Position and stiffness control of an antagonistic variable stiffness actuator with input delay using super-twisting sliding mode control

Motor dynamics in antagonistic variable stiffness actuator (AVSA) is generally disregarded in control system design. This ignorance can lead to an inaccurate system model, affecting the performance of the closed-loop system. The motor dynamics can be modeled as an input-delay in actuator model. In this paper, the motor dynamics is modeled as the input time-delay for an AVSA for the first time. The stiffness of AVSA is a nonlinear function of system states; thus, stiffness tracking for an AVSA is a challenging task. Specifically, many of the existing delay compensation controllers cannot be used for stiffness tracking when the model contains input delay. To handle this issue, a nonlinear transformation is introduced and a super-twisting sliding mode control is then utilized to reach position and stiffness tracking simultaneously. Prediction-based feedback is involved together with some disturbance observers for estimating the external disturbance to compensate for the input time-delay. Simulation results show that the proposed design approach is successful in position and stiffness tracking and simultaneously in attenuating the external disturbance effect.


Introduction
As the main source of mechanical power, actuators are one of the most important components of any robotic system . Traditional rigid robots comprise of actuators with possible gearbox for altering delivered force/torque to the end effector. This feature enables rigid robots for high accuracy positioning applications, such as industrial robots [38]. When the robots are required to work in collaboration with humans, using rigid robots have various disadvantages such as safety in contact with humans, backdrivability and efficiency [4,47].
New alternatives introduced in the literature due to the limitations of conventional rigid robots, [9,35]. The main concept of these designs is to include a compliant hardware in the actuator structure to adjust the impedance of the end effector (in addition to position) can be adjusted [6]. Stiffness control received considerable attention between impedance effects (damping and stiffness) in the past decades and many different structures have been introduced for position and stiffness control. Inspired by human muscles, constant stiffness springs were included between the driving motor and the actuator link to add compliant behavior to the actuator. Series elastic actuators (SEAs), which could guar-antee constant stiffness at the end effector by employing a passive compliance between the high impedance actuator and the load, were introduced in this concept [22]. The main drawback of the SEA lies in its constant stiffness; therefore, variable stiffness actuators (VSAs) were introduced to cover this limitation. Many different structures are found in the VSA category including serial variable stiffness actuator (SVSA) [5,15,43,44], antagonistic variable stiffness actuator (AVSA) [7,29,39], reconfigurable variable stiffness actuator (RVSA) [51], mechanical-rotary variable impedance actuator (MeRIA) [25,26,30], cable-driven variable stiffness actuator (CDVSA) [41], binary-controller variable stiffness actuator (BcVSA) [13], VSA based on rotary flexure hinges [24], VSA based on parallel-assembledfolded serial leaf spring [2], variable stiffness mechanism based on preloaded curved beams [49] and vsaUT-II [10], to name a few. RVSA is a novel torquecontrollable variable stiffness actuator which that can address the desired torque and stiffness targets through a pulley block. The reconfigurable property of RVSA enables its realization of a wide stiffness range. MeRIA, a variable impedance actuator developed for rehabilitative applications, can perform joint position and stiffness control tasks independently. CDVSA comprises servo motors, bearing frames, transmission shaft, base house, flexible beams and some cable-driven mechanisms producing a continuously tunable stiffness arm for safe physical human-robot interactions. The basic design principle of BcVSA is based on a stiffnessvarying mechanism comprising a motor, some inline clutches and three torsional springs connected to the load and the motor shaft using two planetary sun gear trains developed for safe human-robotic interaction. Flexure hinges have also been utilized to design a new variable stiffness structure where output position and stiffness can be controlled independently. The stiffness of VSA can be adjusted by changing the effective length of parallel-assembled-folded serial leaf springs to achieve a low minimal stiffness. Another structure of VSA employs a linear variable stiffness mechanism (LVSM). The basic idea is to connect two lateral curved beams with an axial spring, and the obtained stiffness can be modified by changing the preload adjustment of the curved beams.
In SVSA mechanism, two actuation units are used for one link, where the first unit (usually large) is used for position tracking and the second unit (usually small) is responsible for stiffness adjustment. A lever mech-anism is generally used for moving a sliding component to change the stiffness of the output shaft through a small secondary motor and a remarkably powerful primary motor is utilized to rotate the entire structure for position tracking. By contrast, two identical motors are used in AVSA mechanism connected to the link through two similar springs. When two motors are rotated in the same direction, the link position can be controlled; when two motors are turned in antagonistic directions, then the link stiffness can be adjusted. This behavior imitates human actuation of an upper arm with two motors mimicking biceps and triceps muscles. Other structures of VSAs exhibit the same behavior as of SVSA and AVSA with different configurations and actuation units. The aforementioned issues of rigid robots can be solved on the basis of the VSA structure. VSAa are far safer than rigid robots in collaboration with humans; they are also efficient for some applications such as rehabilitation purposes [31,40,48], and backdrivable against external force/torques. Numerous works on position and stiffness control of AVSA are available. The Feedback linearization method has been used in [37] to control a twisted spring actuator with antagonistic configuration. A novel asymmetric antagonistic actuation unit is developed and controlled in [39] with improved energy efficiency. In [12], two antagonistic configurations are introduced and sliding mode control together with linear extended state observer has been used to control the position and stiffness. Robust adaptive control of AVSA is considered in [27] in the presence of uncertainties and disturbances. Identification and cascaded control of an AVSA used for a bio-inspired robot actuator is investigated in [29] using PID controllers. Decoupled nonlinear adaptive control of an AVSA with pneumatic actuators is considered in [45] for position and stiffness tracking tasks. The performance of sliding mode and model predictive controls is compared in position and stiffness tracking of inflatable soft robots with antagonistic structure [3].
Some elastic structures which are enabled by pneumatic actuators [45], biphasic media [28] and piezoelectric actuator [50], have been introduced; however, the majority of these structures employ electromechanical actuators as the source of mechanical power. The motors used in VSA units have their dynamics that is not considered in the design procedure in almost all results reported in the literature. Notably, motor dynamics is not considered even for other VSA types, such as SVSA, in the literature. All the works in the lit-erature assume an ideal model without considering the effect of motor dynamics. Ignorance of motor dynamics will lead to unstructured model uncertainty; thus, the model-based design of control system would result in low performance closed-loop systems (if not unstable). Instead of integrating motor dynamics to the VSA model, which may increase the complexity of the model for design purposes, assuming the motor dynamics as a transport delay in control input is possible. In practical implementations, obtaining the time-constant of the step response of the motors and computing the timedelay using this time-constant is possible. To the best of authors' knowledge there is no reported work on robust control of a VSA with input time-delay. In this paper, input time-delay will be added to an AVSA model and a robust super-twisting sliding mode controller (STSMC) will be designed on the basis of disturbance observers (DOs) for simultaneous tracking of desired position and stiffness. Any nonlinear robust control method can be utilized in the proposed design; however, STSMC has been chosen in this study. STSMC has all the advantages of SMC, including robustness to the disturbances, with the additional benefit of a continuous structure, which avoids the chattering phenomenon. In addition, the performance of STSMC can be tuned by adjusting fairly low number of parameters. Furthermore, the stability analysis for the STSMC is straightforward. Two general methods are used in the literature for dealing with input time-delay: static (memory-less) and dynamic (predictive) approaches [8,20]. The static method uses state feedback structure for feedback and transform the input delay to the state delay while in predictive method, and the prediction vector of the system is obtained and used for the feedback. The computational cost of the prediction-based approach is higher than the static one; however, the predictive approach can handle longer time-delays [17]. Thus, predictionbased delay compensation will be used in this paper. The main contributions of this paper can be summarized as follows.
• The effect of motor dynamics is considered in AVSA dynamics as input time-delay; • STSMC is used for robust control of AVSA to attenuate external disturbance together with position and stiffness tracking.
The rest of this paper is organized as follows: Sect. 2 presents the delayed model of AVSA with equivalent quadratic torsion spring (EQTS) and equivalent expo-

Notation
Through this paper, the following notations will be used. Let x (n) (t) = d n x dt n represents the (n)-th time derivative of x(t). R n and R + denotes the ndimensional Euclidean space and the set of all positive real numbers, respectively. The superscript "T " stands for matrix transposition. . and |.| stand for the Euclidean norm and absolute value respectively. λ min (P) and λ max (P) are used to show the minimum and maximum eigenvalues of matrix P, respectively.

AVSA model
A schematic view of the AVSA is illustrated in Fig. 1. Several different versions of ENTS-based VSAs have been introduced in [11]; however, the ideal control inputs with no time-delay are considered in these models. The dynamic model of equivalent nonlinear torsional spring (ENTS)-based variable stiffness actuator with input-delay is given by [12]: where M is the inertia of the output link, D q is the damping coefficient of the output link, E g = mgd is the gravity effect acting on the output link, m is the equivalent mass of the link, g is the gravitational acceleration of the Earth, d is the distance from the rotating axis of the output link to its center of mass, τ ext is the unknown external disturbance, J m is the inertia of the elastic actuation unit, D m is the damping coefficient of the elastic actuation unit, τ is the input time-delay, τ α and τ β are input torques of the elastic actuation units with a rotation angle of α and β, respectively; q is the angular position of the output link, ϕ(α, β, q) is the combined effect of the elastic actuating torques acting on the output link, φ(α, q) and ψ(β, q) are elastic actuating torques acting on the output link. For additional information on the mechanical design and details of the ENTS-based AVSA configuration, the readers are referred to [11,12].
The numerical values of model parameters for AVSA (1) are presented in Table 1. Dynamic model (1) and model parameters in Table 1 reveal that two antagonistic actuation units (two motors) are the same; therefore, assuming equal input delay τ for both is reasonable.
Two different configurations for ENTS-based AVSA are introduced in [12]: EQTS and EETS. The elastic torques of actuation units are quadratic functions of system variables (angular displacement) in EQTS configuration; however, in EETS configuration, these torques are exponential function of angular displacement. Two methods have generally been used in the literature to obtain the real stiffness. Most of the works derive a mathematical model for the stiffness of the proposed design and use it as the real stiffness. The quality of the measurement then depends on the accuracy of the stiffness model [29]. Alternatively, creating an angular displacement in the actuator shaft through an external force/torque and obtaining an approximation of the stiffness by dividing external force/torque by angular displacement is possible. For example, [21] has used this method to obtain impedance of the human ankle during locomotion. The stiffness model, which is obtained for EETS and EQTS configurations in [12], is used in this paper.
The generated torque of a quadratic torsion spring considering angular displacement θ is given by τ = aθ 2 , where a is a constant [12]. Considering the effect of pretensions α 0 and β 0 and by computing torques of both sides of antagonistic VSA, the elastic torques of AVSA in EQTS configuration are given by: where a = 0.48765 is a constant and α 0 = β 0 = 2.3 rad are initial amount of pretension of the EQTS AVSA. Corresponding joint stiffness for AVSA in EQTS configuration using elastic torques in (2) is given by: where σ 0 is the initial stiffness of the link. The general relation for the torque of an exponential torsion spring is τ = ae bθ − a with a and b as two constants and θ as the angular displacement [12]. Considering the effect of pretensions α 0 and β 0 and by computing torques of both sides of antagonistic VSA, the elastic torques of AVSA in EETS configuration are given by: where a = 0.1753 and b = 1 are constants. Similarly, joint stiffness for AVSA in EETS configuration using elastic torques in (4) is given by: Comparing (3) and (5), stiffness is the linear and nonlinear functions of state variables for EQTS and EETS configuration, respectively. Nonlinear stiffness function in EETS configuration leads to some challenges in tracking control problem.

Problem formulation
The dynamic model of AVSA with EQTS and EETS configurations was presented in the previous section. In this section, the tracking control problem will be formulated and related results will be investigated. Defining state variables of the original system as y 1 = q, y 2 =q, y 3 = α, y 4 =α, y 5 = β and y 6 =β and state vector as Y = y 1 y 2 y 3 y 4 y 5 y 6 T , the model (1) can be written in the following state space form: Two main tasks must be accomplished for the AVSA system: position and stiffness tracking. Assuming desired link position q d (t) and desired stiffness σ d (t), the following link position and stiffness errors are defined.
The goal is to drive the error variable in (7) to zero asymptotically. Stiffness is a nonlinear function of state variables regarding EETS-based AVSA (5). Thus, the proposed methods in [18,19,23,32,33,36] cannot be used for delay compensation and tracking control of the AVSA system. A new design approach with nonlinear transformation and some DOs for the transformed system will be introduced to handle the aforementioned issue. Suppose error variables in (7) as nonlinear transformation of original state variables. Output errors will be differentiated until the appearance of control inputs.
Regarding (2) and (4), the elastic torques are different for EQTS and EETS configurations therefore, the process of differentiating will lead to different set of equations. The dynamics of the transformed (error) variables will be given in the following subsections.

EQTS configuration
For EQTS configuration, the derivation of error variables leads to the following dynamical equations: x (4) where nonlinear functions in (8) are given by: Differential Equations in (8) revealed that the relative degrees of the link position and stiffness errors are 4 and 2, respectively. Notably all nonlinear functions φ, ϕ, and ψ should be replaced from (2). The original system model is affine considering control inputs. Thus, the transformed model (8) is also affine in control inputs. New control input variables u 1 (t −τ ) and u 2 (t −τ ) can be decoupled considering the original control inputs τ α (t − τ ) and τ β (t − τ ) of the system. This feature considers new control inputs u 1 (t − τ ) and u 2 (t − τ ) for control system design. Finally, the original control inputs from the obtained control inputs u From (9), the determinant of G(Y ) is zero when α 0 + β 0 + α + β = 0, which coincides with zero stiffness (Eq. (3)). In this case, no coupling is observed between the EQTS and the output shaft. To avoid this condition, the stiffness is assumed to always have a nonzero value and initial values of EQTS angles (α 0 and β 0 ) are some positive constants. Therefore, desired stiffness is never zero that is a rational assumption. Notably, the same issue is discussed in [12].

EETS configuration
The same derivation is conducted for EETS configuration. By differentiating error variables (7) using elastic torques (4), where nonlinear functions and new disturbances are given by: In the above equations, φ, ϕ, and ψ should be replaced from (4). Similar to the EQTS case, control inputs can also be decoupled for the EETS case and the original control inputs can be obtained from new control inputs. The condition for this process is an invertible G(Y ). The determinant of the matrix G(Y ) for the EETS case is: The determinant of G(Y ) in (11) shows that G(Y ) is never singular and is invertible for any state value.

Prediction vectors
The dynamics of the original and transformed systems was obtained in the previous subsection. For predictionbased control of AVSA, the first step is to acquire prediction vectors of the original and transformed systems in the time-delay horizon. By shifting the original system model (6) τ units of future time, we get: Integrating (11) from t − τ to t, we have: The prediction vector of the original system is defined asP y (t) = Y (t + τ ). Therefore, (13) can be written in the following form: Future values of the disturbance d y (ξ + τ ) for ξ ∈ [t − τ, t] are unknown; thus, computing the exact value of the prediction vector is impossible (14). One possible approximation of the prediction vector can be obtained by replacing the disturbance term with its estimation: If the disturbance can be estimatedd y (t) = d y (t), then P y (t) = P y (t) can be concluded from (14) and (15).
Owing the absence of difference, one of the models in (8) or (10) can be used to obtain prediction vector for transformed dynamics. Therefore, without loss of generality, by shifting error dynamics (10) we have: Similarly, the exact prediction vector of transformed states (16) cannot be obtained; hence, an approximation can be computed by replacing the disturbance term with its estimation: The comparison of (16) and (17) reveals that if disturbance terms d 1 (t) and d 2 (t) can be estimated correctly, then approximate prediction vectors will approach their real valuesp 1 (t) andṗ 2 (t) =ṗ 2 (t), respectively. (17) are different from widely used formulas in the literature [32,33]. More specifically, the approximate prediction vector appears under integral in (17), whereas the prediction vector of the original system is used under the integral of new prediction vector in [32,33]. The current definition avoids limiting assumptions on system nonlinearities, such as the Lipschitz condition assumed in [32,33], sector bounded condition in [16], and some customized conditions used in [1,14].

Remark 1 Prediction vectors in
The final goal is to have zero tracking errors for link position and joint stiffness. If control input is designed such that all prediction variables and their derivatives vanish asymptotically; then, lim t→∞ p (i) All disturbance terms can be estimated asymptotically then, we can conclude that: The predictions of error variables will approach to zero asymptotically based on (19). The derivatives of prediction variables in (17) are obtained as shown below to guarantees zero tracking error. (20) Substituting error derivatives (10) in (20) yields: The stability analysis is conducted asymptotically; therefore, evaluating all relations at infinity is reasonable: lim t→∞ p (4) If the disturbance terms can be estimated, then the following equations are satisfied: Finally by shifting (23) τ units of time back, we get: which is the same as error dynamics in (10). This finding confirms that if the disturbances can be estimated asymptotically and control inputs can be designed such that the prediction variables and their derivatives approach zero asymptotically, then the dynamics of prediction variables coincides with their dynamics. Hence, zero prediction variables guarantee zero tracking errors. In the next section, the STSMC approach and DOs will be introduced to accomplish the two tasks. Before going further, the following assumption is necessary.

Main results
In this section, the prediction-based approach will be used to compensate for input time-delay using super-twisting sliding mode control. Before designing STSMC, a set of DOs will be introduced to estimate disturbances.

Disturbance observers
Regarding the prediction vector of the original system (15), future values of external disturbance is required.
In addition, future values of new disturbances d 1 (t) and d 2 (t) are required.

Theorem 2
The following DOs are considered. (30) with gains 0 < L 1 < π (2τ ) and 0 < L 2 < π (2τ ). The estimations of disturbances obtained from these DOs approach to the real values of disturbances asymptotically in time-delay horizon lim t→∞d Proof By defining disturbance estimation errors as d i (t +τ ) =d i (t +τ )−d i (t +τ ) for i = 1, 2 and differentiating them using (30)the following can be obtained: Substituting (8) and (30) in (31) yields: According to Assumption 1, (32) can be rewritten as: The stability of (33) is equivalent to that of the following scalar delayed differential equation: Similarly, the stability results in [8]revealed that (34) is stable for DO gains L y2 ∈ (0, π (2τ )) and estimation errors of disturbances will vanish asymptotically. Hence, the proof is complete.

STSMC design
In the previous subsection, a set of DOs was introduced to estimate future values of disturbances; hence, the first task is accomplished. An STSMC is then proposed to drive prediction variables to zero, which ensures that the position and stiffness tracking errors vanish asymptotically. Differentiating (17), the following is obtained: (35) Substituting error derivatives from (8) for EQTS configuration or from (10) for EETS configuration in (35) yields: The following sliding surfaces for prediction variables are defined as follows: where λ 11 , λ 12 , λ 13 , and λ 2 are design parameters determining the location of poles of the characteristic equation. Differentiating sliding surfaces (37) gives: (38) Substituting prediction derivatives (36) in (38) and solvingṡ 1 = 0 andṡ 2 = 0, equivalent controls are obtained as follows.
Instead of the traditional sliding mode controller, which is subject to chattering phenomenon [42], STSMC is used [46]: where equivalent controls u 1eq (t) and u 2eq (t) are given by (39) and K 1 , K 2 , W 1 , W 2 are STSMC gains to be determined later.
As discussed in Sect. 3, after obtaining new control inputs u 1 (t−τ ) and u 2 (t−τ ), the original control inputs can be easily obtained by u y (t −τ ) (8) and (10) for EQTS and EETS configurations, respectively.
Theorem 3 Consider AVSA model (6) with position and stiffness tracking errors (7) together with Assumption 1. For any given positive gains K 1 , K 2 , W 1 , W 2 and any initial condition, STSMC (40) together with DOs (25) and (30) drives tracking and disturbance estimation errors to zero asymptotically, and the sliding surfaces will vanish in the following finite time: Proof The derivations of the following proof are similar for both tracking errors. Thus, subscript i is used to combine repeated terms and subscripts i = 1 and i = 2 represent the variables related to the position and stiffness tracking, respectively. Substituting (40) in (38), the following second-order system is obtained: Defining (41) can be rewritten as follows: Consider the following Lyapunov candidate function for i-th prediction variable: where P is a positive definite matrix. Differentiating (43) gives: Using new variables μ 1 and μ 2 , (44) can be rewritten as: The substitution of (45) in the derivative of Lyapunov function (43) then yields: [34] proved that matrix A i is Hurwitz for K i > 0 and W i > max ḋ i (t) . In this paper, no previous assumptions have been made on the upper bound of disturbance and its derivative, except for vanishing at infinity. The derivative of the disturbance term (and hence its estimate) is assumed to approach zero at infinity; thus, W i is not required to be more than a positive value. Assumption 1, concludes that matrix A i is Hurwitz at infinity for any positive STSMC gains K i > 0 and W i > 0. Therefore, for any given positive definite matrix P > 0, a positive definite matrix Q > 0 can be found such that: (43) and (47)deduced that μ 1 (t r ) = μ 2 (t r ) = 0; hence, s i (t r ) = 0 with t r as a finite reaching time. Zero sliding surfaces in (37) guarantees that prediction variables and their derivatives vanish asymptotically, which fulfills the second task. The reaching time will then be subsequently derived. From (43), the following inequalities are satisfied: The following can be written using (48): Knowing −μ T Qμ < −λ min (Q) μ 2 , the upper bound of Lyapunov derivative (47) is obtained as: Knowing − μ 2 ≤ −V i λ min (P) from (48), the upper bound of (50) is: From (49), the following is obtained: Substituting (52) in (50), the upper bound of the Lyapunov function will be: Integrating (53) from 0 to t ri : From inequality (54), finite reaching time to sliding surface is: which completes the proof.

Remark 2
Larger values of observer gains in (25) and (30) generally lead to fast disturbance estimation. However, their marginal values may lead to divergent estimation because of inaccurate numerical calculation.
Starting from a small positive quantity and increasing it until the performance is acceptable is reasonable. STSMC gains K 1 , K 2 , W 1 , and W 2 in (40) are only required to be positive. Large quantities for these gains may improve disturbance rejection; however, substantially large values are not suggested because of the numerical calculations. In addition, control parameters λ 11 , λ 12 , λ 13 , λ 2 can be used to adjust the poles of characteristic equation, thereby altering the reaching time.
If the poles are selected close to (far from) the imaginary axis, then the reaching phase would be slow (fast).

Simulation results
In this section, the proposed design will be applied to the EQTS-and EETS-based AVSA presented in [12], and the results will be compared with the methods introduced in [12] and [3]. [12] presented a robust controller, which is a combination of linear extended state observer (LESO), SMC, and input saturation compensation (ISC) for robust tracking control of the AVSA subject to saturation limit on the control inputs. An SMC controller, which is introduced for a model without time-delay, has been applied to an inflatable soft robot with antagonistic structures in [3]. The AVSA model in [12] includes no time-delay. However the input time-delay is added to the EQTS-and EETSbased AVSA, and the simulations will be conducted in the presence of time-delay. For simulation purpose, STSMC controllers (40) with equivalent controllers (39) and sliding surfaces (37) along with DOs (25) and (30) using model parameters in Table 1 are utilized. Assume zero initial condition for state variables and zero initial function for both control inputs in t ∈ [−τ, 0]. By choosing poles of the characteristic equation for the reaching phase of STSMC as pol 11 = −10, pol 12,13 = −10 ± 1i for the first sliding surface and pol 2 = −10 for second sliding surface, the coefficients of the characteristic equations are obtained as λ 11 = 30, λ 12 = 301, λ 13 = 1010, and λ 2 = 10. STSMC and DO gains are assumed to be K 1 = K 2 = W 1 = W 2 = 400 and L 1 = L 2 = L y2 = 10, respectively. The following external disturbance is considered for simulation: 35 17 ≤ t ≤ 20 (56) where the units of torques and times are in Nm and second, respectively. The final simulation time is 20 s and the sample time is supposed to be 1 ms. In addition, input time-delay is supposed to be τ = 25 ms.
To demonstrate the simulation results of the proposed STSMC, SMC [3], and LESO+SMC+ISC methods introduced in [12], the phrases "STSMC", "Best2020" and "GuoTian2018" are used in figure legends, respectively. Figure 2 shows the states of AVSA with EQTS configuration for all methods. LESO+SMC+ISC states are subject to oscillation due to the imposed input delay and far position from the desired values. Meanwhile, no oscillation is observed for the STSMC approach. Figure 2 also reveals that traditional SMC fails to stabilize some of the states because input time-delay is ignored in design procedure.
Position and stiffness tracking response of three methods with EQTS configurations are shown in Fig. 3. The figure reveals that LESO+SMC+ISC is not unsuccessful in tracking link position and stiffness because of input delay, whereas the STSMC design has small tracking errors for both. The tracking error of the stiffness is acceptable for the SMC approach; however, the tracking error of the position is divergent. This condition may be originated from the fact that the relative degree of the position error is fairly large (four) but it is two for the stiffness error. This increases the complexity of the position tracking for the controller rather than stiffness tracking. Figures 4 and 5 represent the sliding surfaces and control inputs of three methods for EQTS-based AVSA, which confirms the oscillatory behavior of the LESO+SMC+ISC method and divergence of position channels for the SMC method. The sliding surfaces for a stable closed-loop system should vanish. However, this condition is not accomplished for the SMC controller as shown in Fig. 4.
Predictions of the states obtained by (16) are shown in Fig. 6, confirming acceptable predictions of the states.
Predictions of the first and second error variables are shown in Figs Fig. 7 shows that prediction of third-order error derivative x  1 (t) is worse than the predictions for lower order derivatives of position errors. This finding can be a consequence of using the system model for obtaining error derivatives in (8), which may be insufficiently accurate and also numerical calculations of error derivatives in the simulation.
Finally, the disturbances and their estimations are shown in Fig.9. The lower plot shows that DO (25) can successfully follow external disturbance. In EQTSbased AVSA, the second disturbance in (8) is zero but the first one includes derivatives of external dis-5372 A. Javadi, R. Chaichaowarat  turbance. The external disturbance in (56) is discontinuous at t = 5s, 10s, 17s; thus, real value of d 1 (t) in Fig. 9 has large values in these time instances.
Considering the EETS configuration for AVSA, the stiffness relation is a nonlinear function of link and motors positions (5); hence, input delay compensation is challenging. In the current scenario, input time-delay is assumed to be τ = 15 ms but other design parame-ters and desired trajectories are the same as the previous case. The simulation results for EETS configuration are shown in Figs. 10, 11, 12, 13, 14, 15, 16 and 17. Thus, the time-delay of τ = 15 ms is the margin of stability for the LESO+SMC+ISC method from Fig. 10 because almost all angles and angular velocities start to oscillate. However, the STSMC method has reasonable results. Traditional SMC has already   Fig. 11 but the LESO+SMC+ISC approach has oscillatory behavior for stiffness tracking. Notably, small tracking errors occur when the tracking error in other channel has not converged. For example, from t = 10s to t = 11s with a large tracking error in position channel, a small stiffness tracking error can be observed in 11 which vanishes as soon as the tracking error goes to zero. For the EQTS-based configuration, SMC is also unsuccessful in position tracking for EETS configuration. This is expected from divergent state trajectories of SMC in Fig. 10.  Sliding surfaces and control inputs for EETS configurations are shown in Figs. 12 and 13, respectively. As expected for STSMC, sliding surfaces approach zero almost everywhere except in the presence of a tracking error, while the second sliding surface is almost always nonzero for LESO+SMC+ISC. Furthermore, control inputs for LESO+SMC+ISC exhibit oscillatory behavior. The SMC controller fails to drive first sliding surface to zero, but the second sliding surface vanishes at the steady state.
Predictions of the states shown in Fig. 14 are remarkably close to those of the AVSA. Position and stiffness   Simulation results showed that when motor dynamics as an input time-delay is considered in the AVSA model, conventional methods such as traditional SMC in [3] and LESO+SMC+ISC in [12], cannot handle the time delay. In this case, delay compensation should be included in the design procedure along with the disturbance attenuation and trajectory tracking. Fur-5376 A. Javadi, R. Chaichaowarat   (7) originating from nonlinear relation of stiffness (5) imposes a serious challenge in delay compensation and tracking control of the AVSA system. Existing methods for delay compensation of nonlinear systems in the literature can only handle linear error equations [32,33]. In addition, the proposed approaches impose limiting conditions of system nonlinearities, which may not be the case in some applications. In this paper, a unified predictive approach was used for delay compensation to attenuate the external disturbance effect on tracking errors through STSMC. The proposed approach is applied to the AVSA system with EQTS and EETS configurations in this study. However, this approach can be utilized

Conclusion
A new predictive robust STSMC was introduced for delay compensation and disturbance attenuation of an AVSA with input time-delay. Prediction vectors of the original system and error variables were approximated using estimations of disturbance terms. A set of DOs was introduced to estimate future values of disturbances in time-delay horizon. Assuming that the timederivative of external disturbance vanishes at infinity, the proposed STSMC approach together with DOs can  estimate disturbances and also drive tracking errors to zero asymptotically. Simulation results illustrated the efficacy and usefulness of the proposed design in delay compensation and position and stiffness tracking of AVSA in the presence of input delay and external disturbance. We plan to apply theoretical findings on prac-tical AVSA hardware in the future works to verify the performance of the proposed design.
authors would also like to thank Dr. Guohui Tian for his help and support in the simulation and comparison phase of the research.
Funding The authors declare that no funds, grants, or other support were received during the preparation of this manuscript.
Data availability Data sharing is not applicable to this article as no datasets were generated or analyzed during the current study.