A Novel Optimization Framework for High Dynamics Point-to-Point Direct Drive Motion Control System With a New Type of Surrogate Model

A new optimization framework for a high-dynamic point-to-point direct drive motion control system (HDPDMS) is proposed. The conventional system optimization approach considers all design parameters simultaneously, resulting in a high-dimensional search space and extensive computation. In contrast, the proposed framework uses a new DDM surrogate model that establishes a correlation between the key DDM characteristic parameters to decouple the whole optimization process. It begins with a system-level optimization to identify suitable driver types, motion profile design parameters, and characteristic parameters of the direct drive motors (DDMs) by the new surrogate model. Bayesian optimization then determines the DDM design parameters corresponding to the identified characteristic parameters. Once the DDM surrogate model is built, the proposed framework achieved the desired HDPDMS design in just 1 hour, saving 98.6% of computation time compared to the traditional approach. Additionally, multi-objective optimization and Gaussian process regression prediction intervals were employed to obtain a suitable training dataset and input range for the surrogate model, resulting in a 99.8% reduction in computation resources compared to the traditional DDM surrogate model. Through completing three unique motion task optimizations and creating a prototype, the optimization framework was proven effective, demonstrating the potential of this novel method.


I. INTRODUCTION
High-dynamic point-to-point direct drive motion control systems (HDPDMS) have wide industrial applications such as high-speed pick-and-place drives, die bonders, and integrated circuit sorting machines (Fig. 1). An HDPDMS rotates at a specific angle with fast acceleration and deceleration in The associate editor coordinating the review of this manuscript and approving it for publication was Philip Pong . milliseconds and then rotates to the next position repeatedly. In the production process, a system may work continuously for more than 20 hours a day with a very high unit per hour (UPH) [1]. Consequently, optimizing the motion control system for higher efficiency is important.
In a typical direct drive motion control system, the loadto-motor inertia ratio is often greater than 20:1. The moment inertia of a direct drive motor (DDM), therefore, does not significantly impact the design requirements for continuous balance between overall cost and system reliability necessitates multi-objective optimization, further compounding the computational demands. Moreover, during the initial design phase, the motion task requires additional adjustments, further contributing to the already substantial computational workload.
The multi-level optimization framework [2], [3] divides the system into two levels: the motor level and the control level, as shown in Fig. 3. The optimization process involves two levels, starting with optimization for steady-state performances like torque density and efficiency at the motor level, and obtaining characteristic parameters such as resistance, inductance, and flux-linkage for control-level optimization to improve dynamic performances like current rising time and overshoot. This method reduces the computation burden at each level but only works well if the control design parameters do not affect the steady performance parameters of the motor. It is suitable for traction motor systems, for example, in electric vehicles, but proves challenging for HDPDMS as the design parameters of the motion profile and driver influence design requirements of DDM, such as continuous torque, peak torque, and efficiency.
Numerous analytical magnetic field models have been proposed to reduce the computation burden [8], offering the advantage of high generality [6], [18]. For a fixed motor topology, a wide range of design parameters can be changed to obtain different characteristic parameters of the motor. However, these models are limited in their ability to consider the magnetic saturation influence in soft magnetic materials. Analytical or semi-analytical models have a tradeoff between precision and speed to take magnetic saturation into account [7], [19]. In the case of an HDPDMS, careful consideration of the saturation effect is required for calculating the peak torque, a task that cannot be easily accomplished with fast analytical or semi-analytical models.
The surrogate model is a predictive model trained on information from the sampling points in a training dataset [9], [14], [20]. Once the training is complete, the surrogate model can make accurate predictions quickly and can handle different motion task adjustments to find precise optimization results. However, the 'curse of dimensionality' [12] is challenging for surrogate models. The size of the training dataset increases exponentially with the dimension of the design parameters, making it impractical for the high number of the HDPDMS's design parameters. Therefore, reducing dimensionality is essential for surrogate models. One way is to select critical design parameters based on expert knowledge or perform a sensitivity analysis [21] to identify important design parameters for the surrogate models. However, sensitivity analysis is only effective in a local region with limited ranges for design parameters. Another approach is the sequential optimization method (SOM) [22], [23], which focuses computation resources on the more promising regions with a high probability of containing the optimal solution, similar to Bayesian optimization. However, its effectiveness is limited to a local region, and if the motion task changes, the training dataset needs to be recalculated for the SOM.
In summary, the methods discussed previously have limitations that make them unsuitable for quick and accurate optimization of HDPDMS for different motion tasks with limited computational resources.
After analyzing the traditional optimization framework (Fig. 2), we find that the system-level optimization only requires characteristic parameters of the DDM and driver. The driver's characteristic parameters are fixed once the driver type is selected, leaving the optimization of the DDM's characteristic parameters -such as the phase resistance R s , phase synchronous inductance L s , flux-linkage generated by the permanent-magnet (PM) ψ f , and DDM inertia J DDM to be performed. If we can establish a surrogate model that describes the relationships between these characteristic parameters, such as ψ f = S(R s , L s , J DDM ), we can perform a system-level optimization to determine the required characteristic parameters of the DDM. Following this, a component-level optimization can be conducted to identify the corresponding design parameters of the DDM that meet the required characteristic parameters. These fundamental ideas of the new optimization framework are illustrated in Fig. 4. Since the dimension of characteristic parameters is typically smaller than that of design parameters, the surrogate model relieves the 'curse of dimensionality' problem and enables quick and accurate system-level optimizations for different motion tasks. As the new surrogate model takes the DDM characteristic parameters as inputs, rather than the design parameters as in traditional surrogate models, the primary challenges of this approach are constructing . The fundamental concepts of the new optimization framework. First, we construct a surrogate model that describes the correlation among the characteristic parameters of DDM. Using this surrogate model, we can perform system-level optimization to determine the required characteristic parameters of DDM. Finally, we conduct a component-level optimization to identify the corresponding design parameters of the DDM that meet the required characteristic parameters. an appropriate surrogate model for the DDM characteristic parameters, effectively utilizing the surrogate model for system-level optimization, and swiftly retrieving the design parameters of the DDM.
In this article, we define the HDPDMS optimization problem in section II. In section III, we introduce the overview of the new optimization framework. A detailed process of the new optimization is given in section IV. In section V, we provide a comparison between the traditional method and the new method. The prototype and experiment are shown in section VI. Finally, the conclusions are given in section VII.

II. THE DEFINITION OF THE HDPDMS OPTIMIZATION A. POINT-TO-POINT MOTION TASK AND CONTROL
This article focuses on the point-to-point motion task, which involves rotating a load from position A to position B within a specified time using a DDM. Specifically, we aim to rotate a load with a moment of inertia J load by a specific angle θ m within a motion time of t m . After reaching the target position, the machine waits for a dwell time of t d before moving on to the next position. An S-curve [24] is used to generate the trajectory with two design parameters: the jerk time ratio α s and the constant speed time ratio β s , which are shown in Fig. 5. The dynamics model of the S-curve is described by where ω m is the rotational speed, α m is the rotational acceleration, and j m is the rotational jerk. In our case, the range of α s is set from 0.01 to 0.25, and the range of β s is set from 0.01 to 0.96. They must also satisfy the inequality 4α s + β s ≤ 1.
The HDPDMS control diagram [25] is shown in Fig. 6. The motion control system comprises three feedback loops: a position loop, a velocity loop, and a current loop. The corresponding loop receives the motion profile generator's VOLUME 11, 2023 60061 Authorized licensed use limited to the terms of the applicable license agreement with IEEE. Restrictions apply. position, velocity, and acceleration feedforward commands. The position and velocity loops use classical proportional (P) / proportional-integral (PI) controllers, and several lowpass filters are used to reduce high-frequency noise in the velocity loop. The control diagram shown in Fig. 6 uses the I d = 0 control strategy, so the d-axis current is set to zero and ignored. A current controller and space vector pulse width modulation (SVPWM) is employed to drive the motor. Since the motion task is precisely defined and disturbances are minimal, feedforward control dominates this motion control, with feedback control mainly serving to correct unexpected disturbances. In the HDPDMS optimization, we concentrate on the design parameters of the feedforward control, specifically the jerk time ratio α s and constant speed time ratio β s in the S-curve motion profile generator.
The dynamics model of the DDM [25] and load is described by the DDM characteristic parameters, like the phase resistance R s , phase synchronous inductance L s , fluxlinkage generated by the PM ψ f , and the inertia of the DDM where the u d , u q , i d , and i q are the voltages and currents in the d-axis and q-axis.

B. DDM AND DRIVER
In addition to the motion task and motion control, we must design the DDM and select the proper driver. For the DDM, we use a surface-mounted PM synchronous motor (SPMSM) because it has a high ratio between peak torque and inertia.   Therefore, it suits well for high dynamics point-to-point motion. Straight teeth are used for easier manufacture and a higher slot-filling factor. The topology and all design parameters of the frameless motor are shown in Fig. 7. The optimization design parameters are listed in Table 1, and the fixed design parameters are shown in Table 2. The design parameters of the housing, bearings, and encoder of the DDM are derived from the design parameters of the frameless motor. The whole structure of the DDM is shown in Fig. 8. For the magnetic field models of the DDM, except the FEM model, a subdomain model is used to reduce the computation resource of optimization [26]. The subdomain model is a quick and precise semi-analytical model if the magnetic saturation effect is negligible [18]. A thermal network model is used to calculate the temperature of the DDM [27], [28].
We select the driver from a commercial driver library. The library gives the bus voltage V bus , the output continuous current I c driver , the output peak current I p driver , and the cost of the driver C driver (Table 3).

C. HDPDMS OPTIMIZATION OBJECTIVES AND CONSTRAINTS
A cost indicator and a reliability indicator are taken as the optimization objectives of the HDPDMS optimization. The cost indicator is used to measure the approximate cost of the HDPDMS and operation cost during the operation. It is defined by where C driver and C DDM are the cost of the driver and motor, respectively. C operation is the electricity cost of the motion system during the operation time.
Another reliability indicator R is used to measure the reliability of the motion system by the minimum margin for the bus voltage, continuous and peak current of the driver, and the temperature of the winding. We use a negative reliability indicator R. As the optimization involves a minimization, where , U bus is the required bus voltage amplitude of the motion system, , I c is the required continuous current amplitude of the motion system, , I p is the required peak current amplitude of the motion system, and , T winding is the average temperature of the winding and T max is the maximum tolerable temperature of the winding. In our case, T max is 100℃. All of these margins must have negative values to ensure the safe operation of the system, and demagnetization of the PMs must be considered for long-term reliability. These parameters are referred to as the constraints of HDPDMS and are discussed in more detail in the following section. IV-A2.

III. OVERVIEW OF THE NEW OPTIMIZATION FRAMEWORK
A new optimization framework is provided (Fig. 4). The novel optimization framework is based on the DDM characteristic parameters surrogate models. In this section, we address three crucial issues related to this new optimization framework: building the surrogate model of DDM characteristic parameters, utilizing the surrogate model for system-level optimization, and swiftly retrieving the design parameters of the DDM.

A. THE CONSTRUCTION OF THE NEW SURROGATE MODEL
First, we need to select the DDM characteristic parameters in the HDPDMS optimization. For example, the phase resistance R s , phase synchronous inductance L s , flux-linkage generated by the PM ψ f , and the inertia of the DDM J DDM are selected as the DDM characteristic parameters. Second, we need to collect the training dataset for the surrogate models. However, obtaining a training dataset for the surrogate model of DDM characteristic parameters is not as straightforward as it is for the traditional surrogate model, which takes design parameters as inputs. The reason is that the characteristic parameters of the DDM are interdependent, and not all combinations of them exist in reality. For instance, it is impossible to find a DDM with a high value of fluxlinkage generated by the PM ψ f , along with a low value of inertia of DDM J DDM . Therefore, direct sampling of characteristic parameters is not possible.
To address this issue, we set DDM characteristic parameters as optimization objectives in a multi-objective optimization of DDM. The optimization results, located on the Pareto front, comprise the DDM optimization dataset containing physical correlation information. We use this dataset as a training set to build a surrogate model that takes R s , L s , and J DDM as inputs and produces ψ f as a single output. Since the Pareto front exhibits unique properties, the surrogate model ψ f = S(R s , L s , J DDM ) is an injective function that can be constructed using Gaussian process regression (GPR) [11] to map the inputs to the output. VOLUME 11, 2023 The GPR is a non-parametric machine learning algorithm [11]. The advantage of the GPR is that it can not only get the mean value of the prediction but also the variance of the prediction [12].
For a typical regression problem, we have a training dataset (X , y) with m sampling points, where X is an m×n matrix that includes the m inputs with n-dimensional features, and where y is an m-dimensional vector which includes the m outputs with a single response. The GPR aims to return the prediction value f * corresponding to a new input X * based on the training dataset. We assume that y and f * satisfy the multi-variables Gaussian distribution in: where K (X , X ) is the covariance function and the variance σ 2 n of the Gaussian noise is used to model the random noise in the data. In our case, we use the Matern 5/2 covariance function [11]. Every element k ij in K (X , X ) is defined in: where r = x i − x j is the norm between the x i and x j , and σ f and σ l are hyper-parameters of the GPR. The mean value and variance of the prediction equations [11] are:

B. THE UTILIZATION OF THE NEW SURROGATE MODEL IN THE OPTIMIZATION OF THE SYSTEM-LEVEL
Using the mean prediction from (7), we can predict the fluxlinkage generated by the PM ψ f depending on the inputs: the phase resistance R s , phase synchronous inductance L s , and the inertia of the DDM J DDM . They can be designated as the optimization variables in the system-level optimization. However, we can not use the arbitrary value of characteristic parameters R s , L s , and J DDM . Because the inputs of the surrogate model have an acceptable range. In the case of a traditional surrogate model with design parameters as inputs, determining the acceptable ranges is straightforward, as the design parameters' search space is a box space with minimum and maximum values. However, the input space of the new surrogate model is a projection of the Pareto front onto the characteristic parameters' space. For example, in Fig. 9, a Pareto front of the peak torque T p , copper loss P c , and inertia of the DDM J DDM is shown in Fig. 9 (a). This Pareto front is used to build a surrogate model T p = S(P c , J DDM ) to predict the peak torque T p by the copper loss P c and inertia of DDM J DDM . The input space of this surrogate FIGURE 9. Input space of the surrogate model whose inputs are characteristic parameters. The Pareto front between the peak torque, copper loss, and inertia of DDM is shown on the left. The corresponding input space is a projection from the Pareto front to the copper loss and inertia of DDM space.
model is determined by the projection from this Pareto front to the P c and J DDM space. In Fig. 9 (b), the input space of this surrogate model is irregular. Therefore, we need to find the boundary of input space. The prediction variance of the GPR by (8) is used to find the boundary of the input space. Fig. 10 is used as a simple illustration that provides better visualization and an intuitive explanation. This function shape is similar to the Pareto front of the torque and copper loss in the SPMSM. A random noise ε is added to model real data noise. A 95% prediction interval is defined by the f * ±1.96σ m . From Fig. 10, we observe that the 95% prediction intervals at the two input boundaries are larger than the middle range of the input. The range of the 95% prediction intervals P is computed in Fig. 11. Statistical analysis is performed on P for the training dataset. Its mean value µ and standard deviation σ are obtained. Threshold 1 and threshold 2 are set as µ + σ and µ + 2σ to determine the input space's boundary. In this research, µ + σ is used as the threshold for better robustness of optimization results.
This method works well when the sampling points on the Pareto front are uniform. If the sampling points on the Pareto is nonuniform, like the situation in Fig. 12, this method fails in some place, as shown in Fig. 13. To avoid this problem,   some methods are used in section IV-A to get the uniform Pareto front.

C. RECOVERY OF THE DESIGN PARAMETERS OF DDM
Finally, we need to recover the design parameters of the corresponding DDM with the best characteristic parameters in the system-level optimization. At the system level, we use the DDM characteristic parameters to directly optimize with the new type of surrogate model. The surrogate model only exhibits a correlation between the characteristic parameters of the DDM and does not incorporate any FIGURE 13. The range of the 95% prediction interval ( P) for the uniform sampling. Some P values are bigger than threshold 1 or 2, even in places where the input is away from the two sides. Thresholds 1 and 2 cannot determine the boundary of the input of the GPR. information regarding the corresponding design parameters. Such information is stored in the DDM optimization dataset. With the assistance of the DDM optimization dataset, we use Bayesian optimization [12] to recover the design parameters of the corresponding DDM. The DDM optimization dataset provides the initial samples to build the surrogate model to relieve the cold-start problem of Bayesian optimization. Bayesian optimization makes full use of information from the results during the optimization process to guide the optimization search, which reduces the computation resource.

D. THE NEW OPTIMIZATION FRAMEWORK OF THE HDPDMS
The new optimization framework is shown in Fig. 14 based on the above ideas.
Firstly, we perform multi-objective optimization to obtain the Pareto front of the DDM characteristic parameters. Then, we uniformly sample the optimization results on the Pareto front and store them as the DDM optimization dataset.
Secondly, we build surrogate models of DDM characteristic parameters using the DDM optimization dataset. It is important to note that this surrogate model is used to build the mapping between different characteristic parameters of the DDM and not between the design and characteristic parameters of the DDM as in traditional surrogate models referenced in [3], [12], and [20].
Thirdly, we perform system-level optimization using the DDM surrogate models and dynamics model of HDPDMS. With the assistance of the DDM surrogate model, this system optimization is quick and precise, and we can easily adjust the driver and motion task to check the optimization results.
Finally, based on the optimization results obtained above, we select a desired design to recover the design parameters of the DDM. Using the DDM optimization dataset and Bayesian optimization, we can quickly recover the design parameters of the DDM and perform the final validation check of the entire system design.
The next section will provide a detailed and specific method for performing the above optimization process.

IV. OPTIMIZATION PROCESS OF THE NEW FRAMEWORK A. COMPONENT-LEVEL OPTIMIZATION AND SURROGATE MODEL OF THE DDM 1) SELECTION OF THE CHARACTERISTIC PARAMETERS OF THE DDM
First, we select the characteristic parameters of the DDM as the optimization objectives. In (2), four characteristic parameters, phase resistance R s , phase synchronous inductance L s , flux-linkage generated by the PM ψ f , and the inertia of the DDM J DDM , are used to describe the dynamics model of the DDM. The flux-linkage generated by the PM ψ f should be a function of the current in the q-axis for the SPMSM to consider the magnetic saturation. Hence, the peak torque T p , peak current I p motor , continuous torque T c , and continuous current I c motor are selected as the characteristic parameters to replace the ψ f . In this article, we set the peak current I p motor to three times the continuous current I c motor . The corresponding torques are peak torque T p and continuous torque T c . The cost of the DDM C DDM and copper loss P c are also needed to compute the cost and reliability indicators in the systemlevel optimization. Therefore they are added to the list of characteristic parameters.
These nine characteristic parameters have some redundant information. The continuous current I c motor = P c 3R s is related to P c and R s . The peak current I p motor is three times higher than I c motor . To reduce the computational burden of multi-objective optimization, we have selected the most important characteristic parameter among the rest of the seven parameters. Since point-to-point motion control systems are primarily limited by their peak torque and not their continuous torque, we have chosen peak torque T p as the key characteristic parameter.
In an SPMSM, the electromagnetic dynamics are determined by the synchronous phase inductance L s and the phase resistance R s of the motor's stator windings. The synchronous phase inductance L s plays a more crucial role than the phase resistance R s in determining the response speed of the electromagnetic dynamics in an SPMSM. The minimum jerk time t jerk min = α min T m is used to evaluate the minimum time from the start to the maximum torque T max in the motion control. A smaller minimum jerk time t jerk min corresponds to a quicker dynamic response ability of the motor. For the SPMSM, the I d = 0 current control strategy is used. The speed of the motor is relatively low from the start to the t min jerk and the EMF can be neglected during the current rising time. Based on the above analysis, the q-axis voltage in (2) where U q is the drivers's maximum voltage on the q-axis. Based on this equation, the minimum jerk time t min jerk can be derived from when the maximum torque T max in the motion control is reached, where the K t is the motor's torque constant.
The maximum torque T max should be less than the peak torque T p , and the voltage drop on the resistance should be less than the maximum voltage on the q-axis U q , if the proper driver is used. Therefore, We use the Taylor series for the function f (x) = (1 + x) λ centered at x = 0, where λ ∈ C and |x| < 1: to approximate the minimum jerk time by (11). By selecting the first two series, we obtain Based on this equation, the t min jerk does not depend on the phase resistance R s . Therefore, the synchronous phase inductance L s is the more important characteristic parameter that reflects the response speed of the electromagnetic dynamics in an SPMSM.
In conclusion, the five important characteristic parameters T p , L s , J DDM , P c and C DDM are selected.

2) MULTI-OBJECTIVES OPTIMIZATION OF DDM
Based on the above analysis, characteristic parameters T p , L s , J DDM , P c , and C DDM are selected as the optimization objectives. The resistance and PMs are subject to temperature effects, with peak torque T p and copper loss P c in particular being sensitive to temperature. During DDM optimization, we aim to maintain the winding temperature at around 75℃. As the torque, the loss, and the resistance are all dependent on temperature, it is essential to maintain a consistent temperature for the DDM optimization dataset. Based on the dataset, this allows for the calculation of designs with varying temperatures during system-level optimization, assuming that the motor's equivalent thermal resistance remains constant when the winding temperature is between 50℃ and 100℃. To verify this assumption, we made an experiment to measure the equivalent thermal resistance. Figure 15 shows the relationship between copper loss and the temperature difference between the winding and the environment. For this experiment, the ambient temperature of the experiment was 28.3℃. The slope of the curve is the equivalent thermal resistance of this motor. It is close to a constant at different temperatures.
Based on the above analysis, the constraint for the winding temperature range has been established. The lower and upper constraints of the winding temperature are set at 70℃ and 80℃, respectively. The constant equivalent thermal resistance assumption is used to correct the T p and P c to keep the winding temperature equal to 75℃. The k PM is used to do the thermal correction for the PMs in where T w , T PM , and T a are the temperature of the winding, PMs, and environment, respectively. The variable α PM is the thermal coefficient of the remanence in the PM which is equal to 0.11% in our case. The variable k current is used to do the thermal correction for the current by where I 75℃ and I T w are the current when winding temperatures are equal to 75℃ and T w , respectively. The phase resistances R T w and R 75℃ are obtained from R T = (1+α(T − T a ))R T a at the different winding temperatures, where α = 0.00385 is the temperature coefficient of copper resistivity when the ambient temperature T a is 25℃. Equation (13) is derived from based on the constant equivalent thermal resistance assumption.
In our case, we assume that the peak torque T p is proportional to the current and remanence of the PM when the current and temperature is changed in a small range. Through this assumption, the peak torque T p is correct by the k PM and k current : where T o p is the original value of the peak torque when the motor is at ambient temperature.
In the same way, the copper loss P c can be derived from where P o c is the original value of the copper loss when the temperature of the motor is at ambient temperature.
The demagnetization is set as a constraint to ensure the reliability of the DDM. A current four times the value of the continuous current is used to check the demagnetization of the PM. For the cost consideration, the PM volume ratio upper α PM = V max PM V Motor is set as the constraint, where V max PM is the maximum volume of the PM and V Motor is the volume of the motor. In our case, the α PM is equal to 0.074.
The cost of the motor encompasses a variety of factors, such as material costs, labor expenses, manufacturing expenses, and auxiliary fixtures. To simplify the computation, we assume that the cost of the motor is directly proportional to its volume, considering the PM volume ratio constraints. In this study, we adopt a price of 920 RMB per liter as the unit price for the motor volume. Note that this cost function can vary based on the manufacturing company and the market, but it does not affect the primary conclusions of this article.
The phase resistance R s and synchronous phase inductance L s are related to the total turns of winding N w . In the same motor, the relationships of the phase resistance and synchronous phase inductance with different winding turns are and where R so and L so are the original resistance and inductance with the original total turns of winding N o .
In the DDM optimization, we fix the total turns of winding N w as N o and compute the phase resistance and inductance as R so and L so . Then, we can use (16) and (17) to compute phase resistance R s and synchronous phase inductance L s , when we change the total turns of winding N w in the systemlevel optimization. Therefore, N w is removed from the design parameters in the DDM optimization. The remaining 9 design parameters in Table 1 are used for DDM optimization.
The bi-criterion evolutionary optimization algorithm (BCE-IBEA) [29] is used. It combines the indicator-based and non-dominated optimization algorithm advantages to FIGURE 16. DDM optimization framework. The pre-optimization with the quick subdomain is performed first. The next optimization with FEM inherits the pre-optimal results as the initial design. The BCE-IBEA [29], NRBF, and robust filters [26] are used to get uniform sampling on the 5D Pareto front to generate the DDM optimization dataset.
give a better diversity of the results in the Pareto front. A preoptimization uses a subdomain model for the magnetic field computation to get the pre-optimal results. They are further optimized using the FEM for magnetic field computation. This method reduces the computation resource by about 70% [26]. The population is 210, and the number of generations is 400 and 100 for the optimization using the subdomain model and FEM, respectively. Most of the time is spent on the 100 generations optimization with the FEM model. Its computation resource is 210 × 100 = 21 000 cases. A single computation case involves computing the magnetic and thermal models for a single DDM design. The niche-radius-based filter (NSBF) and robust filter [26] are used to get uniform sampling on the Pareto front based on the optimization results from the different generations. Finally, the DDM optimization dataset is obtained. It includes the design parameters and the corresponding optimization objectives T p , L s , J D , P c , and C DDR , such characteristic parameters. The framework of the DDM optimization is shown in Fig. 16.

3) SURROGATE MODELS OF CHARACTERISTIC PARAMETERS OF THE DDM
After the muti-objectives optimization of the DDM, the surrogate models of the characteristic parameters are built based on the uniform sampling on the Pareto front. The theory of GPR and simple 2D Pareto front examples were shown in sections III-A and III-B. We expand these ideas for the important DDM characteristic parameters to get the 5D Pareto front and corresponding surrogate models. The framework of the surrogate models of DDM characteristic parameters is shown in Fig. 17.
The peak torque T p is selected as the output of the surrogate model, and the remaining four characteristic parameters are set as the inputs of the surrogate model in The phase resistance R s and continuous torque T c surrogate models are also built for system-level optimization. Firstly, we compute the corresponding variables R s and T c for the optimal designs uniformly spread on the Pareto front. We apply thermal corrections to these variables, just as we do for P c and T p . Finally, we use the five important characteristic parameters as inputs and set R s and T c as outputs to build surrogate models using GPR: The prediction error distributions of the surrogate models for the variables T p , R s , and T c are shown in Fig. 18. The precision of the T p and T c surrogate models is high. The average errors are about 0.8% and 0.4%, respectively. The precision of the R s surrogate model is lower. Its average error of it is about 3.5%. Nevertheless, it does not play a critical role in system-level optimization.

B. SYSTEM-LEVEL OPTIMIZATION FOR HDPDMS
After the completion of surrogate models for the DDM characteristic parameters, the optimization of HDPDMS at the system level is performed. The framework of the system-level optimization for the HDPDMS is shown in Fig. 19.

1) HDPDMS OPTIMIZATION PROCESS
The optimization variables of the HDPDMS include synchronous phase inductance L s , the inertia of DDM J D , copper loss P c for a winding at 75℃, the volume of the DDM C DDR , the jerk time ratio α, the constant speed time ratio β, and the winding turns of the DDM N w . Using the surrogate models of DDM characteristic parameters, the peak torque T p , continuous torque T c , and phase resistance R s are computed in less than 1 ms. To begin, we use the dynamics model of HDPDMS to calculate both the current and voltage curves. We derive the continuous current from the current curve to determine the loss. Next, by assuming a constant thermal resistance, we determine the winding temperature and refine the loss through multiple iterations. Ultimately, the HDPDMS is optimized by considering both its cost indicator and reliability indicator defined in (3) and (4). The price of the different drivers is shown in Table 3. The electricity cost of the motion system C operation is based on the following assumption. The motion system operates 20 hours a day and works for three years with 1 RMB / kWh.
The current and voltage margins of the driver are set as constraints to keep the feasibility of the motion system. The maximum winding temperature is set to 100℃ to keep the safety of the DDM. A winding lower temperature of 50℃ is also set as the constraint to reduce the search space. This also ensures that the assumption that equivalent thermal resistance is constant when winding is from 50℃ to 100℃ can be used. The 95% prediction interval of the T p surrogate model should be less than the threshold defined in section III-B. . The optimization variables in the HDPDMS optimization include the DDM characteristic parameters, motion curve design parameters, and winding turns. Eight independent optimization processes are computed in parallel to decrease the effect of the randomness of the evolutionary optimization algorithm NSGA-II. If the optimization results do not satisfy the design requirements, the next optimization inherits the optimal results and performs further optimization searches.
We established it as a constraint to ensure the existence of the DDM with these characteristic parameters in reality.
The NSGA-II [30] is selected as the optimization algorithm. The population of every generation is 30, and the number of generations is 150. Eight independent optimization processes are computed parallelly to decrease the effect of the randomness of the evolutionary optimization algorithm. Then, all the datasets from the optimization processes are combined and sampled uniformly on the Pareto front. Finally, the motion designer can check the optimization results. If the results are unsatisfactory, the next optimization process can use these optimal results as the initial design to do a further optimization search until the optimization results satisfy the requirement.

2) HDPDMS OPTIMIZATION PARETO FRONTS
With the assistance of the surrogate models for the characteristic parameters of the DDM, the system-level optimization of the HDPDMS can be done quickly. Approximately five iterations of the optimization process above are required to obtain the convergence Pareto front for a particular motion task and driver within 15 minutes. All computation was made on an AMD 3700X with eight cores which runs at 4.2 GHz. Three different motion tasks and three different drivers are selected to do the optimization. The three motion tasks and three drivers are shown in Tables 3 and 4. Motion task 1 is typical for the HDPDMS commonly used in industry. Motion task 2 is a relatively quick motion with a light load and long dwell time. Motion task 3 is a relatively low motion with a heavy load and short dwell time.
The Pareto fronts of the different motion tasks with the different drivers are shown in Figs 20, 22, and 21.

FIGURE 20.
The optimization results for motion task 1 are presented for three drivers, where the driver with a higher volume is associated with a greater reliability margin and cost. Driver A provides the lowest cost indicator for reliability margins of less than 10%. Driver B offers the best design solutions for reliability margins between 10% and 38%. However, if the reliability margin needs to be greater than 40%, driver C is the preferred option.

FIGURE 21.
The optimization results for motion task 2 using three different drivers show that the volume of drivers significantly impacts the performance of relatively quick motions. Notably, the Pareto front varies across the different driver types. Driver A is the recommended option for instances where the reliability margin is less than 12%. In cases where the reliability margin falls between 12% to 35%, driver B yields the best candidate designs. However, driver C should be selected for situations where a reliability margin of more than 40% is required.
Based on the results of the above experiments, we find that the proposed new system optimization gets good Pareto fronts for different motion tasks and drivers. Every motion task needed approximately 15 minutes to get the optimization Pareto front. The motion system designers can select the desired HDPDMS design from the Pareto front based on their expert experience.

C. DESIGN PARAMETERS RECOVERY
In the system-level optimization, a desired HDPDMS design is selected in the Pareto front. The design parameters of the DDM are needed to recover for this HDPDMS design. In this FIGURE 22. The motion task 3 optimization results for three different drivers demonstrate that the relatively low motion is not sensitive to the volume of drivers. All the drivers can provide suitable designs when the reliability margin is within the range of 5% to 40%. However, driver A is preferred for achieving the smallest cost indicator designs.
step, we recover the design parameters of the DDM with the assistance of the DDM optimization datasets and Bayesian optimization. The framework of the recovery process is shown in Figs 23 and 24.

1) RECOVERY OF THE DESIGN PARAMETERS WITH THE ASSISTANCE OF THE DDM OPTIMIZATION DATASET
The selected DDM is called the expected design. The DDM characteristic parameters in the expected design can be set as a vector V expected = [T p , L s , J DDM , P c , C DDR ]. The characteristic parameters of every DDM in the DDM optimization dataset can be set as a vector V n DDM , where n is the serial number of DDM in the dataset.
The distance d = | | can be computed. Then, the DDM with the minimum d can be found, called the recovered design. This recovery method is called the 'closest point'. Finally, the whole motion system can be validated with the recovered design parameters to get the corresponding cost C recovered and reliability indicator R recovered . In this step, we do not use the constant equivalent thermal resistance assumption. The validation is performed based on the design parameters of the recovered design by the iterative computation with magnetic, thermal, and motion control dynamics models. The difference in the temperature of the winding between this computation and the last computation is given by where T w is the temperature of the winding and T o w is the temperature of the winding in the last computation. Finally, the optimization objectives of the recovery design are output until the T is less than 0.1%.
The optimization objectives of the expected design and recovery design are set as a vector | is less than 0.02, we output this recovered design. Otherwise, we use Bayesian optimization to do the refined optimization.

2) REFINE-OPTIMIZATION WITH BAYESIAN OPTIMIZATION
Based on the recovered design from the above process, we can do the refined optimization to reduce the distance D. Design parameters recovery framework. To recover the design parameters, we use the DDM optimization dataset to identify the DDM with the characteristic parameters closest to the target design. The validity of the recovered design parameters is assessed through iterative computations involving magnetic, thermal, and motion control dynamics models. If the distance between the recovered and target design parameters, denoted as D, is less than 2%, the recovered design parameters are considered acceptable and output. However, if D exceeds 2%, we use Bayesian optimization to refine the recovered design parameters.

FIGURE 24.
To train the local surrogate model of the distance D, we use the closest designs with the expected designs from the DDM optimization as the training dataset. The expected improvement infill criteria [12] is employed in combination with the genetic algorithm (GA) [31] to determine new samples and update the surrogate model iteratively until the maximum number of samples is reached.
The Bayesian optimization is selected to do the refine optimization. First, we get the closest 30 designs by sorting the distance d in the DDM optimization dataset. The distance D of these 30 designs is computed. The GPR is selected to build the surrogate model between the design parameters of DDM and distance D. The expected improvement infill criteria [12] are used to determine the next sampling. The genetic algorithm (GA) [31] is selected, and 100 generations with eight populations are used to optimize to get the maximum expected improvement. Then, the new sampling is added to the training dataset, and update the surrogate model of distance D until the number of the new sampling reaches 30. We recovered the design parameters of DDM within about 15 minutes.

3) RECOVERED DESIGNS RESULTS
As an example, the Pareto front of motion task 1 and driver C are selected to recover the design parameters. We sampled eight points on the original Pareto front and recovered their design parameters. The result is shown in Fig. 25. From the figure, we can find that the Pareto front with the closest point method is close to the original Pareto front. All of the Pareto front with Bayesian optimization are better than the Pareto front with the closest point method and are closer to the original Pareto front.

A. COMPARISON WITH THE TRADITIONAL OPTIMIZATION FRAMEWORK
A comparison result between the new and traditional optimization framework is performed. The traditional HDPDMS optimization framework is shown in Fig. 26.
The traditional optimization framework used the same range of design parameters as those of the new framework, as seen in Table 1. The optimization objectives and constraints were also the same as those outlined in section IV. VOLUME 11, 2023   Motion task A was selected, and three optimizations were performed using different drivers. For a fair comparison, the traditional optimization framework used the same computation resource of 240 generations and 30 populations with the optimization algorithm NSGA-II. This computation resource was equivalent to 21 600 cases (calculated as 240 × 30 × 3), comparable to the 21 000 cases needed by the new optimization framework to obtain the DDM optimization dataset. The optimization results of the new and traditional optimization frameworks are depicted in Fig. 27. The new optimization framework showed better convergence and a wider coverage range compared to the traditional framework.
The new optimization framework needs 72 hours to collect the DDM optimization dataset. After completing the optimization dataset, the new optimization framework generates the Pareto front for one driver type in just 15 minutes and takes an additional 15 minutes to recover the design parameters of the desired DDM. For a single optimization task, which involves optimizing three types of drivers and recovering design parameters, the total time is included. The HDPDMS optimization is completed within 1 hour using the new framework. In contrast, the traditional optimization framework takes approximately 72 hours to accomplish the same optimization task. For a single optimization task, the new optimization framework can save about 98.6% computation time. The benefits of the new optimization framework become increasingly apparent as the number of motion tasks increases. This will be a powerful tool for the initial design phase requiring numerous adjustments to the motion control system.
Compared to the traditional optimization framework, which combines all the design parameters, models, and constraints to do the optimization, the comprehensive new optimization framework shown in Fig. 28 allows for a separate optimization process at the component level and system level. This is made possible by the new surrogate model of characteristic parameters that decouples the component-level and system-level optimizations. In the component-level optimization, the focus is solely on optimizing DDM to achieve better characteristic parameters without considering the influence of the driver and motion profile. In system-level optimization, the surrogate models provide a physical correlation of the DDM characteristic parameters, enabling the use of characteristic parameters for HDPDMS optimization directly. Compared to the traditional framework, the new framework uses fewer design parameters, models, and constraints in each optimization level, reducing the optimization burden and producing better optimization results, as shown in Fig. 27.

B. COMPARISON WITH THE TRADITIONAL SURROGATE MODEL
If the traditional surrogate model is used for the DDM, the inputs are the design parameters of DDM, and the outputs are characteristic parameters which include T p , T c , L s , and demagnetization. The four surrogate models need to be built, respectively. The dimension of the design parameters of DDM is nine. If the full factorial sampling with the six samples on every dimension is performed, the total size of the training dataset is 6 9 = 10 077 696 cases. The required number of samples significantly exceeds the computational resource of a new surrogate model, which is only about 21 000 cases. The new optimization framework saves 99.8% of computation resources. The SPMSM represents the simplest topology among PM motors. More complex topologies of PM motors require additional design parameters, exacerbating the computational resource issue.
The new surrogate model is a new form of dimension reduction strategy. However, unlike traditional methods outlined in Section I, which directly reduce the number of design parameters, the new surrogate model achieves dimension reduction by transforming the search space from the design parameters space to the characteristic parameters space through the optimization of DDM. Specifically, the search space is reduced from nine design parameters to five characteristic parameters in our case.
Traditional dimension reduction strategies based on design parameters depend on the motor topology. When the motor topology changes, a new dimension reduction strategy must be developed to select different design parameters. In contrast, our new dimension reduction strategy is tailored to a specific application, such as HDPDMS. As long as the application remains unchanged, the selection of the important characteristic parameters remains consistent, regardless of the motor topology. A similar optimization process can be performed for different motor topologies to optimize the HDPDMS, even if the design parameters vary. Thus, this method can be extended to other common motor topologies, such as using different rotor topologies (e.g., Halbach array and spoke array) or using various stator topologies, including different slot shapes and tooth tips.
When viewed from the perspective of the design parameter space, the traditional surrogate model allocates computational resources uniformly across the entire design parameter space. As a result, a considerable amount of computation resources are used on many redundant and unnecessary designs. Traditional dimensional reduction strategies involve replacing the complete design parameter space with a smaller subspace of design parameters that contain the most relevant information. However, the computational resources are still allocated uniformly across the entire subspace. The new optimization framework employs optimization techniques to concentrate computational resources on the most relevant and useful regions of the design parameter space for better characteristic parameters, thereby minimizing wastage and conserving computation resources.
Based on the above analysis, the Performance comparison of the proposed approach and traditional optimization framework and surrogate model is shown in the Table 5.

VI. PROTOTYPE AND EXPERIMENT
We designed three DDMs corresponding to three different motion tasks, with motion tasks 1 and 3 using driver C and motion task 2 using driver B. We selected the designs that met the reliability indicators of -20% as shown in Figs 20,22,and 21. Light loads and quick motion for motion task 2 called for a long and slender motor, while heavy loads and slow motion for motion task 3 required a short and pancake motor. The 3D view of these three frameless motors is depicted in Fig. 29.
To verify the effectiveness of our new optimization framework and the precision of the models used in the optimization process, we constructed a prototype for motion  29. From left to right, the frameless motors correspond to three different motion tasks and vary in size, particularly for the motors in motion tasks 2 and 3. The need for light loads and quick motion results in a long and slender motor for motion task 2, while the need for heavy loads and slow motion leads to a short and stout motor for motion task 3. All the motors were generated using the same DDM optimization dataset and surrogate model through the optimization process described in section IV. task 1. The prototype, load, and experiment platform can be viewed in Fig. 30. To complete the motion task, we used driver C to drive the DDM with the load. We also tested the temperature of the winding using a PT-100 fixed to the winding of the DDM, and the temperature of the housing using a thermocouple. The optimal design parameters of the prototype are presented in Table 6. Additionally, we established the motion profile parameters for completing motion task 1, as listed in Table 7. Figure 31 illustrates the motion process, where each position involves a rotation angle of 18 • and a motion time of 20 ms. Following the motion, a dwell time of 40 ms is allocated for chip sorting or inspection. Figure 32 displays the current loop regulation with an update frequency of 20 kHz. We adjusted the bandwidth of the current loop to approximately 680 Hz with a 63 • phase margin. Figure 33 shows the position and velocity loop regulation with an update frequency of 2 kHz. We adjusted the bandwidth of the position loop to approximately 80 Hz with a 40 • phase margin.   Figure 34 presents the velocity and current curves during the motion time. From the figure, we can observe that the current tracks the current command, while the velocity feedback experiences a delay of 0.5 ms due to the 2 kHz update frequency of the velocity loop. The maximum current observed is approximately 25.2 A, which still has a 16% peak current margin with respect to the driver's 30 A peak current capacity. Similarly, the continuous current observed is approximately 11.6 A, which still has a 22% continuous current margin with respect to the driver's 15 A continuous current capacity.     During the test, the ambient temperature was 20℃, and the winding temperature rose approximately 51℃. We assumed that the temperature rise of the winding would be the same at an ambient temperature of 25℃. The temperature of the winding would be 76℃, which is within the 24% margin of the upper limit of 100℃. This temperature was lower than our predicted FIGURE 34. Velocity and current curve during the motion time. The current and velocity can track the current command and reference velocity, while the velocity feedback experiences a delay of 0.5 ms due to the 2 kHz update frequency of the velocity loop. The whole motion time is 20 ms which satisfies the motion task 1 requirement.

FIGURE 35.
Temperatures rising of winding and housing. The test ambient temperature was about 20℃, and the temperature rise of the winding was about 51℃, which was smaller than our prediction (55℃) due to the heat dissipation from the mechanical load part, which is not taken into account in our thermal model. value of 80℃ due to our thermal model not accounting for heat dissipation from the mechanical load part in Fig. 30, which increased the heat dissipation surface and lowered the winding temperature. This error was acceptable due to the simplification of the thermal model. The prototype was able to perform well for motion task 1, as shown by the prototype test results on the Pareto front in Fig. 36. Due to VOLUME 11, 2023 manufacturing and measurement limitations, the prototype's loss is greater than that of the theoretical model, resulting in slightly higher cost indicators. The prototype's reliability indicator was 16% due to the higher peak current, which is lower than the optimal design reliability indicator of 20% from the new optimization framework but still higher than the reliability indicator of 12% obtained from the traditional optimization framework. These results validate the effectiveness of the new optimization framework.

VII. CONCLUSION
This article presents a novel optimization framework for HDPDMS using surrogate models of characteristic parameters, which allows for quick and precise optimization of the entire system with limited computation resources. Unlike traditional optimization frameworks that combine all design parameters, models, and constraints, this new framework enables a separate optimization process at both the component level and system level through the use of surrogate models of characteristic parameters. During each optimization level, the component-level and system-level optimizations employ fewer design parameters, models, and constraints, effectively reducing the optimization burden and yielding better optimization results. Once the surrogate models are built, the framework could find the desired HDPDMS design for a specific motion task in just 1 hour, saving about 98.6% computation time compared to traditional methods. This tool is particularly useful for the initial design phase, which requires numerous adjustments to the motion control system. Moreover, the new surrogate model uses optimization to centralize computation resources on the most useful search space, resulting in a 99.8% reduction in computation resources required to build the training dataset, compared to the traditional surrogate model, whose inputs are design parameters with full factorial sampling in our case. This new approach can be seen as a dimension reduction strategy that transfers the search space from the design parameters to the characteristic parameters of the components, as opposed to traditional dimension reduction strategies that only reduce the number of design parameters.
The new optimization framework provided is versatile. Firstly, it is not limited to any particular motor topology, making it unnecessary to conduct a sensitivity analysis and manually select important design parameters of the motor as required by traditional surrogate models. This means the method can be easily applied to different motor topologies for the HDPDMS. Secondly, the framework can absorb various optimization strategies to further reduce computation resources in both component-level and systemlevel optimizations. For instance, a pre-optimization using a quick analytical model can be applied during componentlevel optimization. Surrogate models can be substituted with alternative regression methods if those methods can offer sufficient prediction accuracy and can determine the boundaries of the input space. Similarly, the recovery method for design parameters, such as the inverse surrogate model [32], [33], can also be replaced. This method establishes a mapping between the characteristic parameters and the design parameters. In future research, we will investigate the impact of various methods within this innovative optimization framework.