Scholarly article on topic 'Adaptive robust trajectory tracking control of a parallel manipulator driven by pneumatic cylinders'

Adaptive robust trajectory tracking control of a parallel manipulator driven by pneumatic cylinders Academic research paper on "Mechanical engineering"

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Academic research paper on topic "Adaptive robust trajectory tracking control of a parallel manipulator driven by pneumatic cylinders"

Advances in

Mechanical

Special Issue Article Engineering

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

Ce Shang1, Guoliang Tao1 and Deyuan Meng2

Advances in Mechanical Engineering 2016, Vol. 8(4) 1-15 © The Author(s) 2016 DOI: 10.1177/1687814016641914

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®SAGE

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

Date received: 13 January 2016; accepted: 29 February 2016 Academic Editor: Zheng Chen

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-n 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

'State Key Laboratory of Fluid Power and Mechatronic Systems, Zhejiang University, Hangzhou, China

2School of Mechatronic Engineering, China University of Mining and

Technology, Xuzhou, China

Corresponding author:

Ce Shang, State Key Laboratory of Fluid Power and Mechatronic Systems,

Zhejiang University, Hangzhou 310027, China.

Email: czesh@zju.edu.cn

|(ccj \V Creative Commons CC-BY: This article is distributed under the terms of the Creative Commons Attribution 3.0 License

(http://www.creativecommons.Org/licenses/by/3.0/) which permits any use, reproduction and distribution of the work without further permission provided the original work is attributed as specified on the SAGE and Open Access pages (https://us.sagepub.com/en-us/nam/ open-access-at-sage).

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 sys-tems.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.

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

MX = (pa - kafb)Aa Ff = AfSf (X) + bx

ka = Ab/Aa

Ff - Fl +fn +/o

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

Ti = tY^V , fbai = 0.8077—

XPbalJ

d—i gRi- T ■ —---— (miinTs - mioutTi)-

———- + —-Q! + din + ~i0

Vi dt Vi Q

Figure 1. Schematic diagram and physical picture of single cylinder position control system.

Table 1. Key symbols used in the model and controller.

Symbol

Aa, Ab Va, Vb V0a, V0b Ta, Tb Pa, Pb

fn, fa dn, da

Sf (x)

Wiin, W iout

ai, n t

[zi, Z2, Z3]

Description

Mass of the piston rod and load

Piston area of each chamber

Volume of each chamber

Dead volume of each chamber

Air temperature of each chamber

Air pressure of each chamber

Area fixing ratio for one-side rod cylinder

Modeling force error and disturbance

Modeling flow rate error and disturbance

Amplitude of friction model

Stribeck curve

Load force applied on the piston Damping coefficient Input, output mass flow rate Heat exchange rate of each chamber Unknown parameters factor Fading and normalized factor Adaptive function Error vector First-level virtual input Expected first-level virtual input Second-level virtual input Expected second-level virtual input

where heat exchange rate Qi = hShi(x)(Ts — Ti), heat exchange area Shi(x) = 2A,- + pD(L/2 + x), and chambers volumes V = V0i + A,(L/2 + x). h is the thermal conductivity of air and inner surface of cylinder chamber, Shi(x) is the area of heat conduction, and ps is the air source pressure. R is the ideal gas constant and l is linear flow ratio that has the constant value 1.4. D is the cylinder chambers' diameter and L is the stroke. Ts 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

Figure 2. 3-RPS parallel platform structure.

of the valve core can be ignored. The valve model in equation (3) is established to calculate the mass flow under different input conditions

> A(u)CdCaPu/pT~uPd/Pu <Pr qm = : A(u)CdCaCbPu=/ ffiuPr <Pd/ Pu<1 { A(u)CdCa ffiPffipTul < Pd/Pu < 1

Ca (g/R)(2 / (g + 1))(g + 1)/(g-1)

Cb = \l 1 - (Pd/Pu C = (Pd/Pu - Pr ) L

Cc " (1 -1) V1

Cd = 1.099 - 0.1075Pd / P,

Pr )2/(1 - Pr )2

[(1 - Pr)

[(1 - Pr).

where Pd and Pu are downstream pressure and upstream pressure, respectively, qm is the calculated mass flow rate, pr is critical pressure ratio, and Tu 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)

( x 1 = X2 _ Mx2 = Aa(x3 - kaX4) - 0^2 - 02Sf (X2)

+ 03 + ~

—R —A

H—Va

(m ainfs m aoutTa ) ~^^x2x3 Va

— -1

+ "W Qa + 04 + ~a0

H—Va

—R —Ab

—— (mbinTs - mboutTb) - -rr-x2x4 Hp Vb Vb

— - 1 •

+ WIT Q b + 05 + ~b0

The status variables vector X is given by

X = [xi, x2, x3, x4]T = [x, x,—a/Hp, Pb /H—Y

Aa = AaHp

U = [01, 02, 03, 04, 05]T

= [b, Af, -Fl +fn, dan, dbn\T

U = satuM (Prö (Ct)) 0(0) e Be

satu (★) = SaH, Sa =

1 ||*||< 0M

M/||*||||*||>em

where yM is the maximum parameter updating speed determined in advance. According to the definitions above, the adaptive law has the properties below

( 0(t) e Be = {0 : 0min < 0 < 0max}

[0 - e]T [C- 1Pre(Ct) - t]< 0

|0 (t)|<e M

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

The matrix of adaptive law C is described as follows

CiUifvTfCi when l max (C (t)) < PMi

1 + nuf CiUif and ||Pr0(C,-ti)|| < 0Mi 0 otherwise

1 + Viuf TiUif

Uif ei

where Hp = 105 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

where pMi is the upper limit of ||C,-(t)||, while 1max(Cf(i)) is the maximum eigenvalue of C,-(t). ai is the fading factor and vi is the normalized factor. Obviously, it is ensured that C,-(t) < pMiI, 8t. According to the model established in equation (4), the last three equations can be rewritten in the linear regression form and then multiplied by Gf as shown in equation (10)

' y1f = Gf [Aa(x3 - kax4) - Mx2] = 01x2f + 02Sf (x2) - 03 1f y2f = Gf [yR(niainTs - maoutTa) / {HpVa)

-—Aax2x3/Va + (— - 1)^a/(H^Va) - x3]

= - 041f

y3f = Gf [—R(rhbinTs - mboutTb)/{HpVb)

-—Abx2x4/Vb + (— - 1)Qb/(HpVb) - x4] = - 05 1f

where B0 is the bound of the parameters vector U. Pr0 is the standard projection mapping. C is the matrix of adaptive law and t is the adaptive function. This projection mapping has been described in Goodwin and Mayne's37 and Yao and Tomizuka's19 works. sat_ (H) is the saturation function, which is used to limit the updating speed of the parameters. The saturation function is defined as

where Gf is a stable linear time-invariant (LTI) filter that has the form

(tfs + 1)( s2 + 2 jwfs + w2)

where vf, tf, and j are all constants. Obviously, Gf is a third-order transfer function.

Defining yif = ufUs, where = [01,02,03]T, U2s = [04]T, and U3s = [05]T, the regression vectors are Vf = [X2f, Sff — 1f^ vf = [—1f], and uif = [—1f]. The error of estimation can be written in the form of

ei = yif — yif = Vv yis — Vv Uis = Vf &is (12)

The equation above is the standard parameter estimation model. Least square method (LSM) can be used to obtain the estimated vector U.

Adaptive robust controller design

According to the system model built in equation (4), the elements of the error vector is defined as [zi, z2, z3]T and expected first-level input pe is shown in equation (13)

zi = xi - XdZ2 = zi + k\z\

Pe = X3 - kaXAZ3 = Pe - Ped Ped = Peda + Pedsi + Peds2 Peda = [01X2 + 02<S/(X2) - 03

+ M(€id - kiZi)]/^a .Pedsi = - k2Z2/ /AaPeds2 = - ¿2Z2 /(4^2^^a)

i 2 V2 = - Mz2

V2 = AaZ2Z3 - k2Z2 + Z2[AaPeds2 + UiX2 + ~~2S/(X2) - ~3 + /0]

It is not difficult to prove that ^eds2 satisfies the conditions in equation (17)

Z2[~iX2 + ~2S/(X2) - ~3 + Â]< h2

Z2AaPeds2 < 0

V2 < AaZ2Z3 - k2Z2 + h2

size of this domain can be controlled by k2 and h2. In order to satisfy this requirement, the derivation of z3 has to be calculated

/gAa gAb \ g - i •

qe -l "vX2X3 + -vbbX2X4kaJ + H~vaQa

Qbka + 04 - 05ka + ~a0 - ~boka - >edc - .Ped«

(mbi„Ts - mboutTb)-—— (mai„Ts - mao«tTa)

where k1 is defined as the feedback gain of the tracking position error z1.

To calculate the expected pressure input, Ped is separated into three parts as shown in equation (13). Peda is the model compensation part, Peds1 is the proportional feedback part of z2 and Peds2 is used as the robust feedback part to inhibit the affection of model uncertainties. k2 is defined as the feedback gain of z2. h2, h2 are used to control the value of robust feedback part Peds2.

To choose property values of h2 and 0iM, they must satisfy the conditions shown in equation (14)

h2 > \U'M 1 IN + |0M2||S/(X2)| + \9'M 3 | +fmax (14) UMi 0i max 0i min

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

where k3 is defined as the feedback gain of pressure level error z3. To calculate the expected mass flow, qed is separated into three parts as shown in equation (19). qeda is the model compensation part, qeds1 is the proportional feedback part of z3, and qeds2 is used as the robust feedback part to inhibit the affection of model uncertainties. h3, h3 are used to control the value of robust feedback part qeds2

qed = qeda + qeds1 + qeds2

- f gAa gAb

qeda = - Aaz2 + X2X3 + —— X2X4ka

\ Va Vb

g 1 g 1

Qa + ir^T Qbka - 04 + 05ka

+ db0ka + pedc

qedsi = - k3Z3qeds2 = - ^Z3/ (4%) ¿3 > [|0Mi I I X2 I + I 0M2 ||S/(X2)| + I 0M3 I

i dped

^,/max]

+ I 0Mi I + I 0M i I + da

+ db max

Define another semi-positive definite function

The derivation of V2 can be expressed as equation (16) according to the definitions in equation (13)

V3 = V2 + 2 z2

According to equation (19), it is also obvious that qeds2 satisfies

Z3 [qeds2 - ~4 + 05 - ~a0 + ~b0 - Ped«]< h3 Z3 qeds2 < 0

Supposing qe = qed and take the similar calculation to V3 as the process for V2, then the derivation of V3 satisfies

The equations above indicate that when z3 = 0, z2 will converge exponentially to a spherical domain. The

V3 < - k2z2 + h2 - k3Z23 + h3 < - bV3 + h b = min{2k2 / M, 2k3}

h = h2 + h3

which means that the solution of V3 is

V3(t) < e—btV3(0)+ h [1 — e—bt] (23)

It also indicates that the upper limit of error vector

= [Z2, Z3]T is

l|2< e-ßt\\z<

+ ^ [1 - e-bt 1 ß L J

A(u) =

Kqipu,Pd, Ts) = <

Hpqed/ (jRTsKq(ps,Ts) /Va + gRTbKq(pb,po, Tb)ka/ Vb), qed>0 Hpqed/(-gRTaKq(pa,po, Ts)/Va — jRTsKq(ps,pb, Tb)ka/ Vb), qed <0 CdCaPu/ ffiPd/Pu < b CdCaCbPu/PTu, b <Pd/Pu <1 SdCaCcPu / 1 < Pd /Pu < 1

(Kpucd + KG ) , when |uc — um| <udz ucd = (uc — um) /10

where ucd, um, uc, and udz are normalized control values, middle control value, real output voltage, and dead zone value, respectively. KA to KG are constants that describe the mass-flow properties of the MPYE valve. According to the definitions and calculations above, finally the real output voltage uc is

This result indicates that z2 and z3 converge exponentially to a spherical domain, whose size can be controlled by k2, k3, h2, and h3 ■ Due to z2 = Z1 + k1z1, it is clearly that z1 is bounded according to final value theorem of Laplace transform theory.

After the expected mass flow qed 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

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

' KAuld + KBucd + KC when |uc — um| > udz A(u) = J (uc) = (KDucd + Ke )

- r—1

J—1(A(u))

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.

According to the mechanism structure shown in Figure 2, using Tait-Bryan angles X(a)Y(j6)Z(g) to describe the coordinate transformation, the transformation matrix is given by

cßcy easy + sasßcg sasg — casßcy

— cß sy sß

cacy — sasßsy —sacß cy sa + casßsy cacß

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

where s stands for sin, c stands for cos. (Xbo, Ybo, Zbo) is the origin point location of moving board in the fixed board coordinate. The restriction functions are written as follows

AYbo = rb( cos b sin g)

AXbo = (cos b cos g + sin a sin b sin g — cos a cos g)

g = — arctan

sin a sin b cos a + cos b

where ra and rb are the lengths of OAAi and OBBi. In the current structure, the joint points Ai and Bi are all located on the same radius circle on each plane, and OA and OB are the centroids of the equilateral triangles AiA2A3 and BjB2B3, respectively. Points Ai, Bi, OA, and OB are in the same plane.

Given the posture xp = [a,b,AZbo}T, the lengths of all actuators can be written in the vector form

T = A TBp Li BT psi

_A P _ Ar Bp PRi BRxyz p si

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

Pl = 1l + fp + dn + do

HpV ia

(miain Ts miaout Tia )

gaR HpVib

F _ _ gaAia . gaAib .

F pi Jr xiPia Jr xipib

(mibinTs — miboutTib)

Via Vib

g a — 1 Q — g a — 1

Hp Via ia Hp Vib

where Lt is the vector of cylinders' lengths. Use li = | Li | — L0i to calculate the actual lengths of the piston rods. L0i is the original length of the cylinder and li is the actual position of each piston rod. Based on the discussion above, the transition functions from workspace xp(a,b,z) to joint space xd(l\, l2, l3) = [xi,x2,x3]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

Mex = AaPL — bx — AfSfi x) — Fl +fn + f0 (31)

where x = [x1, x2, x3]T is the desired position curve of each cylinder. Aa = HpAa, Me = diagfm1, m2, m3g, mi is the equivalent mass load of each piston rod. Pl = [pli,PL2,PL3]T, PLi = Pia — kaPib, ka is the area factor for the cylinder chambers within the piston rod. pL is the first-level virtual input, which has the physical meaning of equivalent pressure differences vector. Fl = [Fli, FL2, FL3] is the load matrix applied to cylinder _ piston rods. fn = [fni, fn2, fn3]T and fo = fo1, fo2, ffo3]T represent the disturbance and modeling error on each axis.

where qL = [qn, qL2, qL3] , which stands for the second-level virtual input, is used as the desired flow rate to calculate the control voltages of the proportional valves. Fp = [Fp1,Fp2,Fp3]T is the external work vector. dn = [dni, dn2, dn3]T and do = [do\, d02, d03]T are the influence caused by the disturbance and modeling error of air flow, respectively. fo and do 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 FLi 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

Table 2. Initial values for single cylinder control experiments.

Symbol

Description

UmaxUmin

C|(0), C2(0), C3(0)

Initial value of parameter estimation vector Boundary of estimated parameters Initial configuration of adaptive matrices Upper limit of ||C,(t)||/ = 1,2,3 Estimation speed limitations

[125,20,20,0,0]T [500, 150, 100]T, [0,0, —100]T diagf 100, 100, 100}, [ I00]T, [ I00]T [1000, 100, I00]T, [ 100]T, [I00]T

[l0, 10, 10]T, [I0]T, [I0]T

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.

Tracking performance for the trajectory 0.09sin(pi*t)

x 10"3 Tracking error with parameters estimation, for the trajectory 0.09sin(pi*t)

above and converted by real-time workshop, the actual performance of the algorithms could be easily acquired.

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

Figure 4 demonstrates the controller performance of a desired curve 0.09 sin(pt) + 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 pa, pb and the output value uc of the system built in equation (4) are shown in Figure 5.

Figure 6 gives the comparison of different control algorithms with the desired sine curve 0.09 sin(0.5pt). 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.

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).

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 x1, pressures pa,pb, as well as the air source pressure ps are all gathered into the controller. When the desired curve in workspace of xp = [a, 3, AZbo]T is given, the length vector xd 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.

Figure 5. System status: pressure in chambers (Pa, Pb) (upper panel) and control output data uc (lower panel).

Table 3. System constant parameters used in the model and Table 4. New initial value configurations for 3-RPS manipulator controllers. posture control experiments.

The pressure curves of two chambers, for the trajectory 0.09sin(pi*t)

The output voltage curve for the trajectory 0.09sin(pi*t)

Symbol Description Value

Ta. rb Lengths of OAOi. OBB; 0.332 m, 0.336 m

Lc Original length of cylinder 0.494 m

ka Area ratio Aa/Ab 0.899

n Polytropic factor 1.35

g Specific heat of air 1.4

R Universal gas constant 286.9

D Cylinder diameter 0.063 m

L Piston stroke 0.2 m

pr Critical pressure ratio 0.29

l Linear flow pressure ratio 0.999

Hp Configuration factor I05

Aa Section area of chamber A 31. I72e~4 m2

Ab Section area of chamber B 28.030e 4 m2

Va Chamber dead volume 0.02 L

a Fading factor 0.1

n Normalized factor 0.1

ki, k2, k3 Gains of three levels 70. 70. 60

^ h3 Configuration constants 4. 10

h2, h3 Configuration constants 100. 150

Tf, vf, j Filter factors 50. 100. !

Um, Udz Center voltage. dead zone 5.02 V. 0.68 V

M Load mass 3.3kg

The other mechanical components and sensors are noted in the right picture of Figure 3.

For the first step of control experiment, only one of these three axes is activated while the other two are set

Symbol Value

0(0) [325,20,80,0,0]T

Umax, Umin [500, 150, 500]T, [0,0, -!000]T

Me [13.0,9.0,9.0]' (kg)

as zero. The posture tracking results of the roll (a) axis is shown in Figure 8 with the trajectory of 0.32 sin(1.57t) (rad) in workspace. Another two experiments on pitch and heave axes were also executed with the trajectories of 0.28 sin(1.57t) (rad) and 0.09 sin(1.57t) (m), respectively. Meanwhile, the lengths of all rods in joint space (L1, L2, L3) are shown below the error curve in Figure 8, which are also very close to the reference sine-waves.

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, a, b, 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

PID Control Tracking Error of 3RPS

x 10"3 Tracking error without parameters estimation, for the trajectory 0.09sin(0.5pi*t)

<0 o o.

Tracking error with parameters estimation, for the trajectory 0.09sin(0.5pi*t)

Figure 6. Comparison of different control algorithms: (a) PID, (b) SMC, (c) DRC, and (d) ARC. 0.25-Hz reference sine-wave trajectory.

Table 5. Single cylinder control error analysis and algorithms comparison.

Algorithm Trajectory frequency (Hz) Trajectory expression (m) eF (mm) lleILs(mm) 11 e 11 rms = Am (%)

PID 0.25 0.09 sin(l.57t) 10.3 4.52 5.02

SMC 0.25 0.09 sin(l.57t) 2.03 0.98 1.08

DRC 0.25 0.09 sin(l.57t) 1.65 0.79 0.88

ARC 0.5 0.09 sin(3.14t) 1.90 0.866 0.96

ARC 0.25 0.09 sin(l.57t) 0.93 0.424 0.47

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

Figure 7. Parameter estimation results of the single cylinder control, from U| to U5.

Figure 8. Posture tracking results, roll axis, 0.32 sin(l .57t).

red and black lines. The comparison indicated that the proposed controller improves the posture control performance of pneumatic manipulator significantly.

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

Figure 9. Compound posture reference trajectory tracking experiment result. From top to bottom, they are roll axis, pitch axis, and heave axis.

Roll angle tracking of 3RPS

Pitch angle tracking of 3RPS

Up-down tracking of 3RPS

Table 6. Error indexes of one-axis-motion case for 3-RPS manipulator.

Axis er lleILs I|e|Ls/Am (%)

Roll 0.0029 rad 0.0015 rad 0.47

Pitch 0.0050 rad 0.0017 rad 0.61

Heave 1.07 mm 0.39mm 0.44

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.

In order to quantify the performance, error indexes are calculated by the data in last 10 s. The expressions of maximum tracking error eF 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/ Am is calculated for comparison, where Am is the amplitude of the trajectory

eP = max{|e | }| Tf— 10 <t < Tf

Mrms = \JTO if—10 e2dt

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

Roll angle tracking of 3RPS

Pitch angle tracking of 3RPS

Heave tracking error of 3RPS

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.

Table 7. Error indexes of compound trajectories for 3-RPS manipulator of all axes under different control methods.

Algorithm Axis er lleILs l|e|lrms/Am (%)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 —0.7I

Heave 1.20mm 0.48 mm —0.53

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

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

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.

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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