Adaptive robust trajectory tracking control of a parallel manipulator driven by pneumatic cylinders

Abstract

Due to the compressibility of air, non-linear characteristics, and parameter uncertainties of pneumatic elements, the position control of a pneumatic cylinder or parallel platform is still very difficult while comparing with the systems driven by electric or hydraulic power. In this article, based on the basic dynamic model and descriptions of thermal processes, a controller integrated with online parameter estimation is proposed to improve the performance of a pneumatic cylinder controlled by a proportional valve. The trajectory tracking error is significantly decreased by applying this method. Moreover, the algorithm is expanded to the problem of posture trajectory tracking for the three-revolute prismatic spherical pneumatic parallel manipulator. Lyapunov’s method is used to give the proof of stability of the controller. Using NI-CompactRio, NI-PXI, and Veristand platform as the realistic controller hardware and data interactive environment, the adaptive robust control algorithm is applied to the physical system successfully. Experimental results and data analysis showed that the posture error of the platform could be about 0.5%–0.7% of the desired trajectory amplitude. By integrating this method to the mechatronic system, the pneumatic servo solutions can be much more competitive in the industrial market of position and posture control.

Keywords

Pneumatic servo control parallel mechanism adaptive robust control trajectory tracking non-linear analysis online parameter estimation

Introduction

It is well known that there are three major driving types of position servo controls, which are electric, hydraulic, and pneumatic. Due to the linear characteristics and stiffness advantage, higher precision can be obtained using electric or hydraulic cylinders and valves when comparing with the pneumatic ones. But since the compressed air is a low-cost and clean energy, it is still attracting researchers to study on pneumatic position servo control and improve the precision. Compared with feedback linearization,^1,2 robust H-∞ control,³ self-tuning control,⁴ sliding mode control (SMC),⁵ and fuzzy control,⁶ M Smaoui et al.⁷ proposed a robust controller based on back-stepping method and obtained better performance.

With the developments of parallel mechanism theories, the parallel platforms driven by electric motors or hydraulic cylinders have been successfully industrialized, especially in the fields of motion simulation and robotic applications, such as 6-degrees of freedom (DOFs) Stewart platform and Delta robot. Some of these platforms were driven by pneumatic elements. K Grewal et al.⁸ established a pneumatic parallel system used linear quadratic Gaussian (LQG) controller. The control schemes experimental results showed LQG giving slightly better performance.⁹ M Ramsauer et al.¹⁰ proposed a pneumatically driven Stewart platform used as fault detection device. J Pradipta et al.¹¹ used immersion and invariance method with an additional friction compensator applied for motion control. Generally, the control errors were no smaller than 5%. B Andrievsky et al.¹² described the design features of pneumatic platform for driving simulator while using SMC and switching valves. Simulation results of SMC was about 5% and the experimental results of on-off logic control was about 20%. However, for the most ordinary pneumatic cylinders with typical friction properties and their applications, which are used most widely in the industrial systems, some studies still have to be done about the servo control algorithm.

Meanwhile, some insufficient DOF systems which have the freedoms of roll, pitch, and heave have been used in the motion simulators and virtual reality equipment industry. One of them is three-revolute prismatic spherical (3-RPS) manipulator and lots of extensive studies have been done in the past years.^13–15 For the dynamic modeling and control methods studies, G Pfreundschuh et al.¹⁶ presented the mechanics, design, proportional-derivative (PD) control, and experiment results for a 3-RPS platform. YM Cheng and YS Chen¹⁷ investigated an angle trajectory tracking of a 3-DOF pneumatic motion platform, and fuzzy system was used in the trajectory pre-compensation. The angle trajectory following error was about 10%. However, the $5 % - 10 %$ relative position/angle tracking error for a manipulator driven by pneumatic cylinders makes it impossible to compete with hydraulic or electric ones.

Adaptive robust control algorithm was first proposed in 1990s by B Yao and M Tomizuka.^18,19 Then, this theory has been fully developed by the presented indirect adaptive robust control (ARC),²⁰ integrated direct/indirect ARC,²¹ and synthesis-based ARC.²² The applications of these algorithms are quite successful on the studies about linear motor high accuracy control,^23,24 hydraulic systems,²⁵ and vehicle active suspension systems.^26,27 In the field of pneumatic servo control system, X Zhu et al.^28–30 and G Tao and H Zuo³¹ used pneumatic muscle to obtain a much higher performance by applying ARC. The modeling research of pneumatic muscle also attracts some researchers to establish a more accurate model of this element.³² However, the pneumatic muscle can only bear one-directional tensile force, which means it cannot be used in heavy load conditions under compression forces. Recently, D Meng et al.³³ proposed an adaptive robust method to control a low-friction pneumatic cylinder with 25 mm diameter and 1/8′ proportional valve, which got an adequate performance. Indirect ARC can improve the parameter estimation accuracy and direct/indirect ARC can also improve the transient performance with the fast switching output feature for proportional valves with dead zones. However, for the pneumatic manipulator used as the motion simulator, the vibration caused by the switching characteristic might be a disadvantage. So, to overcome the strong non-linearities and parameters uncertainties of pneumatic manipulator driven by cylinders and inhibit the probabilities, indirect adaptive robust control should be an effective approach in this case.

In this article, an adaptive robust controller is proposed to achieve much higher precision based on the back-stepping method. First, the modeling of the pneumatic servo system of single cylinder is introduced. Then, online parameter estimation methods for single cylinder system are discussed in the next section, as well as the proof of stability. In section “Parallel system structure and analysis,” the kinematic analysis of the 3-RPS mechanism is demonstrated and the controller is expanded to the conditions of parallel trajectory control. Experimental results on different conditions are shown in sections “Single cylinder control experiments”, “3-RPS manipulator posture control experiments,” and “Performance analysis and conclusion.”

Modeling of single cylinder system

The single cylinder servo system considered in this article is shown in Figure 1. A cylinder attached with a position sensor is connected to a proportional valve. The controller hardware is based on NI-cRIO system. In order to simplify the descriptions of modeling and controller design, some of the key symbols are listed in Table 1.

Figure 1.

Schematic diagram and physical picture of single cylinder position control system.

Table 1.

Key symbols used in the model and controller.

Symbol	Description
M	Mass of the piston rod and load
$A_{a}$ , $A_{b}$	Piston area of each chamber
$V_{a}$ , $V_{b}$	Volume of each chamber
$V_{0 a}$ , $V_{0 b}$	Dead volume of each chamber
$T_{a}$ , $T_{b}$	Air temperature of each chamber
$p_{a}$ , $p_{b}$	Air pressure of each chamber
$k_{a}$	Area fixing ratio for one-side rod cylinder
$f_{n}, {\tilde{f}}_{0}$	Modeling force error and disturbance
$d_{n}, {\tilde{d}}_{0}$	Modeling flow rate error and disturbance
$A_{f}$	Amplitude of friction model
$S_{f} (\overset{\cdot}{x})$	Stribeck curve
$F_{L}$	Load force applied on the piston
b	Damping coefficient
${\overset{\cdot}{m}}_{iin}$ , ${\overset{\cdot}{m}}_{iout}$	Input, output mass flow rate
${\overset{\cdot}{Q}}_{i}$	Heat exchange rate of each chamber
$θ$	Unknown parameters factor
$α_{i}, ν_{i}$	Fading and normalized factor
$τ_{i}$	Adaptive function
$[z_{1}, z_{2}, z_{3}]^{T}$	Error vector
$p_{e}$	First-level virtual input
$p_{ed}$	Expected first-level virtual input
$q_{e}$	Second-level virtual input
$q_{ed}$	Expected second-level virtual input

According to the coordinate established in Figure 2, equation (1) can be used to describe the piston motion and Stribeck friction model

\begin{array}{l} M \overset{\cdot\cdot}{x} = (p_{a} - k_{a} p_{b}) A_{a} - F_{f} - F_{L} + f_{n} + {\tilde{f}}_{0} \\ F_{f} = A_{f} S_{f} (\dot{x}) + b \dot{x} \\ k_{a} = A_{b} / A_{a} \end{array}

(1)

Figure 2.

3-RPS parallel platform structure.

According to Carneiro and De Almeida’s³⁴ and Meng et al.’s³⁵ research results, considering the thermodynamic process in the chamber as polytropic process is more accurate

{\begin{matrix} T_{i} = T_{s} {(\frac{p_{i}}{p_{b a l}})}^{\frac{n - 1}{n}}, p_{b a l} = 0.8077 p_{s} \\ \frac{d p_{i}}{d t} = \frac{γ R}{V_{i}} ({\dot{m}}_{i i n} T_{s} - {\dot{m}}_{i o u t} T_{i}) - i = a, b \\ \frac{γ p_{i}}{V_{i}} \frac{d V_{i}}{d t} + \frac{γ - 1}{V_{i}} {\dot{Q}}_{i} + d_{i n} + {\tilde{d}}_{i 0} \end{matrix}

(2)

where heat exchange rate ${\overset{\cdot}{Q}}_{i} = h S_{hi} (x) (T_{s} - T_{i})$ , heat exchange area $S_{hi} (x) = 2 A_{i} + π D (L / 2 + x)$ , and chambers volumes $V_{i} = V_{0 i} + A_{i} (L / 2 + x)$ . h is the thermal conductivity of air and inner surface of cylinder chamber, $S_{hi} (x)$ is the area of heat conduction, and $p_{s}$ is the air source pressure. R is the ideal gas constant and $λ$ is linear flow ratio that has the constant value 1.4. D is the cylinder chambers’ diameter and L is the stroke. $T_{s}$ is the air source temperature. Since the natural frequency of MPYE valve is much higher than the bandwidth of pneumatic servo system, the dynamic features of the valve core can be ignored. The valve model in equation (3) is established to calculate the mass flow under different input conditions

\begin{matrix} q_{m} = {\begin{matrix} A (u) C_{d} C_{a} P_{u} / \sqrt{T_{u}} P_{d} / P_{u} \leq p_{r} \\ A (u) C_{d} C_{a} C_{b} P_{u} / \sqrt{T_{u}} p_{r} < P_{d} / P_{u} < λ \\ A (u) C_{d} C_{a} C_{c} P_{u} / \sqrt{T_{u}} λ \leq P_{d} / P_{u} \leq 1 \end{matrix} \\ C_{a} = \sqrt{(γ / R) {(2 / (γ + 1))}^{(γ + 1) / (γ - 1)}} \\ C_{b} = \sqrt{1 - {(P_{d} / P_{u} - p_{r})}^{2} / {(1 - p_{r})}^{2}} \\ C_{c} = \frac{(P_{d} / P_{u} - p_{r})}{(1 - λ)} \sqrt{1 - {[\frac{(λ - p_{r})}{(1 - p_{r})}]}^{2}} \\ C_{d} = 1.099 - 0.1075 P_{d} / P_{u} \end{matrix}

(3)

where $P_{d}$ and $P_{u}$ are downstream pressure and upstream pressure, respectively, $q_{m}$ is the calculated mass flow rate, $p_{r}$ is critical pressure ratio, and $T_{u}$ is the upstream air temperature. The whole model of a single cylinder controlled by proportional valve can be written in the form of equation (4)

{\begin{matrix} {\overset{\cdot}{x}}_{1} = x_{2} \\ M {\overset{\cdot}{x}}_{2} = {\bar{A}}_{a} (x_{3} - k_{a} x_{4}) - θ_{1} x_{2} - θ_{2} S_{f} (x_{2}) \\ + θ_{3} + {\tilde{f}}_{0} \\ {\overset{\cdot}{x}}_{3} = \frac{γ R}{H_{p} V_{a}} ({\overset{\cdot}{m}}_{ain} T_{s} - {\overset{\cdot}{m}}_{aout} T_{a}) - \frac{γ A_{a}}{V_{a}} x_{2} x_{3} \\ + \frac{γ - 1}{H_{p} V_{a}} {\overset{\cdot}{Q}}_{a} + θ_{4} + {\tilde{d}}_{a 0} \\ {\overset{\cdot}{x}}_{4} = \frac{γ R}{H_{p} V_{b}} ({\overset{\cdot}{m}}_{bin} T_{s} - {\overset{\cdot}{m}}_{bout} T_{b}) - \frac{γ A_{b}}{V_{b}} x_{2} x_{4} \\ + \frac{γ - 1}{H_{p} V_{b}} {\overset{\cdot}{Q}}_{b} + θ_{5} + {\tilde{d}}_{b 0} \end{matrix}

(4)

The status variables vector X is given by

\begin{matrix} X & = {[x_{1}, x_{2}, x_{3}, x_{4}]}^{T} = {[x, \overset{\cdot}{x}, p_{a} / H_{p}, p_{b} / H_{p}]}^{T} \\ {\bar{A}}_{a} & = A_{a} H_{p} \\ θ & = {[θ_{1}, θ_{2}, θ_{3}, θ_{4}, θ_{5}]}^{T} \\ = {[b, A_{f}, - F_{L} + f_{n}, d_{an}, d_{bn}]}^{T} \end{matrix}

(5)

where $H_{p} = 10^{5}$ is used to convert the unit of pressure into 100 kPa. It is helpful in the process of controller debugging and adjusting.

Controller design of single cylinder system

Online parameter estimation

Based on the research of online parameter estimation and back-stepping design,^19,36 The adaptive law of parameters is given by

\overset{\cdot}{\hat{θ}} = sa t_{{\overset{\cdot}{θ}}_{M}} (P r_{\hat{θ}} (Ψ τ)) \hat{θ} (0) \in B_{θ}

(6)

where $B_{θ}$ is the bound of the parameters vector $θ$ . $P r_{\hat{θ}}$ is the standard projection mapping. $Ψ$ is the matrix of adaptive law and $τ$ is the adaptive function. This projection mapping has been described in Goodwin and Mayne’s³⁷ and Yao and Tomizuka’s¹⁹ works. $sa t_{{\overset{\cdot}{θ}}_{M}} (★)$ is the saturation function, which is used to limit the updating speed of the parameters. The saturation function is defined as

sa t_{{\overset{\cdot}{\hat{θ}}}_{M}} (★) = s_{a} ★, s_{a} = {\begin{matrix} 1 ‖ ★ ‖ \leq {\overset{\cdot}{\hat{θ}}}_{M} \\ {\overset{\cdot}{\hat{θ}}}_{M} / ‖ ★ ‖ ‖ ★ ‖ > {\overset{\cdot}{\hat{θ}}}_{M} \end{matrix}

(7)

where ${\overset{\cdot}{\hat{θ}}}_{M}$ is the maximum parameter updating speed determined in advance. According to the definitions above, the adaptive law has the properties below

{\begin{matrix} \hat{θ} (t) \in B_{θ} = {\hat{θ} : θ_{\min} \leq \hat{θ} \leq θ_{\max}} \\ \forall t \\ {[\hat{θ} - θ]}^{T} [Ψ^{- 1} P r_{\hat{θ}} (Ψ τ) - τ] \leq 0 \\ \forall τ \\ | \overset{\cdot}{\hat{θ}} (t) | \leq {\overset{\cdot}{θ}}_{M} \\ \forall t \end{matrix}

(8)

which means no matter how the adaptive function is designed, the estimated parameters vector will always be inside the known closed set $B_{θ}$ and the update rate of estimated parameters are all limited.

The matrix of adaptive law $Ψ$ is described as follows

\begin{matrix} {\overset{\cdot}{Ψ}}_{i} = {\begin{matrix} α_{i} Ψ_{i} - \frac{Ψ_{i} φ_{if} φ_{if}^{T} Ψ_{i}}{1 + ν_{i} φ_{if}^{T} Ψ_{i} φ_{if}} & \begin{matrix} when λ_{\max} (Ψ_{i} (t)) \leq ρ_{Mi} \\ and ‖ P r_{\hat{θ}} (Ψ_{i} τ_{i}) ‖ \leq {\hat{θ}}_{Mi} \end{matrix} \\ 0 & otherwise \end{matrix} \\ τ_{i} = \frac{1}{1 + ν_{i} φ_{if}^{T} Γ_{i} φ_{if}} φ_{if} ε_{i} \end{matrix}

(9)

where $ρ_{Mi}$ is the upper limit of $‖ Ψ_{i} (t) ‖$ , while $λ_{\max} (Ψ_{i} (t))$ is the maximum eigenvalue of $Ψ_{i} (t)$ . $α_{i}$ is the fading factor and $ν_{i}$ is the normalized factor. Obviously, it is ensured that $Ψ_{i} (t) \leq ρ_{Mi} I, \forall t$ . According to the model established in equation (4), the last three equations can be rewritten in the linear regression form and then multiplied by $G_{f}$ as shown in equation (10)

{\begin{matrix} \begin{matrix} y_{1 f} = G_{f} [{\bar{A}}_{a} (x_{3} - k_{a} x_{4}) - M {\overset{\cdot}{x}}_{2}] \\ = θ_{1} x_{2 f} + θ_{2} S_{f} (x_{2}) - θ_{3} 1_{f} \end{matrix} \\ \begin{matrix} y_{2 f} = G_{f} [γ R ({\overset{\cdot}{m}}_{ain} T_{s} - {\overset{\cdot}{m}}_{aout} T_{a}) / (H_{p} V_{a}) \\ - γ A_{a} x_{2} x_{3} / V_{a} + (γ - 1) {\overset{\cdot}{Q}}_{a} / (H_{p} V_{a}) - {\overset{\cdot}{x}}_{3}] \\ = - θ_{4} 1_{f} \end{matrix} \\ \begin{matrix} y_{3 f} = G_{f} [γ R ({\overset{\cdot}{m}}_{bin} T_{s} - {\overset{\cdot}{m}}_{bout} T_{b}) / (H_{p} V_{b}) \\ - γ A_{b} x_{2} x_{4} / V_{b} + (γ - 1) {\overset{\cdot}{Q}}_{b} / (H_{p} V_{b}) - {\overset{\cdot}{x}}_{4}] \\ = - θ_{5} 1_{f} \end{matrix} \end{matrix}

(10)

where $G_{f}$ is a stable linear time-invariant (LTI) filter that has the form

G_{f} = \frac{ω_{f}^{2}}{(τ_{f} s + 1) (s^{2} + 2 ξ ω_{f} s + ω_{f}^{2})}

(11)

where $ω_{f}$ , $τ_{f}$ , and $ξ$ are all constants. Obviously, $G_{f}$ is a third-order transfer function.

Defining $y_{if} = φ_{if}^{T} θ_{is}$ , where $θ_{1 s} = [θ_{1}, θ_{2}, θ_{3}]^{T}$ , $θ_{2 s} = [θ_{4}]^{T}$ , and $θ_{3 s} = [θ_{5}]^{T}$ , the regression vectors are $φ_{1 f}^{T} = [x_{2 f}, S_{ff} (x_{2}), - 1_{f}]$ , $φ_{2 f}^{T} = [- 1_{f}]$ , and $φ_{3 f}^{T} = [- 1_{f}]$ . The error of estimation can be written in the form of

ε_{i} = {\hat{y}}_{if} - y_{if} = φ_{if}^{T} {\hat{θ}}_{is} - φ_{if}^{T} θ_{is} = φ_{if}^{T} {\tilde{θ}}_{is}

(12)

The equation above is the standard parameter estimation model. Least square method (LSM) can be used to obtain the estimated vector $\hat{θ}$ .

Adaptive robust controller design

According to the system model built in equation (4), the elements of the error vector is defined as $[z_{1}, z_{2}, z_{3}]^{T}$ and expected first-level input $p_{e}$ is shown in equation (13)

\begin{matrix} z_{1} = x_{1} - x_{d} z_{2} = {\overset{\cdot}{z}}_{1} + k_{1} z_{1} \\ p_{e} = x_{3} - k_{a} x_{4} z_{3} = p_{e} - p_{ed} \\ p_{ed} = p_{eda} + p_{eds 1} + p_{eds 2} \\ p_{eda} = [{\hat{θ}}_{1} x_{2} + {\hat{θ}}_{2} S_{f} (x_{2}) - {\hat{θ}}_{3} \\ + M ({\overset{\cdot\cdot}{x}}_{1 d} - k_{1} z_{1})] / {\bar{A}}_{a} \\ p_{eds 1} = - k_{2} z_{2} / {\bar{A}}_{a} p_{eds 2} = - h_{2}^{2} z_{2} / (4 η_{2} {\bar{A}}_{a}) \end{matrix}

(13)

where $k_{1}$ is defined as the feedback gain of the tracking position error $z_{1}$ .

To calculate the expected pressure input, $p_{ed}$ is separated into three parts as shown in equation (13). $p_{eda}$ is the model compensation part, $p_{eds 1}$ is the proportional feedback part of $z_{2}$ and $p_{eds 2}$ is used as the robust feedback part to inhibit the affection of model uncertainties. $k_{2}$ is defined as the feedback gain of $z_{2}$ . $h_{2}, η_{2}$ are used to control the value of robust feedback part $p_{eds 2}$ .

To choose property values of $h_{2}$ and $θ_{iM}$ , they must satisfy the conditions shown in equation (14)

\begin{matrix} h_{2} \geq | θ_{M 1} | | x_{2} | + | θ_{M 2} | | S_{f} (x_{2}) | + | θ_{M 3} | + f_{max} \\ θ_{Mi} = θ_{i max} - θ_{i min} \end{matrix}

(14)

To prove the stability of controller, a semi-positive definite function is defined as equation (15)

V_{2} = \frac{1}{2} {Mz}_{2}^{2}

(15)

The derivation of $V_{2}$ can be expressed as equation (16) according to the definitions in equation (13)

\begin{matrix} {\overset{\cdot}{V}}_{2} = {\bar{A}}_{a} z_{2} z_{3} - k_{2} z_{2}^{2} + z_{2} [{\bar{A}}_{a} p_{eds 2} + {\tilde{θ}}_{1} x_{2} + \\ {\tilde{θ}}_{2} S_{f} (x_{2}) - {\tilde{θ}}_{3} + {\tilde{f}}_{0}] \end{matrix}

(16)

It is not difficult to prove that $p_{eds 2}$ satisfies the conditions in equation (17)

\begin{matrix} z_{2} [{\tilde{θ}}_{1} x_{2} + {\tilde{θ}}_{2} S_{f} (x_{2}) - {\tilde{θ}}_{3} + {\tilde{f}}_{0}] \leq η_{2} \\ z_{2} {\bar{A}}_{a} p_{eds 2} \leq 0 \\ {\overset{\cdot}{V}}_{2} \leq {\bar{A}}_{a} z_{2} z_{3} - k_{2} z_{2}^{2} + η_{2} \end{matrix}

(17)

The equations above indicate that when $z_{3} = 0$ , $z_{2}$ will converge exponentially to a spherical domain. The size of this domain can be controlled by $k_{2}$ and $η_{2}$ . In order to satisfy this requirement, the derivation of $z_{3}$ has to be calculated

\begin{array}{l} {\dot{z}}_{3} = q_{e} - (\frac{γ A_{a}}{V_{a}} x_{2} x_{3} + \frac{γ A_{b}}{V_{b}} x_{2} x_{4} k_{a}) + \frac{γ - 1}{H_{p} V_{a}} {\dot{Q}}_{a} \\ - \frac{γ - 1}{H_{p} V_{b}} {\dot{Q}}_{b} k_{a} + θ_{4} - θ_{5} k_{a} + {\tilde{d}}_{a 0} - {\tilde{d}}_{b 0} k_{a} - {\dot{p}}_{e d c} - {\dot{p}}_{e d u} \\ q_{e} = \frac{γ R}{H_{p} V_{b}} ({\dot{m}}_{b i n} T_{s} - {\dot{m}}_{b o u t} T_{b}) - \frac{γ R}{H_{p} V_{a}} ({\dot{m}}_{a i n} T_{s} - {\dot{m}}_{a o u t} T_{a}) \end{array}

(18)

where $k_{3}$ is defined as the feedback gain of pressure level error $z_{3}$ . To calculate the expected mass flow, $q_{ed}$ is separated into three parts as shown in equation (19). $q_{eda}$ is the model compensation part, $q_{eds 1}$ is the proportional feedback part of $z_{3}$ , and $q_{eds 2}$ is used as the robust feedback part to inhibit the affection of model uncertainties. $h_{3}, η_{3}$ are used to control the value of robust feedback part $q_{eds 2}$

\begin{matrix} q_{ed} = q_{eda} + q_{eds 1} + q_{eds 2} \\ q_{eda} = - {\bar{A}}_{a} z_{2} + (\frac{γ A_{a}}{V_{a}} x_{2} x_{3} + \frac{γ A_{b}}{V_{b}} x_{2} x_{4} k_{a}) \\ - \frac{γ - 1}{H_{p} V_{a}} {\overset{\cdot}{Q}}_{a} + \frac{γ - 1}{H_{p} V_{b}} {\overset{\cdot}{Q}}_{b} k_{a} - θ_{4} + θ_{5} k_{a} - {\tilde{d}}_{a 0} \\ + {\tilde{d}}_{b 0} k_{a} + {\overset{\cdot}{p}}_{edc} \\ q_{eds 1} = - k_{3} z_{3} q_{eds 2} = - h_{3}^{2} z_{3} / (4 η_{3}) \\ h_{3} \geq [| θ_{M 1} | | x_{2} | + | θ_{M 2} | | S_{f} (x_{2}) | + | θ_{M 3} | \\ + f_{max}] | \frac{1}{M} \frac{\partial p_{ed}}{\partial x_{2}} | + | θ_{M 1} | + | θ_{M 1} | + d_{a max} + d_{b max} \end{matrix}

(19)

Define another semi-positive definite function

V_{3} = V_{2} + \frac{1}{2} z_{3}^{2}

(20)

According to equation (19), it is also obvious that $q_{eds 2}$ satisfies

\begin{matrix} z_{3} [q_{eds 2} - {\tilde{θ}}_{4} + {\tilde{θ}}_{5} - {\tilde{d}}_{a 0} + {\tilde{d}}_{b 0} - {\overset{\cdot}{p}}_{edu}] \leq η_{3} \\ z_{3} q_{eds 2} \leq 0 \end{matrix}

(21)

Supposing $q_{e} = q_{ed}$ and take the similar calculation to $V_{3}$ as the process for $V_{2}$ , then the derivation of $V_{3}$ satisfies

\begin{array}{l} {\dot{V}}_{3} \leq - k_{2} z_{2}^{2} + η_{2} - k_{3} z_{3}^{2} + η_{3} \leq - β V_{3} + η \\ β = \min {2 k_{2} / M, 2 k_{3}} \\ η = η_{2} + η_{3} \end{array}

(22)

which means that the solution of $V_{3}$ is

V_{3} (t) \leq e^{- β t} V_{3} (0) + \frac{η}{β} [1 - e^{- β t}]

(23)

It also indicates that the upper limit of error vector $z = [z_{2}, z_{3}]^{T}$ is

‖ z (t) ‖^{2} \leq e^{- β t} ‖ z (0) ‖ + \frac{2 η}{β} [1 - e^{- β t}]

(24)

This result indicates that $z_{2}$ and $z_{3}$ converge exponentially to a spherical domain, whose size can be controlled by $k_{2}$ , $k_{3}$ , $η_{2}$ , and $η_{3}$ . Due to $z_{2} = {\overset{\cdot}{z}}_{1} + k_{1} z_{1}$ , it is clearly that $z_{1}$ is bounded according to final value theorem of Laplace transform theory.

After the expected mass flow $q_{ed}$ is gotten in equation (19), based on the flow rate equations in equation (3), the area of the proportional valve port can be calculated by the inverse flow-rate functions

\begin{matrix} A (u) = {\begin{matrix} \begin{matrix} H_{p} q_{ed} / (γ R T_{s} K_{q} (p_{s}, p_{a}, T_{s}) / V_{a} \\ + γ R T_{b} K_{q} (p_{b}, p_{0}, T_{b}) k_{a} / V_{b}), q_{ed} > 0 \end{matrix} \\ \begin{matrix} H_{p} q_{ed} / (- γ R T_{a} K_{q} (p_{a}, p_{0}, T_{s}) / V_{a} \\ - γ R T_{s} K_{q} (p_{s}, p_{b}, T_{b}) k_{a} / V_{b}), q_{ed} < 0 \end{matrix} \end{matrix} \\ K_{q} (p_{u}, p_{d}, T_{s}) = {\begin{matrix} C_{d} C_{a} P_{u} / \sqrt{T_{u}}, P_{d} / P_{u} \leq b \\ C_{d} C_{a} C_{b} P_{u} / \sqrt{T_{u}}, b < P_{d} / P_{u} < λ \\ C_{d} C_{a} C_{c} P_{u} / \sqrt{T_{u}}, λ \leq P_{d} / P_{u} \leq 1 \end{matrix} \end{matrix}

(25)

By the determination of experiment results, the relation between control voltage and valve area can be expressed as

\begin{matrix} A (u) = J (u_{c}) = {\begin{matrix} K_{A} u_{cd}^{2} + K_{B} u_{cd} + K_{C} \\ when | u_{c} - u_{m} | \geq u_{dz} \\ \frac{(K_{D} u_{cd} + K_{E})}{(K_{F} u_{cd} + K_{G})} \\ when | u_{c} - u_{m} | < u_{dz} \end{matrix} \\ u_{cd} = (u_{c} - u_{m}) / 10 \end{matrix}

(26)

where $u_{cd}$ , $u_{m}$ , $u_{c}$ , and $u_{dz}$ are normalized control values, middle control value, real output voltage, and dead zone value, respectively. $K_{A}$ to $K_{G}$ are constants that describe the mass-flow properties of the MPYE valve. According to the definitions and calculations above, finally the real output voltage $u_{c}$ is

u_{c} = J^{- 1} (A (u))

(27)

Parallel system structure and analysis

Since this 3-RPS system mainly aims at the industrial market motion simulation equipment, such as in four-dimensional (4D) movie cinema, there are some guidelines to choose the proper elements: (1) to realize the ability to lift the load as heavy as two persons, considering as 200 kg, we choose FESTO DNC-63-200-P as the cylinder, which is a basic and common cylinder production; (2) to satisfy the large flow rate in the occasion of fast moving and heavy acceleration, FESTO MPYE-5-1/4-010B is chosen as the directional valve to control the piston position; and (3) one resistance position sensor and two pressure sensors are used to convert status variables to voltage values. This 3-RPS system uses NI-PXI RT system as the controller hardware, as shown in Figure 3. The control algorithm is written in MATLAB C language S-function and compiled by real-time workshop. NI provides a software named as Veristand which could link the dll model built by MATLAB and the I/O ports on PXI data acquisition cards.

Figure 3.

3-RPS pneumatic manipulator, signal configurations, and controller structure.

According to the mechanism structure shown in Figure 2, using Tait-Bryan angles $X (α) Y (β) Z (γ)$ to describe the coordinate transformation, the transformation matrix is given by

{}_{B}^{A}R = [\begin{matrix} c β c γ & - c β s γ & s β \\ c α s γ + s α s β c γ & c α c γ - s α s β s γ & - s α c β \\ s α s γ - c α s β c γ & c γ s α + c α s β s γ & c α c β \end{matrix}]

(28)

where s stands for $\sin$ , c stands for $\cos$ . $(X_{bo}, Y_{bo}, Z_{bo})$ is the origin point location of moving board in the fixed board coordinate. The restriction functions are written as follows

{\begin{matrix} {{}^{A}Y}_{b o} = r_{b} (\cos β \sin γ) \\ {{}^{A}X}_{b o} = \frac{r_{b}}{2} (\cos β \cos γ + \sin α \sin β \sin γ - \cos α \cos γ) \\ γ = - \arctan \frac{\sin α \sin β}{\cos α + \cos β} \end{matrix}

(29)

where $r_{a}$ and $r_{b}$ are the lengths of $O_{A} A_{i}$ and $O_{B} B_{i}$ . In the current structure, the joint points $A_{i}$ and $B_{i}$ are all located on the same radius circle on each plane, and $O_{A}$ and $O_{B}$ are the centroids of the equilateral triangles $A_{1} A_{2} A_{3}$ and $B_{1} B_{2} B_{3}$ , respectively. Points $A_{i}$ , $B_{i}$ , $O_{A}$ , and $O_{B}$ are in the same plane.

Given the posture $x_{p} = [α, β,^{A} Z_{bo}]^{T}$ , the lengths of all actuators can be written in the vector form

L_{i} =_{B}^{A} T^{B} P_{si} -^{A} P_{Ri} =_{B}^{A} {R_{xyz}}^{B} P_{si} +^{A} P_{bo} -^{A} P_{Ri}

(30)

where $L_{i}$ is the vector of cylinders’ lengths. Use $l_{i} = | L_{i} | - L_{0 i}$ to calculate the actual lengths of the piston rods. $L_{0 i}$ is the original length of the cylinder and $l_{i}$ is the actual position of each piston rod. Based on the discussion above, the transition functions from workspace $x_{p} (α, β, z)$ to joint space $x_{d} (l_{1}, l_{2}, l_{3}) = [x_{1}, x_{2}, x_{3}]^{T}$ are finally established.

According to the structure of the 3-RPS pneumatic platform and single cylinder servo system model, the dynamic model in joint space can be written as follows

M_{e} \overset{\cdot\cdot}{x} = {\bar{A}}_{a} p_{L} - b \overset{\cdot}{x} - A_{f} S_{f} (\overset{\cdot}{x}) - F_{L} + f_{n} + {\tilde{f}}_{0}

(31)

where $x = [x_{1}, x_{2}, x_{3}]^{T}$ is the desired position curve of each cylinder. ${\bar{A}}_{a} = H_{p} A_{a}$ , $M_{e} = diag {m_{1}, m_{2}, m_{3}}$ , $m_{i}$ is the equivalent mass load of each piston rod. $p_{L} = [p_{L 1}, p_{L 2}, p_{L 3}]^{T}$ , $p_{Li} = p_{ia} - k_{a} p_{ib}$ , $k_{a}$ is the area factor for the cylinder chambers within the piston rod. $p_{L}$ is the first-level virtual input, which has the physical meaning of equivalent pressure differences vector. $F_{L} = [F_{L 1}, F_{L 2}, F_{L 3}]$ is the load matrix applied to cylinder piston rods. $f_{n} = [f_{n 1}, f_{n 2}, f_{n 3}]^{T}$ and ${\tilde{f}}_{o} = [{\tilde{f}}_{o 1}, {\tilde{f}}_{o 2}, {\tilde{f}}_{o 3}]^{T}$ represent the disturbance and modeling error on each axis.

The thermodynamic process in the cylinder chambers can be modeled as equation (32)

\begin{matrix} {\overset{\cdot}{p}}_{L} = q_{L} + F_{p} + d_{n} + {\tilde{d}}_{0} \\ q_{Li} = \frac{γ_{a} R}{H_{p} V_{ia}} ({\overset{\cdot}{m}}_{iain} T_{s} - {\overset{\cdot}{m}}_{iaout} T_{ia}) - \\ \frac{γ_{a} R}{H_{p} V_{ib}} ({\overset{\cdot}{m}}_{ibin} T_{s} - {\overset{\cdot}{m}}_{ibout} T_{ib}) \\ F_{pi} = - \frac{γ_{a} A_{ia}}{V_{ia}} {\overset{\cdot}{x}}_{i} p_{ia} - \frac{γ_{a} A_{ib}}{V_{ib}} {\overset{\cdot}{x}}_{i} p_{ib} \\ + \frac{γ_{a} - 1}{H_{p} V_{ia}} {\overset{\cdot}{Q}}_{ia} - \frac{γ_{a} - 1}{H_{p} V_{ib}} {\overset{\cdot}{Q}}_{ib} \end{matrix}

(32)

where $q_{L} = [q_{L 1}, q_{L 2}, q_{L 3}]^{T}$ , which stands for the second-level virtual input, is used as the desired flow rate to calculate the control voltages of the proportional valves. $F_{p} = [F_{p 1}, F_{p 2}, F_{p 3}]^{T}$ is the external work vector. $d_{n} = {[d_{n 1}, d_{n 2}, d_{n 3}]}^{T}$ and ${\tilde{d}}_{o} = {[d_{o 1}, d_{o 2}, d_{o 3}]}^{T}$ are the influence caused by the disturbance and modeling error of air flow, respectively. ${\tilde{f}}_{o}$ and ${\tilde{d}}_{o}$ are all the non-linear uncertainty parts in the model.

Figure 3 shows the basic structure of a 3-RPS pneumatic manipulator and the realistic plant. Based on the single adaptive control model estimation vectors configurations, the load parameter, which causes the biggest difference from single system to a parallel coupled system, has been considered into the online estimation. It could be inferred that the load variations caused by the parallel mechanism structure would be calculated online and the influence on the performance would be decreased. Due to the load parameters $F_{Li}$ are also bounded, the controller designed in Figure 3 is obviously stable according to the analysis in sections above.

Single cylinder control experiments

Constant values mentioned in Table 3 were used in all the models and controllers. Initial value configurations of the variables, vectors, and matrices of the controller are shown in Table 2. The experimental configuration of single cylinder is shown in Figure 1. Using NI-cRIO 9022 as real-time controller and data acquisition cards attached on it, running the controller model described above and converted by real-time workshop, the actual performance of the algorithms could be easily acquired.

Table 2.

Initial values for single cylinder control experiments.

Symbol	Description	Value
$\hat{θ} (0)$	Initial value of parameter estimation vector	$[125, 20, 20, 0, 0]^{T}$
$θ_{max} θ_{min}$	Boundary of estimated parameters	$[500, 150, 100]^{T}$ , $[0, 0, - 100]^{T}$
$Ψ_{1} (0), Ψ_{2} (0), Ψ_{3} (0)$	Initial configuration of adaptive matrices	$diag {100, 100, 100}, [100]^{T}, [100]^{T}$
$ρ_{Mi}$	Upper limit of $‖ Ψ_{i} (t) ‖ i = 1, 2, 3$	$[1000, 100, 100]^{T}, [100]^{T}, [100]^{T}$
${\overset{\cdot}{θ}}_{M}$	Estimation speed limitations	$[10, 10, 10]^{T}, [10]^{T}, [10]^{T}$

The experiment results are shown in Figure 4 in the condition of 0.5 Hz sine-wave desired trajectory.

Figure 4.

Single cylinder tracking performance with 0.5-Hz sine-wave reference trajectory. Upper panel: tracking results (red: desired curve; blue: measured curve); lower panel: tracking error results.

Figure 4 demonstrates the controller performance of a desired curve $0.09 \sin (π t) + 0.1 (m)$ . The tracking error is becoming smaller and smaller while the unknown parameters are being identified with the help of online parameter estimator. The maximum control error can be as low as about 1.7 mm as shown in that figure. The status variables $p_{a}$ , $p_{b}$ and the output value $u_{c}$ of the system built in equation (4) are shown in Figure 5.

Figure 5.

System status: pressure in chambers ( $P_{a}$ , $P_{b}$ ) (upper panel) and control output data $u_{c}$ (lower panel).

Figure 6 gives the comparison of different control algorithms with the desired sine curve $0.09 \sin (0.5 π t)$ . Errors of the proportional–integral–derivative (PID), SMC, determined robust control (DRC) (i.e. the back-stepping designed controller with perfect parameters setting without online estimation) and ARC controller are shown from top to bottom.

Figure 6.

Comparison of different control algorithms: (a) PID, (b) SMC, (c) DRC, and (d) ARC. 0.25-Hz reference sine-wave trajectory.

Obviously, PID controller has the largest tracking error since the method is non-model-based, and SMC method based on back-stepping method improves the precision significantly to about 2 mm for maximum. With the adoption of robust feedback parts shown in equation (19) and online parameters estimation method, the maximum tracking error of ARC finally becomes about 1 mm.

Figure 7 demonstrates the parameters estimation result. It is clear that as the parameters converge to their actual values, the tracking performance becomes better and better, which is also shown in Figure 6(d).

Figure 7.

Parameter estimation results of the single cylinder control, from ${\hat{θ}}_{1}$ to ${\hat{θ}}_{5}$ .

3-RPS manipulator posture control experiments

The scheme of 3-RPS control system and the physical equipment are shown in Figure 3. As mentioned in the former sections, the platform has 3-DOFs which are roll, pitch, and heave. Similar to the single cylinder experiment, each one’s piston position $x_{1}$ , pressures $p_{a}, p_{b}$ , as well as the air source pressure $p_{s}$ are all gathered into the controller. When the desired curve in workspace of $x_{p} = [α, β,^{A} Z_{bo}]^{T}$ is given, the length vector $x_{d}$ would be calculated by inverse kinematics module in the controller and used as the reference curve in joint space. Initial values of 3-RPS system that are shown in Table 3 are applied in the 3-RPS controller and some of the values are updated since the change of load, shown in Table 4. Comparing with the single cylinder condition, the manipulator situation has about three times consumption of calculation for the multi-axis model. So the controller was changed to PXI system which could easily run at more than 5 kHz. In this experiment, the controller frequency was set as 1 kHz. The other mechanical components and sensors are noted in the right picture of Figure 3.

Table 3.

System constant parameters used in the model and controllers.

Symbol	Description	Value
$r_{a}$ , $r_{b}$	Lengths of $O_{A} O_{i}$ , $O_{B} B_{i}$	$0.332 m, 0.336 m$
$L_{0}$	Original length of cylinder	$0.494 m$
$k_{a}$	Area ratio $A_{a} / A_{b}$	0.899
n	Polytropic factor	1.35
$γ$	Specific heat of air	1.4
R	Universal gas constant	286.9
D	Cylinder diameter	0.063 m
L	Piston stroke	0.2 m
$p_{r}$	Critical pressure ratio	0.29
$λ$	Linear flow pressure ratio	0.999
$H_{p}$	Configuration factor	$10^{5}$
$A_{a}$	Section area of chamber A	$31.172 e^{- 4} m^{2}$
$A_{b}$	Section area of chamber B	$28.030 e^{- 4} m^{2}$
$V_{0 i}$	Chamber dead volume	0.02 L
$α_{i}$	Fading factor	0.1
$ν_{i}$	Normalized factor	0.1
$k_{1}, k_{2}, k_{3}$	Gains of three levels	70, 70, 60
$η_{2}, η_{3}$	Configuration constants	4, 10
$h_{2}, h_{3}$	Configuration constants	100, 150
$τ_{f}, ω_{f}, ξ$	Filter factors	50, 100, 1
$u_{m}, u_{dz}$	Center voltage, dead zone	5.02 V, 0.68 V
M	Load mass	3.3 kg

Table 4.

New initial value configurations for 3-RPS manipulator posture control experiments.

Symbol	Value
$\hat{θ} (0)$	$[325, 20, 80, 0, 0]^{T}$
$θ_{max}, θ_{min}$	$[500, 150, 500]^{T}, [0, 0, - 1000]^{T}$
$M_{e}$	$[13.0, 9.0, 9.0]^{T} (kg)$

For the first step of control experiment, only one of these three axes is activated while the other two are set as zero. The posture tracking results of the roll $(α)$ axis is shown in Figure 8 with the trajectory of $0.32 \sin (1.57 t) (rad)$ in workspace. Another two experiments on pitch and heave axes were also executed with the trajectories of $0.28 \sin (1.57 t) (rad)$ and $0.09 \sin (1.57 t) (m)$ , respectively. Meanwhile, the lengths of all rods in joint space $(L_{1}, L_{2}, L_{3})$ are shown below the error curve in Figure 8, which are also very close to the reference sine-waves.

Figure 8.

Posture tracking results, roll axis, $0.32 \sin (1.57 t)$ .

Furthermore, a set of compound trajectories are used as the desired trajectories in workspace. While these inputs are acting on all of the axes in workspace, that is, $α$ , $β$ , and z are all in the motion status. It is a more universal working status of the 3-RPS system.

Figure 9 is the result of the tracking performance of the compound trajectories. These two lines indicate that the real posture of the platform is also very close to the desired curves. The ARC control errors of three axes are shown by the solid blue lines in Figure 10. Meanwhile, the posture performance under different algorithms is also compared in this figure by the dotted red and black lines. The comparison indicated that the proposed controller improves the posture control performance of pneumatic manipulator significantly.

Figure 9.

Compound posture reference trajectory tracking experiment result. From top to bottom, they are roll axis, pitch axis, and heave axis.

Figure 10.

Tracking error for three dimensions in workspace under different algorithms. From top to bottom, they are roll axis, pitch axis, and heave axis. Red dotted line: SMC; black dotted line: DRC; blue solid line: ARC.

Performance analysis and conclusion

As the results shown in the figures above, the effectiveness of the adaptive robust method that used to raise the control performance for ordinary pneumatic cylinder is verified by the experiments. Based on the online parameter estimation, the tracking error becomes smaller and smaller as the estimated values converge to the actual numbers, which is shown clearly in Figures 6 and 7. Table 5 provides comparisons about the performance with different algorithms and different frequency with the proposed ARC controller for single cylinder experiments. It is clear to conclude that the method proposed in this article made a big progress on pneumatic servo control. However, there are also some uncertainties or unmodeled parts, which influence the performance more significantly as trajectory frequency becoming higher. It can also be noted that the estimation still works well in the parallel platform posture control, as shown in Figures 8 –10. The precision analyses of 3-RPS manipulator are shown in Tables 6 and 7.

Table 5.

Single cylinder control error analysis and algorithms comparison.

Algorithm	Trajectory frequency (Hz)	Trajectory expression (m)	$e_{F} (mm)$	$‖ e ‖_{rms} (mm)$	${‖ e ‖}_{r m s} / A_{m} (%)$
PID	0.25	$0.09 \sin (1.57 t)$	10.3	4.52	5.02
SMC	0.25	$0.09 \sin (1.57 t)$	2.03	0.98	1.08
DRC	0.25	$0.09 \sin (1.57 t)$	1.65	0.79	0.88
ARC	0.5	$0.09 \sin (3.14 t)$	1.90	0.866	0.96
ARC	0.25	$0.09 \sin (1.57 t)$	0.93	0.424	0.47

PID: proportional–integral–derivative; SMC: sliding mode control; DRC: determined robust control; ARC: adaptive robust control.

Table 6.

Error indexes of one-axis-motion case for 3-RPS manipulator.

Axis	$e_{F}$	$‖ e ‖_{rms}$	${‖ e ‖}_{r m s} / A_{m} (%)$
Roll	0.0029 rad	0.0015 rad	0.47
Pitch	0.0050 rad	0.0017 rad	0.61
Heave	1.07 mm	0.39 mm	0.44

Table 7.

Error indexes of compound trajectories for 3-RPS manipulator of all axes under different control methods.

Algorithm	Axis	$e_{F}$	$‖ e ‖_{rms}$	${‖ e ‖}_{r m s} / A_{m} (%)$ ^a
SMC	Roll	0.0087 rad	0.0035 rad	$~ 1.09$
	Pitch	0.0108 rad	0.0047 rad	$~ 1.68$
	Heave	3.31 mm	1.19 mm	$~ 1.32$
DRC	Roll	0.0057 rad	0.0026 rad	$~ 0.81$
	Pitch	0.0066 rad	0.0027 rad	$~ 0.96$
	Heave	2.35 mm	0.96 mm	$~ 1.07$
ARC	Roll	0.0042 rad	0.0019 rad	$~ 0.59$
	Pitch	0.0049 rad	0.0020 rad	$~ 0.71$
	Heave	1.20 mm	0.48 mm	$~ 0.53$

SMC: sliding mode control; DRC: determined robust control; ARC: adaptive robust control.

The maximum displacement from middle point is used here as a substitution for a compound motion curve’s amplitude.

In order to quantify the performance, error indexes are calculated by the data in last 10 s. The expressions of maximum tracking error $e_{F}$ and root mean square error $‖ e ‖_{rms}$ are shown in equation (33). To give a general effectiveness evaluation criterion of the controller, relative error $‖ e ‖_{rms} / A_{m}$ is calculated for comparison, where $A_{m}$ is the amplitude of the trajectory

\begin{matrix} e_{F} = max {| e |} |_{T_{f} - 10 \leq t \leq T_{f}} \\ {‖ e ‖}_{rms} = \sqrt{\frac{1}{10} \int_{T_{f} - 10}^{T_{f}} e^{2} dt} \end{matrix}

(33)

Overall in this study, using ARC controller proposed in this article, the precision of a 3-RPS platform driven by pneumatic cylinders was increased significantly to about 0.5%–0.7% (relative error). This controller has successfully overcome the disadvantages of the parameters uncertainty and non-linear characteristics of pneumatic system. The results of the performance satisfied the precision demand for a motion simulator used in 4D movie cinema. However, there were also some challenges to face when considering more about the real working conditions. In the experiments, the platform has not been working in the situation of fast moving and heavy load. Meanwhile, in the controller design for 3-RPS manipulator, the equivalent mass on each piston rod was considered as constant, which was varying actually. This was compensated by the robust part and parameters uncertainties estimation method. However, the online load estimation might fail to identify the actual value while the desired trajectory was changing quickly. The research on how to prevent the disadvantages brought by the large and fast load variation still needs to be done in the future. Meanwhile, an embedded controller hardware solution must be proposed to substitute NI-cRIO/PXI systems used in laboratory for an industrial production.

Footnotes

Academic Editor: Zheng Chen

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This paper is supported by National Natural Science Foundation of China (No. 51375430).

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