M. Jafarinasab, M. Keshmiri, H. Azizan, M. Danesh Mechanical Engineering Department of Isfahan University of Technology

Isfahan, Iran Emails: m.jafarinasab@me.iut.ac.ir

mehdik@cc.iut.ac.ir h.azizan@me.iut.ac.ir danesh@cc.iut.ac.ir

Abstract—A novel six-DOF parallel manipulator is introduced. Kinematics analysis is discussed and using Lagrange equations for constrained systems the full dynamic modeling is performed. Sliding mode control as a class of very special nonlinear control is proposed to control the position of the manipulator in the presence of parametric uncertainties. Stability of the system is guaranteed via Lyapunov approach. An intensive series of simulation studies has been fulfilled to show the abilities of the proposed controller for trajectory tracking even in the case of large parameter variations.

I. INTRODUCTION Nowadays the parallel manipulators have found a

variety of applications due to their advantages over the serial ones such as, better accuracy, higher rigidity and higher load-to-weight ratio. One of the most popular parallel manipulator is the Gough Stewart mechanism introduced in 1965 by D. Stewart [1]. This kind of the Stewart platform is activated by hydraulic actuators. Hence practical usage of this kind has been generally in the area of low speed and high load conditions such as motion base of vehicle and flight simulators, motion bed for machine tool and such like. In the case of the manipulator high speed reactions, it is needed that the electrical actuators can be used instead of hydraulic ones. In the last two decades most of the research has been particularly aimed at Gough Stewart platform with prismatic hydraulic actuators and the rotary type is less studied.

The 6-DOF parallel manipulator with rotary actuators was initially introduced by Hunt [2] in 1983. Thereafter several species of this kind of Parallel manipulator are suggested such as the prototypes constructed by Sarkissian [ 3 ], Zamanov [4] and Mimura [5 ], where they have a few differences in linkage and joint configurations. A more recent and commercial design has been introduced by Servos & Simulation Inc. [6].

It is well-known that the Stewart platform, due to its inclusion of several closed loop structures, is highly complicated mechanism to make kinematic and dynamic analysis. In addition, the rotary type of the Stewart platform has a more complex dynamic system than the typical one. Moreover, the Stewart platform usually suffers from

matched and unmatched uncertainties and also external disturbances due to its various applications. All these issues make the control of the 6-DOF parallel manipulator with rotary actuators a challenging problem.

To the best of the authors’ knowledge the control of a 6-DOF parallel manipulator with rotary actuators has not been studied yet. Though, pending the span of last two decades several control strategies have been developed to control the parallel robots and specially the typical Stewart platform. An adaptive control scheme [7,8] in which the controller gains are regulated by the adaptation law was proposed in 1993. Lee [9 ] used inverse dynamics with approximate dynamics to control a Stewart platform in 2003. Sliding mode controller with sliding perturbation observer is suggested by Sung [ 10 ] in 2004. Some important properties of dynamics of the Stewart platform are derived and exploited to develop a sliding mode controller by Huang [11] in 2005. Iqbal [12] used the mass uncertainties to calculate the upper bounds of perturbation in order to design a robust sliding mode controller. In 2008 Qiang [13] transformed the dynamic model by means of defining one operation point for the manipulator and designed a sliding mode controller based on the simplified dynamics.

In this paper, first, we introduce a novel 6-DOF parallel manipulator with rotary actuators which has some differences in linkage and joint configurations with respect to the conventional designs. Then, due to the several attractive superior properties of SMC, such as good control performance even in the case of nonlinear systems, applicability to MIMO systems, fast transient response and insensitivity to parameter changes and external disturbances without the need for system identification, a sliding mode controller is employed to control the position of the 6-DOF parallel manipulator.

The reminder of this article is organized as follows. Section II introduces the mechanism of the manipulator. Dynamics modeling of the system is presented in Section III. Section IV deals with the design of sliding mode controller. Simulation study has been carried out and the results are discussed in Section V. Finally, some

Sliding Mode Control of a Novel 6-DOF Parallel Manipulator with Rotary Actuators

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conclusions are made in Section VI.

II. MECHANISM STRUCTURE

As shown in Fig. 1, the system is basically made up of two platforms. The base platform is fixed to the ground and is connected to the moving platform through six legs. Each leg includes two links with a universal connection in between. Also, the lower link is connected to the base platform through a revolute joint and the upper is connected to the moving platform through a spherical joint.

The generalized coordinates used for dynamic modeling of the manipulator are summarized in the following. P P P, aX Y nd Z : The coordinates describing

translation of the moving platform. θ ,φ and ψ : Euler parameters of the moving

platform. i : The motors angles.

and ii : The universal joints angles.

Fig. 1. The 6-DOF parallel manipulator with rotary actuators

III. DYNAMICS MODELING

To design a model based controller, one of the crucial steps is modeling of the systems dynamics. Here, for brevity a summary of kinematics and dynamics analysis is presented and discussed.

A. KINEMATICS ANALYSIS

Generally, the exact configuration of 6-DOF parallel manipulators can be determined in two ways. First, defining position and orientation of the moving platform by means of six parameters ( P P PX Y Z θ ,, , , φ ,ψ ) called task space

coordinates and the second is defining positions of the actuators through the variables ( i ) named joint space

coordinates. The inverse kinematics is mathematical treating the problem of describing the joint space coordinates in terms of task space variables and the forward kinematics is to do vice versa.

The inverse kinematics of parallel manipulators is straightforward and it can be solved analytically while that of the forward kinematic problem plays an important role for control and unfortunately it is difficult to be solved due to the nonlinearity and complexity of the system.

Nevertheless, there are analytical solutions [14,15,16] and numerical ones [17] for the forward kinematics of parallel manipulators. The analytic solutions are too complicated since high-order polynomial equations should be solved. In fact there exists no general closed-form solution of the forward kinematics. Numeric solutions like Newton-Raphson method are simple but they take more calculation time and also sometimes converge to the wrong solution due to inappropriate initial value selection.

For simulation purposes a useful method to solve the inverse and forward kinematics is to integrate the differential form of the constraint equations as following.

Defining the vector of generalized coordinates

P P P 1 2 3 4 5 6

1 2 3 4 5 6 1 2 3 4 5 6

, , , , , , , , ,

[X Y Z θ ,φ ,ψ,

, , , , , , , , ], , , ,

T

q (1)

six independent closed-loop vector equations called constraint equations should be ever satisfied. They can be written in algebraic form as

( ) 0, 1,...,18.iF i q (2)

Differentiating these algebraic equations with respect to time will lead to the differential constraint equations,

,Jq 0 (3)

where, J is the 18×24 constraint Jacobian matrix. Separating q to the independent generalized coordinate vector aq and the dependent generalized coordinate

vector qd , the constraint equation is rewritten as

0.aa d

d

qJ J

q

(4)

Rearranging (4) for the dependent vector results in

1 .d d a a q J J q (5)

Differentiating (5) with respect to time yields a useful equation used in dynamics analysis.

-1( ).d d a a q J J q Jq (6)

Integrating (5) for a defined initial condition will lead to the solution of the inverse and forward kinematic problems depending on the choice of independent coordinates. In this paper the goal is to design the controller in task space. Hence, the vector q is separated as follows,

P P P

1 2 3 4 5 6 1 2 3 4 5 6 1 2 3 4 5 6

X Y Z θ φ ψ

[

,

. ]

T

a

T

q

qd (7)

B. DYNAMICS ANALYSIS

Several methods such as Lagrange equations [ 18 ], Newton-Euler [19], virtual work based methods and Kane method can be used to model dynamic behavior of parallel manipulators. Here, Lagrange equations, which are well structured and clearly expressed, are applied to derive dynamic equations of motion. It should be noted that the

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mass properties of all links are considered in modeling. Constructing the kinetic and potential energy of the

moving platform, the upper and lower links in terms of generalized coordinates and their derivatives also applying Lagrange equations for constrained systems, motion equations of the system can be expressed by

( ) ( , ) ,T M q q h q q J (q)ρ B(q) (8)

where ( )M q is an 24×24 inertia matrix, ( , )h q q is a 24×1

vector containing the Coriolis, centrifugal and gravitational terms, ρ is the 18×1 vector of Lagrange multipliers

corresponding to the constraint force, B is the 24×6 actuating torque coefficient matrix and τ is the 6×1 actuating torque vector.

The dynamic equations (8) are a set of 24 dependent differential equations. It is possible to reduce them to a set of 6 independent differential equations by means of eliminatingρ . There exist several ways to eliminateρ . For

computational purposes, it is efficient to use the orthogonal complement of the constraint Jacobian matrix J. The orthogonal complement of J (i.e. cJ ) is the set of vectors which are orthogonal to each vector in J. Therefore, the product of J and cJ vanishes.

c c T T or ( ) . JJ 0 J J 0 (9)

Multiplying (8) by c T( )J from the left, the vector of

Lagrange multipliers is eliminated and independent dynamic equations can be written as

( ) ( , ) , M q q h q q Bτ (10)

where 6 24 6 1 6 6, and M h B are derived as follows.

( ) , ( ) , ( )c T c T c T M J M h J h B J B (11)

Yet, there exist second derivative of all generalized coordinates ( q ) in the independent equations (10).

Separating q to aq and dq , then rewriting (10) yields

( ) ( ) ( , ) .a a d d M q q M q q h q q B(q)τ (12)

Using (6) and substituting dq , gives the final form of the

independent motion equations as the followings,

( ) ( , ) ( ) ,a M q q h q q B q τ (13)

where, 6 6 6 1 6 6, and M h B are defined by

1 1, ,a d d a d d M M M J J h h M J Jq B B (14)

For designing the sliding mode controller forward dynamics analysis is needed. Simultaneous integration of (3) and (13) for a given initial condition will lead to solve forward dynamic problem.

IV. CONTROLLER DESIGN

Sliding mode control approach comprises of two steps; the first is the reachability phase and the second is sliding

phase. In reachability phase, states are being driven to a stable manifold, called sliding surface by means of appropriate control law. Thereafter, in sliding phase, states should stay on the surface while sliding to an equilibrium point.

The tracking control problem in task space is to find a

control law such that given a desired trajectory desq , the

tracking error q tends to zero where,

.a des q q q (15)

Let us define the sliding surface as

, S Λq q (16)

where Λ is a 6×6 diagonal positive definite matrix. Keeping S equal to zero by an appropriate control law will lead to asymptotic stability of the system. Lyapunov direct method could be used to obtain the control law that stabilizes the system.

Consider the Lyapunov function candidate as

1

.2

TV S S (17)

Time derivative of (17) will lead to

,TV S S (18)

in which the term S is given by

.a des S Λq q q (19)

Solving (13) for aq and substituting in (19) results in

1

.des S Λq q M ( h Bτ) (20)

Defining the control signal as

1ˆ sgn( ), τ τ B MK S (21)

where τ̂ is defined as

1ˆ des

τ B M q -Λq h (22)

will cause

sgn( ), S K S (23)

and respectively,

sgn( ) 0.T V S K S (24)

Hence, according to the Lyapunov theory the control law (21) will result in a stable closed loop system. In practice, the control law (21) cannot be used because of containing the term sgn( )S which results in high frequency oscillations, called chattering, and it is replaced by a continuous approximation. The choice of the controller parameters plays an important role to have a good performance of the overall system. Several simulations have been down to choice these parameters properly and the results are presented in the next section.

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V. SIMULATION STUDY & DISCUSSION

To depict the abilities of the proposed controller in trajectory tracking even in the case of large parameter variations an intensive series of simulation studies has been fulfilled on a 6-DOF parallel manipulator with rotary actuators. The parameters of the simulation model are extracted from a prototype built in IUT Dynamics and Robotics Center. Table I shows the manipulator specifications. The parameters of controller are chosen to

be 6 610 , 0.5, Λ E 6 6100 K E where, E is the

identity matrix. The simulation studies are presented in 3 parts. The first

part, a circular motion is supposed to be followed. Via this study the translational motion tracking of the controller is tested. In part 2 the controller performance in rotational motion of the moving platform is considered. Finally, the controller robustness against parameter uncertainties has been checked in part 3.

TABLE I: MANIPULATOR PARAMETER VALUES

Symbol Quantity Values

Mp Mass of the moving platform 6 kg

m Mass of the lower links 0.243 kg

m' Mass of the upper links 0.383 kg

l Length of the lower links 0.31 m

l' Length of the upper links 0.53 m

5.1 RESULTS OF SLIDING MODE CONTROLLER FOR CIRCULAR

MOTION TRACKING

The desired trajectories for the circular motion tracking are shown in table II. It is clear that a circular motion in x-y plant is supposed while the motion along the z direction is periodic. Fig. 2 shows the desired circle and the real motion of the manipulator. Obviously, the manipulator trajectory has covered the desired one. Fig. 3 represents tracking errors of P P P, aX Y nd Z . The steady state errors are less

than 0.3 mm. It should be noted that the tracking errors of θ ,φ and ψ have been almost zero and are not shown

here. TABLE II: CIRCULAR MOTION TRAJECTORIES

Task space parameters Desired Trajectories

PX 0.3cos(πt) (m)

PY 0.3sin(πt) (m)

PZ 0.55+0.2sin(πt/2) (m)

θ 0 rad

φ 0 rad

ψ 0 rad

5.2 RESULTS OF SLIDING MODE CONTROLLER FOR

ROTATIONAL MOTION TRACKING

The desired trajectories for rotational motion tracking are shown in table III. These trajectories are chosen somehow that include the reachable workspace in rotational motion of

the manipulator. Fig. 4 shows the tracking errors of θ ,φ and ψ in which the steady state errors are less than

2×10-5 rad. Also, tracking errors of P P P, aX Y nd Z have

been almost zero.

TABLE III: ROTATIONAL MOTION TRAJECTORIES

TASK SPACE PARAMETERS DESIRED TRAJECTORIES

PX 0 m

PY 0 m

PZ 0.55 m

θ π/6sin(πt/2) rad

φ π/6sin(πt/2) rad

ψ π/6sin(πt/2) rad

5.3 ROBUSTNESS STUDY OF SLIDING MODE CONTROLLER

One of the main goals of designing a sliding mode controller for the discussed manipulator is it's robustness against uncertainties. Therefore controller robustness against large parameter variations, especially the mass properties of the moving platform, is considered here. In this regard, simulations have been performed for different masses of the moving platform, from the nominal mass 6 kg up to 60 kg, and the results are shown in Fig. 5 to Fig. 10. In these simulations the system is forced to depart from the initial state [0 0 0.55 0 0 0] of the task space parameters to the desired state [0.3 0.3 0.8 π/6 π/6 π/6].

After scrutinizing the controller performance via the 3 steps of simulations the following aspects of the controller can be stated: provides fast transient response, no over shoot for the constant desired inputs even in the case of parameter uncertainties, the maximum trajectory tracking error is less than 0.3 mm and 2×10-5 rad for translation and rotation respectively, in addition possesses strong robustness.

Fig. 2. The desired and real circular motions

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Fig. 3. Tracking errors of , anX Y ZP d P P in circular motion

Fig. 4. Tracking errors of θ ,φ and ψ in rotational motion

Fig. 5. Errors of XP for different parameters of the manipulator

Fig. 6. Errors of YP for different parameters of the manipulator

Fig. 7. Errors of ZP for different parameters of the manipulator

Fig. 8. Errors of θ for different parameters of the manipulator

Fig. 9. Errors of φ for different parameters of the manipulator

Fig. 10. Error of ψ for different parameters of the manipulator

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VI. CONCLUSIONS

In this paper a new kind of six-DOF parallel manipulator was introduced. The main differences of this kind with the conventional designs were about the linkage and joints configuration. Kinematic analysis performed via integrating the differential constraint equations. Full dynamical model of the system was obtained using Lagrange formulation for the constrained systems. Due to the robot uncertainties and external disturbances, sliding mode controller was proposed for position control of the manipulator. Simulation studies show the effectiveness of the designed controller even in the case of large parameter variations of the moving platform.

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