Data-Driven Multi-Objective Controller Optimization for a Magnetically-Levitated Nanopositioning System
Xiaocong Li, Haiyue Zhu, Jun Ma, Tat Joo Teo, Chek Sing Teo, Masayoshi Tomizuka, Tong Heng Lee
DData-Driven Multi-Objective Controller Optimization for aMagnetically-Levitated Nanopositioning System
Xiaocong Li,
Member, IEEE , Haiyue Zhu,
Member, IEEE , Jun Ma,
Member, IEEE , Tat Joo Teo,
Member, IEEE ,Chek Sing Teo,
Member, IEEE , Masayoshi Tomizuka,
Life Fellow, IEEE, and Tong Heng Lee
Accepted final version.
To appear in
IEEE/ASME Transactions on Mechatronics , DOI: 10.1109/TMECH.2020.2999401 ©2020 IEEE. Personal use of this material is permitted.Permission from IEEE must be obtained for all other uses, in any current or future media, including reprinting/republishing this material for advertising or promotional purposes,creating new collective works, for resale or redistribution to servers or lists, or reuse of any copyrighted component of this work in other works.
Abstract —The performance achieved with traditional model-based control system design approaches typically relies heavilyupon accurate modeling of the motion dynamics. However,modeling the true dynamics of present-day increasingly complexsystems can be an extremely challenging task; and the usuallynecessary practical approximations often renders the automationsystem to operate in a non-optimal condition. This problem canbe greatly aggravated in the case of a multi-axis magnetically-levitated (maglev) nanopositioning system where the fully float-ing behavior and multi-axis coupling make extremely accurateidentification of the motion dynamics largely impossible. On theother hand, in many related industrial automation applications,e.g., the scanning process with the maglev system, repetitivemotions are involved which could generate a large amount ofmotion data under non-optimal conditions. These motion dataessentially contain rich information; therefore, the possibilityexists to develop an intelligent automation system to learn fromthese motion data, and to drive the system to operate towardsoptimality in a data-driven manner. Along this line then, thispaper proposes a data-driven model-free controller optimizationapproach that learns from the past non-optimal motion data toiteratively improve the motion control performance. Specifically,a novel data-driven multi-objective optimization approach isproposed that is able to automatically estimate the gradient andHessian purely based on the measured motion data; the multi-objective cost function is suitably designed to take into accountboth smooth and accurate trajectory tracking. In the work here,experiments are then conducted on the maglev nanopositioningsystem to demonstrate the effectiveness of the proposed method,and the results show rather clearly the practical appeal ofour methodology for related complex robotic systems with noaccurate model available.
Index Terms —Learning-based control, robot learning, data-driven optimization, iterative feedback tuning, magnetic levita-tion, robot control, precision motion control, nanopositioning.
I. I
NTRODUCTION T HE magnetically-levitated nanopositioning technique [1],[2] is a promising solution for ultra-clean or vacuumprecision motion applications due to its excellent characteris-tics such as multi-axis mobility, ultra-precision, large motion
This work is supported by Collaborative Research Project U18-R-030SU under SIMTech-NUS Joint Lab on Precision Motion Systems (U12-R-024JL). (Corresponding author: Haiyue Zhu.)
X. Li, H. Zhu and C. S. Teo are with the Mechatronics Group,Singapore Institute of Manufacturing Technology, Singapore 138634. (e-mail: li [email protected], zhu [email protected],[email protected]).J. Ma and M. Tomizuka are with the Department of MechanicalEngineering, University of California, Berkeley, CA 94720, USA (e-mail:[email protected], [email protected]).T. J. Teo and T. H. Lee are with the Department of Electrical andComputer Engineering, National University of Singapore, Singapore 117583(e-mail: [email protected], [email protected]). stroke, contact- and dust-free usage, etc. However, due toits fully floating feature, the maglev nanopositioning systemrequires sophisticated motion control in all its six Degree-of-Freedom (DOF) to even simply stabilize at a constant position.The advanced multi-axis positioning and trajectory trackingfurther require high-performance precision motion controltechniques [3]–[7] to reject the internal/external disturbancesand eliminate the coupling effects between axes. Traditionally,such precision motion control systems are designed and op-timized based on the model (when available) obtained fromthe first principle or system identification, i.e., model-basedapproach [8]–[11]. However, obtaining an accurate model forthe multi-axis maglev nanopositioning system is challengingand time-consuming; and the model obtained is typically oftennot adequately representative of the true dynamics, e.g., thecoupling between axes is often not taken into account. Toaddress this often-occurring general issue, it is a notable trendwhere learning-based methods are increasingly being exploredin literature, wherein model parameters are not preciselyknown, and yet the appropriately optimal control performancecan be obtained [12]–[14]. This data-driven methodologyenables learning from available signals in the past, and alsoprevailing, non-optimal control settings to achieve a significantperformance improvement for various cases of real-worldmechatronic systems [15]–[18].Data-driven controls are essentially developed based onthe concept that machines can improve their performanceby learning from previous executions of the same or similartasks, in a way that closely resembles how humans learn. Apromising trend in data-driven controls is deep reinforcementlearning [19], wherein a neural network policy is trained basedon real-world motion data as well as appropriate simulations.In addition to this end-to-end approach, deep neural networkscan also be used for trajectory tracking in many roboticapplications [20]. However, the limitation of these neuralnetwork based approaches is the requirement for a massiveamount of training data. Also, the uncertainty in the importantsystem stability issue due to the black-box nature of neuralnetworks often becomes a concern, especially for safety-critical applications. Apart from the neural network basedapproaches, [21] proposed a novel model-less feedback controldesign for soft robotics, and it was further extended to thehybrid position/force control problem in [22]. The proposedapproaches in these works allow the manipulators to interactwith several constrained environments safely and stably, andthen generate a model-less feedback control policy from theseinteractions. It is worthwhile to note that these works aremainly focused on kinematic-model-free control instead of a r X i v : . [ ee ss . S Y ] J u l ynamic-model-free control, i.e., the Jacobian is unknown andempirically estimated; therefore, challenges in dynamic controlremain.It is worthwhile to note that many industrial processes suchas scanning, pick-and-place, welding, and assembly, involverepetitive motions; therefore, less computationally expensivelearning approaches can be pursued. For instance, the Itera-tive Learning Control (ILC) is a data-driven method that isused widely in precision machines [23], [24] and robotics[25]–[28]. It makes use of the repetitive tracking error datagathered in previous cycles to improve the performance ofthe system in subsequent cycles in a feedforward manner.Thus, it is essentially a feedforward learning approach ratherthan feedback learning; nevertheless it can serve as a veryuseful complement to an existing feedback controller. In[29], the authors proposed a novel Gaussian process basedfeedback controller optimization algorithm with applicationsto quadrotors. This approach models the cost function as aGaussian process and explores the new controller parameterswith a safe performance guarantee. This enables automatic andsafe optimization in repetitive robotic tasks without humanintervention. However, while greatly effective especially inguaranteeing safety, the convergence is relatively slow as ittakes about 30 iterations to converge.The Iterative Feedback Tuning (IFT) methodology is oneof the approaches in the class of fast-converging data-drivencontroller optimization algorithms [30]. Conceptually similarto the other approaches, it makes use of the actual motiondata to estimate the cost function gradient without relyingon the system model. In addition, the Hessian of the costfunction can be estimated to speed up the convergence. Theestimated gradient and Hessian are subsequently used in theGauss-Newton optimization procedure to iteratively obtain theoptimal controller parameters. This IFT approach has beenwidely used in many applications such as path-tracking controlof industrial robots [17], [31], ultra-precision wafer stage [32],[33], flow control over a circular cylinder [34] and compliantrehabilitation robots [18], etc. Extensions of the IFT idea toother types of controller includes iterative dynamic decouplingcontrol [35], disturbance observer sensitivity shaping [36],iterative feedforward tuning [37], [38], and 3-DOF controllertuning [39], [40] etc. However, most of the existing workfocused mainly on accurate tracking and did not take smoothtracking into account. In fact, in semiconductor manufacturingand many other robotic applications, both accurate and smoothtrajectory tracking are required [41], [42], and this challengeremains unsolved. Hence, the contribution of this paper isto propose a learning-based controller optimization algorithmto enable smooth and accurate tracking in repetitive tasks asillustrated schematically and conceptually in Fig. 1. To the bestof our knowledge, this work is the first feedback controlleroptimization method to take into account both accurate andsmooth tracking in a data-driven manner. Furthermore, it isworthwhile to note that the optimization process is both data-efficient and fast-converging.This paper is organized as follows. In Section II, a brief de-scription of the magnetically-levitated nanopositioning systemis provided. Then, in Section III, the proposed multi-objective Controller
Magnetically-LevitatedStage
Cost FunctionEvaluation
Data-DrivenMulti-ObjectiveOptimization
NewControllerParameterCurrentControllerParameter ru y + - n - Typical Feedback Control Loop
Fig. 1. Overview of the data-driven multi-objective controller optimizationalgorithm. The algorithm iteratively updates the controller parameters basedon the actual motion data from the previous iteration to minimize the costfunction value in an iterative and model-free manner.Fig. 2. Schematics of the square-coil-based magnetically-levitated planarnanopositioning system. controller optimization algorithm is described and analyzed indetail. In Section IV, experimental work is conducted based onthe magnetically-levitated nanopositioning system to show theeffectiveness of the proposed algorithm. Finally, conclusionsare drawn in Section V.II. M
AGNETICALLY -L EVITATED N ANOPOSITIONING S YSTEM
In this section, the magnetically-levitated planar nanoposi-tioning system (which is the typical prototype application ofour data-driven controller optimization approach) is first illus-trated, including its working principles and associated overallcontrol scheme. The design objective of our magnetically-levitated planar nanopositioning system is to enable 6-DOFmotion with low system complexity and high energy effi-ciency. For large-stroke applications, the stroke expandabilityis also important as well as the affordability to simultaneouslyoperate multiple motion translators. The schematic design ofthe implemented magnetically-levitated planar nanoposition-ing system for this work is illustrated in Fig. 2. Although thesquare coil array in Fig. 2 is covered here in a small areafor evaluation, such a square coil based design allows suitablyunlimited planar motion stroke as long as the coils spread ig. 3. Top view and side view of the forcer in the square-coil-basedmagnetically-levitated stage design. over. Notably, this system adopts the tiled square coil array foractuation, which shows the comparative advantages in controlcomplexity and energy efficiency as it only requires 8-phasefor 6-DOF motion control and coils far away can be activelyswitched off to save energy [43]. Furthermore, the interferencebetween coils is minimized at the maximum extent by usingthe square coil arrangement, so that multiple translators arefeasible by individually controlling each or set of coils.From Fig. 2, it can be seen that the 6-DOF motion isachieved by the combined force from Forcer 1 to 4, where eachforcer can provide a vertical levitation force and horizontalthrust force. As illustrated in Fig. 3, the moving part of oneforcer is a Halbach permanent magnet array and the stationarypart is a square coil array grouped into two phases. Due to theperiodic arrangement of magnetization directions indicated inFig. 3, the Halbach array generates an almost ideal sinusoidalmagnetic field in both X and Z axes except the magnet endeffects. This is not achievable with the normal magnet arraywidely adopted in 1-DOF linear motors [43]. From the sideview of Fig. 3, each square coil is divided into three segments,i.e., S l , S m , and S r . The current directions in S l and S r areopposite and the magnetic field directions for S l and S r arealso opposite, so that S l and S r generate identical force inboth X and Z axes. S m contributes zero force in two axes dueto its current direction.The force generation on a single square coil can be ex-pressed via the relative location between coil and magnetarrays ( x, z ) as F cx ( x, z ) = K x ( x, z ) I and F cz ( x, z ) = K z ( x, z ) I , where I is the current magnitude, and K x ( x, z ) and K z ( x, z ) are defined as, K x ( x, z ) = C f e − γz (cid:16) −√ τ sin( γx )+ 2 αγ sin( γx ) cos( γτ β ) (cid:17) ,K z ( x, z ) = C f e − γz (cid:16) −√ τ cos( γx )+ 2 αγ cos( γx ) cos( γτ β ) (cid:17) , (1)where C f > is a force constant, τ is the geometrical dimension as indicated in Fig. 3, γ is the spatial wave numberwith γ = π/ τ , and α and β are two constants numericallyidentified, α = 1 . and β = 2 . . Therefore, the totalforce generated by the whole forcer in Fig. 3 is expressed viatwo phases of current as, F f ( x, z ) = N (cid:20) K x ( x, z ) K x ( x + 3 τ, z ) K z ( x, z ) K z ( x + 3 τ, z ) (cid:21) I (2)where F f ( x, z ) = [ F fx ( x, z ) F fx ( x, z )] T , I = [ I I ] T , I and I denote the current magnitudes in Phase 1 and Phase 2,respectively. N is the number of effective coils in each phase,where N = 4 for the case in Fig. 3, and denotes Φ K ( x, z ) = (cid:20) K x ( x, z ) K x ( x + 3 τ, z ) K z ( x, z ) K z ( x + 3 τ, z ) (cid:21) . (3)In order to control the 6-DOF motion, the globalforce/torque given by the controller needs to be allocated tofour forcers. For each forcer, such local force is generatedthrough energizing the two-phase current I i = [ I i I i ] T oneach forcer. According to (2), I i = Φ K ( x, z ) − F f i /N. (4)Therefore, the controllability of the square coil magnetically-levitated system design is based on the invertibility of Φ K ( x, z ) . It is noted that det (cid:16) Φ K ( x, z ) (cid:17) = K x ( x, z ) K z ( x + 3 τ, z ) − K z ( x, z ) K x ( x + 3 τ, z )= (cid:16) C f e − γz (cid:17) (cid:32)(cid:16) −√ τ sin( γx ) + 2 αγ sin( γx ) cos( γτ β ) (cid:17) + (cid:16) −√ τ cos( γx ) + 2 αγ cos( γx ) cos( γτ β ) (cid:17) (cid:33) = (cid:16) C f e − γz (cid:17) (cid:16) √ τ − αγ cos( γτ β ) (cid:17) . (5)Since γ = π/ τ , and thus γτ = π/ , it can be seen thatas long as cos( β + π/ (cid:54) = √ π/ α , det (cid:16) Φ K ( x, z ) (cid:17) > indicates that Φ K ( x, z ) is full rank and invertible, withthe values of the position ( x, z ) not affecting this property.Numerically, as the values of α and β are known, it is thusdirect to verify that the condition cos( β + π/ (cid:54) = √ π/ α ismet, which shows that (4) has no singularity and the 6-DOFmotion is fully controllable. The 6-DOF sensing is achievedvia three channels of laser interferometers ( x , x , y ) andthree channels of capacitive sensors ( z , z , z ) as indicated inFig. 2. With the measured 6-axis state-variables, each DOF canthus be closed-loop controlled as Single-Input Single-Output(SISO) systems and ready for the deployment of the algorithmin Section III.III. D ATA -D RIVEN M ULTI -O BJECTIVE O PTIMIZATION
As noted earlier, certain important precision motion systemssuch as the maglev nanopositioning system emphasize therequirement for smooth and accurate tracking in terms ofcontrol performance. To achieve these objectives, both theracking accuracy and control signal variation needs to betaken into account concurrently in the optimization. Hence,the overall cost function in this paper is defined as J ( i ρ ) = w e ( i ρ ) T · e ( i ρ ) (cid:124) (cid:123)(cid:122) (cid:125) J e + w ˙ u ( i ρ ) T · ˙ u ( i ρ ) (cid:124) (cid:123)(cid:122) (cid:125) J ˙ u , (6)where i ρ is the controller parameter vector in the i th iteration,and J ( i ρ ) is the total cost function consisting of the trackingrelated cost function J e and control variation related costfunction J ˙ u . Here, w is the weighting for the trackingperformance wherein e ( i ρ ) is the tracking error measured inthe i th iteration; w is the weighting for the control variationwherein u ( i ρ ) is the control input and ˙ u ( i ρ ) is the variationof control input. Thus consider the typical feedback controlsystem for the magnetically-levitated system as in Fig. 1,where a fixed structure controller C ( s ) is used for motioncontrol and can be expressed as C ( s, ρ ) = ρ T ¯ C ( s ) . (7)Here, ρ is a vector of the controller parameters to be optimizedand ¯ C ( s ) is a vector of parameter independent transfer func-tions. We can now formulate the data-driven multi-objectiveoptimization problem as: Problem 1.
Assume the motion system is unknown and con-trolled by a fixed structure controller C ( s, ρ ) in (7) ; use onlythe closed-loop experimental data to determine the parametervector ρ that minimizes the multi-objective cost function J ( ρ ) (6) , i.e., to find ρ (cid:63) = arg min ρ J ( ρ ) . (8) A. Gradient Calculation and Estimation
With equation (6), the gradient of the cost function J ( i ρ ) with respect to the parameter in the i th iteration i ρ can bederived as ∇ J ( i ρ ) = 2 w [ ∇ i e ( i ρ )] T · i e ( i ρ )+ 2 w [ ∇ i ˙ u ( i ρ )] T · i ˙ u ( i ρ ) , (9)and the Hessian of the cost function can be approximated as ∇ J ( i ρ ) = 2 w [ ∇ i e ( i ρ )] T · ∇ i e ( i ρ )+ 2 w [ ∇ i ˙ u ( i ρ )] T · ∇ i ˙ u ( i ρ ) . (10)The purpose of obtaining the gradient and the Hessian of thecost function is to apply the Newton’s optimization algorithm[44]: i + ρ = i ρ − i γ ( ∇ J ( i ρ )) − ∇ J ( i ρ ) . (11)where i + ρ is the updated parameter value for iteration i + and i γ is the step size at iteration i . From (9) and (10), theNewton’s optimization algorithm requires ∇ i e ( i ρ ) , ∇ i ˙ u ( i ρ ) , i e ( i ρ ) and i ˙ u ( i ρ ) . i e ( i ρ ) and i ˙ u ( i ρ ) can be obtained directlyfrom the sensor measurement and the control software. How-ever, ∇ i e ( i ρ ) and ∇ i ˙ u ( i ρ ) cannot be obtained directly andhave to be estimated with the input-output data collected from the closed-loop experiments. The gradient of the tracking errorcan be derived as: ∇ i e ( i ρ ) = − P ∂C ( i ρ ) ∂ i ρ [1 + P C ( i ρ )] · r = − P ∂C ( i ρ ) ∂ i ρ P C ( i ρ ) · i e ( i ρ ) . Inspired by the IFT approach [30], ∇ i e ( i ρ ) can then beobtained by setting i e ( i ρ ) as the new reference r in the“special” experiment, and we have ∇ i e ( i ρ ) = − ∂C ( i ρ ) ∂ i ρ · C ( i ρ ) · y s , (12)where y s denotes the position measurement for this experi-ment. Apart from ∇ i e ( i ρ ) , the gradient of i ˙ u ( i ρ ) can also bederived as ∇ i ˙ u ( i ρ ) = ∂C ( i ρ ) ∂ i ρ [1 + P C ( i ρ )][1 + P C ( i ρ )] · ˙ r − P ∂C ( i ρ ) ∂ i ρ C ( i ρ )[1 + P C ( i ρ )] · ˙ r = ∂C ( i ρ ) ∂ i ρ
11 +
P C ( i ρ ) · ˙ e (13) ∇ i ˙ u ( i ρ ) can be estimated with the same special experimentby feeding in i e ( i ρ ) as the reference r ∇ i ˙ u ( i ρ ) = ∂C ( i ρ ) ∂ i ρ · C ( i ρ ) · ˙ u s , (14)where u s denotes the control input of this special exper-iment. Notice that ∇ i e ( i ρ ) and ∇ i ˙ u ( i ρ ) can be estimatedsolely based on the experimental data. In addition, i e ( i ρ ) and i ˙ u ( i ρ ) can be directly obtained or calculated based on thesensor measurement and control software. Hence, the gradient ∇ J ( i ρ ) and Hessian ∇ J ( i ρ ) of the cost function can alsobe estimated according to (9) and (10). It should be noted, aswill be discussed in Section III-B and III-C, that an additionalnormal experiment needs to be conducted in order to obtainan unbiased estimate of the gradient when the measurementnoise is taken into consideration. B. Data Collection
To make the data-driven optimization procedure clearer, allthe experiments needed and data to be collected within a singleiteration are listed below. • Experiment I: Normal experiment. r = r, (15) y = P C ( i ρ )1+ P C ( i ρ ) · r − P C ( i ρ ) · n , (16) e = P C ( i ρ ) · r + P C ( i ρ ) · n . (17) • Experiment II: Special experiment. r = e , (18) y s = y = P C ( i ρ )1+ P C ( i ρ ) · e − P C ( i ρ ) · n , (19) u s = u = C ( i ρ )1+ P C ( i ρ ) · e + C ( i ρ )1+ P C ( i ρ ) · n . (20) • Experiment III: Normal experiment. r = r, (21) e = P C ( i ρ ) · r + P C ( i ρ ) · n . (22) u = C ( i ρ )1+ P C ( i ρ ) · r + C ( i ρ )1+ P C ( i ρ ) · n . (23)he bold right superscript refers to the experiment indexwithin a single iteration. In Experiment I, the normal op-eration with, e.g., a S-curve trajectory, is conducted while y is measured and used to generate e as the reference ofExperiment II. In Experiment II, measurement of y s and u s is taken and it is then used to obtain ∇ i e ( i ρ ) and ∇ i ˙ u ( i ρ ) according to (12) and (14). In Experiment III, measurementof e and u is taken and used to calculate the cost functiongradient ∇ J ( i ρ ) . The complete data-driven multi-objectiveoptimization algorithm can be summarized in Algorithm 1 .It is worth noting that, similar to the IFT and many otheralgorithms inspired by the IFT, there is no strong guarantee(proofs) for robust stability throughout the iterations, due to thelack of the system model. Hence, as also suggested in [45], weshall use cautious updates, i.e., use small step-sizes, especiallyduring the first iterations.
Algorithm 1
Data-Driven Multi-Objective Controller Opti-mization Algorithm1) Set the iteration number i = 0 and select the initialcontroller parameter ρ .2) Conduct Experiment I and measure the output y andtracking error e .3) Evaluate the cost function J ( i ρ ) . Stop if the cost functionvalue is satisfactory. Otherwise, proceed to Step 4.4) Conduct Experiment II and measure the output y fromthis special experiment.5) Obtain ∇ i e ( i ρ ) and ∇ i ˙ u ( i ρ ) according to (12) and (14)respectively.6) Conduct Experiment III and measure e and u .7) Compute ∇ J ( i ρ ) as well as ∇ J ( i ρ ) according to (9) and(10), where i e ( i ρ ) , i ˙ u ( i ρ ) are obtained from ExperimentIII and ∇ i e ( i ρ ) , ∇ i ˙ u ( i ρ ) are obtained from Step 5.8) Execute the Gauss-Newton algorithm (11), and update thecontroller parameters.9) Set the iteration number i ← i + 1 and proceed to Step2. C. Unbiasedness of the Gradient Estimation
The cost function gradient is estimated using the closed-loop experiment data, so the measurement noises can poten-tially lead to errors during this estimation. For this stochasticapproximation method to work, the gradient estimation has tobe unbiased, mathematically E { est [ ∇ J ( i ρ )] } = ∇ J ( i ρ ) . (24)To prove the unbiasedness, we have the following assumptions: Assumption 1.
Noises n in different experiments are indepen-dent from each other. Assumption 2.
Noises n are zero mean, weakly stationaryrandom variables. Theorem 1.
For the motion system under the feedback controlconfiguration as shown in Fig. 1, with Assumption 1 andAssumption 2, the estimation of the gradient of the costfunction J in (6) is unbiased. Proof. From (12), the estimated gradient of e is given byest [ ∇ i e ( i ρ )] = − ∂C ( i ρ ) ∂ i ρ · P P C ( i ρ ) · e − ∂C ( i ρ ) ∂ i ρ · P [1 + P C ( i ρ )] · n + ∂C ( i ρ ) ∂ i ρ · C ( i ρ ) ·
11 +
P C ( i ρ ) · n = ∇ i e ( i ρ ) + w e , (25)where w e ≡ − ∂C ( i ρ ) ∂ i ρ · P [1 + P C ( i ρ )] · n + ∂C ( i ρ ) ∂ i ρ · C ( i ρ ) ·
11 +
P C ( i ρ ) · n . (26)Notice that w e contains noises from Experiment I and Exper-iment II and e contains only the noises from Experiment III.With Assumption 1 and Assumption 2, we have E [ w eT · e ( i ρ )] = E [ w eT ] · E [ e ( i ρ )] , (27)and E [ w eT ] = 0 . (28)Similar results can be obtained for ˙ u from (14). The expec-tation of the estimation of the cost function gradient can bederived as follows E { est [ ∇ J ( i ρ )] } =2 w E { est [ ∇ e T ( i ρ )] e ( i ρ ) } + 2 w E { est [ ∇ ˙ u T ( i ρ )] ˙ u ( i ρ ) } =2 w E [ ∇ e T ( i ρ ) e ( i ρ )] + 2 w E [ w eT · e ( i ρ )]+ 2 w E [ ∇ ˙ u T ( i ρ ) ˙ u ( i ρ )] + 2 w E [ w ˙ uT · ˙ u ( i ρ )]= ∇ J ( i ρ ) + 0 · E [ e ( i ρ )] + 0 · E [ ˙ u ( i ρ )]= ∇ J ( i ρ ) . (29)This completes the proof of the Theorem.From the proof, it can be noticed that Experiment III isindeed necessary in order to guarantee the unbiasedness of costfunction gradient estimation. If the data from Experiment Iwere used, i.e., e ( i ρ ) and ˙ u ( i ρ ) instead of e ( i ρ ) and ˙ u ( i ρ ) ,the same noise would exist in both est [ ∇ e T ( i ρ )] and e ( i ρ ) (aswell as in est [ ∇ ˙ u T ( i ρ )] and ˙ u ( i ρ ) ). This would lead to a biasedestimation for the cost function gradient and it is exactly thereason why Experiment III is needed.IV. E XPERIMENTAL V ALIDATION
This section documents the experimental results of using theproposed data-driven optimization algorithm for the maglevnanopositioning system as a case study. A National Instru-ments (NI) PXI-8110 real-time controller is used with twoFPGAs (NI PXI-7854R and 7831R) to provide the necessaryinput/output (I/O) functions. Two Trust TA320 and two TA115linear current amplifiers are utilized to power up the eight-phase coils. The sampling frequency is 5 kHz, and the current
ABLE IO
VERVIEW OF THE CONTROLLER PARAMETERS FOR Y AND X AXIS
Parameters Y Axis X AxisBefore Optimization After Optimization Before Optimization After Optimization K p
30 25 . . T i .
002 2 . × − .
002 3 . × − T d . . × − . . × − TABLE IIO
VERVIEW OF THE COST FUNCTIONS FOR Y AND X AXIS
Cost functions Y Axis X AxisBefore Optimization After Optimization Before Optimization After OptimizationTotal cost J . × . × . × . × Tracking cost J e . × . × . × . × Control variation cost J ˙ u . × . × . × . × Controller
RoboticAutomationSystem
Cost FunctionEvaluation
Data-DrivenOptimization
NewControllerParameterCurrentControllerParameter ru y +- n - LaserInterferometer Coil Array LaserInterferometerMirror Maglev Stage
Fig. 4. Magnetically-levitated nanopositioning system used in the experimen-tal validation.
Time (s) P o s i t i o n ( mm ) Time (s) V e l ( mm / s ) Time (s) A cc ( mm = s ) -1000100 Time (s) J e r k ( mm = s ) -200002000 Time (s) Sn a p ( mm = s ) -505 Fig. 5. Fourth-order S-curve motion profile used in the real-time experiment. -1 M ag n i t ud e ( d B ) InitialOptimized
Frequency (Hz) -1 P h a s e ( D e g ) -90090 Fig. 6. Y axis controller C ( s, ρ ) comparison before and after the optimizationin the frequency domain. Iteration K p Y axis parametersIteration T i -3 Iteration T d -4 Iteration K p X axis parametersIteration T i -3 Iteration T d -4 Fig. 7. Y axis and X axis controller parameter convergence diagram. teration T o t a l C o s t Y axis cost functionIteration T r a c k i n g c o s t Iteration C o n t r o l v a r i a t i o n c o s t Iteration T o t a l c o s t X axis cost functionIteration T r a c k i n g c o s t Iteration C o n t r o l v a r i a t i o n c o s t Fig. 8. Y axis and X axis cost function convergence diagram. Top: OverallCost. Middle: Cost related to tracking error. Bottom: Cost related to controlsignal variation
Time (s) T r a c k i n g e rr o r ( mm ) -0.08-0.06-0.04-0.0200.020.040.06 InitialOptimized
Time (s) C o n t r o l s i g n a l v a r i a t i o n ( A / s ) -200-150-100-50050100 InitialOptimized
Fig. 9. Y axis tracking error and control signal variation comparison beforeand after the data-driven multi-objective optimization. limit for each phase of the coil arrays is set as 1.2 A. TheRenishaw fiber optic laser interferometers (Model: RLU10) areused for sensing of horizontal positions with a count resolutionof 40 nm, and Lion Precision capacitive sensors (Model:CPL290 controller with C18 heads) are used for sensing ofvertical positions with a root mean square resolution of 150nm. The magnetically-levitated system including its actuationand sensor system are shown in Fig. 4, and its designedworking range is 30 mm ×
30 mm × Time (s) T r a c k i n g e rr o r ( mm ) -3 -1.5-1-0.500.511.5 InitialOptimized
Time (s) C o n t r o l s i g n a l v a r i a t i o n ( A / s ) -8-6-4-202468 InitialOptimized
Fig. 10. X axis tracking error and control signal variation comparison beforeand after the data-driven multi-objective optimization.
Time (s) P o s i t i o n ( mm ) -3 -3-2-10123 InitialOptimized
Fig. 11. X axis disturbance rejection performance comparison under 1 Hzsinusoidal disturbance.
The motion profile used in the experiment is a fourth-orderS-curve which is particularly suitable for precision motioncontrol [46]. In order to meet the requirement of smoothmotion, the profile is defined up to the fourth order withlimited jerk and snap. The position trajectory as well as itsvelocity, acceleration, jerk (time derivative of acceleration)and snap (time derivative of jerk) are plotted in Fig. 5.The magnetically-levitated system is controlled by a feedbackcontroller in LabVIEW designed according to the typical pro-portionalintegralderivative (PID) structure, as the PID controlis essentially the most widely adopted control structure inthe industry. Nevertheless, it is pertinent at this juncture toalso point out that the data-driven multi-objective optimizationalgorithm proposed here is also applicable to other types offeedback controllers, as long as that it can be expressed in therather common and standard form of (7). Here specifically, theontrol input u ( t ) is u ( t ) = K p ( e ( t ) + 1 T i (cid:90) t e ( t (cid:48) ) dt (cid:48) + T d de ( t ) dt ) , (30)and the feedback controller can be written in the form of (7)as C ( s, ρ ) = ρ T ¯ C ( s ) = (cid:2) K p K p /T i K p T d (cid:3) /ss . (31)The goal of the data-driven optimization is to find out thecontroller parameters that provide a smooth and accuratetracking of the motion profile in Fig. 5, i.e., minimizing thecost function J ( ρ ) in (6). Note that during the optimizationprocess, no a priori dynamic model information is needednor will the algorithm attempt to build a model throughsystem identification. To start with, the initial set of controllerparameters ρ is designed based on the loop shaping method in[47] with a second order model (neglecting the nonlinearitiesand higher order dynamics) and further fine-tuned to providea decent but non-optimized control performance, as in Table I.It is worth noting, however, that loop shaping is a model-basedmethod one can choose to use for the controller initializationbut it is by no means necessary when there are no modelsavailable. In such cases, one shall simply tune the controllermanually to achieve a decent performance and then rely on theproposed data-driven algorithm for performance optimization.The weightings are set as w = 10 and w = 1 so that thecost function values for the tracking error and control signalvariation are on the same scale (the tracking error has a muchsmaller numerical value compared with the control signalvariation). Nevertheless, we can still adjust the weightingsaccording to the requirement of the motion system, i.e., furtherimprovement on the accuracy or motion smoothness.Despite the fact that the magnetically-levitated system iscapable of conducting 6-DOF motion, we consider here onlythe X-Y plane motion because it is most commonly usedin semiconductor manufacturing [48], [49]. Yet nevertheless,even in this application scenario, it is still the situation wherethe fully floating behavior and multi-axis coupling make ex-tremely accurate identification of the motion dynamics largelyimpossible, so that traditional model-based approaches wouldencounter great difficulties in being properly successfullydeployed here. In high precision semiconductor manufactur-ing applications, it is often required to conduct a series ofrepetitive motions [50], [51] on one of the axes. Meanwhile,in order to guarantee the accuracy of highly complex semi-conductor circuit patterns, the tracking error from both Yand X axes needs to be minimized. Also, smooth motionshould be ensured by minimizing the control input variationfor both axes and using a higher-order S-curve trajectory. Byusing the proposed data-driven multi-objective optimization in Algorithm 1 , the control parameters in both Y and X axesare iteratively optimized as shown in Table I and the controlperformance in terms of the cost function can be significantlyimproved as shown in Table II. In addition, a comparisonof the optimized controller with the initial controller in thefrequency domain is plotted in Fig. 6. One major advantage of this data-driven approach is its fast convergence rate becauseit takes into account not only the gradient ∇ J ( i ρ ) of the costfunction but the Hessian ∇ J ( i ρ ) . From Fig. 7 and Fig. 8,we can observe that both the controller parameters and costfunction value converge within only four iterations. Note thatthe tracking cost J e or control variation cost J ˙ u alone mayincrease in some iteration, e.g., J ˙ u of the Y axis in the st iteration, but the total cost J always decreases iteration byiteration. The time-domain performance improvement for Yaxis before and after the data-driven optimization is plottedin Fig. 9, showing a significant reduction in both the trackingerror and control signal variation; the root-mean-square (RMS)tracking errors are respectively . mm and . mm.Here, the tracking error peaks, e.g., at . s of the initialresult, could be due to the laser interferometer signal lossor computational delays. Meanwhile, the tracking error andcontrol signal variation of the X axis are also reduced asshown in Fig. 10. The RMS tracking errors for X axis arerespectively . × − 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 ismoving. After . s, the vibration still exists and this is due tothe fact the stage is fully floating with little damping and has todeal with the disturbances from the force coupling. To furtherdemonstrate the disturbance rejection performance, Fig. 11shows the X axis position measurement comparison underthe effect of a Hz sinusoidal disturbance. From all theseexperimental results, we can see that the proposed approachis certainly effective and able to provide the appropriateoptimized trajectory tracking in terms of both accuracy andsmoothness, and the convergence rate is also suitably fastenough (only 4 iterations in our experiment) for practicalapplicability. V. C
ONCLUSION
In this paper, we present a data-driven multi-objectiveoptimization algorithm for repetitive motion tasks, whereno a priori model information is available. The proposedalgorithm is applied to a multi-axis magnetically-levitatedsystem which is difficult to model accurately due to its fullyfloating behavior and multi-axis coupling. By making use ofthe rich information contained in the actual motion data under,say, the prevailing non-optimal conditions, the algorithm canprovide fast, efficient and effective controller optimization forthe system to operate towards optimality in a data-drivenand iterative manner. A well-designed cost function is statedand specified, which takes both smooth and accurate trackinginto account and the optimization process can be completedwithin a few iterations. Our experimental results show that themotion performance of the maglev nanopositioning system isenhanced significantly and could certainly meet the stringentrequirement of present-day high-performance precision motionapplications.For future work, we believe applications of the proposedalgorithm certainly is not be limited to such a multi-axismagnetically-levitated system only, and its potential can befurther exploited and deployed in other robotic systems (es-entially especially those that are challenging to accuratelymodel, e.g. quadrotors, legged robots, and soft robots, etc.).R
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Xiaocong Li (S’14–M’17) received the B.Eng. de-gree from the Department of Electrical and Com-puter Engineering, National University of Singapore,in 2013, and the Ph.D. degree in electrical engineer-ing from the NUS Graduate School for IntegrativeSciences and Engineering, National University ofSingapore, in 2017.He is a Research Scientist with the MechatronicsGroup, Singapore Institute of Manufacturing Tech-nology (SIMTech), Agency for Science, Technologyand Research (A*STAR), where he is currentlyleading a Collaborative Research Project (CRP) with the National Universityof Singapore on data-driven controls. Since 2018, he has served as a memberof the Thesis Advisory Committee for NUS Graduate School for IntegrativeScience and Engineering. His current research interest is focused on makingsense of data for controls through learning as well as its applications torobotics and precision motion systems.
Haiyue Zhu (S’13–M’17) received the B.Eng. de-gree in automation from the School of ElectricalEngineering and Automation and the B. Mgt. degreein business administration from the College of Man-agement and Economics, Tianjin University, Tianjin,China, in 2010, and the M.Sc. and Ph.D. degreesfrom the National University of Singapore (NUS),Singapore, in 2013 and 2017, respectively, both inelectrical engineering.He is currently a Scientist in Singapore Institute ofManufacturing Technology (SIMTech), Agency forScience, Technology and Research (A*STAR). His current research interestsinclude intelligent mechatronic and robotic systems, etc.
Jun Ma (S’15-M’18) received the B.Eng. degreewith First Class Honours in electrical and elec-tronic engineering from the Nanyang Technologi-cal University, Singapore, in 2014, and the Ph.D.degree in electrical and computer engineering fromthe National University of Singapore, Singapore, in2018.From 2018 to 2019, he was a Research Fellowwith the Department of Electrical and ComputerEngineering, National University of Singapore, Sin-gapore. In 2019, he was a Research Associate withthe Department of Electronic and Electrical Engineering, University CollegeLondon, London, U.K. He is currently a Visiting Scholar with the Departmentof Mechanical Engineering, University of California, Berkeley, Berkeley,CA, USA. His research interests include control and optimization, precisionmechatronics, robotics, and medical technology.He was a recipient of the Singapore Commonwealth Fellowship in Inno-vation.
Tat Joo Teo (M’08) received the B. Eng. degree inmechatronics engineering from Queensland Univer-sity of Technology, Australia, in 2003 and the Ph.D.degree from Nanyang Technological University, Sin-gapore, in 2009.He was a research scientist with the SingaporeInstitute of Manufacturing Technology, Singapore,from 2009 to 2018. He was a visiting scientistwith Massachusetts Institute of Technology, USA, in2016, and also served on the engineering faculty atthe National University of Singapore and NewcastleUniversity in Singapore. His research interest is to explore the fundamentalsof Newtonian mechanics, solid mechanics, kinematics, and electromagnetismto develop high precision mechatronics or robotic systems for micro-/nano-scale manipulation and bio-medical applications.Dr. Teo has published over 50 peer-reviewed articles and has 4 patentsgranted. In 2013, he received the IECON Best Paper Award in the theoryand servo design category. In 2014, he became the first Singaporean to winthe R&D 100 Award, which is the most prestigious international award fortechnologically-significant products.
Chek Sing Teo (S’04-M’08) completed his Ph.D.degree at the National University of Singapore in2008, under the Agency for Science Technologyand Research (A-STAR) Scholarship Scheme, work-ing on ”Accuracy Enhancement for High PrecisionGantry Stage”. His research interests are in the ap-plication of advanced control techniques to precisionmechatronic system and instrumentation; to enhanceperformance in motion control and measurement.His current work includes using mechatronics stiff-ness to reduce jerk reaction in high speed motionstage and sensor placement for adaptronics. He is currently working inthe Singapore Institute of Manufacturing Technology (SIMTech) leading thePrecision Mechatronics Team within the Mechatronics Group, as well as theco-Director of the SIMTech-NUS Precision Motion System joint lab. asayoshi Tomizuka (M’86-SM’95-F’97-LF’17)received the B.S. the and M.S. degrees in mechanicalengineering from the Keio University, Tokyo, Japan,and the Ph.D. degree in mechanical engineeringfrom the Massachusetts Institute of Technology inFebruary 1974.In 1974, he joined the faculty of the Department ofMechanical Engineering at the University of Califor-nia at Berkeley, where he currently holds the Cheryland John Neerhout, Jr., Distinguished ProfessorshipChair. His current research interests are optimal andadaptive control, digital control, motion control, and their applications torobotics and vehicles.He served as Program Director of the Dynamic Systems and ControlProgram of the Civil and Mechanical Systems Division of NSF (2002-2004). He served as Technical Editor of the ASME Journal of DynamicSystems, Measurement and Control, J-DSMC (1988-93) and Editor-in-Chiefof the IEEE/ASME Transactions on Mechatronics (1997-99). He is a LifeFellow of the ASME and IEEE, and a Fellow of International Federationof Automatic Control (IFAC) and the Society of Manufacturing Engineers.He is the recipient of the ASME/DSCD Outstanding Investigator Award(1996), the Charles Russ Richards Memorial Award (ASME, 1997), the RufusOldenburger Medal (ASME, 2002), the John R. Ragazzini Award (2006), andthe Richard Bellman Control Heritage Award (2018).