about each rod axes.
opto-electronic system has an accuracy of greater than 0.45 mm at the depth of field used (system documentation). It was determined that reproducibility of position and orientation measurements were better than 0.5 cm and 0.8°, respectively. 2.2. Protocol The apparatus was used to evaluate the performance of the two transmitters in a large open room typical of an institutional setting (11.6 m]9.9 m, 3.9 m ceiling height, metal reinforced concrete floors supported by 18 in I beams).
[R ]"[R ] [R ] [E] . 3 4 or [E]"[R ]T[R ]T[R ], 4 3 .
CA
BD
1 3 + E !1 , ii 2 i/1 where R , R , R and E are the direction cosine matrices . 3 4 of the measured orientation, known rotation, source misalignment and the angular measurement error, respectively (Goldstein, 1950). Position error was defined as the vector distance between the known and measured positions after correcting the measured values for transmitter alignment. The errors were divided into three groups: (i) field distortion, (ii) random noise, and (iii) orientation dependent biases. The definitions for each type of error as well as the values calculated to quantify them are presented in Table 1. A single factor analysis of variance (ANOVA) was performed at each point to determine whether error values were dependent upon sensor orientation. error"cos~1
J.S. Day et al. / Journal of Biomechanics 31 (1998) 957—961
959
Table 1 Definition of errors and correction strategies Type of error
Definition
Calculated value
Correction strategy
Field distortion
Average error encountered over multiple orientation
Average error encountered in 12]200 measurements
Calibration procedure: Correction based on X,Y, Z position
Random noise
Deviation of measured values in a fixed position and orientation
Standard deviation of error for 200 measurements
Data smoothing of filtering
Orientation dependent biases
Deviation of average error for multiple orientations
Standard deviation of 12 average errors calculated as average of 200 measurements
Calibration procedure: Correction based on orientation at each X, Y, Z position — complex
3. Results
4. Discussion
For the standard transmitter, measurements were taken in four slots (maximum distance 2.8 m). At greater distances the signal quality deteriorated rapidly. For the long-range transmitter, measurements were taken in eight slots (maximum distance 5 m). Evaluation at greater distances was curtailed due to space constraints and large errors. Field distortion, the largest error, tended to be large near the floor and increased as the distance between the sensor and transmitter increased. Both transmitters experienced position errors of less than 3% of the total distance for measurements that were not near the floor and had distances of less than 1.2 m. At these positions angular errors were less than 2°. As the distance increased, errors increased rapidly. Measurements from both transmitters were affected by similar magnitudes of field distortion with the long-range transmitter indicating slightly better performance as the distance increased. The effects of field distortion are illustrated in Figs. 2a and 3. The amount of noise increased with distance. The performance of the standard transmitter deteriorated appreciably at distances greater than 1.5 m, while the LR transmitter operated at a much greater distance. While the standard transmitter experienced noise levels of up to 5%, the noise levels did not exceed 1% for the long-range transmitter (Fig. 2b). Orientation related effects were defined as the difference in the measurement error for different orientations at the same point. This became apparent at distances of 120 and 300 cm for the standard and long-range transmitters, respectively, increasing with distance (Fig. 2c). At smaller distances, they were negligible for the long-range transmitter and accounted for less than 1% of the position measurement for the standard transmitter. The results of the ANOVA (p(0.05) indicated high probability of a relation between orientation and the magnitude of the error.
The largest limitation of the method involved positioning the transmitter with respect to the reference frame. Measurements taken by the Fastrak' were used to position the transmitter and align it with a sensor placed in the calibration frame. At a distance of 60 cm the expected error for the standard transmitter is 1—2% ( McKellop et al., 1993; Milne et al., 1995; Zoghi et al., 1992). Day et al. (1996) used an identical test setup as in the present study. After manipulating the data to remove errors due to source misalignment, position errors for the standard source were less than 2% of the measured distance at distances up to 75 cm. It was assumed that the longrange transmitter would behave in a similar manner during alignment. Errors induced because of this limitation affect the field distortion but not the noise or orientation biases. The results of the study indicate that use of the longrange transmitter improved the performance of the system within the tested range. The increased signal strength extended the operating range by a factor of approximately 2 to 3 with respect to noise and orientation errors. However, an extremely large distortion was experienced at increased distances (Fig. 3). Field distortion, the largest source of measurement error, increased with the distance and was the largest when measurements were taken near the floor. The large amount of distortion near the floor was probably the result of the metal reinforcement of the floor. Raab et al. (1979) stated that magnetic interference due to fixed metal could be minimized by ensuring that metallic objects are at least twice as far away from the transmitter as the sensor. This may not be practical when measuring motion in a large volume. Raab et al. (1979) also stated that because field distortion is constant at any given position, compensation is possible. Fig. 3 illustrates that field distortion tended to ‘curve’ the measured space. It is recommended that a calibration procedure be used to correct this curvature.
960
J.S. Day et al. / Journal of Biomechanics 31 (1998) 957—961
Fig. 2. Effect of the three types of error for the standard transmitter (ST) and long-range transmitter (LR) over the test range. (a) Field distortion errors. Error values were calculated by averaging all of the 200]12 measurements for each position. (b) Noise. The value plotted represents the standard deviation of the 200 calculated error values at each fixed position and orientation. (c) Orientation effects. The deviation of the 12 mean position and angular errors are plotted. The value plotted represents the variation in the mean measurement error associated with changes in sensor orientation at a fixed position.
J.S. Day et al. / Journal of Biomechanics 31 (1998) 957—961
961
In conclusion, it was determined that use of the longrange transmitter resulted in decreased noise and orientation errors. However, because of the large amount of distortion of the magnetic field, it is recommended to calibrate these devices when used to measure in large volumes.
Acknowledgements This research has been supported by Dupont Canada (Kingston Site), NSERC-CRD d661-001-95 and URIF dQU27-005.
References
Fig. 3. Graphical representation of average measurement error. The position and orientation at each true and measured point are represented by ‘sailboats’. The dark sailboats are located at the true position and orientation. The lighter sailboats are located at the average measured position. Position errors are indicated by the difference in position of the error boats from the true boats. Average measured angular errors are represented by the tilt of the sailboats from vertical. (a) Measurement errors in the range common to both measurement systems. (b) Top view of distortion encountered by the long-range transmitter over the entire range.
While calibration could remove the effects of field distortion, error correction is confounded by noisy measurements and orientation dependent biases. Random noise is probably introduced to the system as a result of a combination of ambient field noise and internal electronic noise. Filtering or averaging techniques may be employed to smooth noisy data. However, orientation dependent biases are very difficult to correct and it is recommended not to use the system at distances where orientation dependent biases are significant.
An, K.N., Jacobsen, M.C., Gerlund, L.J., Chao, E.Y.S., 1988. Application of a magnetic tracking device to kinesiological studies. Journal of Biomechanics 21, 613—620. Day, J.S., Dumas, G.A., Murdoch, D.J., 1996. Evaluation of the Polhemus Fastrak for the collection of biomechanical kinematic data. In: Proceedings of the Ninth Biennial Conference and Symposia of the Canadian Society for Biomechanics, Simon Fraser University, Burnaby, British Columbia, Canada, pp. 338—339. Dolan, P., Adams, M.A., 1993. Influence of lumbar and hip mobility on the bending stresses acting on the lumbar spine. Clinical Biomechanics 8, 185—192. Goldstein, H., 1950. Classical Mechanics, Addison-Wesley, Reading, MA. Johnson, G.R., Anderson, M., 1990. Measurement of three-dimensional shoulder movement by an electromagnetic sensor. Clinical Biomechanics 5, 131—136. McKellop, H., Hoffman, R., Sarmiento, A., Bin, L.U., Ebramzadeh, E., 1993. Control of motion of tibial fractures with use of a functional brace or external fixator. Journal of Bone Joint Surgery 75A, 1019—1025. Milne, A.D., Chess, D.G., Johnson, J.A., King, G.J.W., 1995. Accuracy of an electromagnetic tracking device: A study of the optimal operating range and metal interferecnce. Journal of Biomechanics 29, 79—793. Pearcy, M.J., 1992. Twisting mobility of the human back in flexed postures. Spine 18, 114—119. Raab, F.H., Blood, E.B., Steiner, T.C., Jones, H.R., 1979. Magnetic position and orientation tracking system. IEEE Transactions in Aerospace Elector Systems AES15, 709—718. Zoghi, M., Hefzy, M.S., Fu K.C., Jackson W.T., 1992. A three dimensional morphometrical study of the distal human femur. Journal of Engineering and Medicine 206, 147—157.