Soft robotics represents a new generation of robotic research, which provides numerous advantages, such as being lightweight, safe for close contact with humans, and environmentally friendly, as well as having low cost in terms of materials, construction time, and power (Neppalli and Jones, 2007; Trivedi et al., 2008; Al-Ibadi et al., 2020). In addition to the general advantages of soft robotics, soft actuators, such as contraction and extension pneumatic muscle actuators (PMAs), have their benefits when compared with the traditional electrical and mechanical actuators. Moreover, there is a high ratio of force to the actuator weight, in most cases a 100 newtons for several 100 g (Tondu and Lopez, 2000; Al-Ibadi et al., 2017, 2018a; Yang et al., 2019), but on the other hand, due to the softness, low stiffness, and hysteresis, the PMA shows a high degree of non-linearity and adds more challenges to controlling such types of actuators (Wang et al., 2017; Giannaccini et al., 2018; Teramae et al., 2018).

The performances of soft robots provide infinite degrees-of-freedom (DoF) motions, such as elongation, contraction, bending, shrinkage, and rotation. Furthermore, different designs and actuation techniques give unique behaviors (Manti et al., 2016; Al-Ibadi et al., 2018b; George Thuruthel et al., 2018), and the value mechanism and the high rubber material non-linearity of the PMA make the control process difficult and rule out simple controllers. Therefore, to overcome these difficulties, the high robust control has to be considered (Tondu and Lopez, 2000; Leephakpreeda, 2011). Numerous types of control strategies were used to control the position and force of the PMA. Among them, a linear proportional–integral–derivative (PID) controller has been used in Andrikopoulos et al. (2011), Shen et al. (2015), and Chan et al. (2020). Four PID controllers have been used to control the orientation of the ankle rehabilitation robot of four PMAs, one controller for each actuator (Meng et al., 2017). Adaptive pole placement techniques for positioning PID controllers were applied in Bowler (1996). A sliding mode control was used in Cai and Yamaura (1997) and Carbonell et al. (2001), and a fuzzy sliding mode controller, which is trained by a neural network for single dimensional PMA, is used in Chiang and Chen (2017) and Chiang and Chen (2018). Fuzzy PID was used in Balasubramanian and Rattan (2003), and a fuzzy PD controller and an integration controller were used in Chan et al. (2003). Tracking control with hysteresis compensation was done by PID (Schreiber et al., 2011). A series combination of PID controllers and an artificial neural network (ANN)—non-linear PID—was used in Thanh and Ahn (2006) for physical rehabilitation by using multijoint actuate based on pneumatic muscles. Similar techniques have been used in Andrikopoulos et al. (2014).

This article aims to provide an efficient, simple structure, controller system to be used for various soft robotic systems. For that purpose, a parallel controller structure is proposed by using a neural network (NN) controller and a proportional (P) controller. This structure provides a fast and accurate response to track the soft pneumatic robot systems. The proposed controller has been used to control the position of single actuators, the bending angles of the self-bending contraction actuator (SBCA), and a human–robot shared control system to show the efficiency of the proposed controller for different robot behaviors and applications.

The order of this paper has been organized as follows: Section non-linear PID controller shows the idea of non-linear PID; Section other controller approaches describes several approaches to control the soft pneumatic systems. The proposed controller structure is presented in Section parallel neural network proportional controller together with its applications.

The PID controller has been one of the most important strategies used in industrial applications due to its simplicity and robustness. The need for variable efficient controller performance in operating conditions or parameters in the environment is often beyond the abilities of linear PID controllers (Su et al., 2005). Moreover, the high non-linearity of the PMAs makes the PID controller insufficient to solve this complex control problem. To improve the performance of linear PID to control the performances of PMA, numerous techniques have been utilized to enhance the performance and robustness of the PID controller by using the self-tuning method of general predictive control, fuzzy logic, and neural networks (Cervantes and Alvarez-Ramirez, 2001; Duan et al., 2004). Figure 1 shows the non-linear PID by connecting it serially to the ANN.

A multilayer ANN is used for three inputs, one neuron in one hidden layer and one output neuron, with

where θ_ref and θ are the setpoints and the actual output for each joint, respectively; ΔT is the sampling time; z is the Z-transform operator; K_p, K_I, and, K_D are PID constants that have to be modified to find the optimal value; f (x) is a sigmoid function; and u(k) is the controller output.

Other work was done in Anh (2010) using the same PID-ANN technique with added bias input to the hidden and output neurons.

The inverse control strategy for PMA motion control was presented in Kang et al. (2013) and Kang et al. (2014). By using this idea, they were able to define an inverse kinematic (IK) model for control application. Furthermore, they assumed that the dynamics of the system could be ignored because the speed of these types of actuators is low. Meanwhile, Nakamura and Shinohara (2007) presented the controller system according to the mathematical model of PMA, which drives the inverse relationship between both the position and force of the PMA and the pressure input where P is the function of L and F.

The fuzzy control based on bang-bang control strategy is used in Leephakpreeda (2011) with a combination of proportional control to adjust the system output around the desired points either for the length of the contraction force. Figure 2 shows a diagram of this control system. In this method, the author used the pulse width modulation (PWM) technique as a variable time on–off controller to adjust the air valve outlets. The model-based statics controller has been utilized in Camarillo et al. (2009) for a 5-DoF-per-section model by formulating IK. The most frequently used of IK-based static controls uses the constant curvature (CC) approximation (Hannan and Walker, 2003).

On the other hand, model-free approaches for control of soft robots are quite a novel field and provide a wide range of possibilities. The early utilization of this method has been proposed in Giorelli et al. (2013) and for a 2- and a 3-DoF cable-driven soft manipulator (Giorelli et al., 2015a,b). The main idea for this controller system is applying a closed-loop control system with an effective sensory feedback system.

The parallel neural network proportional (PNNP) controller is suggested in this section. The NARMA-L2 neural network control system has been utilized. The structure of nine neurons has been chosen in a single hidden layer, three delayed controlled signal outputs, and two delayed plant outputs. The NN has been trained by trainlm for 100 Epochs. The mean square error (MSE) for the training, testing, and validating data is about 10⁻⁷. The NN controller system provides good performance; nonetheless, the PMA system is too slow, and it needs a fast controller to track its behavior. To enhance the speed of the controller system, a proportional controller has been used in parallel to the NN controller. While the NN controller provides high precision, the P controller offers a high-speed response. As a result, the structure of the PNNP controller provides efficient performances in terms of precision and speed. The structure of the controller is shown in Figure 3.

The reference model states the required target such as length, position, bending angle, and so on. Moreover, since the air pressure in single or multiple PMA defines the system performances, G1 can either be the inverse kinematics of the plant and, in this case, the proposed controller will adjust the pressure p, or be equal to 1 and, in this case, the controller system will track the error in the output y.

The error e can be defined as follows:

or

p_r and y_r are the reference (setpoint) for the pressure and the system's output, respectively.

The controller outputs u1 and u2 represent the duty cycle of the PWM signal for the NN controller and P controller, respectively, where

The NARMA-L2 NN controller output u1 can be defined as

where f() and g() are approximated using neural networks, and

where n and m are equal to 2 and 3, respectively, according to the proposed controller structure.

While the proportional controller output has been defined as

The PWM signal controls the airflow for the valve output in fill and vent directions. Therefore, two PNNP controllers are required: one to control the airflow in the fill direction and the other controls the venting process.

Depending on the error, the proposed controller activates either the filling controller (positive error) or the venting controller (negative error). On the other hand, two possible methods are used to train the NN. The first method is using an approximate function between the output and the duty cycle as in (11):

In the case of the pressure controller of the single or multiple actuators, y^* and y₀ represent the air pressure in the actuator p and the initial pressure in the actuator, respectively, x is the maximum applied pressure p_max (p_max is subject to the actuator size and material), and u^* is the training duty cycle of the NN controller.

In most cases,

In order to prevent a continuously applied voltage (100% duty cycle) on the solenoid valve, we chose 98% as the maximum operating duty cycle. Formula (11) provides an acceptable linear performance of the PMA pressure at variable duty cycles.

Alternatively, the actual relationship between the output and the duty cycle can be found experimentally as follows.

A contraction actuator of 30 cm in length and 1.7 cm in diameter is chosen. A source of 600 kPa is used to apply air pressure to this actuator via a solenoid valve by different duty cycles ranging from 0 to 100% within 1 s. Firstly, a 10% duty cycle is selected, the air pressure is measured by a pressure sensor, and then the venting process is activated. This process is repeated for 20, 30…, and 100%, respectively. The result of this experiment is shown in Figure 4. The trained line to these data is utilized for training the NN.

Formula (11) is used due to the similarity in the performances of the NN and to decrease the complexity of the control system. Moreover, the PNNP controller is tracking the desired behavior online; therefore, the controller is adjusting the duty cycle to minimize the error.

The PNNP controller shows efficient performances when it is applied to soft pneumatic systems. In this section, the PNNP is compared with several other controller approaches from the literature to show the advantages of the proposed controller system. Table 2 lists the main characteristics of numerous controllers.

Table 2 shows that the performance of the PNNP controller system in terms of speed response, accuracy, and applications (complexity of the pneumatic system) is higher than other chosen researches.

The high non-linear behaviors of the pneumatic muscle actuator require fast response and high accuracy control systems. In this article, a parallel structure of the neural network controller and the proportional controller is presented to control single extensor PMA and single SBCA, respectively, at different load values. For further validation of the PNNP controller, an interaction between a human and a bidirectional continuum arm has been designed, and the controller system shows a valuable tracking to the human hand. The results illustrated the efficiency of using the parallel structure to increase precision and decrease the tracking time.

The results show that the venting time is more than the filling time due to the non-linear behavior of the PMA such as hysteresis and the air pressure difference inside and outside the actuator. Furthermore, increasing the load for the presented pneumatic systems does not have any effect on the resulting performances. Nonetheless, the PNNP decreases the required air pressure for the extensor PMA at a higher load to decrease the extension ratio, while the proposed controller increases the applied air pressure for the single SBCA and the unidirectional continuum arm to raise the bending angle. The actuators are tested at pressure up to 600 kPa, but the maximum air pressure has been set at 500 kPa for safe working.

All datasets presented in this study are included in the article/Supplementary Material.

Written informed consent was obtained from the individual(s) for the publication of any potentially identifiable images or data included in this article.

AA-I designed, performed the experiments, and wrote the paper. AA-I and SD analyzed the data. SN-M edited the paper. All authors have read and agreed to the published version of the manuscript.

The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.