© C. Guo et al., Published by EDP Sciences, 2026

1 Introduction

Robots have taken a more prominent stage in service and manufacturing industry, since they became autonomous and less dependent from the human labor [1]. For robots with high flexibility, their driving systems or actuations are the key components, and directly influence the robot's motion accuracy and dynamic response speed, energy efficiency, and operational reliability in future factories [2,3]. To ensure the performance and reliability of these actuation systems, rigorous performance characterization and quality control protocols are essential, as highlighted in recent metrology research [4,5].

The world's leading robot systems have utilized advanced control algorithms and precision joint drive technology to address key challenges in the field of drive system performance and achieve complex motion capabilities. Atlas developed by Boston Dynamics adopts high torque density servo motors and precision harmonic reducer as the core, combined with advanced algorithms to achieve dynamic balance and complex motion control [6]. MIT'Cheetah robot simulates the elastic drive mechanism of animal tendons through the biomimetic joint structure to optimize energy storage and release efficiency [7]. Tesla'Optimus relies on a combination of lightweight planetary gears and high-efficiency motors. This design considers torque output and energy consumption control within a compact modular framework, significantly enhancing the smoothness of movement and the practicality of joints in complex scenarios [8].

Beyond the technical practices of these cutting-edge robot prototypes, the driving methods of robot joints can be classified into three categories, including pneumatic drive, hydraulic drive, and electric drive in view of system implementation. Among these, electric drive has become the dominant way for robotic joints due to its irreplaceable technological advantages, primarily its perfect alignment with the rigid requirements, namely “high precision, high reliability, compactness, and low energy consumption”. Leveraging these inherent advantages, electric drive has demonstrated superior performance in terms of reliability, efficiency, and control accuracy, making it the preferred choice for robot joint.

By integrating a motor and gear set, modular joints have been developed for electrically driven robot joints. These include direct-drive actuators, semi-direct-drive actuators, and tandem elastic actuators. Direct-drive actuators are robotic joint drives that consist solely of motors, featuring a straightforward mechanical structure and commendable control performance. However, their output torque density is constrained by the motors [9]. In contrast, the semi-direct-drive actuator includes a motor with high output torque density paired with a reducer that has a low reduction ratio. This configuration offers precise torque control, rapid response, and excellent shock resistance, although it exhibits slightly reduced positioning accuracy [10]. The tandem elastic actuator consists of a motor, a high reduction ratio gearbox, and an elastomer, which offers high output density but exhibits low stiffness and a weak torque response [11]. In the confined spaces of robotic joints, permanent magnet(PM) synchronous motors prove effective when paired with medium- and high-reduction-ratio gearboxes, thereby achieving torque amplification [12,13]. These motors have become the promising technological path for joint actuation in robots, with multiple advantages of compact design, high operational efficiency, and minimal torque fluctuations.

Accurate measurement characteristics of key performance indicators (such as torque output and efficiency) are prerequisites for high-quality motor design [14]. The evaluation of these indicators usually depends on advanced modeling technologies, such as finite element analysis (FEA), which itself requires careful uncertainty quantification to ensure the credibility of the results [15,16]. Therefore, as the core tool of electromagnetic field numerical calculation, FEA plays an irreplaceable role in motor optimization design [17]. By precisely characterizing the magnetic field distribution and nonlinear material properties [15], FEA enables comprehensive insights into electromagnetic characteristics (e.g., flux density saturation, magnetic line distortion) [16], which in turn facilitates the optimization of design parameters to enhance torque density and efficiency [17]. Existing research has validated FEA's value in motor design. For example, Tran et al. [18] developed a magnetic chain theoretical model for robotic joint drives, clarified structural parameter-performance relationships through coupled electromagnetic-thermal simulations [19], and verified design validity via 3D transient FEA. However, such studies primarily focus on single-parameter validation rather than multi-objective performance enhancement under spatial constraints.

Combined by FEA, multi-objective optimization has become efficient approach for optimizing high-performance motors [20,21]. Relevant studies have shown that embedding domain knowledge into optimization frameworks can effectively improve the quality of configuration optimization for robotic systems, a strategy that provides valuable references for motor parameter optimization [22]. Yet PM DC motors still suffer from torque ripple(TR) and energy losses caused by cogging torque(CT) and commutation issues, which degrade robot efficiency, motion smoothness, and control accuracy [23,24]. Existing solutions remain fragmented—most studies either optimize electromagnetic structures (e.g., winding configurations, pole shapes [25,26]) or improve control algorithms in isolation, while structural-control integration under space limitations is rarely addressed [27–30]. In contrast, this work proposes an integrated optimization framework that combines FEA-driven structural refinement with multi-objective algorithmic optimization. Unlike conventional fragmented approaches, our method dynamically couples electromagnetic field simulation results with multi-objective optimization objectives (torque density, efficiency, and TR), enabling iterative optimization of key parameters such as stator/rotor geometry and winding distribution under strict spatial constraints. This integrated strategy aims to achieve concurrent improvements in multiple performance metrics, contributing to the research on holistic optimization of robot joint motors within confined spaces.

This article proposes a design framework for an external rotor brushless DC motor that integrates structural innovation and multi-objective optimization, aiming to break through the inherent contradiction between high torque density and manufacturing feasibility in robot drive systems. The core innovation lies in the use of a multi-module, spliced external rotor structure, which simplifies the manufacturing process through split-slot winding and modular stator tooth design. At the same time, the magnetic circuit symmetry optimization reduces the slot torque by 38.89%, from 0.036 N · m to 0.022 N · m. Combining electromagnetic theory modeling and ANSYS Maxwell finite element simulation, the quasi Newton multi-objective optimization algorithm is introduced to iteratively optimize key parameters such as pole arc coefficient, air gap length, and permanent magnet thickness, ultimately achieving peak efficiency improvement to 95.06%, iron loss reduction of 16.36%, from 26.9 W to 22.5 W, copper loss(CuL) stability at 12.8 W, air gap magnetic flux density reduction of 8.33%, from 0.72 T to 0.66 T to avoid magnetic saturation risk. This integrated structure optimization strategy effectively solves the limitations of single-parameter optimization in traditional designs. Simulation and prototype testing show that torque pulsation can be controlled within 24% to 26%, which is lower than the threshold requirement of 30% for robot drive systems. It provides a new path for quantitative verification of high-performance design of robot joint motors in compact spaces.

The paper is organized as follows: Section 2 establishes the structural model of the external rotor motor, clarifies the basic design parameters, analyzes the torque composition, and examines the impact mechanism of key parameters on performance. Section 3 implements multi-objective parameter optimization based on the quasi-Newton method to determine the optimal combination of structural parameters. Section 4 evaluates the static and transient electromagnetic characteristics of the motor through finite element simulation to verify the optimization effect. Section 5 conducts prototype experimental testing, compares simulation and measured data to verify the reliability of the design. Section 6 summarizes the research results and looks forward to future optimization directions.

2 Structural design and model of external rotor motor for robotic joints

2.1 Determination of design dimension for motor

A multi-module spliced direct drive external rotor brushless DC motor is designed for robot joint drive, balancing static load capacity and dynamic response speed. Based on research on advanced robots from Boston Dynamics and Agility Robotics Digit, the rated speed is set to 260 r/min (within the typical 200∼300 r/min range) to meet demands for compact structure and high torque output. The maximum speed accommodates short-term overloads from sudden actions (e.g., rapid obstacle avoidance), preventing frequent motor stalling, with its relationship to rated speed defined by the following equation:

$$n_{\max }=\beta n_N$$(1)

where n_max is the maximum speed of the motor, n_N the motor's rated speed, and β is the expansion of the constant power meter coefficient. This coefficient is typically assumed to be between 2 and 4, with β=3 employed in this instance.

The material is a PM N35SH, with temperature stability from 20 °C to 150 °C meeting the robot's continuous operation needs. Core parameters as rated power (500 W) and torque (18.36 N · m) balance actuation demands and practicality, which are chosen based on static/dynamic loads in robot joint, Their insufficiency causes lag/stalling, and excess increases size/energy use. Suitable torque, derived from T=9550 P/n, ensures output to overcome resistance, where inadequacy impairs accuracy, and excess raises manufacturing costs/heat risks. These parameters lay a ground for subsequent design. Other basic design parameters based on rated rotational speed are listed in Table 1.

It is essential to recognize that the motor load consists of two components: electrical load(EL) A and magnetic load(ML) B_δ. The EL is primarily associated with the current density and line load of the motor's stator winding, while the ML is mainly related to the magnetic flux density within the motor. These loads significantly impact the mechanical characteristics, efficiency, torque, and other performance parameters of the motor. The following deduction is made concerning the relationship between its primary dimensions and the EL:

$$C_A=D^2{L}^2=\frac{6.1}{{\alpha }_{\text{i}}{K}_{\phi }{K}_{w}A{B}_{\delta }}\cdot \frac{P}{n_N}$$(2)

where C_A is the motor constant; D is the armature diameter (cm); L is the stator core(SC) length (cm); P is the calculated power; η_N is the rated speed; K_φ is the magnetic field waveform coefficient when the waveform is a sinusoidal wave for 1.11; K_W is the winding coefficient, typically around 0.95; α_i=0.7 is an empirical value for small PM motors, which can balance the utilization of main magnetic flux and the difficulty of processing. But the simulation results show that when α_i=0.7, the magnetic density of the stator teeth reached 0.48 T (close to 30% of the saturation threshold of 1.6 T for silicon steel sheets), indicating a risk of magnetic saturation, generally ranging from 0.7 to 0.9, here take 0.7; A is the EL, with small PM motors usually takes the value of 30-150 A/cm, and a value of 150 A/cm; B_δ is the ML, typically between 0.8 T and 0.9 T. The final result is D²L²≈719.7 cm³.

The outer rotor of DC brushless motor may experience deformation when subjected to force. If the size of air gap(AG) is not properly calibrated, it can lead to friction between the outer rotor and the stator. A small AG will increase the difficulty of assembly, but the Excessive AG will increase magnetic resistance, resulting in a decrease of ET by>10%. So, 1 mm is the balance point between magnetic performance and manufacturing feasibility. In actual assembly, positioning fixtures are used to ensure AG uniformity, with tolerances controlled within ± 0.05 mm, which can suppress torque pulsation. Therefore, this paper establishes the initial AG length L_δ set at 1 mm.

Thus, the outer diameter of the motor rotor is set as 100 mm, with a rotor core(RC) thickness of 2.5 mm and a PM thickness of 2.5 mm. The inner diameter of the outer rotor can be calculated using equation (3):

$$D_{Ri}=D_{Ro}-2(h_m+h_R)$$(3)

where D_Ri is the outer rotor inner diameter, D_Ro is the outer rotor outer diameter, h_m is the PM thickness, and h_R is the RC thickness. The outer rotor inner diameter D_Ri of the motor is 90 mm, calculated by equation (3). The initial size of AG is 1 mm, and the armature diameter D is set to be 88 mm. The effective length of the SC is then approximated to be 93 mm, calculated by equation (2).

Finally, the number of pole pairs is set to 12, and the number of slots is set to 27. The schematic diagram of the three-phase winding connection for the outer rotor DC brushless motor is shown in Figure 1.

Fig. 1 Three-phase winding connection.

2.2 Outer rotor motor structure design

Based on the basic design parameters and dimensional relationships determined in Section 2.1, the structural design of main component of the outer rotor motor is further refined to ensure the coordination of electromagnetic performance and manufacturing feasibility. As the magnetic field source of the motor, the PM structure directly affects the distribution of AGMD and the magnitude of electromagnetic torque(ET). To match the 24-pole and 27-slot configuration (with the least common multiple of poles and slots NL = 216) determined in Section 2.1, a surface-mounted layout with symmetrical 24-pole distribution (pole pairs Np = 12) is adopted. This pole-slot matching helps to reduce CT by optimizing the interaction between the stator and rotor magnetic fields. The pole arc length is initially set according to the pole-arc coefficient(PAC) α_i = 0.7 specified in Section 2.1, while the PM thickness h_m = 2.5 mm is derived from magnetic circuit analysis to ensure the AGMD B_δ = 0.72 T, which meets the ML requirement in equation (2). Using N35SH NdFeB PM with radial magnetization ensures the magnetic field direction is perpendicular to the AG, effectively reducing magnetic leakage losses and maintaining the magnetic density stability required for torque output.

As a critical part for transmitting ET, the SC is designed as a 27-slot semi-closed structure consistent with the slot number defined in Section 2.1. A slot width of 3 mm is used to mitigate CT fluctuations caused by the slot effect, with a tooth-to-slot width ratio of 1.2 to balance magnetic flux distribution and mechanical strength. To reduce eddy current losses(ECL) under alternating magnetic fields, the SC is made of 50W350 non-oriented silicon steel sheets, with a lamination coefficient of 0.95, a core length L_s = 93 mm, and a lamination thickness of 0.35 mm. The stator yoke height is set to 12 mm to ensure magnetic circuit balance, with the yoke flux density controlled below 1.2 T to avoid magnetic saturation. The RC is fabricated from low-carbon steel (Q235) through integral processing, ensuring sufficient mechanical strength while reducing magnetic resistance to match the rotor inner diameter (90 mm) and outer diameter (100 mm). The winding adopts a three-phase centralized structure, with 28 turns per slot and a wire cross-sectional area of 0.5 mm². This design reduces harmonic electromotive force and improves the stability of ET, complementing the pole-slot configuration to achieve the rated torque of 18.36 N · m.

These structural designs, based on the torque analysis model and parameter matching in Section 2.1, achieve a balance between ET and CT through rational coordination of key parameters, providing an initial benchmark for the parameter optimization in Section 3.

2.3 Calculation of PM motor for CT

The instantaneous torque of a PM motor is composed of both ET and CT, where ET determines output capability and CT is one of the main sources inducing torque pulsation. In this section, the design parameters of the motor's structure are determined by electromagnetic theory and simulation analysis, achieving a balance between torque performance and manufacturing feasibility. Based on theoretical analysis, the instantaneous torque is expressed as equation (4):

$$T=T_{cog}+T_{em}$$(4)

where T is the instantaneous torque of the PM motor, T_cog is the CT and T_em is the ET.

In the coordinate system, the expression of the ET of the PM motor is shown in equation (5):

$$T_{\text{em}}=N_p\left[\left({\psi }_{fd}{i}_q-{\psi }_{fq}{i}_d\right)+\left(L_d-L_q\right)i_d{i}_q\right]$$(5)

where N_p is the number of pole pairs of the motor, i_d, and i_q are the components of the stator current at d and q, respectively; L_d and L_q are the inductances of the stator windings at the d and q axes, respectively; and Ψ_fd and Ψ_fq are the components of the flux linkage of the PM at the d and q axes, respectively.

According to the energy method, the CT of the motor can be defined as the negative derivative of the magnetic field energy W with respect to the relative position angle α of the stator-rotor at power failure:

$$T_{cog}\left(\alpha \right)=-\frac{\partial w}{\partial \alpha }$$(6)

where W is the magnetic field energy stored in the motor at power failure; α is the relative position angle of the stator-rotor.

This can be used to derive the defining equation for the CT when the change in magnetic field energy in the PM and core is neglected:

$$T_{cog}\left(\alpha \right)=-\frac{\partial }{\partial \alpha }\left[\frac{L_s}{2{\mu }_0}\cdot \frac{1}{2}\left(R_2^2-R_1^2\right)\cdot {\displaystyle \underset{V}{\int }}{G}^2\left(\theta \right)B^2\left(\theta ,\alpha \right)dV\right]$$(7)

where G(θ) is the relative AG permeability function; B(θ, α) is the remanent magnetic density function of the equivalent slotless motor; µ₀, R₁, R₂, and L_s are the AG permeance, stator inner diameter, rotor outer diameter and core axial length, respectively.

The Fourier expansion of G(θ) and B(θ, α) gives the following equation:

$$G\left(\theta \right)={\displaystyle \sum _{n=0}^{\infty }}{G}_{n{N}_s}\cos n{N}_s$$(8)

$$B^2\left(\theta ,\alpha \right)={\displaystyle \sum _{n=0}^{\infty }}{B}_{n{N}_p}\cos n{N}_p\left(\theta +\alpha \right)$$(9)

where G_nNs and B_nNp are the coefficients in the Fourier expansion equation; N_s and N_p are the number of stator slots and the number of PM poles of the PM motor, respectively.

Finally, by orthogonally transforming the trigonometric function, the expression for the T_cog can be derived:

$$T_{cog}\left(\alpha \right)=\frac{L_{ef}\pi }{4{\mu }_0}\left(R_2^2-R_1^2\right){\displaystyle \sum _{n=1}^{\infty }}n{N}_L{G}_{n{N}_L}{B}_{n{N}_L}\sin n{N}_L\alpha$$(10)

where N_L is the least common multiple of N_s and N_p; L_ef is the SC length.

From equation (10), it can be concluded that the primary factors influencing the CT include the pole-slot fit, PAC, the shape of the PM, slot geometry, SC length, stator inner diameter, and rotor outer diameter. This model provides a theoretical foundation for subsequent optimization efforts.

3 Optimization of motor structure for high efficiency and low loss

3.1 The influence of key parameters on motor performance

The energy conversion process of the motor primarily occurs at the AG. An insufficient AG can lead to increased core loss(CL), resulting in overheating and noise, while an excessively large AG can increase magnetic reluctance and reduces magnetic flux, consequently decreasing both torque and efficiency [31–34]. Therefore, the size of the AG significantly influences the motor's performance. Without altering other parameters of the motor, the AG length is simulated by varying it from 0.5 mm to 4 mm using the Optimetrics module of ANSYS Maxwell. The results are shown in Figure 2 to illustrate the relationship between motor efficiency and torque. It is evident that the outer rotor structure ensures that the local magnetic density remains uniformly distributed for different AG length. This consistency is due to the motor's magnetic circuit remaining in a non-saturated state, which contributes to its excellent stability. Simultaneously, as the AG lengthens, the magnetic reluctance increases, leading to a corresponding decrease in AG density. This results in a rise in CuL, which is analogous to the decline in CL. Consequently, the curve in Figure 2 is relatively smooth, indicating that the impact of AG length on motor efficiency is minimal.

The PAC is defined as the ratio of the pole arc width of the motor pole to the pole pitch. This coefficient directly influences the distribution of magnetic flux density in the motor's AG and the magnitude of the ET. Optimizing the PAC can improve the distribution of magnetic flux density in the motor AG, thereby reducing iron and CuL and enhancing motor efficiency. The PAC typically ranges from 0.6 to 0.9. An electromagnetic analysis method is utilized to obtain the curves of AG flux density(AGFD), CT, and average armature current as functions of the PAC, as illustrated in Figures 3 and 4.

As illustrated in Figure 3, the overall trend of the air-gap flux density exhibits a non-monotonic change characterized by an initial increase followed by a decrease. This behavior is primarily attributed to the dynamic balance between the optimization of the main flux path and the effects of magnetic saturation. It is evident that when the PAC increases from 0.6 to the critical value of 0.72, the effective coverage area of the PM expands, significantly enhancing the magnetic potential source and leading to a continuous increase in AG magnetism. However, when the PAC exceeds 0.72, the magnetism of the stator teeth surpasses the saturation threshold due to pole expansion. This results in a sudden decrease in the magnetic permeability of the magnetic circuit, which restricts the increase in the main flux. Consequently, the magnetism in the AG begins to decline. In the range of 0.65 to 0.75, the fifth and seventh space harmonics are periodically superimposed due to variations in phase difference. Simultaneously, as the PAC approaches 0.68, the geometric alignment between the stator slot openings and the pole edges induces permeability harmonic resonance. These two factors synergistically contribute to the anomalous spikes in AGFD observed at specific PAC.

In Figure 4, there is a fluctuations which is primarily influenced by two phase modulation of the permeability and the remanence harmonics. As the PAC varies, the phase difference changes periodically, causing specific harmonic components (e.g., the 5th and 7th harmonics) during the integration process. This interaction results in fundamental amplitude fluctuations of the CT. Additionally, the combined effects of these harmonics contribute to the non-monotonic fluctuation pattern observed in the curve due to resonance coupling.

It was concluded that there is a positive correlation between AGFD and torque. Therefore, the AGFD must not be too low to achieve high torque, CT is a type of TR and unique to PM motors. As shown in Figures, CT may cause torque fluctuations, increase control difficulty and reduce accuracy, intensify mechanical wear, and lead to greater energy loss. Additionally, an increase in armature current significantly exacerbates the motor's overheating issue, which, in turn, affects performance and can even result in permanent damage to the PM.

Fig. 2 Influence of AG length on efficiency.

Fig. 3 AGFD for different PAC.

Fig. 4 CT for different PAC.

3.2 Structural optimization for motor based on Quasi-Newton method

The above analysis demonstrate that the size of the AG influences the magneto-density harmonic phase relationship by altering the distribution of magnetoresistance. Concurrently, the PAC predominates over the characteristics of the magneto-density distribution through the dynamic balance between optimizing the main flux path and the effects of magnetic saturation. The strong coupling among parameters, combined with the necessity for multi-objective optimization—such as enhancing efficiency, reducing losses, and minimizing TR—renders traditional univariate experimental methods ineffective for achieving a globally optimal solution. Given the numerous significant independent variables that affect motor performance (see Sect. 2.3), the application of mathematical optimization is essential. Optimization algorithms for motor parameters include the genetic algorithm, the equivalent magnetic circuit method, Taguchi's method, and Newton's method [35–38].

After a comprehensive evaluation of the available options, this paper proposes the use of the Quasi-Newton method as an iterative approach for optimizing motor parameters. The effectiveness of the Quasi-Newton method is highlighted by its robust convergence properties, which facilitate the rapid identification of precise solutions when initialized from appropriate values. This, in turn, enhances the responsiveness of the real-time control system.

The essence of the proposed Quasi-Newton method lies in the continuous updating of the search direction to approximate the Hessian matrix through iterative processes, thereby facilitating convergence toward the optimal solution. The process of each iteration involves updating the next point, x_k+1, using the current point, x_k, and the search direction, p_k. The iteration formula is as follows:

$$x_{k+1}=x_k+{\alpha }_k{p}_k$$(11)

where α_k is the step size; x_k is the motor structure parameter vector; p_k is the search direction.

The search direction, p_k, is determined by the following equation (12):

$$p_k=-H_k\nabla f\left(x_k\right)$$(12)

where ▽f(x_k) is the gradient vector of the objective function f(x) at x_k; H_k is the approximation matrix to the Hessian matrix at the kth iteration. It is important to note that different proposed Quasi-Newton algorithms have different ways of updating H_k.

In the process of motor optimization, the proposed Quasi-Newton matrix H_k is updated according to this formula at each iteration. Subsequently, the search direction p_k is adjusted to gradually converge towards the optimal solution of the motor's structural parameters, thereby achieving the objectives of improving efficiency and reducing CT.

The selection of optimization variables is closely related to the degree of design freedom, which is defined as the number of design variables. As the number of variables increases, the degrees of freedom in design also expand, resulting in a greater number of available optimization paths. This, in turn, facilitates the achievement of the optimization goals. However, it is important to note that an excess of variables is not necessarily advantageous. A high-dimensional optimization design objective function can lead to an increase in the scale of calculations, greater computational complexity, and reduced efficiency. Therefore, it is essential to select a reasonable number of variables.

Combining the torque analysis model outlined in Section 2.2, the following parameters have been identified as optimized design variables: (a) polar arc coefficient; (b) stator outer diameter; (c) rotor outer diameter; (d) rotor inner diameter; (e) SC length; and (f) PM thickness. For them, the range of variations are demonstrated in Table 2.

The observed performance trade-off — improved efficiency and reduced CL at the expense of increased CuL and thermal load—results directly from the electromagnetic redesign. To reduce cogging torque and avoid saturation, the optimization reduced the PM thickness and adjusted the pole arc coefficient, which in turn decreased the air-gap flux density. Maintaining the required output torque necessitated a slight increase in armature current, directly leading to higher CuL. This compromise was strategically justified. For robotic actuation, high torque density and low torque ripple were prioritized as the primary objectives. The increase in thermal load was deemed acceptable, as it remains within the motor's safe operational limits. Meanwhile, the gains in efficiency and CL reduction provide substantial benefits for battery life and thermal management, thus outweighing the manageable rise in copper-related heating.

After determining the design variables and optimization method, optimization is performed using the ANSYS RMxprt module. Initially, the variable parameters and simulation steps are integrated into Optimetrics, after which the simulation objects are defined in the Calculations section. Finally, by conducting the optimized design calculations within the RMxprt module, a comparison between the initial optimized motor and the optimized motor parameters can be obtained, as shown in Table 3.

Optimization model involve the objective function prioritizes efficiency maximization, with secondary goals of minimizing CL and the TR coefficient. Design variables include the six key structural parameters listed in Table 2, and the model is constrained by variable boundaries (Tab. 2) and performance requirements (e.g., efficiency ≥ 90%).

The solution process begins with initialization—starting from preliminary design parameters (x₀ = [0.7,88,100, 90,93,2.5]), convergence criteria (gradient norm <10⁻⁴) and a maximum of 50 iterations are set. Simulation-driven parameter updates are enabled via the interface between ANSYS Maxwell and MATLAB, where each set of variables x_k triggers automatic motor model updates, mesh division, and output of performance indicators such as efficiency, loss, and TR. The iterative core involves approximating the Hessian matrix using the BFGS formula (H_k), with the search direction p_k calculated as p_k =-H_k▽f(x_k) while maintaining positive definiteness to optimize directionality. Step size α_k is determined through backtracking search, and variables are updated as x_k+1 = x_k + α_kp_k. Upon meeting convergence criteria, optimal parameters x* are output, with performance stability verified via independent simulations.

Following the multi-objective optimization of the key structural parameters for the motor using the proposed Quasi-Newton method, the comparative data listed in Table 3 demonstrate that the optimized model has achieved significant improvements in efficiency, reduced losses, and enhanced torque characteristics. Although some parameters (e.g., stator outer diameter and rotor outer diameter) are only slightly adjusted, their synergistic effect effectively balances electromagnetic performance through the reconstruction of the magnetic circuit. The polar arc coefficient has been optimized from 0.7 to 0.605, resulting in an 18.75% reduction in the magnetic flux density of the stator teeth and a concomitant decrease in CT by 38.89%. This validates the efficacy of the Fourier coefficient amplitude reduction theory as outlined in equation (10). Furthermore, a 4.8% reduction in the thickness of the PM leads to a 30.16% decrease in the magnetic flux density of the rotor yoke. This, combined with a moderate 8.33% reduction in air-gap magnetic density(AGMD), helps mitigate the risk of magnetic saturation. Additionally, it reduces CL by 16.36% through the judicious distribution of magnetic potential. Although the optimized efficiency increase of 0.41% may seem negligible, it actually reflects a delicate trade-off between CuL (with a thermal load increase of 6.5%) and CL, illustrating the Pareto-optimal characteristic of multi-objective optimization. These optimization results provide more reasonable parameter benchmarks for subsequent electromagnetic field (EMF) simulations, and the nonlinear coupling relationship between the variables further confirms the necessity of employing the proposed Quasi-Newton method discussed in Section 3.2, which establishes a structural foundation for the accuracy of the FEA in Section 4.

According to the above structural optimization results, the multi-module structure design of the motor is carried out. In this design, the direct-drive, low-speed outer rotor brushless DC motor is considerably large, which restricts the applicability of conventional manufacturing methods. In order to overcome this problem, the split-slot winding is utilized in this design. The coil span is set as one stator tooth, and the structure of the SC is designed as multi-module splicing. Each stator tooth acts as a module with a separate winding coil. Each module can be manufactured individually and then assembled, as shown in Figures 5 and 6.

The multi-module splicing structure simplifies manufacturing but requires tight assembly tolerances to maintain magnetic symmetry, as recommended in relevant industry guidelines and supported by finite-element analyses. In practice, stator tooth positioning is typically controlled within a few hundredths of a millimeter to limit TR caused by air-gap unevenness that disrupts flux distribution. End-face parallelism is also kept within small deviations per meter of axial length to reduce axial magnetic leakage. These tolerances are maintained using dedicated positioning fixtures with high-precision laser alignment systems, capable of sub-micron repeatability in advanced manufacturing setups.

This design can simplify manufacturing process, improve the space coefficient of the coil, and increases the output torque of the motor. The rotor structure consists of magnetic poles and a RC, with the magnetic poles affixed to the inner surface of the RC. The shaft is connected to the base via rolling bearings.

Fig. 5 Multi-module spliced stator with split-slot windings.

Fig. 6 Multi-module outer rotor BLDC motor assembly.

4 Performance assessment of motor based on electromagnetic analysis

4.1 Electromagnetic analysis model

Using the optimized structural parameters, a systematic analysis of the motor's electromagnetic characteristics is below. This analysis will verify the electromagnetic performance of the multi-module spliced structure and evaluate the effectiveness of the optimization. In this section, an accurate electromagnetic analysis model is constructed using the finite element method based on the optimization parameters determined in Section 3. Multiple simulation types will be conducted under static magnetic field conditions, both with and without load, utilizing the ANSYS Maxwell platform. The focus will be on analyzing the flux density distribution, reverse electromotive force characteristics, and loss mechanisms of the optimized motor. This analysis will provide theoretical support for the subsequent TR analysis and prototype testing. This progressive research, from structural optimization to the verification of electromagnetic characteristics, not only assesses the validity of parameter optimization but also reveals the complex electromagnetic coupling mechanisms within the motor.

As previously established, the fundamental dimensions of the motor must be determined alongside the optimized relevant parameters in ANSYS Electronics RXMprt to create the basic model of the motor (see Fig. 7). Subsequently, a two-dimensional model of the motor will be developed using ANSYS Maxwell software (see Fig. 8). To expedite the simulation calculations, it is essential to utilize only one-third of the motor model. For reliable finite element simulation, key settings in ANSYS Maxwell are specified based on prior mesh sensitivity studies and domain knowledge — Air-gap regions, critical for field accuracy, use adaptive meshing with a minimum element size of 0.1 mm. The SC and PM are meshed with element sizes of 0.5 mm and 1 mm, respectively, balancing accuracy and computational cost (resulting in approximately 250,000 elements total). Physically consistent boundary conditions are applied — zero tangential H-field (magnetic insulation) on the outer rotor surface and periodic boundaries on symmetric sections. Convergence is rigorously monitored, terminating when the energy error falls below the recommended threshold of 1e⁻⁶ or a maximum of 50 iterations is reached.

Fig. 7 The motor model established by RMxprt.

Fig. 8 The two-dimensional model in Maxwell.

4.2 Magnetic leakage assessment based on static magnetic field analysis

A static magnetic field is produced by a constant current or a PM, with both its strength and direction remaining unchanged over time at any given location. In a static magnetic field, the lines of magnetic induction form closed curves and do not originate or vanish spontaneously. When the motor is idling—meaning it is not connected to any external load while the power is on—its internal magnetic field can be approximated as a static magnetic field. The magnetic leakage of the motor can be assessed by analyzing the flux distribution of this static magnetic field. The distribution of the motor's magnetic field is illustrated in Figures 9 and 10.

Figures 9 and 10  illustrate an uneven distribution of magnetic flux lines within the internal magnetic poles of the motor. The slotted windings results in a non-uniform area of the magnetic poles corresponding to the stator teeth, and a limited number of closed loops of magnetic flux lines that do not completely traverse the stator teeth. Consequently, there is some magnetic leakage in the motor. However, it is not significant. The quantification of magnetic leakage, represented by the magnetic leakage coefficient, allows for the local approximation of the motor as a piece of PM. Four points are selected at the midpoints of the two sides and the midpoint of the AG to calculate the motor's magnetic leakage coefficient. This approximation, which yields a value of 1.25 as outlined in equation (13), indicates that the magnetic leakage within the motor is negligible and can be disregarded.

$${\sigma }_1=\frac{{\phi }_m}{{\phi }_{\delta }}=\frac{\left|m_1-m_2\right|\cdot L_{ef}}{\left|m_3-m_4\right|\cdot L_{ef}}=\frac{\left|m_1-m_2\right|}{\left|m_3-m_4\right|}$$(13)

where the total magnetic flux of the PM (Wb) is denoted by φ_m, while the main magnetic flux of the AG (Wb) is denoted by φ_δ. The effective length of the magnetic core (mm) is denoted by L_ef, Additionally, the values of the four-point vector magnetic potential (Wb/m) are denoted by m₁, m₂, m₃, and m₄.

The flux density diagram of the external rotor motor is presented in Figure 11, where varying colors represent different values of flux density. The maximum flux density is estimated to be 1.58 T. The static magnetic field and the air-gap magnetic chain waveform are illustrated in Figure 12. It is evident that there is no magnetic saturation within the motor, with the maximum flux density occurring at the tips of the stator teeth.

The results of the static magnetic field analysis demonstrate that the optimized multi-module spliced structure offers significant advantages in terms of electromagnetic performance. The magnetic flux density cloud diagram indicates that the maximum magnetic flux density is concentrated at the tip of the stator tooth and does not exceed the saturation threshold of the silicon steel sheet. This finding verifies that the optimization of the PAC effectively controls the saturation of the magnetic circuit. The magnetic leakage coefficient, calculated using equation (13), is 1.25, which is approximately 18% lower than that of the traditional outer rotor structure. This reduction indicates that the multi-module splicing process effectively suppresses end magnetic leakage, which is closely related to the optimal distribution of magnetic potential achieved through the optimization of the thickness of the PM discussed in Section 3.2. The AG magnetic chain waveform exhibits a quasi-sinusoidal distribution characteristic, with a third harmonic distortion rate of only 4.7%. This finding confirms the positive impact of the pole-slot configuration (24 poles and 27 slots) on harmonic suppression. Notably, the magnetic density uniformity index of the stator teeth reaches 0.92, and the local magnetic density variance does not exceed 12%. This indicates that the sub-module manufacturing process has not introduced significant asymmetry in the magnetic circuit. These results validate the effectiveness of parameter optimization at the static magnetic field level, establishing a robust electromagnetic foundation for subsequent transient analysis and demonstrating the engineering feasibility of the multi-module splicing structure in maintaining the symmetry of the magnetic circuit.

Fig. 9 The magnetic field of motor.

Fig. 10 Local amplification magnetic chain cloud diagram.

Fig. 11 Static magnetic field flux density.

Fig. 12 Static magnetic field AG magnetic chain waveforms.

4.3 Torque assessment based on transient analysis

In situations where the motor operates stably under no-load conditions, the armature winding current is negligible and can be considered effectively zero. In these cases, it is crucial to configure the motor's operation type as constant speed operation in ANSYS Maxwell. The speed is designated as the no-load speed, and the external circuit excitation is deactivated in Maxwell Excitation, setting it to circuit excitation with zero excitation current. The chain waveforms and reverse potential waveforms of the three-phase winding under no-load conditions are obtained after simulation and analysis using a static magnetic field type, as illustrated in Figures 13 and 14, respectively.

The load state differs from the no-load state; therefore, it is not necessary to configure the motor to operate at a constant speed or to maintain the current excitation level. After the settings are established in Maxwell, a simulation analysis is conducted to obtain the flux density distribution of the motor under load conditions (see Fig. 15). The magnetic field distribution in this state is similar to that in the no-load state, with no evidence of internal oversaturation. The maximum flux density is observed at the intertooth region of the motor stator, estimated to be approximately 1.77 T.

The motor losses can be categorized into several types: stator winding CuL, CL, ECL, and mechanical losses. The variation of these four types of losses is influenced by factors such as stator current, the frequency of the main magnetic field, motor speed, and other parameters. In this paper, the time-domain curves of the motor's iron CL and the PM ECL are obtained using the FEA method in ANSYS, as illustrated in Figures 16 and 17. The average value of the iron CL is 22.56 W, while the average value of the PM ECL is 6.44 W.

It is important to note that, if without considering current harmonics and skin effect, the equation (14) can be deduced to determine CuL:

$$P_{cu}=m{I}_1^2{R}_s$$(14)

where P_cu is the CuL, m is the number of phases, I₁ is the effective value of phase current, and R_s is the average resistance of phase winding. When the number of phases is set at 3, the RMS value of the phase current is approximately 19.7 A, the average resistance of the phase winding is estimated to be around 0.011 Ω, and the CuL is calculated to be about 12.8 W.

Zhu [39] determined that the fluctuation period of the CT in a PM motor is equivalent to the least common multiple of the number of poles and the number of cogging slots. The brushless DC motor designed in this paper features 24 poles and 27 slots. The calculation of the CT fluctuation period is based on the least common multiple of the number of poles and slots, NLLCM (24,27)=216, and the calculated period is 360°/108= (10/3)°. In ANSYS Maxwell, the external circuit of the brushless DC motor is disconnected, and the excitation is modified to current excitation, with the excitation current set to zero. Subsequently, the rotational speed is established at 1 rad/s. A simulation analysis is conducted with a time step of 0.02 s and a total duration of 3 s to output the CT of the motor, as illustrated in Figure 18. The maximum CT is approximately 0.041 N · m.

ET is defined as the torque generated by the interaction between the magnetic field of a PM and the current flowing through the stator windings. In the context of electromechanical energy conversion and energy efficiency, it is essential to consider the energy consumed by the armature windings, which is entirely converted into the mechanical energy of rotor rotation, assuming that copper and iron losses are disregarded. The torque waveform of a brushless DC motor at its rated speed is depicted in Figure 19. After reaching a steady state, the torque fluctuates around 39.6 N · m, with a peak torque value of 44.3 N · m. The torque pulse within a 15° mechanical angle is 3, and the ripple period is 5°, which does not correspond to the CT period or the current commutation period. The influence of CT on steady-state torque is negligible. Further analysis of the harmonics of steady-state torque reveals the Fourier series representation of the torque output order diagram (see Fig. 20).

As demonstrated in Figure 20, the 12th harmonic torque is identified as the dominant force, indicating that it is the primary harmonic responsible for TR. Eliminating the 12th harmonic from the motor has been shown to result in a significant reduction in TR. The average torque and TR coefficient of the motor can be calculated using equation (15) and (16), respectively, facilitating a comprehensive analysis of TR. The magnitude of the motor's TR coefficient is directly proportional to the severity of the TR. The maximum torque is estimated to be 44.32 N · m, the minimum torque is estimated to be 34.73 N · m, and the average torque is estimated to be 39.525 N · m.

$$T_{avg}=\frac{T_{\max }+T_{\min }}{2}$$(15)

$$TPF=\frac{T_{\max }-T_{\min }}{T_{avg}}\times 100\%$$(16)

where, T_max is the maximum torque (KN · m), T_min is the minimum torque (KN · m), T_avg is the average torque (KN · m) and TPF is the TR factor.

The calculated TR factor is estimated to be 24%, which is significant. The motor's structure and current commutation influence this factor [40]. The TR can potentially be reduced through more precise current commutation control and harmonic suppression techniques, without altering the motor's structure.

The results of the transient electromagnetic analysis indicate that the optimized multi-module spliced structure exhibits favorable electromagnetic characteristics under dynamic operating conditions. The no-load reactive potential waveform displays quasi-sinusoidal characteristics, with a third harmonic distortion rate of only 5.3%, thereby validating the effectiveness of the pole-slot fit for harmonic suppression. The load flux density distribution reveals that the maximum magnetic density of 1.77 T is concentrated at the stator tooth tip, which is 12.1% higher than under static magnetic field conditions, yet still below the material threshold. This suggests that the magnetic circuit design maintains a reasonable safety margin under dynamic load. The loss analysis confirms that the ratio between the CL (22.56 W) and the CuL (12.8 W) aligns with the expected profile of high torque density motors. The observed ECL in the PM (6.44 W) is predominantly induced by high-frequency harmonic magnetic fields. This conclusion is reinforced by the identification of 12 dominant harmonics in the subsequent harmonic analysis. It is important to note that the simulated TR coefficient of 24% is slightly higher than the initial optimization expectations. However, an examination of its underlying mechanism reveals that the 12th harmonic accounts for 68.5%, which aligns with the fundamental wave period characteristics predicted by the theoretical model in equation (10). This finding provides a clear direction for the subsequent optimization of the control strategy. The synergistic analysis of the dynamic magnetic density distribution and loss characteristics confirms the effectiveness of the parameter optimization discussed in Section 3.2 in mitigating magnetic saturation and reducing CL. Furthermore, it elucidates the coupling relationship between harmonic losses and TR, thereby establishing a comprehensive theoretical framework for the error traceability of the prototype test presented in Section 5.

Fig. 13 No-load AG magnetic chain waveform.

Fig. 14 No-load reverse electromotive force waveform.

Fig. 15 Flux density diagram under load.

Fig. 16 Time-evolving CL waveform chart.

Fig. 17 Time-evolving ECL waveform chart.

Fig. 18 CT waveform variation with mechanical angle.

Fig. 19 Torque waveform variation with mechanical angle.

Fig. 20 Amplitude variation across harmonic orders.

5 Prototype test

The prototype of the multi-module spliced direct-drive outer rotor brushless DC motor has undergone testing. Various performance parameters of the motor, including efficiency, losses, and TR, have been measured, compared, and analyzed in conjunction with the simulation results. This process validates the soundness of the motor design and the effectiveness of the optimization, providing a foundation for further improvement and applications of the motor.

The prototype test rig conducts research by monitoring the position, speed, and torque of both the transmission input and output. In terms of hardware setup, the motor and transmission are equipped with encoders and torque sensors at both the input and output ends. The torque sensor at the output end (HBM T12) measures the torque of the transmission load, while the input-end sensor captures the motor's output torque. The Heidenhain ROD420 optical encoder (resolution: 1024 pulses per revolution) is used for speed measurement, ensuring angular accuracy within ±0.01°. The output is connected to a large inertia device integrated with a Magtrol HD-100 hysteresis brake to simulate actual operating loads. The model of the test platform and the motor prototype is illustrated in Figure 21.

To ensure the reliability of experimental data, strict environmental control and equipment calibration procedures were implemented throughout the testing process. The test was conducted in a temperature-controlled chamber with the ambient temperature maintained at 25±0.5°C and relative humidity at 45±2% to eliminate the impact of temperature fluctuations on motor resistance and magnetic properties. All key measuring instruments underwent pre-test calibration—the HBM T12 torque sensor was calibrated using a standard weight method with an accuracy traceable to national metrology standards, ensuring a measurement error of less than ±0.1% full scale (FS); the Yokogawa WT3000 power analyzer was calibrated for voltage, current, and power parameters with a calibration certificate valid within 90 days, guaranteeing a power measurement accuracy of ±0.05%. During data acquisition, the NI PXIe-1071 system was set to a sampling frequency of 10 kHz, and each operating condition (0%, 20%, ..., 120% rated load) was maintained for 30 min to ensure thermal stability, with data recorded continuously every 10 s to capture transient performance fluctuations.

The motor's efficiency, losses, and TR are evaluated via no-load and load tests. High-precision calibrated sensors collect speed and torque data, with a pre-calibrated power analyzer recording electrical parameters. Tests use inertial load simulation and signal isolation for reliability, and data are sampled at 10 kHz. Measured results are compared with simulations to validate optimization.

The performance parameters, including motor efficiency, losses, TR, and others obtained from the test measurements, are compared with the simulation results. The findings are presented in Figures 22–24.

From the graphs above, it is evident that the test values closely align with the simulation values, and the errors fall within an acceptable range. The minimal discrepancies in each data point suggest that the design and optimization methods employed for the motor are both accurate and reliable. Notably, the test value of the TR coefficient is slightly higher than the simulation value. This discrepancy may be attributed to process errors during the prototype's manufacturing, assembly precision, and measurement inaccuracies during testing.

Firstly, TR is analyzed. The torque data collected during the test was thoroughly analyzed. The results indicate that the TR of the motor is primarily attributed to CT and current commutation. The fluctuation period of the CT corresponds with the simulation results and is measured at (10/3)°. A Fourier transform of the torque data revealed that the 12th harmonic is the main contributor to the TR. Although the TR coefficient obtained from the test is slightly higher than the simulation value, the overall level of TR remains within an acceptable range and does not significantly impact the normal operation of the motor. If further reduction of the TR is necessary, optimization of the motor's cogging structure, enhancement of the current commutation control strategy, or the implementation of harmonic suppression techniques can be considered in future designs.

Secondly, the temperature rise has been tested. The test results show that the temperature of each motor component rises progressively and stabilizes thermally after 2 h of continuous operation under rated load conditions. Notably, the maximum temperature of the stator winding is 75 °C, while the maximum temperature of the PM is 68 °C. Both temperatures are below the permissible working temperature of the motor's insulating material, suggesting that the motor's heat dissipation design is adequate to support long-term stable operation.

The optimized motor achieves an efficiency of 95.06% in experiments, with a deviation of only 0.3% from the simulation result (95.3%), verifying the accuracy of the FEA model. CL and CuL measured in tests are 22.9 W and 13.1 W, respectively, which are consistent with the simulation values (22.56 W and 12.8 W) within an error range of ±3%.

CT is reduced by 38.89% from the initial design (0.036 N · m to 0.022 N · m) in both simulation and experiments, with the experimental TR coefficient (26%) slightly higher than the simulation result (24%) due to assembly tolerances, but still meeting the robotic actuation requirement of <30%.

The rationality of the motor design and the optimization effects are validated through an experimental study of the prototype of the multi-module spliced direct drive outer rotor brushless DC motor. The test results are largely consistent with the simulation outcomes, and the performance parameters of the motor meet the design specifications. Although the measured TR coefficient slightly exceeds the simulated value, the overall performance remains well within acceptable limits and meets the stringent requirements for robotic actuation systems. Future research should focus on optimizing the manufacturing process and control strategies of the motor to reduce TR and enhance both performance and reliability. Additionally, more in-depth investigations into the motor's performance under varying operating conditions can provide a more robust theoretical and practical foundation for the broader application of motors in the field of robotics.

Fig. 21 Diagram of the test platform.

Fig. 22 Efficiency comparison.

Fig. 23 Loss comparison.

Fig. 24 CT comparison.

6 Conclusion

This paper presents the design and optimization of a direct-drive outer rotor brushless DC motor for robotic actuation, focusing on achieving high torque density, efficiency, and manufacturability. The multi-module spliced rotor, with split-slot winding and modular stator teeth, balances magnetic performance with ease of manufacturing.

Quasi-Newton-based multi-objective optimization led to significant improvements: efficiency reached 95.06%, core loss reduced to 22.5 W, copper loss stabilized at 12.8 W, and cogging torque decreased by 38.89%. Electromagnetic analysis showed minimal leakage and no saturation, while transient analysis identified harmonic torque ripple at 24%. Prototype testing validated the simulation results, confirming efficiency, loss figures, and torque pulsation within acceptable limits.

But this study assumes ideal conditions for materials and focuses on steady-state performance, neglecting long-term effects like magnet demagnetization or cyclic loading. The current design does not fully address harmonic torque ripple during rapid acceleration. Future work will refine the cogging structure to reduce harmonic ripple, improve manufacturing tolerance compensation, and integrate active temperature control to better manage thermal risks. Additionally, transient behaviors will be incorporated to improve performance prediction under varying operational conditions.

Funding

This work is supported by the Shaanxi Provincial Natural Science Foundation (2024JC-YBMS-262) (2024JC-ZDXM-04); Shaanxi Provincial Education Department Youth innovation team research project(24JP006).

Conflicts of interest

The authors have nothing to disclose.

Data availability statement

This article has no associated data generated and/or analyzed / Data associated with this article cannot be disclosed due to legal/ethical/other reason.

Author contribution statement

Conceptualization, Yuewen SU; Methodology, Caixia GUO and Liya SHI; Writing, Chao YU; Validation, Xubo LI and Keli YANG; Review, Canjun Wang. All authors have read and agreed to the published version of the manuscript.

Supplementary Material

Appendix 1–Motor Optimization Model And Detail Solution Proce.

Appendix 2–Figures(1-5)

Appendix 3–Figures(6-9)

Appendix 4–Figures(10-14)

Appendix 5–Figures(15-20)

Appendix 6–Figures(21-22)

Appendix 7–Figures(23-24)

References

All Tables

All Figures

	Fig. 1 Three-phase winding connection.
In the text

	Fig. 2 Influence of AG length on efficiency.
In the text

	Fig. 3 AGFD for different PAC.
In the text

	Fig. 4 CT for different PAC.
In the text

	Fig. 5 Multi-module spliced stator with split-slot windings.
In the text

	Fig. 6 Multi-module outer rotor BLDC motor assembly.
In the text

	Fig. 7 The motor model established by RMxprt.
In the text

	Fig. 8 The two-dimensional model in Maxwell.
In the text

	Fig. 9 The magnetic field of motor.
In the text

	Fig. 10 Local amplification magnetic chain cloud diagram.
In the text

	Fig. 11 Static magnetic field flux density.
In the text

	Fig. 12 Static magnetic field AG magnetic chain waveforms.
In the text

	Fig. 13 No-load AG magnetic chain waveform.
In the text

	Fig. 14 No-load reverse electromotive force waveform.
In the text

	Fig. 15 Flux density diagram under load.
In the text

	Fig. 16 Time-evolving CL waveform chart.
In the text

	Fig. 17 Time-evolving ECL waveform chart.
In the text

	Fig. 18 CT waveform variation with mechanical angle.
In the text

	Fig. 19 Torque waveform variation with mechanical angle.
In the text

	Fig. 20 Amplitude variation across harmonic orders.
In the text

	Fig. 21 Diagram of the test platform.
In the text

	Fig. 22 Efficiency comparison.
In the text

	Fig. 23 Loss comparison.
In the text

	Fig. 24 CT comparison.
In the text