Robust control algorithm using time delay estimation for speed mode of twisted string actuator

Mechanism and Machine Theory 146 (2020) 103733

Contents lists available at ScienceDirect

Mechanism and Machine Theory journal homepage: www.elsevier.com/locate/mechmachtheory

Short communication

Robust control algorithm using time delay estimation for speed mode of twisted string actuator Hyoryong Lee, Hyunchul Choi, Joowon Park, Sukho Park∗ Department of Robotics Engineering, Daegu Gyeongbuk Institute of Science and Technology, (DGIST), Daegu 42988, Republic of Korea

a r t i c l e

i n f o

Article history: Received 12 September 2019 Revised 29 October 2019 Accepted 24 November 2019

Keywords: Disturbance Payload Robustness Time delay control Time delay estimation Twisted string actuator

a b s t r a c t Twisted string actuators (TSAs) have been used where conversion of the rotational motion of a motor into a translatory motion by twisting two strings to control the length of the actuator is needed, e.g., in robot applications. Speed mode TSA (SM-TSA) improves the translatory motion achieved in previous TSAs by adding a shaft between two strings. However, the nonlinear response of the translatory displacements remains a problem. Modeling has been one approach, but payload changes or disturbances make it difficult to solve the nonlinear response of SM-TSA through modeling. Here, proportional–integral–derivative (PID) control and time delay control (TDC) in SM-TSA are evaluated as feedback mechanisms during translatory displacements. By following the desired trajectory through PID control and TDC under conditions of payload changes and spring disturbances in SM-TSA and evaluating tracking results, we show that both control methods can provide somewhat precise positional control in SM-TSA even if there are payload changes or spring disturbances. However, TDC shows smaller tracking errors and yields more robust performance against payload changes and disturbances compared to PID control. © 2019 Elsevier Ltd. All rights reserved.

1. Introduction Tendon actuation using a mechanically translatory reciprocating motion of wires has been widely used for applications including steering flexible endoscopes, driving minimally invasive surgical robots, actuating the movement of artificial arms and fingers, and wearable robots [1–4]. In general, the translatory reciprocating motion of wires in tendon actuation is realized by electric motors and winches [5]. However, because the motor and the winch must be aligned, and the wire for the tendon actuation should be connected in a perpendicular direction to the winch, tendon actuation can have design limitations. Recently, a twisted string actuator (TSA) was proposed, where the translatory reciprocating motion is realized based on the principle that length decreases or increases when two strings that are linearly aligned are twisted or untwisted [6,7]. Speed-mode TSA (SM-TSA) is a type of TSA that produces a large translatory displacement of the twisted string through rotation of a motor by adding a shaft between two twisted strings. The conversion ratio of the twisted-string translatory displacement and the rotation of the motor in SM-TSA can be adjusted by changing the diameter and length of the shaft [8,9]. However, one limitation of TSAs is that the rotation of the motor is not linearly proportional to the resulting translational displacement of the twisted strings. To overcome this limitation, several studies have compensated for the nonlinear ∗

Corresponding author. E-mail address: [email protected] (S. Park).

https://doi.org/10.1016/j.mechmachtheory.2019.103733 0094-114X/© 2019 Elsevier Ltd. All rights reserved.

2

H. Lee, H. Choi and J. Park et al. / Mechanism and Machine Theory 146 (2020) 103733

displacement of TSAs by accurate modeling 7,10,11, and various models of TSAs and SM-TSAs have been proposed [10,11]. We also proposed an enhanced SM-TSA modeling method by improving previously proposed modeling methods and analyzing the twisting tendency of the twisted strings in SM-TSAs [12]. Meanwhile, there were many kinds of research on the controls based on system modeling [13,14]. However, because there is a big difference between the modeling and the actual SM-TSA when payload changes or disturbance exists, it is very difficult to apply the modeling-based controls to SM-TSA system. Therefore, under conditions where disturbances such as payload changes exist, feedback control is necessary to follow a desired trajectory in SM-TSAs. M. Hosseini proposed a finger actuated by a TSA where the force was measured by force sensor and the finger was controlled by the measured force [15]. S. H. Jeong developed a robotic hand using TSAs where a new optical force sensor was proposed, and the robotic hand was controlled by an optical force sensor [16]. In these studies, they adopted force feedback controls for the finger and the robotic hand using TSAs. In this paper, however, we will study a feedback control to follow the desired trajectory in an SM-TSA using a displacement sensor. Generally, proportional–integral–derivative (PID) control methods are widely used because they have a simple structure, yet their control is precise. However, in SM-TSAs, nonlinear responses can be increased by payload changes or external disturbances, and PID control might exhibit poor tracking accuracy. Therefore, we adopted a time delay control (TDC), which is known to be robust control algorithm. Generally, TDC has a simple algorithm structure, does not require a real-time computation for nonlinearities and uncertainties of the control targeted system, and does not need a parameter estimation, unlike an adaptive control. And, TDC can indirectly estimate and compensate for nonlinearities such as payload changes or external disturbances through a time delay estimation, and thus can maintain robust control performances [17–19]. There were no reports on the application of robust controllers to TSA or SM-TSA. Therefore, in this paper, we firstly applied TDC, one of the robust controllers, to SM-TSA with large nonlinearities and uncertainties. This paper is organized as follows: the experimental setup of SM-TSA and control algorithms (PID control and TDC) are described in Section 2; the experimental results of using PID control and TDC for the desired trajectory of a sinusoidal waveform in SM-TSA with payload changes or external disturbances is presented in Section 3; the results are discussed in Section 4; and our conclusions are proposed in Section 5. 2. Materials and methods 2.1. Force mode and speed mode of twisted string actuator Generally, TSA uses one or two strings, where one end is fixed at an actuation unit and the other end is connected to a motor that rotates the strings. When the motor rotates, the strings are twisted and the distance between the actuation unit and the motor decreases. Hence, the rotation of the motor generates the displacement of TSA and controls the TSA’s stiffness [4,6]. TSAs can have two actuation modes: force mode and speed mode. First, a force mode TSA (FM-TSA) does not use a shaft along the axis on which the strings are twisted, and thus the translatory displacement is caused by directly twisted strings connected to the actuation part and the motor. FM-TSA, which is a general TSA structure, can generate a relatively large actuation force because the displacement of the actuation part compared to the rotation of the motor is small. These characteristics enable a FM-TSA to have precise position control [10]. Second, a speed mode TSA (SM-TSA) has a circular shaft with a constant diameter along the axis on which the strings of TSA are twisted. The SM-TSA can generate a large displacement in a short time because the displacement of the actuation part compared to the rotation of the motor is large, but the actuation force is relatively small [11]. Depending on the environment, the FM-TSA or SM-TSA mode can be used, or a dual-mode TSA that combines both modes can be used [8,9]. However, TSAs typically have a nonlinear relationship between the rotation of the motor and the translatory displacement of the actuation unit, which can be a major disadvantage of TSAs [12]. In this study, we focused on SM-TSAs in which the displacement of the TSA can vary greatly owing to payload changes, external disturbances, and the friction between the shaft and the strings. In addition, we introduced feedback control algorithms to compensate for the nonlinearity of SM-TSAs. 2.2. Experimental setup A diagram illustrating the SM-TSA used in this study is shown in Fig. 1(a). In the SM-TSA system, as the motor rotates, the two strings connected to the motor are twisted on the shaft and the end of the strings connected to the actuation unit moves translationally. To evaluate the effects of payload changes in SM-TSA, as shown in Fig. 1(b), movable blocks corresponding to payload changes can be stacked on the actuation unit of the SM-TSA. To evaluate external disturbances, a spring with a different elastic modulus can be connected to the actuation unit of the SM-TSA, as shown in Fig. 1(c). The detailed description of the SM-TSA system is as follows. A pair of strings (Spider Wire, Pure Fishing, USA, d = 0.45 mm) were used as the twisted string of the SM-TSA. A geared DC motor (39872, Maxon Motor, Swiss) and a controller (EPOS2 70/10, Maxon Motor, Swiss) were used to control the rotational motion of SM-TSA in the torque mode. A DAQ board (USB-6341, National Instruments, USA) and a LabVIEW program were used to control the rotational speed and direction of the motor. A coupling (SRB-22C-5X5, Sungil Machinery, Korea) was used to fix the pair of strings firmly and to

H. Lee, H. Choi and J. Park et al. / Mechanism and Machine Theory 146 (2020) 103733

3

Fig. 1. (a) Experimental set up and enlarged scheme of geared DC motor and string twisted area, (b) scheme for twisted string according to load change, and (c) scheme for twisted string according to spring constant change with specific load (1kg, which is the weight of LM guide with other parts in this paper) in SM-TSA.

connect the motor to the twisting shaft. The two strings were twisted and wound along the twisting zone of a stainlesssteel shaft with a diameter of 8 mm and a length of 25 mm. The size of the shaft was chosen from the best experimental result in our previous paper [12]. The shaft was connected to the center hole of the coupling. The fixed frame had two holes with a diameter of 1 mm placed symmetrically to the center hole of the fixed frame so that the strings could only make a translatory reciprocating motion through them. When the strings are twisted in the twisting zone of the shaft, the other ends of the strings are connected to the linear motion (LM) guide (SXRZ42-280, MISUMI, Japan) to measure the displacement of the actuation unit of the SM-TSA accurately. Because the movable block and additional parts of the LM guide act as a payload of approximately 10 N, we set the lowest payload as 10 N. A ring attached to the lower part of the movable block in the actuation unit allows the spring to be connected to the actuation unit. As the actuation unit moves upwards and downwards, the spring acts as an external disturbance. The translatory displacement of the SM-TSA was measured in

4

H. Lee, H. Choi and J. Park et al. / Mechanism and Machine Theory 146 (2020) 103733

Fig. 2. (a) Block diagram of PID control and (b) block diagram of robust control using time delay estimation in SM-TSA.

real time by the laser displacement sensor (HL-G112-A-C5, Panasonic, Japan) and processed by the LabVIEW program using a data acquisition system (USB-6341, National Instruments, USA). 2.3. Control algorithms for SM-TSA In the SM-TSA, the translatory displacement of the actuation unit exhibits extreme nonlinearity because of its intrinsic nonlinearity and extrinsic factors (frictional force, payload changes, external disturbances, etc.). In this study, to overcome the nonlinearity of the SM-TSA and to execute a precise tracking control of the SM-TSA along the desired trajectory, two control algorithms (PID control and TDC) were applied, as shown in Fig. 2. We selected the PID control algorithm because of its frequent use, and the TDC algorithm because it is considered a robust controller. The goal was to compare the control methods’ performance in controlling SM-TSA translatory displacement under conditions of payload changes or external disturbances. First, as shown in Fig. 2(a), an error (e) is generated from the difference between the desired trajectory (xd ) and the current position (x) of the SM-TSA measured from the laser sensor, and PID control algorithm is calculated by using proportional gain (KP ), integral gain (KI ), and differential gain (KD ), as follows:

u(t ) = KP e(t ) + KI ∫ e(t )dt + KD e˙ (t )

(1)

In this study, we selected the PID gains (KP , KI , KD ) based on the minimum values of payload and external disturbance for stability in the control of the SM-TSA system and performed the experiments with the SM-TSA system with payload changes and external disturbance changes.

H. Lee, H. Choi and J. Park et al. / Mechanism and Machine Theory 146 (2020) 103733

5

Second, TDC is known as a robust control technique. By using time delay estimation, TDC estimates and compensates for the uncertainties caused by unclear parameters, model uncertainties, and disturbances and obtains the desired response [17–19]. In actual SM-TSA systems, there are various uncertainties such as frictional force, payload changes, and external disturbance changes. Therefore, it is difficult to model the uncertainties and to control the SM-TSA precisely because of these uncertainties and the inherent nonlinearity of the SM-TSA. Based on Fig. 1(b) and (c), the dynamic equation of the SM-TSA system can be simply described, as follows:

mx¨ + Fv + Fg + Fk + Ff = Fs = ks (θ )u

(2)

where m and x¨ denote the effective mass and the acceleration of the actuation unit, respectively. And Fv , Fg , Fk , and Ff denote the viscosity force, the gravitational force by the payload, the external disturbance force generated by the spring, and all frictional force occurring in the SM-TSA system, respectively. Here, Ff includes Ff sha f t−spring , which is the frictional force between the shaft and the twisted strings, and Ff hole−spring , which is the frictional force generated in the hole through which the twisted strings pass in the fixed frame. Because Ff sha f t−spring is changed whenever the strings are twisted, it is one of the causes of increasing uncertainty and nonlinearity in the SM-TSA system. Fs is the force of the strings that pulls the actuation unit in the opposite direction of the gravitational force while the motor rotates. In this study, since the motor is controlled by the torque mode, the input of the motor is given as the current value and Fs is proportional to the current input value u (Fs = ks (θ ) · u), where ks (θ ) is a variable value as the string rotates in the SMTSA and becomes a function indicating a nonlinear proportional relationship between the pulling force (Fs ) and the motor torque (u). Therefore, for the nonlinear relationship, we adopted the nonlinear proportional variable (ks (θ )) and constructed the relational equation of Fs for u. To derive the TDC law formula for the SM-TSA system, Eq. (2) is divided by mass m and is rewritten as

x¨ +

Fv + Fg + Fk + Ff ks ( θ ) = u m m

(3)

(θ ) where ksm is defined as b and its estimated value is defined as bˆ . Therefore, after bˆ u is added to both sides of Eq. (3), it can be described as

x¨ +





Fv + Fg + Fk + Ff + bˆ − b u = bˆ u m

(4)

Here, since bˆ is the estimated value of b, all uncertainty including the unclear parameters in Eq. (4) are defined as F (x, x˙ , u )

F (x, x˙ , u ) =





Fv + Fg + Fk + Ff + bˆ − b u m

(5)

Based on Eqs. (4) and (5), the uncertainty term can be rewritten as

F (x(t ), x˙ (t ), u(t )) = bˆ u(t ) − x¨ (t )

(6)

Assuming that the uncertainty term in Eq. (6) is changed slowly, it can be described as

F (x(t ), x˙ (t ), u(t )) ≈ F (x(t − L ), x˙ (t − L ), u(t − L )) ≈ bˆ u(t − L ) − x¨ (t − L ) as L → 0

(7)

where the last term denotes the time delay estimation for the uncertainty. Therefore, based on the time delay estimation, the following TDC algorithm could be generated:

  (t ) = u(t − L ) − bˆ −1 x¨ (t − L ) + bˆ −1 x¨d + 2ζ ωn e˙ + ωn2 e

(8)

where ζ and ωn denote the damping coefficient and natural frequency of the desired closed loop dynamics, respectively. Through the TDC input, the nonlinear dynamic system of Eq. (2) can have the following error dynamics

e¨ + 2ζ ωn e˙ + ωn2 e = 0

(9)

Here, it is important to determine the appropriate ζ and ωn to obtain stable error dynamics. Therefore, for the stability of the SM-TSA system with the payload changes and the external disturbances, TDC gains (bˆ −1 , ζ , ωn ) are selected at the smallest payload and spring disturbance in a manner similar to how PID control was executed. 3. Results In this section, we applied the two control algorithms (PID control and TDC) to SM-TSA with the payload changes and the external disturbances and evaluated whether the SM-TSA system could follow a desired trajectory using the control algorithms. In this study, we selected the following sinusoidal wave continuous function as a desired trajectory:



xd (t ) = 10 sin

π

π

20

2

t−



+1

(10)

6

H. Lee, H. Choi and J. Park et al. / Mechanism and Machine Theory 146 (2020) 103733

Table 1 RMS and Maximum error obtained from results of payload change from 1kg to 5kg in SM-TSA. (Unit of RMS and Max. Error is mm). Control methods

PID TDC

Load 1 kg

2 kg

3 kg

4 kg

5 kg

RMS Error

Max. Error

RMS Error

Max. Error

RMS Error

Max. Error

RMS Error

Max. Error

RMS Error

Max. Error

0.25 0.19

0.36 0.29

0.33 0.23

0.53 0.36

0.39 0.28

0.63 0.48

0.45 0.35

0.78 0.58

0.52 0.41

0.89 0.69

where the unit of time (t) is seconds and the unit of displacement (xd ) is millimeter. The gains of the controllers used in the experiment were chosen so that the maximum errors of the two controllers were similar for the smallest payload of 1 kg, as follows. The gains of the PID control are KP = 16, KI = 0.3, and KD = 0.6, and the gains of the TDC are bˆ −1 = 1.9, ζ = 1.0 and ωn = 21.8. In particular, among the various gain values of TDC, the gain value that affects the stability of the overall system is bˆ −1 . When TDC was applied as a controller, bˆ −1 is determined by the following equation for the stability of the overall system [20].

|I − b · bˆ −1 | < 1

(11)

As the mass m of the payload decreases, the value of b increases and the maximum value of bˆ −1 was decreased by Eq. (11), which increases the possibility of the system’s instability. Therefore, to secure the stability of the overall system in the range of payloads used in this paper, we conducted a simple stability analysis to select the value of bˆ −1 for the smallest payload of 1 kg. In order to determine the value of bˆ −1 , the SM-TSA system was simplified to a system consisting of inertia term and control input. And the acceleration and b value of the system were estimated using the input 0.4volt or less, which is the range of input value used in the actual experiment. As a result, the range of b value was 0 < b < 0.46, and the range of bˆ −1 which guarantees the stability of the overall system was 0 < bˆ −1 < 4.35 by Eq. (11). Based on this range, the value of bˆ −1 = 1.9 was selected and confirmed that the stability is secured for all payloads in the experiments. For the payload changes and the external disturbances, the experiments were performed using the same gains of PID control and TDC, where the sampling time was about 5ms. 3.1. Tracking accuracy in SM-TSA with payload changes First, we applied payload changes ranging from 1 to 5 kg, as shown in Fig. 1(b). We compared the trajectory tracking accuracy in SM-TSA systems with PID control and with TDC, and the results are shown in Fig. 3(a). The SM-TSA system with PID control initially appears to follow the desired trajectory precisely, but steady-state errors increase as the payload increases, as shown in the enlarged graph of Fig. 3(a). However, as shown in Fig. 3(b), the SM-TSA system with TDC demonstrates superior tracking accuracy for the desired trajectory, and the tracking error increases (that occur as the payload increases) were smaller with TDC than with PID control, as shown in the enlarged graph of Fig. 3(b). The tracking errors between the desired trajectory and the actual trajectory in the SM-TSA system with PID control and with TDC are shown in Fig. 3(c) and (d), respectively. Both PID control and TDC show tracking errors of less than 1 mm. However, TDC generally shows smaller errors than PID control. Statistics for the tracking errors of the SM-TSA system with PID control and TDC are shown in Fig. 3(e) and (f), respectively. Specifically, Fig. 3(e) shows the root mean square error (RMSE) of the tracking error and Fig. 3(f) shows the maximum error of the filtered tracking errors. The statistical data are summarized in Table 1. Overall, TDC exhibits better tracking accuracy than PID control, although the tracking errors of both PID control and TDC tend to increase slightly as the payload increases. The statistical graphs in Fig. 3(e) and (f) show that as payload increases, the tracking error increase of TDC is smaller than that of PID control. Therefore, it is confirmed that the SM-TSA system with TDC has a more robust tracking accuracy than the SM-TSA system with PID control. 3.2. Tracking accuracy in SM-TSA with spring disturbances Second, we applied external disturbance changes by the spring in Fig. 1(c). We compared the trajectory tracking accuracy of the SM-TSA system with PID control and TDC. Trajectory tracking was measured with no spring attached and with two springs having different spring constants (Low k: 0.17 N/mm, High k: 0.36 N/mm). To clearly confirm the effect of the external disturbances, the same experimental conditions used in Section 3.1 (payload 3 kg, PID control and TDC gain values, sampling time) were applied to the SM-TSA system. Fig. 4(a) shows the trajectory tracking accuracy of the SM-TSA system with PID control and external disturbance changes, where it was confirmed that tracking accuracy gradually decreased as the external disturbances increased. On the other hand, Fig. 4(b) shows the trajectory tracking accuracy of the SM-TSA system with TDC and external disturbance changes, where we found that the tracking accuracy did not change significantly despite the external disturbance change. Fig. 4(c) and (d) present the tracking errors between the desired trajectory and the actual trajectory of the SM-TSA system with PID control and TDC, respectively. Similarly, Fig. 4(e) and (f) show the statistics for the tracking errors of the SM-TSA system with PID control and TDC, respectively. The statistical data are summarized in Table 2.

H. Lee, H. Choi and J. Park et al. / Mechanism and Machine Theory 146 (2020) 103733

7

Fig. 3. Is results of load change from 1 kg to 5 kg in SM-TSA, (a) sinusoidal trajectory and experimental results combined graph using PID control, (b) sinusoidal trajectory and experimental results combined graph using TDC, (c) error graph for results in PID control, (d) error graph for results in TDC, and (e) root mean square error derived from (c) and (d), respectively, and (f) maximum error obtained from filtered line.

8

H. Lee, H. Choi and J. Park et al. / Mechanism and Machine Theory 146 (2020) 103733

Fig. 4. Is results of spring constant change (without (w/o), low k, and high k spring, respectively) in SM-TSA, (a) sinusoidal trajectory and experimental results combined graph using PID control, (b) sinusoidal trajectory and experimental results combined graph using TDC, (c) error graph for results in PID control, (d) error graph for results in TDC, (e) root mean square error derived from (c) and (d), respectively, and (f) maximum error obtained from filtered line. (low k = 0.17 N/mm and high k = 0.36 N/mm).

H. Lee, H. Choi and J. Park et al. / Mechanism and Machine Theory 146 (2020) 103733

9

Table 2 RMS and Maximum error obtained from results of spring constant change (without (w/o), low k, and high k spring, respectively) with 3kg payload in SM-TSA Control methods

PID TDC

Spring type w/o Spring

Low k Spring

High k Spring

RMS Error (mm)

Max. Error(mm)

RMS Error (mm)

Max. Error (mm)

RMS Error (mm)

Max. Error (mm)

0.39 0.28

0.63 0.48

0.42 0.29

0.72 0.47

0.45 0.30

0.79 0.49

Overall, TDC exhibits superior tracking accuracy than PID control for each of the three external disturbances conditions (w/o spring, low k spring, and high k spring). Therefore, we found that the tracking errors of the SM-TSA with PID control change as the external disturbance changes, but those of the SM-TSA system with TDC remain nearly unchanged. Therefore, it is confirmed that the SM-TSA with TDC shows more robust tracking accuracy with external disturbance changes than the SM-TSA system with PID control. 4. Discussion In this study, we introduced feedback control to avoid the degradation of the actuation performance caused by the nonlinearity of the SM-TSA system, where PID control and TDC with a time delay estimation were applied to the SM-TSA system. We implemented payload changes and external disturbances in the SM-TSA system and compared the trajectory tracking of the two control methods for the desired trajectory of a sinusoidal waveform. As a result, we found that tracking errors with TDC are much smaller than with PID control. Compared with PID control, TDC had relatively small increases in tracking errors for payload changes or external disturbance changes in the SM-TSA system. In other words, despite the influence of the friction between the shaft and the string, the change of the payload, and the disturbance caused by the external spring, we confirmed that TDC shows robust and precise tracking accuracy through time delay estimation in the SM-TSA system. Although the experimental results demonstrated that TDC has more robust performance than PID control, in the experiment where the payload was increased, the tracking error of TDC increased. We surmise that because the gains of the TDC were set for the lowest payload and the small external disturbance to reduce the instability of the SM-TSA system caused by the payload changes and the external disturbance changes in the experiment. And, to implement TDC accurately, the time delay estimation of the TDC must be accurately performed. To achieve this, a sufficiently short time delay is required. In this study, the time delay of approximately 5ms was used because of the limitations of the control system. Therefore, if the gains of the TDC are finely tuned and the accuracy of the time delay estimation is improved, it is expected that the increase of tracking errors in the SM-TSA system with TDC and payload changes can be greatly reduced. In addition, from the trajectory tracking responses of the SM-TSA in Figs. 3 and 4, we found that the tracking error in the upward region of the SM-TSA is greater than that in the downward region. It is expected that the tracking error in the upward region of the SM-TSA can be slightly increased because the payload moves in the opposite direction of the gravitational force, but the tracking error in the downward region of the SM-TSA can be decreased because the payload moves in the same direction of the gravitational force. To reduce the tracking error in the upward region of the SM-TSA, the gain value is generally increased, but the increased gain can make instability in the downward region of the SM-TSA. Therefore, more precise gain tuning is necessary for a better trajectory tracking of SM-TSA. In conclusion, it was confirmed that TDC shows better tracking accuracy and robustness for SM-TSA under the conditions of payload changes and external disturbance changes than PID control. Therefore, if a robust control method such as TDC is applied to a TSA application such as wearable robots or artificial arms, we expect that more precise and stable control performance will be possible for various uncertainties (payload changes and external disturbance, etc.). 5. Conclusion In this paper, we applied the PID control and TDC methods to a SM-TSA system in which nonlinearities such as friction, payload changes, and external disturbance changes were introduced. Tracking accuracy tests for the desired trajectory of the sinusoidal waveform were performed with these two SM-TSA systems. The results indicate that compared to the SM-TSA system with PID control, the SM-TSA system with TDC has superior tracking accuracy and more robust control performance against payload changes and external disturbance changes. In the future, it is expected that the control performance of TSA systems with nonlinearity and uncertainty, and the various fields using such TSA systems, will be further improved if TDC as a robust control algorithm is applied to the TSA. Declaration of Competing Interest The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

10

H. Lee, H. Choi and J. Park et al. / Mechanism and Machine Theory 146 (2020) 103733

Acknowledgment This research was supported by the DGIST R&D Program of the Ministry of Science and ICT (19-RT-01) and the Korea Health Technology Development R&D Project through the Korea Health Industry Development Institute (KHIDI), funded by the Ministry of Health & Welfare, Republic of Korea (grant number: HI19C0642). Supplementary material Supplementary material associated with this article can be found, in the online version, at doi:10.1016/j.mechmachtheory. 2019.103733. References [1] D. Ji, T.H. Kang, S. Shim, S. Lee, J. Hong, Wire-driven flexible manipulator with constrained spherical joints for minimally invasive surgery, Int. J. Comput. Assist. Radiol. Surg. (2019) 1–13. [2] H. Dong, E. Asadi, C. Qiu, J. Dai, I.-M. Chen, Grasp analysis and optimal design of robotic fingertip for two tendon-driven fingers, Mech. Mach. Theory 130 (2018) 447–462. [3] L. Gerez, J. Chen, M. Liarokapis, On the development of adaptive, tendon-driven, wearable Exo-gloves for grasping capabilities enhancement, IEEE Robot. Autom. Lett. 4 (2) (2019) 422–429. [4] B. Suthar, M. Usman, H. Seong, I. Gaponov, J.H. Ryu, Preliminary study of twisted string actuation through a conduit toward soft and wearable actuation, 2018 IEEE International Conference on Robotics and Automation (ICRA), IEEE, 2018. [5] W. Kraus, M. Kessler, A. Pott, Pulley friction compensation for winch-integrated cable force measurement and verification on a cable-driven parallel robot, 2015 IEEE International Conference on Robotics and Automation (ICRA), IEEE, 2015. [6] G. Palli, C. Natale, Chris May, Claudio Melchiorri, Thomas Würtz, Modeling and control of the twisted string actuation system, IEEE/ASME Trans. Mech. 18 (2) (2013) 664–673. [7] H. Singh, D. Popov, I. Gaponov, J.H. Ryu, Twisted string-based passively variable transmission: concept, model, and evaluation, Mech. Mach. Theory 100 (2016) 205–221. [8] S.H. Jeong, Y.J. Shin, K.S. Kim, Design and analysis of the active dual-mode twisting actuation mechanism, IEEE/ASME Trans. Mech. 22 (6) (2017) 2790–2801. [9] S.H. Jeong, K.S. Kim, A 2-speed small transmission mechanism based on twisted string actuation and a dog clutch, IEEE Robot. Autom. Lett. 3 (3) (2018) 1338–1345. [10] I. Gaponov, D. Popov, J.H. Ryu, Twisted string actuation systems: a study of the mathematical model and a comparison of twisted strings, IEEE/ASME Trans. Mech. 19 (4) (2014) 1331–1342. [11] L. Hua, X. Sheng, X. Zhu, High-performance transmission mechanism for robotic applications, Mech. Mach. Theory 105 (2016) 176–184. [12] H. Lee, H. Choi, S. Park, Accurate modeling and nonlinearity compensation in the speed mode of a twisted string actuator, Mech. Mach. Theory 137 (2019) 53–66. [13] W.H. Chen, J. Yang, L. Guo, S. Li, Disturbance-observer-based control and related methods—an overview, IEEE Trans. Ind. Electron. 63 (2015) 1083–1095. [14] D. Zhang, B. Wei, A review on model reference adaptive control of robotic manipulators, Annu. Rev. Control 43 (2017) 188–198. [15] M. Hosseini, A. Sengul, Y. Pane, J.D. Schutter, H. Bruyninckx, ExoTen-Glove: a force-feedback haptic glove based on twisted string actuation system, 2018 27th IEEE International Symposium on Robot and Human Interactive Communication (RO-MAN), IEEE, 2018. [16] S.H. Jeong, H.J. Lee, K.R. Kim, K.S. Kim, Design of a miniature force sensor based on photointerrupter for robotic hand, Sens. Actuators A 269 (2018) 444–453. [17] P.H. Chang, J.W. Lee, A model reference observer for time-delay control and its application to robot trajectory control, IEEE Trans. Control Syst. Technol. 4 (1996) 2–10. [18] K. Youcef-Toumi, Y. Sasage, J. Ardini, S.Y. Huang, The application of time delay control to an intelligent cruise control system, in: IEEE In 1992 American Control Conference, 1992, pp. 1743–1747. [19] Y.X. Wang, D.H. Yu, Y.B. Kim, Robust time-delay control for the DC–DC boost converter, IEEE Trans. Ind. Electron. 61 (2014) 4829–4837. [20] E. Fridman, U. Shaked, An improved stabilization method for linear time-delay systems, IEEE Trans. Autom. Control 47 (2002) 1931–1937.