Traditional industrial robots are designed to perform continuous and inflexible tasks. To this end, rigid and non-back drivable actuators are employed to achieve position or velocity control. Stiff actuators, however, may cause severe injuries when these robots come in contact with humans in a dynamic working environment [1] . Therefore, considerable efforts have been made to develop compliant solutions that enable robots and human workers to share the same workspace [2] . Such solutions make it possible for robots to achieve a combination of high precision and repeatability with flexibility and versatility, which are significant to current manufacturing techniques that emphasize interoperability [3] .
Series elastic actuators (SEAs) have attracted extensive interest owing to their low output impedance, impact resistance, energy storage, etc. [4, 5] . These actuators are constructed by introducing compliant components between the actuator and load. Compared with traditional stiff actuators, the introduction of passive compliance can enhance safe human-robot collaboration in industrial manufacturing. In recent years, SEAs have been successfully applied to industrial robots such as IIWA (developed by KUKA) [6] and Baxter (developed by Rethink Robotics) [7] .
However, to achieve active compliance, the limitation introduced by the adoption of elasticity must be overcome [4] . To this end, various impedance/admittance control methods have been developed to afford increased SEA performance. Impedance control differs from admittance control: the former uses an inner force control loop, whereas the latter uses an inner position control loop [5] . Vallery et al. [8] proposed a cascade proportional-integral-derivative (PID)-based control method in which the bandwidth limitation was discussed. A recent attempt has been made to manage the disturbance and uncertainty by disturbance observers (DOBs). Oh et al. [4] developed a DOBbased force controller with feedback compensation and
Traditional industrial robots are designed to perform continuous and inflexible tasks. To this end, rigid and non-back drivable actuators are employed to achieve position or velocity control. Stiff actuators, however, may cause severe injuries when these robots come in contact with humans in a dynamic working environment [1] . Therefore, considerable efforts have been made to develop compliant solutions that enable robots and human workers to share the same workspace [2] . Such solutions make it possible for robots to achieve a combination of high precision and repeatability with flexibility and versatility, which are significant to current manufacturing techniques that emphasize interoperability [3] .
Series elastic actuators (SEAs) have attracted extensive interest owing to their low output impedance, impact resistance, energy storage, etc. [4, 5] . These actuators are constructed by introducing compliant components between the actuator and load. Compared with traditional stiff actuators, the introduction of passive compliance can enhance safe human-robot collaboration in industrial manufacturing. In recent years, SEAs have been successfully applied to industrial robots such as IIWA (developed by KUKA) [6] and Baxter (developed by Rethink Robotics) [7] .
However, to achieve active compliance, the limitation introduced by the adoption of elasticity must be overcome [4] . To this end, various impedance/admittance control methods have been developed to afford increased SEA performance. Impedance control differs from admittance control: the former uses an inner force control loop, whereas the latter uses an inner position control loop [5] . Vallery et al. [8] proposed a cascade proportional-integral-derivative (PID)-based control method in which the bandwidth limitation was discussed. A recent attempt has been made to manage the disturbance and uncertainty by disturbance observers (DOBs). Oh et al. [4] developed a DOBbased force controller with feedback compensation and
Traditional industrial robots are designed to perform continuous and inflexible tasks. To this end, rigid and non-back drivable actuators are employed to achieve position or velocity control. Stiff actuators, however, may cause severe injuries when these robots come in contact with humans in a dynamic working environment [1] . Therefore, considerable efforts have been made to develop compliant solutions that enable robots and human workers to share the same workspace [2] . Such solutions make it possible for robots to achieve a combination of high precision and repeatability with flexibility and versatility, which are significant to current manufacturing techniques that emphasize interoperability [3] .
Series elastic actuators (SEAs) have attracted extensive interest owing to their low output impedance, impact resistance, energy storage, etc. [4, 5] . These actuators are constructed by introducing compliant components between the actuator and load. Compared with traditional stiff actuators, the introduction of passive compliance can enhance safe human-robot collaboration in industrial manufacturing. In recent years, SEAs have been successfully applied to industrial robots such as IIWA (developed by KUKA) [6] and Baxter (developed by Rethink Robotics) [7] .
However, to achieve active compliance, the limitation introduced by the adoption of elasticity must be overcome [4] . To this end, various impedance/admittance control methods have been developed to afford increased SEA performance. Impedance control differs from admittance control: the former uses an inner force control loop, whereas the latter uses an inner position control loop [5] . Vallery et al. [8] proposed a cascade proportional-integral-derivative (PID)-based control method in which the bandwidth limitation was discussed. A recent attempt has been made to manage the disturbance and uncertainty by disturbance observers (DOBs). Oh et al. [4] developed a DOBbased force controller with feedback compensation and
Traditional industrial robots are designed to perform continuous and inflexible tasks. To this end, rigid and non-back drivable actuators are employed to achieve position or velocity control. Stiff actuators, however, may cause severe injuries when these robots come in contact with humans in a dynamic working environment [1] . Therefore, considerable efforts have been made to develop compliant solutions that enable robots and human workers to share the same workspace [2] . Such solutions make it possible for robots to achieve a combination of high precision and repeatability with flexibility and versatility, which are significant to current manufacturing techniques that emphasize interoperability [3] .
Series elastic actuators (SEAs) have attracted extensive interest owing to their low output impedance, impact resistance, energy storage, etc. [4, 5] . These actuators are constructed by introducing compliant components between the actuator and load. Compared with traditional stiff actuators, the introduction of passive compliance can enhance safe human-robot collaboration in industrial manufacturing. In recent years, SEAs have been successfully applied to industrial robots such as IIWA (developed by KUKA) [6] and Baxter (developed by Rethink Robotics) [7] .
However, to achieve active compliance, the limitation introduced by the adoption of elasticity must be overcome [4] . To this end, various impedance/admittance control methods have been developed to afford increased SEA performance. Impedance control differs from admittance control: the former uses an inner force control loop, whereas the latter uses an inner position control loop [5] . Vallery et al. [8] proposed a cascade proportional-integral-derivative (PID)-based control method in which the bandwidth limitation was discussed. A recent attempt has been made to manage the disturbance and uncertainty by disturbance observers (DOBs). Oh et al. [4] developed a DOBbased force controller with feedback compensation and achieved improvements in both accuracy and system robustness. In a real-world application, the estimation of the external force that applies to the end effector of SEA is essential to achieve compliance. To this end, as reported in [9] , an external force observer (EFO) was developed by decoupling the load dynamics. A low-pass filter was introduced to make the model realizable. However, the application performance in impedance/admittance control was not discussed.
Although the control accuracy of SEAs has been well investigated, studies that focus on the time-delay problem are considerably few. Communication, filtering, and computation can induce extended time delays as well as amplitude decay [10] . Time delays can cause inaccurate actuator response, which may lead to tracking errors and performance losses; it is also perilous [11, 12] . A number of compensation methods have been investigated to cope with such a problem. A Smith predictor was developed by Budolak et al. [11] to eliminate the position drift in a haptic teleoperation. Kim and Bae [13] provided a model-based time-delay controller to demonstrate the zero-impedance control in an exoskeleton. Another effective method was developed by Phillips and Spencer [14] ; a feedforward compensator was designed to cancel the system dynamics based on an inverse model. The aforementioned methods require prior system information to determine necessary coefficients, which can be problematic in real-world cases especially for hydraulic SEAs [15] . Chae et al. accordingly developed an adaptive time series (ATS) compensator that does not require user-defined parameters and system model [15] . It was then applied to an approximate SEA system to achieve a real-time force control [16] ; however, only a static circumstance with a fixed end effector was considered. Furthermore, similar to the study in [9] , the performance in impedance/admittance application was not elaborated.
This study aims to design a low-delay admittance controller for hydraulic SEAs using minimal information regarding the system as well as the environment. To this end, the ATS compensator along with a high-precision admittance control scheme is employed. This scheme is developed based on a DOB-based position controller proposed by the authors [17] . In this study, we further adopt the EFO for the original algorithm and provide an admittance control method for real-world applications. Two ATS compensators are then applied to countervail the time delay in both load position control and EFO. Different from that in load position control, the compensator designed for the EFO employs a fixed set of parameters to cope with the unknown external force in practical applications. The novel low-delay admittance control method with the ATS compensator (ATS-admittance control) is made independent of hydraulic dynamics. Additional system information, except for load mass and spring stiffness that can be easily acquired, is unnecessary. Simulation experiments confirm that a shorter time delay and a lower amplitude decay can be obtained by using the compensator for the time delay. The reduction in phase lag and amplitude error in the estimation of external force and position tracking enables faster and more accurate response, thus enhancing the safety of human-robot interaction in industrial manufacturing.
The remainder of this paper is organized as follows. The proposed low-delay admittance control method is discussed in Section 2. The simulation experiments employed to verify the performance of the proposed method are described in Section 3, whereas results and discussion are presented in Section 4. The conclusions drawn from the study are finally summarized in Section 5.
In this section, a novel admittance control method with highprecision and short time-delay is developed. The ATS compensators are introduced to countervail both the time delay and amplitude error in the actuator control and external force estimation. First, the EFO is employed for the original position controller, and thereafter the holistic admittance control scheme is developed. The details of the improved ATS-based method are presented below.
The authors have previously proposed a DOB-based position control method with feedback compensation for hydraulic SEAs, which has been proven to achieve a higher tracking accuracy than traditional methods [17] . It is designed independent of hydraulic dynamics, thereby benefiting system robustness and practical applications. This method is developed based on a known external force. The estimation of this external force, however, can be problematic in real-world cases. Oh et al. proposed an EFO that decouples the load dynamics [9] to obtain the accurate force. Based on previous works, an EFO-based admittance control scheme for hydraulic SEAs is proposed, as presented in Fig. 1 .
Compared to the original position controller, the admittance control loop and EFO are adopted. The admittance model from measured external force, Fm, to position variations, Δl, is represented by Pk(s) as follows:
where Dds and Kd denote the desired damping and stiffness, respectively; Ph(s) represents the hydraulic dynamics from current I to flow Qa; Pl(s)= DL/Da denotes the load dynamics from cylinder extension Da to load position DL, whereas Pln(s) is the nominal plant; Dd represents the desired load position. The details can be found in the corresponding paper [17] . The external observer can be developed based on inverse dynamics as follows:
where Fm denotes the measured external force via an external force observer. A second-order Butterworth filter, Q(s), with a cut-off frequency of 15 Hz is added to make it realizable and eliminate high-frequency noise, whereas a mean filter is employed to obtain a smoother curve. The position variation, Δl, is thus calculated based on the estimated force via the admittance model, and input to the load target position. Provided that SEA can follow the target position, the actuator can achieve the desired stiffness. The use of filters, however, introduces time delay and amplitude decay [16] , thereby degrading the control performance. Furthermore, tracking error and time delay introduced by the actuator also exist, as shown in Fig. 2 . Such influence can adversely affect the SEA performance and increase the risk of injury especially when human-robot contact occurs.
The ATS was proposed by Chae et al. [16] to achieve a realtime hybrid simulation of a servo-hydraulic system. It does not require any previous system information, and the compensator parameters self-adapt according to historical information. Such features make it an appropriate method to improve the control performance of hydraulic SEAs because of nonlinearities in hydraulic dynamics. In this study, the ATS is applied to achieve a low-delay admittance control for hydraulic SEAs. Two ATS compensators are designed for countervailing the load position control and the external force observer. The structure of the ATS-based admittance control method is schematically shown in Fig. 3.   Fig. 3 . The schematic of the proposed low-delay admittance control method.
First, an ATS-based SEA position controller is developed. Provided that the load position output (DL) contains a time delay (τp) and amplitude error (Lp) compared to the desired input position (Dd), the output at time t is obtained as follows.
This equation can be approximated using Taylor series:
To match the actuator output (DL) with the desired input (Dd), the compensated input, Dc, should be as follows.
To improve efficiency and eliminate the noise introduced by high-order terms, a second-order system is applied (i.e., n = 2). The coefficient matrix, Ap = [a0, a1, a2], can therefore be identified by making a set of historical compensated input position, Dc, match the actual output, DL, over a previous jΔt time period via the least-square method, where Δt represents the sampling cycle. The coefficient, Ap, at time k can be derived as follows: 
where Dd k = [Dd k Ḋd k D d k ]; Dd k represents the desired load position at time k. The coefficient matrix, Ap, will therefore be updated for every sampling cycle based on previous input and output load positions. This procedure begins after an initial time, jΔt. Such a compensation method in the discrete time domain is referred to as the ATS compensator. The ATS compensator is also applied to the external force observer. An improved EFO with an ATS compensator (ATS-EFO) is accordingly proposed.
Compared to the actual external force, Fr, if the estimated force, Fm, has a constant time delay, τf, and an amplitude decay, Lf, then the following is obtained.
Different from the position control, however, the actual external force is difficult to be obtained in real-world applications. Thus, the parameters of EFO are not able to be updated by using historical information. In this case, a set of fixed parameters, Af = [a0f, a1f, a2f], is therefore employed for compensation without feedback. The compensated force at time k is therefore derived as follows:
where Fm k = [Fm k Ḟm k F̈m k ]; Fm k represents the measured force by using the conventional external force observer at time k.
Simulation experiments are performed to evaluate the performance of the improved admittance control method by using MATLAB 2017b software. The control scheme is developed based on Figs. 1 and 3 . The definitions and corresponding values of hydraulic dynamics for the simulation are found in [17] . In the experiments, the following are implemented.
1) Position-tracking test under non-contact condition. The target position is set using a sine wave with a frequency and form of 5 Hz and 40sin(10πt), respectively. 2) Stiffness-tracking test under contact condition with varying external force. The target stiffness is set to 5000 N/mm, and the external force is set using a sine wave with a frequency of 5 Hz and form of 200sin(10πt). The damping is set to 0. The external force is set to zero in the non-contact condition, which denotes the situation when the SEAs move freely without any contact with the environment (Fr = 0). In this case, the SEAs track the target position. The contact condition, however, represents the situation when the SEAs come in contact with the environment (Fr ≠ 0) . This makes the SEAs track the desire stiffness according to the measured external force. These two tests are designed to evaluate SEA performance on tracking accuracy and safety insurance.
In order to evaluate the effectiveness of the proposed method, the non-compensated admittance control method is selected for comparison. In the second stiffness-tracking test, the external force is applied to the load after the actuator first reaches an equilibrium position to simulate a sudden contact. The sampling rate is set to 1000 Hz (Δt = 0.001 s), while the control parameter values of the two compared methods are listed in Table 1 . We select the sampling rate and control parameters based on a self-built experimental platform, the details can be found in [17] . The proposed method requires two sets of specific parameters: one set of initial parameters, [a0, a1, a2], for the actuator control and one set of fixed parameters, [a0f, a1f, a2f], for the external force observer. In this study, based on a recommendation, the ranges of [a0, a1, a2] are set to (0.1, 2), (0, 0.1), and (0, 0.005), respectively [15] . We also want to evaluate the response rate of the ATS compensator. Therefore, different from the original study, the initial value of [a0, a1, a2] is set to [1, 0, 0], which indicates that there is no compensation at the beginning. Moreover, to ensure a quick response, no limit is set to the change rate in each step. The number of rows in Xr is set to 250 (j = 250), indicating that the first parameter change begins at 0.25 s. On the other hand, it is necessary to determine the coefficient [a0f, a1f, a2f] in advance. The identification procedure is discussed in the subsequent section. 
To determine the values of ATS-EFO parameters, [a0f, a1f, a2f], a series of external forces are applied to the actuator. The input is a combined set of data including two different waves: a large amplitude sine wave with a 4-Hz frequency and form of 100sin(8πt) as well as a small amplitude Gaussian white noise with a 10-Hz frequency and an amplitude of 10. Figure 4 shows the combined actual input force and estimated force obtained by using the EFO. The parameters [a0f, a1f, a2f] are determined by using the method of least squares as discussed in Section 2 based on the data shown in Fig. 4 , where, a0f = 1.015, a1f = 0.051 s, and a2f = 0.001 s 2 . The time delay can thus be calculated by a1f/a0f = 50.24 ms.
In order to maintain consistency and present clearer figures, we select the wave peak in the fifth period to calculate the time delay. The reason is that the SEAs need to achieve a low-delay tracking in less than 1000 ms in the real-world applications. From Fig. 1 , the actual time delay is therefor found to be 51 ms, which is consistent with the value estimated by the proposed method. The identified parameters are applied to develop the ATS-EFO.
In this section, the maximum value, the average value and time delay are selected as the evaluation criteria. The experimental time for each test is set to 3000 ms. Therefore, the maximum and average values are calculated using 3000 sampling points according to the sampling rate. Figure 5 shows the position-tracking results under the noncontact condition. It can be observed from the figure that the non-compensated method produces a more considerable error and longer time delay. Figures. 6 and 7 show the tracking errors and values of time-varying parameters, [a0, a1, a2], during the test, respectively. It can be observed from Fig. 6 that the tracking errors gradually decrease by using the proposed method. This is because the proposed parameter-adaptive mechanism enables the three parameters (a0, a1, and a2) to change according to the demands, as shown in Fig. 7 . The compensation for the time delay is particularly beneficial to the overall performance. The details listed in Table 2 indicate a 75.6% reduction (from 14 to 3 ms) in time delay. A smaller overshoot, which can be beneficial to the interaction safety, can also be achieved through the proposed method. Figure 8 shows the results of external force estimation. It is clear from this figure that the estimated force obtained by the non-compensated method has a large amplitude decay and time delay. Such negative presentations are introduced by the lowpass and mean filters, which can adversely affect the stiffnesstracking performance. However, the proposed ATS-EFO can provide a more accurate estimation via the compensation for the amplitude decay and the phase lag. This conclusion is further proven by the list in Table 3 , which indicates that the proposed method can reduce the maximum absolute error by 76.5% (from 260.99 to 61.31 mm). Although the method yields a 6-ms phase lead, it can still improve the practical application by providing a faster response. It is also observed from Fig. 8 that the ATS-EFO can introduce a slight initial vibration. Such vibration is introduced by the differential computation. In early experiments, we found that the vibrations were eliminated after we removed the differential part. However, it can be suppressed rapidly, and the improvements to the overall performance validate the proposed method. Figures 9 and 10 present the position and stiffness-tracking performance under contact conditions, respectively. The target position is calculated based on the actual force and target stiffness, indicating that the desired stiffness can be achieved provided that the actuator displacement can follow the target position. It can be observed from Fig. 9 that the proposed ATSbased method allows an immediate reaction to the external force compared with the non-compensated method. From Fig.  10 , it is observed that the stiffness-tracking error obtained by the proposed method gradually decreases, whereas the conventional method produces larger and periodic errors. This is consistent with the position-tracking results under the noncontact condition. The variation of parameters [a0, a1, a2] tends to reduce the position-tracking error, thus leading to a higher Table 5 indicates that the proposed method aids in reducing 68.9% (from 2948.50 to 917.84 N/mm) of the maximum error and 82.9% (from 552.06 to 94.16 N/mm) of the average error in the stiffness-tracking test. It is also observed from the lists in Tables 3, 4 , and 5 that the overall time-delay resulting from the use of the conventional method is 63 ms, which is derived from both force estimation (48 ms) and position tracking (14 ms). The results obtained by the proposed method, however, are not strictly consistent with the foregoing conclusion. The reason is that parameters [a0, a1, a2] can be time-varying in different tests. It is further observed that the tracking error and actuator delay are mainly caused by the ill-estimated force. This validates the necessity of improving the external force observer. The experiments thus verify the effectiveness and superiority of the proposed method. The tracking accuracy can be significantly improved and the time delay can be considerably reduced in stiffness tracking. This method also provides a potential approach to accurately estimate the external force in real-time response. 
In this study, a novel admittance control method with timevarying delay compensation that considers the external force estimation is proposed for hydraulic SEAs. First, the noncompensated admittance control scheme based on the external force observer (EFO) is provided. Thereafter, the ATS compensators are employed to enhance both the positiontracking and force-estimation performance. Based on the improved position controller and ATS-based external force observer (ATS-EFO), the low-delay admittance control method is developed. The advantages of the proposed method over the non-compensated approach are elaborated via the simulation experiment results.
Based on these results, the use of the proposed method can effectively reduce 75.6% of the time delay in position tracking while achieving an 82.9% improvement to the stiffness tracking accuracy on average. We also observe that the time delay introduced by force estimation has a more detrimental effect than that caused by position tracking and may be effectively countervailed by the developed compensators. Although a slight vibration is introduced by the proposed method at the early stage of force estimation, the overall control performance is remarkably improved with the reductions in time delay and amplitude error. Further, the proposed method can be easily applied with no user-defined parameters. All the results demonstrate that the proposed low-delay admittance control method can be potentially applied to SEAs to achieve a safe human-robot collaboration in the industrial setting. A future study is planned to conduct further tests to evaluate the influence of vibration on SEA performance. The applicability of the proposed method is also needed to be verified in realworld cases.