1. Introduction
In recent years, high-compliant and low-cost pneumatic artificial muscles (PAMs) have been widely implemented in rehabilitation systems [
1,
2,
3,
4]. PAMs are shortened in the longitudinal direction and enlarged in the radial direction when being inflated, and they will turn back to their initial form when being completely deflated. PAMs act similar to the human muscle, e.g., the longer muscles produce bigger force and vice versa. Furthermore, these pneumatic muscles are also inherently compliant, which makes them suitable for applying in human-robotic systems. In comparison with the motorized actuators, PAMs are lightweight and have a high power-to-weight ratio. In addition to the aforementioned advantages, the PAM-based applications also have inherent drawbacks, such as very high nonlinearity and uncertainty, and slow response in force generation. These drawbacks make it difficult to model and control PAMs.
Using a nonlinear mathematical model to describe the nonlinear characteristic of the PAMs is the most common choice of researchers. In 2003, D. B. Reynolds et al. introduced a three-elements model of PAM, which consists of a contractile (force-generating) element, spring element, and damping element in parallel [
5]. Using this type of model, K. Xing et al. developed the sliding mode control (SMC) based on a nonlinear disturbance observer to improve the tracking performance of a single PAM-mass system [
6]. A boundary layer augmented SMC and its modified versions have also been developed for both antagonistic configuration of PAMs and robot orthosis actuated by PAMs [
4,
7,
8,
9,
10,
11,
12]. However, the procedure to identify this model’s parameters remains complicated with at least two separate experiments: one experiment for determining spring (
K) and contractile (
F) coefficients and another experiment for estimating damping (
B) coefficient. Each experiment must be carried out in three steps [
6]. Besides, the parameters of the damping (
B) coefficient must be obtained by measuring the load’s acceleration, which is very sensitive to external noise. For this reason, it is difficult to obtain the model’s parameters with high accuracy.
To deal with hysteresis of PAMs, many hysteresis models have been proposed recently, e.g., Maxwell-slip model [
13], Prandtl–Ishlinskii model [
14], and Preisach model [
15]. In these reports, the dynamic characteristic of PAMs was described by an equivalent pressure/length hysteresis model. The obtained models were used in the feedforward term of the cascade position control scheme for hysteresis compensation. The inner loop of the controllers was designed to regulate the inside pressure of the muscles. The outer loops were designed to deal with the nonlinearity of the PAMs characteristic. Both of the loops use PID-based control strategy. Consequently, some authors continued to develop the modified hysteresis model for both single PAM-mass system and PAMs in antagonistic configuration [
16,
17]. However, they mainly focused on modelling of PAMs. Only the trajectory-tracking experiments with low frequency, e.g., up to
Hz, were conducted in literature. Furthermore, enhanced PID control methods, which were most widely used in these studies, could not deal with hysteresis of PAMs.
Another common way to identify the model of PAM-based actuator is the grey-box experiment method [
18,
19,
20,
21]. In 2015, to deal with uncertain nonlinearity of PAMs, Dang Xuan Ba et al. introduced a grey-box experimental model, which consisted of uncertain, unknown, and nonlinear terms. Based on the built-in model, the authors employed a sliding mode control strategy [
18] and an integrated intelligent nonlinear control approach [
19] for the tracking purpose. The control performance was significantly improved, and the system could track the
amplitude sinusoidal signal with
Hz frequency. The grey-box method was also reported by Robinson et al. in 2016 [
20] and by L. Cveticanin et al. in 2018 [
21]. The relationships angle/torque and force/pressure were thoroughly investigated in the wide range of pressure. However, only the mathematical model was considered and verified in [
20]. The low rate of desired trajectories was tracked in [
21].
The mechanism-based model [
22,
23] is another method in which the behaviour of PAMs was described based on their physical properties: length, diameter, and volume of PAMs, etc. However, as most of nonlinear models mentioned above, these types of models also require a complex procedure to derive the model parameters.
To obtain the model of PAMs in a more simple way with a good enough accuracy, the linear mathematical model has recently been applied to approximate the characteristic of PAMs [
24,
25,
26,
27,
28]. In these studies, the uncertain nonlinearity of PAM was considered as the system disturbance and solved by extended state observer (ESO) together with an active disturbance rejection controller (ADRC). The control performance is considerably improved with
of sinusoidal reference signal frequency.
In this research, a discrete-time fractional order integral sliding mode control (DFISMC) is employed to improve tracking performance of an antagonistic actuator driven by PAMs. First, a linear discrete-time second order plus dead time (SOPDT) model is chosen to describe the nonlinear dynamic behaviour of the antagonistic actuator. In this approximation, a nonlinear term in characteristics of PAM is considered as a disturbance. Then, the DFISMC is designed based on the fractional order integral (FOI) calculus and a disturbance observer (DSO) for the trajectory tracking purpose. Finally, the effectiveness of the proposed control technique is confirmed through multi-scenario experiments. The proposed method shows many advantages in both mathematical model and control technique. The linear discrete-time SOPDT can approximate the behaviour of antagonistic actuator at a good accuracy. Besides, the identification procedure is also simplified. By employing the FOI function together with a DSO, the DFISMC controller is able to reduce the “chattering” problem in sliding mode control systems. In addition, the proposed is designed in a discrete-time domain, therefore, it can be easily implemented by any digital control system.
3. Control Design
Recently, SMC has been employed for designing the controller for PAMs or systems powered by PAMs [
4,
6,
7,
8,
9,
10,
11]. SMC is able to provide highly accurate tracking performance with a bounded error; however, “chattering” problem is a big challenge that SMC must overcome. SMC is a suitable control approach for PAM-based systems to deal with their uncertain, nonlinear and time varying characteristics. In this research, we addressed a DFISMC to improve the tracking performance of the antagonistic actuator powered by PAMs. The fractional order integral is implemented together with disturbance observer to deal with the “chattering” problem.
Figure 3 illustrates the block diagram of the proposed control system.
We consider the following fractional integral sliding surface:
where
is the tracking error with the desired trajectory
, and
is the integral of the tracking error with fractional order
and integral gain
.
can be calculated as follows:
and
at the initial state. Please refer to
Appendix A for details about fractional integral approximation. We also obtain
From (
6)–(
8), we can obtain
Therefore,
where
is one-step-ahead tracking error, which can be computed from the SISO model of the actuator in (
4) as
where
is one step ahead of the desired trajectory, which is considered to be known when apply the model to a specific application. In (
4), disturbance
is unknown and needs to be estimated. In this study, one-step delayed technique was used to estimate
. This technique is based on the following assumptions:
Assumption 1. Sampling time was sufficiently small and system disturbance is bounded, so the difference between two consecutive time samples is also bounded, i.e.,where is the thickness boundary layer. It means there always exist constants A and B, ∀ , such that The aforementioned assumption was based on the Taylor expansion described in
Appendix B.
Estimation
of disturbance
can be computed based on (
4) as
where
Hence, the error of estimation
is
Finally, the one-step-ahead tracking error (
11) can be expressed by
When substituting
in (
11) and
into (
10), we can obtain
Disturbance estimation error
is unknown in practice; however, it is very small and bounded by assumption 1. Control signal
can be obtained by solving the reaching law
with the absence of
as follows:
Adjusting integral gain and fractional order integral may improve performance of the control system.
5. Discussion and Conclusions
This paper proposed an advanced SMC control strategy for PAMs in antagonistic configuration. First, the discrete-time SOPDT is chosen to describe the dynamic behaviour of the antagonistic actuator. The chosen model demonstrated a good approximation of nonlinear characteristics of the actuator: the root mean square errors between estimated and measured values are less than . Based on the built-in model, an DFISMC controller, which employed a fractional order integral of tracking error together with a disturbance observer, was proposed for the tracking purpose. The implemented approximation of FOI and DO was able to reduce the “chattering” phenomenon, which often occurs in SMC implementations. The reduction of the “chattering” phenomenon is very important for applications of the PAMs in rehabilitation robot field. Finally, multi-scenario experiments were carried out to compare the tracking performances between the DFISMC and the conventional DSMC.
In comparison with the three-elements model [
5], hysteresis model [
13,
14,
15], and mechanism-based model [
22,
23], the identification procedure of the proposed method is simplified. Besides, this procedure does not need to measure the load’s acceleration, which is very sensitive to noise. Experiments show that, in comparison with DSMC, the DFISMC was able to significantly enhance the tracking performance of
amplitude sinusoidal signals with frequency up to 1.0 Hz. In particular, when the actuator drove a load of m = 2.5 kg, the RMSTEs of DFISMC were about two times less than those of conventional DSMC in most of the desired trajectory frequencies. For example, with a frequency of 0.2 Hz, the RMSTEs are
and
for DFISMC and DSMC, respectively. The proposed controller achieves a performance comparable to the experimental results with similar configuration and desired trajectory in [
23,
24]. In [
23], when tracking a 0.4 Hz frequency and 5
sinusoidal signal, the residual error amplitude is 0.5
equivalent to 10%. When tracking a 0.5 Hz frequency and 20
amplitude sinusoidal signal, the RMSTE of the DFISMC is 1.47
, equivalent to 7.35% of amplitude. This result was better than the one in [
24], in which a sinusoidal signal with 40
amplitude and frequency 0.25 Hz is used as a desired trajectory. The experiments also show that the proposed controller can track a human-gait pattern with the MTE of less than 6
. This result is in accordance with the commercial gait training system LOKOMAT [
30], in which the MTE is 15
. It is shown that the built-in model and proposed controller can be applied in robot gait training system. Furthermore, the proposed DFISMC is designed in discrete-time domain, so that it is convenient for implementing in any digital industrial controller, e.g., the NI instrument in this research.
In summary, this paper presents the control of an antagonistic actuator powered by PAMs. The dynamic behaviour of the antagonistic actuator is described by a discrete-time SOPDT model, which requires a simpler identification procedure. The DFISMC controller based on a DSO and the approximated FOI is used to improve the tracking performance. The implementation of DSO and FOI also helps the system reduce the “chattering” phenomenon. The experimental results illustrate the applicability of the proposed model and controller to a robotic gait training system with a human-gait pattern trackable ability. Future work will involve the impedance control of the antagonistic actuator to increase applicability of PAMs in the field of rehabilitation. The impedance of the actuator can be regulated by manipulating the nominal pressure of two PAMs. To integrate the impedance controller into the system, the relationship between the actuator compliance and nominal pressure would be considered and modelled in future work.