Error analysis of a robot positioning system based on biplanar navigation technology

Background: Some problems have been found in the clinical application of biplanar navigation technology. In particular, conical distortions caused by the X-ray image intensifier and deleterious effects on the navigation image caused by respiratory motion are occasionally ignored by surgeons. This paper discusses possible errors in the practical application of biplanar navigation technology and performs an analysis of the experimental correction for the conical distortion caused by X-rays. Methods: Five patients with unstable fractures of the pelvic posterior ring were included in this study. In four of these cases, the patient received a sacroiliac screw placement using biplanar navigation technology, while one received a respiratory motion test. The standard pelvic inlet and outlet positions were used for positioning precision verification. Using the images and manual tests, the point positioning error was measured and analyzed, and the image distortion was corrected by means of a geometric transformation mapping. A circle was fitted to the corrected lines, and the radius of this circle was used to evaluate the correction. Results: The preoperative planning agreed with the sacroiliac screw placement performed using biplanar navigation technology (representing precise positioning) in only one of the four cases. The other three cases presented deviations. Because of the possibility to cut out the nailing channel, the positioning operation was converted to a manual method. The breathing motion test showed that the pelvis can vibrate back and forth with breathing movement, especially the sacral promontory, with a maximum amplitude above 5 mm. The positioning error of the points in space with the biplanar technology was within 2 mm (entrance point, 1.60±0.28 mm; exit point, 1.28±0.43 mm). The corrected image curvature was significantly reduced, and the error can be greatly reduced. Conclusions: Conical distortion error and respiratory motion can affect the accuracy of biplanar navigation systems.

years, CAS technology has shown some outstanding characteristics, and its technical advantages have become increasingly apparent. However, computer technology emanates from the engineering field and may suffer from many limitations when applied to the clinical field.
In the present medical field, CAS technology can be largely divided into soft tissue manipulation robots and hard tissue navigation robots. Soft tissue robots act as extensions of the doctors' hands and eyes, mainly using endoscopic technology. They use optical and mechanical technology to perform surgery using minimally invasive access, and are used in thoracic surgery, general surgery, urology, gynecology, and laryngology, in combination with laparoscopy, thoracoscopy, arthroscopy and hysteroscopy (such as the DaVinci and Zeus surgical robot) [2][3][4][5][6][7]. The technology in these surgical robots is relatively mature, and they have been available on the market for many years. Hard-tissue robots make use of the high contrast of certain body tissues on radiological imaging, such as that of bone on X-ray-based imaging, to conduct image processing and reconstruction, and include biplanar navigation robots and spine and joint surgery robots [8][9][10].
For minimally invasive operations in orthopedics, omni-directional and multi-angle perspectives are needed to determine the nailing channel and to avoid the iatrogenic injury of important soft tissue such as the spinal cord and the neurovasculature in front of the pelvic posterior ring [10,11]. In such minimally invasive procedures, the patient and surgeon may be exposed to a relatively large radiation dose.
CAS technology can digitally reconstruct the bone structure through image processing technology, allowing the important soft tissues to be protected by using the navigation technology to determine the nailing channel both preoperatively and intraoperatively. In the practical application of CAS technology, small errors may come from many sources. Sometimes the accumulated error reaches the level of image drift, namely errors between the virtual position and posture and the actual ones in the skeleton, and it may be difficult to attract the attention of the operator to such problems. Once image drift occurs, the system should be reregistered, and the whole procedure should be recommenced from the beginning, which can greatly prolong the operation time [10]. If the image drift is ignored, there is a great possibility of serious iatrogenic damage to the patient.
In clinical work, a biplanar positioning robot system is frequently used to locate the sacroiliac nailing channel. However, the operative procedure is subject to several problems. First, when the positioning images are acquired by the biplanar robot, there are typically differences due to individual differences, projection angle, and the subjective intention of the operators. Second, the X-rays emitted by the X-ray tube are not parallel, but are in a conical configuration. The dispersion extent of the X-rays also differs according to the different projection planes, and the images of complex structures can be distorted [12][13][14]. Finally, as abdominal pressure fluctuates in association with respiratory movement, the spatial posture of the pelvis can show a slight anteversion at the end of inhalation compared with the end of exhalation, and this posture can oscillate at any time during the operation process.
In this paper, we analyzed the sources of error in biplanar navigation technology through the establishment of a model, to help clinical workers to further understand the technology and its application to clinical work.
Materials And Methods 1. Clinical research 1.1 Patient group Five patients were enrolled in this study, all of whom suffered from a high-energy injury. Four cases underwent sacroiliac fixation, including B1-type (tile classification) sacroiliac joint separation, with three cases being on the right side and one on the left side. The other patient was evaluated for respiratory oscillation.

Data Collection
For the patients undergoing sacroiliac fixation, the biplanar navigation system was used for sacroiliac fixation, with standard procedures for positioning, filming, registration, planning, and fixation placement. Standard pelvic inlet and outlet films were used for positioning (as shown in Fig. 1-4).
During the positioning procedure using the biplanar navigation system, the inlet and outlet films are intermittently viewed and compared with those of the preoperative planning images. If there is no deviation, the fixation procedure is continued. If there is deviation, the incidence of the iatrogenic injury is evaluated according to the operator's experience. If there is unlikely to be any serious consequence and the positioning is acceptable, the fixation procedure is continued. However, if an iatrogenic injury may occur, the fixation procedure is transferred to a manual method.
In the evaluation of respiratory oscillation, the patient was maintained under general anesthesia and the endotracheal intubation, and image intensifier were kept still. Five consecutive fluoroscopies were taken at the inlet and outlet position at intervals of 2-3 s. Variations in the positions of characteristic anatomical structures (such as the sacrum front, sacrum promontory, and the great notch of ischia) were observed.

The Error Calibration Experiment
A rectangular block of 50 × 80 mm was manufactured from Polyvinyl chloride (PVC), as shown in Steel balls of a diameter of 2.5 mm were evenly distributed over the correction plate as feature points ( Fig. 5-(1)), which was fixed on the end of the image intensifier ( Fig. 5-(2)). The steel balls could be clearly imaged on fluoroscopy ( Fig. 5-(3)). By measuring the original coordinates of the feature points, a distortion-correction model was built based on the ideal distribution of the feature points, and this model was mapped to the ideal image through geometric transformation (Fig. 6).
All five images were corrected by this method, and the lengths of AD, DG, BE, EH, CF, and FI before and after correction were calculated. For the images before and after correction, nine axis pixels of points A-I were selected. Using SolidWorks software(SolidWorks® Premium 2017 × 64 version SP2.0, Dassault Systemes), three feature points on each Kirschner wire (ABC, DEF, GHI) were fitted to a circle, and the radius of each circle was calculated, with these radii being used to evaluate the correction. The larger the radius is, the smaller the curvature is, and the closer the points were to being on a straight line.

Data analysis of the sacroiliac fixation procedures
The biplanar robot was used to locate the sacroiliac nailing channel for the four patients, and the individual results are described below.

Case 1
The guide pin location shown on the pelvic outlet film demonstrated little variation, which was acceptable ( Fig Although there was some deviation from the preoperative planning, the actual nailing channel was acceptable for the clinical work, and thus it was not adjusted, and the operation was continued.
Case 2 8 In this case, the pelvic projection angle had a little rotation, such as on the outlet positioning film ( Fig. 2.1), which had a certain clockwise rotation, while some had an anti-clockwise rotation ( Fig. 2

.2).
In the whole process, the position of the projection center was relatively fixed (Fig. 2

The Error Model Verification Experiment
Using the biplanar method, the spatial positioning error results for the K-wire tip are shown in Table 1.
The positioning accuracy of the system was within 2 mm (1.60 ± 0.28 mm for the entrance point; 1.28 ± 0.43 mm for the exit point).  Table 2 Distance between AD, DG, BE, EH, CF, FI before and after correction 150 200 250 300 350mm from the receiver respectively For the images before and after correction, three axis pixels were selected on each line (ABC, DEF and GHI) and were fitted to a circle with the measuring tools of the SolidWorks software in the five groups of images. The corresponding radius of each circle was then calculated. The results are shown in Fig. 8 and Table 3. Table 3 Before and after correction, 3 axial pixels were fitted into a circle with SolidWorks measurement tools and the radii of the circus were used to evaluate the correction

Conical distortion drift caused by the technical defects of the image intensifier
Because of the influence of the image intensifier structure, the imaging principle, imaging environment and the projection can lead to a larger image distortion than that present when an ordinary camera is used. The traditional X-ray fluoroscopy system consists of an X-ray transmitter and a cylindrical receiver (Fig. 12 = 9) that contains the XRII system. In the imaging process, the X-ray beam is emitted from the X-ray source and passes through the sample or the human body, and is then converted into visible light through the XRII system, which is projected onto the CCD camera.
The image distortions include a pillow-shaped distortion caused by the spherical nature of the input screen surface of the X-Ray Image Intensifier (XRII) system, S-shaped distortion caused by the terrain and the environmental magnetic field, and local distortion caused by the installation and manufacturing errors (Fig. 5-a ~ c).  Fig. 9), and vice versa (B, 'B' point in Fig. 9). When a Kirschner pin is filmed with the fluoroscopy system, the end that is close to the spherical tube will present a wider image than the other one. Second, all the points on the same (perpendicular to the Z-axis) plane would have the same dispersion trend, and the projection image would not present an obvious distortion. This could be called the dispersion feature for the point conical distortion. The point spatial errors shown in the error model verification experiment (Table 1) are in accordance with the dispersion features for the point conical distortion, and there is no obvious spatial distortion error.
There are two types of dispersion features for the line conical distortion. First, when a line AB (Fig. 12) is coplanar with the Z-axis, i.e., they intersect or are parallel, the line can be presented as a straight line on the screen. This situation is shown in Fig. 9. When the longitudinal axis of the Kirschner pin passes through the Z-axis, they are coplanar, and the projection of the Kirschner pin can be presented as a straight line on the screen, which is the first type of dispersion feature for the line conical distortion.
With the second type of dispersion feature, when the line AB and Z-axis are neither parallel nor intersect (that is, they are noncoplanar), an irregular arc would appear on the screen (Fig. 9). There are two causes for this situation. First, the dispersion trend of the points from different planes of the image intensifier is different. Second, because of the influence of the XRII system, the dispersion trend decreases as the distance between the points on the line AB and Z-axis decreases. The image formed by the line on the screen would form an arc to the projection center (Z-axis, Fig. 9). This is the second type of dispersion feature for the line conical distortion.
This kind of image distortion is rather small and can be easily ignored. In the biplanar navigation system, the errors of the two positioning films can overlap, which might have a significant effect on the operation results.
The plane dispersion features of a solid structure caused by conical distortion When the biplanar navigation system was used for nailing positioning, the channel structure for the screw placement was always irregular as the nailing channel was positioned for the four patients, but the channel structures could be idealized as cylinders. When the channel structure is horizontally placed on the projection plane, a relatively standard image can be formed, as the dispersion trend of each point is the same, conforming to the dispersion feature for the point conical distortion.
When the cylinder CD is obliquely placed in the projection area of the image intensifier (Fig. 9), the Zaxis passes through its center, which is in accord with the first type of dispersion feature for line conical distortion. Apart from the C end being a little wider than the D end, the overall dispersion trend of the structure shows little change. When the cylinder CD is translated along the Y-axis as one the degree of freedom movement, the projection image would change as shown in the difference from Fig. 9-1 to Fig. 9-3. However, the projection image of the sideline of cylinder CD would change the arc direction during the translation (from Fig. 9-1 to Fig. 9-3 First of all, we emphasize that no matter how precise the correction of the image distortion is, the corrected lines AC, DH, and GI can infinitely approach to a straight line, but can never be corrected to a real straight line, and therefore the curvature of the line can well reflect the correction extent of the distortion. That is, the larger the radius is, the smaller the curvature is, and the closer the corrected line is to a real straight line. As shown in Table 3, the radius of the curve representing each line obviously increases after correction, and therefore, as the curvature decreases, the lines become closer to a straight line. The closer to the Z-axis (line DH), the larger the radius of the fitted circle is, the smaller its curvature is, and the higher the extent of the distortion correction. This is the exact presentation of the second type of dispersion feature for the line conical distortion. For example, the curvatures of the line DH before and after correction are both smaller than those of lines AC and GI on both sides. It should be noted that: 1) all the images are distorted in the projection process; 2) no matter how precise and rigorous the technology is for the image correction, the distortion cannot be thoroughly corrected to an ideal straight line, and can just be corrected to be infinitely close to one.
The clinical characteristics and the avoidance principle of the conical distortion drift During the minimally invasive pelvic fixation procedure, the nailing channel is rarely selected as the projection center. For each projection, the spatial position and posture of the nailing channel is subject to the second type of dispersion features for the line conical distortion. In particular, the safe nailing channel for sacroiliac screws has a special angle.
How can we minimize the error caused by this kind of distortion (Fig 12)? 1.
Try to locate the whole projection structure on the same plane perpendicular to the Z-axis, which can be in accordance with the dispersion features for the point conical distortion. The dispersion trend of all parts is basically the same, and there is no obvious image distortion for the projection image.

2.
If the position and posture of the projection structure are not in accordance with the dispersion features for the point conical distortion, it is better to locate the longitudinal axis of the projection structure so that it is coplanar with the Z-axis of the image intensifier so that the longitudinal axis direction of the projection structure is consistent with the normal structure. This situation would then be in accord with the first type of dispersion feature for the line conical distortion.

3.
If the position and posture of the projection structure are not in accord with the above two features, only the second type of dispersion feature for the line conical distortion is present, and the operators' experience is needed to avoid error.
The effect of the respiratory motion on the navigation precision As respiratory oscillation can cause abdominal pressure to fluctuate, the pelvis has a slight pronation and caudal displacement at the end of inhalation. In the example in this study (Fig. 7), it was found that the caudal displacement of the sacral promontory was more than 5 mm, which would not be acceptable in clinical practice, and might cause serious iatrogenic consequences. Therefore, this is a special drift error, and the real-time position and posture of a target such as the pelvis could oscillate at any time. After the marker is registered, the target's spatial position and posture registered in the computer is fixed. The respiratory oscillation would then form a random error, which can be called the random drift.
If the operator does not pay attention to this problem during the navigation procedure, the biplanar positioning films may be taken as the patient is at the end of inhalation. When the nailing placement is at the end of expiration, the following pelvic oscillation may cause the fixation channel to go through a similar oscillation, and there is the possibility of an unforeseen injury.
However, this problem can easily be solved. During the navigation procedure, the patient is under general anesthesia and endotracheal intubation. Three time phases are chosen for the inlet and outlet positioning films being taken and guide pin penetration. At these three time phases, the ventilator is suspended for a short time under the premise of security at a special respiratory phase (such as the end of the inhalation), then the pelvis is located at the same spatial position and posture and the influences for the respiratory motion can be limited in the special respiratory phase and the operation would result in no serious iatrogenic consequences.

Conclusions
Although these two kinds of errors can also be seen in the traditional application of fluoroscopy, realtime perspective verification may not be affected by these errors. However, during the biplanar navigation procedure, these errors might affect the surgical results. The errors from both situations are relatively small, and the errors in the navigation operation are the cumulative results of them.
Therefore, they may be difficult to separate. Because of the technical limitation of the image intensifier, there is no possibility to completely eliminate these kinds of errors. However, by standardizing the whole operation, the errors can be minimized.

Availability of data and materials
The datasets used and analyzed during the current study are available from the corresponding author on reasonable request.

Competing interests
The authors declare that they have no competing interests. all the data. The study will not be published elsewhere in any language without the consent of the copyright owners.

Authors' contributions
ZLH, TPF and TJD made a contribution in designing the study and providing critical revisions to this article. DHL and HN were responsible in writing the article. YGX and LHS made a contribution in collecting the data. ZGZ and ZSW made a contribution in collecting the data. All authors read and approved the final manuscript.          Each image was corrected, and the results are shown.

Figure 9
The presentations of the dispersion feature for the point conical distortion, two types of the line conical distortion.

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