Innovative variable stiffness and variable damping magnetorheological actuation system for robotic arm positioning

Abstract

Overshoot and long settling time are two common problems of the positioning control for robotic arms. To solve the positioning control problems, an innovative variable stiffness and variable damping (VSVD) magnetorheological (MR) actuation system for robotic arms was designed, prototyped and evaluated in this paper. The system can reduce the overshoot and settling time of the robotic arm with less energy consumption by controlling the stiffness and damping of its VSVD unit. A robotic arm with the VSVD actuation system was developed and prototyped. In order to evaluate the performance of the system, a step route and a customised route were designed for the robotic arm system to trace. Under these two routes, the positioning control performances of the VSVD robotic arm were evaluated numerically and experimentally with the control modes of uncontrolled, VD, VS and VSVD, respectively. Both the numerical and experimental results demonstrated that the VSVD control mode works best in general with less overshoot, settling time and energy consumption, indicating that the proposed VSVD actuation system can serve as a good candidate to solve the positioning control problems of robotic arms.

Keywords

Robotic arm actuation system variable stiffness and variable damping magnetorheological dampers positioning control overshoot settling time

1. Introduction

Robotic arms are programmed machines with similar functions to a human arm (Gautam et al., 2017). They either are complete machines or work as individual robot parts of a larger and more complex piece of equipment. With the capability of executing repetitive procedures rapidly, accurately and efficiently, robotic arms have been widely applied in industrial production, manufacturing and assembly, such as circuit board assembly, automotive production lines, pick-and-place applications.

A typical robotic arm consists of joints, articulations and manipulators. Motors are installed in the joints to provide the driving force. Through the cooperative action of joints, a robotic arm can have either rotational or translational motion to fulfil different tasks. Large overshoot (the difference between the maximum arm position and the preset arm position) and long settling time (time period from the initial to the time at which the oscillation is within $\pm 1 %$ of the preset position) are two common issues in the positioning process of a robotic arm (Al-Qahtani et al., 2017; Elsisi et al., 2021; Jiang et al., 2015), which may be caused by poor rigid connection, low stiffness elements and heavy loads. For instance, when a robotic arm is commanded to rotate from 0° to 10° (the preset position), it may perform a large overshoot beyond 10°, and then oscillate around the preset position before the final settling down. The large overshoot and long settling time adversely affect the robotic arm’s accuracy and efficiency, thus they should be reduced to improve the performance.

One approach to reduce the overshoot and settling time of a robotic arm is developing advanced controllers for the motor. For instance, Hamid et al. (2016) developed a neuro-optimal controller for an industrial robotic crane to stabilise the system by reducing the overshoot, undershoot and settling time. To decrease the overshoot and steady error as well as improve the tracking performance for a robotic manipulator, Nikdel et al. (2017) designed an adaptive backstepping control based on the Lyapunov theory. Iqbal (2019) modelled a six DOF robotic arm and compared its tracking performances under H∞ and model predictive control. Other control methods have also been developed in Bhattacharyya et al. (2017), Calderon et al. (2017), Nguyen et al. (2020), and Talib et al. (2013). The advanced controllers can achieve satisfying positioning performance of a robotic arm in most scenarios via controlling the output of actuators. However, when a robotic arm is heavy loaded or overloaded, the maximum output torque of the actuator may be inadequate to effectively impede the trend of overshoot and long settling time; in this case, the positioning performance of the controller will be compromised by the limitation of the maximum output torque of the actuator. In addition, the motor normally has a low torque-to-weight and torque-to-volume ratio, which further limits the positioning performance in heavy load and overload scenarios via advanced controller (Viau et al., 2017).

Hence, introducing magnetorheological (MR) dampers into the existing actuator and controller system can be a good solution for positioning control of a robotic arm in heavy load and overload scenarios. MR dampers (Zhu et al., 2012) are smart material-based devices whose damping can be regulated via controlling the magnetic field generated by electromagnets. With the merits of fast response, large controllable damping force, and low power consumption, MR dampers have been widely applied to solve engineering problems (Ahamed et al., 2018). In terms of the positioning control, small damping can ensure a fast response of the system in high-speed operating conditions, and large damping can be applied to reduce the motor’s overshoot and oscillation around the preset position during the settling time. Liam et al. (2019) integrated MR brakes into the links of a robotic arm. The effectiveness of the MR brakes to control the position and behaviour of the flexible joints of the manipulator was numerically validated. Nagai et al. (2011) investigated an MR brake with an adjustable viscosity coefficient to improve the position and vibration control performance of an artificial muscle manipulator. Tomori et al. (2013) developed a one-degree-of-freedom manipulator using an artificial muscle and an MR brake to control the vibration and overshoot of the manipulator. Sapiński et al. (2018) reported a magnetorheological damper-based positioning system. The overshoot and settling time of the system were reduced by the controlling of an MR damper. More works can be found in (Bai et al., 2011; Jolly, 2001; Yadmellat et al., 2014).

However, for the abovementioned positioning control system of robotic arms based on MR dampers, only damping controllability has been investigated so far. Apart from variable damping, the characteristic of variable stiffness/impedance has also been applied to the research of robotic arms. Barrett et al. (2017) incorporated variable stiffness joints into a robotic arm mounted on a ground rover to improve the compliant interaction with the environment. In addition to the joints, variable stiffness has been introduced in the link of robotic arms as well. She et al. (2016) prototyped a robotic arm with tunable stiffness to increase the safety of human-robot interaction, the stiffness of which is controlled by morphing the shape of the flexible robotic beam. An DLR hand arm system with variable stiffness actuation was developed (Grebenstein et al., 2011, 2012). The variable stiffness design in the DLR hand arm system is able to temporarily store energy during the process of collisions and release the stored energy to regain kinetic energy, which enhances the robustness, dynamics and safety of the hand arm system. In terms of positioning control of a robotic arm, the concept of variable stiffness can also play an important role. The explanation is detailed as follow. The joint stiffness should be hard during the rotation direction changing; this is because a hard stiffness unit can be twisted to store energy during the deceleration of the robotic arm and apply the stored energy to actuate the robotic arm during the acceleration mode, so as to save energy consumption. Besides, the spring unit can provide additional torque to reduce the overshoot. During the operation with constant speed, however, the joint’s stiffness should be controlled to be soft to avoid impeding the movement of the robotic arm. Thus, the stiffness of the joint should be controlled to be soft during constant-speed operation and be hard during the deceleration and acceleration. Based on the above analysis, integrating both the stiffness variation and the damping variation to the positioning control of a robotic arm system by introducing a variable stiffness and variable damping unit can further improve its positioning performance with smaller overshoot, shorter settling time and less energy consumption. Hence, this paper innovatively proposed a VSVD actuation system followed by its numerical and experimental positioning evaluation. The rest of the article is organised as follows. Section 2 presents the mechanical structure of the proposed actuation system for the robotic arm and its working principle. Then, the positioning performance of the robotic arm system is evaluated numerically in Section 3. Experimental tests are also presented in Section 4 to further evaluate the positioning performance. Finally, the conclusion is drawn in Section 5.

2. The proposed actuation system of a robotic arm and its working principle

2.1. System configuration

Figure 1 presents the proposed VSVD actuation system for a robotic arm. The robotic arm was made of aluminium extrusions. A 10 kg mass was fixed to one end of the arm, representing a heavy load carried by the arm. The mass’s weight was chosen based on the maximum power and torque of the actuator. With this arrangement, a robotic arm suffering overshoot and oscillation around settling position due to overload can be demonstrated. The robotic arm is driven by a servo motor and a gearbox (gear ratio: 20:1) under the unmovable plate. Between the unmovable plate and the robotic arm, A VSVD unit based on magnetorheological dampers was installed to regulate the system’s stiffness and damping. A computer communicates with a myRIO via a Labview programme. Torque commands are passed to the servo motor through myRIO and an AC servo driver to drive the robotic arm. Besides, current commands are also sent to the amplifiers for regulating the currents applied to the VSVD unit to control its damping and stiffness. The servo motor has a built-in encoder that can record both real-time angular position and torque output signals of the motor and send these signals back to the computer.

The control block diagram of the system is presented in Figure 2. For a conventional positioning control system, only the controller for the servo motor is working. As the most common control algorithm used in industry, a proportional-integral-derivative (PID) control with friction compensation is used for the motor control unit. After a designed route is prescribed, the motor control unit will calculate the real-time motor torque required for the servo motor based on the position error between the designed position signal and the real-time position feedback signal. As the sum of the calculated PID control torque and the friction compensation torque, motor torque drives the robotic arm. In this control mode, the robotic arm is controlled by the motor alone. In contrast, for the proposed actuation system, a VSVD control unit that can provide additional damper torque also works in parallel with the motor control unit. Therefore, the output torque acting on the arm is the combination of the motor torque and the damper torque, so the robotic arm is controlled by both the motor unit and the VSVD control unit. It is noted that the specific equations and parameters of the motor control unit and VSVD control unit are presented in Section 3.1 in detail.

With the VSVD unit, a better positioning control performance can be realised. When the robotic arm works on the conditions of constant velocity or acceleration, small damping is required to ensure a fast response of the system. In contrast, large damping can help under the condition of deceleration. Besides, for a robotic arm carried with a heavy load, overshoot will happen when moving from one position and stop at another because of the inertia of the heavy load. In this case, large damping can be applied after the overshoot to reduce the amount of it. Besides, large stiffness can reduce overshoot because the system becomes stiffer. Large stiffness can save kinematics energy during the deceleration of the robotic arm and apply it to accelerate the robotic arm when required as well. While during the constant-speed operation of the robotic arm, the stiffness of the joint is controlled to be small to avoid movement obstruction.

2.2. VSVD unit

As shown in Figure 3, the VSVD unit consists of an internal damper unit that controls the damping and an external damper unit that controls the stiffness. The internal damper casing is supposed to be fixed to the unmovable base and an aluminium part of the external damper. With a small permeability, this aluminium part works as a magnetic insulator between the internal and the external damper. Hence, no magnetic interference will occur between these two dampers, and they can be controlled separately without interference. Meanwhile, other aluminium parts are also utilised to optimise the magnetic flux distribution. The shaft with the internal damper rotor integrated connects the gearbox with the motor on the bottom side, and the plate on the upper side. The plate works as an output, and it connects to the robotic arm. A cylindrical rubber spring made of silicone rubber (4601A/B, Barnes crop.) connects the plate and the external damper casing. Two electromagnetic coils (0.5 mm copper wire, 180 turns each) are wound on the internal damper rotor in opposite directions to enhance the magnetic field distribution. The applied current on these two coils is designated as the internal damper current $I_{1}$ . Besides, an external damper coil (0.5 mm copper wire, 200 turns) is seated in the external damper casing, and the corresponding current applied to it is designated as external damper current $I_{2}$ . For both the internal and external dampers, magnetorheological Fluids (MRF-132GD, Lord crop.) is filled in the 0.8 mm gaps (black chambers) of the VSVD unit. One gap is located between the internal damper rotor and casing, and the other gap is located between the external damper disc and casing. All the parts in the yellow block cannot rotate, while other parts are rotatable.

Without current given to both the internal damper and the external damper ( $I_{1} = I_{2} = 0 A$ ), the internal damper rotor, plate, rubber spring, and external damper casing can rotate freely against the fixed parts. Apart from the motor torque, the initial torque of the internal damper is added to the torque output. At the same time, the rubber spring is slightly twisted to overcome the initial torque of the external damper so that the external damper casing can rotate following the rotation of the plate. In this case, the output torque acting on the arm is the motor torque plus the initial torques of the internal and external damper. If $I_{1} > 0 A$ , the damping torque of the internal damper will increase as the result of magnetic flux density increases in the internal damper, so the VSVD unit’s damping torque will increase. If $I_{2} > 0 A$ , the damping torque of the external damper will increase, and the rubber spring will make a large twist to yield the external damper’s torque. In this case, a large spring torque will be added to the output torque and the stiffness of the VSVD unit increases as a result. As no magnetic interference will occur between these two dampers, the damping and stiffness of the VSVD unit can be controlled by I₁ and I₂ independently without interference.

To characterise the VSVD unit, a test platform similar to Figure 1 was built, and the robotic arm wasn’t installed. A sinusoidal signal with an amplitude of 10° and a frequency of 0.5 Hz was selected as the test input. In the damping variability test, I₁ = 0, 1.0, 2.0, 3.0 A and I₂ = 0 A were selected as the currents input. Similarly, in the stiffness variability test, I₂ = 0, 1.0, 2.0, 3.0 A and I₁ = 0 A were selected as the currents input. The displacement and the torque of the damper were recorded by the built-in encoder of the servo motor. With the above arrangement, the damping and stiffness variabilities were tested.

Figure 4(a) and (b) present the experimental results of the variable damping and variable stiffness tests of the VSVD unit, respectively. As shown in Figure 4(a), the maximum torque output of the VSVD unit increased 3.50 times from 9.54 to 33.47 N m as I₁ increased from 0 to 3.0 A. The damping, denoted by the torque-displacement loop, also increases with the increase of I₁. To quantify the damping, the equivalent damping was calculated by referring to equation (1) in Deng et al. (2019). It is found that the equivalent damping increased 3.34 times from 20.64 to 68.98 N m s/rad with the increase of I₁. From the stiffness variability test result in Figure 4(b), it is seen that the maximum output torque increased from 9.54 to 60 N m as the increase of I₂. The equivalent stiffness is defined as the difference ratio of torque to displacement. Take the results of ‘I₁ = 0 A, I₂ = 1.0 A’ as an example, the equivalent stiffness under this condition is represented by the torque-displacement ratio of line AB in Figure 4(b). Calculated by referring to equation (2) in Deng et al. (2019), the equivalent stiffness of the VSVD unit increased 6.71 times from 48.50 to 325.63 N m/rad as the increase of I₂ from 0 to 3.0 A.

2.3. Evaluation routes

Two routes were designed to evaluate the performance of the proposed actuation system. One is a step route (Figure 5(a)): a 10° step is commanded at the initial moment $t = 0 s$ . The arm moves from 0° to 10°, and be settled at 10°. The step route can test the response of the proposed actuation system under an instant command. To evaluate the positioning control performance of the system under a back-and-forth movement, a customised route as shown in Figure 5(b) was also designed; this route is based on a sinusoid wave with an amplitude of 10° and a frequency of 1.3 Hz. 2-second dwells are added at the positions of 10° and −10°. Under this route, the robotic arm moves back-and-forth between −10 and 10°, following half-sinusoid waves with 2 s of dwells at the position of $\pm 10$ °. The half-sinusoid waves allow the system to accelerate and decelerate gradually during the back-and-forth movement.

For the step route, VSVD will be turned on once the real-time position reaches the preset position of 10° (I₁ = I₂ = 3.0 A), after which the overshoot will happen. For the customised route, VSVD is turned on once the real-time position reaches 10° and −10°, namely the dwell periods (I₁ = I₂ = 3.0 A). In the periods of sinusoid waves, the VSVD is turned off to avoid obstructing movement. It is noted that the motor control unit works in the whole period of movement to drive the arm.

3. Numerical evaluation of the positioning control system

In this section, the model of the positioning control system and the phenomenological model of the VSVD unit are firstly introduced in Section 3.1. Then, the simulation results of positioning control under the two designed routes are presented and analysed in Section 3.2.

3.1. System modelling and parameter identification

The governing equation for the robotic arm positioning system is expressed as

J {\overset{\cdot\cdot}{φ}}_{1} (t) = 20 T_{m} (t) - T_{VSVD} (t) sign ({\overset{\cdot}{φ}}_{1} (t)) - T_{f} sign ({\overset{\cdot}{φ}}_{1} (t))

(1)

where $J = 1.0875 kg m^{2}$ represents the moment of inertia of the rotational parts. $T_{m} (t)$ is the output torque of the motor determined by PID control and friction compensation. It is magnified 20 times by the 20-time gearbox before being transferred to the shaft. It is known that the gear flexibility, friction, and loading have an influence on the gearbox’s efficiency. To simplify the model, the gearbox efficiency was ignored. Namely, a 20-time gearbox was assumed to both amplify the torque and diminish the rotation angle by 20 times. $T_{VSVD} (t)$ is the torque output of the VSVD unit. $T_{f}$ represents the friction of the sealing components such as O-rings and bearings, and its value was assumed as 4.235 N m based on experimental experience. $φ_{1} (t)$ is the position of the shaft and it can represent the real-time position of the robotic arm. As both $T_{VSVD}$ and $T_{f}$ are in the reverse direction of the robotic arm’s movement, they are multiplied with ‘ $- sign ({\overset{\cdot}{φ}}_{1} (t))$ ’.

For the motor output $T_{m} (t)$ , it is determined by a PID controller and a friction compensation:

T_{m} (t) = T_{PID} (t) + T_{fc} (t), and T_{m} (t) \leq T_{m_max}

(2)

where $T_{PID} (t)$ and $T_{fc} (t)$ are the output torque of the PID controller and the friction compensation, respectively. $T_{m_max}$ is the output limitation of the motor. Because the continue rated torque for the motor used in this research is 1.3 N m and that it can perform the peak torque of 1.5 N m for a short period, $T_{m_max}$ is set as 1.5 N m.

$T_{PID} (t)$ is calculated by:

T_{PID} (t) = k_{p} φ_{error} (t) + k_{i} \int_{0}^{t} φ_{error} (t) dt + k_{d} \frac{d (φ_{error} (t))}{dt}

(3)

where $k_{p} = 1.2$ , $k_{i} = 0.001$ , and $k_{d} = 0.001$ are parameters of the PID controller. These parameters are determined by experimental experiences. $φ_{error} (t) = φ_{design} (t) - φ_{1} (t)$ denotes the displacement error between the design position and the real-time position of the robotic arm. The real-time position is acquired by the feedback signal from the built-in encoder.

$T_{fc} (t)$ is calculated by:

T_{fc} (t) = k_{f} {\overset{\cdot}{φ}}_{1} (t) and T_{fc} (t) \leq T_{fc_max}

(4)

where $k_{f}$ and $T_{fc_max}$ are the friction coefficient and the assumed maximum friction, respectively. Based on experiment experience, $k_{f}$ was set as 0.003 and $T_{fc_max}$ was set as 0.2 N m.

The mathematical model of the VSVD unit, as shown in Figure 6, was also built to describe its behaviour.

$φ_{1}$ and $φ_{2}$ in the figure represent the angular displacement of the internal damper shaft and external damper casing, respectively. The output torque of VSVD unit $T_{VSVD} (t)$ is the sum of the internal damper torque $T_{in} (t)$ and the spring torque $T_{s} (t)$ :

T_{VSVD} (t) = T_{in} (t) + T_{s} (t)

(5)

To model the internal damper unit, Bouc-Wen model (Spencer et al., 1997; Wang and Liao, 2011) is chosen for its versatility, simplicity, and accuracy, and $T_{in} (t)$ is written as

T_{in} (t) = c_{in} (I_{1}) {\overset{\cdot}{φ}}_{1} (t) + α_{in} (I_{1}) z (t)

(6)

where $c_{in} (I_{1})$ is the viscous coefficient and $α_{in} (I_{1})$ is a scaling factor. Their relationship to $I_{1}$ will be confirmed in the following parameter identification. $z (t)$ is an evolutionary variable representing a function of the time history of the displacement, and its governing equation is expressed as

\overset{\cdot}{z} (t) = - γ | {\overset{\cdot}{φ}}_{1} (t) | z | z | - β {\overset{\cdot}{φ}}_{1} (t) z (t)^{2} + A {\overset{\cdot}{φ}}_{1} (t)

(7)

where $A$ , $β$ , and $γ$ are parameters used to describe the hysteresis behaviour.

The rubber spring served as a stiffness unit is characterised by a spring unit $k_{s}$ and a damping coefficient $c_{s}$ , so the spring torque $T_{s} (t)$ is calculated by:

T_{s} (t) = k_{s} (φ_{1} (t) - φ_{2} (t)) + c_{s} ({\overset{\cdot}{φ}}_{1} (t) - {\overset{\cdot}{φ}}_{2} (t))

(8)

where $φ_{1} (t) - φ_{2} (t)$ means the displacement difference of the plate and the external damper casing. As the rubber spring was installed between the plate and the external damper casing, it also represents the twisted angle of the rubber spring. Accordingly, ${\overset{\cdot}{φ}}_{1} (t) - {\overset{\cdot}{φ}}_{2} (t)$ indicates the twisting velocity.

When the plate rotates with the shaft, the rubber spring will be twisted at the first stage. At this stage, the spring torque is smaller than the yield torque of the external damper ( $T_{s} (t) < T_{ex_yield} (I_{2})$ ), so the external damper casing keeps still ( ${\overset{\cdot}{φ}}_{2} (t) = 0$ ). As the increase of the twisting angle, the spring torque will increase until it yields the external damper torque ( $T_{s} (t) = T_{ex_yield} (I_{2})$ ) and an equilibrium state is reached. At this equilibrium state, the spring torque remains the same as the external damper torque. The external damper casing will also rotate at the same velocity as the plate ( ${\overset{\cdot}{φ}}_{2} (t) = {\overset{\cdot}{φ}}_{1} (t)$ ) and the twisting angle of the rubber spring and the spring torque will not change. This equilibrium is held until the shaft changes its rotation direction. The external damper casing will keep still again until the external damper torque is yield. It is stated that the process mentioned above is a simplified description of the physical phenomenon and factors such as the inertial of the external casing are neglected for simplicity. As the relationship between $T_{ex_yield} (I_{2})$ and $I_{2}$ can be established by the following parameter identification, the external damper is not modelled. To sum up, the governing equations for $φ_{2}$ is built as:

\begin{matrix} If ks | φ_{1} (t) - φ_{2} (t) | < T_{ex_yield} (I_{2}) \\ {\overset{\cdot}{φ}}_{2} (t) = 0 \end{matrix},

(9)

\begin{matrix} Else if ks | φ_{1} (t) - φ_{2} (t) | = T_{ex_yield} (I_{2}) \\ {\overset{\cdot}{φ}}_{2} (t) = {\overset{\cdot}{φ}}_{1} (t) \end{matrix},

(10)

To characterise and predict the behaviour of the VSVD unit, parameters in (4)–(10) were identified using the parameter estimation tool of Simulink based on Matlab 2016b with the default nonlinear least-squares method and trust-region reflective algorithm. This tool can find the best set of parameters to match the modelled results with the experimental data of variable damping and vibrable stiffness tests shown in Figure 4. Identified constant parameters are given in Table 1, and identified variable parameters are provided in Table 2. It is observed that $α_{in} (I_{1})$ and $c_{in} (I_{1})$ increase with the increase of $I_{1}$ , and $T_{ex_yield} (I_{2})$ increases with the increase of $I_{2}$ .

Table 1.

Identified constant parameters (parameter/value).

A	0.34847
γ (rad⁻²)	9467.8
$c_{s}$ (N m s/rad)	3.258
β (rad⁻²)	441.53
$k_{s}$ (N m/rad)	391.90

Table 2.

Identified variable parameters.

Variable damping (I₂ = 0 A)
Current	I ₁ = 0 A	I ₁ = 1 A	I ₁ = 2 A	I ₁ = 3 A
$α_{in}$ (N m/rad)	492.88	2424.2	3548.6	4327.7
$c_{in}$ (N m s/rad)	1.0369	1.3695	1.8523	2.3648
$T_{ex_yield}$ (N m)	2.680	2.680	2.680	2.680
Variable stiffness (I₁ = 0 A)
Current	I ₂ = 0 A	I ₂ = 1 A	I ₂ = 2 A	I ₂ = 3 A
$α_{in}$ (N m/rad)	492.88	492.88	492.88	492.88
$c_{in}$ (N m s/rad)	1.0369	1.0369	1.0369	1.0369
$T_{ex_yield}$ (N m)	2.680	26.133	46.524	52.228

As shown in Figure 7, the modelling results (solid lines) obtained by using the established mathematic model with the identified parameters can fit the experimental data (dash lines) well in both variable damping (Figure 7(a)) and variable stiffness (Figure 7(b)) cases.

3.2. Numerical evaluation results

Following the mathematical modelling, the performances of the positioning control under the designed step and customised routes were evaluated by mathematical simulations using Matlab/Simulink. Four control modes were compared to prove the effectiveness of VSVD: (1) ‘Uncontrolled’ means the positioning control is achieved only by the motor control unit; (2) ‘VD’ means the internal damper that controlling the damping is working; (3) ‘VS’ means the external damper that controlling the stiffness is working; and (4) ‘VSVD’ means both of the internal and external dampers that controlling damping and stiffness are working. It is noted that the motor control unit is always working for all four control modes to drive the robotic arm.

On-off control of damping and stiffness is used for these four control modes. In terms of the step route evaluation, when the real-time position reaches the preset position: 10°, maximum stiffness (I₂ = 3 A) and maximum damping (I₁ = 3 A) will be applied in the VS and VD modes, respectively, and both of maximum stiffness and damping (I₁ = I₂ = 3 A) will be applied in the VSVD modes. It is noted that minimum stiffness and damping (I₁ = I₂ = 0 A) are applied in the period when the real-time position raise from 0° to 10° (before the overshoot) to ensure a fast response. In terms of the customised route evaluation, when the real-time position reaches +10° and −10°, maximum stiffness (I₂ = 3 A) and maximum damping (I₁ = 3 A) will be applied in the VS and VD modes, respectively, and both of maximum stiffness and damping (I₁ = I₂ = 3 A) will be applied in the VSVD mode. However, both the stiffness and damping will keep their minimum values (I₁ = I₂ = 0 A) over the periods of sinusoid signals to avoid obstruction.

The simulation result of the step route is illustrated in Figure 8. Figure 8(a) demonstrates the time history of the robotic arm position, which is represented by the shaft position $φ_{1} (t)$ , and Figure 8(b) plots the motor output torque $T_{m} (t)$ . Besides, the calculated motor output power $P (t)$ using the following equation is presented in Figure 8(c).

P (t) = | 20 T_{m} (t) (φ_{1} (t) - φ_{1} (t - 1 / f_{s})) f_{s} |

(11)

where $f_{s} = 2000 Hz$ is the sampling rate. Twenty (20) represents the magnification factor of the gearbox. $(φ_{1} (t) - φ_{1} (t - 1 / f_{s}))$ means the displacement difference of the robotic arm in the sampling period 1/ $f_{s}$ . As each data group of the time, arm position and motor torque are recorded in every sampling period, the real-time motor power can be calculated as the energy consumed in each period divided by the time period.

The uncontrolled mode was chosen as an example to illustrate the positioning control process. As shown in Figure 8(a) and (b), when the robotic arm moved from 0 (t = 0 s) to 10 degree (t = 0.131 s), the motor provided its maximum output torque (1.5 N m) in the positive direction to drive the robotic arm to move towards the preset position. After the overshoot occurred at t = 0.131 s, the motor torque changed to negative with maximum output (−1.5 N m) to pull back the arm to the preset position. Because the maximum output torque of the motor was limited as 1.5 N m and the arm carried a heavy load, large overshoot of 3.98° was observed. Then, both the arm position and motor torque oscillated, with their amplitudes gradually reduced to zero. Finally, the arm position and the motor torque were settled at t = 0.729 s, with the motor power settled as well. It is seen that the time histories of the arm position, motor torque and motor power were the same for all the four control modes before t = 0.131 s because only the motor control unit was working in this period. In contrast, these three values are different for the four control modes after t = 0.131 s when different control modes are applied. If VD control works after t = 0.131 s, damping torque that always works in the reverse direction of the movement will act on the system, which helps to reduce the overshoot and the settling time. If VS control works after t = 0.131 s, spring torque always pointed to the preset position (10°) will increase the stiffness of the system. Therefore, the overshoot is reduced.

The overshoot, settling time, and energy consumed by the motor for all the control modes of the simulation results are summarised in Table 3. In addition, the reduction proportions compared to the uncontrolled mode of these indexes are also provided in the table. A positive proportion value denotes a better performance than the uncontrolled mode; otherwise, a negative proportion value indicates a worse performance. The energy worked by the motor was acquired by integrating the real-time output power $P (t)$ over time. Because the energy differences occurred after t = 0.131 s, the energy consumed after t = 0.131 s was also presented together with that in the whole period.

Table 3.

Comparison of simulation results under the step route.

	Uncontrolled	VD	VS	VSVD
Overshot (degree)	3.98	3.21	2.52	2.21
Reduction proportion compared to the uncontrolled mode	NA	19.35%	36.68%	44.47%
Settling time (s)	0.729	0.594	0.945	0.611
Reduction proportion compared to the uncontrolled mode	NA	18.52%	−29.63%	16.19%
Energy consumed after $t = 0.310 s$ (J)	6.0001	3.1372	5.1327	2.1267
Energy consumed in the whole period (J)	10.7175	7.8546	9.8501	6.8441
Reduction proportion compared to theuncontrolled mode (after /whole)	NA	47.71%/26.71%	14.46%8.09%	64.56/36.14%

Compared with the uncontrolled mode, the overshoot was reduced in the modes of VD (19.35%) and VS (36.68%) and VSVD had the largest overshoot reduction proportion of 44.47%. It is noticed that the VS mode increased the settling time by 29.63% because more oscillation was induced by large stiffness. As VSVD control consists of VD control in addition to VD control, the VSVD mode performed a longer settling time than the VD mode. However, the VSVD mode still reduced 16.19% settling time than the uncontrolled mode. In terms of energy consumption, the VSVD mode had the largest energy reduction proportions of 64.56% if only the energy worked after t = 0.131 s is considered, and of 36.14% in the whole period. It is concluded from the above analyses that the performance of the positioning system was improved in the modes of VD, VS and VSVD and the VSVD mode worked best among them under the step function excitation.

The positioning control performance under the customised route was also simulated, and the results are presented in Figure 9. The figures in the left column are the results over the whole period of the customised route. To further analyse the results at the peaks, the response of the second top peak from 6.5 to 7.3 s is also enlarged and illustrated in the right column.

It is seen from Figure 9(a) that the first top peak had the largest overshoot over the whole period. Besides, the last period when the robotic arm returned to zero also had a large overshoot for all the control modes because only the motor control unit was working in this period. As shown in Figure 9(d), the overshoot was reduced at the second top peak in modes of VD and VS. The VSVD mode performed the smallest overshoot, even though the VD mode had very comparable performance. Regarding the motor output torque and the output power (Figure 9(e) and (f)), results before the overshoot ( $t \approx 6.851 s$ ) were almost the same, since only the motor control unit was working in this period. After the overshoot, the VSVD mode performed the best with smaller motor torque and power.

The average overshoot, settling time, energy consumed by the motor in the whole working period and their reduction proportion compared to the uncontrolled mode are listed in Table 4. Because the last re-zero movement was the same for all modes, it was excluded in the calculation. It is seen that the VSVD mode had the best performance in terms of average overshoot (40.77% reduced), average settling time (27.64% reduced) and the energy consumed by the motor (8.16% reduced). VD and VS modes also illustrated improvements in these three indexes.

Table 4.

Comparison of simulation results under the customised route.

	Uncontrolled	VD	VS	VSVD
Average overshot (°)	1.989	1.344	1.504	1.178
Reduction proportion compared to the uncontrolled mode	NA	32.43%	24.38%	40.77%
Average settling time (s)	0.796	0.618	0.851	0.576
Reduction proportion compared to the uncontrolled mode	NA	22.36%	−6.91%	27.64%
Total energy (J)	43.418	41.082	42.485	39.875
Reduction proportion compared to the uncontrolled mode	NA	5.38%	2.15%	8.16%

4. Experimental evaluation of the positioning control system

Following the numerical evaluation, the positioning control performance of the proposed actuation system was also accessed by experimental tests. Figure 10 shows the established experimental platform according to the configuration of Figure 1. The gearbox, servo motor, and AC servo driver were under the table, so they are not demonstrated. With this experimental setup, the positioning control performance of the robotic arm system was evaluated under the step and the customised routes, respectively. The test results and analyses are presented in this section.

Under the step route, the time history results of the robotic arm position, motor torque output and the calculated power output are presented in Figure 11; the overshoot, settling time, the energy consumed by the motor and their reduction proportions compared to the uncontrolled mode are summarised in Table 5.

Table 5.

Comparison of experimental results under the step route.

	Uncontrolled	VD	VS	VSVD
Overshot (°)	3.937	3.051	2.463	2.008
Reduction proportion compared to the uncontrolled mode	NA	22.50%	37.44%	49.00%
Settling time (s)	0.7085	0.3010	1.0905	0.4230
Reduction proportion compared to the uncontrolled mode	NA	57.52%	−53.92%	40.30%
Energy consumed after t = 0.127 s (J)	4.4423	2.4640	3.2242	1.5248
Energy consumed in the whole period (J)	9.7549	7.8198	8.5806	6.8530
Reduction proportion compared to theuncontrolled mode (after/whole)	NA	44.53%/19.84%	27.42%/12.04%	65.68%/29.74%

Before the overshoot happened at 0.127 s, the results of robotic arm position, motor torque output and motor output power of the four modes were almost the same in all figures. After 0.127 s, VSVD performed the smallest overshoot of 2.008 with a reduction of 49% compared to the uncontrolled mode, and a relative small settling time of 0.4230 s with a reduction of 40.30% compared with the uncontrolled mode (Table 5). The VS mode had a smaller overshoot but a longer settling time than the uncontrolled mode. As VSVD control contains VS control in addition to VD control, it had less reduction proportion of settling time than VD control because VS induced extra settling time. Regarding the energy consumed by motor, VD and VS modes reduced 19.84% and 12.04% of energy in the whole period, respectively; the VSVD mode had the largest reduction proportion of 29.74%. If only consider the period after the preset position in which different control modes were applied, the VSVD mode can reduce 65.68% of the energy of the uncontrolled mode. In addition, VS and VD modes were also improved in these three indexes.

The experimental results under the customised routes are presented in Figure 12. Different control modes were applied in the dwell period where overshoots happened, while only the motor control worked in the periods of sinusoid waves for all modes. Experimental results in the whole time period are shown in the left column, and the results of the second top peak from 6.5 to 7.3 s are illustrated in the right column. The average overshoot, average settling time and the total energy consumed by the motor are summarised in Table 6.

Table 6.

Comparison of experimental results under the customised route.

	Uncontrolled	VD	VS	VSVD
Average overshoot (°)	1.724	1.046	1.309	0.989
Reduction proportion compared to the uncontrolled mode	NA	39.33%	24.07%	42.63%
Average settling time (s)	0.998	0.701	1.048	0.694
Reduction proportion compared to the uncontrolled mode	NA	29.76%	−1.36%	30.46%
Energy (total) (J)	43.046	40.901	42.557	39.153
Reduction proportion compared to the uncontrolled mode	NA	4.98%	1.14%	9.04%

As shown in Figure 12(a), the overshoot at the first top and bottom peaks were larger than other peaks. The VSVD mode had the smallest overshoot as well as the oscillation during the settling time at the second top peak, given that the results are very close for all modes (Figure 12(d)); it also had the smallest motor output torque (Figure 12(e)) and motor output power (Figure 12(f)) after the overshoot. It is observed from Table 6 that both VD and VS modes can reduce the overshoot, and the VSVD mode works best among them with a 42.63% reduction of the overshoot compared with the uncontrolled mode. Meanwhile, the VSVD mode also had the largest settling time reduction of 30.46%. It is noticed that the VS mode had a smaller overshoot but a longer settling time than the uncontrolled mode. The VD mode also showed improvement in the overshoot and settling time. Compared with the uncontrolled mode, the VSVD mode saved 9.044% of energy, which is the best among them.

Compared with the results under step route in Table 5, less improvement of the VSVD mode control over the VD mode is observed in the results under customised route in Table 6. This is because the uncontrolled robotic arm suffered less overshoot under the customised route (1.724°) than under the step route (3.937°), leaving less room for improvement via VS and VD control under the customised route. It should also be noticed that only the energy consumed after overshoot at ±10° can be reduced by applying the VD and VS control under the customised route, while other large proportions of energy consumed in the ranges between ±10° cannot; thereby, the value of energy reduction proportions of all control modes in Table 6 are not significant.

5. Conclusion

An innovative VSVD actuation system for a robotic arm is studied in this paper. The VSVD unit of the system is capable of varying damping and stiffness by 3.34 and 6.71 times, respectively. To evaluate the positioning control performance of the system, a step route and a customised route were designed for the robotic arm to trace. The control modes of uncontrolled, VD, VS and VSVD have been numerically and experimentally evaluated under these two routes. The results showed that the VD mode can reduce the overshoot, settling time and energy consumption compared with the uncontrolled mode; the VS mode can reduce the overshoot and energy consumption but will induce a larger settling time; the VSVD mode has the best positioning control performance in general among all the control modes. Specifically, for the step route, the VSVD control is numerically proved to be able to reduce the overshoot by 44.47%, settling time by 16.19% and energy consumption by 36.14% compared with the uncontrolled mode. The experimental results also validated a 49.0% reduction of overshoot, 40.30% reduction of the settling time and 29.74% reduction of the energy cost by using VSVD control. Similarly, for the customised route, the mathematical simulation indicated a 40.77% reduction of overshoot, 27.64% reduction of settling time and 8.16% reduction of energy cost by using VSVD control. Experimental results also proved that VSVD control could reduce overshoot by 42.63%, settling time by 30.46% and energy consumption by 9.04%. The positioning control performance of the robotic arm was significantly improved by the designed VSVD actuation system with smaller overshoot, shorter settling time and less energy consumption. Based on the results of this paper, it is concluded that the proposed VSVD actuation is capable of solving the positioning control problems of the robotic arms.

Footnotes

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 disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This research is supported by the Australian Research Council Linkage Grant (No. LP190100603) and the Faculty PhD scholarships of the University of Wollongong.

ORCID iDs

Lei Deng

Shuaishuai Sun

Wenxing Li

Shiwu Zhang

Weihua Li

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