Control of a worm-wheel gear driven rescue robot with friction compensation and its experimental verification

Abstract

This paper presents the control of a rescue robot driven by the worm-wheel gear transmission. In the modeling process, the load-dependent friction of the worm-wheel gear is considered, and the governing equations for static and dynamic analyses are formulated. Especially we examine the dependency of break-in joint torques on the loading torque and directionality of motion. The friction parameters of the worm-wheel gear of a physical rescue robot are identified through experimental investigation. A friction compensation controller is then designed based on the modeling results and experimental operating conditions. And the designed controller is applied to a dual-arm rescue robot to validate its effectiveness.

Keywords

Rescue robot worm-wheel gear transmission friction modeling friction compensation control

Introduction

The worm-wheel gears are one of the most promising options in the electro-mechanical systems for high transmission ratio and energy efficiency.^1–4 Contrary to the other types of transmission elements such as bevel and spur gears, where ample space, a large number of gears, and a complex gear train are required to provide the desired gear ratio, the worm-wheel gear requires small space and a single gear set to generate the large transmission ratio.^5–8 As a result, worm-wheel gears allow the use of high velocity motors in robotics applications involving slow motion without redirecting the motor torque along a complex gear train.⁹ Another potential benefit of a worm-wheel gear is its unique property of self-locking or non-backdrivability that allows it to be driven on the side of input and not back driven on the load side.^10–14 Therefore, actuators do not need to stay powered all the time to hold or lift the loads in a particular position. These promising features make the worm-wheel gear transmission a desirable candidate for several robotic applications, including rescue operations.^15,16 Compared to the other rescue or field robots such as jumping, snake, and four-flipper robots,^17–21 which, used for searching, scanning, and video recording tasks in the disaster areas, are lightweight and can handle small payload, the worm-wheel gear driven rescue robot can hold and manipulate the heavy or oversized payload.

In the worm-wheel gear, the relative movement between the threads of worm and wheel gears is purely sliding, so the performance of worm-wheel gear driven systems are supposed to be affected by the complicated frictional interactions between the worm and wheel gears.^22,23 The friction causes nonlinearities that can result in undesired stick-slip motion, tracking errors, and instability in the control of the system.^2,24 Unfortunately, there exists only a handful of works reported to the research community to address the relevant issues regarding worm-wheel gear driven systems. A friction model based on a wedge-like planar transmission was introduced by Dohring et al.,²⁵ to address the friction losses of worm-wheel gear driven robots. This friction model consists of implicit parameters in the dynamic equations, although these parameters cannot be directly applied in the controller design. Then, velocity lead–lag and quantitative feedback controllers were proposed in May et al.,^26,27 respectively. These are linear controllers and may not be robust enough to deal with the system’s uncertainties. After that, Yeh and Wu²⁸ proposed a sliding mode controller for friction compensation of the worm-wheel gear, which requires the accurate modeling of the system, knowledge of uncertainty bounds, and worm-wheel gear manufacturing parameters such as worm pitch, pressure angle, coefficient of kinetic frictions, and many other parameters. Their study thoroughly investigated different worm and wheel gear engagement configurations. Despite this study’s significant contributions, it lacked consideration of several crucial issues that arise in practice. First, the response of internal force and coefficient of friction for different loading and velocities is not investigated despite their direct effect on the worm gear’s motion. Second, they perceived the worm-wheel gearbox as a switched system given the friction force dependence on the operating conditions, but the design of control with this assumption has not been proposed. Third, friction torque may cause dynamic switching due to uncertain external factors, and this aspect also has not been considered in controller design. In Homaeinezhad and Adineh²⁹ the sensorless torque algorithm based on kinetic motion/friction realization is obtained for all kinetic or kinematic configurations of worm-wheel gear. A new method is proposed in Homaeinezhad et al.³⁰ for worm-wheel gear that is robust to dynamic switching, but it results in a highly sensitive control algorithm that limits its use in real-time mechanical systems. Besides other shortcomings of the previous studies, they were only validated for a customized single-joint worm-wheel gear and may not be directly used to control multi-jointed systems driven by worm-wheel gears due to the high inertia and weight.

The present study is devoted to addressing these undiscovered issues, with examination of a new controller based on the experimental estimation of the worm gear friction parameters, with unknown model uncertainties and external disturbance. More specifically, the main contributions of this paper are:

To estimate the worm gear friction parameters of a rescue robot through experimental investigation;

To propose a friction compensation controller by using the experimentally estimated parameters;

To verify the performance of the proposed controller through experimental investigation.

The paper is organized as follows. In Section “Dual-arm rescue robot driven by worm-wheel gear,” our dual-arm rescue robot driven by worm-wheel gear is introduced; the modeling and control design for the dual-arm rescue robot is covered in Section “Modeling and controller design”; Section “Experimental validation” addresses the effectiveness of the proposed controller. Finally concluding remarks are provided in Section “Conclusion.”

Dual-arm rescue robot driven by worm-wheel gear

Figure 1 shows the dual-arm rescue robot driven by worm-wheel gears developed by the Korea Institute of Machinery and Materials (KIMM) that is under investigation for possible applications for rescue in the field.^28,31 (The dual arms are to be attached to a vehicle to move around the field.) This dual-arm rescue robot can carry a payload of approximately 70 kg by the both arms that reach about 95 cm when the arms are fully stretched. It consists of 10 degrees of freedom (DOFs) in total, which are equally distributed among two arms. Each arm of this rescue robot has 5 DOFs and is comprised of pitch $(L 1 / R 1)$ and roll $(L 2 / R 2)$ joints for the shoulder, pitch $(L 3 / R 3)$ and yaw $(L 4 / R 4)$ joints for the elbow, and $(L 5 / R 5)$ joint for the wrist as shown in Figure 1(b), where $L_{i}$ and $R_{i}$ for $(i = 1, 2, \dots, 5)$ , respectively, represent the i-th joint of the left and right arms. The $R 1$ , $R 2$ , and $R 3$ joints of the right arm of rescue robot are made up of a motor (Panasonic/MSMD041S1T) and gear train system (which includes a worm-wheel gear [20:1] and a harmonic drive [80:1] as shown in the Figure 2). The $R 4$ and $R 5$ joints of the right arm are composed of a motor (Sanyo Denki/R2EA06020F) and gear train system (which consists of a harmonic gear [160:1], a spiral bevel gear [1:2], and a timing pulley belt [1.5:1]). The DH (Denavit-Hartenberg) parameters of the right arm of the dual-arm rescue robot are shown in Table 1 along with the coordinate systems in Figure 3. Note the left arm DH parameters are the same as right arm, except that the direction signs are different.

Figure 1.

Dual-arm rescue robot driven by worm-wheel gear: (a) hardware diagram and (b) schematic diagram.

Figure 2.

Worm-wheel gear actuation module.

Table 1.

DH-parameters of the right arm of rescue robot.

Joints	d (mm)	$θ$ (rad)	a (mm)	$α$ (rad)
R1	0	$θ_{1} + π / 2$	0	$π / 2$
R2	55	$θ_{2} + π$	400	$π / 2$
R3	0	$θ_{3} + π / 2$	−30	$π / 2$
R4	350	$θ_{4}$	40	$- π / 2$
R5	0	$θ_{5} + π / 2$	200	0

Figure 3.

Coordinate system of the right arm of dual-arm rescue robot.

The $R 1$ , $R 2$ , and $R 3$ joints, which are driven by worm-wheel gear, are mainly responsible for holding or lifting the payload. The worm-wheel gear transmission has pivotal advantages in the rescue robots: capability of heavy load handling and ability to sustain the load without actuators’ support. However, the friction of the worm-wheel gear transmission leads the control of the worm-wheel gear driven robot as a substantial challenge, which is a cost to pay. To resolve this issue, a good friction model and a faithful controller for the dual-arm rescue robot should be provided or developed, where the worm-wheel gears parameters and uncertainties are unknown. In the proceeding sections, governing equations of motion, friction parameters estimation, and friction compensation controller for the dual arm rescue robot are presented.

Modeling and controller design

Friction modeling of worm-wheel gear

Figure 4 presents the schematic diagram of a worm-wheel gear transmission, where $τ_{m}$ and $τ_{l}$ donate the motor torque and the load torque, respectively. $τ_{m}$ actively drives the worm, whereas $τ_{l}$ , a reverse force on the worm by the wheel gear, prevents the motion by the $τ_{m}$ . Referring to Mohan and Shunmugam,⁷ the worm and wheel gears engagement can either be left sided or right sided, as shown in Figure 4(a) and (b), respectively. In the left sided engagement case, the left side of the tooth of wheel gear drives the worm gear, while the right side of the tooth of wheel gear drives the worm gear in the right sided engagement case.

Figure 4.

Schematic of the worm and wheel gears engagement: (a) left side engagement and (b) right side engagement.

Figure 5 shows the schematic force diagram of the former two cases of engagement, where the force exerted on the worm by the wheel gear consists of a friction force $F^{f}$ and a normal force $F^{n}$ . In this diagram, the worm and wheel-gear axis are parallel to the $Y$ -axis and $X$ -axis, respectively, and a right-handed coordinate system is employed; $α$ is the angle between the x-axis and $F^{f}$ ; and $F^{n}$ is normal to the pitch helix of the worm gear with an angle $ϕ_{n}$ . Referring to the case of left sided engagement (Figure 5(a)), the force exerted on the worm can be decomposed into three components $F_{x}$ , $F_{y}$ , and $F_{z}$ , as

F_{x} = F^{n} \cos ϕ_{n} \sin α - F^{f} \cos α,

(1)

F_{y} = - F^{n} \cos ϕ_{n} \cos α - F^{f} \sin α,

(2)

F_{z} = - F^{n} \sin ϕ_{n},

(3)

Figure 5.

Force diagram of the worm-wheel gear engagement: (a) left side engagement and (b) right side engagement.

among which $F_{x}$ is the only component of the force that involves in driving the worm. Thus, the dynamics of the worm can be written as

I_{w} {\overset{\cdot\cdot}{θ}}_{w} = τ_{m} + (F^{n} \cos ϕ_{n} \sin α - F^{f} \cos α) r_{w},

(4)

where $I_{w}$ , $θ_{w}$ , and $r_{w}$ donate the inertia, rotation angle, and radius of the worm, respectively. On the other hand, $F_{y}$ is the only component of force that can generate a driving torque on the wheel gear whose dynamics can be written as

I_{g} {\overset{\cdot\cdot}{θ}}_{g} = τ_{l} + (F^{n} \cos ϕ_{n} \cos α + F^{f} \sin α) r_{g},

(5)

where $I_{g}$ , $θ_{g}$ , and $r_{g}$ indicate the inertia, rotation angle, and radius of the wheel gear, respectively.

Similar force analysis can be carried out for the case with right side engagement (Figure 5(b)). The resulting dynamic equations for the worm and wheel can be written as

I_{w} {\overset{\cdot\cdot}{θ}}_{w} = τ_{m} + (- F^{n} \cos ϕ_{n} \sin α - F^{f} \cos α) r_{w},

(6)

and

I_{g} {\overset{\cdot\cdot}{θ}}_{g} = τ_{l} + (F^{n} \cos ϕ_{n} \cos α - F^{f} \sin α) r_{g} .

(7)

Note that the only difference in the dynamic equations of the left and right sided engagements is the different direction of the friction force $F^{f}$ and a normal force $F^{n}$ .

Self-locking condition: For the worm-wheel gear, the static state (i.e. self-locking) occurs, if the relation between $τ_{m}$ and $τ_{l}$ satisfies the following condition^28,32:

- μ_{s} \leq μ = \frac{\cos ϕ_{n} (τ_{m} r_{g} \cos α + τ_{g} r_{w} \cos α)}{τ_{m} r_{g} \sin α - τ_{g} r_{w} \cos α} = \frac{F^{f}}{F^{n}} \leq μ_{s},

(8)

where $μ_{s}$ indicates the coefficient of static friction and $μ$ is the ratio of $F^{f}$ to $F^{n}$ . Note (8) has been derived by employing the conditions ${\overset{\cdot\cdot}{θ}}_{w} = 0$ and ${\overset{\cdot\cdot}{θ}}_{g} = 0$ into (6) and (7), respectively. The physical meaning behind (8) is that the maximum frictional torque in the motor varies by the load side torque. If the robotic joints are built with the worm-wheel gears, varying payloads or self configuration surely affects the amount joint friction. However, even if the load side torque is known to us, it is challenging, in practice, to determine the maximum frictional motor torque in most cases, because $μ_{s}$ and other gear parameters are not available to the end users. This gives us the necessity to experimentally determine how the frictional motor torque changes as the load side torque varies, which we shall discuss in the next section.

Dynamic condition: It is said that the worm gear is in the dynamic condition when it is not in the self-locking state. If so, the friction between worm and wheel gears could be written as

F^{f} = μ_{s} sgn ({\overset{\cdot}{θ}}_{g}) F_{n},

(9)

where $sgn (\cdot)$ denotes the signum function that outputs 1 for non-negative input and −1 for negative input, and the coefficients of kinetic and static frictions are assumed to be equal as $μ_{s}$ . Moreover, the angular displacements of the worm and wheel gears are linearly related by

θ_{w} = λ θ_{g},

(10)

where $λ$ denotes the gear ratio between the worm and wheel gears. Now plugging (9) and (10) into (4) and (5), the normal force $F_{n}$ can be solved in terms of $τ_{m}$ and $τ_{l}$ , which roughly takes a form of convex combination like $F_{n} = A τ_{m} + B τ_{l}$ , where $A$ and $B$ are some constants made from the physical parameters of the gear system. Now, for ${\overset{\cdot}{θ}}_{g} > 0$ , if we put the above back into the dynamic equation, we obtain

τ_{m} = {\begin{matrix} (I_{w} λ + K_{1} I_{g}) {\overset{\cdot\cdot}{θ}}_{g} - K_{1} τ_{l} if load driven \\ (I_{w} λ + K_{2} I_{g}) {\overset{\cdot\cdot}{θ}}_{g} - K_{2} τ_{l} if motor driven \end{matrix}

(11)

where

K_{1} = \frac{(\cos ϕ_{n} \sin α - μ_{s} \cos α) r_{w}}{(\cos ϕ_{n} \cos α + μ_{s} \sin α) r_{g}},

and

K_{2} = \frac{(\cos ϕ_{n} \sin α + μ_{s} \cos α) r_{w}}{(\cos ϕ_{n} \cos α - μ_{s} \sin α) r_{g}} .

The dynamic equation for ${\overset{\cdot}{θ}}_{g} \leq 0$ is obtained simply by replacing $μ_{s}$ with $- μ_{s}$ in (11).

As compared to the value of wheel inertia $I_{g}$ , the worm inertia $I_{w}$ is negligibly small. So, the dynamic model can be further simplified as

τ_{m} = {\begin{matrix} K_{1} I_{g} {\overset{\cdot\cdot}{θ}}_{g} - K_{1} τ_{l} if load driven \\ K_{2} I_{g} {\overset{\cdot\cdot}{θ}}_{g} - K_{2} τ_{l} if motor driven \end{matrix}

(12)

From above, four possible dynamic modes exists depending on the direction of the joint and the left or right engagement of the worm-wheel gear. Therefore, it is better for precise control to employ a proper mode of the dynamics based on the current state and the loading condition.

Note that the rescue robot is not intended for high speed tasks. Thus, the inertial reaction among the joints can be neglected, so that most part of $τ_{l}$ consists of the effective gravitational torque generated by the robot’s own weight and externally applied torque $τ^{ext} = J^{T} F$ , where $J$ denotes the manipulator Jacobian matrix of the robot arm and $F$ is an external force applied at the end-effector.

Worm gear friction parameters estimation

In the quasi static condition, where the velocity and acceleration is negligibly small, the worm-wheel gear, $τ_{m}$ and $τ_{l}$ can be related from (12) as follows.

τ_{m} = - K_{i} τ_{l}

(13)

where $K_{i}$ (i.e. $K_{1}$ or $K_{2}$ ) is selected based on the engagement side. This motor torque accounts for the combined effect of the external load and worm-wheel gear friction. We shall call this quasi static torque simply the friction torque, which depends on the load. As mentioned before, gear parameters and friction parameters in between the worm and wheel involve in $K_{i}$ . In general it is difficult to know the value of $K_{i}$ because friction coefficients are not provided by the manufacturer, and micro data such as $α$ and $ϕ_{n}$ are not likely to be available. Thus, we may have to obtain the value of $K_{i}$ experimentally.

For this purpose, we deployed the rescue robot for experimental investigation. For a fixed $τ_{l}$ , a pulse type motor torque is provided to each joint of the robot and is gradually increased until the joint starts to rotate. It is supposed that the break-in torque for which the joint starts to rotate satisfies the relation (13). For reliability, the break-in torque should be measured multiple times. If the procedure to find the break-in torque is repeated by increasing the external load $τ_{l}$ , any conventional linear regression technique³³ gives the estimate for $K_{i}$ .

Note the load dependent friction model (13) does not have a constant offset friction although most practical mechanical joints are likely to possess. The joint with a worm-wheel gear can also be one of those. So, for a better regression accuracy, a modified model for the quasi static joint torque can be used as

τ_{m} = - K_{i} (τ_{l} - τ_{os}),

(14)

where $τ_{os}$ denotes a constant effective load that serves as a frictional offset.

Figure 6 shows the measured break-in torques for the first three joints in the right arm (i.e. $R 1$ , $R 2$ , and $R 3$ ) in both upward and downward directions while the arm maintains the stretched-out configuration. We performed this experiment multiple times with different external loads for each joint.

Figure 6.

Experimental results for the break-in torque of dual-arm rescue robot joints $R 1$ , $R 2$ , and $R 3$ with varying external load: (a) joint R1-upward, (b) joint R1-downward, (c) joint R2-upward, (d) joint R2-downward, (e) joint R3-upward, and (f) joint R3-downward.

Where the blue dots $(*)$ indicate the break-in torque of joints corresponding to varying external load and the red line indicates the first-order regression model (14) obtained through the least square solution. From the data and fitted lines, the linear regression model suits well in describing the friction of the joints with worm-wheel gears. The obtained least square solutions for $K_{i}$ and $τ_{os}$ are shown in Table 2. From the data, we can see that the characteristic of the friction torque differs much with the direction of movement. This may be explained by the fact that the sign of $μ_{s}$ within $K_{i}$ becomes switched if the direction of the velocity changes.

Table 2.

Identification of linear regression model coefficient by least square method.

Joint	Direction	$K$	$τ_{os}$
R1	Upward	0.2588	26.1219
R1	Downward	−0.6730	48
R2	Upward	0.2593	−4.9351
R2	Downward	−0.9163	44.6126
R3	Upward	0.2967	15.9730
R3	Downward	−0.8740	22.6801

Controller design

The friction compensation controller of each joint is proposed by using the linear regression model in (13) as follows.

τ_{m} = K I_{g} {\overset{\cdot\cdot}{θ}}_{g}^{des} - K (τ_{inv} - τ_{os}) + K_{p} e_{m} + K_{v} {\overset{\cdot}{e}}_{m}

(15)

where $K$ denotes the correct value of $\pm K_{1}$ or $\pm K_{2}$ depending on the direction of velocity and left/right engagement of the joint; $τ_{inv}$ denotes the inverse joint torque that is required to produce a given desired robot motion; $K_{p}$ and $K_{v}$ denote the proportional and derivative (PD) controller gains, respectively; and $e_{m} = θ_{g}^{des} - θ_{g}$ refers to joint position error measured by the joint encoder with $θ_{g}^{des}$ being the desired joint angle. $τ_{friction} = K (τ_{inv} - τ_{os})$ denotes the torque calculated for the frictional force compensation. The controller takes a PD + load compensation form including friction. However, an important difference with similar conventional forms of controllers is the possibility that $K$ is not a fixed constant but a switching value.

The inverse torque of the robot, $τ_{inv}$ , is computed by using the widely known dynamics of the articulated robots.

τ_{inv} = M (q^{des}) {\overset{\cdot\cdot}{q}}^{des} + N (q^{des}, {\overset{\cdot}{q}}^{des}) + G (q^{des}),

(16)

where $M (\cdot)$ , $N (\cdot)$ , and $G (\cdot)$ , respectively, denote the mass matrix, nonlinear torque including Coriolis and centripetal terms, and gravitational torques; and $q^{des}$ denotes the desired joint torque of the robot. The whole terms in the left side of the equation implies the joint torques before entering the worm-wheel gear system. The inverse torque is computed based on the computer-aided design (CAD) model of the robot system.

Experimental validation

The experimental investigation aimed to verify the effectiveness of our proposed friction compensation control scheme. The performance of the proposed controller was evaluated by using the following performance evaluation index:

P = 100 | \frac{e_{\max}}{θ_{g}^{des} (t_{f})} |,

(17)

where $e_{\max}$ implies the maximum tracking error during the desired task for time duration $t_{f}$ and $θ_{g}^{des} (t_{f})$ , serving as the normalizing factor, is the travel range of joint motion.

Experimental setup

The dual-arm rescue robot shown in Figure 1 was employed for the experiments. As mentioned earlier, the first three joints, $L 1 / R 1$ , $L 1 / R 2$ , and $L 3 / R 3$ , of the rescue robot arms, which are mainly responsible for holding or lifting the payload, are driven via the worm-wheel gears. Thus, for the experimentation, responses of $R 1$ , $R 2$ , and $R 3$ joints of the right arm were examined.

For the software implementation, we patched Real-Time eXtension (RTX) over Windows XP and deployed it on an IBM PC (Intel Core i5 CPU). RTX from IntervalZero is a real-time extension for Windows and enables the timer to interrupt with the highest priority, with a maximum latency of only 12 µs. The operating frequency of the control algorithm was 1000 Hz. The effectiveness of the proposed controller was verified by laboratory tests. In the future, field experiments in an open field will be conducted.

Tracking control for dual-arm rescue robot

To demonstrate the effectiveness of the proposed friction compensation control scheme, we compared the tracking performance of the proposed controller in (15) with that of the conventional proportional and derivative control without any additional action (PD-only) controller, which is of the form: $τ_{m} = K_{p} e_{m} + K_{v} {\overset{\cdot}{e}}_{m}$ . The dual-arm rescue robot has to perform the rescue operations in the human robot coexisting environment, so an integral control action was avoided to prevent the risk of error accumulation that might cause a control saturation. The PD-only controller by nature has no ability to cope with the friction in the joints driven by worm-wheel gear. The control parameters for the PD-only controller were set to be the same as of the proposed controller.

A set of experiments was conducted to evaluate the performance of the proposed controller in both upward and downward directions of the joints motion, in which the joints of the dual-arm rescue robot had to follow a reference trajectory that traveled $5 °$ from a configuration ( $- 35 °$ , $45 °$ , $75 °$ ) to ( $- 40 °$ , $50 °$ , $80 °$ ) and the vice versa, as shown in Figure 7(a) and (b). The worm-wheel gear driven joints usually have high friction coefficient at low speed,³⁴ so in this set of experiments, we set the traveling time as low as 15 s to analyze the performance of the proposed controller at low speed. The results of trajectory tracking with the PD-only controller and the proposed controller, for the upward and downward direction of joint motion are shown in Figures 8 and 9, respectively. The proportional and the velocity gains were chosen as $K_{p} = 70, 000, K_{v} = 20, 000$ for $R 1$ joint, $K_{p} = 50, 000, K_{v} = 9, 500$ for $R 2$ joint, and $K_{p} = 15, 000, K_{v} = 9, 500$ for $R 3$ joint for the both controllers. (Note that the sign and value of the friction coefficient $K$ in (15), as commented in Section “Modeling and controller design,” were chosen in real time by considering the movement direction and the leading torque (input torque vs external load)). For each case of joint motion, the trajectory tracking error is shown first, and then the actual driving input torque $τ_{m}$ and the torque calculated for the frictional force compensation $τ_{friction}$ are presented in the red and black colors.

Figure 7.

Configurations of the dual-arm rescue robot. (a and b) initial/terminal configurations for a small-range trajectory and (c and d) initial/terminal configurations for a large-range trajectory: (a) config( $- 35 °$ , $45 °$ , $75 °$ ), (b) config( $- 40 °$ , $50 °$ , $80 °$ ), (c) config( $0 °$ , $10 °$ , $0 °$ ), and (d) config( $- 90 °$ , $100 °$ , $90 °$ ).

Figure 8.

Experimental results using the PD-Only and proposed controllers for 5° upward joint motion: Trajectory tracking and Required Torque for Joints R1 (a to d), R2 (e to h), and R3 (i to l) with PD-Only (left) and Proposed (right) Control Techniques.

Figure 9.

Experimental results using the PD-Only and proposed controllers for 5° downward joint motion: Trajectory tracking and Required Torque for Joints R1 (a to d), R2 (e to h), and R3 (i to l) with PD-Only (left) and Proposed (right) Control Techniques.

Referring to the Figures 8 and 9, the overall tracking performance of the PD-only controller fell significantly behind the proposed controller. In most of the joint responses, the tracking errors by the PD-only controller were larger by an order than those by the proposed controller. The main reason for this behavior might be the uncontrolled friction in the worm-wheel gear driven joints. Any increase in controller gains of PD-only controller to minimize the trajectory tracking error tended to excite the system instability. Once the proposed friction compensation controller was applied, the overall control tracking performance was improved. The performance evaluation indexes indicated the effectiveness of the proposed controller; they were improved by 5.1, 4, and 7 times during the upward movements and 2.5, 2.3, and 7.7 times during the downward movements for the joints $R 1$ , $R 2$ , and $R 3$ of dual-arm rescue robot, respectively. The experimental results of the PD-only controller showed the significant difference between the input torque and the frictional force compensation torque in most of the cases, as shown in Figures 8 and 9, which caused the sudden increase in tracking error. On the other hand, the results of proposed controller demonstrated that the difference between the actual drive input torque and the frictional force compensation torque was reduced, which resulted in the reduction in peak of the trajectory tracking error. After all, the friction compensation scheme of the proposed controller seemed effective in reducing the tracking error and input torque variation.

Next, we carried out another set of experiments to evaluate the performance of the proposed controller with the same experimental setting, except reference trajectory was extended to $90 °$ from a configuration ( $0 °$ , $10 °$ , $0 °$ ) upward to ( $- 90 °$ , $100 °$ , $90 °$ ) for 15 s and then downward to the original configuration for 15 s, as shown in Figure 7(c) and (d). As summarized in Figures 10 and 11, the outcome of experiments was very similar to the experimental results from the previous set of experiments. Naturally, the joint tracking errors by the PD-only controller were poorer than those by the proposed controller, because of the lack of joint friction compensation. The performance evaluation indexes demonstrated the effectiveness of proposed controller as which were improved by 7.6, 2, and 3.7 times during the upward movements and 1.5, 2.6, and 0.8 times during the downward movements, of the joints $R 1$ , $R 2$ , and $R 3$ of dual-arm rescue robot, respectively. The tracking performance was enhanced in the case of large-range experiments from the case of small-range experiments, even if the travel range and speed of motion were increased. This result seemed to be due to that the worm gear friction played more adversely in the slow motion. The experimental results of the PD-only controller showed the input torque started from zero level and sluggishly grew to compensate for the tracking error. On the contrary, the input torque by the proposed controller started from non-zero values that cancel off the joint friction.

Figure 10.

Experimental results using the PD-Only and proposed controllers for 90° upward joint motion: Trajectory tracking and Required Torque for Joints R1 (a to d), R2 (e to h), and R3 (i to l) with PD-Only (left) and Proposed (right) Control Techniques.

Figure 11.

Experimental results using the PD-Only and proposed controllers for 90° downward joint motion: Trajectory tracking and Required Torque for Joints R1 (a to d), R2 (e to h), and R3 (i to l) with PD-Only (left) and Proposed (right) Control Techniques.

A large amount of additional experiments was conducted to further compare the averaged performance of the proposed controller with that of the PD-only control different travel ranges, speeds, and directions of joints motion. The performance evaluation indexes of experimental tests with the PD-only controller and the proposed friction compensation controller are shown in Figure 12. The experimental results showed the effectiveness of the proposed controller with improved performance evaluation indexes in all the cases due to the worm gear friction compensation. Among the results, better performance enhancement tended to be obtained when the robot motion was given in the upward direction than in the downward direction. Specifically, the improvement in the performance evaluation indexes of joints motion was relatively small in some cases, such as the tracking results of $R 3$ joint presented in Figure 11(i) and (j) and $R 1$ joint shown in Figure 12(d). These specific cases may be judged to be the result of the structural specificity such as irregularity of lubricant conditions in motion/stationarity and mechanical defects in the joint transmission of the dual-arm rescue robot, but the general tendency does not change. We hope that mechanical refinement in the near future would help secure the better homogeneity of the experimental results.

Figure 12.

Performance evaluation indexes of experimental tests with the PD-only controller (red color) and the proposed friction compensation controller (blue color): (a) travel range $5 °$ /speed (5 s)/upward direction, (b) travel range $5 °$ /speed (5 s)/downward direction, (c) travel range $90 °$ /speed (30 s)/upward direction, and (d) travel range $90 °$ /speed (30 s)/downward direction.

Hence, these experimental results show that the proposed friction compensation controller is effective and capable of reducing the trajectory tracking error in worm-wheel gear driven joints of dual-arm rescue robot.

Conclusion

Previous friction compensation controller for worm-wheel gear driven system required accurate modeling of the system, knowledge of uncertainty bounds, and worm-wheel gear parameters such as pressure angle, coefficient of kinetic frictions, worm radius, wheel gear radius, speed ratio, and the steady state torque ratio. However, it may be challenging to find this information about the commercially or off the shelf available worm-wheel gear. This paper introduced a friction compensation control for the dual-arm rescue robot driven by commercially available worm-wheel gears, with unknown worm-wheel gear friction parameters and uncertainties. This paper has three distinct contributions: (i) the worm-wheel gear friction parameters of a worm-wheel gear driven dual-arm rescue robot were identified through experimental investigation, (ii) a friction compensation controller was then designed based on the modeling results and experimental operating conditions, (iii) the proposed control technique was applied to a dual-arm rescue robot and experimental results indicated the proposed control system exhibits significant improvement in trajectory tracking control in both slow and fast motions.

Footnotes

Handling Editor: Chenhui Liang

Declaration of conflicting interests

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

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This work has been supported partly by the Energy Efficiency and Resources of the Korea Institute of Energy Technology Evaluation and Planning (KETEP) grant (No. 20202020800020) funded by the Ministry of Trade, Industry and Energy (MOTIE), partly by the National Research Foundation of Korea (NRF) grant funded by the Korea government (NRF-2021R1A2C1095085), and partly by the Korea University grant.

Consent to participate

The authors agree to participate.

Consent for publication

The authors consent to publish.

ORCID iD

Muhammad Shoaib

Availability of data and material

All data used in this work will be presented upon reasonable request.

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