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Neural network fuzzy sliding mode control of pneumatic muscle actuators
Engineering Applications of Artificial Intelligence 65 (2017) 68–86
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Engineering Applications of Artificial Intelligence journal homepage: www.elsevier.com/locate/engappai
Neural network fuzzy sliding mode control of pneumatic muscle actuators Chia-Jui Chiang *, Ying-Chen Chen National Taiwan University of Science and Technology, No.43, Sec. 4, Keelung Rd., Daan Dist., Taipei 106, Taiwan, ROC
a r t i c l e
i n f o
Keywords: Pneumatic muscle actuators Intelligent control Fuzzy sliding mode control Neural network
a b s t r a c t The pneumatic muscle actuator (PMA) is one of the most promising actuators especially for the applications that require greater proximity between the humans and the robots. Fast and precise control of PMA, however, is difficult to achieve due to the compressibility of the air and the elasticity of the PMA. In order to achieve accurate and consistent tracking performance of a one axis PMA actuated manipulator over considerably wide ranges of frequency and stroke, an intelligent adaptive control algorithm is developed in this paper. The adaptive learning is enabled by a neural network in which the control gains to a fuzzy sliding mode controller (FSMC) and an integrator are adjusted to minimize the tracking error. Experimental results show that the proposed control strategy achieves fast, accurate and consistent performance tracking sinusoidal reference trajectories up to 1 Hz in frequency and close to the extreme stroke of the PMA actuated manipulator with the compressed air regulated to 4 bar. Results also show that the proposed control strategy, with a more aggressive learning for the control gain to the FSMC, achieves satisfactory performance tracking a trapezoidal reference trajectory. © 2017 Elsevier Ltd. All rights reserved.
1. Introduction In recent years, the robotics industry is developing rapidly. The robot actuators include electric motors, hydraulic cylinders and pneumatic actuators. The electric motors are characterized with excellent accuracy and controllability, but suffer from the possibility of sparking (Caldwell et al., 1995). The hydraulic actuators deliver large power with minimal backlash. Their main drawback lies in its lack of cleanness (Caldwell et al., 1995). The pneumatic actuators provide quick response and cleanness with low cost and are therefore widely used in industrial applications. The pneumatic muscle actuator (PMA), shown in Fig. 1, is one of the most promising pneumatic actuation systems for new types of industrial robots (Caldwell et al., 1995). The PMA consists of a rubber tube surrounded by a braided mesh with inextensible threads (Shen, 2010). As the rubber inner tube is inflated, the PMA is contracted (Tondu and Lopez, 2000). The advantages of PMA include high power-to-weight and power-to-volume ratios, cleanness, ease of maintenance, pliability, inherent safety, low cost and ready availability. As a result, the PMA is potentially one of the most promising actuators for the applications that require greater proximity between the humans and the robots (Caldwell et al., 1995; Tondu and Lopez, 2000). Due to the compressibility of the air and the elasticity of the PMA, however, it is difficult to achieve fast and precise control of the PMA (Thanh and Ahn, 2006a).
In order to realize satisfactory control performance, many control methods have been proposed to solve these challenging problems in controlling the PMA. The complex and nonlinear dynamics of the PMA make it a challenging and yet appealing system for model-based control design. Among previous approaches, Repperger and his team used a second order nonlinear differential equation to describe the PMA dynamics and based on which a gain scheduling tracking controller is developed (Repperger et al., 1999). John H. Lilly et al. developed adaptive tracking controllers and sliding mode tracking controllers for PMA actuated manipulators (Lilly, 2003; Lilly and Yang, 2005). In Aschemann and Schindele (2008), a cascaded sliding mode control algorithm is developed for a linear axis driven by a pair of pneumatic muscles. Shen (2010) derived a physics-based nonlinear model and based on which a sliding mode control approach is applied to achieve robustness against model uncertainties and disturbances. A switching model predictive control approach is employed in Andrikopoulos et al. (2013) based on linearized models at various operating conditions. In Pujana-Arrese et al. (2010), a nonlinear model is developed based on Dymola/Modelica and validated against experimental data. The nonlinear model is then linearized and based on which linear controllers are designed and compared. In Ganguly et al. (2012), on–off solenoid valves are used to control the PMA pressures in the inner loop and a PID controller is augmented in the outer loop to achieve position tracking of
* Corresponding author.
E-mail addresses: [email protected] (C.-J. Chiang), [email protected] (Y.-C. Chen). http://dx.doi.org/10.1016/j.engappai.2017.06.021 Received 4 October 2016; Received in revised form 20 May 2017; Accepted 25 June 2017 0952-1976/© 2017 Elsevier Ltd. All rights reserved.
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Engineering Applications of Artificial Intelligence 65 (2017) 68–86
network is applied to control the robot joints by continuously modeling every joint and generating a joint inverse using the back-propagation algorithm. In Balasubramanian and Rattan (2003), the PMA is modeled as a mass–spring–damper system and the coefficients for the spring and damper are represented by a nonlinear fuzzy model. Based on the model, a linearizing controller is developed and a PID controller is augmented in the outer loop before the force command is fed into a fuzzy inverse model to obtain the pressure command. In Chan et al. (2003), a control strategy with an incremental fuzzy logic controller in place of the proportional term in a conventional PID controller is developed based on a neuro-fuzzy model of the PMA. In Chang and Lilly (2013), an adaptive fuzzy PD controller is augmented with an integral controller and the fuzzy PD consequent singleton locations are dynamically adjusted by a fuzzy inverse model with the interior pressure of the PMA as one of its inputs. In Ahn and Nguyen (2007), the PID control gains are switched based on the external load estimate obtained from a learning vector quantization neural network so as to achieve precise position control under various external loads. In Thanh and Ahn (2006b), the PID control gains are adjusted by neural networks and the tracking performance of a PMA actuated two-axes robot is examined up to 0.2 Hz. In Anh and Ahn (2011), the authors of Thanh and Ahn (2006b) further added bias weightings to the PID-based control structure so as to achieve precise tracking control when the two-axes robot is subjected to various loads. In Xie and Jamwal (2011), the PID feedback controller is augmented with a feedforward controller developed based on a fuzzy model and the control performance is examined at various load conditions. In Anh (2010), a feedforward/feedback control structure is adopted and each sub-controller is developed based on fuzzy logic. The tracking results of a PMA actuated robot up to 0.2 Hz are shown. In Andrikopoulos et al. (2014), extra degrees of freedom are incorporated into a PIDbased control structure and the tracking performance is examined up to 0.25 Hz. In an effort to achieve accurate and consistent tracking performance of a PMA actuated manipulator over considerably wide ranges of frequency and stroke, an intelligent adaptive control algorithm is proposed in this paper. The adaptive learning capability is provided by a neural network in which the control gains to a fuzzy sliding mode controller (FSMC) and an integrator are adjusted, based on the gradient descent method and the back propagation algorithm, to minimize the tracking error. Experimental results with a one axis PMA actuated manipulator show that the proposed control strategy achieves accurate and consistent performance tracking sinusoidal reference trajectories up to 1 Hz in
Fig. 1. Pneumatic muscle actuator.
Fig. 2. Pneumatic muscle actuated manipulator.
a single degree of freedom manipulator. In Minh et al. (2010), hysteresis compensation is implemented in a cascaded control strategy to improve the position tracking performance of a single PMA-mass system. On the other hand, the intelligent control methods have been proposed to control the PMA without detailed modeling of the complex dynamics of the PMA. In Hesselroth et al. (1994), the neural network learning algorithm is employed for position control of a PMA actuated five-joint robot. In Iskarous and Kawamura (1995), the neuro-fuzzy
Fig. 3. Structure of the PMA system.
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Engineering Applications of Artificial Intelligence 65 (2017) 68–86
Fig. 4. Operating range of the PMA actuated manipulator supplied with compressed air regulated to 4 bar.
Fig. 5. Block diagram of the neural network fuzzy sliding mode integral (NNFSMI) controller.
Fig. 6. Block diagram of neural network.
frequency and close to the extreme stroke of the PMA actuated manipulator with the compressed air regulated to 4 bar. Results also show that the proposed control strategy, with a more aggressive learning for the control gain to the fuzzy sliding mode controller (FSMC), achieves satisfactory performance tracking a trapezoidal reference trajectory. The rest of this paper is organized as follows. Section 2 describes experimental setup of the one axis pneumatic muscle actuated robot arm used in the study. Section 3 shows the adaptive control algorithm developed in the paper. Section 4 examines the performance of the
Fig. 7. The shapes of sigmoid function.
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Fig. 8. Triangular input membership function and output singletons.
Fig. 9. Experimental results of the NNFSMI tracking a sinusoidal trajectory of 0.05 Hz in frequency and ±20◦ in amplitude with 𝜂𝑢 = 0.001 and 𝜂𝑖 = 0.00005. 71
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Engineering Applications of Artificial Intelligence 65 (2017) 68–86
Fig. 10. Experimental results of the FSMI and NNFSMI tracking a sinusoidal trajectory of 0.05 Hz in frequency and ±20◦ in amplitude with 𝜂𝑢 = 0.01 and 𝜂𝑖 = 0.0005.
PMA actuated manipulator tracking various sinusoids and a trapezoid. Section 5 concludes this paper and points out possible extensions of this work.
in and out of the two PMAs and thus the rotating motion. The rotating angle 𝜃 is measured using an encoder (Nemicon HES-2048-2MD) with resolution of 2048 pulses per revolution and the digital signal decoding is carried out using a decoder IC (HTCL-2020). The controller is implemented in a PC with Advantech PCI-1710 board and the sampling time used in the experiments is 10 ms. In order to examine the operating range of the PMA actuated manipulator, open loop test is conducted. Fig. 4 shows that by switching the command to the proportional control valve the manipulator gradually reaches its extremes (+28.56◦ and −28.52◦ respectively) as one of the muscles is inflated to 4 bar while the other is deflated to atmospheric pressure.
2. Experimental setup Fig. 2 shows a one axis PMA actuated manipulator with its schematic diagram shown in Fig. 3. The architecture consists of two pneumatic muscle actuators (FESTO MAS-20-200N) with 20 mm internal diameter and 200 mm nominal length. The manipulator arm weights around 104 g and the length 𝑙 is 235 mm. The pneumatic muscle actuated system are supplied with compressed air regulated to 4 bar. A proportional control valve (Festo MPYE-5-1/8-CF-010-B) is used to control the flow 72
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Engineering Applications of Artificial Intelligence 65 (2017) 68–86
Fig. 11. Experimental results of the FSMI and NNFSMI (with and without 𝐺𝑢 tuning) tracking a sinusoidal trajectory of 0.25 Hz in frequency and ±20◦ in amplitude.
3. Adaptive control algorithm
3.1. Fuzzy sliding mode control (FSMC)
The block diagram of the adaptive control algorithm is shown in Fig. 5. Fuzzy logic-based controller is chosen to cope with the complex and nonlinear dynamics of the pneumatic muscle actuators. A sliding surface is adopted to reduce the dimension of the input space and the number of fuzzy rules (Kim and Lee, 1995; Tzafestas and Rigatos, 1999). An integrator is augmented to further improve the tracking performance at steady state. Neural network provides an adaptive learning algorithm that updates the control parameters and achieves accurate and consistent tracking performance of the PMA actuated manipulator over considerably wide ranges of frequency and stroke.
The conventional fuzzy logic control theory involves fuzzification, a fuzzy rule base, a fuzzy inference engine and defuzzification. Conventionally, the fuzzy rule base depends both on the error 𝑒 and error difference Δ𝑒, resulting in complicated fuzzy rules and membership functions. To reduce the dimension of the input and thus the number of fuzzy rules, a fuzzy sliding surface 𝑠 = 0 is introduced with the variable 𝑠 defined as follows Kim and Lee (1995) and Tzafestas and Rigatos (1999). 𝑠(𝑘) = 𝛼𝑒(𝑘) + Δ𝑒(𝑘) 73
(1)
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Fig. 12. Zoomed views of the responses during transient (left) and at steady state (right) tracking a sinusoidal trajectory of 0.25 Hz in frequency and ±20◦ in amplitude with and without 𝐺𝑢 tuning.
frequency and stroke, a neural network algorithm is adopted for its powerful capability of learning, adaptivity and tackling nonlinearity (Thanh and Ahn, 2006b). The neural network is trained by the back propagation algorithm to minimize the tracking error. Fig. 6 shows the block diagram of the proposed neural network. The inputs to the neural network are the output of the FSMC, 𝑢𝑓 , and ∑ the numerical integral of tracking error, 𝑒𝑖 (𝑘) = 𝑘𝑛=1 𝑒(𝑛)Δ𝑡, where Δ𝑡 is the sampling time. The control output is obtained from a nonlinear sigmoid function (Yamada and Yabuta, 1992).
where the tracking error 𝑒(𝑘) = 𝜃𝑟 (𝑘) − 𝜃(𝑘) is the difference between the reference angle 𝜃𝑟 and the measurement of the angle 𝜃, Δ𝑒(𝑘) = 𝑒(𝑘) − 𝑒(𝑘 − 1) is the error difference and 𝛼 is a strictly positive constant which determines the bandwidth of the sliding mode control law and is typically limited by the mechanical properties of the system, time delay in the actuators and the available computing power (Slotine and Li, 1991). In practice, the tuning of this single scalar is often conducted experimentally. The control objective is then to force the system into the sliding surface so that tracking error is reduced. The normalized input variable 𝑆 to the fuzzification process is obtained by multiplying a scaling factor 𝐺𝑠 to the sliding variable 𝑠: 𝑆 = 𝐺𝑠 ⋅ 𝑠. The choice of number of fuzzy sets is generally a balance between precision and transparency (Babuska, 2000). Depending on the application of the pneumatic muscles, the number of fuzzy sets per variable used in the literature can be as small as 2 and as large as 17 (Anh and Ahn, 2011; Balasubramanian and Rattan, 2003; Chan et al., 2003; Chang and Lilly, 2013; Iskarous and Kawamura, 1995; Xie and Jamwal, 2011). In this paper, in order to achieve accurate and consistent tracking performance of the PMA actuated manipulator over considerably wide ranges of frequency and stroke, the universe of the variable 𝑆 is partitioned into thirteen fuzzy sets [𝑁𝑉 𝐵𝑠 𝑁𝐵𝑠 𝑁𝑀𝑠 𝑁𝑆𝑀𝑠 𝑁𝑆𝑠 𝑁𝑉 𝑆𝑠 𝑍𝑂𝑠 𝑃 𝑉 𝑆𝑠 𝑃 𝑆𝑠 𝑃 𝑆𝑀𝑠 𝑃 𝑀𝑠 𝑃 𝐵𝑠 𝑃 𝑉 𝐵𝑠 ] characterized by triangular membership functions. Similarly, the fuzzy sets for the control output are defined as [𝑁𝑉 𝐵𝑢 𝑁𝐵𝑢 𝑁𝑀𝑢 𝑁𝑆𝑀𝑢 𝑁𝑆𝑢 𝑁𝑉 𝑆𝑢 𝑍𝑂𝑢 𝑃 𝑉 𝑆𝑢 𝑃 𝑆𝑢 𝑃 𝑆𝑀𝑢 𝑃 𝑀𝑢 𝑃 𝐵𝑢 𝑃 𝑉 𝐵𝑢 ]. The linguistic variables represented by those labels of fuzzy sets are defined in Table A.1. The fuzzy inference implements fuzzy control rules by using the single input to single output mapping. Finally, the defuzzification process is conducted based on the height method (also called center average defuzzifier) to obtain the control output 𝑢𝑓 of the fuzzy sliding mode controller (FSMC). ∑ ℎ𝑦 (2) 𝑢𝑓 (𝑘) = ∑ 𝑖 𝑖 ℎ𝑖
𝑢 = 𝑓 (𝑥) =
2𝐴(1 − 𝑒−𝜇𝑥 ) 𝜇(1 + 𝑒−𝜇𝑥 )
(3)
where the parameters 𝐴 and 𝜇 determine the shape of the sigmoid function. Fig. 7 shows the shapes of the sigmoid function with a constant gain 𝐴 = 1.25 and various values of parameter 𝜇. Note that the constant gain A represents the slope of the sigmoid function in the linear region close to the origin and needs to be tuned for optimum convergence properties. The tuning of the parameter 𝜇, on the other hand, depends on the operating range of the actuation system as the parameter 𝜇 varies the saturation level of the sigmoid function. The input to the sigmoid function is defined by 𝑥(𝑘) = 𝐺𝑢 (𝑘)𝑢𝑓 (𝑘) + 𝐾𝑖 (𝑘)𝑒𝑖 (𝑘)
(4)
where the parameters 𝐺𝑢 and 𝐾𝑖 are the control gain to the FSMC and the integral gain respectively. To obtain the control gains that minimize the tracking error, an energy function 𝐽 (𝑘) is chosen as follows. 1 1 𝑊 𝑒2 (𝑘) + 𝑊2 𝑠2 (𝑘) (5) 2 1 2 where the parameters 𝑊1 and 𝑊2 are the weightings on the tracking error 𝑒(𝑘) and the sliding variable 𝑠(𝑘) respectively. The control gains 𝐺𝑢 and 𝐾𝑖 are then updated to minimize the energy function 𝐽 in Eq. (5). In order to achieve direct and fast convergence for parameters 𝐺𝑢 and 𝐾𝑖 , gradient descent method is adopted due to its simplicity, fast computation per iteration and low data storage requirement compared to other search techniques (Boyd and Vandenberghe, 2004). 𝐽 (𝑘) =
where ℎ𝑖 is the fuzzy membership weighting and 𝑦𝑖 is the center point of each fuzzy set. 3.2. Neural network
𝜕𝐽 (𝑘) 𝜕𝐺𝑢 𝜕𝐽 (𝑘) 𝐾𝑖 (𝑘 + 1) = 𝐾𝑖 (𝑘) − 𝜂𝑖 𝜕𝐾𝑖 𝐺𝑢 (𝑘 + 1) = 𝐺𝑢 (𝑘) − 𝜂𝑢
In order for the PMA actuated manipulator to achieve accurate and consistent tracking performance over considerably wide ranges of 74
(6)
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Fig. 13. Experimental results of the FSMI and NNFSMI tracking a sinusoidal trajectory of 0.5 Hz in frequency and ±20◦ in amplitude with 𝜂𝑢 = 0.001 and 𝜂𝑖 = 0.00005.
where the parameters 𝜂𝑢 and 𝜂𝑖 are the learning rates that determine the
where each individual term is derived based on Eqs. (1), (3), (4) and
convergence speed. The back propagation algorithm is then realized by
(5).
applying chain rule.
𝜕𝐽 (𝑘) 𝜕𝑠(𝑘) 𝜕𝑒(𝑘) 𝜕𝜃(𝑘) 𝜕𝑢(𝑘) 𝜕𝑥(𝑘) 𝜕𝐽 (𝑘) = 𝜕𝐺𝑢 𝜕𝑠 𝜕𝑒 𝜕𝜃 𝜕𝑢 𝜕𝑥 𝜕𝐺𝑢 𝜕𝐽 (𝑘) 𝜕𝐽 (𝑘) 𝜕𝑒(𝑘) 𝜕𝜃(𝑘) 𝜕𝑢(𝑘) 𝜕𝑥(𝑘) = 𝜕𝐾𝑖 𝜕𝑒 𝜕𝜃 𝜕𝑢 𝜕𝑥 𝜕𝐾𝑖
𝜕𝐽 (𝑘) 𝜕𝑠(𝑘) 𝜕𝐽 (𝑘) = 𝑊2 𝑠(𝑘); = 𝑊1 𝑒(𝑘); =𝛼 𝜕𝑠 𝜕𝑒 𝜕𝑒 𝜕𝑒(𝑘) 𝜕𝜃(𝑘) 𝜕𝑥(𝑘) = −1; = 𝛽; = 𝑢𝑓 (𝑘) 𝜕𝜃 𝜕𝑢 𝜕𝐺𝑢 𝜕𝑥(𝑘) 𝜕𝑢(𝑘) 4𝐴𝑒−𝜇𝑥 = 𝑒𝑖 (𝑘); = 𝜕𝐾𝑖 𝜕𝑥 (1 + 𝑒−𝜇𝑥 )2
(7)
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(8)
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Fig. 14. Zoomed views of the responses during transient (left) and at steady state (right) tracking a sinusoidal trajectory of 0.5 Hz in frequency and ±20◦ in amplitude with 𝜂𝑢 = 0.001 and 𝜂𝑖 = 0.00005.
where the parameter 𝛽 is defined as follows ⎧ ⎪𝛽 = 1, ⎨ ⎪𝛽 = −1, ⎩
In Section 4.1, the performance of NNFSMI and FSMI tracking sinusoidal reference trajectories of various frequencies is examined. In Section 4.2, the performance of NNFSMI and FSMI tracking sinusoidal reference trajectories of various strokes is examined. In Section 4.3, the performance of NNFSMI and FSMI tracking a sinusoidal reference trajectory of mixed frequencies is examined. In Section 4.4, the performance of NNFSMI and FSMI tracking a trapezoidal reference trajectory is examined.
Δ𝜃 ≥0 Δ𝑢 Δ𝜃 if <0 Δ𝑢 if
combining Eqs. (6)–(8), the resulting learning law is summarized as follows 4𝐴𝑒−𝜇𝑥 𝐺𝑢 (𝑘 + 1) = 𝐺𝑢 (𝑘) + 𝜂𝑢 𝑊2 𝛼𝛽𝑠(𝑘) 𝑢𝑓 (𝑘) (1 + 𝑒−𝜇𝑥 )2 (9) −𝜇𝑥 4𝐴𝑒 𝐾𝑖 (𝑘 + 1) = 𝐾𝑖 (𝑘) + 𝜂𝑖 𝑊1 𝛽𝑒(𝑘) 𝑒𝑖 (𝑘). (1 + 𝑒−𝜇𝑥 )2
4.1. Tracking sinusoidal reference trajectories of various frequencies In this section, the closed-loop results with and without the neural network, NNFSMI and FSMI respectively, are compared tracking sinusoidal reference trajectories ranging from 0.05 Hz to 1 Hz. Fig. 9 shows the experimental results when the NNFSMI is applied to track a sinusoidal trajectory of 0.05 Hz in frequency and ±20◦ in amplitude with the learning parameters 𝜂𝑢 = 0.001 and 𝜂𝑖 = 0.00005. The results show that even though the tracking performance seems acceptable, the control parameters 𝐺𝑢 (𝑘) and 𝐾𝑖 (𝑘) have never converged to the optimum values within the time span of 120 s as the reference input of lower frequency provides a relatively weak persistent-excitation (PE) condition. Therefore, when tracking a sinusoidal reference trajectory of lower frequency, a more aggressive learning needs to be adopted so that the control parameters can converge within the restricted time span displayed in this paper. Fig. 10 shows that a more aggressive learning within 𝜂𝑢 = 0.01 and 𝜂𝑖 = 0.0005 results in fast convergence accompanied by the oscillatory behavior in the first few seconds. At steady state, the maximum error of the FSMI occurs when the manipulator is changing direction, resulting from the hysteresis effect of the PMAs. The NNFSMI, on the other hand, is able to eliminate those errors by quickly adjusting the command to the proportional control valve. Specifically, the maximum error achieved by the NNFSMI is less than 0.09◦ at steady state
4. Experimental results In this section, the tracking performance of the adaptive control algorithm developed in Section 3 is examined over considerably wide ranges of frequency and stroke. The fuzzy rule matrices are [−1.0 −0.8 −0.6 −0.4 −0.2 −0.1 0.0 0.1 0.2 0.4 0.6 0.8 1.0] for the control input and [−1.0 −0.7 −0.5 −0.3 −0.15 −0.07 0.0 0.07 0.15 0.3 0.5 0.7 1.0] for the control output. Fig. 8 shows the resulting triangular input membership function and the output singletons. The parameters used in the neural network fuzzy sliding mode integral (NNFSMI) controller throughout this section are 𝛼 = 1, 𝐴 = 1.25, 𝜇 = 0.8, 𝐺𝑠 = 0.5, 𝑊1 = 1, 𝑊2 = 1, 𝐺𝑢 (0) = 0.3 and 𝐾𝑖 (0) = 0.03. To illustrate the effectiveness of the learning algorithm, the closed-loop results with and without the neural network are compared. Without the neural network, constant control parameters 𝐺𝑢 (𝑘) = 𝐺𝑢 (0) and 𝐾𝑖 (𝑘) = 𝐾𝑖 (0) are used in the fuzzy sliding mode integral (FSMI) controller. The principles for the tuning of control parameters 𝛼, 𝐴 and 𝜇 are introduced in Section 3 whereas the initial values 𝐺𝑢 (0) and 𝐾𝑖 (0) are obtained by trial-and-error through experiments with the FSMI controller. 76
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Fig. 15. Experimental results of the FSMI and NNFSMI tracking a sinusoidal trajectory of 1 Hz in frequency and ±20◦ in amplitude with 𝜂𝑢 = 0.001 and 𝜂𝑖 = 0.00005.
tracking the sinusoidal trajectory of 0.05 Hz. Fig. 10 also shows that the value of the input variable 𝑆 to the fuzzification process ranges from NSM to PSM during transient and stays between NVS and PVS at steady state. Fig. 11 shows the experimental results tracking a sinusoidal trajectory of 0.25 Hz in frequency and ±20◦ in amplitude. The more aggressive learning parameters 𝜂𝑢 = 0.01 and 𝜂𝑖 = 0.0005 are again applied in the NNFSMI in order for the control parameters 𝐺𝑢 (𝑘) and 𝐾𝑖 (𝑘) to converge to the optimum values within the restricted time span (60 s) displayed in the paper. Fig. 11 shows that the NNFSMI achieves significantly better tracking performance than the FSMI, especially
when the manipulator is changing direction. Specifically, the maximum error achieved by the NNFSMI is less than 0.54◦ at steady state tracking the sinusoidal trajectory of 0.25 Hz. The range of the input variable 𝑆 to the fuzzification process stretches between NVB and PVB during transient and is bounded between NSM and PSM at steady state. Note that when the NNFSMI is applied the FSMC gain 𝐺𝑢 is increased by about 30% whereas the integral gain 𝐾𝑖 is raised by more than 100%. The NNFSMI relies more on the integral gain tuning to compensate for the hysteresis effect as the sinusoidal input excites the PMA-actuated system continuously. The necessity of fine tuning both control parameters in tracking a sinusoidal trajectory is further investigated by setting the 77
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Fig. 16. Zoomed views of the responses during transient (left) and at steady state (right) tracking a sinusoidal trajectory of 1 Hz in frequency and ±20◦ in amplitude with 𝜂𝑢 = 0.001 and 𝜂𝑖 = 0.00005.
valve and thus achieves significantly better tracking performance. The maximum error achieved by the NNFSMI is close to 1◦ at steady state tracking the sinusoidal trajectory of 0.5 Hz. Fig. 13 also shows that, while tracking the reference trajectory of 0.5 Hz, the range of the input variable 𝑆 to the fuzzification process stretches between NVB and PVB during transient and is bounded between NM and PM at steady state. Fig. 15 shows the experimental results comparing the FSMI and NNFSMI tracking a sinusoidal trajectory of 1 Hz in frequency and ±20◦ in amplitude. The same moderate learning parameters 𝜂𝑢 = 0.001 and 𝜂𝑖 = 0.00005 are used. Fig. 16 shows the zoomed views of the responses during transient (left) and at steady state (right) while tracking the same trajectory as in Fig. 15. Figs. 15 and 16 show that the tracking error can reach close to 10◦ when FSMI is applied, resulting from the hysteresis effect of the PMAs while the manipulator is changing direction. The NNFSMI, on the other hand, is able to intelligently adjust the control gains 𝐺𝑢 and 𝐾𝑖 and produce faster control command to the proportional control valve resulting in significantly better tracking performance especially at steady state. Specifically, the maximum error achieved by the NNFSMI tracking the sinusoidal trajectory of 1 Hz is reduced to less than 1.83◦ at steady state. Fig. 15 also shows that, when tracking the reference trajectory of 1 Hz, which is the extreme input frequency we have tested, the value of the input variable 𝑆 to the fuzzification process exceeds the limits of NVB and PVB while the manipulator is changing direction. Fig. 17 summarizes the steady state control performance achieved by the FSMI and NNFSMI tracking sinusoidal trajectories of ±20◦ in amplitude ranging from 0.05 Hz to 1 Hz in frequency. The rate of parameter convergence in the neural network learning algorithm proposed in this paper depends on the persistence of excitation from the reference input. The reference input of lower frequencies (less than about 0.35 Hz) provides a relatively weak persistent-excitation (PE)
learning parameter 𝜂𝑢 = 0 while keeping 𝜂𝑖 = 0.0005. Fig. 11 shows that as the control gain 𝐺𝑢 is fixed at 0.3, larger integral gain 𝐾𝑖 is rendered by the neural network to compensate for the absence of the FSMC control gain tuning. The zoomed views of the responses shown in Fig. 12 indicate that the absence of the FSMC control gain tuning (𝜂𝑢 = 0) results in longer settling time both during transient (left) and at steady state (right) while tracking the sinusoidal trajectory of 0.25 Hz. Specifically, the mean absolute errors (Hoaglin et al., 1983) achieved by the NNFSMI with and without the FSMC gain tuning are about 0.09◦ and 0.12◦ respectively. Therefore, even though while tracking a sinusoidal trajectory the NNFSMI relies more on the integral gain tuning, the fine tuning of the FSMC control gain is still needed for shorter settling time and improved tracking performance. Fig. 13 shows the experimental results comparing the FSMI and NNFSMI tracking a sinusoidal trajectory of 0.5 Hz in frequency and ±20◦ in amplitude. At higher frequency the moderate learning parameters 𝜂𝑢 = 0.001 and 𝜂𝑖 = 0.00005 are chosen as the reference input provides a strong persistent-excitation (PE) condition to the neural network learning algorithm for the control parameters to converge within the restricted time span displayed in the paper. Fig. 14 shows the zoomed views of the responses during transient (left) and at steady state (right) while tracking the same trajectory as in Fig. 13. As can be seen from Figs. 13 and 14, significantly larger tracking error is observed when the FSMI control strategy is applied, especially when the manipular is changing direction. The learning mechanism of the NNFSMI, on the other hand, is able to adjust the control gains 𝐺𝑢 and 𝐾𝑖 based on Eq. (9) and significantly reduce the tracking error. Fig. 14 shows that the control commands to the proportional control valve and thus the tracking errors resulted from FSMI and NNFSMI are quite similar in the beginning. The NNFSMI, however, is able to quickly adjust the control gains and generate faster control command to the proportional control 78
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Fig. 17. Comparison of the control performance achieved by the FSMI and NNFSMI at steady state tracking sinusoidal trajectories of ±20◦ in amplitude ranging from 0.05 Hz to 1 Hz in frequency. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
condition and thus more aggressive learning parameters 𝜂𝑢 = 0.01 and 𝜂𝑖 = 0.0005 are applied in order for the control parameters to converge within the restricted time span (60 s) displayed in this paper. The reference input of higher frequencies (from about 0.35 to 1 Hz), on the other hand, produces a strong PE condition for the control parameter convergence and thus moderate learning parameters 𝜂𝑢 = 0.001 and 𝜂𝑖 = 0.00005 are used to alleviate the oscillatory behavior during transient. Fig. 17 shows that the tracking precision and control effort attained by the FSMI at steady state is comparable to that of the NNFSMI while tracking a sinusoidal trajectories of lower frequencies. The tracking precision achieved by the FSMI at steady state, however, deteriorates dramatically while enormous control effort is required tracking sinusoidal trajectories of higher frequencies. Specifically, at the higher frequencies the hysteresis effect results in substantial growth of the tracking error, leading to saturation of the fuzzy sliding mode controller (FSMC) as evidenced in Fig. 17 by the out-of-bound variable
𝑆 generated by the FSMI (red lines) at frequencies higher than 0.5 Hz. As the FSMC is saturated, the error further extends and as a result immense control effort is demanded by the integral action. The NNFSMI, on the other hand, significantly reduces the tracking errors at steady state while consuming much less control effort tracking sinusoidal trajectories of higher frequencies via quickly adjusting the control gains and generating faster control command that inhibits the FSMC from being saturated as evidenced in Fig. 17 by the ranges of the variable 𝑆 generated by the NNFSMI (blue lines). Moreover, the fact the variable 𝑆 shown in Fig. 17 spanning over various ranges of fuzzy sets at different frequencies validates that our choice of 13 fuzzy sets is appropriate in order to achieve consistent and accurate tracking performance over the frequency range from as low as 0.05 Hz to as high as 1 Hz. 79
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Fig. 18. Experimental results of the FSMI and NNFSMI tracking a sinusoidal trajectory of 0.1 Hz in frequency and ±15◦ in amplitude with 𝜂𝑢 = 0.01 and 𝜂𝑖 = 0.0005.
results comparing the FSMI and NNFSMI tracking a sinusoidal trajectory of 0.1 Hz in frequency and ±20◦ in amplitude. The maximum errors at steady state achieved by the FSMI and NNFSMI are about 0.62◦ and 0.19◦ respectively. The variable 𝑆 extends to the range between NM and PM during transient and still manages to stay between NMN and PVS at steady state. These steady state results are also summarized in Fig. 17. Fig. 20 shows the experimental results when the amplitude of the sinusoidal reference trajectory is increased to ±25◦ , which is the extreme stroke we have tested. Note that ±25◦ is quite close to the extremes (+28.56◦ and −28.52◦ respectively) of the PMA actuated manipulator supplied with compressed air regulated to 4 bar, as stated in Section 2. As the PMA actuated manipulator reaches close to its extremes, the impact from the stiffness difference between the two PMAs becomes
4.2. Tracking sinusoidal reference trajectories of various strokes In this section, the FSMI and NNFSMI are compared tracking sinusoidal reference trajectories of 0.1 Hz in frequency and ±15◦ , ±20◦ and ±25◦ in amplitude. Based on the conclusion drawn in Section 4.1, the more aggressive learning parameters 𝜂𝑢 = 0.01 and 𝜂𝑖 = 0.0005 are adopted in the lower frequency range. Fig. 18 shows the experimental results comparing the FSMI and NNFSMI tracking a sinusoidal reference trajectory of 0.1 Hz in frequency and ±15◦ in amplitude. The maximum errors at steady state achieved by the FSMI and NNFSMI are about 0.47◦ and 0.16◦ respectively. The variable 𝑆 varies between NM and PM during transient and remains between NVS and PVS at steady state. Fig. 19 shows the experimental 80
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Fig. 19. Experimental results of the FSMI and NNFSMI tracking a sinusoidal trajectory of 0.1 Hz in frequency and ±20◦ in amplitude with 𝜂𝑢 = 0.01 and 𝜂𝑖 = 0.0005.
apparent. This difference is compensated by the asymmetric control signals to the proportional control valve shown in Fig. 20. The resulting maximum steady state errors achieved by the FSMI and NNFSMI are about 0.8◦ and 0.25◦ respectively. The variable 𝑆 spans about a range similar to the ±20◦ case from NB to PB during transient whereas it reaches slightly outside the range of NVS and PVS at steady state.
where 𝑓1 = 0.1 Hz, 𝑓2 = 0.3 Hz, 𝑓3 = 0.5 Hz and Δ𝑡 is the sampling time. In other words, the reference trajectory is a sum of three sinusoids with initial condition 𝜃𝑟 (0) = 0◦ . As the reference input of mixed frequencies provides a strong PE condition, the moderate learning parameters 𝜂𝑢 = 0.001 and 𝜂𝑖 = 0.00005 are adopted. Fig. 21 shows the FSMI and NNFSMI tracking a sinusoidal trajectory of mixed frequencies. The maximum error of the FSMI at steady state occurs when the manipulator is changing direction toward another extreme (either from +20◦ to −20◦ or the other way around), due to the hysteresis effect of the PMAs. The NNFSMI, on the other hand, effectively reduces those errors by intelligently adjusting the control gains 𝐺𝑢 and 𝐾𝑖 based on the neural network learning algorithm. Fig. 21 also shows that the value of
4.3. Tracking a sinusoidal reference trajectory of mixed frequencies In this section, we compare the performance of the FSMI and NNFSMI tracking a sinusoidal reference trajectory of mixed frequencies. 𝜃𝑟 (𝑘) = 15 sin(2𝜋𝑓1 𝑘Δ𝑡) + 10 sin(2𝜋𝑓2 𝑘Δ𝑡) + 5 sin(2𝜋𝑓3 𝑘Δ𝑡)
(10) 81
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Fig. 20. Experimental results of the FSMI and NNFSMI tracking a sinusoidal trajectory of 0.1 Hz in frequency and ±25◦ in amplitude with 𝜂𝑢 = 0.01 and 𝜂𝑖 = 0.0005.
the input variable 𝑆 exceeds the limit of PVB during transient and is bounded between NM and PM at steady state.
moving away from a stationary position, a more aggressive learning (𝜂𝑢 = 0.08) for the control gain to the FSMC (𝐺𝑢 ) needs to be applied in order to achieve optimum tracking performance. The integral gain (𝐾𝑖 ), on the other hand, is updated with a modest learning parameter (𝜂𝑖 = 0.00003) for the steady state tracking performance. Due to the hysteresis effect of the PMAs, the maximum error of the FSMI occurs every time when the manipulator starts moving away from a stationary position followed by gradually diminished oscillation. The NNFSMI, on the other hand, is able to reduce the peak of the error and the oscillation. The value of variable 𝑆 reaches NM and PM when the manipulator starts moving away from a stationary position. Note that when the NNFSMI is applied to track the trapezoidal input, the FSMC gain 𝐺𝑢 is raised by
4.4. Tracking a trapezoidal reference trajectory So far we have shown that the proposed control strategy NNFSMI achieves accurate and consistent performance tracking sinusoidal reference trajectories over considerably wide ranges of frequency and stroke. As an extension to these promising results, the performance of NNFSMI tracking a trapezoidal reference trajectory is examined in this section as shown in Fig. 22. As the trapezoidal input only excites the neural network learning system at the instances when the manipulator starts 82
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Fig. 21. Experimental results of the FSMI and NNFSMI tracking a sinusoidal trajectory of mixed frequencies with 𝜂𝑢 = 0.001 and 𝜂𝑖 = 0.00005.
stroke even when the PMA actuated manipulator reaches close to its extremes. The fuzzy sliding mode control strategy provides robustness with reduced number of fuzzy rules whereas the integral controller is augmented to further improve the steady state tracking performance. The neural network enables adaptive learning and automatic tuning of the control gains to the fuzzy sliding mode controller (FSMC) and the integrator so that minimum tracking error is achieved under different circumstances. Experimental results show that the neural network learning algorithm effectively achieves consistent tracking of sinusoidal reference trajectories up to 1 Hz in frequency and close to the extreme stroke of the PMA actuated manipulator with the compressed air regulated to 4 bar. In order for the control parameters 𝐺𝑢 and
more than 100% whereas the integral gain 𝐾𝑖 is barely changed by about 6%. In contrast to the dependence on the integral gain (𝐾𝑖 ) tuning while tracking continuously varying sinusoidal inputs in the previous sections, the NNFSMI relies mainly on the FSMC gain (𝐺𝑢 ) tuning to reduce the settling time right after the manipulator moves into or away from a stationary position while tracking a trapezoidal input. 5. Conclusions The neural network fuzzy sliding mode integral (NNFSMI) controller developed in this paper achieves fast, accurate and consistent tracking performance over considerably wide ranges of frequency and 83
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Fig. 22. Experimental results of the FSMI and NNFSMI tracking a trapezoidal trajectory of 0.05 Hz in frequency and ±20◦ in amplitude with 𝜂𝑢 = 0.08 and 𝜂𝑖 = 0.00003.
network learning system at the instances when the manipulator starts moving away from a stationary position, a more aggressive learning for the control gain to the FSMC and a modest learning for the integral gain need to be used. The NNFSMI depends mainly on the FSMC gain tuning to quickly compensate for the hysteresis effect when the manipulator is suddenly commanded to start or stop moving in tracking a trapezoidal input. In the future, the intelligent adaptive control algorithm can be applied to PMA actuated manipulators with higher degrees of freedom.
𝐾𝑖 to converge within the restricted time span shown in this paper, more aggressive learning parameters are adopted for sinusoidal inputs of lower frequencies whereas moderate learning parameters are chosen for sinusoidal inputs of higher frequencies. The neural network learning algorithm significantly reduces the error tracking sinusoidal inputs of higher frequencies while consuming much less control effort at steady state by inhibiting the FSMC from being saturated. The sinusoidal input excites the system continuously and as a result the NNFSMI relies more on the integral action (integral gain tuning) to compensate for the hysteresis effect when the PMA actuated manipulator is changing direction. The fine tuning of the control gain to the FSMC is added to further shorten the settling time and improve the overall performance tracking the sinusoidal input. On the contrary, in order to achieve satisfactory performance tracking a trapezoidal input, which only excites the neural
Acknowledgments This work is funded by the Ministry of Science and Technology, Taiwan, under grant MOST 104-2628-E-011-013-MY2. The authors would like to thank Prof. Y.T. Wang form NTUST for the fruitful discussions. 84
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Engineering Applications of Artificial Intelligence 65 (2017) 68–86 Table A.1 List of parameters and their values, if constant. Parameters
Definition
Value
𝛼 𝛽 𝜂𝑖
Weighting on the tracking error Parameter representing the term 𝜕𝜃 𝜕𝑢 Learning rate of the integral gain 𝐾𝑖 (sinusoid, <0.35 Hz) (sinusoid, ≥0.35 Hz) (trapezoid) Learning rate of the control gain 𝐺𝑢 (sinusoid, <0.35 Hz) (sinusoid, ≥0.35 Hz) (trapezoid) Rotating angle of the manipulator arm, degree Reference angle of the manipulator arm, degree Parameter in the sigmoid function Gain in the sigmoid function Scaling factor of the FSMC Control gain to the FSMC Integral gain Energy function for the gradient descent method Negative big Negative medium Negative small medium Negative very big Negative very small Positive big Positive medium Positive small medium Positive very big Positive very small Weighting on the tracking error in the energy function Weighting on the sliding variable 𝑠 in the energy function Zero Tracking error, degree Numerical integral of the tracking error 𝑒, degree-s Error difference, degree The fuzzy membership weighting Length of the manipulator arm, mm Sliding variable Control command to the proportional control valve, V FSMC output Center point of each fuzzy set
1.0
𝜂𝑢
𝜃 𝜃𝑟 𝜇 𝐴 𝐺𝑠 𝐺𝑢 𝐾𝑖 𝐽 𝑁𝐵 𝑁𝑀 𝑁𝑆𝑀 𝑁𝑉 𝐵 𝑁𝑉 𝑆 𝑃𝐵 𝑃𝑀 𝑃 𝑆𝑀 𝑃𝑉 𝐵 𝑃𝑉 𝑆 𝑊1 𝑊2 𝑍𝑂 𝑒 𝑒𝑖 Δ𝑒 ℎ𝑖 𝑙 𝑠 𝑢 𝑢𝑓 𝑦𝑖
Appendix
0.0005 0.00005 0.00003 0.01 0.001 0.08
0.8 1.25 0.5 0.3 0.03
1.0 1.0
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Iskarous, M., Kawamura, K., 1995. Intelligent control using a neuro-fuzzy network. In: IEEE International Conference on Human Robot Interaction and Cooperative Robots, Vol. 3. IEEE, Pittsburgh, PA, pp. 350–355. Kim, S.W., Lee, J.J., 1995. Design of a fuzzy controller with fuzzy sliding surface. Fuzzy Sets and Systems 71 (3), 359–367. Lilly, J.H., 2003. Adaptive tracking for pneumatic muscle actuators in bicep and tricep configurations. IEEE Trans. Neural Syst. Rehabil. Eng. 11 (3), 333–339.
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Contents lists available at ScienceDirect
Engineering Applications of Artificial Intelligence journal homepage: www.elsevier.com/locate/engappai
Neural network fuzzy sliding mode control of pneumatic muscle actuators Chia-Jui Chiang *, Ying-Chen Chen National Taiwan University of Science and Technology, No.43, Sec. 4, Keelung Rd., Daan Dist., Taipei 106, Taiwan, ROC
a r t i c l e
i n f o
Keywords: Pneumatic muscle actuators Intelligent control Fuzzy sliding mode control Neural network
a b s t r a c t The pneumatic muscle actuator (PMA) is one of the most promising actuators especially for the applications that require greater proximity between the humans and the robots. Fast and precise control of PMA, however, is difficult to achieve due to the compressibility of the air and the elasticity of the PMA. In order to achieve accurate and consistent tracking performance of a one axis PMA actuated manipulator over considerably wide ranges of frequency and stroke, an intelligent adaptive control algorithm is developed in this paper. The adaptive learning is enabled by a neural network in which the control gains to a fuzzy sliding mode controller (FSMC) and an integrator are adjusted to minimize the tracking error. Experimental results show that the proposed control strategy achieves fast, accurate and consistent performance tracking sinusoidal reference trajectories up to 1 Hz in frequency and close to the extreme stroke of the PMA actuated manipulator with the compressed air regulated to 4 bar. Results also show that the proposed control strategy, with a more aggressive learning for the control gain to the FSMC, achieves satisfactory performance tracking a trapezoidal reference trajectory. © 2017 Elsevier Ltd. All rights reserved.
1. Introduction In recent years, the robotics industry is developing rapidly. The robot actuators include electric motors, hydraulic cylinders and pneumatic actuators. The electric motors are characterized with excellent accuracy and controllability, but suffer from the possibility of sparking (Caldwell et al., 1995). The hydraulic actuators deliver large power with minimal backlash. Their main drawback lies in its lack of cleanness (Caldwell et al., 1995). The pneumatic actuators provide quick response and cleanness with low cost and are therefore widely used in industrial applications. The pneumatic muscle actuator (PMA), shown in Fig. 1, is one of the most promising pneumatic actuation systems for new types of industrial robots (Caldwell et al., 1995). The PMA consists of a rubber tube surrounded by a braided mesh with inextensible threads (Shen, 2010). As the rubber inner tube is inflated, the PMA is contracted (Tondu and Lopez, 2000). The advantages of PMA include high power-to-weight and power-to-volume ratios, cleanness, ease of maintenance, pliability, inherent safety, low cost and ready availability. As a result, the PMA is potentially one of the most promising actuators for the applications that require greater proximity between the humans and the robots (Caldwell et al., 1995; Tondu and Lopez, 2000). Due to the compressibility of the air and the elasticity of the PMA, however, it is difficult to achieve fast and precise control of the PMA (Thanh and Ahn, 2006a).
In order to realize satisfactory control performance, many control methods have been proposed to solve these challenging problems in controlling the PMA. The complex and nonlinear dynamics of the PMA make it a challenging and yet appealing system for model-based control design. Among previous approaches, Repperger and his team used a second order nonlinear differential equation to describe the PMA dynamics and based on which a gain scheduling tracking controller is developed (Repperger et al., 1999). John H. Lilly et al. developed adaptive tracking controllers and sliding mode tracking controllers for PMA actuated manipulators (Lilly, 2003; Lilly and Yang, 2005). In Aschemann and Schindele (2008), a cascaded sliding mode control algorithm is developed for a linear axis driven by a pair of pneumatic muscles. Shen (2010) derived a physics-based nonlinear model and based on which a sliding mode control approach is applied to achieve robustness against model uncertainties and disturbances. A switching model predictive control approach is employed in Andrikopoulos et al. (2013) based on linearized models at various operating conditions. In Pujana-Arrese et al. (2010), a nonlinear model is developed based on Dymola/Modelica and validated against experimental data. The nonlinear model is then linearized and based on which linear controllers are designed and compared. In Ganguly et al. (2012), on–off solenoid valves are used to control the PMA pressures in the inner loop and a PID controller is augmented in the outer loop to achieve position tracking of
* Corresponding author.
E-mail addresses: [email protected] (C.-J. Chiang), [email protected] (Y.-C. Chen). http://dx.doi.org/10.1016/j.engappai.2017.06.021 Received 4 October 2016; Received in revised form 20 May 2017; Accepted 25 June 2017 0952-1976/© 2017 Elsevier Ltd. All rights reserved.
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network is applied to control the robot joints by continuously modeling every joint and generating a joint inverse using the back-propagation algorithm. In Balasubramanian and Rattan (2003), the PMA is modeled as a mass–spring–damper system and the coefficients for the spring and damper are represented by a nonlinear fuzzy model. Based on the model, a linearizing controller is developed and a PID controller is augmented in the outer loop before the force command is fed into a fuzzy inverse model to obtain the pressure command. In Chan et al. (2003), a control strategy with an incremental fuzzy logic controller in place of the proportional term in a conventional PID controller is developed based on a neuro-fuzzy model of the PMA. In Chang and Lilly (2013), an adaptive fuzzy PD controller is augmented with an integral controller and the fuzzy PD consequent singleton locations are dynamically adjusted by a fuzzy inverse model with the interior pressure of the PMA as one of its inputs. In Ahn and Nguyen (2007), the PID control gains are switched based on the external load estimate obtained from a learning vector quantization neural network so as to achieve precise position control under various external loads. In Thanh and Ahn (2006b), the PID control gains are adjusted by neural networks and the tracking performance of a PMA actuated two-axes robot is examined up to 0.2 Hz. In Anh and Ahn (2011), the authors of Thanh and Ahn (2006b) further added bias weightings to the PID-based control structure so as to achieve precise tracking control when the two-axes robot is subjected to various loads. In Xie and Jamwal (2011), the PID feedback controller is augmented with a feedforward controller developed based on a fuzzy model and the control performance is examined at various load conditions. In Anh (2010), a feedforward/feedback control structure is adopted and each sub-controller is developed based on fuzzy logic. The tracking results of a PMA actuated robot up to 0.2 Hz are shown. In Andrikopoulos et al. (2014), extra degrees of freedom are incorporated into a PIDbased control structure and the tracking performance is examined up to 0.25 Hz. In an effort to achieve accurate and consistent tracking performance of a PMA actuated manipulator over considerably wide ranges of frequency and stroke, an intelligent adaptive control algorithm is proposed in this paper. The adaptive learning capability is provided by a neural network in which the control gains to a fuzzy sliding mode controller (FSMC) and an integrator are adjusted, based on the gradient descent method and the back propagation algorithm, to minimize the tracking error. Experimental results with a one axis PMA actuated manipulator show that the proposed control strategy achieves accurate and consistent performance tracking sinusoidal reference trajectories up to 1 Hz in
Fig. 1. Pneumatic muscle actuator.
Fig. 2. Pneumatic muscle actuated manipulator.
a single degree of freedom manipulator. In Minh et al. (2010), hysteresis compensation is implemented in a cascaded control strategy to improve the position tracking performance of a single PMA-mass system. On the other hand, the intelligent control methods have been proposed to control the PMA without detailed modeling of the complex dynamics of the PMA. In Hesselroth et al. (1994), the neural network learning algorithm is employed for position control of a PMA actuated five-joint robot. In Iskarous and Kawamura (1995), the neuro-fuzzy
Fig. 3. Structure of the PMA system.
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Fig. 4. Operating range of the PMA actuated manipulator supplied with compressed air regulated to 4 bar.
Fig. 5. Block diagram of the neural network fuzzy sliding mode integral (NNFSMI) controller.
Fig. 6. Block diagram of neural network.
frequency and close to the extreme stroke of the PMA actuated manipulator with the compressed air regulated to 4 bar. Results also show that the proposed control strategy, with a more aggressive learning for the control gain to the fuzzy sliding mode controller (FSMC), achieves satisfactory performance tracking a trapezoidal reference trajectory. The rest of this paper is organized as follows. Section 2 describes experimental setup of the one axis pneumatic muscle actuated robot arm used in the study. Section 3 shows the adaptive control algorithm developed in the paper. Section 4 examines the performance of the
Fig. 7. The shapes of sigmoid function.
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Fig. 8. Triangular input membership function and output singletons.
Fig. 9. Experimental results of the NNFSMI tracking a sinusoidal trajectory of 0.05 Hz in frequency and ±20◦ in amplitude with 𝜂𝑢 = 0.001 and 𝜂𝑖 = 0.00005. 71
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Fig. 10. Experimental results of the FSMI and NNFSMI tracking a sinusoidal trajectory of 0.05 Hz in frequency and ±20◦ in amplitude with 𝜂𝑢 = 0.01 and 𝜂𝑖 = 0.0005.
PMA actuated manipulator tracking various sinusoids and a trapezoid. Section 5 concludes this paper and points out possible extensions of this work.
in and out of the two PMAs and thus the rotating motion. The rotating angle 𝜃 is measured using an encoder (Nemicon HES-2048-2MD) with resolution of 2048 pulses per revolution and the digital signal decoding is carried out using a decoder IC (HTCL-2020). The controller is implemented in a PC with Advantech PCI-1710 board and the sampling time used in the experiments is 10 ms. In order to examine the operating range of the PMA actuated manipulator, open loop test is conducted. Fig. 4 shows that by switching the command to the proportional control valve the manipulator gradually reaches its extremes (+28.56◦ and −28.52◦ respectively) as one of the muscles is inflated to 4 bar while the other is deflated to atmospheric pressure.
2. Experimental setup Fig. 2 shows a one axis PMA actuated manipulator with its schematic diagram shown in Fig. 3. The architecture consists of two pneumatic muscle actuators (FESTO MAS-20-200N) with 20 mm internal diameter and 200 mm nominal length. The manipulator arm weights around 104 g and the length 𝑙 is 235 mm. The pneumatic muscle actuated system are supplied with compressed air regulated to 4 bar. A proportional control valve (Festo MPYE-5-1/8-CF-010-B) is used to control the flow 72
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Fig. 11. Experimental results of the FSMI and NNFSMI (with and without 𝐺𝑢 tuning) tracking a sinusoidal trajectory of 0.25 Hz in frequency and ±20◦ in amplitude.
3. Adaptive control algorithm
3.1. Fuzzy sliding mode control (FSMC)
The block diagram of the adaptive control algorithm is shown in Fig. 5. Fuzzy logic-based controller is chosen to cope with the complex and nonlinear dynamics of the pneumatic muscle actuators. A sliding surface is adopted to reduce the dimension of the input space and the number of fuzzy rules (Kim and Lee, 1995; Tzafestas and Rigatos, 1999). An integrator is augmented to further improve the tracking performance at steady state. Neural network provides an adaptive learning algorithm that updates the control parameters and achieves accurate and consistent tracking performance of the PMA actuated manipulator over considerably wide ranges of frequency and stroke.
The conventional fuzzy logic control theory involves fuzzification, a fuzzy rule base, a fuzzy inference engine and defuzzification. Conventionally, the fuzzy rule base depends both on the error 𝑒 and error difference Δ𝑒, resulting in complicated fuzzy rules and membership functions. To reduce the dimension of the input and thus the number of fuzzy rules, a fuzzy sliding surface 𝑠 = 0 is introduced with the variable 𝑠 defined as follows Kim and Lee (1995) and Tzafestas and Rigatos (1999). 𝑠(𝑘) = 𝛼𝑒(𝑘) + Δ𝑒(𝑘) 73
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Fig. 12. Zoomed views of the responses during transient (left) and at steady state (right) tracking a sinusoidal trajectory of 0.25 Hz in frequency and ±20◦ in amplitude with and without 𝐺𝑢 tuning.
frequency and stroke, a neural network algorithm is adopted for its powerful capability of learning, adaptivity and tackling nonlinearity (Thanh and Ahn, 2006b). The neural network is trained by the back propagation algorithm to minimize the tracking error. Fig. 6 shows the block diagram of the proposed neural network. The inputs to the neural network are the output of the FSMC, 𝑢𝑓 , and ∑ the numerical integral of tracking error, 𝑒𝑖 (𝑘) = 𝑘𝑛=1 𝑒(𝑛)Δ𝑡, where Δ𝑡 is the sampling time. The control output is obtained from a nonlinear sigmoid function (Yamada and Yabuta, 1992).
where the tracking error 𝑒(𝑘) = 𝜃𝑟 (𝑘) − 𝜃(𝑘) is the difference between the reference angle 𝜃𝑟 and the measurement of the angle 𝜃, Δ𝑒(𝑘) = 𝑒(𝑘) − 𝑒(𝑘 − 1) is the error difference and 𝛼 is a strictly positive constant which determines the bandwidth of the sliding mode control law and is typically limited by the mechanical properties of the system, time delay in the actuators and the available computing power (Slotine and Li, 1991). In practice, the tuning of this single scalar is often conducted experimentally. The control objective is then to force the system into the sliding surface so that tracking error is reduced. The normalized input variable 𝑆 to the fuzzification process is obtained by multiplying a scaling factor 𝐺𝑠 to the sliding variable 𝑠: 𝑆 = 𝐺𝑠 ⋅ 𝑠. The choice of number of fuzzy sets is generally a balance between precision and transparency (Babuska, 2000). Depending on the application of the pneumatic muscles, the number of fuzzy sets per variable used in the literature can be as small as 2 and as large as 17 (Anh and Ahn, 2011; Balasubramanian and Rattan, 2003; Chan et al., 2003; Chang and Lilly, 2013; Iskarous and Kawamura, 1995; Xie and Jamwal, 2011). In this paper, in order to achieve accurate and consistent tracking performance of the PMA actuated manipulator over considerably wide ranges of frequency and stroke, the universe of the variable 𝑆 is partitioned into thirteen fuzzy sets [𝑁𝑉 𝐵𝑠 𝑁𝐵𝑠 𝑁𝑀𝑠 𝑁𝑆𝑀𝑠 𝑁𝑆𝑠 𝑁𝑉 𝑆𝑠 𝑍𝑂𝑠 𝑃 𝑉 𝑆𝑠 𝑃 𝑆𝑠 𝑃 𝑆𝑀𝑠 𝑃 𝑀𝑠 𝑃 𝐵𝑠 𝑃 𝑉 𝐵𝑠 ] characterized by triangular membership functions. Similarly, the fuzzy sets for the control output are defined as [𝑁𝑉 𝐵𝑢 𝑁𝐵𝑢 𝑁𝑀𝑢 𝑁𝑆𝑀𝑢 𝑁𝑆𝑢 𝑁𝑉 𝑆𝑢 𝑍𝑂𝑢 𝑃 𝑉 𝑆𝑢 𝑃 𝑆𝑢 𝑃 𝑆𝑀𝑢 𝑃 𝑀𝑢 𝑃 𝐵𝑢 𝑃 𝑉 𝐵𝑢 ]. The linguistic variables represented by those labels of fuzzy sets are defined in Table A.1. The fuzzy inference implements fuzzy control rules by using the single input to single output mapping. Finally, the defuzzification process is conducted based on the height method (also called center average defuzzifier) to obtain the control output 𝑢𝑓 of the fuzzy sliding mode controller (FSMC). ∑ ℎ𝑦 (2) 𝑢𝑓 (𝑘) = ∑ 𝑖 𝑖 ℎ𝑖
𝑢 = 𝑓 (𝑥) =
2𝐴(1 − 𝑒−𝜇𝑥 ) 𝜇(1 + 𝑒−𝜇𝑥 )
(3)
where the parameters 𝐴 and 𝜇 determine the shape of the sigmoid function. Fig. 7 shows the shapes of the sigmoid function with a constant gain 𝐴 = 1.25 and various values of parameter 𝜇. Note that the constant gain A represents the slope of the sigmoid function in the linear region close to the origin and needs to be tuned for optimum convergence properties. The tuning of the parameter 𝜇, on the other hand, depends on the operating range of the actuation system as the parameter 𝜇 varies the saturation level of the sigmoid function. The input to the sigmoid function is defined by 𝑥(𝑘) = 𝐺𝑢 (𝑘)𝑢𝑓 (𝑘) + 𝐾𝑖 (𝑘)𝑒𝑖 (𝑘)
(4)
where the parameters 𝐺𝑢 and 𝐾𝑖 are the control gain to the FSMC and the integral gain respectively. To obtain the control gains that minimize the tracking error, an energy function 𝐽 (𝑘) is chosen as follows. 1 1 𝑊 𝑒2 (𝑘) + 𝑊2 𝑠2 (𝑘) (5) 2 1 2 where the parameters 𝑊1 and 𝑊2 are the weightings on the tracking error 𝑒(𝑘) and the sliding variable 𝑠(𝑘) respectively. The control gains 𝐺𝑢 and 𝐾𝑖 are then updated to minimize the energy function 𝐽 in Eq. (5). In order to achieve direct and fast convergence for parameters 𝐺𝑢 and 𝐾𝑖 , gradient descent method is adopted due to its simplicity, fast computation per iteration and low data storage requirement compared to other search techniques (Boyd and Vandenberghe, 2004). 𝐽 (𝑘) =
where ℎ𝑖 is the fuzzy membership weighting and 𝑦𝑖 is the center point of each fuzzy set. 3.2. Neural network
𝜕𝐽 (𝑘) 𝜕𝐺𝑢 𝜕𝐽 (𝑘) 𝐾𝑖 (𝑘 + 1) = 𝐾𝑖 (𝑘) − 𝜂𝑖 𝜕𝐾𝑖 𝐺𝑢 (𝑘 + 1) = 𝐺𝑢 (𝑘) − 𝜂𝑢
In order for the PMA actuated manipulator to achieve accurate and consistent tracking performance over considerably wide ranges of 74
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Fig. 13. Experimental results of the FSMI and NNFSMI tracking a sinusoidal trajectory of 0.5 Hz in frequency and ±20◦ in amplitude with 𝜂𝑢 = 0.001 and 𝜂𝑖 = 0.00005.
where the parameters 𝜂𝑢 and 𝜂𝑖 are the learning rates that determine the
where each individual term is derived based on Eqs. (1), (3), (4) and
convergence speed. The back propagation algorithm is then realized by
(5).
applying chain rule.
𝜕𝐽 (𝑘) 𝜕𝑠(𝑘) 𝜕𝑒(𝑘) 𝜕𝜃(𝑘) 𝜕𝑢(𝑘) 𝜕𝑥(𝑘) 𝜕𝐽 (𝑘) = 𝜕𝐺𝑢 𝜕𝑠 𝜕𝑒 𝜕𝜃 𝜕𝑢 𝜕𝑥 𝜕𝐺𝑢 𝜕𝐽 (𝑘) 𝜕𝐽 (𝑘) 𝜕𝑒(𝑘) 𝜕𝜃(𝑘) 𝜕𝑢(𝑘) 𝜕𝑥(𝑘) = 𝜕𝐾𝑖 𝜕𝑒 𝜕𝜃 𝜕𝑢 𝜕𝑥 𝜕𝐾𝑖
𝜕𝐽 (𝑘) 𝜕𝑠(𝑘) 𝜕𝐽 (𝑘) = 𝑊2 𝑠(𝑘); = 𝑊1 𝑒(𝑘); =𝛼 𝜕𝑠 𝜕𝑒 𝜕𝑒 𝜕𝑒(𝑘) 𝜕𝜃(𝑘) 𝜕𝑥(𝑘) = −1; = 𝛽; = 𝑢𝑓 (𝑘) 𝜕𝜃 𝜕𝑢 𝜕𝐺𝑢 𝜕𝑥(𝑘) 𝜕𝑢(𝑘) 4𝐴𝑒−𝜇𝑥 = 𝑒𝑖 (𝑘); = 𝜕𝐾𝑖 𝜕𝑥 (1 + 𝑒−𝜇𝑥 )2
(7)
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Fig. 14. Zoomed views of the responses during transient (left) and at steady state (right) tracking a sinusoidal trajectory of 0.5 Hz in frequency and ±20◦ in amplitude with 𝜂𝑢 = 0.001 and 𝜂𝑖 = 0.00005.
where the parameter 𝛽 is defined as follows ⎧ ⎪𝛽 = 1, ⎨ ⎪𝛽 = −1, ⎩
In Section 4.1, the performance of NNFSMI and FSMI tracking sinusoidal reference trajectories of various frequencies is examined. In Section 4.2, the performance of NNFSMI and FSMI tracking sinusoidal reference trajectories of various strokes is examined. In Section 4.3, the performance of NNFSMI and FSMI tracking a sinusoidal reference trajectory of mixed frequencies is examined. In Section 4.4, the performance of NNFSMI and FSMI tracking a trapezoidal reference trajectory is examined.
Δ𝜃 ≥0 Δ𝑢 Δ𝜃 if <0 Δ𝑢 if
combining Eqs. (6)–(8), the resulting learning law is summarized as follows 4𝐴𝑒−𝜇𝑥 𝐺𝑢 (𝑘 + 1) = 𝐺𝑢 (𝑘) + 𝜂𝑢 𝑊2 𝛼𝛽𝑠(𝑘) 𝑢𝑓 (𝑘) (1 + 𝑒−𝜇𝑥 )2 (9) −𝜇𝑥 4𝐴𝑒 𝐾𝑖 (𝑘 + 1) = 𝐾𝑖 (𝑘) + 𝜂𝑖 𝑊1 𝛽𝑒(𝑘) 𝑒𝑖 (𝑘). (1 + 𝑒−𝜇𝑥 )2
4.1. Tracking sinusoidal reference trajectories of various frequencies In this section, the closed-loop results with and without the neural network, NNFSMI and FSMI respectively, are compared tracking sinusoidal reference trajectories ranging from 0.05 Hz to 1 Hz. Fig. 9 shows the experimental results when the NNFSMI is applied to track a sinusoidal trajectory of 0.05 Hz in frequency and ±20◦ in amplitude with the learning parameters 𝜂𝑢 = 0.001 and 𝜂𝑖 = 0.00005. The results show that even though the tracking performance seems acceptable, the control parameters 𝐺𝑢 (𝑘) and 𝐾𝑖 (𝑘) have never converged to the optimum values within the time span of 120 s as the reference input of lower frequency provides a relatively weak persistent-excitation (PE) condition. Therefore, when tracking a sinusoidal reference trajectory of lower frequency, a more aggressive learning needs to be adopted so that the control parameters can converge within the restricted time span displayed in this paper. Fig. 10 shows that a more aggressive learning within 𝜂𝑢 = 0.01 and 𝜂𝑖 = 0.0005 results in fast convergence accompanied by the oscillatory behavior in the first few seconds. At steady state, the maximum error of the FSMI occurs when the manipulator is changing direction, resulting from the hysteresis effect of the PMAs. The NNFSMI, on the other hand, is able to eliminate those errors by quickly adjusting the command to the proportional control valve. Specifically, the maximum error achieved by the NNFSMI is less than 0.09◦ at steady state
4. Experimental results In this section, the tracking performance of the adaptive control algorithm developed in Section 3 is examined over considerably wide ranges of frequency and stroke. The fuzzy rule matrices are [−1.0 −0.8 −0.6 −0.4 −0.2 −0.1 0.0 0.1 0.2 0.4 0.6 0.8 1.0] for the control input and [−1.0 −0.7 −0.5 −0.3 −0.15 −0.07 0.0 0.07 0.15 0.3 0.5 0.7 1.0] for the control output. Fig. 8 shows the resulting triangular input membership function and the output singletons. The parameters used in the neural network fuzzy sliding mode integral (NNFSMI) controller throughout this section are 𝛼 = 1, 𝐴 = 1.25, 𝜇 = 0.8, 𝐺𝑠 = 0.5, 𝑊1 = 1, 𝑊2 = 1, 𝐺𝑢 (0) = 0.3 and 𝐾𝑖 (0) = 0.03. To illustrate the effectiveness of the learning algorithm, the closed-loop results with and without the neural network are compared. Without the neural network, constant control parameters 𝐺𝑢 (𝑘) = 𝐺𝑢 (0) and 𝐾𝑖 (𝑘) = 𝐾𝑖 (0) are used in the fuzzy sliding mode integral (FSMI) controller. The principles for the tuning of control parameters 𝛼, 𝐴 and 𝜇 are introduced in Section 3 whereas the initial values 𝐺𝑢 (0) and 𝐾𝑖 (0) are obtained by trial-and-error through experiments with the FSMI controller. 76
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Fig. 15. Experimental results of the FSMI and NNFSMI tracking a sinusoidal trajectory of 1 Hz in frequency and ±20◦ in amplitude with 𝜂𝑢 = 0.001 and 𝜂𝑖 = 0.00005.
tracking the sinusoidal trajectory of 0.05 Hz. Fig. 10 also shows that the value of the input variable 𝑆 to the fuzzification process ranges from NSM to PSM during transient and stays between NVS and PVS at steady state. Fig. 11 shows the experimental results tracking a sinusoidal trajectory of 0.25 Hz in frequency and ±20◦ in amplitude. The more aggressive learning parameters 𝜂𝑢 = 0.01 and 𝜂𝑖 = 0.0005 are again applied in the NNFSMI in order for the control parameters 𝐺𝑢 (𝑘) and 𝐾𝑖 (𝑘) to converge to the optimum values within the restricted time span (60 s) displayed in the paper. Fig. 11 shows that the NNFSMI achieves significantly better tracking performance than the FSMI, especially
when the manipulator is changing direction. Specifically, the maximum error achieved by the NNFSMI is less than 0.54◦ at steady state tracking the sinusoidal trajectory of 0.25 Hz. The range of the input variable 𝑆 to the fuzzification process stretches between NVB and PVB during transient and is bounded between NSM and PSM at steady state. Note that when the NNFSMI is applied the FSMC gain 𝐺𝑢 is increased by about 30% whereas the integral gain 𝐾𝑖 is raised by more than 100%. The NNFSMI relies more on the integral gain tuning to compensate for the hysteresis effect as the sinusoidal input excites the PMA-actuated system continuously. The necessity of fine tuning both control parameters in tracking a sinusoidal trajectory is further investigated by setting the 77
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Fig. 16. Zoomed views of the responses during transient (left) and at steady state (right) tracking a sinusoidal trajectory of 1 Hz in frequency and ±20◦ in amplitude with 𝜂𝑢 = 0.001 and 𝜂𝑖 = 0.00005.
valve and thus achieves significantly better tracking performance. The maximum error achieved by the NNFSMI is close to 1◦ at steady state tracking the sinusoidal trajectory of 0.5 Hz. Fig. 13 also shows that, while tracking the reference trajectory of 0.5 Hz, the range of the input variable 𝑆 to the fuzzification process stretches between NVB and PVB during transient and is bounded between NM and PM at steady state. Fig. 15 shows the experimental results comparing the FSMI and NNFSMI tracking a sinusoidal trajectory of 1 Hz in frequency and ±20◦ in amplitude. The same moderate learning parameters 𝜂𝑢 = 0.001 and 𝜂𝑖 = 0.00005 are used. Fig. 16 shows the zoomed views of the responses during transient (left) and at steady state (right) while tracking the same trajectory as in Fig. 15. Figs. 15 and 16 show that the tracking error can reach close to 10◦ when FSMI is applied, resulting from the hysteresis effect of the PMAs while the manipulator is changing direction. The NNFSMI, on the other hand, is able to intelligently adjust the control gains 𝐺𝑢 and 𝐾𝑖 and produce faster control command to the proportional control valve resulting in significantly better tracking performance especially at steady state. Specifically, the maximum error achieved by the NNFSMI tracking the sinusoidal trajectory of 1 Hz is reduced to less than 1.83◦ at steady state. Fig. 15 also shows that, when tracking the reference trajectory of 1 Hz, which is the extreme input frequency we have tested, the value of the input variable 𝑆 to the fuzzification process exceeds the limits of NVB and PVB while the manipulator is changing direction. Fig. 17 summarizes the steady state control performance achieved by the FSMI and NNFSMI tracking sinusoidal trajectories of ±20◦ in amplitude ranging from 0.05 Hz to 1 Hz in frequency. The rate of parameter convergence in the neural network learning algorithm proposed in this paper depends on the persistence of excitation from the reference input. The reference input of lower frequencies (less than about 0.35 Hz) provides a relatively weak persistent-excitation (PE)
learning parameter 𝜂𝑢 = 0 while keeping 𝜂𝑖 = 0.0005. Fig. 11 shows that as the control gain 𝐺𝑢 is fixed at 0.3, larger integral gain 𝐾𝑖 is rendered by the neural network to compensate for the absence of the FSMC control gain tuning. The zoomed views of the responses shown in Fig. 12 indicate that the absence of the FSMC control gain tuning (𝜂𝑢 = 0) results in longer settling time both during transient (left) and at steady state (right) while tracking the sinusoidal trajectory of 0.25 Hz. Specifically, the mean absolute errors (Hoaglin et al., 1983) achieved by the NNFSMI with and without the FSMC gain tuning are about 0.09◦ and 0.12◦ respectively. Therefore, even though while tracking a sinusoidal trajectory the NNFSMI relies more on the integral gain tuning, the fine tuning of the FSMC control gain is still needed for shorter settling time and improved tracking performance. Fig. 13 shows the experimental results comparing the FSMI and NNFSMI tracking a sinusoidal trajectory of 0.5 Hz in frequency and ±20◦ in amplitude. At higher frequency the moderate learning parameters 𝜂𝑢 = 0.001 and 𝜂𝑖 = 0.00005 are chosen as the reference input provides a strong persistent-excitation (PE) condition to the neural network learning algorithm for the control parameters to converge within the restricted time span displayed in the paper. Fig. 14 shows the zoomed views of the responses during transient (left) and at steady state (right) while tracking the same trajectory as in Fig. 13. As can be seen from Figs. 13 and 14, significantly larger tracking error is observed when the FSMI control strategy is applied, especially when the manipular is changing direction. The learning mechanism of the NNFSMI, on the other hand, is able to adjust the control gains 𝐺𝑢 and 𝐾𝑖 based on Eq. (9) and significantly reduce the tracking error. Fig. 14 shows that the control commands to the proportional control valve and thus the tracking errors resulted from FSMI and NNFSMI are quite similar in the beginning. The NNFSMI, however, is able to quickly adjust the control gains and generate faster control command to the proportional control 78
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Fig. 17. Comparison of the control performance achieved by the FSMI and NNFSMI at steady state tracking sinusoidal trajectories of ±20◦ in amplitude ranging from 0.05 Hz to 1 Hz in frequency. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
condition and thus more aggressive learning parameters 𝜂𝑢 = 0.01 and 𝜂𝑖 = 0.0005 are applied in order for the control parameters to converge within the restricted time span (60 s) displayed in this paper. The reference input of higher frequencies (from about 0.35 to 1 Hz), on the other hand, produces a strong PE condition for the control parameter convergence and thus moderate learning parameters 𝜂𝑢 = 0.001 and 𝜂𝑖 = 0.00005 are used to alleviate the oscillatory behavior during transient. Fig. 17 shows that the tracking precision and control effort attained by the FSMI at steady state is comparable to that of the NNFSMI while tracking a sinusoidal trajectories of lower frequencies. The tracking precision achieved by the FSMI at steady state, however, deteriorates dramatically while enormous control effort is required tracking sinusoidal trajectories of higher frequencies. Specifically, at the higher frequencies the hysteresis effect results in substantial growth of the tracking error, leading to saturation of the fuzzy sliding mode controller (FSMC) as evidenced in Fig. 17 by the out-of-bound variable
𝑆 generated by the FSMI (red lines) at frequencies higher than 0.5 Hz. As the FSMC is saturated, the error further extends and as a result immense control effort is demanded by the integral action. The NNFSMI, on the other hand, significantly reduces the tracking errors at steady state while consuming much less control effort tracking sinusoidal trajectories of higher frequencies via quickly adjusting the control gains and generating faster control command that inhibits the FSMC from being saturated as evidenced in Fig. 17 by the ranges of the variable 𝑆 generated by the NNFSMI (blue lines). Moreover, the fact the variable 𝑆 shown in Fig. 17 spanning over various ranges of fuzzy sets at different frequencies validates that our choice of 13 fuzzy sets is appropriate in order to achieve consistent and accurate tracking performance over the frequency range from as low as 0.05 Hz to as high as 1 Hz. 79
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Fig. 18. Experimental results of the FSMI and NNFSMI tracking a sinusoidal trajectory of 0.1 Hz in frequency and ±15◦ in amplitude with 𝜂𝑢 = 0.01 and 𝜂𝑖 = 0.0005.
results comparing the FSMI and NNFSMI tracking a sinusoidal trajectory of 0.1 Hz in frequency and ±20◦ in amplitude. The maximum errors at steady state achieved by the FSMI and NNFSMI are about 0.62◦ and 0.19◦ respectively. The variable 𝑆 extends to the range between NM and PM during transient and still manages to stay between NMN and PVS at steady state. These steady state results are also summarized in Fig. 17. Fig. 20 shows the experimental results when the amplitude of the sinusoidal reference trajectory is increased to ±25◦ , which is the extreme stroke we have tested. Note that ±25◦ is quite close to the extremes (+28.56◦ and −28.52◦ respectively) of the PMA actuated manipulator supplied with compressed air regulated to 4 bar, as stated in Section 2. As the PMA actuated manipulator reaches close to its extremes, the impact from the stiffness difference between the two PMAs becomes
4.2. Tracking sinusoidal reference trajectories of various strokes In this section, the FSMI and NNFSMI are compared tracking sinusoidal reference trajectories of 0.1 Hz in frequency and ±15◦ , ±20◦ and ±25◦ in amplitude. Based on the conclusion drawn in Section 4.1, the more aggressive learning parameters 𝜂𝑢 = 0.01 and 𝜂𝑖 = 0.0005 are adopted in the lower frequency range. Fig. 18 shows the experimental results comparing the FSMI and NNFSMI tracking a sinusoidal reference trajectory of 0.1 Hz in frequency and ±15◦ in amplitude. The maximum errors at steady state achieved by the FSMI and NNFSMI are about 0.47◦ and 0.16◦ respectively. The variable 𝑆 varies between NM and PM during transient and remains between NVS and PVS at steady state. Fig. 19 shows the experimental 80
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Fig. 19. Experimental results of the FSMI and NNFSMI tracking a sinusoidal trajectory of 0.1 Hz in frequency and ±20◦ in amplitude with 𝜂𝑢 = 0.01 and 𝜂𝑖 = 0.0005.
apparent. This difference is compensated by the asymmetric control signals to the proportional control valve shown in Fig. 20. The resulting maximum steady state errors achieved by the FSMI and NNFSMI are about 0.8◦ and 0.25◦ respectively. The variable 𝑆 spans about a range similar to the ±20◦ case from NB to PB during transient whereas it reaches slightly outside the range of NVS and PVS at steady state.
where 𝑓1 = 0.1 Hz, 𝑓2 = 0.3 Hz, 𝑓3 = 0.5 Hz and Δ𝑡 is the sampling time. In other words, the reference trajectory is a sum of three sinusoids with initial condition 𝜃𝑟 (0) = 0◦ . As the reference input of mixed frequencies provides a strong PE condition, the moderate learning parameters 𝜂𝑢 = 0.001 and 𝜂𝑖 = 0.00005 are adopted. Fig. 21 shows the FSMI and NNFSMI tracking a sinusoidal trajectory of mixed frequencies. The maximum error of the FSMI at steady state occurs when the manipulator is changing direction toward another extreme (either from +20◦ to −20◦ or the other way around), due to the hysteresis effect of the PMAs. The NNFSMI, on the other hand, effectively reduces those errors by intelligently adjusting the control gains 𝐺𝑢 and 𝐾𝑖 based on the neural network learning algorithm. Fig. 21 also shows that the value of
4.3. Tracking a sinusoidal reference trajectory of mixed frequencies In this section, we compare the performance of the FSMI and NNFSMI tracking a sinusoidal reference trajectory of mixed frequencies. 𝜃𝑟 (𝑘) = 15 sin(2𝜋𝑓1 𝑘Δ𝑡) + 10 sin(2𝜋𝑓2 𝑘Δ𝑡) + 5 sin(2𝜋𝑓3 𝑘Δ𝑡)
(10) 81
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Engineering Applications of Artificial Intelligence 65 (2017) 68–86
Fig. 20. Experimental results of the FSMI and NNFSMI tracking a sinusoidal trajectory of 0.1 Hz in frequency and ±25◦ in amplitude with 𝜂𝑢 = 0.01 and 𝜂𝑖 = 0.0005.
the input variable 𝑆 exceeds the limit of PVB during transient and is bounded between NM and PM at steady state.
moving away from a stationary position, a more aggressive learning (𝜂𝑢 = 0.08) for the control gain to the FSMC (𝐺𝑢 ) needs to be applied in order to achieve optimum tracking performance. The integral gain (𝐾𝑖 ), on the other hand, is updated with a modest learning parameter (𝜂𝑖 = 0.00003) for the steady state tracking performance. Due to the hysteresis effect of the PMAs, the maximum error of the FSMI occurs every time when the manipulator starts moving away from a stationary position followed by gradually diminished oscillation. The NNFSMI, on the other hand, is able to reduce the peak of the error and the oscillation. The value of variable 𝑆 reaches NM and PM when the manipulator starts moving away from a stationary position. Note that when the NNFSMI is applied to track the trapezoidal input, the FSMC gain 𝐺𝑢 is raised by
4.4. Tracking a trapezoidal reference trajectory So far we have shown that the proposed control strategy NNFSMI achieves accurate and consistent performance tracking sinusoidal reference trajectories over considerably wide ranges of frequency and stroke. As an extension to these promising results, the performance of NNFSMI tracking a trapezoidal reference trajectory is examined in this section as shown in Fig. 22. As the trapezoidal input only excites the neural network learning system at the instances when the manipulator starts 82
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Engineering Applications of Artificial Intelligence 65 (2017) 68–86
Fig. 21. Experimental results of the FSMI and NNFSMI tracking a sinusoidal trajectory of mixed frequencies with 𝜂𝑢 = 0.001 and 𝜂𝑖 = 0.00005.
stroke even when the PMA actuated manipulator reaches close to its extremes. The fuzzy sliding mode control strategy provides robustness with reduced number of fuzzy rules whereas the integral controller is augmented to further improve the steady state tracking performance. The neural network enables adaptive learning and automatic tuning of the control gains to the fuzzy sliding mode controller (FSMC) and the integrator so that minimum tracking error is achieved under different circumstances. Experimental results show that the neural network learning algorithm effectively achieves consistent tracking of sinusoidal reference trajectories up to 1 Hz in frequency and close to the extreme stroke of the PMA actuated manipulator with the compressed air regulated to 4 bar. In order for the control parameters 𝐺𝑢 and
more than 100% whereas the integral gain 𝐾𝑖 is barely changed by about 6%. In contrast to the dependence on the integral gain (𝐾𝑖 ) tuning while tracking continuously varying sinusoidal inputs in the previous sections, the NNFSMI relies mainly on the FSMC gain (𝐺𝑢 ) tuning to reduce the settling time right after the manipulator moves into or away from a stationary position while tracking a trapezoidal input. 5. Conclusions The neural network fuzzy sliding mode integral (NNFSMI) controller developed in this paper achieves fast, accurate and consistent tracking performance over considerably wide ranges of frequency and 83
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Engineering Applications of Artificial Intelligence 65 (2017) 68–86
Fig. 22. Experimental results of the FSMI and NNFSMI tracking a trapezoidal trajectory of 0.05 Hz in frequency and ±20◦ in amplitude with 𝜂𝑢 = 0.08 and 𝜂𝑖 = 0.00003.
network learning system at the instances when the manipulator starts moving away from a stationary position, a more aggressive learning for the control gain to the FSMC and a modest learning for the integral gain need to be used. The NNFSMI depends mainly on the FSMC gain tuning to quickly compensate for the hysteresis effect when the manipulator is suddenly commanded to start or stop moving in tracking a trapezoidal input. In the future, the intelligent adaptive control algorithm can be applied to PMA actuated manipulators with higher degrees of freedom.
𝐾𝑖 to converge within the restricted time span shown in this paper, more aggressive learning parameters are adopted for sinusoidal inputs of lower frequencies whereas moderate learning parameters are chosen for sinusoidal inputs of higher frequencies. The neural network learning algorithm significantly reduces the error tracking sinusoidal inputs of higher frequencies while consuming much less control effort at steady state by inhibiting the FSMC from being saturated. The sinusoidal input excites the system continuously and as a result the NNFSMI relies more on the integral action (integral gain tuning) to compensate for the hysteresis effect when the PMA actuated manipulator is changing direction. The fine tuning of the control gain to the FSMC is added to further shorten the settling time and improve the overall performance tracking the sinusoidal input. On the contrary, in order to achieve satisfactory performance tracking a trapezoidal input, which only excites the neural
Acknowledgments This work is funded by the Ministry of Science and Technology, Taiwan, under grant MOST 104-2628-E-011-013-MY2. The authors would like to thank Prof. Y.T. Wang form NTUST for the fruitful discussions. 84
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Engineering Applications of Artificial Intelligence 65 (2017) 68–86 Table A.1 List of parameters and their values, if constant. Parameters
Definition
Value
𝛼 𝛽 𝜂𝑖
Weighting on the tracking error Parameter representing the term 𝜕𝜃 𝜕𝑢 Learning rate of the integral gain 𝐾𝑖 (sinusoid, <0.35 Hz) (sinusoid, ≥0.35 Hz) (trapezoid) Learning rate of the control gain 𝐺𝑢 (sinusoid, <0.35 Hz) (sinusoid, ≥0.35 Hz) (trapezoid) Rotating angle of the manipulator arm, degree Reference angle of the manipulator arm, degree Parameter in the sigmoid function Gain in the sigmoid function Scaling factor of the FSMC Control gain to the FSMC Integral gain Energy function for the gradient descent method Negative big Negative medium Negative small medium Negative very big Negative very small Positive big Positive medium Positive small medium Positive very big Positive very small Weighting on the tracking error in the energy function Weighting on the sliding variable 𝑠 in the energy function Zero Tracking error, degree Numerical integral of the tracking error 𝑒, degree-s Error difference, degree The fuzzy membership weighting Length of the manipulator arm, mm Sliding variable Control command to the proportional control valve, V FSMC output Center point of each fuzzy set
1.0
𝜂𝑢
𝜃 𝜃𝑟 𝜇 𝐴 𝐺𝑠 𝐺𝑢 𝐾𝑖 𝐽 𝑁𝐵 𝑁𝑀 𝑁𝑆𝑀 𝑁𝑉 𝐵 𝑁𝑉 𝑆 𝑃𝐵 𝑃𝑀 𝑃 𝑆𝑀 𝑃𝑉 𝐵 𝑃𝑉 𝑆 𝑊1 𝑊2 𝑍𝑂 𝑒 𝑒𝑖 Δ𝑒 ℎ𝑖 𝑙 𝑠 𝑢 𝑢𝑓 𝑦𝑖
Appendix
0.0005 0.00005 0.00003 0.01 0.001 0.08
0.8 1.25 0.5 0.3 0.03
1.0 1.0
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