THE INTERNATIONAL JOURNAL OF MEDICAL ROBOTICS AND COMPUTER ASSISTED SURGERY Int J Med Robotics Comput Assist Surg (2014) Published online in Wiley Online Library (wileyonlinelibrary.com) DOI: 10.1002/rcs.1620
ORIGINAL ARTICLE
Accuracy of a hexapod parallel robot kinematics based external fixator† Maximilian Faschingbauer1*‡ Hinrich J. D. Heuer1‡ Klaus Seide1,3 Robert Wendlandt2 Matthias Münch3 Christian Jürgens1,4 Rainer Kirchner4 1
Berufsgenossenschaftliches Unfallkrankenhaus (Trauma Hospital) Hamburg; Department for Trauma Surgery, Orthopaedics and Sportstraumatology; Hamburg, Germany
2
University Medical Center Schleswig-Holstein, Campus Luebeck; Biomechanics Laboratory; Luebeck, Germany
3
Berufsgenossenschaftliches Unfallkrankenhaus (Trauma Hospital) Hamburg; Laboratory for Biomechanics; Hamburg, Germany
Abstract Background Different hexapod-based external fixators are increasingly used to treat bone deformities and fractures. Accuracy has not been measured sufficiently for all models. Methods An infrared tracking system was applied to measure positioning maneuvers with a motorized Precision Hexapod® fixator, detecting threedimensional positions of reflective balls mounted in an L-arrangement on the fixator, simulating bone directions. By omitting one dimension of the coordinates, projections were simulated as if measured on standard radiographs. Accuracy was calculated as the absolute difference between targeted and measured positioning values. Results In 149 positioning maneuvers, the median values for positioning accuracy of translations and rotations (torsions/angulations) were below 0.3 mm and 0.2° with quartiles ranging from 0.5 mm to 0.5 mm and 1.0° to 0.9°, respectively.
4
University Medical Center Schleswig-Holstein, Campus Luebeck; Clinic for Musculoskeletal Surgery; Luebeck, Germany
* Correspondence to: Maximilian Faschingbauer, Berufsgenossenschaftliches Unfallkrankenhaus (Trauma Hospital) Hamburg, Department for Trauma Surgery, Orthopaedics and Sportstraumatology, Bergedorfer Strasse 10, D-21033 Hamburg, Germany. E-mail: m.faschingbauer@ buk-hamburg.de †
This study was conducted at the Berufsgenossenschaftliches Unfallkrankenhaus (Trauma Hospital) Hamburg, Germany, Department for Trauma Surgery, Orthopaedics and Sportstraumatology ‡
Equally contributed Accepted: 27 August 2014
Copyright © 2014 John Wiley & Sons, Ltd.
Conclusions The experimental setup was found to be precise and reliable. It can be applied to compare different hexapod-based fixators. Accuracy of the investigated hexapod system was high. Copyright © 2014 John Wiley & Sons, Ltd. Keywords
hexapod; external fixator; accuracy
Introduction Complex multidimensional deformities of extremities remain a challenge for the orthopedic surgeon. However, the advent of ring external fixators, popularized by the Russian surgeon Gavril A. Ilizarov 1, as well as systematical deformity analysis and correction planning techniques, developed by Paley and colleagues (2,3), brought great improvements in this area. While there were independent French and Russian patent applications for adding hexapod parallel robot kinematics to external ring fixators during the 1980s, none of these constructs have ever been applied clinically (4,5). In the 1990s, Taylor in the USA (Taylor Spatial FrameTM, TSF, Smith & Nephew, Memphis, TN, USA) (6) and Seide in Germany (Precision Hexapod®, Litos, Ahrensburg, Germany; Figure 1) (7) independently succeeded in introducing
M. Faschingbauer et al.
Figure 1. Precision Hexapod® mounted on a plastic tibia utilizing wire-fixated Ilizarov ring systems (courtesy of Litos®, Ahrensburg, Germany, reprinted with permission). (A). Drawing of the Precision Hexapod® coordinate system showing the defined nomenclature of the six spatial degrees of freedom movements (B). The origin of the coordinate system is defined at the center of the reference ring, in this case the lower ring
hexapod kinematics as advancements to Ilizarov ring fixators into clinical practice. Such ring-systems of the hexapod-based fixators are mounted on the bone segments through wires or bone screws, and connected by six telescopic linear lengthening elements called distractors or struts, which are linked to the rings through special joints. Lengthening or shortening these six distractors leads to positioning or correction maneuvers of the ring systems in the six spatial degrees of freedom. Figure 2 shows a clinical case of a deformity correction of a boy’s right lower leg using the Precision Hexapod® external fixator. Hexapod external fixators can be used to correct extremity deformities, reduce fractures, and to slowly mobilize and reduce joint contractures. Stability of the system is maintained at all times, which leads to painless correction or reduction maneuvers in clinical setups (8). This, as well as the fact that one can easily alter correction plans at any time without remounting of fixator parts or returning to the operation room, is a frequently cited advantage of hexapod-based external fixators (9). Hexapod-based fixators require computer software to calculate the distractor adjustment parameters in order to achieve a high degree of accuracy (10). Many studies in recent years have reported on the successful use of hexapod-based external fixators in clinical practice for various indications. In some of these studies, the authors define the achieved clinical accuracy Copyright © 2014 John Wiley & Sons, Ltd.
as the amount of remaining bone deformity (angulation and torsion in degrees, and translation in millimeters or percentage of dislocation) (11–24). In other studies, the standard joint angle system established by Paley and colleagues (2,3) is used as the reference to report clinical accuracy (9,25–38). For the Precision Hexapod® system, median values are between 1 mm and 3 mm remaining translational deformity and 1° to 3° remaining angulation deformity, with maximum values of 5 mm to 14 mm and 3° to 18°, respectively (22–24). Maximum values exceeding 5° of remaining angulation deformity are only observed in a very few cases (22–24). For the TSF system, mean values are between 1.3 mm and 2 mm or 0% to 9% remaining translational deformity/dislocation and 0.5° to 3° remaining angulation deformity, with maximum values ranging from 3 mm to 5 mm or 6% to 40% and 2.5° to 30°, respectively (9,11–21,25–38). Furthermore, maximum values exceeding 5° of remaining angulation deformity only occur very rarely (9,11–21,25–38). Besides the clinical literature, there are insufficient reports on systematic measurements of the accuracy of the clinically established hexapod-based external fixators in experimental setups. Using coordinate-measuring machines, accuracy is reported with values ranging from 0.4 mm to 2 mm for translations, and 0.4° to 0.7° for angulations (6,8,24,39). However, exact descriptions of applied methods, study design, and, more importantly, a Int J Med Robotics Comput Assist Surg (2014) DOI: 10.1002/rcs
Accuracy of a hexapod parallel robot kinematics based external fixator
Figure 2. Case example: Clinical photographs and corresponding radiographs of an angular deformity with shortening in the lower leg of a 14-year-old boy (A and B), with fixator in place, before (C and D) and after the correction (E), as well as the radiographic result after correction and removal of the fixator (F). Copyright © 2014 John Wiley & Sons, Ltd.
systematic method for conducting measurements are lacking. Even the corresponding manufacturer manuals of the two above-mentioned established systems are missing information regarding accuracy measurements. In addition to the results referred to above, there are only two further publications in established medical journals dealing with accuracy of hexapod-based external fixators in more detail. Simpson and colleagues report the development of a method to increase accuracy of the TSF system in clinical usage (40). They introduced preoperative virtual 3D correction planning and trials based on 3D computed tomography (CT) imaging, as well as intraoperative tracking of anatomy and true fixator positioning through navigation. This led to postoperative computed real correction planning based on the captured data. The resulting accuracy in twenty experiments with sawbones showed mean correction errors of 0.3 mm total lengthening (range 2.5 mm to 3 mm) and 1.8° total rotation/angulation (range 0.8° to 4.4°). In contrast, Tang et al. reported on the development of a new computerassisted motorized hexapod-based fracture reduction system that aids fracture reduction based on 3D-CT-imaging and the concept of mirror symmetry of the human body (41). That is, they used the non-fractured contralateral bone as a template. Based on ten experiments with bovine femurs, these authors reported a reduction accuracy with average values and standard deviations of 1.3 mm ± 0.7 mm for axial deflection, 1.2 mm ± 0.4 mm for translational deflection, 2.3° ± 1.8° for angulation, and 2.8° ± 0.9° for rotation. Nevertheless, Tang and colleagues failed to describe the exact measuring methods. They did, however, point out that the hexapod system’s high number of mechanical parts influences the reduction accuracy and should therefore be the focus of future improvements. Their hexapod construct was similar to existing hexapod-based fixator systems. Innovations of the hexapod system by Tang et al. compared with existing systems were the use of 3D-CT data with the contralateral bone as a template, as well as localizing the fixator parameters with 3D-CT imaging through location balls at the fixator rings, all for computer correction planning. Given that further hexapod-based external fixator systems of other manufacturers, such as the TL-HEX TrueLok Hexapod SystemTM (TL-HEX, Orthofix® International Orthopedics, Verona, Italy) (42), are pushing into a growing market, and in the absence of any documented accuracy measurements, there is a need for development of a reliable but easy to use measuring setup. Such a setup should be based on the standard clinical use of hexapodbased external fixator systems to minimize friction between theoretical development and practical implementation. Clinically, preoperative deformity/fracture analysis is performed on radiographs in two perpendicular planes showing a projection of the deformity/displacement, Int J Med Robotics Comput Assist Surg (2014) DOI: 10.1002/rcs
M. Faschingbauer et al.
and, to the results, clinical examination findings are added, and in special cases CT scans are also used. Then the hexapod adjustment parameters are calculated with the manufacturer´s software, and the corrections are performed following a gradual correction plan. Consecutive standard radiographs in two planes and clinical examinations, as well as CT scans in special cases, monitor the correction and final result. The approach of Simpson et al. was of a rather extensive nature for just measuring mechanical accuracy. Tang et al. did not clarify how mechanical accuracy was calculated, and therefore, a reproduction of this method for other systems is difficult. Consequently, the objective of this work was to develop an easy to use and reproducible laboratory setup of high precision to measure and compare positioning accuracy of hexapod-based external fixator systems. Optical tracking was used for the measurement, as it is a well-established and readily available method in clinical and technical applications. The experiment was intended to reproduce the clinical application as close as possible, while at the same time, deliberately excluding errors related to performing radiographic examinations and clinical planning by an orthopedic surgeon, as the focus was on the characteristic parameters of the fixator system itself. Besides some possible variation for the mechanical mounting and the threaded rods of the controlling elements, the main focus of such a test is the software, which is specific for the system. Software errors could result from the transfer of mechanical data into the robot kinematic calculation and the transformation of input projection data describing the bone positions into the coordinate system of the robot kinematics, as well as programming errors of the robotic kinematics’ algorithm.
The accuracy of the Precision Hexapod® (Litos, Ahrensburg, Germany) was measured as an example application.
Materials and Methods Experimental measuring setup The proposed experimental setup for measuring the mechanical accuracy of hexapod-based fixator systems is displayed in Figure 3. The experiments conducted were based on the Precision Hexapod®. The tested fixator system was mounted as follows. A standard Ilizarov ring as stationary lower ring with a diameter of 160 mm was firmly mounted on a stone plate. Six motorized distractors were used. These were developed at the Trauma Hospital Hamburg, Germany 8, and are functionally equivalent to standard manual distractors (Litos, Ahrensburg, Germany), except for a motor and gearbox mounted at the side (Figure 3). With these motors, a large number of positioning maneuvers can be conducted in a short period of time without interfering with manipulations, thus eliminating a possible source of error. Six standard ball-joint adapters (Litos, Ahrensburg, Germany) were used to connect the stationary lower ring and the mobile upper ring to the distractors. Calculations of hexapod positioning parameters were conducted with the standard software ‘Hexapod Calculator 2009 C1’ (Figure 4(B) and 4(C), sold with the Precision Hexapod, on a notebook personal computer running Windows® (Microsoft® Corporation, Redmond, WA, USA) in versions 95 to 8 (version used was Windows® XP). The
Figure 3. Measuring setup with indicated Cartesian coordinate system (x/y/z axes). The infrared camera of the tracking system (1) detects the reflective marker balls (3) at the hexapod fixator assembled with motor distractors (4). Movements are controlled by a notebook personal computer (2) via an electric control unit (5). The hexapod calculation software (2a) and the detection software of the tracking system (2b) are running side-by-side (2). For initialization, the distractor lengths are measured with a special digital gauge (7). Emergency stop switch (6). Copyright © 2014 John Wiley & Sons, Ltd.
Int J Med Robotics Comput Assist Surg (2014) DOI: 10.1002/rcs
Accuracy of a hexapod parallel robot kinematics based external fixator
Figure 4. The hexapod calculation software and tracking software are running simultaneously on the notebook personal computer. Motor control module (A). Standard hexapod software with screen showing ring, joint and distractor (D1–D6) parameters (B), and standard hexapod software with screen showing calculation input and calculation output of distractor adjustments (d1–d6) for the movement ‘20 mm right’ (C). Right and left camera images (D). Tracking data output showing the coordinates of the marker balls (f0, f1, f2, f3, f4) (E). Remark: A fulcrum is not specified in this example, as it is mathematically irrelevant for translational movements.
motorized distractors were controlled by a hardware unit connected to the notebook via USB (Figure 3), as well as a software motor control module, which was isolated from the calculation software, and therefore, had no influence on it (Figure 4(B) and 4(C)). An infrared camera tracking system (infiniTrack-RB, Atracsys LLC, Renens, Switzerland) with associated software (Figures 3 and 4(D), (E)), usually applied to detect surgical tools in the operating room, was utilized as a measuring device. It was connected via a firewire card (Delock, Berlin, Germany) to the notebook personal computer. All of the software ran simultaneously with the hexapod software components (Figures 3 and 4). Three reflective marker balls (fiducials; f0, f1, and f2) were mounted in a pseudo-L-configuration on an aluminum plate on the mobile upper ring as shown in Figure 5. There was no fiducial placed at the vertex (fE) of the Lconfiguration, as it caused detection errors in the tracking software. For calculation purposes, the coordinates of point fE were determined by extrapolation along the straight line through f0 and f1. The extrapolation calculations were based on the defined mounting distances of the fiducials shown in Figure 5. In order to register the origin of the hexapod coordinate system (F0) with the tracking system, two fiducials (m1 and m2) were mounted on the stationary lower ring (Figures 4 and 6). In measuring sequences where the accuracy of the fulcrum position was to be investigated (sequence 7, 8 and 10), an additional fiducial (fp) was mounted on the mobile upper ring in the chosen location of the investigated fulcrum (Figure 6). In clinical situations, bones are usually Copyright © 2014 John Wiley & Sons, Ltd.
Figure 5. Close-up view of the marker plate mounted at the upper, mobile ring with the marker balls f0, f1, and f2 in ‘L-configuration’. At the vertex fE of the ‘L’, no marker ball could be mounted due to detection errors by the tracking system.
fixed/located near the center of the corresponding fixator rings. In clinical cases of angulation deformities without translation deformities, the fulcrum usually lies somewhere near the junction of the virtually or really broken proximal and distal parts of the mechanical or anatomical axes (Figures 1 and 2). Consequently, fp was named F1, F2, or F2.1 in the different sequences according to its example position between the two rings. The positions of F1, F2, or F2.1 in relation to F0 were entered into the hexapod software to define the fulcrum, and subsequently, to be considered in the calculation of the calculating positioning parameters. The aim was to measure Int J Med Robotics Comput Assist Surg (2014) DOI: 10.1002/rcs
M. Faschingbauer et al.
Figure 6. Hexapod system with markerplate (fiducials f0, f1, f2) to measure translations, torsion, and angulations. Fiducials f 3 (m2) and f4 (m1) are used to detect the center of the static/reference ring (F0). The fiducial fp is placed at the fulcrum position (here in position F2, in further sequences in positions F1 or F2.1). If no fulcrum is specified, angulations/torsions are conducted around F0 as the origin of the hexapod coordinate system. Starting position (A). Position after torsion and angulation around F2 (B).
the exact location of the fiducial, which would be stationary, while the upper ring rotates/angulates around it or moves a specific distance in a certain direction if translational movement was combined with the rotation. The whole setup was assembled using a water level and set-square. Two fiducials marked the anterior–posterior orientation of the stationary lower ring so that coordinate systems of the hexapod (Figure 1B) and tracking software (Figure 3) matched the axes. The orientation of the stationary lower ring was re-evaluated by checking the consecutively measured coordinates of the tracking system before every measurement.
Labeling in the software The labeling in the software ‘Hexapod Calculator 2009’ corresponds to the clinical situation in which the observer is standing in front of the patient, to whose leg the hexapod fixator is mounted (Figures 1 and 2). The movement labeling corresponds to the movement of the mobile ring with respect to the reference ring (Figure 1). Translational movements of the mobile ring in the hexapod software of ‘anterior’/’posterior’ correspond to x-axis movements of the tracking software, ‘up’/’down’ to the y-axis, and ‘right’/’left’ to z-axis movements. Rotational movements of the mobile ring around the x-axis of the tracking software correspond to movements ‘right up’/’right down’ of the hexapod software (the right side of the ring moving up or down), around the y-axis to ‘anterior right’/‘anterior left’ (a point anterior on the ring moving right or left), and around the z-axis to ‘anterior up’/ ‘anterior down’ (the anterior side of the ring moving up or down) Copyright © 2014 John Wiley & Sons, Ltd.
movements. Rotational movements (torsions/angulations) are specified in the software in reference to a fulcrum, the coordinates of which are also described as ‘anterior’/‘posterior’, ‘right’/‘left’, and ‘up’/‘down’ with reference to the center of the reference ring, i.e., the origin of the hexapod coordinate system (also see Figure 1). If no fulcrum is specified, the Hexapod Calculator software uses the origin of the hexapod coordinate system as the fulcrum (Figure 1). In clinical cases, the fulcrum has to be set to the measured vertex of the bone deformity, which Paley et al. labeled ‘center of rotation of angulation’ (CORA) (2,3).
Confirming the accuracy of the tracking system The accuracy of the tracking system was confirmed and the experimental setup was calibrated using a modified digital caliper gauge (Profitexx® GmbH, Hagen a. T. W., Germany; manufacturer statement: metering range 0.00–300.00 mm, error limit 0.02 mm; www.profitexx. de) holding two fiducials. The measurement error for the setup was determined by setting these two fiducials to certain distances along the three axes of the tracking coordinate system. The caliper gauge was aligned with the axes of the tracking system by balancing out the coordinates of the two axes that were not measured. The manufacturer-stated accuracy of the tracking system is 0.36 mm root mean square (RMS) with a maximum range 400–2000 mm (working volume width: 1337 mm; working volume height: 1076 mm; http://atracsys.com). Int J Med Robotics Comput Assist Surg (2014) DOI: 10.1002/rcs
Accuracy of a hexapod parallel robot kinematics based external fixator
Measuring sequences A high but reasonable number of positioning maneuvers, covering different Cartesian directions, were executed and measured both individually and in combination. The directions and extent of the chosen movements included clinically as well as technically relevant and possible maneuvers. Furthermore, the extent of maneuvers was limited to the range possible without replacement of fixator elements throughout a measuring sequence, thus eliminating a potential source of error. To include the most important variables with a possible influence on accuracy, measuring sequences for three different ring sizes, and three different fulcrum positions and movements, in two extreme positions of the system, were investigated. One hundred and forty-nine measurements, distributed over ten measuring sequences, were conducted. At the beginning of each sequence, the system was initialized by entering ring sizes, joint adapter positions, and initial distractor lengths into the hexapod calculation software. Next, the desired movement parameters, as well as fulcrum position and final orientation, were entered. The software calculated the lengthening/shortening to be performed for the six distractors. The motor control software module executed the positioning maneuver automatically. Screenshots with all components of the hexapod and tracking software were taken before and after each positioning maneuver, showing the calculation software input for the desired movement and the initial and final positions of the mounted fiducials (Figures 3 and 4). During each measuring sequence, individual movements along all three Cartesian axes or around these were executed and measured first, followed by a set of simultaneous movements. In sequences one to six, the accuracy for three different ring sizes (diameters: 100 mm, 160 mm, 180 mm) was evaluated. In sequences two, four, six, and nine, no specific fulcrum was entered into the calculation software for rotational (torsion/angulation) movements, i.e. the software used a fixed fulcrum in the center of the lower ring system (F0). In sequences seven and eight, the fulcrum position accuracy of specified fulcrums (F1 and F2, Figure 6(A)) was investigated, as well as the accuracy of the rotational (torsion/angulation) movements with respect to it. Sequence nine was designed to measure the accuracy of rotational (torsion/angulation) movements with the fixator rings far apart (high position) and close together (low position), simulating extreme system positions. The distance of the starting high position and the starting low position was 60 mm. In sequence ten, the accuracy of combined translational and rotational (torsion/angulation) movements with Copyright © 2014 John Wiley & Sons, Ltd.
respect to a specified fulcrum (F2.1, Figure 6(A)) and fulcrum translation accuracy was measured covering all six spatial degrees of freedom. In Table 1, the detailed parameters of the specific measuring sequences are summarized.
Evaluation Screenshots with all components of the hexapod and tracking software were taken before and after each positioning maneuver showing the calculation software input for the desired movement, and the initial, as well as final, positions of the mounted fiducials (Figures 3 and 4). The measured fiducial coordinates and the hexapod software input data were transferred from the screenshots into Excel® worksheets (Microsoft® Corporation, Redmond, WA, USA), in which calculations were conducted. The absolute difference between target movement and the measured movement was determined as the accuracy measure. This was done analogous to measuring angles and distances in clinical setups on standard radiographs. For translational movements, as well as fulcrum positions, the positioning differences were determined along the three coordinate axes. If no movement was meant to be executed along a coordinate axis, 0 mm was taken as the target movement. The mean of the measured positions of the three fiducials f0, f1, and f2 was calculated. For rotational movements, projections of straight lines through two fiducials, simulating clinical radiographic projection of the bone (f0 and f1) or foot (fE and f2) axis, onto each of the three planes of the coordinate system (to be interpreted clinically as an anterior–posterior, a lateral, and a virtual cranio–caudal radiograph, the latter being related to torsional bone movements) were calculated. Projection simulation was performed by omitting one coordinate, which resulted in projection lines on the three spatial planes. Using trigonometric principles, direction angles in the three planes were calculated with respect to the axis of the coordinate system, and thus the absolute differences between these describe the movements. The absolute angular differences between measured movements and given movements were determined as accuracy measurement values. This was done, as in clinical situations, for all planes separately, so that values for the three rotational movement directions (‘right up’/‘right down’, ‘anterior right’/‘anterior left’, ‘anterior up’/‘anterior down’) were determined. To evaluate the results statistically, median values and quartile ranges of the absolute differences were determined for every sequence and the six directions of movement. Int J Med Robotics Comput Assist Surg (2014) DOI: 10.1002/rcs
(+/ ) 5 and 10 (+/ ) 5 and 10 (+/ ) 5, 10, and 15 (+/ ) 5 and 10 (+/ ) 5 and 10 -
(+/ ) 10&10, 10&20, 20&40, 5&10/20 (+/ ) 8
(+/ ) 15 (+/ ) 10 and 20 (+/ ) 10 and 20 (+/ ) 10, 15, and 20 (+/ ) 10 and 20 (+/ ) 10 and 20 (+/ ) 10 and 20 (+/ ) 10 and 20 (+/ ) 10 and 20 (+/ ) 10 and 20 (+/ ) 10, 20, and 30 (+/ ) 10, 20, and 30 (+/ ) 30 and 60 F0 F0 F0 F1 F2 F0 F2.1 160/160 160/160 100/160 100/160 180/160 180/160 160/160 160/160 160/160 160/160 14 14 15 12 15 12 12 13 27 15
Translation (mm) Translation (mm) Sequence
1 2 3 4 5 6 7 8 9 10
Rotation (Torsion/ Angulation) (°) & Translation (mm) Rotation (Torsion/ Angulation) (°) Rotation (Torsion/ Angulation) (°) Fulcrum position Ring size (mm) upper/lower No. of movements
Table 1. Parameters of the measuring sequences
Single movement amounts with respect to the three axes
Simultaneous movement amountswith respect to all three axes
M. Faschingbauer et al.
Copyright © 2014 John Wiley & Sons, Ltd.
Results Confirming the accuracy of the tracking system Fifteen measurements, five along each coordinate axis, were conducted to confirm the accuracy of the tracking system in the setup. Along the x-axis, the mean measured absolute difference was 0.02 mm (standard deviation 0.11 mm). Along the y-axis, the mean measured absolute difference was 0.20 mm (standard deviation 0.20 mm), and along the z-axis the mean absolute difference was 0.21 mm (standard deviation 0.14 mm). No major deviations were observed, and the accuracy of measurements was consistent with the manufacturer´s statements.
Measuring sequences Ten measuring sequences with a sum of 149 individual or simultaneous movements were calculated, executed, measured, and evaluated. Overall median absolute differences of target movements and measured movements (accuracy) were between 0.25 mm and 0.1 mm, and between 0.14° and 0.21°, with quartiles ranging from 0.47 mm to 0.47 mm, and 1.02° to 0.94°. That is, with
ORIGINAL ARTICLE
Accuracy of a hexapod parallel robot kinematics based external fixator† Maximilian Faschingbauer1*‡ Hinrich J. D. Heuer1‡ Klaus Seide1,3 Robert Wendlandt2 Matthias Münch3 Christian Jürgens1,4 Rainer Kirchner4 1
Berufsgenossenschaftliches Unfallkrankenhaus (Trauma Hospital) Hamburg; Department for Trauma Surgery, Orthopaedics and Sportstraumatology; Hamburg, Germany
2
University Medical Center Schleswig-Holstein, Campus Luebeck; Biomechanics Laboratory; Luebeck, Germany
3
Berufsgenossenschaftliches Unfallkrankenhaus (Trauma Hospital) Hamburg; Laboratory for Biomechanics; Hamburg, Germany
Abstract Background Different hexapod-based external fixators are increasingly used to treat bone deformities and fractures. Accuracy has not been measured sufficiently for all models. Methods An infrared tracking system was applied to measure positioning maneuvers with a motorized Precision Hexapod® fixator, detecting threedimensional positions of reflective balls mounted in an L-arrangement on the fixator, simulating bone directions. By omitting one dimension of the coordinates, projections were simulated as if measured on standard radiographs. Accuracy was calculated as the absolute difference between targeted and measured positioning values. Results In 149 positioning maneuvers, the median values for positioning accuracy of translations and rotations (torsions/angulations) were below 0.3 mm and 0.2° with quartiles ranging from 0.5 mm to 0.5 mm and 1.0° to 0.9°, respectively.
4
University Medical Center Schleswig-Holstein, Campus Luebeck; Clinic for Musculoskeletal Surgery; Luebeck, Germany
* Correspondence to: Maximilian Faschingbauer, Berufsgenossenschaftliches Unfallkrankenhaus (Trauma Hospital) Hamburg, Department for Trauma Surgery, Orthopaedics and Sportstraumatology, Bergedorfer Strasse 10, D-21033 Hamburg, Germany. E-mail: m.faschingbauer@ buk-hamburg.de †
This study was conducted at the Berufsgenossenschaftliches Unfallkrankenhaus (Trauma Hospital) Hamburg, Germany, Department for Trauma Surgery, Orthopaedics and Sportstraumatology ‡
Equally contributed Accepted: 27 August 2014
Copyright © 2014 John Wiley & Sons, Ltd.
Conclusions The experimental setup was found to be precise and reliable. It can be applied to compare different hexapod-based fixators. Accuracy of the investigated hexapod system was high. Copyright © 2014 John Wiley & Sons, Ltd. Keywords
hexapod; external fixator; accuracy
Introduction Complex multidimensional deformities of extremities remain a challenge for the orthopedic surgeon. However, the advent of ring external fixators, popularized by the Russian surgeon Gavril A. Ilizarov 1, as well as systematical deformity analysis and correction planning techniques, developed by Paley and colleagues (2,3), brought great improvements in this area. While there were independent French and Russian patent applications for adding hexapod parallel robot kinematics to external ring fixators during the 1980s, none of these constructs have ever been applied clinically (4,5). In the 1990s, Taylor in the USA (Taylor Spatial FrameTM, TSF, Smith & Nephew, Memphis, TN, USA) (6) and Seide in Germany (Precision Hexapod®, Litos, Ahrensburg, Germany; Figure 1) (7) independently succeeded in introducing
M. Faschingbauer et al.
Figure 1. Precision Hexapod® mounted on a plastic tibia utilizing wire-fixated Ilizarov ring systems (courtesy of Litos®, Ahrensburg, Germany, reprinted with permission). (A). Drawing of the Precision Hexapod® coordinate system showing the defined nomenclature of the six spatial degrees of freedom movements (B). The origin of the coordinate system is defined at the center of the reference ring, in this case the lower ring
hexapod kinematics as advancements to Ilizarov ring fixators into clinical practice. Such ring-systems of the hexapod-based fixators are mounted on the bone segments through wires or bone screws, and connected by six telescopic linear lengthening elements called distractors or struts, which are linked to the rings through special joints. Lengthening or shortening these six distractors leads to positioning or correction maneuvers of the ring systems in the six spatial degrees of freedom. Figure 2 shows a clinical case of a deformity correction of a boy’s right lower leg using the Precision Hexapod® external fixator. Hexapod external fixators can be used to correct extremity deformities, reduce fractures, and to slowly mobilize and reduce joint contractures. Stability of the system is maintained at all times, which leads to painless correction or reduction maneuvers in clinical setups (8). This, as well as the fact that one can easily alter correction plans at any time without remounting of fixator parts or returning to the operation room, is a frequently cited advantage of hexapod-based external fixators (9). Hexapod-based fixators require computer software to calculate the distractor adjustment parameters in order to achieve a high degree of accuracy (10). Many studies in recent years have reported on the successful use of hexapod-based external fixators in clinical practice for various indications. In some of these studies, the authors define the achieved clinical accuracy Copyright © 2014 John Wiley & Sons, Ltd.
as the amount of remaining bone deformity (angulation and torsion in degrees, and translation in millimeters or percentage of dislocation) (11–24). In other studies, the standard joint angle system established by Paley and colleagues (2,3) is used as the reference to report clinical accuracy (9,25–38). For the Precision Hexapod® system, median values are between 1 mm and 3 mm remaining translational deformity and 1° to 3° remaining angulation deformity, with maximum values of 5 mm to 14 mm and 3° to 18°, respectively (22–24). Maximum values exceeding 5° of remaining angulation deformity are only observed in a very few cases (22–24). For the TSF system, mean values are between 1.3 mm and 2 mm or 0% to 9% remaining translational deformity/dislocation and 0.5° to 3° remaining angulation deformity, with maximum values ranging from 3 mm to 5 mm or 6% to 40% and 2.5° to 30°, respectively (9,11–21,25–38). Furthermore, maximum values exceeding 5° of remaining angulation deformity only occur very rarely (9,11–21,25–38). Besides the clinical literature, there are insufficient reports on systematic measurements of the accuracy of the clinically established hexapod-based external fixators in experimental setups. Using coordinate-measuring machines, accuracy is reported with values ranging from 0.4 mm to 2 mm for translations, and 0.4° to 0.7° for angulations (6,8,24,39). However, exact descriptions of applied methods, study design, and, more importantly, a Int J Med Robotics Comput Assist Surg (2014) DOI: 10.1002/rcs
Accuracy of a hexapod parallel robot kinematics based external fixator
Figure 2. Case example: Clinical photographs and corresponding radiographs of an angular deformity with shortening in the lower leg of a 14-year-old boy (A and B), with fixator in place, before (C and D) and after the correction (E), as well as the radiographic result after correction and removal of the fixator (F). Copyright © 2014 John Wiley & Sons, Ltd.
systematic method for conducting measurements are lacking. Even the corresponding manufacturer manuals of the two above-mentioned established systems are missing information regarding accuracy measurements. In addition to the results referred to above, there are only two further publications in established medical journals dealing with accuracy of hexapod-based external fixators in more detail. Simpson and colleagues report the development of a method to increase accuracy of the TSF system in clinical usage (40). They introduced preoperative virtual 3D correction planning and trials based on 3D computed tomography (CT) imaging, as well as intraoperative tracking of anatomy and true fixator positioning through navigation. This led to postoperative computed real correction planning based on the captured data. The resulting accuracy in twenty experiments with sawbones showed mean correction errors of 0.3 mm total lengthening (range 2.5 mm to 3 mm) and 1.8° total rotation/angulation (range 0.8° to 4.4°). In contrast, Tang et al. reported on the development of a new computerassisted motorized hexapod-based fracture reduction system that aids fracture reduction based on 3D-CT-imaging and the concept of mirror symmetry of the human body (41). That is, they used the non-fractured contralateral bone as a template. Based on ten experiments with bovine femurs, these authors reported a reduction accuracy with average values and standard deviations of 1.3 mm ± 0.7 mm for axial deflection, 1.2 mm ± 0.4 mm for translational deflection, 2.3° ± 1.8° for angulation, and 2.8° ± 0.9° for rotation. Nevertheless, Tang and colleagues failed to describe the exact measuring methods. They did, however, point out that the hexapod system’s high number of mechanical parts influences the reduction accuracy and should therefore be the focus of future improvements. Their hexapod construct was similar to existing hexapod-based fixator systems. Innovations of the hexapod system by Tang et al. compared with existing systems were the use of 3D-CT data with the contralateral bone as a template, as well as localizing the fixator parameters with 3D-CT imaging through location balls at the fixator rings, all for computer correction planning. Given that further hexapod-based external fixator systems of other manufacturers, such as the TL-HEX TrueLok Hexapod SystemTM (TL-HEX, Orthofix® International Orthopedics, Verona, Italy) (42), are pushing into a growing market, and in the absence of any documented accuracy measurements, there is a need for development of a reliable but easy to use measuring setup. Such a setup should be based on the standard clinical use of hexapodbased external fixator systems to minimize friction between theoretical development and practical implementation. Clinically, preoperative deformity/fracture analysis is performed on radiographs in two perpendicular planes showing a projection of the deformity/displacement, Int J Med Robotics Comput Assist Surg (2014) DOI: 10.1002/rcs
M. Faschingbauer et al.
and, to the results, clinical examination findings are added, and in special cases CT scans are also used. Then the hexapod adjustment parameters are calculated with the manufacturer´s software, and the corrections are performed following a gradual correction plan. Consecutive standard radiographs in two planes and clinical examinations, as well as CT scans in special cases, monitor the correction and final result. The approach of Simpson et al. was of a rather extensive nature for just measuring mechanical accuracy. Tang et al. did not clarify how mechanical accuracy was calculated, and therefore, a reproduction of this method for other systems is difficult. Consequently, the objective of this work was to develop an easy to use and reproducible laboratory setup of high precision to measure and compare positioning accuracy of hexapod-based external fixator systems. Optical tracking was used for the measurement, as it is a well-established and readily available method in clinical and technical applications. The experiment was intended to reproduce the clinical application as close as possible, while at the same time, deliberately excluding errors related to performing radiographic examinations and clinical planning by an orthopedic surgeon, as the focus was on the characteristic parameters of the fixator system itself. Besides some possible variation for the mechanical mounting and the threaded rods of the controlling elements, the main focus of such a test is the software, which is specific for the system. Software errors could result from the transfer of mechanical data into the robot kinematic calculation and the transformation of input projection data describing the bone positions into the coordinate system of the robot kinematics, as well as programming errors of the robotic kinematics’ algorithm.
The accuracy of the Precision Hexapod® (Litos, Ahrensburg, Germany) was measured as an example application.
Materials and Methods Experimental measuring setup The proposed experimental setup for measuring the mechanical accuracy of hexapod-based fixator systems is displayed in Figure 3. The experiments conducted were based on the Precision Hexapod®. The tested fixator system was mounted as follows. A standard Ilizarov ring as stationary lower ring with a diameter of 160 mm was firmly mounted on a stone plate. Six motorized distractors were used. These were developed at the Trauma Hospital Hamburg, Germany 8, and are functionally equivalent to standard manual distractors (Litos, Ahrensburg, Germany), except for a motor and gearbox mounted at the side (Figure 3). With these motors, a large number of positioning maneuvers can be conducted in a short period of time without interfering with manipulations, thus eliminating a possible source of error. Six standard ball-joint adapters (Litos, Ahrensburg, Germany) were used to connect the stationary lower ring and the mobile upper ring to the distractors. Calculations of hexapod positioning parameters were conducted with the standard software ‘Hexapod Calculator 2009 C1’ (Figure 4(B) and 4(C), sold with the Precision Hexapod, on a notebook personal computer running Windows® (Microsoft® Corporation, Redmond, WA, USA) in versions 95 to 8 (version used was Windows® XP). The
Figure 3. Measuring setup with indicated Cartesian coordinate system (x/y/z axes). The infrared camera of the tracking system (1) detects the reflective marker balls (3) at the hexapod fixator assembled with motor distractors (4). Movements are controlled by a notebook personal computer (2) via an electric control unit (5). The hexapod calculation software (2a) and the detection software of the tracking system (2b) are running side-by-side (2). For initialization, the distractor lengths are measured with a special digital gauge (7). Emergency stop switch (6). Copyright © 2014 John Wiley & Sons, Ltd.
Int J Med Robotics Comput Assist Surg (2014) DOI: 10.1002/rcs
Accuracy of a hexapod parallel robot kinematics based external fixator
Figure 4. The hexapod calculation software and tracking software are running simultaneously on the notebook personal computer. Motor control module (A). Standard hexapod software with screen showing ring, joint and distractor (D1–D6) parameters (B), and standard hexapod software with screen showing calculation input and calculation output of distractor adjustments (d1–d6) for the movement ‘20 mm right’ (C). Right and left camera images (D). Tracking data output showing the coordinates of the marker balls (f0, f1, f2, f3, f4) (E). Remark: A fulcrum is not specified in this example, as it is mathematically irrelevant for translational movements.
motorized distractors were controlled by a hardware unit connected to the notebook via USB (Figure 3), as well as a software motor control module, which was isolated from the calculation software, and therefore, had no influence on it (Figure 4(B) and 4(C)). An infrared camera tracking system (infiniTrack-RB, Atracsys LLC, Renens, Switzerland) with associated software (Figures 3 and 4(D), (E)), usually applied to detect surgical tools in the operating room, was utilized as a measuring device. It was connected via a firewire card (Delock, Berlin, Germany) to the notebook personal computer. All of the software ran simultaneously with the hexapod software components (Figures 3 and 4). Three reflective marker balls (fiducials; f0, f1, and f2) were mounted in a pseudo-L-configuration on an aluminum plate on the mobile upper ring as shown in Figure 5. There was no fiducial placed at the vertex (fE) of the Lconfiguration, as it caused detection errors in the tracking software. For calculation purposes, the coordinates of point fE were determined by extrapolation along the straight line through f0 and f1. The extrapolation calculations were based on the defined mounting distances of the fiducials shown in Figure 5. In order to register the origin of the hexapod coordinate system (F0) with the tracking system, two fiducials (m1 and m2) were mounted on the stationary lower ring (Figures 4 and 6). In measuring sequences where the accuracy of the fulcrum position was to be investigated (sequence 7, 8 and 10), an additional fiducial (fp) was mounted on the mobile upper ring in the chosen location of the investigated fulcrum (Figure 6). In clinical situations, bones are usually Copyright © 2014 John Wiley & Sons, Ltd.
Figure 5. Close-up view of the marker plate mounted at the upper, mobile ring with the marker balls f0, f1, and f2 in ‘L-configuration’. At the vertex fE of the ‘L’, no marker ball could be mounted due to detection errors by the tracking system.
fixed/located near the center of the corresponding fixator rings. In clinical cases of angulation deformities without translation deformities, the fulcrum usually lies somewhere near the junction of the virtually or really broken proximal and distal parts of the mechanical or anatomical axes (Figures 1 and 2). Consequently, fp was named F1, F2, or F2.1 in the different sequences according to its example position between the two rings. The positions of F1, F2, or F2.1 in relation to F0 were entered into the hexapod software to define the fulcrum, and subsequently, to be considered in the calculation of the calculating positioning parameters. The aim was to measure Int J Med Robotics Comput Assist Surg (2014) DOI: 10.1002/rcs
M. Faschingbauer et al.
Figure 6. Hexapod system with markerplate (fiducials f0, f1, f2) to measure translations, torsion, and angulations. Fiducials f 3 (m2) and f4 (m1) are used to detect the center of the static/reference ring (F0). The fiducial fp is placed at the fulcrum position (here in position F2, in further sequences in positions F1 or F2.1). If no fulcrum is specified, angulations/torsions are conducted around F0 as the origin of the hexapod coordinate system. Starting position (A). Position after torsion and angulation around F2 (B).
the exact location of the fiducial, which would be stationary, while the upper ring rotates/angulates around it or moves a specific distance in a certain direction if translational movement was combined with the rotation. The whole setup was assembled using a water level and set-square. Two fiducials marked the anterior–posterior orientation of the stationary lower ring so that coordinate systems of the hexapod (Figure 1B) and tracking software (Figure 3) matched the axes. The orientation of the stationary lower ring was re-evaluated by checking the consecutively measured coordinates of the tracking system before every measurement.
Labeling in the software The labeling in the software ‘Hexapod Calculator 2009’ corresponds to the clinical situation in which the observer is standing in front of the patient, to whose leg the hexapod fixator is mounted (Figures 1 and 2). The movement labeling corresponds to the movement of the mobile ring with respect to the reference ring (Figure 1). Translational movements of the mobile ring in the hexapod software of ‘anterior’/’posterior’ correspond to x-axis movements of the tracking software, ‘up’/’down’ to the y-axis, and ‘right’/’left’ to z-axis movements. Rotational movements of the mobile ring around the x-axis of the tracking software correspond to movements ‘right up’/’right down’ of the hexapod software (the right side of the ring moving up or down), around the y-axis to ‘anterior right’/‘anterior left’ (a point anterior on the ring moving right or left), and around the z-axis to ‘anterior up’/ ‘anterior down’ (the anterior side of the ring moving up or down) Copyright © 2014 John Wiley & Sons, Ltd.
movements. Rotational movements (torsions/angulations) are specified in the software in reference to a fulcrum, the coordinates of which are also described as ‘anterior’/‘posterior’, ‘right’/‘left’, and ‘up’/‘down’ with reference to the center of the reference ring, i.e., the origin of the hexapod coordinate system (also see Figure 1). If no fulcrum is specified, the Hexapod Calculator software uses the origin of the hexapod coordinate system as the fulcrum (Figure 1). In clinical cases, the fulcrum has to be set to the measured vertex of the bone deformity, which Paley et al. labeled ‘center of rotation of angulation’ (CORA) (2,3).
Confirming the accuracy of the tracking system The accuracy of the tracking system was confirmed and the experimental setup was calibrated using a modified digital caliper gauge (Profitexx® GmbH, Hagen a. T. W., Germany; manufacturer statement: metering range 0.00–300.00 mm, error limit 0.02 mm; www.profitexx. de) holding two fiducials. The measurement error for the setup was determined by setting these two fiducials to certain distances along the three axes of the tracking coordinate system. The caliper gauge was aligned with the axes of the tracking system by balancing out the coordinates of the two axes that were not measured. The manufacturer-stated accuracy of the tracking system is 0.36 mm root mean square (RMS) with a maximum range 400–2000 mm (working volume width: 1337 mm; working volume height: 1076 mm; http://atracsys.com). Int J Med Robotics Comput Assist Surg (2014) DOI: 10.1002/rcs
Accuracy of a hexapod parallel robot kinematics based external fixator
Measuring sequences A high but reasonable number of positioning maneuvers, covering different Cartesian directions, were executed and measured both individually and in combination. The directions and extent of the chosen movements included clinically as well as technically relevant and possible maneuvers. Furthermore, the extent of maneuvers was limited to the range possible without replacement of fixator elements throughout a measuring sequence, thus eliminating a potential source of error. To include the most important variables with a possible influence on accuracy, measuring sequences for three different ring sizes, and three different fulcrum positions and movements, in two extreme positions of the system, were investigated. One hundred and forty-nine measurements, distributed over ten measuring sequences, were conducted. At the beginning of each sequence, the system was initialized by entering ring sizes, joint adapter positions, and initial distractor lengths into the hexapod calculation software. Next, the desired movement parameters, as well as fulcrum position and final orientation, were entered. The software calculated the lengthening/shortening to be performed for the six distractors. The motor control software module executed the positioning maneuver automatically. Screenshots with all components of the hexapod and tracking software were taken before and after each positioning maneuver, showing the calculation software input for the desired movement and the initial and final positions of the mounted fiducials (Figures 3 and 4). During each measuring sequence, individual movements along all three Cartesian axes or around these were executed and measured first, followed by a set of simultaneous movements. In sequences one to six, the accuracy for three different ring sizes (diameters: 100 mm, 160 mm, 180 mm) was evaluated. In sequences two, four, six, and nine, no specific fulcrum was entered into the calculation software for rotational (torsion/angulation) movements, i.e. the software used a fixed fulcrum in the center of the lower ring system (F0). In sequences seven and eight, the fulcrum position accuracy of specified fulcrums (F1 and F2, Figure 6(A)) was investigated, as well as the accuracy of the rotational (torsion/angulation) movements with respect to it. Sequence nine was designed to measure the accuracy of rotational (torsion/angulation) movements with the fixator rings far apart (high position) and close together (low position), simulating extreme system positions. The distance of the starting high position and the starting low position was 60 mm. In sequence ten, the accuracy of combined translational and rotational (torsion/angulation) movements with Copyright © 2014 John Wiley & Sons, Ltd.
respect to a specified fulcrum (F2.1, Figure 6(A)) and fulcrum translation accuracy was measured covering all six spatial degrees of freedom. In Table 1, the detailed parameters of the specific measuring sequences are summarized.
Evaluation Screenshots with all components of the hexapod and tracking software were taken before and after each positioning maneuver showing the calculation software input for the desired movement, and the initial, as well as final, positions of the mounted fiducials (Figures 3 and 4). The measured fiducial coordinates and the hexapod software input data were transferred from the screenshots into Excel® worksheets (Microsoft® Corporation, Redmond, WA, USA), in which calculations were conducted. The absolute difference between target movement and the measured movement was determined as the accuracy measure. This was done analogous to measuring angles and distances in clinical setups on standard radiographs. For translational movements, as well as fulcrum positions, the positioning differences were determined along the three coordinate axes. If no movement was meant to be executed along a coordinate axis, 0 mm was taken as the target movement. The mean of the measured positions of the three fiducials f0, f1, and f2 was calculated. For rotational movements, projections of straight lines through two fiducials, simulating clinical radiographic projection of the bone (f0 and f1) or foot (fE and f2) axis, onto each of the three planes of the coordinate system (to be interpreted clinically as an anterior–posterior, a lateral, and a virtual cranio–caudal radiograph, the latter being related to torsional bone movements) were calculated. Projection simulation was performed by omitting one coordinate, which resulted in projection lines on the three spatial planes. Using trigonometric principles, direction angles in the three planes were calculated with respect to the axis of the coordinate system, and thus the absolute differences between these describe the movements. The absolute angular differences between measured movements and given movements were determined as accuracy measurement values. This was done, as in clinical situations, for all planes separately, so that values for the three rotational movement directions (‘right up’/‘right down’, ‘anterior right’/‘anterior left’, ‘anterior up’/‘anterior down’) were determined. To evaluate the results statistically, median values and quartile ranges of the absolute differences were determined for every sequence and the six directions of movement. Int J Med Robotics Comput Assist Surg (2014) DOI: 10.1002/rcs
(+/ ) 5 and 10 (+/ ) 5 and 10 (+/ ) 5, 10, and 15 (+/ ) 5 and 10 (+/ ) 5 and 10 -
(+/ ) 10&10, 10&20, 20&40, 5&10/20 (+/ ) 8
(+/ ) 15 (+/ ) 10 and 20 (+/ ) 10 and 20 (+/ ) 10, 15, and 20 (+/ ) 10 and 20 (+/ ) 10 and 20 (+/ ) 10 and 20 (+/ ) 10 and 20 (+/ ) 10 and 20 (+/ ) 10 and 20 (+/ ) 10, 20, and 30 (+/ ) 10, 20, and 30 (+/ ) 30 and 60 F0 F0 F0 F1 F2 F0 F2.1 160/160 160/160 100/160 100/160 180/160 180/160 160/160 160/160 160/160 160/160 14 14 15 12 15 12 12 13 27 15
Translation (mm) Translation (mm) Sequence
1 2 3 4 5 6 7 8 9 10
Rotation (Torsion/ Angulation) (°) & Translation (mm) Rotation (Torsion/ Angulation) (°) Rotation (Torsion/ Angulation) (°) Fulcrum position Ring size (mm) upper/lower No. of movements
Table 1. Parameters of the measuring sequences
Single movement amounts with respect to the three axes
Simultaneous movement amountswith respect to all three axes
M. Faschingbauer et al.
Copyright © 2014 John Wiley & Sons, Ltd.
Results Confirming the accuracy of the tracking system Fifteen measurements, five along each coordinate axis, were conducted to confirm the accuracy of the tracking system in the setup. Along the x-axis, the mean measured absolute difference was 0.02 mm (standard deviation 0.11 mm). Along the y-axis, the mean measured absolute difference was 0.20 mm (standard deviation 0.20 mm), and along the z-axis the mean absolute difference was 0.21 mm (standard deviation 0.14 mm). No major deviations were observed, and the accuracy of measurements was consistent with the manufacturer´s statements.
Measuring sequences Ten measuring sequences with a sum of 149 individual or simultaneous movements were calculated, executed, measured, and evaluated. Overall median absolute differences of target movements and measured movements (accuracy) were between 0.25 mm and 0.1 mm, and between 0.14° and 0.21°, with quartiles ranging from 0.47 mm to 0.47 mm, and 1.02° to 0.94°. That is, with
