 Research
 Open Access
Nonorthogonal onestep calibration method for robotized transcranial magnetic stimulation
 He Wang^{1}View ORCID ID profile,
 Jingna Jin^{1},
 Xin Wang^{1},
 Ying Li^{1},
 Zhipeng Liu^{1} and
 Tao Yin^{1, 2}Email authorView ORCID ID profile
https://doi.org/10.1186/s1293801805709
© The Author(s) 2018
 Received: 9 April 2018
 Accepted: 26 September 2018
 Published: 1 October 2018
Abstract
Background
Robotized transcranial magnetic stimulation (TMS) combines the benefits of neuronavigation with automation and provides a precision brain stimulation method. Since the coil will normally remain unmounted between different clinical uses, hand/eye calibration and coil calibration are required before each experiment. Today, these two steps are still separate: hand/eye calibration is performed using methods proposed by Tsai/Lenz or Floris Ernst, and then the coil calibration is carried out based on the traditional TMS experimental step. The process is complex and timeconsuming, and traditional coil calibration using a handheld probe is susceptible to greater calibration error.
Methods
A novel onestep calibration method has been developed to confirm hand/eye and coil calibration results by formulating a matrix equation system and estimating its solution. Hand/eye calibration and coil calibration are performed to confirm the pose relationships of the marker/end effector ‘X’, probe/end effector ‘Y’, and robot/world ‘Z’. First, the coil is fixed on the end effector of the robot. During the onestep calibration process, a marker is mounted on the top of the coil and a calibration probe is fixed at the actual effective position of the coil. Next, the robot end effector is moved to a series of random positions ‘A’, the tracking data of marker ‘B’ and probe ‘C’ is obtained correspondingly. Then, a matrix equation system AX = ZB and AY = ZC can be acquired, and it is computed using a leastsquares approach. Finally, the calibration probe is removed after calibration, while the marker remains fixed to the coil during the TMS experiment. The methods were evaluated based on simulation data and on experimental data from an optical tracking device. We compared our methods with two classical methods: the QR24 method proposed by Floris Ernst and the handheld coil calibration method.
Results
The new methods outperform the QR24 method in the aspect of translational accuracy and performs similarly in the aspect of rotational accuracy, the total translational error decreased more than fifty percent. The new approach also outperforms traditional handheld coil calibration of navigated TMS systems, the total translational error decreased three to fourfold, and the rotational error decreased six to eightfold. Furthermore, the convergence speed is improved 16 to 27fold for the new algorithms.
Conclusion
These results suggest that the new method can be used for hand/eye and coil calibration of a robotized TMS system. Two complex steps can be simplified using a leastsquares approach.
Keywords
 Transcranial magnetic stimulation
 TMS
 Medical robots and systems
 Hand/eye and coil calibration
 Leastsquares approach
Background
Transcranial magnetic stimulation (TMS) is a noninvasive and painless method for stimulating the cerebral cortex nerve [1–3]. Based on the principle of electromagnetic induction, an electric current is created on the cerebral cortex using a magnetic coil that is manually placed on top of the patient’s head. Recently, singlepulse and repetitive TMS have been used in clinical study for the therapy of mental disease [4–9]. However, TMS is still not widely promoted because its therapeutic effect changes between subjects [10, 11].
Recently, two types of robotic TMS systems have been reported and shown to improve stimulation accuracy [15–18]. First, a robotic TMS system has been developed using a general industrial robot [15–17]. A sixjointed industrial robot (Adept Viper s850) was used in a robotic TMS system designed by Lars et al. [17]. For this kind of robotic system, the coil was mounted to the robot’s end effector, a Polaris Spectra infrared tracking system was used for navigation. After calibration, the magnetic coil is placed quite precisely and directly over a selected target region by the robot. However, the safety of this kind of robotic system has been queried, because the system is equipped with actuators selected for highspeed motions like any other industrial robot [18]. Second, a dedicated robotic system for TMS has been designed based on mechanical architecture and a control strategy [18]. A sevenjointed dedicated robot is designed from the kinematic scheme of the mechanism, including the arm, the prismatic joint, and the wrist [19]. The design and control of the robotic system optimizes the safety of the procedure, and allows the force applied between the robot and the head to be controlled. Robotguided neuronavigated TMS has been used in some recent studies of psychiatric diseases [19–21].
The handheld coil calibration method typically suffers from larger calibration errors; a translational error of 3 mm is acceptable for handheld coil calibration [27–30]. In addition, the traditional hand/eye calibration method was reported separately by Shiu and Ahmad [23, 24] and Tsai and Lenz [25, 26]. Matrix algebra and the special properties of homogeneous matrices were used for determining the unknown matrices mentioned above. A review of these calibration algorithms was proposed by Wang et al. [31]. All these methods require that orthogonal homogeneous unknown matrices can be found. To overcome this limitation, the QR24 method, which uses three different variations on the basis of a naïve leastsquares solution of the equation system, was developed for simultaneous hand/eye calibration [22]. However, all the hand/eye calibration methods mentioned above are used for determining two unknown pose matrices. For a robotized navigated TMS system, there are three unknown pose matrices, which need to be solved. The leastsquares method can be applied to get unknown data and to minimize the error sum of squares between the gotten data and the actual data. For this reason, it is not unreasonable to assume that the simultaneous calibration of two steps in a naïve leastsquares solution of the equation system might result in improved accuracy and efficiency.
In order to simplify the two complex calibration steps and improve the accuracy, this report presents an innovative approach for simultaneous hand/eye and coil calibration. A matrix equation system AX = ZB and AY = ZC was acquired at different robot positions, as shown in Fig. 2D. A linear equation system was obtained based on the matrix equation system, and a leastsquares solution was calculated for determining ‘X’, ‘Y’, and ‘Z’. The feasibility and effectiveness of the method were demonstrated by comparing with two classical calibration methods: the QR24 method and the handheld coil calibration method.
Methods
Synchronous hand/eye and coil calibration
Generally, the position of the robot’s end effector in Eq. 2 is chosen randomly in a sphere of radius r. The rotation angles of yaw, pitch, and roll for each position are also selected randomly between ± d degrees.
To use these matrices, orthonormalization is the final and most important step, because ‘X’, ‘Y’ and ‘Z’ are not orthogonally solved in the QR36 method. Without this step, the homogeneous coordinate transformations obtained from ‘X’, ‘Y’ and ‘Z’ would be incorrect. In this paper, the singular value decomposition (SVD) method was used for orthonormalization. For a position N obtained from a tracking device, for which we want to calculate the position in robotic coordinates, we compute ZN to obtain a nonorthogonal matrix; next, the SVD of ZN is calculated as UΣV^{T} = ZN. Finally, the orthonormalized ZN can be calculated with (ZN)^{⊥ }= UV^{T}.
Calibration errors
Data acquisition
Simulated and optical tracking data were obtained to estimate the accuracy and robustness of the QR36 method. To compare the quality of our method with the QR24 and handheld methods, the QR24 method and the QR36 method were performed on the same dataset. The results are presented in the following section.
Simulated data
Data from optical tracking device
Results
Evaluation and implementation
The algorithms were evaluated on a standard personal computer (Intel core E7500 CPU with 4 GB of RAM, running 64bit Windows 10 OS). The algorithms were all implemented in MATLAB 2010a. The performance period was less than 0.1 s for the QR24 and QR36 algorithms.
Calibration errors
The QR36 algorithm was performed using poses P_{1},…,P_{n} for n = 5,…,250. The rotational and translational errors were computed for each remaining testing pose P_{251},…,P_{500} using Eqs. 9 and 10, and the average of all 250 testing poses was determined. The QR24 algorithm was calculated on the marker and the probe separately using poses P_{1},…,P_{n} for n = 5,…,250. The rotational and translational errors of the marker and the probe were computed for each remaining testing pose P_{251},…,P_{500} using Eqs. 12 and 13, and the average of all 250 testing poses was determined. Then, error E_{z} and θ_{z} were determined for each remaining testing pose P_{251},…,P_{500}, and the average of all 250 testing poses was determined.
Simulated results
Clearly, using more than 20 poses hardly influences the translational and rotational errors of the QR36 method. However, using the QR24 method to calibrate the marker and probe separately could create two different calibration matrices Z. This can create a new calibration error e_{z}, which significantly influences the translational error of the QR24 algorithm. In the simulated data, the resulting error came very close to the error obtained using the correct matrices, and remained within 1%. The minimum total translational error decreased 59.54% for the simulated data. The minimum total translational error e_{t} of the QR24 method was 0.3419 mm, with an e_{z} of 0.1267 mm, while the minimum total error of the QR36 method was 0.2143 mm. In other words, without error e_{z}, the minimum total errors of the QR24 method and the QR36 method are approximately equal; the same result was obtained from the data for the 25th, median, and 75th errors, but not the maximum total error. Insufficient sample data produced an extremely large error e_{z}; therefore, our method is more useful for a small data set. Overall, both convergence speed and precision of the translational error were improved for the QR36 method compared with the QR24 method. However, although the QR36 method outperformed the QR24 method regarding the rotational error of θ_{t}, the effect of θ_{z} was very slight. The θ_{z} of the QR24 method only had a 0.78% influence on the minimum total rotational error.
Error statistics of the calibration algorithms using matrices on the simulated data
Alg  Para  Min  25th  Med  75th  Max 

QR24  Translational error (mm)  
e_{trans}[O_{p}]  0.1071  0.1075  0.1074  0.1088  0.5134  
e_{trans}[O_{M}]  0.1081  0.1085  0.1081  0.1138  0.7211  
e _{ z}  0.1267  0.2059  0.2735  0.5078  5.9305  
e _{ t}  0.3419  0.4219  0.4889  0.7305  7.1649  
P _{ n}  203  147  182  73  6  
Rotational error (°)  
θ _{ p}  0.0814  0.0814  0.0812  0.0818  0.0839  
θ _{ m}  0.0847  0.0847  0.0851  0.0842  0.1024  
θ _{ z}  0.0013  0.0043  0.0056  0.0079  0.0609  
θ _{ t}  0.1675  0.1704  0.1719  0.1739  0.2472  
P _{ n}  168  173  97  245  7  
QR36  Translational error (mm)  
e_{trans}[O_{p}]  0.1066  0.1068  0.1071  0.1088  0.3004  
e_{trans}[O_{M}]  0.1076  0.1079  0.1084  0.1118  0.3203  
e _{ z}  0  0  0  0  0  
e _{ t}  0.2143  0.2148  0.2155  0.2206  0.6207  
P _{ n}  202  158  133  64  6  
Rotational error (°)  
θ _{ p}  0.0815  0.0812  0.0815  0.0813  0.0861  
θ _{ m}  0.0844  0.0848  0.0848  0.0853  0.1064  
θ _{ z}  0  0  0  0  0  
θ _{ t}  0.1659  0.1661  0.1663  0.1665  0.1925  
P _{ n}  203  107  142  69  6 
Experimental results
Error statistics of the calibration algorithms using matrices on the experimental data
Alg  Para  Min  25th  Med  75th  Max 

QR24  Translational error (mm)  
e_{trans}[O_{p}]  0.3263  0.3296  0.3272  0.3331  1.4110  
e_{trans}[O_{M}]  0.2787  0.2845  0.2787  0.2854  1.1929  
e _{ z}  0.3301  0.5024  0.5935  0.6995  5.7679  
e _{ t}  0.9353  1.1166  1.1995  1.3181  8.3719  
P _{ n}  247  109  200  81  5  
Rotational error (°)  
θ _{ p}  0.1882  0.1889  0.1886  0.1883  0.2157  
θ _{ m}  0.1584  0.1590  0.1583  0.1583  0.1825  
θ _{ z}  0.0005  0.0003  0.0030  0.0069  0.0832  
θ _{ t}  0.3466  0.3482  0.3500  0.3536  0.4815  
P _{ n}  176  88  190  127  5  
QR36  Translational error (mm)  
e_{trans}[O_{p}]  0.3256  0.3261  0.3277  0.3353  0.6580  
e_{trans}[O_{M}]  0.2781  0.2797  0.2807  0.2873  0.5803  
e _{ z}  0  0  0  0  0  
e _{ t}  0.6037  0.6059  0.6084  0.6226  1.2384  
P _{ n}  250  227  132  73  6  
Rotational error (°)  
θ _{ p}  0.1880  0.1883  0.1885  0.1893  0.2074  
θ _{ m}  0.1583  0.1584  0.1584  0.1586  0.1674  
θ _{ z}  0  0  0  0  0  
θ _{ t}  0.3464  0.3467  0.3470  0.3480  0.3749  
P _{ n}  141  128  217  68  7 
Convergence properties of the calibration algorithms using matrices on the simulated and experimental data
Data group  Algorithm  Error type  Total errors  P _{ n} 

Simulation  QR24  Translation  < 0.4 mm  118 
< 0.3 mm  None  
Rotation  < 0.2°  10  
< 0.17°  28  
QR36  Translation  < 0.4 mm  7  
< 0.3 mm  14  
Rotation  < 0.2°  6  
< 0.17°  22  
Experiment  QR24  Translation  < 1 mm  247 
< 0.7 mm  None  
Rotation  < 0.4°  10  
< 0.35°  66  
QR36  Translation  < 1 mm  9  
< 0.7 mm  27  
Rotation  < 0.4°  6  
< 0.35°  57 
Workspace sizes
Handheld coil calibration
To compare the quality of our method with the handheld coil calibration method, two operators with similar experience of TMS participated in this experiment. The QR36 method and handheld calibration method were performed by the two operators. The translational error e_{trans}[O_{p}] and rotational error θ_{p} of the probe, which were defined in Eqs. 9 and 10, were estimated to evaluate the calibration accuracy.
A set of five additional data acquisition experiments using the robotic TMS system in a workspace of 300 mm (10°) were performed by each operator. For the experimental procedure, refer to the experimental data acquisition paragraph in the Methods section. For each experiment, the probe was remounted on the coil by the operator, and 500 measurements (A_{i}, B_{i}, C_{i} _{i = 1,…,500}) were collected, with the end effector moving to 500 random positions around the first position P_{0}. Next, the robot was moved to the first position P_{0}, and the probe was unloaded from the coil and held at the bottom centre labels of the coil by the operator. An additional measurement (B_{a}, C_{a}) was acquired with an optical tracking device. The acquired 500 measurements and the additional measurement were defined as the dataset of each experiment.
As shown in Fig. 6, both the mean values and the standard deviations of the rotational and translational errors significantly decreased when the QR36 method was used; the total translational error decreased three to fourfold, and the rotational error decreased six to eightfold. For the two operators, the minimum total translational errors for QR36 method were decreased to 0.7931 and 0.8361 mm, respectively, as compared with 2.4997 and 2.5354 mm for handheld method. The minimum total rotational errors were decreased to 0.5509° and 0.5624° using QR36 method, as compared with 4.2808° and 3.6202° using handheld method. The rotational error showed greater improvement, probably because there was still a positioning label on the coil for the position probe, whereas there is no method for reducing the perspective error of the operator. Moreover, because the probe was fixed on the coil mechanically and an optimization algorithm was used, no significant differences in either the translational or the rotational errors of the QR36 method were found for different operators. Finally, the QR36 method showed better robustness to choice of operator than the handheld coil calibration method.
Discussion
An innovative approach for synchronous hand/eye and coil calibration has been presented and evaluated. A leastsquares method that has been widely used in medical imaging and robotics was used to estimate the optimal calibration matrix [32, 33]. The approach has been validated with synthetic data and experimental data from an optical tracking device. The calibration effect of our method was compared with that of traditional methods such as QR24 and the handheld coil calibration method.
These results show that QR36 calibration method is advisable for use in the robotized TMS system. We have shown that both the QR24 method and our new QR36 method perform very well, with errors below 1 mm and 0.2°, using both simulated and experimental data. This shows that the two methods can both be used for typical calibration. However, whereas more than 200 different robot positions were required to achieve a calibration error below 1 mm for the QR24 method, we found that fewer than 20 positions were typically required by our method. This means that the QR36 method is especially suitable to handle cases where fewer tracking data are available.
Traditional handeye calibration method proposed by Shiu and Ahmad [23, 24] and Tsai and Lenz [25, 26] calculates the rotational and translational parts of the unknown matrices separately using matrix algebra. Li and Betsis applied a geometric approach and leastsquares solution for handeye calibration [34]. Dual quaternion approach was also used for handeye calibration [35]. All those methods expect that orthogonal homogeneous matrices can be found. Moreover, a robust realtime handeye calibration method was proposed by Lars and Floris present [17]. A marker is attached to the robot’s third link for realtime handeye calibration. However, this method is not as precise as the QR24 algorithm. The total translational errors were 0.88 mm and 1.36 mm for QR24 method and realtime calibration method, respectively [17]. A robot system will not be calibrated perfectly, so orthogonality is not necessary. It is accepted or requested to permit nonorthogonality of the matrices in our calibration method. Our results demonstrated that the calibration method used in this study is more accurate than the classical handeye calibration approaches [23, 25]. In terms of translational accuracy, our method also outperforms the QR24 method. Therefore, the calibration method proposed in this study is more suitable for robotized transcranial magnetic stimulation.
We should point out that the maximum optimal translation errors for five handheld experiments per operator can reach 3 mm, as shown in Fig. 6a. This result is acceptable for a handheld navigated TMS experiment. During the robotassisted TMS stimulation, the head motion is tracked using the marker on the subject’s head and compensated by the robot [18]. But, without robotic assistance and head motion compensation, the relative motion between the subject’s head and the handheld stimulation coil is greater than 3 mm during handheld TMS experiments [30]. Moreover, the calibration error of registration of the subject’s MRI and optical tracking device is also controlled at around 3 mm for handheld navigated TMS experiments [36]. However, for a robotic TMS system, which is designed for highprecision stimulation, the handheld coil calibration method is not suitable [37–39].

It is accepted or requested to permit nonorthogonality of the matrices in our calibration method;

The calibrated robotic TMS system is used in the same space where calibration was carried out;

There are three unknown matrices that need to be solved in the robotic system.
Conclusion
We have developed a new onestep calibration method to acquire three pose relationships from a navigated robotic system with a leastsquares approach. The new method can significantly improve the accuracy of the robotic TMS system. Besides, the convergence speed is improved for the new method, which means that our method is particularly suited to handle fewer tracking data. The capability of the new method has been demonstrated for synchronous calibration and determination of the pose relationship of marker/end effector, probe/end effector, and robot/world for a robotic TMS system, which can be used to perform precision TMS experiment. Finally, for many robotic applications, where three unknown matrices need to be solved, the method presented in this paper provides an alternative solution to the classical approaches. In the future, it will be interesting to quantitatively compare the stimulation effect of robotic and manual techniques. Investigation on patients will then be pursued to evaluate the medical benefits of robotic TMS system.
Declarations
Authors’ contributions
HW and JNJ conceived and designed the study. YL and XW performed the experiments. HW wrote the paper. HW and TY reviewed and edited the manuscript. All authors read and approved the final manuscript.
Acknowledgements
H.W. acknowledges the grant support from PUMC Youth Fund and the Fundamental Research Funds for the Central Universities (2017320025). T.Y. acknowledges grant support from CAMS Initiative for Innovative Medicine (CAMSI2M 2016I2M1004) and the Natural Science Foundation of China (No. 81772003).
Competing interests
The authors declare that they have no competing interests.
Availability of data and materials
Not applicable.
Consent for publication
At the time of their initial briefing, all study participants were informed of the likelihood that the data would be part of a publication.
Ethics approval and consent to participate
Written informed consent was obtained from the subject for the publication of this report and any accompanying, in accordance with the regulations of the local ethics committee (Committee on Health Research Ethics, Chinese Academy of Medical Sciences).
Funding
CAMS Initiative for Innovative Medicine (CAMSI2M 2016I2M1004). Natural Science Foundation of China NSFC (No. 81772003). PUMC Youth Fund and the Fundamental Research Funds for the Central Universities (2017320025).
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Authors’ Affiliations
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