Key Parameters Design of Robot Joint Motor Based on Frozen Permeability

Abstract

Robot joint motors have the characteristics of high torque density, low torque fluctuation, short-term high-overload, and light weight. Conventional design methods only focus on the overload capacity, without precisely analyzing the stator-rotor parameters and the effect of magnetic saturation on load torque fluctuation. Combining with the frozen permeability technology, the effects of structural parameters such as tooth width, slot opening, permanent magnet thickness and pole arc coefficient on the fluctuating component of load torque and ultimate torque are studied. Then, based on the performance requirements of joint motors for quadruped robots, a compact design joint motor for robots was designed to achieve a maximum torque overload capacity of 7.33 times and low torque fluctuation. A design method for a high-overload, low torque fluctuation robot joint motor was obtained, and finally the effectiveness of the proposed method was verified by prototype experiments.

1. Introduction

Robot joint motors (hereinafter referred to as “joint motors”) are characterized by miniaturization, light weight, low speed, high torque, and low torque fluctuation [1]. When the robot is walking, running, jumping, and other actions, the joint motor is often in short-time high-overload operation and the motor torque overload multiplier can reach more than 3–5 times the rated load; at this time the motor is in high saturation operation of the magnetic field, and the magnetic saturation of the motor will lead to increased current harmonics, torque to current ratio coefficient drop, and large torque fluctuations. In order to achieve smooth operation of quadruped robots, it is extremely important to optimize torque fluctuations under overload conditions for articulated motors that are often subjected to high overload operating conditions.

Joint motors often use integrated modules of motor and retarder in order to obtain sufficient torque, and this method can increase torque by reducing the speed of the motor. However, the modular design will increase the overall size and cause difficulties in assembly, while a larger reduction ratio will reduce the response speed of the motor. Therefore, by optimizing the motor stator and rotor parameters to increase the motor output torque, the speed of response can be improved by reducing the reduction ratio, and the overall size of the joint motor can be reduced, while the overall weight can be reduced.

In the design of high overload motor, the traditional design method of electromagnetic scheme can no longer meet the needs of high overload working conditions. The design method of a high overload motor can be obtained by comprehensive analysis of the influence law of stator-rotor structure parameters such as pole-slot combination, magnet thickness, stator split ratio, tooth width on output torque [2,3], and the temperature rise variation law under different operating conditions [4]. Y. K. Cao et al. of Wuhan University [5] combined the characteristics of quadruped robot joint motors, and comprehensively analyzed and compared the effects of four slot-pole combinations on motor performance from the perspectives of average torque, cogging torque, winding factor, back-emf, air-gap flux density, and unbalance force. Scholars from the Chinese Academy of Sciences analyzed the effects of electrical and magnetic loads on the overload capacity of the motor, as well as the degree of variation of stator current, rotor electric density and iron core magnetic density under overload conditions from the electromagnetic torque of the asynchronous motor, and elucidated the variation law between electromagnetic parameters [6]. In addition, the loss distribution of the motor under overload operation was statistically processed by establishing the copper and iron loss calculation models for the optimal design of asynchronous motor overload capacity enhancement [7]. S. W. Hwang of Korea [8] designed a surface-mounted permanent magnet synchronous motor from the electromagnetic and thermal perspectives based on the lumped parameter thermal network according to the electromagnetic performance and dimensional constraints of wearable robots. S. Zhang and Q. Chen identified the key structural parameters affecting the air-gap flux density and the total harmonic distortion of the output torque by multi-target sensitivity analysis, which improved the topology of the motor and enhanced the electromagnetic characteristics of the motor [9,10]. In terms of reducing motor torque fluctuations, magnetic circuit adjustment [11] and direct torque control [12] minimize torque fluctuations by improving the number of stator slots [13] and stator shape [14]. Overload capability and torque fluctuation are the focal points of joint motors, but how to optimize the motor body design to improve motor output torque while reducing torque fluctuation under overload conditions has been less studied; moreover, due to the mutual coupling of the stator and rotor magnetic fields when the motor is loaded, the conventional calculation method cannot accurately separate the fluctuating component of the load torque.

To address this situation, this paper makes the following contributions to the design methodology of joint motors for quadruped robots with high overload and low torque fluctuation as the design goal:

• Combined with the frozen permeability (hereinafter referred to as FP) method, the influence law of stator-rotor parameters such as tooth width, slot opening, permanent magnet thickness, and pole arc coefficient on load torque fluctuation are summarized to achieve smooth controllability of overload conditions of joint motors;
• Analyze the influence law of stator-rotor parameters such as tooth width, slot opening, permanent magnet thickness, and pole arc coefficient on the ultimate output torque to improve the overload capacity of the joint motor;
• A comprehensive analysis of the motor body design provides a high overload, low torque fluctuation joint motor design method.

2. Fundamental and Model

2.1. Mathematical Model

In terms of motor type selection, permanent magnet synchronous motors (hereinafter referred to as PMSM) are the most commonly used drive motors for quadruped robot joints due to their high efficiency, light weight, and low torque fluctuations. In a surface-mounted permanent magnet synchronous motor, the motor output torque is usually defined as:

$$T=T_{PM}+T_C,$$

(1)

where $T_{PM}$ is permanent magnet torque, $T_C$ is the open circuit cogging torque. From the equation, it can be seen that the main influencing factor of torque fluctuation is the open cogging torque. However, when the motor is in a high overload condition, the counter potential, inductance, and current waveform distortion also affect the torque fluctuation due to the magnetic saturation.

Based on Maxwell’s stress tensor method, the electromagnetic torque of the motor is generated by the tangential force, and if integrated along a circle of radius r, the expression for the total electromagnetic torque in the motor is:

$$T_{em}=\frac{L_{ef}}{{\mu }_0}{\displaystyle \oint r^2{B}_r{B}_{\theta }d\theta },$$

(2)

where ${\mu }_0$ is the vacuum permeability; $L_{ef}$ is the effective axial length of the motor; r is the arbitrary circumference radius in the air gap, $B_r$ and $B_{\theta }$ are the normal and tangential components of the air gap magnetic density, respectively.

The FP technique is based on the principle that the stator-rotor magnetic field enjoys the same permeability distribution, which allows decoupling of the stator-rotor magnetic field. By analogy with (2), the motor load torque fluctuation component during load operation can be calculated by Maxwell’s stress tensor method based only on the loaded permanent magnets (hereinafter referred to as PM).

$$\begin{array}{cc}\hfill T_{\mathrm{c}}\left(\mathrm{load}\right)& =T_{\mathrm{mw}}(FP,PM)\hfill \\ & =\frac{L_{ef}}{{\mu }_0}{\displaystyle \oint r^2{B}_r(FP,PM)B_{\theta }(FP,PM)d\theta }\hfill \end{array},$$

(3)

where $T_{mw}(FP,PM)$ is the calculated torque based on Maxwell’s stress tensor method with a loaded PM field only, $B_r(FP,PM)$ and $B_{\theta }(FP,PM)$ are the normal and tangential air-gap flux density components due to the loaded PM field only.

2.2. Principle of Frozen Permeability

Taking the permanent magnet motor as an example, the principle of FP can be explained with the help of the B-H curve in Figure 1. The working points jointly excited by permanent magnet and armature current, separately excited by permanent magnet, and separately excited by armature current are, respectively, represented by points a, b, and c, and the corresponding magnetic flux densities are, respectively, B_all, B_PM, and B_i. Due to the nonlinearity of soft magnetic materials, $H_{all}=H_i+H_{PM}$, $B_{all}<B_i+B_{PM}$, the load magnetic field component of the motor cannot be linearly decomposed.

Using the FP method, the motor becomes d and e at the operating points where the permanent magnets are separately excited and the armature currents are separately excited, corresponding to flux density B(FP, PM) and B(FP, i). At this point, the three calculation methods are based on the same permeability, $B_{\mathrm{all}}=B(FP,PM)+B(FP,i)$, and the loaded PM and armature magnetic field components are accurately decomposed. In addition, magnetic saturation and cross-coupling are inherently taken into account since ${\mu }_{all}$ varies with the load conditions.

As can be seen from Figure 1, the principle of the frozen permeability method is to linearize the coupled magnetic field and thus achieve separation of the stator-rotor field for the purpose of separating the torque fluctuation components at load. The differences between the motor body design in combination with the frozen permeability method and the conventional design method are shown in

Table 1. The difference between proposal and traditions.

Design Method	Traditional Method	Proposed Method
Magnetic saturation	×	√
Cross-coupling	×	√
Load torque fluctuation	×	√

.

In Table 1, “×” means outside the calculation range of the design method, and “√” means within the calculation range of the design method. As can be seen from Table 1, the difference between the proposed design method and related design methods is reflected in whether the motor scheme design can adequately consider the magnetic saturation and cross-coupling effects on load torque fluctuations under load conditions, which are the key factors to be considered in the design of high-overload, low-fluctuation robot joint motors.

2.3. Main Performance Requirements of Motor

According to the requirements of the whole machine of the quadruped robot, the main parameters of the joint motor studied in this paper are shown in

Table 2. The main performance requirements of motor.

Parameter	Value
Weight	≤380 g
Voltage level	48 V
Motor outer diameter	≤94 mm
Rated torque	2.32 N·m
Peak torque	6.96 N·m
Rated speed	2300 rpm

.

3. Optimization of Parameters

From Equation (2), it can be seen that the motor torque characteristics depend mainly on the integrity of the product of the tangential and normal components of the air gap magnetic density, with the main dimensions of the motor remaining unchanged. The motor tooth width, slot opening, permanent magnet thickness and pole arc coefficient are the main factors affecting the air gap magnetic density. Therefore, it is necessary to analyze the influence law of these key parameters on the torque performance of the motor. Figure 2 shows the key dimensions of the motor, in which L₁ is the tooth width, L₂ is the slot width, and L₃ is the permanent magnet thickness.

3.1. Combination of Slot-Pole

For quadruped robots, there are certain constraints in the design of joint motors. Taking into full consideration the heat dissipation and temperature rise of the motor in a small space, and reasonably choosing the combination of slot-pole of the motor could allow the motor to output a larger torque and greatly reduce the torque fluctuation. The winding factor $k_w$ varies for motors with different slot-pole combinations. A comparison of the winding factors of the common fractional slot concentrated winding motors for joint motors is shown in

Table 3. The winding factor of the fraction slot motor.

Slot Number	Pole Number
	24	26	28	30	32
18	0.866	/	/	/	/
24	/	0.945	0.966	/	0.866
30	/	0.936	0.951	/	0.951
36	0.866	0.87	0.902	0.966	0.945

.

A smaller winding factor can improve the torque overload capacity of PMSM, but from the perspective of reducing the total weight of the motor, the yoke thickness of motors with different slot-pole combinations varies; the higher the number of slot-pole, the thinner the yoke thickness of the motor and the lighter the motor weight. Considering various factors such as motor winding factor, torque fluctuation, axial length and controller frequency, a 36-slot 32-pole motor is a better and feasible choice to achieve high torque, low fluctuation, light weight and meet the size requirements of quadruped robot joints.

3.2. Influence of Structural Parameters on Load Torque Ripple Component

The cogging torque is one of the main factors causing torque fluctuations and is the focus of conventional design methods. However, due to the influence of magnetic saturation, cogging torque is no longer suitable for the study of high overload torque fluctuation of joint motors, and the load torque fluctuation component is more realistic and effective to reflect the torque fluctuation under load conditions.

Considering the limitation of the axial length of the motor, it is not appropriate to reduce the torque fluctuation of the motor by using the skewed pole or skewed slot scheme, and the optimized design of the motor stator-rotor parameters is the preferred choice. However, due to the mutual coupling of stator-rotor magnetic fields, conventional calculation methods cannot accurately separate the stator-rotor magnetic fields and cannot more accurately evaluate the individual effects of stator-rotor parameters on the load torque fluctuation components. The FP method can be used to decouple the stator-rotor magnetic fields and qualitatively analyze the influence law of each parameter on the load torque fluctuation of the motor as shown in Figure 3.

From Figure 3a, it can be seen that the load torque fluctuation component of the motor is negatively correlated with the tooth width. The reason for this is that as the tooth width increases, the magnetic saturation of the stator teeth decreases, and the magnetic flux through the stator tooth tips increases, and thus smaller flux leakage leads to a decrease in the load torque fluctuation component. The trend of the load torque fluctuation component of the motor with the slot opening is shown in Figure 3b. As the slot opening increases, the magnetic leakage generated by the magnetic saturation of the stator tooth tip decreases, so the load torque fluctuation component decreases. The thickness of the permanent magnet is in the range shown in Figure 3c, and the load torque fluctuation component fluctuates within 2.4%. The variation of the load torque fluctuation component with the pole arc coefficient is shown in Figure 3d when the stator parameters are kept constant, and the increase in the pole arc coefficient increases the saturation effect of the rotor magnetic field on the stator tooth tip, which leads to the increase in the load torque fluctuation component.

In order to analyze the output performance of the motor more comprehensively, the variation law of the motor ultimate output torque with the stator and rotor parameters is analyzed below.

3.3. Influence of Structural Parameters on Ultimate Output Torque

The output torque of PMSM is affected by both electrical and magnetic loads, and the conventional motor structure design is not applicable to high-overload motors because of the large armature current. The variation law of the maximum motor torque with the stator-rotor parameters can be obtained by continuously increasing the input current within the voltage constraint as shown in Figure 4.

From Figure 4a, it can be seen that an appropriate increase in tooth width can effectively increase the ultimate output torque of the motor because a larger tooth width can keep the stator core permeability in the linear segment, which is conducive to achieving torque overload. The effect of slot opening on the ultimate output torque of the motor is shown in Figure 4b; with the increase in slot opening, the saturation of the stator tooth tip decreases, which in turn can produce a larger ultimate output torque. The increase in the thickness of the permanent magnet and the pole arc coefficient increases the magnitude of the magnetic potential per pole pair, which increases the magnetic load and thus increases the ultimate output torque.

4. Finite Element Simulation

Through the comprehensive analysis of the above motor stator-rotor structure parameters to realize the high torque overload multiplier, low torque fluctuation, light weight, and miniaturization in order to meet the design requirements of the joint motor, the design scheme of the motor is finally determined as shown in

Table 4. The main parameters of the high overload motor.

Parameter	Value
Slot number	36
Pole number	32
Magnet thickness	3 mm
Pole arc coefficient	0.91
Air gap length	0.5 mm
Tooth width	2.9 mm
Slot opening	2 mm
Stator outer diameter	94 mm
Stator inner diameter	74.5 mm
Axial length	18 mm
Weight	367.1 g

.

The established finite element model is shown in Figure 5, and the no-load and load characteristics, as well as the ultimate output torque of the model, are simulated and analyzed.

4.1. No-Load Characteristic

Firstly, a no-load performance calculation was performed to obtain the back-emf of the motor at 2300 r/min no-load condition as shown in Figure 6a,b, and the no-load cogging torque waveform as shown in Figure 6c.

The phase back-emf amplitude of the motor at 2300 r/min no-load condition is 23.43 V, and the waveform is relatively smooth, and the harmonics are relatively small as seen in the back-emf spectrum. The peak-to-peak value of a no-load cogging torque is 0.0084 N·m, around 0.4% of the rated output torque.

4.2. Rated State Analysis

At the rated condition, the phase current of the motor is 12.4 A RMS, and the simulation results at rated load are presented in Figure 7. Currently, the rated output torque is 2.32 N·m and the maximum value of stator tooth magnetic density is 1.59 T.

In the rated condition, the maximum value of tooth magnetic density of the motor is less than 1.8 T, the stator core works in the linear section, and the output torque at rated current is 2.32 N·m, which can meet the requirements.

Define the torque fluctuation as:

$$T_{ripple}=\frac{T_{\mathit{max}}-T_{\mathit{min}}}{2\ast T_{avg}}\times 100\%,$$

(4)

According to Equation (4), the simulation value of torque fluctuation of motor output torque at rated condition is 0.62%, and the torque fluctuation is small.

4.3. Analysis of Triple Overload State

Under a triple overload condition, the phase current of the motor is 38.5 A RMS. The simulation result of triple overload condition is displayed in Figure 8; currently, the average value of output torque is 6.96 N·m. The back-emf RMS of load condition is 18 V (the phase RMS of end voltage is 27.8 V), which is within the end voltage limit. The maximum value of tooth density in Figure 8c is 1.85 T, at which time the stator core enters the saturated working section, and if the input current is further increased, the output torque growth slows down until it remains constant.

According to Equation (4), the motor output torque fluctuation at triple overload condition is 0.81%, which is 30.6% higher than that at rated condition.

4.4. Comparison

The results of the no-load cogging torque obtained by the conventional calculation method and the rated load and triple overload load torque fluctuation components obtained by combining the FP method are shown in

Table 5. Cogging torque and load torque fluctuation components.

Operating Conditions	Cogging Torque	Load Torque Fluctuation Components
No-load	8.4 mN·m
Rated load		27.2 mN·m
Triple overload		81.9 mN·m

.

It can be seen from Table 5 that the load torque fluctuation component in rated condition has increased significantly compared to the no-load cogging torque, and the increment of load torque fluctuation component in triple overload condition is more obvious compared to the no-load cogging torque as the magnetic saturation degree increases.

It can be seen from Figure 9 that the period becomes 3.75°, which is three times the no-load cogging torque, due to the joint action of the stator tooth tip magnetic circuit saturation by the three-phase armature magnetic field; further, the degree of magnetic saturation is positively correlated with the load torque fluctuation component, which means that the influence law of magnetic saturation on torque fluctuation under overload conditions is verified, and the results support the design of the proposed scheme.

4.5. Limit Output Torque

When the stator phase current RMS value is 135 A, the load back-emf RMS value is 48 V and the load T-I curve is shown in Figure 10.

As can be seen in Figure 10, the T-I curve of PMSM is divided into two stages.

• Linear section, in which the electromagnetic torque varies essentially linearly with the increase in current;
• In the saturation section, when the current exceeds 30 A, the stator core becomes locally magnetically saturated and the motor output torque tends to change flat, and the higher the current, the more severe the saturation.

The results of finite element analysis show that, combined with the frozen permeability method, the stator-rotor parameters are optimized, and the designed 32-pole 36-slot high-overload motor not only meets the main performance requirements of the motor, but also has a torque fluctuation of 0.81% under triple overload conditions, which can guarantee the operational stability of the joint motor under high overload conditions.

5. Experimental Validation

According to the design results, a 32-pole 36-slot robot joint motor with torque high-overload capability at constant speed was fabricated; the physical diagram of the prototype is shown in Figure 11 and an experimental platform was built for experimental verification.

5.1. No-Load Back-EMF Test

The line back-emf waveform of the prototype at 2300 r/min measured using the pair drag method is presented in Figure 12.

The measured no-load back-emf amplitude of the prototype is 22.71 V. The measured opposite potential is 3.07% smaller than the simulated predicted value because of end effects, manufacturing workmanship differences, and measurement errors.

5.2. Load T-I Curve Test

The ultimate torque experimental platform was built as seen in Figure 13, and the load torque test data are displayed in

Table 6. The measurement results of load torque.

Current	Torque
15 A	2.85 N·m
30 A	5.25 N·m
45 A	7.71 N·m
60 A	9.97 N·m
76 A	11.91 N·m
90 A	13.59 N·m
107 A	15.05 N·m
121 A	16.17 N·m
134 A	17.01 N·m

.

Figure 14 lists the T-I curve of the prototype compared to the simulated motor.

As can be seen from Figure 14, the motor load T-I curve is not an ideal straight line (shown by the dashed line in the figure) due to the effect of stator core magnetic density saturation. When the input current of the prototype is 135 A, the maximum output torque reaches 17 N·m, achieving a torque overload of 7.33 times, and the experimental results are basically consistent with the simulation results.

The effectiveness of the FP method has been confirmed in the article [15]. The shortcoming of the method is that the load torque fluctuation components cannot be separated accurately by an experiment based on the limitations of the current equipment technology.

6. Summary

According to the requirement of high torque and low fluctuation of a quadruped robot joint motor, this paper improves the output torque by changing the stator-rotor parameters to weaken the magnetic saturation of core, and then separates the load torque fluctuation components under load conditions by combining with the method of frozen permeability, and qualitatively analyzes the influence law of stator-rotor parameters, such as tooth width and slot opening on the load torque fluctuation components. A 32-pole 36-slot quadruped robot joint motor is designed to achieve a maximum overload multiplier of 7.33 and a torque fluctuation of 0.81% under triple overload conditions, which ensures the smooth operation of the joint motor under overload conditions. The simulation and experimental results show that under the load condition, due to the influence of magnetic saturation, the cogging torque is no longer applicable to the study of torque fluctuation under overload condition. The load torque fluctuation component can more truly and effectively reflect the torque fluctuation under overload condition and the degree of magnetic saturation is positively correlated with the load torque fluctuation component, which can better guide the design of high overload joint motor and, at the same time, it verifies the combination of frozen permeability method to achieve low torque fluctuation. The proposed method also demonstrates the rationality of combining the frozen permeability method to achieve low torque fluctuation and high torque overload multiplier joint motor design, which is of great practical value in engineering. However, the proposed method requires manual point-by-point calculation of motor performance, which is a bit tedious. In the future work plan, the output torque should be further increased while reducing the overall mass of the joint motor module from the perspective of motor structure and material to achieve a lightweight design, while optimizing heat dissipation to provide a complete design system for robot joint motors that can be used as a reference.