Fractional-order modeling and control of pneumatic-hydraulic upper limb rehabilitation training system 1

Abstract

In this paper, by replacing the integral mass flow equation to fractional-order mass flow equation, the fractional-order mathematical model of 2DOF pneumatic-hydraulic upper limb rehabilitation training system is established. A new 2DOF fractional-order fuzzy PID (FOFPID) controller is designed, to provides a new reference for improving the control accuracy of the pneumatic system. In the design of the controller, the weight parameters of the input terms are transformed into the weight parameters of the error, and the input, which are analyzed to improve the accuracy of the controller design. The parameters of the control system are determined by multi-objective particle swarm optimization. To prove the effectiveness of the proposed control method, the experimental research was carried out by building the experimental platform of pneumatic-hydraulic upper limb rehabilitation training system. The results show that the 2DOF FOFPID controller has better performance than other designed controllers under different working conditions.

Keywords

Pneumatic-hydraulic drive rehabilitation training system fractional-order modeling fractional-order fuzzy PID control

1 Introduction

Today, there exists an increasingly serious aging problem and a large number of disabled people. The research and application of rehabilitation training systems will provide some technical means to solve the problem of disability assistance and rehabilitation [1]. At present, the rehabilitation training systems are mainly driven by motor, hydraulic and pneumatic systems [2 –4]. The motor-driven system has the advantages of mature technology, high control precision and simple structure, but the power/quality ratio is low and the dynamic stability is poor [5 –7]. Hydraulic-driven system has the advantages of small inertia, high power/quality, simple structure and stable operation. But it has disadvantages of high noise, high cost and poor flexibility [8 –10]. The pneumatic-driven system has the advantages of lightweight, pollution-free, fast speed and good flexibility. However, due to the influence of compressibility of gas, the system has low stiffness, poor bearing capacity and unstable control performance [11]. If the structural form of the system can be improved in rehabilitation training system, so that the system structure has the form of adjustable damping, it is hopeful to improve the performance of the flexibility of system, and to add scientific theoretical and practical methods to the rehabilitation training systems.

The pneumatic-hydraulic series control structure was first proposed by Harbin institute of technology, and the good position and force tracking control results were obtained through studies [12]. The main idea of the system is to connect the cylinders in series or parallel, adjust the valve installed between chambers of the hydraulic cylinder to change the structural characteristic of the system and improve the stiffness of the system. The power of the system is supported by the pneumatic cylinders, the liquid part is self-circulating (no hydraulic pump station needed), to make the system clean energy and low cost [13]. The pneumatic-hydraulic driving system overcomes the shortcomings of low stiffness of pneumatic system and low flexibility of hydraulic system. It combines the characteristics of flexibility of gas medium with rigidity of liquid medium, which maintains the advantages of flexibility of pneumatic system and provides adjustable damping capacity provided by the hydraulic part [14]. The application of this driving method in rehabilitation training system not only give full play to the advantage of ‘combining rigidity with flexibility’, and get better flexibility to assist effect, but also provide different system resistance by changing the opening size of hydraulic valve without changing the control strategy. The research of this subject has important practical application value for improving the performance of the rehabilitation training system.

Due to the limitation of calculation ability, mathematical models are always approximated to be integer. In recent years, with the rapid development of computer technology, fractional modeling has attracted many scholars in many fields [15 –20]. With the help of a non-integral integrator, a transfer function with fractional derivative is chosen to represent the mobility of the rod. W. Yu researched of fractional modeling and experimental analysis of permanent magnet synchronous motor [21]. The traditional integral order model of PMSM speed system is extended to fractional-order model. Through the experiment of two kinds of PI controllers, the superiority of the proposed fractional-order PMSM speed system model is verified. S.M.S. Bahraini analyzed and calculated the deflection characteristics of the elastic beam by establishing the fractional model [22]. The validity of the present analysis is verified by comparing the results with those found in the literature. H.L. Cao completed the research on the subsection modeling and control of the electrical characteristics of SOFC [23]. The results show that the fractional-order dynamic model has high accuracy for the dynamic description of SOFC electrical characteristics, which lays a solid foundation for the controller based on the accurate model. M. Fabrizio established fractional-order heat conduction model [24]. The fractional derivative is considered to replace the time derivative in the Cattaneo-Maxwell equation. The new model well describes the wave-like distribution in the heat propagation phenomenon. F.B. Pelap establishes a fractional-order earthquake model, which proves that the magnitude of an earthquake depends largely on the fractional derivative [25].

In recent years, some works can be seen in literature of fractional modeling of pneumatic system. In [26], the pneumatic transmission system is described by fractional mathematical model. The fractional-order dynamic model of flapper type pressure pneumatic system with bellows-nozzle-flapper configuration is established. Numerical simulations show the effective application of the theoretical results in a simple example. In [27], the fractional model of pneumatic muscle is established in robot system. In [28], J.L. Battaglia proposes that the air flow is fractional. On this basis, the fractional-order modeling and control method of single degree of freedom pneumatic system is studied in [29]. The performance of these systems was improved by fractional modeling, and the advantages of fractional modeling are obvious.

Many literatures show that the fractional-order PID method is an effective method in control of fractional-order modeling system [30 –36]. In recent years, some scholars use fractional-order PID controller to control 1DOF pneumatic system, the representative is H.P. Ren proposed an online multi-variable multi-objective genetic algorithm to solve the tuning of parameters in FPID controller of 1DOF pneumatic system, the results show the best performance compared to six other methods [29]. However, the fractional-order control method for 2DOF pneumatic system is rare. Similar studies include Sharma designed fractional-order fuzzy logic control technique for a two-link robot [37]. The parameters are optimized by the controller, and the simulation results prove that the fractional fuzzy logic controller outruns PID controllers in all aspects.

The degree of freedom of the controller refers to the number of closed-loop systems that can be modified independently, which is of great significance to the design of effective control strategies. The single degree of freedom controller has only one closed-loop, which makes the system unable to deal with multiple conflicts at the same time. V. Mohan presented 2DOF fractional-order fuzzy PID controller for the 2DOF system, which provided accurate tracking in the presence of noise and disturbance. Simulation results proved the superior performance of the controller compared to other controllers in facets of different operation conditions [38]. But in this work, the weight parameters of the input terms are ignored in the controller. This motivated us to design a new 2-DOF fractional-order fuzzy PID controller, which consider all the weight parameters to improve the accuracy of the controller design.

Fig.1

Structure of the pneumatic-hydraulic upper limb rehabilitation training system.

This project plans to carry out the research work of fractional modeling and control for the pneumatic-hydraulic upper limb rehabilitation training system. The key contributions of this work can be summarized as follows:

In this paper, by replacing the integral mass flow equation to fractional-order mass flow equation, the fractional-order mathematical model of pneumatic-hydraulic upper limb rehabilitation training system is established.

A new 2DOF fractional-order fuzzy PID controller is designed, to provides a new reference for improving the control accuracy of the pneumatic system. In the design of the controller, the weight parameters of the input terms are transformed into the weight parameters of the error, and the input, which are analyzed to improve the accuracy of the controller design.

The benefits of proposed controller are experimented by the pneumatic-hydraulic upper limb rehabilitation training system.

The rest of this paper is organized as follows: In Section 2, we provide the structure and dynamic model of the pneumatic-hydraulic upper limb rehabilitation training system. In Section 3, the research on posture control based on 2DOF fractional-order fuzzy PID algorithms are introduced. In Section 4, the multi-objective particle swarm optimization method is introduced to choose the parameters of the controller. In Section 5, the corresponding experiment illustrate the effectiveness of the method. Finally, we draw the conclusions in Section 6.

2 Pneumatic-hydraulic upper limb rehabilitation training system

2.1 Structure of system

The structure diagrams of the pneumatic-hydraulic upper limb rehabilitation training system is shown in Fig. 1. The system has two degrees of freedom, which both powered by a parallel pneumatic-hydraulic drive system.

The principle diagram of 1DOF pneumatic-hydraulic parallel drive system is shown in Fig. 2. Two cylinders and one hydraulic cylinder are connected in parallel to form the driving structure of the system. The cylinders are powered by air, and the hydraulic cylinder is filled with oil, which is a self-circulation closed system. A hydraulic valve is installed between the two chambers. The output damping of the hydraulic cylinder is controlled by adjusting the opening size of the hydraulic valve. Parallel displacement sensor on both sides of the cylinder are used to measure the cylinder elongation.

Fig.2

Principle diagram of 1DOF pneumatic-hydraulic parallel drive system.

2.2 Mathematical model of pneumatic-hydraulic driving system

The force balance equation of the single DOF pneumatic-hydraulic driving system can be expressed as the following: $M \frac{d^{2} x}{{dt}^{2}} = A_{1} p_{1} - A_{2} p_{2} - b_{L} \frac{dx}{dt} - K_{L} x - F_{L} - F_{h}$ (1) where M denotes the inertial mass of system, x is the position of the piston rod, A₁ (A₂) is effective section area of piston in the left (right) cavity of cylinder, b_L is the viscous damping coefficient, K_L is elastic load stiffness, F_L is the external force to the piston rod, F_h is liquid damping force, and $F_{h} = A_{3}^{2} \frac{\dot{x}}{K_{h}}$ (2) where A₃ is effective area of hydraulic cylinder piston, and $K_{h} = \frac{π d^{4}}{128 γ l}$ (3) in which d is opening area of hydraulic valve, γ is viscosity of oil, l denotes the length of hydraulic oil flowing through slender holes.

The pressure differential equation of the two chambers of the cylinder is ${\begin{matrix} \begin{matrix} c \\ t_{o} \end{matrix} D_{t}^{λ} p_{1} = \frac{k R T}{v_{1}} D_{t}^{λ} m_{1 i (0)} - \frac{k A_{1} p_{1}}{v_{1}} D_{t}^{λ} x \\ \begin{matrix} c \\ t_{o} \end{matrix} D_{t}^{λ} p_{2} = \frac{k R T}{v_{2}} D_{t}^{λ} m_{2 i (0)} + \frac{k A_{2} p_{2}}{v_{2}} D_{t}^{λ} x \end{matrix}$ (4) where t₀ and t are the start time and the end time of the integration, respectively. λ is the fractional-order, p₁ (p₂) is gas pressure of left (right) chamber of pneumatic cylinder, V₁ (V₂) is instantaneous gas volumes in the left (right) chamber of the cylinder, m_1i(o)(m_2i(o)) is mass flow rate of intake or exhaust in left (right) chamber of pneumatic cylinder, k is the entropic adiabatic index of gas, R is gas constant, T represents gas temperature, and $\begin{matrix} D_{t}^{λ} m_{1 (2) i} \\ = {\begin{matrix} p_{s} C_{1} \sqrt{1 - {(\frac{p_{r 1 (2) i} - b_{1}}{1 - b_{1}})}^{2}} (b_{1} ⩽ p_{r 1 (2) i} ⩽ 1) \\ p_{s} C_{1} (p_{r 1 (2) i} ⩽ b_{1}) \end{matrix} \end{matrix}$ $\begin{matrix} D_{t}^{λ} m_{1 (2) o} \\ = {\begin{matrix} p_{1 (2)} C_{2} \sqrt{1 - {(\frac{p_{r 1 (2) o} - b_{2}}{1 - b_{2}})}^{2}} (b_{2} ⩽ p_{r 1 (2) o} ⩽ 1) \\ p_{1 (2)} C_{2} (p_{r 1 (2) o} ⩽ b_{2}) \end{matrix} \end{matrix}$ where $p_{r 1 (2) i} = \frac{P_{1 (2)}}{p_{s}}$ , $p_{r 1 (2) o} = \frac{p_{o}}{p_{1 (2)}}$ , p_s and p_o are air source pressure and ambient gas pressure, respectively. b₁ and b₂ are critical pressure ratio of the left (right) chamber of the cylinder. C₁ and C₂ are sonic conductance in the left (right) chamber of the cylinder.

2.3 Dynamic modeling

The mechanical model of the simplified rehabilitation training system is shown in Fig. 3. It consists of two connecting links with length of L₁ and L₂. The mass centers m₁ and m₂ are located in the center and end of the links, respectively.

Fig.3

2-DOF simplified mechanical diagram of rehabilitation training system.

The Lagrange dynamic equation of the system is: $T = M (θ) \overset{+}{θ} H (θ, \dot{θ}) \dot{θ} + G (θ)$ (5) where $T = {[τ_{1} τ_{2}]}^{T}$ $M (θ) = [\begin{matrix} M_{11} (θ) & M_{12} (θ) \\ M_{21} (θ) & M_{22} (θ) \end{matrix}]$ $H (θ, \dot{θ}) = [\begin{matrix} H_{11} (θ, \dot{θ}) & H_{12} (θ, \dot{θ}) \\ H_{21} (θ, \dot{θ}) & H_{22} (θ, \dot{θ}) \end{matrix}]$ $G (θ) = [\begin{matrix} G_{11} (θ) \\ G_{21} (θ) \end{matrix}]$ $\begin{matrix} M_{11} (θ) = (m_{1} + m_{2}) L_{1 c}^{2} + m_{2} L_{2}^{2} + 2 m_{2} L_{1 c} L_{2} \cos θ_{2} \\ M_{12} (θ) = M_{21} (θ) = m_{2} L_{2}^{2} + m_{2} L_{1 c} L_{2} \cos θ_{2} \\ M_{22} (θ) = m_{2} L_{2}^{2} \\ H_{11} (θ, \dot{θ}) = - 2 m_{2} L_{1 c} L_{2} \sin θ_{2} {\dot{θ}}_{2} \\ H_{12} (θ, \dot{θ}) = - m_{2} L_{1 c} L_{2} \sin θ_{2} {\dot{θ}}_{2} \\ H_{21} (θ, \dot{θ}) = m_{2} L_{1 c} L_{2} \sin θ_{2} {\dot{θ}}_{1} \\ H_{22} (θ) = 0 \\ G_{11} (θ) = (m_{1} + m_{2}) L_{1 c} \sin θ_{1} + m_{2} {gL}_{2} \sin (θ_{1} + θ_{2}) \\ G_{21} (θ) = m_{2} {gL}_{2} \sin (θ_{1} + θ_{2}) \end{matrix}$ where T is the torque matrix, M(θ) is inertia matrix, $H (θ, \dot{θ})$ is Coriolis force and centripetal force matrix, G(θ) is gravity matrix. The values of system parameters measured are listed in Table 1.

Table 1

Description of parameters with their values

Description	Symbols	Value	SI units
Upper arm mass	m ₁	2.1	kg
Lower arm mass	m ₂	3.2	kg
Length of upper arm	L ₁	0.36	m
Length of lower arm	L ₂	0.31	m
Mass center distance	L _1c	–	m
Angle of upper arm	θ ₁	–	rad
Angle of lower arm	θ ₂	–	rad
Torque of upper arm	τ ₁	–	Nm
Torque of lower arm	τ ₂	–	Nm

3 Controller design

3.1 Basic of fractional-order calculus

In this work, the fractional differentiator and integrator are designed by binomially expanding backward difference transformation in discrete domain by operator $s = \frac{(1 - z^{- 1})}{T}$ (T represent sampling time) as:

$\begin{matrix} s^{\pm σ} = T^{\mp σ} \sum_{k = 0}^{\infty} {(- 1)}^{k} \\ \frac{\pm σ (\pm σ - 1) (\pm σ - 2) \dots (\pm σ - k + 1)}{k!} z^{- k} \end{matrix}$ (6)

Denoting discrete-time differentiator/integrator operator with D, and

$\begin{matrix} D^{\pm σ} (z^{- 1}) = T^{\mp σ} \sum_{k = 0}^{M} {(- 1)}^{k} \\ \frac{\pm σ (\pm σ - 1) (\pm σ - 2) \dots (\pm σ - k + 1)}{k!} \end{matrix}$ (7)

To design the feasible fractional-order controller, the Lubich formulation is defined as [27]:

$\begin{matrix} D^{\pm σ} g (nT) = T^{\mp σ} \sum_{k = 0}^{M} {(- 1)}^{k} \\ \frac{\pm σ (\pm σ - 1) (\pm σ - 2) \dots (\pm σ - k + 1)}{k!} g ((n - k) T) \end{matrix}$ (8)

where σ is the order of operator, T represent sampling time, M is number of delay elements which is chosen as 100.

3.2 Fractional-order PID control law

The general equation of classical integer order PID (IPID) controller in frequency domain is defined as follows: $U_{2 DOFPID} (s) = K_{P} (b * R - Y) + \frac{K_{I}}{s} (R - Y)$ $+ \frac{K_{D} N}{1 + \frac{N}{s}} (c * R - Y)$ (9) where K_P,K_I and K_D are proportional, integral and derivative gains, respectively. b and c are the weights of 2DOF controller, R is the reference signal, Y is the output of nonlinear system, N and s are filter coefficients and complex frequency, respectively.

In frequency domain, the non-integer order 2DOF fractional-order PID (FOPID) is expressed as: $U_{2 DOFFPID} (s) = K_{FP} (b * R - Y) + \frac{K_{FI}}{s^{λ}} (R - Y)$ $+ \frac{K_{FD} N}{1 + \frac{N}{s^{μ}}} (c * R - Y)$ (10) where λ > 0 and μ > 0 indicate the fractional orders of integral and derivative, respectively. K_FP,K_FI and K_FD are proportional, fractional integral and fractional derivative gains, respectively.

3.3 Fractional-order fuzzy PID controller design

3.3.1 Fractional-order fuzzy PID algorithm

Equation (10) can be changed as following: $U_{2 DOFFPID} (s) = U_{FPI} (s) + U_{FD} (s)$ (11) where ${\begin{matrix} U_{FPI} (s) = U_{FPI 1} (s) + U_{FPI 2} (s) \\ U_{FPI 1} (s) = (K_{FP} b + \frac{K_{FI}}{s^{λ}}) E (s) \\ U_{FPI 2} (s) = \frac{K_{FD} Nc}{1 + \frac{N}{s^{μ}}} E (s) \\ U_{FD} (s) = (K_{FP} (b - 1) + \frac{K_{FD} N}{1 + \frac{N}{s^{μ}}} (c - 1)) Y (s) \\ E (s) = R (s) - Y (s) \end{matrix}$

Using backward transformation $s = \frac{(1 - z^{- 1})}{T}$ and Equation (6)-Equation (8), Equation (11) can be written in discrete form as: $U_{2 DOFFPID} (z) = U_{FPI 1} (z) + U_{FPI 2} (z) + U_{FD} (z)$ (12) where $U_{FPI 1} (z) = (K_{FP} b + \frac{K_{FI} T^{λ}}{{(1 - z^{- 1})}^{λ}}) E (z)$ (13) $U_{FPI 2} (z) = \frac{K_{FD} Nc {(1 - z^{- 1})}^{μ}}{{(1 - z^{- 1})}^{μ} + {NT}^{μ}} E (z)$ (14)

$\begin{matrix} U_{FD} (z) \\ = (K_{FP} (b - 1) + \frac{K_{FD} N {(1 - z^{- 1})}^{μ}}{{(1 - z^{- 1})}^{μ} + N} (c - 1)) Y (z) \end{matrix}$ (15)

$Δ U_{FPI 1} (z) = U_{FPI 1} (z) \frac{{(1 - z^{- 1})}^{λ}}{T^{λ}}$ (16) $Δ U_{FPI 2} (z) = U_{FPI 2} (z) \frac{{(1 - z^{- 1})}^{μ}}{T^{μ}}$ (17)

$Δ U_{FPI 1} (z) = \frac{K_{FP} b}{T^{λ}} \sum_{k = 0}^{\infty} {(- 1)}^{k} b_{k} z^{- k} E (z) + K_{FI} E (z)$ (18)

$Δ U_{FPI 2} (z) = U_{FPI 2} (z) \frac{{(1 - z^{- 1})}^{μ}}{T^{μ}}$ (19) $Δ U_{FPI 1} (nT) = K_{FP} {bD}^{λ} e (nT) + K_{FI} e (nT)$ (20) $U_{FPI 1} (nT) = D^{- λ} (Δ U_{FPI 1} (nT))$ (21)

Similarly, solving Equation (14) and (15) for control laws give: $U_{FPI 2} (nT) = \frac{K_{FD} {NcD}^{μ}}{D^{μ} + N} e (nT)$ (22)

$U_{FD} (nT) = (K_{FP} (b - 1) + \frac{K_{FD} {ND}^{μ}}{D^{μ} + N} \cdot (c - 1)) y (nT)$ (23)

Then, the fractional-order fuzzy PID algorithm is expressing as:

$\begin{matrix} U_{2 DOFFPID} (nT) \\ = D^{- λ} (Δ U_{FPI 1} (nT)) + \frac{K_{FD} {NcD}^{μ}}{D^{μ} + N} e (nT) \\ + (K_{FP} (b - 1) + \frac{K_{FD} {ND}^{μ}}{D^{μ} + N} \cdot (c - 1)) y (nT) \end{matrix}$ (24)

The structure of 2DOF FOFPID controller for the system is shown in Fig. 4.

3.3.2 Control rule base and fuzzification

Fig.4

The structure of 2DOF FOFPID controller for the pneumatic hydraulic upper limb rehabilitation training system.

Figure 5 shows the definition of input and output membership functions in the universe of discourse. Membership functions namely ‘positive (p)’ and ‘negative(n)’ are considered for input variable e (nT) and r (nT). The output variable u are described by four membership functions, i.e. ‘negative large (nl)’, ‘negative small (ns)’, ‘positive small (ps)’ and ‘positive large (pl)’. Based on the input-output membership function, the following control rules are designed:

Rule1: If e (nT) = e . n and r (nT) = r . p then u = o . ns

Rule2: If e (nT) = e . n and r (nT) = r . n then u = o . nl

Rule3: If e (nT) = e . p and r (nT) = r . n then u = o . ps

Rule4: If e (nT) = e . p and r (nT) = r . p then u = o . pl

Fig.5

Input and output membership function.

3.3.3 Defuzzification

The center mass method is used in this work to defuzzify the fuzzy fractional PID controller, which expressed as [31]:

$u (nT) = \frac{\sum (input membership value * output membership value)}{\sum input membership value}$ (25)

The 20 input combination regions are divided by the input linguistic variables of e (nT) and r (nT) as shown in Fig. 6. The input error signal K_FIe (nT) and the fractional rate of error signal K_FPbr (nT) are plotted. The rules of FOFPID controller Rule1-Rule4 and input regions (IR) are used to evaluate appropriate fuzzy control law of each region.

In region IR2, both K_FIe (nT) and K_FPbr (nT) lie in the range [0, L], Therefore, the condition e . n<0.5, r . p>0.5 can be met for Fig. 5a. Now, Rule1 and the fuzzy logic collectively yield the following:

If e (nT) = e . n and r (nT) = r . p=min{e . n,r . p}=e . n

Therefor Rule1 yields

Rule1:If input = min{e . n,r . p}=e.n then u = o . ns

Similarly, in region IR2, Rule 2-Rule4 yield:

Fig.6

Input regions for FOFPID controller.

Rule2:If input = min{e . n,r . n}=e.n then u = o . nl

Rule3: If input = min{ e.p, r.n} = r.n then u = o . ps

Rule4: If input = min{ e.p, r.p} = r.p then u = o . pl

In region IR2, the output of the defuzzified FOFPID controller can be written by using Equation (25) as:

$u (nT) = \frac{\sum e . n * o . ns + e . n * o . nl + r . n * o . ps + r . p * o . pl}{e . n + e . n + r . n + r . p}$

The input membership functions are derived from the straight-line formula as follows: $e \cdot p = \frac{K_{FI} e (nT) + L}{2 L}, e \cdot n = \frac{- K_{FI} e (nT) + L}{2 L}$

$r \cdot p = \frac{K_{FP} {be}_{r} (nT) + L}{2 L}, r \cdot n = \frac{- K_{FP} {be}_{r} (nT) + L}{2 L}$

The output value of o . ns = - L/2, o . nl = - L,o . ps = L/2,o . pl = L.

The control law for FOFPID in IR2 region is expressed based on the values above as:

$u (nT) = \frac{3 {LK}_{FI} e (nT)}{4 [2 L - K_{FI} e (nT)]} + \frac{K_{FP} {bLe}_{r} (nT)}{4 [2 L - K_{FI} e (nT)]}$ (26)

The control law for FOFPID in region IR1 can be computed similarly, and is found to be same as Equation (26).

In regions IR5 and IR6, where e (nT)<0, the control law for FOFPID controller is expressed as:

$u (nT) = \frac{3 {LK}_{FI} e (nT)}{4 [2 L + K_{FI} e (nT)]} + \frac{K_{FP} {bLe}_{r} (nT)}{4 [2 L + K_{FI} e (nT)]}$ (27)

Combined formula (26) and (27), the control law for regions IR1, IR2, IR5 and IR6 is as follows:

$\begin{matrix} u (nT) = & \frac{3 {LK}_{FI}}{4 [2 L - K_{FI} | e (nT) |]} e (nT) \\ + \frac{K_{FP} bL}{4 [2 L - K_{FI} | e (nT) |]} e_{r} (nT) \end{matrix}$ (28)

Therefore, the fuzzy control rate in (24) is summarized as:

\begin{matrix} Δ u_{FPI 1} (nT) & = \frac{L [3 k_{FI} e (nT) + K_{FP} {be}_{r} (nT)]}{4 [2 L - K_{I} | e (nT) |]} & for IR 1, 2, 5, 6 \\ = \frac{L [3 k_{FI} e (nT) + K_{FP} {be}_{r} (nT)]}{4 [2 L - K_{FP} b | e_{r} (nT) |]} & for IR 3, 4, 7, 8 \\ = [3 L + K_{FP} {be}_{r} (nT)] / 4 & for IR 9, 10 \\ = [3 L + K_{FI} e_{r} (nT)] / 4 & for IR 11, 12 \\ = [- 3 L + K_{FP} {be}_{r} (nT)] / 4 & for IR 13, 14 \\ = [- 3 L + K_{FI} e (nT)] / 4 & for IR 15, 16 \\ = L & for IR 17 \\ = - L / 2 & for IR 18 \\ = - L & for IR 19 \\ = L / 2 & for IR 20 \end{matrix}}

(29)

Fig.7

Photo of the pneumatic hydraulic upper limb rehabilitation training system.

Fig.8

Experimental results of upper arm (a) Step signal of 10° (b) Step signal of 30° (c) Sinusoidal signal of 0.2 Hz (d) Sinusoidal signal of 0.5 Hz.

Fig.9

Experimental results of lower arm. (a) Step signal of 20° (b) Step signal of 40° (c) Sinusoidal signal of 0.2 Hz (d) Sinusoidal signal of 0.5 Hz.

4 Multi-objective particle swarm optimization

The control performance effected directly by the parameters of the controller. It is difficult to adjust the parameters of the fractional-order PID controller because of two more parameters than PID controller. To solve this problem, particle swarm optimization (PSO) is used to optimize the parameters of the controller in this paper.

The idea of PSO algorithm is described mathematically as follows [36]:

In the search space D, each individual is a group of M particles moving at a certain speed. Based on the individual and group moving experience, each particle dynamically adjusting the parameters in the searching process. Hence, the position X_i of the ith particle is determined by (X_i1, X_i2,⋯,X_id). The velocity V_i of the ith particle is composed of (V_i1, V_i2,⋯,V_id) vectors. The optimal value P_i of the ith particle is determined by (P_i1, P_i2,⋯,P_id). The optimal group value P_g is composed of (P_g1, P_g2,⋯,P_gd). Generally, the position and velocity of each particle are taken in the continuous real number space.

The positions and velocities of the particles are as follows: $V_{id}^{k + 1} = V_{id}^{k} + c_{1} r_{1} (P_{id}^{k} - X_{id}^{k}) + c_{2} r_{2} (P_{gd}^{k} - X_{id}^{k})$ $X_{id}^{k + 1} = X_{id}^{k} + V_{id}^{k + 1}$ where 1 ⩽ d ⩽ D, 1 ⩽ i ⩽ M . c₁ and c₂ are cognitive and social term coefficients. r₁ and r₂ are random uniform numbers in the range of [0,1]. $V_{id}^{k}$ and $X_{id}^{k}$ represent the speed and position of the ith particle, which searching k times in group from the d-dimensional space, and update within [- V_max,V_max] and [- X_max,X_max], respectively(V_max and X_max are the maximum speed and position of particles). $V_{id}^{k + 1}$ and $X_{id}^{k + 1}$ represent the speed and position of the ith particle, which searching k + 1 times in group from the D-dimensional space. After searching, the particle position changes to $X_{id}^{k + 1}$ , and the speed changes to $V_{id}^{k + 1}$ . $P_{id}^{k}$ and $P_{gd}^{k}$ are local and global optimal values found by theith particle searching k times in d-dimensional space.

Table 2
Parameter values of the controllers

Parameters Upper arm Parameters Lower arm

PID FPID FOFPID1 FOFPID2 PID FPID FOFPID1 FOFPID2

K_FP1[0,800] 789.65 664.35 197.68 479.76 K_FP2[0,800] 684.23 555.02 103.76 402.12

K_FI1[0,800] 623.76 487.23 342.87 248.70 K_FI2[0,800] 465.13 327.35 241.68 279.47

K_FD1[0,800] 27.86 20.32 32.03 38.98 K_FD2[0,800] 24.69 21.94 17.46 29.67

μ₁[0,1] 1 1 0.985 0.982 μ₂[0,1] 1 1 0.979 0.932

λ₁[0,1] 1 1 0.991 0.941 λ₂[0,1] 1 1 0.989 0.978

b ₁ – – 0.802 0.745 b ₂ – – 0.593 0.587

c ₁ – – – 0.346 c ₂ – – – 0.269

Parameters	Upper arm	Parameters	Lower arm
K_FP1[0,800]	789.65	664.35	197.68	479.76	K_FP2[0,800]	684.23	555.02	103.76	402.12
K_FI1[0,800]	623.76	487.23	342.87	248.70	K_FI2[0,800]	465.13	327.35	241.68	279.47
K_FD1[0,800]	27.86	20.32	32.03	38.98	K_FD2[0,800]	24.69	21.94	17.46	29.67
μ₁[0,1]	1	1	0.985	0.982	μ₂[0,1]	1	1	0.979	0.932
λ₁[0,1]	1	1	0.991	0.941	λ₂[0,1]	1	1	0.989	0.978
b ₁	–	–	0.802	0.745	b ₂	–	–	0.593	0.587
c ₁	–	–	–	0.346	c ₂	–	–	–	0.269

Table 3

Overshoot table of step signal tracking

Overshoot	Upper arm				Overshoot	Lower arm
	PID	FPID	FOFPID1	FOFPID2		PID	FPID	FOFPID1	FOFPID2
10°	21.4%	15.2%	14%	4.1%	20°	24.1%	18.7%	16.3%	9.5%
30°	11%	8.33%	6.8%	2.4%	40°	16.3%	13.5%	12.5%	8.8%

Table 4

Peak time table of step signal tracking

Peak time(s)	Upper arm				Peak time(s)	Lower arm
	PID	FPID	FFOPID1	FOFPID2		PID	FPID	FFOPID1	FOFPID2
10°	0.85	0.82	0.8	0.75	20°	1.31	1.25	1.24	1.2
30°	1.48	1.46	1.44	1.41	40°	1.9	1.86	1.84	1.82

Table 5

Lag time table of sinusoidal signal tracking

Delay time(s)	Upper arm				Delay	Lower arm
	PID	FPID	FFOPID1	FOFPID2	time(s)	PID	FPID	FFOPID1	FOFPID2
0.2Hz	0.026	0.02	0.019	0.018	0.2Hz	0.055	0.048	0.046	0.043
0.5Hz	0.56	0.52	0.5	0.42	0.5Hz	0.61	0.56	0.5	0.47

Table 6

Error table of sinusoidal signal tracking

Tracking error	Upper arm				Tracking	Lower arm
	PID	FPID	FFOPID1	FOFPID2	error	PID	FPID	FFOPID1	FOFPID2
0.2Hz	0.18	0.15	0.14	0.11	0.2Hz	0.2	0.13	0.12	0.1
0.5Hz	2.2	2.05	2.01	1.9	0.5Hz	1.6	1.45	0.15	0.12

5 Experiment

5.1 Experimental Platform

The experiment setup is shown in Fig. 7. The system includes two pneumatic cylinders with 20 mm diameter bore and 300 mm stroke, two hydraulic cylinders with 20 mm diameter, four high speed on/off pneumatic valves, two proportional hydraulic valves, two potentiometers, and an air pump. The nominal operating pressure of the valve is 4.5 bar, the maximum flow rate of the valve under the nominal pressure is 650 L/min. The resolution of the potentiometer is less than 0.01 mm, and the linear error of the potentiometer is less than 0.09% of the whole range. The control voltage of valve is 15 V. The output of the potentiometer is also 15 V.

The controller algorithm is developed in Qt software. The user interface is designed to display the angular displacement curve, error curve, reference signal and parameter setting, etc. The sampling period of the controller is 10 ms. The piston of the cylinder is forced to reach the end point of the champers after each experiment. In this paper, through the mathematical conversion, the linear displacement is converted into the angle experimental results.

5.2 Experiment results

Figure 8 and Figure 9 show the experimental results of the upper and lower arms tracking step and sine signals. Each picture includes four tracking curves with PID, FPID, FOFPID1 and FOFPID2 controller. Here, FOFPID1 and FOFPID2 represent the fractional-order fuzzy PID controller without considering parameter c and the fractional-order fuzzy PID controller with considering parameter c. The tuned parameter values of four controllers are listed in Table 2. According to the experimental results, the tracking performance of the system under different control strategies are shown in Tables 3–6.

Angular tracking curves of 10°and 30°step signal of upper arm with four different controllers are shown in Fig. 8(a) and Fig. 8(b). As can be seen from the figures, compared with the other three controllers, the overshoot of the FOFPID controller reduced by maximum about 150–400%. And the peak time reduced by maximum about 0.12 s.

Angular tracking curves of 0.2 Hz and 0.5 Hz sinusoidal signal of upper arm with four different controllers are shown in Fig. 8(c) and Fig. 8(d). As can be seen from the Table 3 and Table 4, compared with the other three controllers, the delay time of the FOFPID controller reduced by maximum about 350%. And the tracking accuracy is improved maximum about 0.14 s.

The similar conclusion can be drawn from the results of lower arm experiments(see Table 5 and Table 6).

From the experiment results, we draw the conclusion that the system get better performance with the 2DOF fractional-order fuzzy PID controller.

6 Conclusions

This paper proposed a new modeling method to pneumatic-hydraulic upper limb rehabilitation training system with fractional-order theory. A new 2DOF fractional-order fuzzy PID controller is designed, which full considering the weight parameters of the input terms. The experimental results showed that the designed method is more effective compared with other controllers.

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