Algorithm for the torque sensorless worm gearbox servo application based on kinetic motion/friction realization

Abstract

This paper provides an effective systematic procedure for the efficient treatment of worm gearboxes for industrial-grade cases. The accurate and general dynamic model of the worm gearbox is analytically obtained and the effects of the dominant factors, such as the disturbance torque and friction between the teeth and threads, are also addressed. A universal algorithm is proposed for accurate estimation of the disturbance torque that makes no utilization of any designated sensing unit. Numerous experimentations have been conducted to validate the proposed procedure and algorithm. The contributions of this paper are not limited to worm gearboxes but rather are extendable to all other conventional gearboxes that use similar power transmission methods. The gradient patterns of the normal force, the friction force, and the coefficient of friction in worm gearboxes are also identified as a function of the effective elements, such as the output loading torque and angular velocity.

Keywords

Worm gear modeling internal forces detection kinetic motion-friction modeling disturbance torque estimation coefficient of friction analysis

1. Introduction

Worm gearboxes have drawn extensive attention in recent years. They have numerous advantages, such as a high transmission ratio while occupying small spaces, the self-locking phenomenon, and low manufacturing costs relative to their efficiency. Worm gears and some other types of gearbox systems are widely utilized in industrial applications and, in copious cases, they are considered as an adequate and even essential replacement for multiple expensive equipment with notable maintenance costs. The steering system of modern cars is a paragon of this replacement, where expensive hydraulic systems with higher maintenance costs and larger space occupations are substituted with worm gearboxes. It is also noteworthy to mention that in many precise engineering applications, worm gearboxes have also emerged as a preferred choice, such as the gimbal systems carrying mobile satellite antennas; such precise systems were previously driven via hydraulic or pneumatic actuation systems but it is now apparent that worm gearboxes and harmonic drive gearboxes are the preferable options. Although both harmonic drive gearboxes and worm gearboxes are theoretically considered to be good options for precise engineering applications, it must be noted that the latter has a significantly better accuracy-to-price ratio and, henceforth, in practical cases, it is efficient to favor worm gearboxes. From a mechanical point of view, worm gearboxes also demonstrate more desirable characteristics relative to harmonic drive gearboxes, such as occupying small spaces and tolerating larger torques before the teeth–threads begin to undergo significant deformation. Harmonic drive gearboxes also suffer from several other drawbacks, such as the existence of fatigue, hysteresis, and the flexibility of Flex Spline and Circular Spline teeth that cause undesirable vibration and reduced accuracy of the output shaft. These common drawbacks of harmonic drive gearboxes are not present in the worm gearbox. Despite all the advantages discussed for worm gearboxes, it is necessary to state that these systems suffer from a notable amount of power dissipation; henceforth, it is essential to accurately model the dynamics of the worm gearbox and identify the interactions of the internal forces. By introducing a precise model for the dynamical behavior of the worm gear, it is possible to employ such systems in the structure of control systems that require a high level of accuracy. This employment can be observed in some studies,¹ while other studies^2,3 have utilized the worm gearbox setup in a control loop for a precise engineering task.

Two points are of significant importance in the valid and accurate modeling of worm gearboxes. Primarily, when the gearbox experiences conventional loading, which is the case in most applications, the teeth of the gear and the threads of the worm are negligibly deformed and there is no need for considering them as flexible structures. The materials of the teeth and threads are selected to fail and experience fracture in the case of excessive loading, and so the flexibility analysis of the teeth and threads is not important. Nevertheless, many studies have been conducted on the flexibility analysis of threads. Zeng et al.⁴ used continuum mechanics and finite elements methods to offer a detailed analysis. Recent advancements in this field are also concerned with designing gears using polymer materials,⁵ which are known to illustrate fractional behavior.⁶ They can also be discussed dynamically, but more sophisticated tools must be selected for the efficient treatment of materials with fractional-order dynamics.^7–9 Despite the widespread applications of worm gearboxes in industry, the researches conducted on the dynamical analysis of such systems are not properly advanced and cannot provide a foundation for adequate employment of worm gearboxes in precise servo applications. This point is briefly discussed by Yeh and Wu¹⁰ and is mainly concerned with analyzing the worm gearbox, both kinetically and kinematically, which is amongst the very few valid approaches in accurate modeling of the motion of worm gearbox systems. This study provides a detailed analysis of the different engagement configurations of the teeth and threads in the case of correct engagement conditions and also in the case of erroneous engagement conditions, which is completely contingent upon the active interval of the coefficient of friction.

Despite the insightfulness of this study and its multiple contributions to the literature, it lacks extensive discussions on several critical cases that arise in practice and are essential in laying the foundations of this field. The behavior of the coefficient of friction and internal forces of the worm gearbox under different loadings in dissimilar velocities during distinct configurations are not explored in spite of their direct effect on the motion of the system. The appropriate identification of the behavior of the internal forces results in accurate estimation of the disturbance torque, which is not known in real world cases.

In this paper, a standard procedure for universal and accurate modeling of worm gearboxes is proposed and the governing dynamic equations of the system are obtained that are extendable to other conventional gearboxes that utilize the tooth–thread engagement as a basis for power transmission. The gradient patterns of the friction force, coefficient of friction, and the normal force are extensively analyzed and several experiments have been conducted to determine their variation patterns. Subsequently, a general procedure in which the worm gearboxes are recommended to be utilized is highlighted. In addition, a comprehensive discussion on the algorithm for accurate estimation of disturbance torque that is applied to the gearbox has been held. This estimation is closely related to the internal forces of the worm gearbox system. It is important to state that in the modeling process, the conventionally disregarded factors, such as the disturbance torque and the friction between the tooth and its corresponding engaged thread, are also addressed. The main contributions and novelties of this paper can be summarized as follows:

experimental and universal identification of the gradient patterns of the normal force, the friction force, and the coefficient of friction in worm gearboxes as a function of pragmatic and practical elements, namely, output loading torque and angular velocity;

accurate analytical study of the effectiveness level of the disregarded yet significant factors, such as the disturbance torque and the friction of teeth–threads, on the output of the worm gearbox;

introducing a novel procedure for accurate estimation of the disturbance torque without the use of any designated sensing and transducing unit.

Section 2 begins with stating all the assumptions that were employed in the obtainment of the dynamic model of the worm gearbox. Subsequently, the modeling procedure of the worm gearbox is discussed and the governing relations are extracted. In addition, the extracted relations are combined with the relations of the driving actuator. Section 3 is mainly concerned with experimentation and begins with the introduction of the experimental setup that is used to acquire the variation of frictional and normal forces under different kinematic and kinetic conditions. An experimental analysis of the internal forces and the coefficient of friction of the worm gearbox is also presented in this section. Finally, a novel disturbance torque estimation algorithm, based on the conducted experiments and with logical mathematical procedure, is proposed and its error percentage is analyzed.

2. Mathematical modeling of the worm gearbox based on governing kinetic-kinematic schemes

2.1. Modeling assumptions

The following assumptions have been presumed and are highly consistent with practical cases.

In the free body diagram (FBD) of the worm and gear, only friction and normal forces are considered and other negligible forces are ignored.

In practical cases, multiple teeth of the gear and threads of the worm are in contact with each other, but in the analysis provided in this paper, it is assumed that only one tooth and one thread are engaged at every instance. It is also noteworthy to state that the distributed forces between the worm and the gear are modeled as point forces that are represented by $N$ and $μ N$ . The aforementioned forces are exerted to a single point on the tooth–thread that is demonstrated by $A$ in all figures of this paper.

Due to the fact that there exists a small but not negligible backlash, the engagement of the worm and gear occurs in a single side of the tooth–thread (that is, both sides of the tooth–threads are not in contact and one side is always free), but upon a change in the direction of rotation and the loading conditions, the engaged side might change.

The threads of the worm and the teeth of the gear are assumed to be rigid structures and it is assumed that no wear happens between the surfaces of the gear and the worm.

The gearbox employs a hydrodynamic lubrication and the dissipations related to the lubrication are included in friction modeling, where the coefficient of friction is assumed to be independent of temperature in a limited temperature interval.

The friction resulting from the rotation of ball-bearings, holding both the worm and the gear, is neglected in the modeling procedure, but their effects are observed in the friction force during the tests. The ball-bearings do not suffer from any longitudinal or lateral clearances.

During the tests, the torque produced by the connected motor is identical to the torque exerted to the input shaft of the gearbox (there exists no power loss in the torque transmission procedure from the output of the motor to the input of the gearbox) and the coupling connecting the motor and gearbox is considered to be rigid. Consequently, the acceleration of the motor’s output shaft is equal to the acceleration of the worm gearbox’s input.

The worm gearbox specifications are defined in accordance to the standard handbook,¹¹ where the input torque (applied to the worm), disturbance torque (exerted to the gear), mass inertial moment of the worm, mass inertial moment of the gear, worm radius, gear radius, helix angle of the worm, pressure angle, and coefficient of friction are demonstrated as $T_{1}$ , $T_{2}$ , $J_{w}$ , $J_{g}$ , $R_{w}$ , $R_{g}$ , $γ$ , $ϕ_{n}$ , and $μ$ , respectively.

The positive direction of $T_{1}$ is selected as illustrated in Figure 1(a1) and the positive direction of $T_{2}$ is concurrent to the rotation direction of the gear.

Figure 1.

Schematic view of tooth–thread engagement in the resistive mode and worm gearbox free body diagram (FBD): (a1) left-handed (L-H) helix – $T_{1} > 0$ ; (b1) L-H helix – $T_{1} < 0$ ; (c1) right-handed (R-H) helix – $T_{1} < 0$ ; (d1) R-H helix – $T_{1} > 0$ ; (a2, b2, c2, d2) worm’s FBD for each configuration; (a3, b3, c3, d3) gear’s FBD for each configuration.

2.2. Dynamics modeling of worm gearbox systems

Accurate identification of the internal forces of the gearbox is contingent upon a detailed analysis of the dynamics of the system under various loadings. To facilitate and to enhance the accuracy of this analysis, it is recommended to classify the tooth–thread engagement based on the kinematics of the motion and loading conditions. Upon meticulous observations of the force diagram for the worm and gear system, it became apparent that the tooth–thread engagement can be categorized into three general modes. These three modes represent the behavior of the worm gearbox with universality and are named the resistive mode, pseudo-resistive mode, and assistive mode.

Definition 1 (switching torque $T_{s}$ ). $T_{s}$ is a specified torque that depends on the angular velocity of the system and is applied to the gear while being concurrent with the direction of the motion. This means that if $T_{s}$ is exerted on the gear at its specified velocity, the worm and gear will rotate without any engagement and, subsequently, the normal force will be zero. This specified torque corresponds exactly to the point, with respect to the loading condition, that the surfaces of engagement of the teeth–threads are on the verge of being changed.

Based on the fact that whether the worm’s helix is right-handed (R-H) or left-handed (L-H) and also the rotation direction of the worm (clockwise or counter-clockwise) as the driving unit, it is evident that four configurations will be encountered. Combining these configurations with the previously mentioned modes, 12 distinct configurations will be obtained and each of these configurations is extensively discussed in the following section.

2.2.1. Mode 1: resistive mode

In this mode, the disturbance torque that is exerted to the gear opposes the rotation direction of the system. Each of the following configurations is analyzed and the corresponding FBDs of the worm and gear are illustrated in Figures 1(a2), (b2), (c2), (d2), (a3), (b3), (c3), and (d3), respectively. A schematic view is presented in Figures 1(a1), (b1), (c1), and (d1) for each of the configurations.

Using the aforementioned FBDs and by implementing Newton–Euler’s principle, it is possible to derive the governing equations for each of the configurations and it is observed that all the governing equations are identical; that is, only two distinct equations are obtained in this mode (one for the gear and one for the worm). After decomposing the forces in accordance with the direction of rotation, the dynamic equations of the worm and gear are derived as represented by Equations (1) and (2), respectively. It is important to state that the force decomposition has been only performed in the axes that were effective in the analysis of motion and the ineffective forces have been neglected:

+ T_{1} - (R_{w} \cos ϕ_{n} \sin γ) N - (R_{w} \cos γ) μ N = J_{w} {\overset{\cdot\cdot}{θ}}_{w}

(1)

- T_{2} + (R_{g} \cos ϕ_{n} \cos γ) N - (R_{g} \sin γ) μ N = J_{g} {\overset{\cdot\cdot}{θ}}_{g}

(2)

If the system is analyzed statically $({\overset{\cdot\cdot}{θ}}_{g} = {\overset{\cdot\cdot}{θ}}_{w} = 0)$ , Equations (1) and (2) can be rewritten as Equations (3) and (4), respectively, and by eliminating $μ$ , it is possible to extract the normal force as a function of the input torque and disturbance torque. This matter is achieved in Equation (5):

+ T_{1} - (R_{w} \cos ϕ_{n} \sin γ) N - (R_{w} \cos γ) μ N = 0

(3)

- T_{2} + (R_{g} \cos ϕ_{n} \cos γ) N - (R_{g} \sin γ) μ N = 0

(4)

N = \frac{+ T_{1} + (\frac{R_{w}}{R_{g}} \cot γ) T_{2}}{R_{w} \frac{\cos ϕ_{n}}{\sin γ}}

(5)

In this case, as it was expected, upon the existence of input or disturbance torque ( $T_{1} \neq 0$ , $T_{2} \neq 0$ ), the value of the normal force will never be equal to zero.

2.2.2. Mode 2: pseudo-resistive mode ( $| T_{2} | 〈 | T_{s} |$ )

The analysis of this mode, from a dynamic point of view, is similar to the resistive mode but, in this case, the disturbance torque is concurrent with the system’s rotation direction. It is essential to state that, in this case, the magnitude of the disturbance torque is smaller than the switching torque ( $T_{s}$ ) and, subsequently, the engaged tooth has not changed relative to the previous mode. As extensively discussed in Section 2.2.1, four configurations will also emerge in this mode. Figure 2 illustrates the FBDs of the worm and gear. It also demonstrate a schematic view of tooth–thread engagement in the same order of Figure 1.

Figure 2.

Schematic view of tooth–thread engagement in the pseudo-resistive mode and worm gearbox free body diagram (FBD): (a1) left-handed (L-H) helix – $T_{1} > 0$ ; (b1) L-H helix – $T_{1} < 0$ ; (c1) right-handed (R-H) helix – $T_{1} < 0$ ; (d1) R-H helix – $T_{1} > 0$ ; (a2, b2, c2, d2) worm’s FBD for each configuration; (a3, b3, c3, d3) gear’s FBD for each configuration.

In this mode, the FBDs are identical to the previous mode, but owing to the fact that the direction of implementation of $T_{2}$ has been changed, the equations of motion of both the worm and gear will undergo certain changes and are represented by Equations (6) and (7), respectively. It is evident that the only difference between Equations (1) and (2) and Equations (6) and (7) is in the sign of the disturbance torque:

+ T_{1} - (R_{w} \cos ϕ_{n} \sin γ) N - (R_{w} \cos γ) μ N = J_{w} {\overset{\cdot\cdot}{θ}}_{w}

(6)

+ T_{2} + (R_{g} \cos ϕ_{n} \cos γ) N - (R_{g} \sin γ) μ N = J_{g} {\overset{\cdot\cdot}{θ}}_{g}

(7)

Assuming the system to be static, it is possible to eliminate $μ$ from Equations (8) and (9), and Equation (10) can be obtained:

+ T_{1} - (R_{w} \cos ϕ_{n} \sin γ) N - (R_{w} \cos γ) μ N = 0

(8)

+ T_{2} + (R_{g} \cos ϕ_{n} \cos γ) N - (R_{g} \sin γ) μ N = 0

(9)

N = \frac{+ T_{1} - (\frac{R_{w}}{R_{g}} \cot g γ) T_{2}}{R_{w} \frac{\cos ϕ_{n}}{\sin γ}}

(10)

In this case, despite the presence of input and disturbance torques, it is possible to have a normal force equal to zero. By solving $N = 0$ , utilizing Equation (10), it is possible to obtain the following equation:

T_{2} = (\frac{R_{g}}{R_{w}} \tan γ) T_{1}; T_{s} \overset{Δ}{=} (\frac{R_{g}}{R_{w}} \tan γ) T_{1}

(11)

Remark 1. It is evident that if the magnitude of $T_{2}$ surpasses the value specified by Equation (11), the normal force will become negative and that conveys the meaning that the dynamic mode has changed and the FBD is no longer valid.

Remark 2. It is noteworthy to state that both resistive and pseudo-resistive modes can be expressed as a single mode, which is due to a substantial similarity between their FBDs, but for the sake of clarity and with the intentions of disambiguating any equivocation and to also emphasize their different physical interpretations, the aforementioned modes are categorized into two modes named the resistive and pseudo-resistive modes.

2.2.3. Mode 3: assistive mode ( $| T_{2} | > | T_{s} |$ )

In this mode, not only is the direction of the disturbance torque in accordance with the direction of rotation of the system, but also its value is significant and can be considered to be effective at inducing an increase in the angular velocity of the worm. As demonstrated in Figure 3, the engagement surface of the tooth–thread has been changed compared to the previous configurations that were presented in Figures 1 and 2, and the normal and friction forces are now exerted on the other surface, which means that the next thread of the worm is now engaged with the tooth of the gear on the other side. Figure 3 illustrates the FBDs of the worm and gear and a schematic view of tooth–thread engagement in the same order of Figure 1.

Figure 3.

Schematic view of tooth–thread engagement in the assistive mode and worm gearbox free body diagram (FBD): (a1) left-handed (L-H) helix – $T_{1} > 0$ ; (b1) L-H helix – $T_{1} < 0$ ; (c1) right-handed (R-H) helix – $T_{1} < 0$ ; (d1) R-H helix – $T_{1} > 0$ ; (a2, b2, c2, d2) worm’s FBD for each configuration; (a3, b3, c3, d3) gear’s FBD for each configuration.

The governing equations can be obtained as represented in the following equations:

+ T_{1} + (R_{w} \cos ϕ_{n} \sin γ) N - (R_{w} \cos γ) μ N = J_{w} {\overset{\cdot\cdot}{θ}}_{w}

(12)

+ T_{2} - (R_{g} \cos ϕ_{n} \cos γ) N - (R_{g} \sin γ) μ N = J_{g} {\overset{\cdot\cdot}{θ}}_{g}

(13)

Upon conducting a static analysis, it is possible to eliminate $μ$ from Equations (14) and (15) and Equation (16) is obtained:

+ T_{1} + (R_{w} \cos ϕ_{n} \sin γ) N - (R_{w} \cos γ) μ N = 0

(14)

+ T_{2} - (R_{g} \cos ϕ_{n} \cos γ) N - (R_{g} \sin γ) μ N = 0

(15)

N = \frac{- T_{1} + (\frac{R_{w}}{R_{g}} \cot g γ) T_{2}}{R_{w} \frac{\cos ϕ_{n}}{\sin γ}}

(16)

By solving the equation $N = 0$ , it is possible to achieve Equation (17) for $T_{s}$ , which is identical to Equation (11):

T_{2} = (\frac{R_{g}}{R_{w}} \tan γ) T_{1}; T_{s} \overset{Δ}{=} (\frac{R_{g}}{R_{w}} \tan γ) T_{1}

(17)

It can be concluded that this value ( $T_{s}$ ), as defined in Definition 1, is a torque of particular importance for the worm gear dynamics that depends on the angular velocity in an instantaneous manner.

Remark 3. It is essential to state that upon the negligence of this torque, it is unfeasible to obtain a correct force analysis for the internal forces of the worm gear. In fact, the value of the switching torque ( $T_{s}$ ) determines which FBD is correct.

In order to express the dynamics of the worm gear system with universality and generality, Equations (1), (2), (6), (7), (12), and (13) are compressed to Equations (19) and (20) with respect to Equation (17) for $T_{s}$ and by introducing a new variable named $E$ . The compact equations, governing the motion of worm gear dynamics under any type of loading mode (assistive or resistive) are expressed in Equations (18)–(20):

E = [\frac{R_{w}}{R_{g}} \cot γ (T_{2}) - T_{1}] sgn (T_{1})

(18)

+ T_{1} + sgn (E) N R_{w} \cos ϕ_{n} \sin γ - μ N R_{w} \cos γ = J_{w} {\overset{\cdot\cdot}{θ}}_{w}

(19)

+ T_{2} - sgn (E) N R_{g} \cos ϕ_{n} \cos γ - μ N R_{g} \sin γ = J_{g} {\overset{\cdot\cdot}{θ}}_{g}

(20)

Using Equations (19) and (20), it is possible to eliminate $μ$ and obtain a general relationship for the normal force with respect to variables $T_{1}$ , $T_{2}$ , ${\overset{\cdot\cdot}{θ}}_{w}$ , ${\overset{\cdot\cdot}{θ}}_{g}$ , as represented in the following equation:

\begin{matrix} N = [(J_{w} {\overset{\cdot\cdot}{θ}}_{w} - T_{1}) + \frac{R_{w}}{R_{g}} \cot γ (T_{2} - J_{g} {\overset{\cdot\cdot}{θ}}_{g})] \\ \frac{\sin γ}{R_{w} sgn (E) \cos ϕ_{n}} \end{matrix}

(21)

Equation (22) for friction force is obtained by eliminating the normal force term from Equations (19) and (20):

μ N = [(T_{1} - J_{w} {\overset{\cdot\cdot}{θ}}_{w}) + \frac{R_{w}}{R_{g}} \tan γ (T_{2} - J_{g} {\overset{\cdot\cdot}{θ}}_{g})] \frac{\cos γ}{R_{w}}

(22)

It is also noteworthy to state that in order to achieve the numerical values of the coefficient of friction, it is only necessary to combine Equation (21) with Equation (22):

μ = [\frac{(T_{1} - J_{w} {\overset{\cdot\cdot}{θ}}_{w}) + \frac{R_{w}}{R_{g}} \tan γ (T_{2} - J_{g} {\overset{\cdot\cdot}{θ}}_{g})}{(J_{w} {\overset{\cdot\cdot}{θ}}_{w} - T_{1}) + \frac{R_{w}}{R_{g}} \cot γ (T_{2} - J_{g} {\overset{\cdot\cdot}{θ}}_{g})}] \frac{sgn (E) \cos ϕ_{n}}{\tan γ}

(23)

It can be observed from Equations (21)–(23) that normal force, friction force, and the coefficient of friction are functions of $T_{1}$ and $T_{2}$ . To sample the behavior of these three functions, several experiments have been conducted on the setup, which will be comprehensively discussed in Section 3.

There are two cases where the tooth–thread engagement occurs correctly and not erroneously.¹⁰ The first case is when the system operates without the existence of any disturbance torque ( $T_{2} = 0$ , $T_{1} \neq 0$ ) and the second case is when a disturbance torque is applied to the motionless system ( $T_{2} \neq 0$ , $T_{1} = 0$ ).

Remark 4. It is evident that in the case that no disturbance torque is exerted to the system and by considering a quasi-static state ( ${\overset{\cdot\cdot}{θ}}_{w} = {\overset{\cdot\cdot}{θ}}_{g} = 0$ ), a definite value can be obtained for the coefficient of friction that is contingent upon the pressure angle and helix angle of the gearbox. By substituting the gearbox parameters, and by equating $T_{2}$ to zero, it is evident that the variable $E$ is always negative and Equation (24) can be obtained from Equation (23):

μ_{max} = [\frac{(T_{1})}{(- T_{1})}] \frac{sgn (E) \cos ϕ_{n}}{\tan γ} = \cos ϕ_{n} \cot γ

(24)

Remark 5. By equating $T_{1}$ to zero and taking into consideration that $E$ is positive, it is possible to obtain a constant value for the coefficient of friction of the system when the velocity is zero. Due to the fact that the velocity of the system is equal to zero, this state of motion is called static:

μ_{min} = [\frac{\frac{R_{w}}{R_{g}} \tan γ (T_{2})}{\frac{R_{w}}{R_{g}} \cot γ (T_{2})}] \frac{sgn (E) \cos ϕ_{n}}{\tan γ} = \cos ϕ_{n} \tan γ

(25)

By obtaining these two values with respect to the mentioned conditions, the reliable interval for correct tooth–thread engagement is acquired.

In order to directly observe the effects of the disturbance torque on the system, it is essential to combine the kinematics relations of the gearbox system with its kinetics relations. To do so, it is necessary to utilize Equation (26), which relates the angle of the worm with the angle of the gear and also represents the transmission ratio of the worm gearbox.¹² By calculating the first and second derivatives of Equation (26), the relationship between the velocity and acceleration of the worm and gear can also be obtained:

θ_{g} = n (\frac{R_{w}}{R_{g}} \tan γ) θ_{w}

(26)

where n represents the number of threads in the worm ( $n = 1, 2, 3$ ). Using Equation (26), it is possible to combine Equations (19) and (20) and the relationship between the input torque and the acceleration of input/output shaft is obtained as represented in the following equation:

\begin{matrix} T_{1} + (\frac{R_{w}}{R_{g}} \tan γ) T_{2} - (\frac{R_{w}}{\cos γ}) μ N \\ = [J_{w} + \frac{J_{g}}{n} {(\frac{R_{w}}{R_{g}} \tan γ)}^{2}] {\overset{\cdot\cdot}{θ}}_{w} \\ = [\frac{J_{w}}{n} \frac{R_{g}}{R_{w}} \cot γ + J_{g} \frac{R_{w}}{R_{g}} \tan γ] {\overset{\cdot\cdot}{θ}}_{g} \end{matrix}

(27)

A number of points can be perceived from this relation. Primarily, it is evident that the number of the threads does not affect the transmitted disturbance torque. Secondly, in order to mitigate the effects of friction, the helix angle and the radius of the worm should be decreased. This change also directly affects the transmission ratio, so a rational compensation is essential. Computing the effective mass inertial moment in worm gears and their effects on the input/output accelerations are also remarkable.

It is evident from Equation (28) that in quasi-static conditions ( ${\overset{\cdot\cdot}{θ}}_{w} = {\overset{\cdot\cdot}{θ}}_{g} = 0$ ) and upon having no disturbance torque, all the input torque is spent on overcoming the friction force between the components of the worm gearbox systems. As mentioned in the explanations of Equation (25), the coefficient of friction is demonstrated as $μ_{min}$ in quasi-static conditions:

T_{1} = (\frac{R_{w}}{\cos γ}) μ_{min} N

(28)

This particular input torque represents the minimum amount of torque that is essential to initiate the motion of the worm gearbox. The term $μ_{min} N$ describes the static friction force in the gearbox.

2.3. Incorporation of the worm gearbox driving machine model into the mathematical governing equation of worm gearbox motion

Different types of motors are utilized in industrial applications to drive the worm gearbox system. Permanent magnet direct current (PMDC) motors, three-phase synchronous motors, and stepper motors are at the top of the list of the most widely used motors. One of the main advantages of PMDC motors lies in their simplistic launch procedure relative to other motors (such as the three-phase or AC motors). Angle, velocity, and torque control algorithms are also employed on PMDC motors with ease and there is no need to make use of sophisticated procedures.¹³ Due to the conveniences stated above, PMDC motors are employed in the experimental setup of this paper. The experimental details of the PMDC motors are extensively discussed in Section 3. For the sake of clarification and completeness, a brief introduction on the theoretical basis of PMDC motors is provided here that aims to elucidate the part in the experiment that is related to the motor. The relation between the armature current ( $i$ ) and the output torque ( $T_{1}$ ) of the PMDC motor is as depicted by Equation (29), and the equation relating the motor voltage ( $v$ ) of the PMDC motor to its armature current ( $i$ ) and the rotational velocity of the motor output shaft ( $ω$ ) is expressed in Equation (30):

T_{1} = K_{t} i

(29)

\frac{di}{dt} = - \frac{R}{L} i - \frac{K_{e}}{L} ω + \frac{1}{L} v

(30)

where $K_{t}$ , $K_{e}$ , $R$ , and $L$ represent torque constant, velocity constant, resistance, and inductance of the motor’s armature circuit, respectively. The summation of the output torque of the motor and perturbation torque ( $T_{prt}$ ) is linearly related to the acceleration of the output shaft:

J_{eff} {\overset{\cdot\cdot}{θ}}_{w} = T_{1} + T_{prt}

(31)

Utilizing the modeling assumptions, the output torque and acceleration of the motor are equal to the input torque and acceleration of the gearbox. By combining Equations (27) and (29), the coupled dynamics of the motor and gearbox is obtained and is presented in the following equation:

\begin{matrix} {\overset{\cdot\cdot}{θ}}_{w} = \frac{K_{t}}{J_{eff}} i + \frac{1}{J_{eff}} T_{prt}; J_{eff} \overset{Δ}{=} J_{w} + \frac{J_{g}}{n} {(\frac{R_{w}}{R_{g}} \tan γ)}^{2}; \\ T_{prt} \overset{Δ}{=} (\frac{R_{w}}{R_{g}} \tan γ) T_{2} - (\frac{R_{w}}{\cos γ}) μ N \end{matrix}

(32)

where $T_{prt}$ accounts for the disturbance torque applied to the system and also the internal friction force. Here, $J_{eff}$ represents the effective mass inertial moment of the output shaft and ${\overset{\cdot\cdot}{θ}}_{w}$ denotes the angular acceleration of the output shaft of the motor. It can be observed that $T_{prt}$ in Equation (32) is linearly related to the internal friction force ( $μ N$ ) and the disturbance torque ( $T_{2}$ ).

Upon knowing the input torque and measuring the angular acceleration of the input/output shaft, it is only necessary to estimate the disturbance torque and the friction force to obtain the accurate dynamics of the gearbox system without any unmodeled dynamics.

3. Experimentation: a thorough examination of the obtained model and the disturbance estimation algorithm

3.1. Experimental setup description

In order to analyze the behavior of normal and friction forces under different modes of loading, it is essential to perform extensive and distinct tests. The experimental setup of this paper consists of a MOTOVARIO NMRV030 gearbox with three PMDC motors made by ElectroCraft under the commercial labeling of DP20. One motor is installed on the worm as the driving unit and two motors are connected to the output of the gearbox to exert disturbance torque to the gear.

Primarily, it should be noted that with respect to the relation between output torque of the motor $(T_{1})$ and the motor’s armature current, as represented in Equation (29), it is necessary to measure the value of the current to obtain the quantity of the output torque of the motor and correspondingly the input torque exerted to the gearbox (utilizing modeling assumptions). A tachogenerator (which is in fact a miniature PMDC motor) has been used to measure the angular velocity of the motor’s shaft.

A tachogenerator and a two-channel optical encoder with a resolution of 500 pulses/rev are adjusted and attached to the armature of the PMDC motor. Each PMDC motor is independently driven by a BTS7960B driver, where the armature current is measured by a conventional Hall effect sensor. Due to the fact that a considerable number of analogue feedback signals were present in the setup and their presence was inevitable for the appropriate conduction of the experiment, two digitally addressed analogue multiplexers of model SN74HC4051 were used and each of the multiplexers connected eight lines to the microprocessor via analogue to digital converter (ADC) ports. The microprocessor is an ATMEGA2560 with 8 kB SRAM, 4 kB EEPROM and 16 MHz CPU clock. In Figure 4(a), a real depiction of the mechanical/electrical components of the experimentation setup is illustrated and Figure 4(b), which is a pseudo-Altium representation, demonstrates the signal-flow diagram between the mechanical assembly, sensors, drivers and digital/analogue hardware.

Figure 4.

General view of the experimental setup: (a) real depiction of the experimental setup consisting of the mechanical assembly, the electrical components and circuits, and the monitoring PC; (b) signal-flow diagram between the mechanical assembly, sensors, drivers, and digital/analogue hardware (R = 6.8 kΩ). PMDC: permanent magnet direct current.

In this experiment, despite the fact that one motor could produce the disturbance torque, two motors were used and the second motor is added just to increase the power and magnitude of the disturbance torque. By using two motors, it is possible to produce significant perturbations on the motion of the worm as the driving unit.

Table 1 represents the gearbox parameters and their corresponding numerical values, and Table 2 provides readers with the parameters of the PMDC motor (https://www.electrocraft.com/products/). Figure 5 illustrates a section view of a conventional worm gearbox. It demonstrates the worm gears, bearing placement, oil seals, and other components that are visible in the figure.

Table 1.

Gearbox parameters and their corresponding numerical values.

Symbol	Description	Quantity
$R_{w}$	Worm radius	5 mm
$R_{g}$	Gear radius	27 mm
$J_{w}$	Worm mass moment of inertia	$1 \times 10^{- 5}$ kg.m²
$J_{g}$	Gear mass moment of inertia	$110 \times 10^{- 5}$ kg.m²
$γ$	Helix angle	7.7 deg.
$ϕ_{n}$	Pressure angle	20 deg.
$n$	Worm number of threads	2

Table 2.

Motor parameters and their corresponding numerical values.

Symbol	Description	Quantity
$J_{m}$	Output shaft mass moment of inertia	$5 \times 10^{- 5}$ kg.m²
$K_{e}$	Velocity constant	0.167 v/rad/s
$K_{t}$	Torque constant	0.104 N.m/A
$R$	Armature circuit resistance	3.05 Ω
$L$	Armature circuit inductance	2.97 mH

Figure 5.

Section view of a conventional worm gearbox and its internal components (www.motovario.com/eng/).

3.2. Experimental tests

Based on Equations (21)–(23) that were extracted in Section 2.2, all the experiments were conducted on the setup introduced in Section 3.1, in a quasi-static manner and in 17 different constant velocities.

To produce these 17 velocities, the driving motor draws 17 constant currents from the power supply that, based on Equation (29) of the PMDC motor, corresponds to 17 constant yet distinct values for the input torque ( $T_{1}$ ). In order to analyze the effects of the external loading on the values of normal and friction forces, 16 different yet invariant torques ( $T_{2}$ ) have been applied to the gear, both in assistive and resistive modes. In order to signify the effects of the disturbance torque on the worm’s motion, disturbance torques with different yet invariant values, with pre-specified constant angular velocities, have been applied to the gear from the worm. To do so, a velocity control has been designed for the driving motor. In addition to the velocity control, a torque control has been also introduced for the disturbance motors, and the governing control scheme of the aforementioned controllers are discussed in detail by Homaeinezhad and Yaqubi² and Homaeinezhad and Shahhosseini.¹⁴

Remark 6. It is noteworthy to state that due to the fact that in each of the 17 controlled velocities of the system, 16 different yet invariant disturbance torques have been applied to the system, so a total of 272 tests have been conducted in Section 3.1. By saving the output of the voltage and current sensors of the three motors (that leads to the measurement of their torques based on Equation (29)), a significant database is created. These experimentations are different from those that will be conducted in Section 3.3, but they act as a database.

It should be noted that complementary explanations and comprehensive discussions on the details of each figure are provided hereafter.

By utilizing Equation (21), Figure 6 illustrates the linear behavior of the normal force with respect to the disturbance torque. It is notable to state that the magnitude of the velocity does not play an effective role on the normal force, and the parameter of significant importance on the normal force is the applied disturbance torque. To avoid a disorderly illustration, only four sets of data are labeled.

Figure 6.

Variation diagram of the normal force upon systematic alteration of the disturbance torque on the gear at different velocities.

Figure 7 demonstrates the behavior of friction forces that are based on Equation (22). The sampling process is performed as discussed (17 invariant velocities and 16 constant disturbance torques). Figure 7 expresses that in the resistive mode and regardless of the value of velocity, upon an increase in the magnitude of the disturbance torque (exerted on the output of the gearbox), the magnitude of the friction force between the teeth and threads will increase. In addition, it is noteworthy to mention that under a certain predetermined loading, the magnitude of the friction force experiences an increase with a surge in the angular velocity of the system. As expected, in the assistive mode (regardless of the value of the velocity) the system experiences a smaller friction force.

Figure 7.

Variation diagram of the friction force upon systematic alteration of the disturbance torque on the gear at different velocities.

It is evident from Figure 8 that an increase in the velocity of the system will subsequently result in an increase in the value of $T_{s}$ . It can be comprehended from Figure 8 that by increasing the velocity of the system, a larger torque should be exerted on the output of the system to change the surface of the engagement. Due to the fact that increasing the velocity of the system is equivalent to increasing the input torque, the obtained data are plausible.

Figure 8.

Variation diagram of the switching torque upon systematic alteration of the angular velocity.

Figure 9 demonstrates the general behavior of the coefficient of friction for different values of disturbance torques. It can be observed from Figure 9 that upon an increase in the magnitude of the disturbance torque, the value of the coefficient of friction decreases. Using Remark 4 and by substituting the numerical values of the gearbox parameters, it is possible to obtain a maximum value for the coefficient of friction. This value, for the experimental setup of this paper, is obtained to be $μ_{max} = 6.951$ , and any coefficient of friction larger than this value is undoubtedly erroneous and should be considered as an outlier data.

Figure 9.

Variation diagram of the coefficient of friction upon systematic alteration of the disturbance torque on the gear at different velocities.

To better observe the behavior of the coefficient of friction, the part of Figure 9 that corresponds to coefficients of friction less than $μ_{max}$ is magnified and illustrated separately in that figure. The large values acquired for the coefficient of friction from Figure 9 might be surprising and worthy of attention. As these values are obtained at low velocities and via small disturbance torques, using Figure 6, it is rational to expect minuscule normal forces. By dividing the friction force by this minuscule normal force, it is logical to obtain large values for the coefficient of friction. Owing to the fact that none of these large and irrational values are obtained in the resistive mode and due to the fact that, in this mode, the normal force can never be equal to zero (the tooth–thread engagement is always present), this interval is known to be reliable for measuring the coefficient of friction. In this interval the tooth–thread engagement is correct and the case of erroneous engagement never occurs; so, it is reasonable to consider this interval as the adequate interval for data acquisition.

It is possible to filter the data of Figure 9 and equate the values that are larger than $μ_{max}$ to $μ_{\max}$ and similarly equate the values that are smaller than $μ_{\min}$ to $μ_{\min}$ . By substituting the values of Table 1 in Equation (25), $μ_{\min} = 0.127$ can be obtained. Finally, Figure 10 represents the usable data under different velocities and distinct conventional loadings.

Figure 10.

Filtered variation diagram of the coefficient of friction limited by the values of $μ_{max}$ and $μ_{min}$ .

Figure 11 demonstrates the general behavior of the coefficient of friction under different types of disturbance loading and at different velocities. It is noteworthy to state that upon an increase in the velocity of the system (both in positive and negative directions), while maintaining a specific disturbance torque, the coefficient of friction increases. Contrary to the previous case, when the disturbance torque is increased while maintaining a specific velocity, the value of the coefficient of friction decreases. In Figure 11, $ω_{w}$ represents the worm’s velocity. Owing to the fact that all of the examined coefficients of friction are illustrated in Figure 9 and there exists anomalous data for small values of $T_{2}$ (as discussed on Figure 9), from the 16 cases of applied torque to the gearbox only 10 cases were selected. It is observed that the general behavior of the coefficient of friction follows the same pattern regardless of the magnitude of disturbance torque.

Figure 11.

Behavior of the coefficient of friction for different angular velocities of the worm under distinct disturbance torques.

3.3. Accurate estimation of the disturbance torque using the k-nearest neighbors algorithm

In order to obtain a valid and adequate estimation for the disturbance torque, it is vital to utilize accurate sensors and sensing methods. In order to validate the disturbance estimation procedure that was employed in the experiments and for further clarification of the proposed method, an extensive discussion was held in Section 3.1 regarding the correct sensing method.

Based on the velocity control that has been applied to the driving motor (as discussed in Section 3.2), in order to reach a predefined angular velocity $ω'$ , a corresponding current of $i'$ is required, but upon the existence of a disturbance torque, this is no longer valid and the value of $i'$ experiences certain changes. If the disturbance torque causes the gearbox system to operate in the resistive mode, the new value of the current is subsequently larger (as the disturbance torque is resisting the movement and further energy is required to achieve the desired velocity), but if the disturbance torque helps the system to rotate and triggers the assistive mode, the new value of current will experience a decrease. Due to the fact that in quasi-static conditions the rate of change of armature current is equal to zero ( $\frac{di}{dt} = 0$ ), it is possible to reduce Equation (30) to the following equation:

v = Ri + K_{e} ω

(33)

Based on the aforementioned explanations and utilizing Equation (33), it is possible to obtain a one-to-one mapping between [ $v$ , $ω$ , $i$ ] and disturbance torque ( $T_{2}$ ), where $v$ , $ω$ , $i$ are manipulating voltage, rotational velocity, and armature current measured from the driving motor, respectively. Due to the fact that the performance of the installed actuator (on the worm), which is contingent upon the introduced variables of [ $v$ , $ω$ , $i$ ], depends solely on the value of $T_{2}$ , it is possible to utilize interpolation algorithms to estimate the value of $T_{2}$ from [ $v$ , $ω$ , $i$ ] without employing any designated sensing unit. This algorithm is operational in feedback control systems, monitoring algorithms, and fault-detection algorithms. A flowchart of the disturbance estimation algorithm is presented in Figure 12.

Definition 2 (bundle). A bundle is a distinguished subset of the experimental data that is associated with a specific invariant velocity of the system that experiences a range of disturbance torques. The disturbance torques are systematically changed and cover both the positive and negative spectrum. The database of this paper consists of 17 bundles and each bundle has 16 elements and is sorted so the corresponding disturbance torques are monotonically ascending. Each element of the bundle consists of four data, namely the manipulating voltage, rotational velocity, armature current, and the disturbance torque applied to the system.

Figure 12.

Schematic view of the flowchart of the disturbance torque ( $T_{2}$ ) estimation algorithm. KNN: k-nearest neighbors.

In order to adequately interpolate the data, a k-nearest neighbors (KNN) algorithm is used that is reviewed by studies Homaeinezhad et al.¹⁵ and Zhang et al.¹⁶ In this algorithm, it is assumed that the dataset ${[v_{j}, ω_{j}, i_{j}]}_{j = 1}^{272}$ is obtained from the experimentation and forms the database of the KNN algorithm (as stated in Remark 6). At a given instance, the manipulating voltage, rotational velocity of the motor output shaft, and armature current of the driving motor are measured and are demonstrated by $v_{T}, ω_{T}, i_{T}$ , respectively. To adequately organize the database, the bundles are also sorted based on their velocity.

Initially, the Euclidean distance between the measured data from sensors and the elements of the database are calculated as depicted in the following equation:

\begin{matrix} d \overset{Δ}{=} {[\begin{matrix} d_{1} & \dots & d_{272} \end{matrix}]}^{T}; \\ d_{j} = \sqrt{{(v_{T} - v_{j})}^{2} + {(ω_{T} - ω_{j})}^{2} + {(i_{T} - i_{j})}^{2}}; \\ j = 1, \dots, 272 \\ \bar{d} \overset{Δ}{=} {[sort (d, Ascending)]}_{1}^{K = 10} \end{matrix}

(34)

where the dataset $[v_{j}, ω_{j}, i_{j}]$ represents the data from the pre-acquired database. Afterwards, the array $d$ is sorted in an ascending manner and the first $K = 10$ elements of this sorted array are selected and named $\bar{d}$ for convenience. The first element of this sorted array ( $\bar{d}$ ) is extracted as the first candidate for adequate estimation of $T_{2}$ . The bundle number of this element is known and, consequently, due to the fact that each bundle is also sorted based on the corresponding disturbance torque, it is possible to compare the known element with its preceding and succeeding elements in that bundle in the database. This comparison is performed in a component-wise manner and the manipulating voltage, rotational velocity, and armature current of the selected element are compared with the manipulating voltage, rotational velocity, and armature current of the preceding and succeeding elements. If the measured manipulating voltage, rotational velocity, and armature current fall between the two elements (preceding and succeeding), a score is assigned to the element. For example, the measured manipulating voltage, rotational velocity, and armature current fall between the first element of the $\bar{d}$ array and its preceding element in its corresponding bundle in the database. So, the score of this element is three (one for manipulating voltage, one for rotational velocity, and one for armature current). Upon having an element with a score of three, it can be concluded that the measured element is exactly between two elements of the database and, via a linear interpolation, it is possible to estimate the exact magnitude of the disturbance torque. For the case in that the first element of the $\bar{d}$ array could not achieve a score of three, the next element of this array is selected. The length of the $\bar{d}$ array is obtained experimentally and it was observed that an element with a score of three always exists with the selection of such a length.

In order to evaluate the proposed algorithm, 17 distinct tests were conducted on the experimental setup where in each test, the system velocity and the applied disturbance torque were different from the previous test and the database. Figure 13 illustrates the deviation of the estimation disturbance torque from its actual value. Table 3 presents the saved sensor data during experimentation. It is attempted to select an utterly random value for the magnitude of the applied torque (to further render the merits of the proposed method), but it is also noteworthy to state that there existed a rational logic in this torque exertion. For example, it can be mentioned that in samples 2, 4, and 5, the magnitude of the applied torque is similar but the system itself is in completely different velocities (both in magnitude and direction). Also, in samples 3 and 5 (or 12 and 14) the velocity of the system is selected to be approximately similar, but the applied torque is different. It should be noted that even the number of positive torque tests is approximately the same as the number of negative torque tests. This symmetry has also been considered in the direction.

Figure 13.

Deviation of the estimated torques from actual measured disturbance torques exerted on the system.

Table 3.

Experimental data to validate the proposed disturbance estimation algorithm.

Sample number	$v_{T}$ [v]	$ω_{T}$ [ $\frac{rad}{s}$ ]	$i_{T}$ [A]	$T_{2, real}$ [N.m]	$T_{2, est}$ [N.m]	$\| \frac{T_{2, real} - T_{2, est}}{T_{2, real}} \| \times 100$ %
1	−13.11	−131.32	−0.64	−0.84	−0.83	1.19
2	−16.02	−150.07	−1.36	−0.42	−0.39	7.14
3	15.31	119.41	0.52	−0.30	−0.32	6.67
4	−3.82	−17.72	−0.37	−0.40	−0.38	5.00
5	15.36	119.33	0.63	−0.41	−0.39	4.88
6	6.82	28.36	0.74	−0.86	−0.88	2.32
7	−16.18	−134.54	−2.21	−0.07	−0.07	0.00
8	4.39	30.07	0.21	−0.42	−0.40	4.76
9	5.68	45.36	0.19	0.20	0.21	5.00
10	2.20	14.91	0.10	0.45	0.43	4.44
11	−3.40	−22.59	−0.22	0.37	0.39	5.41
12	−11.21	−105.46	−0.74	0.37	0.40	8.11
13	7.99	63.01	0.29	0.65	0.68	4.62
14	−11.06	−107.41	−0.59	0.11	0.12	9.10
15	−5.84	−36.59	−0.62	0.51	0.52	1.96
16	12.51	103.43	0.43	0.64	0.60	6.25
17	10.60	79.73	0.57	0.45	0.49	8.89

Here, $T_{2, real}$ and $T_{2, est}$ represent the real torque exerted on the system during the experiment and the estimated torque using the algorithm, respectively. The final column presents the relative error percentage in each sample. The average relative error percentage of the proposed algorithm is computed to be about 5.04 %.

Using the provided estimation of $T_{2}$ and measuring the input torque, it is possible to calculate the internal forces of the gearbox using Equations (21) and (22) and, subsequently, it is feasible to utilize the control scheme of the PMDC motor, as represented in Equation (27), in control and servo applications.

4. Conclusion

Worm gearboxes are widely used in industrial power transmission systems whose internal dynamics might be considered as a nonlinear function of worm velocity, external torque, and input voltage of the driving PMDC machine. For the implementation of worm gearboxes in high-precision servo applications, the measurement of output shaft loading torque is of critical importance. Due to the fact that torque/force measurement does require extra mechanical mounting and electrical/digital circuitry equipped with several noise and condition filtering components, this paper is devoted to describe a verified technique for torque sensorless internal model estimation of such gearboxes. To this end, the accurate dynamic model of worm gearbox systems, under all kinetic or kinematic configurations, is obtained in this paper. The approach of this paper is universal and it is extendable to other conventional gearbox systems that utilize similar power transmission methods. Several experiments are conducted and, primarily, a database is created that aims to provide the groundwork for accurate estimation of the disturbance torque. It is illustrated that upon integrating this database with a modified KNN algorithm, it is possible to estimate the disturbance torque with acceptable accuracy. The general behavior of the coefficient of friction, normal force, and friction force, upon systematic alteration of the disturbance torque and system velocity, are also presented and these data can act as a guide for further advancement in this topic.

4.1 Future works

By utilizing the proposed algorithm of this paper, the obtained dynamic model of the worm gearbox can be embedded in the structure of a feedback control system. This can make the estimation of the output torque possible without using any torque sensors, whose implementation is always expensive and requires special mounting design and signal conditioners/filters for noise removal. After the determination of the output torque, based on the procedure described in this manuscript, the kinetic model of the worm gearbox is parameterized and identified to be used in the structure of control algorithms. This will eventually lead to more accuracy and precision in the performance of servo systems.

Footnotes

Acknowledgements

The authors of this manuscript are members of the Department of Mechanical Engineering, K. N. Toosi University of Technology, and are not employed by any non-academic government agencies. Also, they are not dependent upon funding or data from any political, military, or sovereignty corporation or organization.

Declaration of conflicting interests

The authors declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The authors received no financial support for the research, authorship, and/or publication of this article.

ORCID iD

MR Homaeinezhad

Author biographies

MR Homaeinezhad, Associate Professor of Mechanical Engineering, received his BSc, MSc, and PhD degrees with the best ranked all in Mechanical Engineering, dynamic systems and control, in 2003, 2005, 2010, respectively from K. N. Toosi University of Technology, Tehran, Iran. Since September 2010, he has been with the Department of Mechanical Engineering, K. N. Toosi University of Technology. His research interests include nonlinear control theory, model predictive control, embedded artificial intelligence, mechatronics, statistical signal analysis/parameter estimation, and automatic decision making (detection and modulation) theory.

S Adineh received his BSc degree in Mechanical Engineering in 2019 from K. N. Toosi University of Technology and is currently studying M. Sc. at Tarbiat Modares University. He has been an undergraduate research assistant at K. N. Toosi Mechatronics Mechanisms Laboratory from 2018 to 2021. He is currently a graduate research associate at Tarbiat Modares Mechatronics Laboratory. His research interests include mechanical mechanisms modeling, robotics and control theories.

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Algorithm for the torque sensorless worm gearbox servo application based on kinetic motion/friction realization

Abstract

Keywords

1. Introduction

2. Mathematical modeling of the worm gearbox based on governing kinetic-kinematic schemes

2.1. Modeling assumptions

2.2. Dynamics modeling of worm gearbox systems

2.2.1. Mode 1: resistive mode

2.2.2. Mode 2: pseudo-resistive mode ( | T 2 | 〈 | T s | )

2.2.3. Mode 3: assistive mode ( | T 2 | > | T s | )

2.3. Incorporation of the worm gearbox driving machine model into the mathematical governing equation of worm gearbox motion

3. Experimentation: a thorough examination of the obtained model and the disturbance estimation algorithm

3.1. Experimental setup description

3.2. Experimental tests

3.3. Accurate estimation of the disturbance torque using the k-nearest neighbors algorithm

4. Conclusion

4.1 Future works

Footnotes

Acknowledgements

Declaration of conflicting interests

Funding

ORCID iD

Author biographies

References

2.2.2. Mode 2: pseudo-resistive mode ( $| T_{2} | 〈 | T_{s} |$ )

2.2.3. Mode 3: assistive mode ( $| T_{2} | > | T_{s} |$ )