I Introduction
The magneticallylevitated nanopositioning technique [48, 32]
is a promising solution for ultraclean or vacuum precision motion applications due to its excellent characteristics such as multiaxis mobility, ultraprecision, large motion stroke, contact and dustfree usage, etc. However, due to its fully floating feature, the maglev nanopositioning system requires sophisticated motion control in all its six DegreeofFreedom (DOF) to even simply stabilize at a constant position. The advanced multiaxis positioning and trajectory tracking further require highperformance precision motion control techniques
[14, 25, 6, 8, 31] to reject the internal/external disturbances and eliminate the coupling effects between axes. Traditionally, such precision motion control systems are designed and optimized based on the model (when available) obtained from the first principle or system identification, i.e., modelbased approach [40, 43, 17, 39]. However, obtaining an accurate model for the multiaxis maglev nanopositioning system is challenging and timeconsuming; and the model obtained is typically often not adequately representative of the true dynamics, e.g., the coupling between axes is often not taken into account. To address this oftenoccurring general issue, it is a notable trend where learningbased methods are increasingly being explored in literature, wherein model parameters are not precisely known, and yet the appropriately optimal control performance can be obtained [12, 44, 29]. This datadriven methodology enables learning from available signals in the past, and also prevailing, nonoptimal control settings to achieve a significant performance improvement for various cases of realworld mechatronic systems [5, 7, 42, 30].Datadriven controls are essentially developed based on the concept that machines can improve their performance by learning from previous executions of the same or similar tasks, in a way that closely resembles how humans learn. A promising trend in datadriven controls is deep reinforcement learning
[13], wherein a neural network policy is trained based on realworld motion data as well as appropriate simulations. In addition to this endtoend approach, deep neural networks can also be used for trajectory tracking in many robotic applications
[20]. However, the limitation of these neural network based approaches is the requirement for a massive amount of training data. Also, the uncertainty in the important system stability issue due to the blackbox nature of neural networks often becomes a concern, especially for safetycritical applications. Apart from the neural network based approaches, [45] proposed a novel modelless feedback control design for soft robotics, and it was further extended to the hybrid position/force control problem in [46]. The proposed approaches in these works allow the manipulators to interact with several constrained environments safely and stably, and then generate a modelless feedback control policy from these interactions. It is worthwhile to note that these works are mainly focused on kinematicmodelfree control instead of dynamicmodelfree control, i.e., the Jacobian is unknown and empirically estimated; therefore, challenges in dynamic control remain.It is worthwhile to note that many industrial processes such as scanning, pickandplace, welding, and assembly, involve repetitive motions; therefore, less computationally expensive learning approaches can be pursued. For instance, the Iterative Learning Control (ILC) is a datadriven method that is used widely in precision machines [3, 51] and robotics [15, 1, 36, 38]. It makes use of the repetitive tracking error data gathered in previous cycles to improve the performance of the system in subsequent cycles in a feedforward manner. Thus, it is essentially a feedforward learning approach rather than feedback learning; nevertheless it can serve as a very useful complement to an existing feedback controller. In [2], the authors proposed a novel Gaussian process based feedback controller optimization algorithm with applications to quadrotors. This approach models the cost function as a Gaussian process and explores the new controller parameters with a safe performance guarantee. This enables automatic and safe optimization in repetitive robotic tasks without human intervention. However, while greatly effective especially in guaranteeing safety, the convergence is relatively slow as it takes about 30 iterations to converge.
The Iterative Feedback Tuning (IFT) methodology is one of the approaches in the class of fastconverging datadriven controller optimization algorithms [11]. Conceptually similar to the other approaches, it makes use of the actual motion data to estimate the cost function gradient without relying on the system model. In addition, the Hessian of the cost function can be estimated to speed up the convergence. The estimated gradient and Hessian are subsequently used in the GaussNewton optimization procedure to iteratively obtain the optimal controller parameters. This IFT approach has been widely used in many applications such as pathtracking control of industrial robots [42, 41], ultraprecision wafer stage [19, 9], flow control over a circular cylinder [33] and compliant rehabilitation robots [30], etc. Extensions of the IFT idea to other types of controller includes iterative dynamic decoupling control [18], disturbance observer sensitivity shaping [24], iterative feedforward tuning [34, 37], and 3DOF controller tuning [23, 22] etc. However, most of the existing work focused mainly on accurate tracking and did not take smooth tracking into account. In fact, in semiconductor manufacturing and many other robotic applications, both accurate and smooth trajectory tracking are required [28, 26], and this challenge remains unsolved. Hence, the contribution of this paper is to propose a learningbased controller optimization algorithm to enable smooth and accurate tracking in repetitive tasks as illustrated schematically and conceptually in Fig. 1. To the best of our knowledge, this work is the first feedback controller optimization method to take into account both accurate and smooth tracking in a datadriven manner. Furthermore, it is worthwhile to note that the optimization process is both dataefficient and fastconverging.
This paper is organized as follows. In Section II, a brief description of the magneticallylevitated nanopositioning system is provided. Then, in Section III, the proposed multiobjective controller optimization algorithm is described and analyzed in detail. In Section IV, experimental work is conducted based on the magneticallylevitated nanopositioning system to show the effectiveness of the proposed algorithm. Finally, conclusions are drawn in Section V.
Ii MagneticallyLevitated Nanopositioning System
In this section, the magneticallylevitated planar nanopositioning system (which is the typical prototype application of our datadriven controller optimization approach) is first illustrated, including its working principles and associated overall control scheme. The design objective of our magneticallylevitated planar nanopositioning system is to enable 6DOF motion with low system complexity and high energy efficiency. For largestroke applications, the stroke expandability is also important as well as the affordability to simultaneously operate multiple motion translators. The schematic design of the implemented magneticallylevitated planar nanopositioning system for this work is illustrated in Fig. 2. Although the square coil array in Fig. 2 is covered here in a small area for evaluation, such a square coil based design allows suitably unlimited planar motion stroke as long as the coils spread over. Notably, this system adopts the tiled square coil array for actuation, which shows the comparative advantages in control complexity and energy efficiency as it only requires 8phase for 6DOF motion control and coils far away can be actively switched off to save energy [50]. Furthermore, the interference between coils is minimized at the maximum extent by using the square coil arrangement, so that multiple translators are feasible by individually controlling each or set of coils.
From Fig. 2, it can be seen that the 6DOF motion is achieved by the combined force from Forcer 1 to 4, where each forcer can provide a vertical levitation force and horizontal thrust force. As illustrated in Fig. 3, the moving part of one forcer is a Halbach permanent magnet array and the stationary part is a square coil array grouped into two phases. Due to the periodic arrangement of magnetization directions indicated in Fig. 3, the Halbach array generates an almost ideal sinusoidal magnetic field in both X and Z axes except the magnet end effects. This is not achievable with the normal magnet array widely adopted in 1DOF linear motors [50]. From the side view of Fig. 3, each square coil is divided into three segments, i.e., , , and . The current directions in and are opposite and the magnetic field directions for and are also opposite, so that and generate identical force in both X and Z axes. contributes zero force in two axes due to its current direction.
The force generation on a single square coil can be expressed via the relative location between coil and magnet arrays as and , where is the current magnitude, and and are defined as,
(1) 
where is a force constant, is the geometrical dimension as indicated in Fig. 3, is the spatial wave number with , and and are two constants numerically identified, and . Therefore, the total force generated by the whole forcer in Fig. 3 is expressed via two phases of current as,
(2) 
where , , and denote the current magnitudes in Phase 1 and Phase 2, respectively. is the number of effective coils in each phase, where for the case in Fig. 3, and denotes
(3) 
In order to control the 6DOF motion, the global force/torque given by the controller needs to be allocated to four forcers. For each forcer, such local force is generated through energizing the twophase current on each forcer. According to (2),
(4) 
Therefore, the controllability of the square coil magneticallylevitated system design is based on the invertibility of . It is noted that
(5)  
Since , and thus , it can be seen that as long as , indicates that is full rank and invertible, with the values of the position not affecting this property. Numerically, as the values of and are known, it is thus direct to verify that the condition is met, which shows that (4) has no singularity and the 6DOF motion is fully controllable. The 6DOF sensing is achieved via three channels of laser interferometers and three channels of capacitive sensors as indicated in Fig. 2. With the measured 6axis statevariables, each DOF can thus be closedloop controlled as SingleInput SingleOutput (SISO) systems and ready for the deployment of the algorithm in Section III.
Iii DataDriven MultiObjective Optimization
As noted earlier, certain important precision motion systems such as the maglev nanopositioning system emphasize the requirement for smooth and accurate tracking in terms of control performance. To achieve these objectives, both the tracking accuracy and control signal variation needs to be taken into account concurrently in the optimization. Hence, the overall cost function in this paper is defined as
(6) 
where
is the controller parameter vector in the
iteration, and is the total cost function consisting of the tracking related cost function and control variation related cost function . Here, is the weighting for the tracking performance wherein is the tracking error measured in the iteration; is the weighting for the control variation wherein is the control input and is the variation of control input. Thus consider the typical feedback control system for the magneticallylevitated system as in Fig. 1, where a fixed structure controller is used for motion control and can be expressed as(7) 
Here, is a vector of the controller parameters to be optimized and is a vector of parameter independent transfer functions. We can now formulate the datadriven multiobjective optimization problem as:
Problem 1.
Iiia Gradient Calculation and Estimation
With equation (6), the gradient of the cost function with respect to the parameter in the iteration can be derived as
(9) 
and the Hessian of the cost function can be approximated as
(10) 
The purpose of obtaining the gradient and the Hessian of the cost function is to apply the Newton’s optimization algorithm [4]:
(11) 
where is the updated parameter value for iteration and is the step size at iteration . From (IIIA) and (IIIA), the Newton’s optimization algorithm requires , , and . and can be obtained directly from the sensor measurement and the control software. However, and cannot be obtained directly and have to be estimated with the inputoutput data collected from the closedloop experiments. The gradient of the tracking error can be derived as:
Inspired by the IFT approach [11], can then be obtained by setting as the new reference in the “special” experiment, and we have
(12) 
where denotes the position measurement for this experiment. Apart from , the gradient of can also be derived as
(13) 
can be estimated with the same special experiment by feeding in as the reference
(14) 
where denotes the control input of this special experiment. Notice that and can be estimated solely based on the experimental data. In addition, and can be directly obtained or calculated based on the sensor measurement and control software. Hence, the gradient and Hessian of the cost function can also be estimated according to (IIIA) and (IIIA). It should be noted, as will be discussed in Section IIIB and IIIC
, that an additional normal experiment needs to be conducted in order to obtain an unbiased estimate of the gradient when the measurement noise is taken into consideration.
IiiB Data Collection
To make the datadriven optimization procedure clearer, all the experiments needed and data to be collected within a single iteration are listed below.

Experiment I: Normal experiment.
(15) (16) (17) 
Experiment II: Special experiment.
(18) (19) (20) 
Experiment III: Normal experiment.
(21) (22) (23)
The bold right superscript refers to the experiment index within a single iteration. In Experiment I, the normal operation with, e.g., a Scurve trajectory, is conducted while is measured and used to generate as the reference of Experiment II. In Experiment II, measurement of and is taken and it is then used to obtain and according to (12) and (14). In Experiment III, measurement of and is taken and used to calculate the cost function gradient . The complete datadriven multiobjective optimization algorithm can be summarized in Algorithm 1. It is worth noting that, similar to the IFT and many other algorithms inspired by the IFT, there is no strong guarantee (proofs) for robust stability throughout the iterations, due to the lack of the system model. Hence, as also suggested in [10], we shall use cautious updates, i.e., use small stepsizes, especially during the first iterations.
IiiC Unbiasedness of the Gradient Estimation
The cost function gradient is estimated using the closedloop experiment data, so the measurement noises can potentially lead to errors during this estimation. For this stochastic approximation method to work, the gradient estimation has to be unbiased, mathematically
(24) 
To prove the unbiasedness, we have the following assumptions:
Assumption 1.
Noises in different experiments are independent from each other.
Assumption 2.
Noises
are zero mean, weakly stationary random variables.
Theorem 1.
Proof.
From (12), the estimated gradient of is given by
(25)  
where
(26)  
Notice that contains noises from Experiment I and Experiment II and contains only the noises from Experiment III. With Assumption 1 and Assumption 2, we have
(27) 
and
(28) 
Similar results can be obtained for from (14). The expectation of the estimation of the cost function gradient can be derived as follows
(29) 
This completes the proof of the Theorem. ∎
From the proof, it can be noticed that Experiment III is indeed necessary in order to guarantee the unbiasedness of cost function gradient estimation. If the data from Experiment I were used, i.e., and instead of and , the same noise would exist in both and (as well as in and ). This would lead to a biased estimation for the cost function gradient and it is exactly the reason why Experiment III is needed.
Iv Experimental Validation
Parameters  Y Axis  X Axis  

Before Optimization  After Optimization  Before Optimization  After Optimization  





Cost functions  Y Axis  X Axis  

Before Optimization  After Optimization  Before Optimization  After Optimization  
Total cost  
Tracking cost 

Control variation cost 
This section documents the experimental results of using the proposed datadriven optimization algorithm for the maglev nanopositioning system as a case study. A National Instruments (NI) PXI8110 realtime controller is used with two FPGAs (NI PXI7854R and 7831R) to provide the necessary input/output (I/O) functions. Two Trust TA320 and two TA115 linear current amplifiers are utilized to power up the eightphase coils. The sampling frequency is 5 kHz, and the current limit for each phase of the coil arrays is set as 1.2 A. The Renishaw fiber optic laser interferometers (Model: RLU10) are used for sensing of horizontal positions with a count resolution of 40 nm, and Lion Precision capacitive sensors (Model: CPL290 controller with C18 heads) are used for sensing of vertical positions with a root mean square resolution of 150 nm. The magneticallylevitated system including its actuation and sensor system are shown in Fig. 4, and its designed working range is 30 mm 30 mm 3 mm according to the coil array length.
The motion profile used in the experiment is a fourthorder Scurve which is particularly suitable for precision motion control [16]. In order to meet the requirement of smooth motion, the profile is defined up to the fourth order with limited jerk and snap. The position trajectory as well as its velocity, acceleration, jerk (time derivative of acceleration) and snap (time derivative of jerk) are plotted in Fig. 5. The magneticallylevitated system is controlled by a feedback controller in LabVIEW designed according to the typical proportional–integral–derivative (PID) structure, as the PID control is essentially the most widely adopted control structure in the industry. Nevertheless, it is pertinent at this juncture to also point out that the datadriven multiobjective optimization algorithm proposed here is also applicable to other types of feedback controllers, as long as that it can be expressed in the rather common and standard form of (7). Here specifically, the control input is
(30) 
and the feedback controller can be written in the form of (7) as
(31) 
The goal of the datadriven optimization is to find out the controller parameters that provide a smooth and accurate tracking of the motion profile in Fig. 5, i.e., minimizing the cost function in (6). Note that during the optimization process, no a priori dynamic model information is needed nor will the algorithm attempt to build a model through system identification. To start with, the initial set of controller parameters is designed based on the loop shaping method in [49] with a second order model (neglecting the nonlinearities and higher order dynamics) and further finetuned to provide a decent but nonoptimized control performance, as in Table I. It is worth noting, however, that loop shaping is a modelbased method one can choose to use for the controller initialization but it is by no means necessary when there are no models available. In such cases, one shall simply tune the controller manually to achieve a decent performance and then rely on the proposed datadriven algorithm for performance optimization. The weightings are set as and so that the cost function values for the tracking error and control signal variation are on the same scale (the tracking error has a much smaller numerical value compared with the control signal variation). Nevertheless, we can still adjust the weightings according to the requirement of the motion system, i.e., further improvement on the accuracy or motion smoothness.
Despite the fact that the magneticallylevitated system is capable of conducting 6DOF motion, we consider here only the XY plane motion because it is most commonly used in semiconductor manufacturing [47, 27]. Yet nevertheless, even in this application scenario, it is still the situation where the fully floating behavior and multiaxis coupling make extremely accurate identification of the motion dynamics largely impossible, so that traditional modelbased approaches would encounter great difficulties in being properly successfully deployed here. In high precision semiconductor manufacturing applications, it is often required to conduct a series of repetitive motions [35, 21] on one of the axes. Meanwhile, in order to guarantee the accuracy of highly complex semiconductor circuit patterns, the tracking error from both Y and X axes needs to be minimized. Also, smooth motion should be ensured by minimizing the control input variation for both axes and using a higherorder Scurve trajectory. By using the proposed datadriven multiobjective optimization in Algorithm 1, the control parameters in both Y and X axes are iteratively optimized as shown in Table I and the control performance in terms of the cost function can be significantly improved as shown in Table II. In addition, a comparison of the optimized controller with the initial controller in the frequency domain is plotted in Fig. 6. One major advantage of this datadriven approach is its fast convergence rate because it takes into account not only the gradient of the cost function but the Hessian . From Fig. 7 and Fig. 8, we can observe that both the controller parameters and cost function value converge within only four iterations. Note that the tracking cost or control variation cost alone may increase in some iteration, e.g., of the Y axis in the iteration, but the total cost always decreases iteration by iteration. The timedomain performance improvement for Y axis before and after the datadriven optimization is plotted in Fig. 9, showing a significant reduction in both the tracking error and control signal variation; the rootmeansquare (RMS) tracking errors are respectively mm and mm. Here, the tracking error peaks, e.g., at s of the initial result, could be due to the laser interferometer signal loss or computational delays. Meanwhile, the tracking error and control signal variation of the X axis are also reduced as shown in Fig. 10. The RMS tracking errors for X axis are respectively mm and mm. Here, the tracking error is much smaller compared with the Y axis, because the X axis is kept stationnary while the Y axis is moving. After s, the vibration still exists and this is due to the fact the stage is fully floating with little damping and has to deal with the disturbances from the force coupling. To further demonstrate the disturbance rejection performance, Fig. 11 shows the X axis position measurement comparison under the effect of a Hz sinusoidal disturbance. From all these experimental results, we can see that the proposed approach is certainly effective and able to provide the appropriate optimized trajectory tracking in terms of both accuracy and smoothness, and the convergence rate is also suitably fast enough (only 4 iterations in our experiment) for practical applicability.
V Conclusion
In this paper, we present a datadriven multiobjective optimization algorithm for repetitive motion tasks, where no a priori model information is available. The proposed algorithm is applied to a multiaxis magneticallylevitated system which is difficult to model accurately due to its fully floating behavior and multiaxis coupling. By making use of the rich information contained in the actual motion data under, say, the prevailing nonoptimal conditions, the algorithm can provide fast, efficient and effective controller optimization for the system to operate towards optimality in a datadriven and iterative manner. A welldesigned cost function is stated and specified, which takes both smooth and accurate tracking into account and the optimization process can be completed within a few iterations. Our experimental results show that the motion performance of the maglev nanopositioning system is enhanced significantly and could certainly meet the stringent requirement of presentday highperformance precision motion applications.
For future work, we believe applications of the proposed algorithm certainly is not be limited to such a multiaxis magneticallylevitated system only, and its potential can be further exploited and deployed in other robotic systems (essentially especially those that are challenging to accurately model, e.g. quadrotors, legged robots, and soft robots, etc.).
References
 [1] (2018) Decentralized trajectory tracking control for soft robots interacting with the environment. IEEE Transactions on Robotics 34 (4), pp. 924–935. Cited by: §I.
 [2] (2016) Safe controller optimization for quadrotors with Gaussian processes. In 2016 IEEE International Conference on Robotics and Automation (ICRA), pp. 491–496. Cited by: §I.
 [3] (2016) Batchtobatch rational feedforward control: from iterative learning to identification approaches, with application to a wafer stage. IEEE/ASME Transactions on Mechatronics 22 (2), pp. 826–837. Cited by: §I.
 [4] (2004) Convex optimization. Cambridge university press. Cited by: §IIIA.
 [5] (2020) Iterative learning of dynamic inverse filters for feedforward tracking control. IEEE/ASME Transactions on Mechatronics 25 (1), pp. 349–359. Cited by: §I.
 [6] (2019) Precision motion control of permanent magnet linear synchronous motors using adaptive fuzzy fractionalorder slidingmode control. IEEE/ASME Transactions on Mechatronics 24 (2), pp. 741–752. Cited by: §I.
 [7] (2019) Finitetime learning control using frequency response data with application to a nanopositioning stage. IEEE/ASME Transactions on Mechatronics. Cited by: §I.
 [8] (2019) Design of a feedforwardfeedback controller for a piezoelectricdriven mechanism to achieve highfrequency nonperiodic motion tracking. IEEE/ASME Transactions on Mechatronics 24 (2), pp. 853–862. Cited by: §I.
 [9] (2016) Constrained iterative feedback tuning for robust control of a wafer stage system. IEEE Transactions on Control Systems Technology 24 (1), pp. 56–66. Cited by: §I.
 [10] (2001) Robust loopshaping using iterative feedback tuning. In 2001 European control conference (ECC), pp. 2046–2051. Cited by: §IIIB.
 [11] (2002) Iterative feedback tuningan overview. International Journal of Adaptive Control and Signal Processing 16 (5), pp. 373–395. Cited by: §I, §IIIA.
 [12] (2017) Datadriven control and learning systems. IEEE Transactions on Industrial Electronics 64 (5), pp. 4070–4075. Cited by: §I.
 [13] (2019) Learning agile and dynamic motor skills for legged robots. Science Robotics 4 (26), pp. eaau5872. Cited by: §I.
 [14] (2020) Six degreesoffreedom directdriven nanopositioning stage using crableg flexures. IEEE/ASME Transactions on Mechatronics 25 (2), pp. 513–525. Cited by: §I.
 [15] (2019) Optimizing the execution of dynamic robot movements with learning control. IEEE Transactions on Robotics 35 (4), pp. 909–924. Cited by: §I.
 [16] (2005) Trajectory planning and feedforward design for electromechanical motion systems. Control Engineering Practice 13 (2), pp. 145–157. Cited by: §IV.
 [17] (2019) Harmonic model and remedy strategy of multiphase pm motor under opencircuit fault. IEEE/ASME Transactions on Mechatronics. Cited by: §I.
 [18] (2019) Databased iterative dynamic decoupling control for precision mimo motion systems. IEEE Transactions on Industrial Informatics. Cited by: §I.
 [19] (2019) Convergence rate oriented iterative feedback tuning with application to an ultraprecision wafer stage. IEEE Transactions on Industrial Electronics 66 (3), pp. 1993–2003. Cited by: §I.
 [20] (2017) Deep neural networks for improved, impromptu trajectory tracking of quadrotors. In 2017 IEEE International Conference on Robotics and Automation (ICRA), pp. 5183–5189. Cited by: §I.
 [21] (2018) Feedforward control with disturbance prediction for linear discretetime systems. IEEE Transactions on Control Systems Technology 27 (6), pp. 2340–2350. Cited by: §IV.
 [22] (2018) Datadriven modelfree iterative tuning approach for smooth and accurate tracking. In 2018 IEEE/ASME International Conference on Advanced Intelligent Mechatronics (AIM), pp. 593–598. Cited by: §I.
 [23] (2017) Databased tuning of reducedorder inverse model in both disturbance observer and feedforward with application to tray indexing. IEEE Transactions on Industrial Electronics 64 (7), pp. 5492–5501. Cited by: §I.
 [24] (2019) Enhanced sensitivity shaping by databased tuning of disturbance observer with nonbinomial filter. ISA transactions 85, pp. 284–292. Cited by: §I.
 [25] (2020) Adaptive extended state observerbased synergetic control for a longstroke compliant microstage with stress stiffening. IEEE/ASME Transactions on Mechatronics 25 (1), pp. 259–270. Cited by: §I.
 [26] (2020) Parameter space optimization towards constrained controller design with application to tray indexing. IEEE Transactions on Industrial Electronics 67 (7), pp. 5575–5585. Cited by: §I.
 [27] (2017) Integrated mechatronic design in the flexurelinked dualdrive gantry by constrained linear–quadratic optimization. IEEE Transactions on Industrial Electronics 65 (3), pp. 2408–2418. Cited by: §IV.
 [28] (2019) Parameter space optimization towards integrated mechatronic design for uncertain systems with generalized feedback constraints. Automatica 105, pp. 149–158. Cited by: §I.
 [29] (2020) Advanced optimization for motion control systems. CRC Press. Cited by: §I.
 [30] (2017) Robust iterative feedback tuning control of a compliant rehabilitation robot for repetitive ankle training. IEEE/ASME Transactions on Mechatronics 22 (1), pp. 173–184. Cited by: §I, §I.
 [31] (2018) Precision position tracking for piezoelectricdriven motion system using continuous thirdorder sliding mode control. IEEE/ASME Transactions on Mechatronics 23 (4), pp. 1521–1531. Cited by: §I.
 [32] (201712) Twophase lorentz coils and linear halbach array for multiaxis precisionpositioning stages with magnetic levitation. IEEE/ASME Transactions on Mechatronics 22 (6), pp. 2662–2672. Cited by: §I.
 [33] (2018) Iterative feedback tuning of the proportionalintegraldifferential control of flow over a circular cylinder. IEEE Transactions on Control Systems Technology. Cited by: §I.
 [34] (2008) Iterative tuning of feedforward controller with force ripple compensation for wafer stage. In 10th IEEE International Workshop on Advanced Motion Control, pp. 234–239. Cited by: §I.
 [35] (2019) Disturbance compensation by reference profile alteration with application to tray indexing. IEEE Transactions on Industrial Electronics 66 (12), pp. 9406–9416. Cited by: §IV.
 [36] Modelbased online learning and adaptive control for a “humanwearable soft robot” integrated system. The International Journal of Robotics Research. External Links: Document, Link, https://doi.org/10.1177/0278364919873379 Cited by: §I.
 [37] (2008) Fixed structure feedforward controller design exploiting iterative trials: application to a wafer stage and a desktop printer. Journal of Dynamic Systems, Measurement, and Control 130 (5), pp. 051006. Cited by: §I.
 [38] (2016) Robust twodegreeoffreedom iterative learning control for flexibility compensation of industrial robot manipulators. In 2016 IEEE International Conference on Robotics and Automation (ICRA), pp. 2381–2386. Cited by: §I.
 [39] (2019) Dynamical model based contouring error positionloop feedforward control for multiaxis motion systems. IEEE Transactions on Industrial Informatics. Cited by: §I.
 [40] (2019) An efficient identification method for dynamic systems with coupled hysteresis and linear dynamics: application to piezoelectricactuated nanopositioning stages. IEEE/ASME Transactions on Mechatronics 24 (1), pp. 326–337. Cited by: §I.
 [41] (2019) Robust cascade pathtracking control of networked industrial robot using constrained iterative feedback tuning. IEEE Access 7, pp. 8470–8482. Cited by: §I.
 [42] (2019) Iterative datadriven fractional model reference control of industrial robot for repetitive precise speedtracking. IEEE/ASME Transactions on Mechatronics. Cited by: §I, §I.
 [43] (2017) Continuous integral terminal thirdorder sliding mode motion control for piezoelectric nanopositioning system. IEEE/ASME Transactions on Mechatronics 22 (4), pp. 1828–1838. Cited by: §I.
 [44] (2015) Databased techniques focused on modern industry: an overview. IEEE Transactions on Industrial Electronics 62 (1), pp. 657–667. Cited by: §I.
 [45] (2014) Modelless feedback control of continuum manipulators in constrained environments. IEEE Transactions on Robotics 30 (4), pp. 880–889. Cited by: §I.
 [46] (2016) Modelless hybrid position/force control: a minimalist approach for continuum manipulators in unknown, constrained environments. IEEE Robotics and Automation Letters 1 (2), pp. 844–851. Cited by: §I.
 [47] (2016) Time optimal contouring control of industrial biaxial gantry: a highly efficient analytical solution of trajectory planning. IEEE/ASME Transactions on Mechatronics 22 (1), pp. 247–257. Cited by: §IV.

[48]
(201908)
Magnetically levitated parallel actuated dualstage (maglevpad) system for sixaxis precision positioning
. IEEE/ASME Transactions on Mechatronics 24 (4), pp. 1829–1838. Cited by: §I.  [49] (2017) Analysis and control of a 6 DOF maglev positioning system with characteristics of endeffects and eddy current damping. Mechatronics 47, pp. 183–194. Cited by: §IV.
 [50] (2016) Design and modeling of a sixdegreeoffreedom magnetically levitated positioner using square coils and 1d halbach arrays. IEEE Transactions on Industrial Electronics 64 (1), pp. 440–450. Cited by: §II, §II.
 [51] (2019) An internal model based iterative learning control for wafer scanner systems. IEEE/ASME Transactions on Mechatronics. Cited by: §I.
Comments
There are no comments yet.