6R instrumented spatial linkages for anatomical joint motion measurement--Part 1: Design

An instrumented spatial linkage for measuring knee joint kinematics

BACKGROUND

In this study, the design and development of a highly accurate instrumented spatial linkage (ISL) for kinematic analysis of the ovine stifle joint is described. The ovine knee is a promising biomechanical model of the human knee joint.

METHODS

The ISL consists of six digital rotational encoders providing six degrees of freedom (6-DOF) to its motion. The ISL makes use of the complete and parametrically continuous (CPC) kinematic modeling method to describe the kinematic relationship between encoder readings and the relative positions and orientation of its two ends. The CPC method is useful when calibrating the ISL, because a small change in parameters corresponds to a small change in calculated positions and orientations and thus a smaller optimization error, compared to other kinematic models. The ISL is attached rigidly to the femur and the tibia for motion capture, and the CPC kinematic model is then employed to transform the angle sensor readings to relative motion of the two ends of the linkage, and thereby, the stifle joint motion.

RESULTS

The positional accuracy for ISL after calibration and optimization was 0.3±0.2mm (mean +/- standard deviation). The ISL was also evaluated dynamically to ensure that accurate results were maintained, and achieved an accuracy of 0.1mm.

CONCLUSIONS

Compared to the traditional motion capture methods, this system provides increased accuracy, reduced processing time, and ease of use. Future work will be on the application of the ISL to the ovine gait and determination of in vivo joint motions and tissue loads.

CLINICAL RELEVANCE

Accurate measurement of knee joint kinematics is essential in understanding injury mechanisms and development of potential preventive or treatment strategies.

Design, Calibration and Validation of a Novel 3D Printed Instrumented Spatial Linkage that Measures Changes in the Rotational Axes of the Tibiofemoral Joint

An accurate axis-finding technique is required to measure any changes from normal caused by total knee arthroplasty in the flexion–extension (F–E) and longitudinal rotation (LR) axes of the tibiofemoral joint. In a previous paper, we computationally determined how best to design and use an instrumented spatial linkage (ISL) to locate the F–E and LR axes such that rotational and translational errors were minimized. However, the ISL was not built and consequently was not calibrated; thus the errors in locating these axes were not quantified on an actual ISL. Moreover, previous methods to calibrate an ISL used calibration devices with accuracies that were either undocumented or insufficient for the device to serve as a gold-standard. Accordingly, the objectives were to (1) construct an ISL using the previously established guidelines,(2) calibrate the ISL using an improved method, and (3) quantify the error in measuring changes in the F–E and LR axes. A 3D printed ISL was constructed and calibrated using a coordinate measuring machine, which served as a gold standard. Validation was performed using a fixture that represented the tibiofemoral joint with an adjustable F–E axis and the errors in measuring changes to the positions and orientations of the F–E and LR axes were quantified. The resulting root mean squared errors (RMSEs) of the calibration residuals using the new calibration method were 0.24, 0.33, and 0.15 mm for the anterior–posterior, medial–lateral, and proximal–distal positions, respectively, and 0.11, 0.10, and 0.09 deg for varus–valgus, flexion–extension, and internal–external orientations, respectively. All RMSEs were below 0.29% of the respective full-scale range. When measuring changes to the F–E or LR axes, each orientation error was below 0.5 deg; when measuring changes in the F–E axis, each position error was below 1.0 mm. The largest position RMSE was when measuring a medial–lateral change in the LR axis (1.2 mm). Despite the large size of the ISL, these calibration residuals were better than those for previously published ISLs, particularly when measuring orientations, indicating that using a more accurate gold standard was beneficial in limiting the calibration residuals. The validation method demonstrated that this ISL is capable of accurately measuring clinically important changes (i.e. 1 mm and 1 deg) in the F–E and LR axes.

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

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.

Validation of a New Method for Finding the Rotational Axes of the Knee Using Both Marker-Based Roentgen Stereophotogrammetric Analysis and 3D Video-Based Motion Analysis for Kinematic Measurements

In a previous paper, we reported the virtual axis finder, which is a new method for finding the rotational axes of the knee. The virtual axis finder was validated through simulations that were subject to limitations. Hence, the objective of the present study was to perform a mechanical validation with two measurement modalities: 3D video-based motion analysis and marker-based roentgen stereophotogrammetric analysis (RSA). A two rotational axis mechanism was developed, which simulated internal-external (or longitudinal) and flexion-extension (FE) rotations. The actual axes of rotation were known with respect to motion analysis and RSA markers within ±0.0006 deg and ±0.036 mm and ±0.0001 deg and ±0.016 mm, respectively. The orientation and position root mean squared errors for identifying the longitudinal rotation (LR) and FE axes with video-based motion analysis (0.26 deg, 0.28 m, 0.36 deg, and 0.25 mm, respectively) were smaller than with RSA (1.04 deg, 0.84 mm, 0.82 deg, and 0.32 mm, respectively). The random error or precision in the orientation and position was significantly better (p=0.01 and p=0.02, respectively) in identifying the LR axis with video-based motion analysis (0.23 deg and 0.24 mm) than with RSA (0.95 deg and 0.76 mm). There was no significant difference in the bias errors between measurement modalities. In comparing the mechanical validations to virtual validations, the virtual validations produced comparable errors to those of the mechanical validation. The only significant difference between the errors of the mechanical and virtual validations was the precision in the position of the LR axis while simulating video-based motion analysis (0.24 mm and 0.78 mm, p=0.019). These results indicate that video-based motion analysis with the equipment used in this study is the superior measurement modality for use with the virtual axis finder but both measurement modalities produce satisfactory results. The lack of significant differences between validation techniques suggests that the virtual sensitivity analysis previously performed was appropriately modeled. Thus, the virtual axis finder can be applied with a thorough understanding of its errors in a variety of test conditions.

Design and evaluation of a new general-purpose device for calibrating instrumented spatial linkages

Because instrumented spatial linkages (ISLs) have been commonly used in measuring joint rotations and must be calibrated before using the device in confidence, a calibration device design and associated method for quantifying calibration device error would be useful. The objectives of the work reported by this paper were to (1) design an ISL calibration device and demonstrate the design for a specific application, (2) describe a new method for calibrating the device that minimizes measurement error, and (3) quantify measurement error of the device using the new method. Relative translations and orientations of the device were calculated via a series of transformation matrices containing inherent fixed and variable parameters. These translations and orientations were verified with a coordinate measurement machine, which served as a gold standard. Inherent fixed parameters of the device were optimized to minimize measurement error. After parameter optimization, accuracy was determined. The root mean squared error (RMSE) was 0.175 deg for orientation and 0.587 mm for position. All RMSE values were less than 0.8% of their respective full-scale ranges. These errors are comparable to published measurement errors of ISLs for positions and lower by at least a factor of 2 for orientations. These errors are in spite of the many steps taken in design and manufacturing to achieve high accuracy. Because it is challenging to achieve the accuracy required for a custom calibration device to serve as a viable gold standard, it is important to verify that a calibration device provides sufficient precision to calibrate an ISL.

The Effects of Femoral Fixed Body Coordinate System Definition on Knee Kinematic Description

Understanding the differences in knee kinematic descriptions is important for comparing data from different laboratories and observing small but important changes within a set of knees. The purpose of this study was to identify how differences in fixed body femoral coordinate systems affect the described tibiofemoral and patellofemoral kinematics for cadaveric knee studies with no hip present. Different methods for describing kinematics were evaluated on a set of seven cadaveric knees during walking in a dynamic knee simulator. Three anatomical landmark coordinate systems, a partial helical axis, and an experimental setup-based system were examined. The results showed that flexion-extension was insensitive to differences in the kinematic systems tested, internal-external rotation was similar for most femoral coordinate systems although there were changes in absolute position, varus-valgus was the most sensitive to variations in flexion axis direction, and anterior-posterior motion was most sensitive to femoral origin location. Femoral coordinate systems that define the sagittal plane using anatomical landmarks and locate the flexion axis perpendicular to the femur’s mechanical axis in the frontal plane were typically similar and described kinematics most consistently.

Development and evaluation of an instrumented linkage system for total knee surgery

The principles and application of total knee surgery using optical tracking have been well demonstrated, but electromagnetic tracking may offer further advantages. We asked whether an instrumented linkage that attaches directly to the bone can maintain the accuracy of the optical and electromagnetic systems but be quicker, more convenient, and less expensive to use. Initial testing using a table-mounted digitizer to navigate a drill guide for placing pins to mount a cutting guide demonstrated the feasibility in terms of access and availability. A first version (called the Mark 1) instrumented linkage designed to fix directly to the bone was constructed and software was written to carry out a complete total knee replacement procedure. The results showed the system largely fulfilled these goals, but some surgeons found that using a visual display for pin placement was difficult and time consuming. As a result, a second version of a linkage system (called the K-Link) was designed to further develop the concept. User-friendly flexible software was developed for facilitating each step quickly and accurately while the placement of cutting guides was facilitated. We concluded that an instrumented linkage system could be a useful and potentially lower-cost option to the current systems for total knee replacement and could possibly have application to other surgical procedures.

Design and demonstration of a new instrumented spatial linkage for use in a dynamic environment: application to measurement of ankle rotations during snowboarding

Joint injuries during sporting activities might be reduced by understanding the extent of the dynamic motion of joints prone to injury during maneuvers performed in the field. Because instrumented spatial linkages (ISLs) have been widely used to measure joint motion, it would be useful to extend the functionality of an ISL to measure joint motion in a dynamic environment. The objectives of the work reported by this paper were to (i) design and construct an ISL that will measure dynamic joint motion in a field environment, (ii) calibrate the ISL and quantify its static measurement error, (iii) quantify dynamic measurement error due to external acceleration, and (iv) measure ankle joint complex rotation during snowboarding maneuvers performed on a snow slope. An "elbow-type" ISL was designed to measure ankle joint complex rotation throughout its range (+/-30 deg for flexion/extension, +/-15 deg for internal/external rotation, and +/-15 deg for inversion/eversion). The ISL was calibrated with a custom six degree-of-freedom calibration device generally useful for calibrating ISLs, and static measurement errors of the ISL also were evaluated. Root-mean-squared errors (RMSEs) were 0.59 deg for orientation (1.7% full scale) and 1.00 mm for position (1.7% full scale). A custom dynamic fixture allowed external accelerations (5 g, 0-50 Hz) to be applied to the ISL in each of three linear directions. Maximum measurement deviations due to external acceleration were 0.05 deg in orientation and 0.10 mm in position, which were negligible in comparison to the static errors. The full functionality of the ISL for measuring joint motion in a field environment was demonstrated by measuring rotations of the ankle joint complex during snowboarding maneuvers performed on a snow slope.

Validation of a calibration technique for 6-DOF instrumented spatial linkages

This paper presents a practical and effective approach to the calibration of instrumented spatial linkages for biomechanical applications. A 6-DOF mechanical linkage with rotational transducers is designed and in-house manufactured for this purpose. In order to assess the validity of the proposed calibration technique and to distinguish between geometrical and electrical parameters uncertainties, high-precision optical encoders are used and calibration is addressed from a kinematic point of view only. The proposed technique is based on a closed-loop method, in which the end segments of the linkage are connected to each other by revolute joints. A parametrical model of the system is formulated using a standard link-to-link transformation matrices approach. Continuous data collection is carried out and a recursive identification of kinematic parameters is implemented by the use of an extended Kalman filter algorithm. Results shows that the proposed technique, despites its simplicity, is effective in improving the accuracy of the system up to its theoretically computed resolution, which limits the performance of the real system.

In vivo elbow biomechanical analysis during flexion: three-dimensional motion analysis using magnetic resonance imaging

The purpose of this article is to evaluate in vivo 3-dimensional kinematics of the elbow joint during elbow flexion. We studied the ulnohumeral and radiohumeral joint noninvasively in 3 elbows in healthy volunteers using a markerless bone registration algorithm. Magnetic resonance images were acquired in 6 positions of elbow flexion. The inferred contact areas on the ulna against the trochlea tended to occur only on the medial facet of the trochlear notch in all of the elbow positions we tested. The inferred contact areas on the radial head against the capitellum occurred on the central depression of the radial head in all of the tested elbow positions except for 135 degrees flexion, where the anterior rim of the radial head articulates with the capitellum.

Calibration and validation of 6 DOFs instrumented spatial linkage for biomechanical applications. A practical approach

A method for both calibration and validation of a 6 DOF electrogoniometer is presented. A 6 Revolute Instrumented Spatial Linkage (6R-ISL) and a three-dimensional digitizer (3DD) were used simultaneously to collect both static and continuous poses of unconstrained or constrained motions. Validation occurred using a calibrated ball-and-socket joint. A parametrical model of the 6R-ISL (i.e. Virtual Goniometer or VG) was designed using a standard multibody system geometry. Two approaches were used to adjust the VG parameters: a parametrical adjustment of the VG linkage geometry, and a functional adjustment of the potentiometer calibration curves (angle-voltage) in a predefined range of motion. After calibration, 6R-ISL accuracy was better than 1 mm and 1 degrees for translation and orientation, respectively. The functional method presented in this paper can be suggested as a practical approach, which allows on-line checking and calibration of 6R-ISL within the specific range of interest of a particular anatomical joint. In addition, improving the potentiometer calibration curves was less time consuming than the parametrical adjustment.

Subject-specific finite element analysis of the human medial collateral ligament during valgus knee loading

The objectives of this study were (1) to develop subject-specific experimental and finite element (FE) techniques to study the three-dimensional stress-strain behavior of ligaments, with application to the human medial collateral ligament (MCL), and (2) to determine the importance of subject-specific material properties and initial (in situ) strain distribution for prediction of the strain distribution in the MCL under valgus loading. Eight male knees were subjected to varus-valgus loading at flexion angles of 0 degrees, 30 degrees, and 60 degrees. Three-dimensional joint kinematics and MCL strains were recorded during kinematic testing. Following testing, the MCL of each knee was removed to allow measurement of the in situ strain distribution and to perform material testing. A FE model of the femur-MCL-tibia complex was constructed for each knee to simulate valgus loading at each flexion angle, using subject-specific bone and ligament geometry, material properties, and joint kinematics. A transversely isotropic hyperelastic material model was used to represent the MCL. The MCL in situ strain distribution at full extension was used to apply in situ strain to each MCL FE model. FE predicted MCL strains during valgus loading were compared to experimental measurements using regression analysis. The subject-specific FE predictions of strain correlated reasonably well with experimentally measured MCL strains (R(2)=0.83, 0.72, and 0.66 at 0 degrees, 30 degrees, and 60 degrees, respectively). Despite large inter-subject variation in MCL material properties, MCL strain distributions predicted by individual FE models that used average MCL material properties were strongly correlated with subject-specific FE strain predictions (R(2)=0.99 at all flexion angles). However, predictions by FE models that used average in situ strain distributions yielded relatively poor correlations with subject-specific FE predictions (R(2)=0.44, 0.35, and 0.33 at flexion angles of 0 degrees, 30 degrees, and 60 degrees, respectively). The strain distribution within the MCL was nonuniform and changed with flexion angle. The highest MCL strains occurred at full extension in the posterior region of the MCL proximal to the joint line during valgus loading, suggesting this region may be most vulnerable to injury under these loading conditions. This work demonstrates that subject-specific FE models can predict the complex, nonuniform strain fields that occur in ligaments due to external loading of the joint.

A method for measuring joint kinematics designed for accurate registration of kinematic data to models constructed from CT data

A method for measuring three-dimensional kinematics that incorporates the direct cross-registration of experimental kinematics with anatomic geometry from Computed Tomography (CT) data has been developed. Plexiglas registration blocks were attached to the bones of interest and the specimen was CT scanned. Computer models of the bone surface were developed from the CT image data. Determination of discrete kinematics was accomplished by digitizing three pre-selected contiguous surfaces of each registration block using a three-dimensional point digitization system. Cross-registration of bone surface models from the CT data was accomplished by identifying the registration block surfaces within the CT images. Kinematics measured during a biomechanical experiment were applied to the computer models of the bone surface. The overall accuracy of the method was shown to be at or below the accuracy of the digitization system used. For this experimental application, the accuracy was better than +/-0.1mm for position and 0.1 degrees for orientation for linkage digitization and better than +/-0.2mm and +/-0.2 degrees for CT digitization. Surface models of the radius and ulna were constructed from CT data, as an example application. Kinematics of the bones were measured for simulated forearm rotation. Screw-displacement axis analysis showed 0.1mm (proximal) translation of the radius (with respect to the ulna) from supination to neutral (85.2 degrees rotation) and 1.4mm (proximal) translation from neutral to pronation (65.3 degrees rotation). The motion of the radius with respect to the ulna was displayed using the surface models. This methodology is a useful tool for the measurement and application of rigid-body kinematics to computer models.

Biomechanical functions of the iliolumbar ligament in L5 spondylolysis

Abiomechanical study of the functions of the iliolumbar ligament in L5 spondylolysis was performed. Five fresh cadaveric specimens were used. The bilateral ilia and sacrum were fixed. Four kinds of pure moments (10 Nm) were applied to the specimens at the top (L4) vertebra: flexion, extension, and right and left axial rotations. The three-dimensional position of the L5 vertebra was measured after serial transections in: (1) the intact condition; (2) bilateral pars interarticulares of L5 transected; (3) anterior bands of the iliolumbar ligaments transected; and (4) posterior bands of the iliolumbar ligaments transected. In L5 spondylolysis, flexion and axial rotation of L5 on S1 are significantly regulated by the anterior and posterior bands of the iliolumbar ligaments (especially by the posterior bands of the ligaments). The integrity of the ligament may determine the stability of the lumbosacral junction and the amount of forward slipping of the L5 vertebra.

Knee joint motion: description and measurement

Knee joint motion has been described in various ways in the literature. These are explained and commented on. Two methods for describing knee joint motion with 6 degrees of freedom (DOF)--Euler angle and the helical axis of motion--are discussed. Techniques to measure joint motion which can either approximate the motion to less than 6 DOF or fully measure the spatial motion are identified. These include electrical linkage methods, radiographic and video techniques, fluoroscopic techniques and electromagnetic devices. In those cases where the full spatial motion is measured, the data are available to describe the motion in simpler terms (or with less DOF) than three rotations with three translations. This is necessary for clinical application and to facilitate communication between the clinician and the engineer.

Accuracy of an electromagnetic measurement device and application to the measurement and description of knee joint motion

The aims of this study were to analyse the accuracy and sensitivity to metals of an electromagnetic motion measurement device and to use it to provide a description of motion of the human knee joint. A calibrated grid and sine bar were used to measure translational and rotational accuracy, which were found to be +/- 0.23 per cent of the step size (translation) and +/- 2.0 per cent of the step size (rotation) within an optimal operational zone of minimal positional error for which the receiver to transmitter separation was between 271 and 723 mm. The presence of multiple receivers was found to have a significant effect on the accuracy only when positioned within 30 mm of one another. Mild steel was found to have a significant detrimental effect when within 150 mm of the transmitter or receiver. A stainless steel bone screw had no effect on the accuracy of the device. A mathematical description of knee joint motion is presented which describes the motion in terms of clinical rotations. The device is a useful tool for measuring joint motion in the operating theatre and laboratory, owing to its accuracy and insensitivity to surgical alloys.

Mechanics of the anterior drawer and talar tilt tests. A cadaveric study of lateral ligament injuries of the ankle

We analyzed the changes in lateral ligament forces during anterior drawer and talar tilt testing and examined ankle joint motion during testing, following an isolated lesion of the anterior talofibular ligament (ATFL) or a combined lesion of the ATFL and calcaneofibular ligament (CFL). 8 cadaver specimens were held in a specially designed testing apparatus in which the ankle position (dorsiflexion-plantarflexion and supination-pronation) could be varied in a controlled manner. Ligament forces were measured with buckle transducers, and joint motion was measured with an instrumented spatial linkage. An anterior drawer test was performed using an 80 N anterior translating force, and a talar tilt test was performed using a 5.7 Nm supination torque with intact ligaments, after sectioning of the ATFL, and again after sectioning of the CFL. The tests were repeated at 10 degrees dorsiflexion, neutral, and 10 degrees and 20 degrees plantarflexion. In the intact ankle, the largest increases in ATFL force were observed during testing in plantarflexion, whereas the largest increases in CFL force were observed in dorsiflexion. Isolated ATFL injury caused only small laxity changes, but a pronounced increase in laxity was observed after a combined CFL and ATFL injury.

Biomechanics of ankle ligament reconstruction. An in vitro comparison of the Broström repair, Watson-Jones reconstruction, and a new anatomic reconstruction technique

We wanted to use biomechanical testing in a cadaveric model to compare the Broström repair, the Watson-Jones reconstruction, and a new anatomic reconstruction method. Eight specimens were held in a specially designed testing apparatus in which the ankle position (dorsiflexion-plantar flexion and supination-pronation) could be varied in a controlled manner. Testing was done with intact ligaments and was repeated after sectioning of the anterior talofibular ligament and the calcaneofibular ligament and after a Broström repair, a Watson-Jones reconstruction, and a new anatomic reconstruction were performed. An anterior drawer test was performed using an anterior translating force of 10 to 50 N, and a talar tilt test was performed using a supination torque of 1.1 to 3.4 N-m. The forces in the anterior talofibular ligament and calcaneofibular ligament were measured with buckle transducers, and tibiotalar motion and total ankle joint motion were measured with an instrumented spatial linkage. The increase in ankle joint laxity observed after sectioning of both the anterior talofibular and calcaneofibular ligaments was significantly reduced by the three reconstructive techniques, although not always to the level of the intact ankle. Joint motion was restricted after the Watson-Jones procedure compared with that in the intact ankle. Unlike the Watson-Jones procedure, the ligament or graft force patterns observed during loading after the Broström repair and the new anatomic technique resembled those observed in the intact ankle.

Optimization technique for the calculation of in vitro three-dimensional vertebral motion

A method for the calculation of translations and Eulerian rotations of an orthogonal axis system with respect to a fixed reference is described with application to the measurement of position in a vertebral motion segment. Kinematic equations were derived to compute the three-dimensional motion of a moving vertebra relative to an adjacent fixed body, without the requirement of a direct physical link between the two bodies. For this calculation, the quadratic error of the lengths of six position vectors was minimized to obtain a mathematically optimal estimate of the translations and rotations. Tests with a rigid model resulted in mean maximum overall system errors of 2.8 percent for the measurement of translation (translations less than 3.5 mm) and 6.1 percent for the measurement of rotations (rotations less than 10 deg) limited by transducer accuracy. The mathematical techniques presented for the quantitative description of rigid body motion, based on the measurement of three reference vectors, may be extended to a broad range of kinematic problems.