A Robust Fuzzy Tracking Control Scheme for Robotic Manipulators with Experimental Verification

The performance of any fuzzy logic controller (FLC) is greatly dependent on its inference rules. In most cases, the closed-loop control performance and stability are enhanced if more rules are added to the rule base of the FLC. However, a large set of rules requires more on-line computational time and more parameters need to be adjusted. In this paper, a robust PD-type FLC is driven for a class of MIMO second order nonlinear systems with application to robotic manipulators. The rule base consists of only four rules per each degree of freedom (DOF). The approach implements fuzzy partition to the state variables based on Lyapunov synthesis. The resulting control law is stable and able to exploit the dynamic variables of the system in a linguistic manner. The presented methodology enables the designer to systematically derive the rule base of the control. Furthermore, the controller is decoupled and the procedure is simplified leading to a computationally efficient FLC. The methodology is model free approach and does not require any information about the system nonlinearities, uncertainties, time varying parameters, etc. Here, we present experimental results for the following controllers: the conventional PD controller, computed torque controller (CTC), sliding mode controller (SMC) and the proposed FLC. The four controllers are tested and compared with respect to ease of design, implementation, and performance of the closed-loop system. Results show that the proposed FLC has outperformed the other controllers.


Introduction
Robots are familiar examples of trajectory-following mechanical systems.Their nonlinearities and strong coupling of the robot dynamics present a challenging control problem, [1-3].Conventional methods of controlling a nonlinear system are based on models, especially in the field of robot control.Many robotic control schemes can be considered as special cases of model-based control called computed torque approach, [4].The basic concept of computed torque is to linearize a nonlinear system, and then to apply linear control theory.Practical implementation of the computed torque and other model based approaches can be found in [5], where the experimental results revealed that the simple PD controller has outperformed the other model based controllers.This is mainly due to the fact that in many dynamic systems the parameters may slowly change or cannot be exactly pre-dicted in advance due to different operating conditions.
Sliding mode controllers (SMCs) were first proposed in early 1950s.Due to their good robustness to uncertainties, SMC has been accepted as an efficient method for robust control of uncertain systems.Being limited only by practical constraints on the magnitude of control signals, the sliding mode controller, in principle, can treat a variety of uncertainties as well as bounded external disturbances, [6].A key step in the design of controllers is to introduce a proper transformation of tracking errors to generalized errors so that an n-order tracking problem can be transformed into an equivalent first-order stabilization problem.Since the equivalent first-order problem is likely to be simpler to handle, a control law may thus be easily developed to achieve the so-called reaching condition.Unfortunately, an ideal sliding mode controller inevitably has a discontinuous switching function.Due to imperfect switching in practice it will raise the issue of chattering, which is highly undesirable.To suppress chattering, a continuous approximation of the discontinuous sliding control is usually employed in the literature.Although chattering can be made negligible if the width of the boundary layer is chosen large enough, the guaranteed tracking precision will deteriorate if the available control bandwidth is limited, [7].A number of works related to sliding mode control of robotic manipulators have been published in [8][9][10][11][12].
Generally speaking, multiple-input multiple-output (MIMO) systems usually have characteristics of nonlinear dynamics coupling.Therefore, the difficulty in controlling MIMO systems is how to overcome the coupling effects between the degrees of freedom.The computational burden and dynamic uncertainty associated with MIMO systems make model-based decoupling impractical for real-time control.Adaptive control has been studied for many decades to deal with constant or slowly changing unknown parameters.Applications include manipulators, ship steering, aircraft control and process control.Although the perfect knowledge of the inertia parameters can be relaxed via adaptive technique, its real practical usefulness is not really clear and the obtained controllers may be too complicated to be easily implemented, [13].Because many design parameters (like learning rates and initialization of the parameters to be adapted) have to be considered in controller construction, most existing methodologies have limitations.Moreover, owing to the different characteristics among design parameters, attaining a complete learning, while considering an overall perfomance goal, is an extremely difficult task.Nevertheless, some experiments have been presented in [14,15].
Fuzzy controllers have demonstrated excellent robustness in both simulations and real-life applications, [16].They are able to function well even when the controlled system differs from the system model used by the designer.A customary for this phenomenon is that fuzzy sets, with their gradual membership property, are less sensitive to errors than crisp sets.Another explanation is that a design based on the "computing with words" paradigm is inherently robust; the designer forsakes some mathematical rigor but gains a very general model which remains valid even when the system's parameters and structure vary.
Otherwise, FLCs consist of a number of parameters that are needed to be selected and configured in prior, i.e. input membership functions, fuzzificztion method, output membership functions, rule base, premises connective, inference method and defuzzification.Optimal tuning of FLCs using genetic algorithms has attracted many authors, [17][18][19].In these papers, however, there are too many parameters involved in the development of FLCs.Furthermore, genetic algorithms cannot be used in real time control applications.In another study similar to the presnt work, i.e. real-time trajectory tracking control of two link robot using fuzzy systems [20], the controller needs 26 parameters to be experimentally selected.Also, the FLC in [21] needs 45 parameters to be tuned.This is beside the huge number of calculations involved in the online computation of the control signals.
In this research paper, we introduce a simple and computationally efficient FLC for MIMO second order systems with application to robotic manipulators.Earlier theoritical investigation of this controller, by the first author, can be found in [22].The controller is stable in the sense of Lyapunov theory of stability and few parameters are needed to be tuned.The approach can be implemented to both tracking and stabilizing control problems.However, in this paper, the emphasis is on the tracking control problem of robotic systems.The performance of the proposed controller is experimentally verified and compared with the conventional PD controller, computed-torque controller (CTC), and sliding mode controller (SMC).
The rest of this paper is organized as follows.Section 2 presents the model based controllers (CTC and SMC) that are used for comparison purposes.The proposed control scheme is introduced in Section 3 and Section 4 describes the experimental setup, the examined trajectories and the performance measures used in the control performance evaluation.The experimental results are demonstrated and discussed in Section 5. Section 6 offers our concluding remarks.

Preliminaries
The dynamic model of an n-joint manipulator can be written as follows: where q is the 1 n  joint angle vector,   u t is the 1 n  input torque vector,

 
M q is the n n  positive definite inertia matrix, is the  matrix representing the centrifugal and Coriolis terms,   f q  is the 1 n  vector of the frictional terms and   g q is the 1 n  vector of the gravity terms.Decoupled or decentralized (also independent) control means that the torque i to be generated by the ith actuator is based only on the value of the position of the ith joint and its time derivative u   , , , where is the actual value of the ith joint coordinate, i q and di is its desired value.The later ( di ) is usually available signal from the robot operating system and is planned in advance.Generally, defining the position error as , (2) can be written as where, again, d e q q   is the difference between the desired joint position vector and the actual one.
Obviously, the assumption of exact knowledge of the robot dynamic model cannot be satisfied in practical cases.Hence, the achievement of the desired tracking performance cannot be guaranteed.For this purpose, it would be desirable to add a term in the controller that compensates for the modeling errors.Several related works can be found in literature which suggests the use of neural networks [23] and neuro-fuzzy systems [24] in-order to compensate for the modeling errors.However, a complete review in this area is out of the scope of this work.In the experimental verification (Section 5), the CTC algorithm has been implemented as it is shown in Figure 1 This approach is widely adopted in industrial settings because of its simplicity (no dynamic model is required, in general) and because of its fault-tolerant feature, since, in case a single joint is affected by a failure, the robot can be retrieved in a safe position by means of the other joints.
The motion control problem of manipulators in joint space can be stated in the following terms.Assume that the joint position q and the joint velocity q are available for measurement.Let the desired joint position d be a differential vector function.We define a motion controller as a controller which determines the actuator torques u in such a way that the following control aim be achieved: q

Sliding Mode Control
In this subsection, the well-developed literature is used to demonstrate the main features and assumptions needed to synthesis a SMC for robotic systems.SMC employs a discontinuous control effort to derive the system trajectories toward a sliding surface, and then switching on that surface.Accordingly, it will gradually approach the control objectives, i.e. keep these trajectories at the origin of the phase plane.The following assumptions are needed to synthesis a SMC.

Computed Torque Control
The computed torque (also called inverse dynamics) technique is a special application of feedback linearization of nonlinear systems.The computed torque controller is utilized to linearize the nonlinear equation of robot motion by cancellation of some, or all, nonlinear terms.For this purpose, the dynamic model of the manipulator is exploited.Taking into account (1) and defining Assumption 1: The matrix

 
M q is positive definite and is bounded by a known positive definite matrix  

M q .
Assumption 2: There exists a known estimate   ˆ, h q q  for the vector function   , h q q  in (5).Now, let us define the linear time-varying surface we can derive the control scheme shown in Figure 1, where p K and v K are user-chosen diagonal matrices.So that the system is decoupled, linearized and the error dynamics is governed by the following expression [23]: is a time-varying linear function.From (1), ( 5) and ( 7), we can get the equivalent controller (also called ideal controller) 0 where   eq u t is equivalently the average value of   u t which maintains the system's trajectories (i.e.tracking errors) on the sliding surface .To ensure that they attain the sliding surface in a finite time and thereafter maintain there, the control torque The role of   ht u t acts to overcome the effects of the uncertainties and bends the entire system trajectories towards the sliding surface until sliding mode occurs.The hitting controller   ht u t is taken as: where and .
To verify the control stability, let us first get an expression for   s t  .Using (1), ( 5), ( 9) and ( 10), the first derivative of ( 7) is: s q t e t t q t q t t q t M q u t h q q t M q u K s Choosing a Lyapunov function and differentiating ( 12) using (11) which provides an asymptotically stable system.Since the parameters of (1) and ( 5) depend on the manipulator structure, it is difficult to obtain completely accurate values for   M q and .In SMC theory, estimated values are usually used in the control context instead of the exact parameters.So that, ( 8)-( 10) can be written as: where

 
M q and are bounded estimates for

 
ˆ, h q q    M q and   , q q  tively.As mentioned earlier mption 2, they are assumed to be known in advance. In at in g mode, the error dynamics is: So th the slidin c and 2 c .In su mary, e sl m arantee the stability in the Lyapunov sense even under parameter variations.As a result, the system trajectories are confining to the sliding surfaces (7).The control law (14) however, shows that the coupling effects have not been eliminated since the control signal for each degree of freedom is dependent on the dynamics of the other degrees of freedom.Independency is usually preferred in practice.Furthermore, to satisfy the existence condition of the sliding modes, a large uncertainty bound should be used.In this case, the controller results in large implementation cost and may lead to chattering efforts which should be avoided in practical implementation.

3
In design of stable controllers.To this end, consider a class of MIMO nonlinear second order systems whose dynamic equation can be expressed as: where   is the state vector and for the continuous feedback model (18) e definite, i.e.   0 Differentiating (19) with respect to time gives where Then the standard results in unov stability theory imply that the dynamic system (18) has a stab uilibriu 20) is 0  along the system trajectories.To achieve this, we have chosen the contro  i x to be p portional to 2 l  u ro i x  .Next, our controller design is achieved if we determine a fuzzy co   u x so that: ntrol where is a positive constant.The results o [26] state that, a fuzzy system that would approximate f Wang (21) ex .To this end, one would consider the state vector   x t and   x t  to be the inputs to the fuzzy system.The output of the fuzzy system is the control u .A possi orm of ontrol rules is: x is (lv) and/or 2 re the (lv) are lingui c values (e.g.p itive, neg ).T se rules consti e the rule base for a Mam dani-type FLC.
In the above formulation, two basic assumptions have been made.They The knowledge of the state vector.It is assumed to be available from mea  The control input, u is proportional to 2 x  .This assumption can be justified for a larg f second rules.O ro zy Tracking Control ry-following echanical systems.Their nonlinearities and strong courder to find a fuzzy controller that would ac e class o order nonlinear mechanical systems, [27,28].For instance, here in robotics, it means that the acceleration of links is proportional to the input torque.These two assumptions represent the basic knowledge about the system which is needed to derive the FLC f course, the exact mathematical model is not needed.
In the coming sub-section, we use this approach to design a PD-type FLC for the tracking control problem of botic systems.

Robotic Fuz
Robots are familiar examples of trajecto m pling of the robot dynamics present a challenging control problem.In practice, the load may vary while performing different tasks, the friction coefficients may change in different configurations and some neglected nonlinearities as backlash may appear.Therefore, the control objective is to design a stable fuzzy controller so that the link movement follows the desired trajectory in spite of such effects.We now apply the approach presented in the previous subsection in o hieve tracking to the robotic system under consideration.To this end, let us choose the following Lyapunov function candidate where again,      d e t q t q t    ,       d e t q t q t        d q t and and   d q t  are vectors locity respectiv of the desired joint ely.Differentiating with position and ve respe ime and g (20) gives

ct to t usin
To enforce asymptotic stability, it is required to find u so that 0 V ee ee in some neighborhood of the equilibr the control u to be proportional to ium of (22).Taking , ( 23) can be ree  written as: where i  is positive constant, (24)  gative and zero control inputs.These rules are simply the fuzzy par on f e, e  nd u which follow directly from the stabilizing conditions of the Lyapunov function, (22).
In concluding words, the presented approach transforms c titi s o of act mathematical quantities to the world of words [16].This combination provides us with a solid analytical basis from which the rules are obtained and justified.
To complete the design, we must specify the membership functions defining the linguistic terms in the se.Here, we use the Gaussian membership functions where and z stands for control variable, the product nd" and center of gravity inferencing.some p in eq 0 z a  for "a g For ositive constants u a , ep a and ev a , the above four rules can be represented by the follow uation: In (25), the inputs are the error in position and the error in velocity and the output is the contr l input of joint i; i.e. it is a PD-type FLC.The followin remarks ar  The FLC in (25)  e in order: stems, o introduce the input variables ( i e and i e  ) to he fuzzy network.Also the fuzzification and defuzzification methods used in this study are not unique; see [28] for other alter x ple, usi membership functions (e.g.triangular, trapezoidal  etc.) will result in a different FLC.However, the FLC in ( 25) is a simple one and the closed form relation between the inputs and the output makes it computationally inexpensive. Only three parameters per each DOF need to be tuned, namely, they are .This greatly simplifies the tuning procedure, since the search space is quite small relative to other works.For instance, the FLC in [21] needs 45 parameters to be tuned for a one DOF system  This controller is inherently bounded since

Experimental Setup: Test Rig, Reference Trajectory and Performance Measures
In this study, we have considered a two link planar robot whose diagrammatic sketch is shown in  2. These inertia parameters have been calculated by simply measuring and weighting the mechanical ele arms.

The Test Rig
The test rig consists of a geared-drive horizontal robot arm with 2 DOF whose rigid links are joined with revolute jo  The potentiometers are one turn (300 degrees) and 1 kΩ.Each potentiometer is coupled to the joint motor.Both potentiometers are supplied by ±5 V, so that each one has a resolution of 0.033 volt/degree.The velocity of each link is obtained by using the position signal and utilizing first order backward differencing technique.
The feedback signals from the potentiometers and the control signals to the motor drives are sent to/from the computer via PCI-DAS6014 AD/DA interface card.The card has a minimum 200 kS/s conversion rate and has an absolute accuracy of 8.984 mV when operates at the range ±10 V.The control program is written in C++ an ecuted at 1 ms sampling rate.Figure 5 shows the closed loop control system.

The Reference Trajectory
During the preliminary evaluation of the proposed FLC, we have examined three trajectories.They are sinusoidal trajectory, linear trajectory with parabolic c results of the latest trajectory.Resu je parabolic blends, can be found in the master thesis of the second author, [29].A cubic polynomial trajectory in the joint space is defined by: where 0 1 2 , , a a a and 3 a are constants determined upon the trajectory constraints.The desired motion of the two joints is identical and starts from zero to 45˚ in 10 seconds.The motion constraints (boundary conditions) are: ( 0) is the joint number.The desired to (26) will be: trajectory according where   d q t is in degrees.

The Performance Measures
While comparing the efficiencies of the ere experimentally tested on the robot arm, we will use king error to quantatively compare the performance results.One measure controllers that w some meaningful measures of the trac ti that will be used is the scalar valued Root Mean Square (RMS) error defined as where is the tracking error.Since data are only discrete time intervals,   e t sent back at 1 N t t with constant sam ling period p 1 j j T t t     for all j; we discretize (28) as   q j denotes     j q t q j T   and f N T T    .To get m nsight o form use the maximum absolute value of the tracking error after two second from the starting time.We name it as as ore i n the tracking per ance, we also max e which is defined The above two measures have been also adopted in [15].

Results a
n, the experiments conducted using four are the conosed FLC, the CTC and e SMC.For the sake of comparison, we ran each The control torque for the proportional-plus-derivative (PD) controller is defined by: position error eq where K P and K D are 2 2  po gonal matrices called the propor e derivative gain matrices, respectively.A traditional with PD c sitive definite dia tional and th problem associated ontrol is that we cannot increase the controller ga of the gains exceed their critical values, the system becomes u performance of the PD controller is restri ins, as much as we want, to improve the controller's performance.When the values nstable.Thus the cted with the values of these gains.
In the experiments, the proportional feedback gains of the PD controller were set to

 
, h q q  were computed on-line using the parameter values presented in Table 2 and the equation of m of two link planar robot which can be found in [4].
For the SMC, the e

 
M q ere se ,   ˆ, h q q  and the hitting control gain K in (14) w t The coefficients of the function  as:  6 shows that the transient period of the PD controller is slightly higher than that of the FLC, Figure 7.The transient phase of the CTC was the longest one as it can be noticed from Figure 8. Figure 10 shows that the SMC was successful in bending the system trajectories toward sliding surfaces and consequently the errors have converged as depicted in Figure 9.
The performance measures are given in Figure 11 and edback controllers rder nonlinear systems.This control scheme has been applied to the control of a two-link robot.It can also be extended to n number of link robots.Experimental results show that the design procedure has been successful in representing the nonlinear dynamics in the control context and resulted in a stable closed-loop control.Robustness of the FLC has been examined via initial position errors.Relative to the conventional PD controller, CTC and SMC, the proposed FLC exhibits the best performance.

References
the case of robotic control, this controller can be regarded as output feedback controller since the joint's joi easured, it can be easily obigure, and are the links lengths; and tio position and velocity are usually the outputs.If the nt velocity is not m tained using a differentiator as shown in Figure 2.

FigureFigure 4 .
The robot has been built at the a ab the ce masses of the two lin The parameter values of the links are given in Table

Figure 2 .
Figure 2. Configuration of the robotic fuzzy control structure (the case of two-link robot).

Figure 3 .Figure 4 .
Figure 3. Schematic diagram of the two-link robot.Table 2. Parameters of the robot arm.Parameter Link 1 Link 2 m mass ( 0.096 kg) 0.471  length

Figure 5 .
Figure 5. Block diagram of the test rig.
strength and weakness of each design.To show robustness, the four controllers have been initiated with initial ual to 10˚, i.e. the robot is at rest, i.e. .This condition nd Discussion n this sectio I controllers are presented.These controller ventional PD controller, the prop th controller with the same initial conditions to analyze the yields an initial position error [ 0.175 0.175] T e   radian.
the base and elbow links, respectively.They have been selected as high as possible without violating the stability of the overall system.With respect to the proposed FLC, the control gains were set these parameters experimentally after few trials.The criteria upon which these values have been chosen is simply the fa C, the cont stest possible convergence of the initial errors.For the CT rol gains according to (for the base and elbow links, respectively.ontrol ve e best possible tracking performance.The matrix These c gains ha achieved th M q and the vector otion stimated values for

Figure 12 .Figure 6 .Figure 7 .Figure 12 .
Figure 6.The desired and actual trajectories of (a) joint one, (b) joint two (b) and (c) the tracking errors of the PD controller.

Table 1 . Fuzzy rules for the tracking controller.
Copyright © 2011 SciRes.ICA They show that, after suitable selection of the tuning parameters, the tracking errors of the four controllers have converged to a close zone around zero in the steady state phase.Figure