Direct Sliding-Mode Controller Design for a 6DOF Stewart Manipulator

transcript

Direct Sliding-Mode Controller Design for a6DOF Stewart Manipulator

S. Iqbal, A. I. Bhatti

Abstract-The focus is on direct sliding-mode control designfor tracking and regulation of a Stewart platform without anyapriori knowledge of the system's mass properties in presence ofnonlinearities. The position and velocities are the major inputsto the controller. The appropriately chosen sliding surface s ER6, composed of system states vectors, proper choice ofreachability gains drive the system to stable manifold and thenslide it to an equilibrium point. It leads us to a control lawwhich clearly deals with nominal performance and robuststability. The resulting dynamic feedback is shown to providechatter-free control. The controller is based on generalizedLyapunov approach and guarantees global asymptotic andexponential convergence. The control performance of theproposed algorithm is verified by computer simulations. Thesesimulations show that system follows the desired trajectory anderrors efficiently converged to zero.

I. INTRODUCTION

T HE parallel link manipulators attract many researchersin the current decade due to its precision, rigidity and

high force-to-load ratio. A Stewart platform is a parallelrobot provides six-degree-of-freedom i.e. roll, pitch, yaw,surge, sway and heave. Its practical usage is for disturbanceisolation, precise machining and flight simulators.The Stewart manipulator consists of top-plate and base-

plate connected by the help of six variable-length electro-mechanical actuators with spherical joints that are used forrotation and translation of the top-plate with respect to thebase-plate as shown in figure 1 and 2. The angular andtranslation motion of the top plate with respect to the baseplate is produced by reducing or extending the actuatorslength. The proper coordination of the actuators lengthenables the top plate to follow the desired trajectory withhigh accuracy. Thus the six inputs to the Stewart platform interm of torque are calculated by controller and provided byhigh speed motors. The outputs of the Stewart platform arethe upper plate's angular and translation positions (in surge,sway, heave, roll, pitch and yaw) sensed by highly precisesensors.

In recent years many people worked on sliding-modefor Stewart platform. Lee and Kim [1] in 1998 presented themodel based sliding mode control for the Stewart platform inpresence of low frequency motion of base-plate as an

Manuscript received December 1, 2006.1. S. Iqbal is a postgraduate student at Center for Advance Studies in

Engineering (CASE) Islamabad Pakistan; (e-mail: siayubi@ yahoo.com).2. A. I. Bhatti is Faculty Member at Center for Advance Studies in

Engineering (CASE) Islamabad Pakistan; (e-mail: aib@case.edu.pk).

unmodeled dynamics of the manipulator. Sliding modecontroller with sliding perturbation observer is suggested [2]by Sung and Lee in 2004. A Sliding-mode control forStewart platform, which can drive motion tracking error tozero asymptotically, has been proposed [3] in 2004 byHuang and Fu. After that same authors exhibited [4] slidingmode back stepping controller for the Stewart platform in2005.

ErrorControlAction Torque RPM Position 0

Figure 1: Block diagram of the Stewart platform

The novelty in this paper is tracker and regulator designfor the Stewart platform, its simplicity and ease of itsimplementation.

Figure 2: The Stewart platform

In this paper we first define a hyper plane as slidingsurface, and then choice of proper reachability gains, whichdrag the system to the sliding surface and ultimately trap thestates to an equilibrium point. Afterward existence of

solution and stability analysis of closed-loop system basedon generalized Lyapunov theorem is done. At the end wedefine a thin boundary layer in the neighborhood of theswitching surface to avoid chattering and smoothing out thecontroller discontinuities.The rest of this paper is structured as follow, kinematics

and dynamics are explained in section II. Section III dealswith direct sliding-mode controller design for regulation andtracking. Simulation results are discussed in session IV.Conclusions are drawn in session V.

II. KINEMATICS AND DYNAMICS

The length of each actuator of Stewart platform for agiven orientation can be finding out by the help of inversekinematics and can be written as [5]:

li =Ra3PtP+ D-B, Vi= 1.. <(2.1)Where Ra,yis a rotation matrix; Pi is the coordinates of top

plate; D is distance vector and Bi is the coordinate of baseplate respectively.The dynamic equations of the Stewart platform

considering all inertial and Coriolis effects is verychallenging to determine, Lebret in [6] developed thedynamic equation using Lagrange method as:

Firstly calculate kinetic and potential energy as a function

of q, where qe R6x] is of the form [x y z a , r]T.K=K(q,q.) ! q.TMq where Me R6x6

2P = P(q)

and then develop Lagrange equation using the formula as

d a K (q, q.)) a K(q,q )+ a P(q)=where t is the applied torque. Finally get the dynamicalequation of Stewart platform as:

M (q)q + C(q, q )q + G(q) = JT(q)u - (2.2)

Where qe R6x] of the form [x y z a r6 j]TME R6X6 is an inertial matrix; VE R6x6 is Coriolis/centripetalmatrix; GE R6x' is vector containing gravity torques; J E R6x6is Jacobean matrix which changes angular velocities intoCartesian velocities and uE R6x' is vector of input signals,respectively. Some relevant properties are as below.

Property 1: M(q) is bounded function if q and q are

bounded. V (q, q>) is a bounded function if q, q and q"are bounded. G( q ) is bounded if q is bounded.Property 2: M is a symmetric and positive definite matrix

for all qe R. Moreover, M - 2C is a skew-symmetricmatrix, such that

xT(M -2C)x=0 Vxe R

The state-space representation of the Stewart platformdynamics can be written as

q; = q2 - (2.3)

q,=M (JTu-Cq2-G) - (2.4)

Fq;] F06x6 I6x6 q + 06xl 1

Lq;] LQ6x6 -M '(q)C(q, q )j q2j L-M-1 (q)G(q)j

M-1 (q)J JT(q)] u(t) - (2.5)

we can also write the above equation asx (t) = f(x(t))+ g (x(t))u(t) - (2.6)

y = h(x) - (2.7)

h(x) =[h(x) h2 (x) h (X) h4 (X) h5 (X) h6 (X)]Twhere

r .Tx= q q ]

f(X)= L06X6 I6x6 06xl 1

06X6 -M '(q)C(q,q)_ L-M '(q)G(q)j

(x) L M 6X6 ]

(x) = III, = Rafiy(q)P+D(q)-Bi

III. DIRECT SLIDING MODE CONTROL DESIGN

In sliding-mode we define a hyper-plane as sliding-surface. This design approach comprises of two components;first is the reachability phase and second is sliding phase. Inreachability phase states are being driven to a stablemanifold by the help of appropriate control law, and insliding phase states are then slide to an equilibrium point.One advantage of this design approach is that the effect ofnonlinear terms which may be construed as a disturbance oruncertainty in the nominal plant has been completelyrejected. Another benefit accruing from this situation is thatthe system is forced to behave as a first-order; thisguarantees that no overshoot will occur when attempting toregulate the system from an arbitrary initial displacement tothe equilibrium point.

A. Regulator DesignLet the sliding surface for Stewart platform is define as

s=Aq1+q2 - (3.1)where A e R6X6 is diagonal positive definite matrix and

qeE R6xi are system's states vectors. We can write the

above equation as

q2 =-Alql + sand also

q; = -A1ql + s - (3.2)

The above system equation is stable if s = 0 and the rate Me + Ce + G = -Mq- - Cqd + jTuof convergence of system is depend upon the diagonal The state space model of the above equation isentries of A matrix. The time derivative of s is

s'=A1q>+q; -(3.3) e, =e2 -(3.17)by equation (2.4) we can get as e; =-M 1 Ce2+ G +F_JTU] (3.18)

s =AAq2 +M (JTU-Cq2G) - (3.4) where F(q ,q>) = Mqd + CqdLet the sliding surface for tracking is define as

To evaluate stability, the Lyapunov candidate function is

v = I sTS (3.5) s = A2e1 +e2 -*(3.19)

taking the time derivative of both side of equation (3.5) where C2e R6,6 is diagonal positive definite matrix ande E 6xi are system's states vector. We can also write the

v' = sTs < (3.6) above equation as

from equation (3.4) we can write equation (3.6) as e= -A2e + s

v' = ST [Alq2 + M-1 (jTU-Cq -G)0 4 (3-7) and alsoW=sT[Ai2M (Juc2 eG)]-e*(3s.7)20

V is negative definite if

Vs > 0

Vs =0-(3.8)Vs <0

stability of the system is ensured if

u = ,6(t)-J TMKdsign(s) - (3.9)where

/3(t)=J-T (Cq22+G-MAq2 )

and Kd > 0, if we put the above control law then equation(3.7) becomes

V = -S Kdsign(s) -*(3. 10)Which is always negative definite.

B. Tracking DesignLet the error dynamic for Stewart platform is define as

e=q-qd ->(3.11)By taking the first and second derivative of equation (3.11)we get

e&= q-qd - (3.12)

e" = q - q - (3.13)

Also we can rewrite equation (3.11) - (3.13) as

q =e+q d (3.14)

q=e+qd -*(3.15)

q" =e +qd -*(3.16)

Where qd is reference signal. Put the values from (3.14)- (3.16) in the dynamical equation (2.2) of the Stewartplatform, we get

1 L2 1 J

The above system equation is stable if s = 0 and the rateof convergence of system is depend upon the diagonalentries of C2 matrix.The time derivative of (3.19) is

s'= A2e>+e;by equation (3.18) we get

s = A2e2 +M 1[JTU - Ce2-G - F (3.21)

To evaluate stability, the Lyapunov candidate function is

v = I sTs (3.22)

taking the time derivative of both side of equation (3.5)

v = s s - (3.23)

from equation (3.4) we can write equation (3.6) as

V =sT[A2e2+M - (JTU-Ce2-GF-)] (3.24)

V is negative definite if

F<0 Vs >0A2e2 +M-1 (JTu -Ce2 -G-F) = 0 Vs =0 (3.25)

>0 Vs <0stability is ensured if

u = ,6(t)-J TMKdsign(s) - (3.26)where

(t)=j-T (Ce2 +G+F-MA2e2)and Kd > 0, if we put the above control law then equation(3.24) becomes

v = ST Kdsign(s) - (3.27)Which is always negative definite.

C. Chattering-free ControllerFor chattering-free controller design, instead of using (3.9)

and (3.26) for regulation and tracking, we define control lawas

u = -J TMKd sat(s)where sat(s) is a saturation function and can be defined asfollow

s(t) if s>

sat(s) =s(t)I1(l)+ l if s <S

which provide a very smooth and chatter-free controlaction.

IV. SIMULATION RESULTS

Simulation has been performed in-order to examine theeffectiveness of proposed controller design. The platformcan perform rotational and translation motion i.e. surge,sway, heave, roll pitch and yaw. Figure 3 and 4 present thesimulation results of regulation and tracking controllers.Results show that the controller achieved its goal withinnominal time.

However highly discontinuous control action shown infigure 5 may not be practically implementable due tohardware limitations.

-2000o

-2000 _o

-20000

Torque Input of each Axis2000

,E 0 X

-20002 4 6 0 2 4 6

Surge Sway2000

E 0 111 III-2000

2 4 6 0 2 4 6Heave

E 0gIl M !I gli z

-20002 4 6 0

2 4 6Yaw

Figure 5: The Sliding-mode Control Action

An inherent discontinuity occurred in the conventionalsliding mode control is reduce by using the chatter-freecontrol. A dead band of 8 =0.01 is introduced for thispurpose. Figure 6 and 7 shows that by implementing achatter-free design, performance is not much degraded and asmoother control action has been achieved as shown infigure 8.

2 3 4 5 6

-0.5 _0 4 5 62

Figure 3: Sliding Mode Regulation

-0.5 _0

Figure 6: Chatter-free Regulation

2 3 4 5 6 -0.5 _o

Figure 4: Sliding Mode Tracking Figure 7: Chatter-free Tracking

2 4 5 6

In summary the controller with chatter-free design provessuperior as compared to conventional sliding-modecontroller in term of nominal performance and controlaction.

Torque Input of each Axis200 500 I0~~~~~~~~0

E -20z

-5000 2 4 6 0 2 4 6

Surge Sway

M64 = M46M66 = IzThe Coriolis and centrifugal matrix C can be written as:

0 00 00 00 0

-K1/3- K22y

Ko +KJIK2a, - K4(4/

O O O00-

-10000 2 4 6 0

-Kla -K3±+K4yK KY

-K40 -K

2 4 6 K1 = Cos ,6 sin/,(I cos2 y+ I sin2 y-I )Roll

K2 = cos2pcosy siny(I, - Iy)

K3 = cosy siny sinf(I, - Iy)J0 Pitch -1UUU00 2 Pi4 6 0 2 4 6 K4 -!L-cosl3(cosy -siny)(cosy±+siny)(I, Iy)

Pitch ~~~~~Yaw2

Figure 8: Chatter-free Control Action

V. CONCLUSION

A charting-free direct sliding-mode controller designapproach is employed successfully for the regulation and

tracking of a multi-input multi-output Stewart platform in

presence of nonlinearities and time-varying uncertainties.Stability analysis based on generalized Lyapunov theory is

performed to guarantee global, asymptotic and exponentialconvergence. In contrast to heuristic based PID tuningmethods, sliding-mode design method performs better in

terms of bandwidth, disturbance rejection and axes

decoupling.

APPENDIX

Here we explore each component of the Stewart platformdynamic equation. The inertial matrix M can be written as:

m OO 0 0 0

O m O 0 0 0

OO m 0 0 0

O O 0 M44 M45 M46

O O 0 M54 M55 0

O O 0 M64 0 M66

K5 = cosy siny(I, - Iy)where Ii represent the moment-of-inertia of the payload.Moreover, the Jacobean matrix J can be derived as:

M44 =Ix CSoS2CoS2y + Iy cos2f3sin2y + I, sin2f3

M45 = (I, - Iy) cos Pcosy siny

M46 = Iz sinPM54 = M45

M55 = Ix sin2y+ Iycos2y

11T 11T uT~ uTR1 P' UTR2P' UTfR3P'Ulx Uly Ufz ufRiP ufR2P' uR3P'

U2X UTy UTz uTR P2 UTR2PP UTR3 P2

uTTU33z

uTR, Pp

uTR, PpP

uTR2PP

u4TR2PP

u'TR1Pp UsTR2PP

uTR3 P3

u4TR3PP

usTR3 PP

u6T u6T uT u R1P6 UR2PP uTR3 P6

Ra = RX (a), R/ = R (,),R = RZ(Y)

Ra = 0 cosa -sina

0 sina cos;acos o s7n

R~~ 0 1 0

sin 0 cosfcosy -Fshr

R7= siy cos r 0

L0 0 1

_K2a, +K4)+K

0 0 0S(i)= O7 -7

_- 0 0_

S(k)= I 0 O

_O O O_

[13] C.C. Nguyen, S.S. Antrazi, Z.L. Zhou, and C.E. Campbell, " AdaptiveControl of a Stewart Platform Based Manipulator," Journal of RoboticSystems, vol.10, No. 5, pp.657-687, 1993.

[14] S. Raghavan and J. K. Hedrick, "Observer design for a class ofnonlinear systems", International Journal of Control, Vol. 59, pp 515-528, 1994.

[15] R.Nair,J.H.Maddocks, "On the Forward Kinematics of ParallelManipulators", International Journal of Robotics Research, Vol.13,No.2,April 1994, pp 171-188

[16] Advani, S.K., "The Kinematic Design of Flight Simulator Motion-Bases", Ph.D. Thesis, Delft University of Technology. DelftUniversity Press, 1998. ISBN 90-407- 1672-2.

[17] K. Liu, M. Fitzgerald, D. Dawson and F. L. Lewis, "Modeling andcontrol of a Stewart Platform manipulator", Proc. of the Symp. ofControl of Systems with Inexact Dynamic Models, Atlanta, GA, 1991,pp 83-89

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ACKNOWLEDGMENT

The authors would like to thank Mr. Saif Ullah, HeadModeling & Simulation Division NESCOM for hisassistance in simulation of the Stewart platform. The authorswould also like to thank the Higher Education Commission(HEC), Pakistan for the financial support of our work.

REFERENCES

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Direct Sliding-Mode Controller Design for a 6DOF Stewart Manipulator

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