Adaptive Fuzzy Computed-Torque Control for Robot Manipulator with Uncertain Dynamics Regular Paper

To  overcome  the  disadvantages  of conventional  computed‐torque  control  and  fuzzy control,  and  to  exploit  their  attractive  features,  this paper proposes  two  types of  adaptive  control  scheme combining  conventional  computed‐torque  control  and different  fuzzy  compensators  for  the  robust  tracking control  of  robotic  manipulators  with  structured  and unstructured uncertainties. Fuzzy compensators based on  feed‐forward  and  feed‐back  are  developed  to compensate  these uncertain dynamics. On  the basis of Lyapunov  stability  theory,  a  tracking  error  limit  is derived  for  the  closed‐loop  control  system  and  the convergence  and  stability  of  the  control  schemes  are proved.  Comparisons  of  their  performances  with conventional  computed‐torque  controllers  under  the condition  of  these  uncertainties  are  carried  out.  The validity of the two types of adaptive control scheme is shown by numerical simulations of a  three‐link rotary robot manipulator.


Introduction
Robot manipulators with high-speed and high-precision trajectory tracking are in urgent demand in advanced manufacturing fields, such as packaging and assembly of micro-electronic products [1].High-speed and highprecision mean that the robot manipulator completes pick-and-place operations with good control performance.However, controller design is subject to structured and unstructured uncertainties, which stem from the strong coupled and nonlinear dynamic system of robotic manipulators.Structured uncertainty is characterized by correct dynamic models which contain uncertain dynamic parameters due to the imprecision of the manipulator link properties, unknown loads, inaccuracies of the torque constants of the actuators, and so on.Unstructured uncertainty is characterized by unstructured dynamics resulting from the presence of high-frequency dynamic modes of the manipulator, nonlinear friction, and so on [2].These uncertainties may have a dramatic effect on a controller's performance and may induce instability.To deal with these uncertainties, some authors [3][4][5] have proposed several adaptive control schemes, where the adaptive sliding mode control is an effective and popular means of dealing with the parameter variations and external disturbances.Kuo [6] also proposed a novel adaptive fuzzy sliding mode control method, where fuzzy rules effectively alleviate the input chattering.In order to compensate the friction force, Ohri [7] presented an adaptive fuzzy compensator based on the sliding mode control.Lin and Hsu [8] discussed a direct adaptive fuzzy sliding mode control method, where a fuzzy compensator is used to tune both the premise and the consequence parts of fuzzy rules.Lu [9] revised the adaptation laws of the algorithm discussed by Lin and Hsu [8] and applied it to robotic manipulators.Li [10] presented a new fuzzy sliding mode controller with a fixed-order compensator based on the theory of sliding mode control and fuzzy control.In order to achieve robustness with respect to external disturbances, unstructured dynamics, and model uncertainty properties of muscle-joint dynamics, Kobravi [11] proposed a robust control framework, which is based on the combination of an adaptive nonlinear compensator with a sliding mode control.Zeng [12] presented a fuzzy sliding mode control strategy based on support vector machines, where a fuzzy control algorithm completes adaptability to system disturbance.Zhu [13] designed a fuzzy support vector machine control strategy to reduce the oscillation of the sliding mode control.Although the sliding mode control has the ability to provide high robustness for control systems, undesirable chattering on the sliding surface due to frequent switching can deteriorate control system performances [14].Another problem of the sliding mode control schemes is that all of the upper and lower bounds of uncertain parameters need to be obtained before the design of the controller.Thus, if the controlled systems have many unknown parameters, the design of the sliding mode controller will become very difficult.
Computed-torque control is one effective model-based control scheme that can offer a large variety of advantages over model-free methods, such as potentially higher tracking accuracy, lower feedback gains, and lower energy consumption [15].However, it has become widely recognized that the tracking performance of this method in high-speed operations is severely affected by structured and unstructured uncertainties [16].Fuzzy logic system has also been widely applied to generate auxiliary joint torques to compensate these uncertainties.Song [17] developed a new approach to solve trajectory tracking problems of robotic manipulators through combing computed-torque control and fuzzy control.Although the fuzzy control part in the approach can approximate uncertainties in robotic manipulator, parameters in the fuzzy control part need to be determined and tuned by a complex Lyapunov equation, which will result in a complex controller design.Chen [18] combined the conventional computed-torque control with a fuzzy logic system to improve trajectory tracking performance of an industrial robot.However, because the fuzzy logic rules in the fuzzy control system are determined by some experimental data and designers' experiences rather than the fuzzy adaptive control law, it is very difficult to ensure stability for the general fuzzy control systems with uncertain dynamics.Mohan and Bhanot [19] presented an investigation into three kinds of hybrid fuzzy control algorithms combined with a conventional computed-torque controller.Although performances of a hybrid controller improves appreciably compared to their respective fuzzy or conventional counterparts, the fuzzy controller used in these algorithms is a lookup table-based controller rather than a self-adaptive fuzzy controller, and therefore presents the disadvantage of having to build up the lookup table, which can be a long process.
To overcome these disadvantages and exploit the attractive features of conventional computed-torque control and fuzzy control, this paper proposes two types of adaptive control schemes combining the conventional computed-torque control and different fuzzy compensators for the robust tracking control of robotic manipulators with constructed and unconstructed uncertainties.The stability and convergence of the tracking error are guaranteed by using the Lyapunov method.The experiment is carried out on the first three links of a PUMA 560 robotic manipulator.Numerical simulations are employed to assess the performance of the two types of adaptive controller.This paper is organized as follows.Section 2 outlines the dynamic modelling of the robot manipulator and the model descriptions of fuzzy logic systems.This is followed in Section 3 by the formulation of the two proposed adaptive controllers.Their performances are verified through computer simulations of a three-link rotary robot manipulator.The results are illustrated in Section 4 along with comparisons between controllers.In Section 5, we conclude with remarks and suggestions for further studies pertaining to the complex control problem.

Problem formulation and preliminaries
The dynamic equation of a n degree-of-freedom manipulator in joint space coordinates is given by: where the vectors , , q q q   are the joint angle, the angular velocity, and the angular acceleration, respectively;   D q is the n n  symmetrical positive definite inertia matrix;   , H q q  is the 1 n  vector of Coriolis, centrifugal torques and gravitational torque;   , F q q  are uncertainties including friction terms and external disturbances, and so on; and  is the 1 n  vector of actuator joint torques.
The robot dynamic Eq. (1) represents a highly nonlinear and coupled system.In most practical cases, the model is not exactly known.Thus, only nominal estimations of the model are available for controller design.Computedtorque control is the most effective approach for robot motion control when a nominal robot dynamic model is available.In this case the control law can be written as where , H q q  , respectively; v K and p K are n n  symmetrical positive definite gain matrices; d q is the desired joint trajectory;   is defined as trajectory tracking error vectors.
Substituting Eq. ( 2) into Eq.( 1) yields the following closed loop tracking error dynamic equation in the statespace form: where Since there are always uncertainties in the robot dynamic model, the ideal error response cannot be achieved in general.The actual system performance is governed by Eq. ( 3), which will result in the computed-torque control not being robust in practice.To improve robustness, we introduce a multi-input and multi-output fuzzy logic controller as a compensator for the uncertainties due to friction, disturbance, or payload variation.In this paper, the final output result of a multi-input and multi-output fuzzy system is expressed by the following definition.
Definition 1: A multi-input and multi-output fuzzy logic system performs a mapping from fuzzy sets in  , based on the fuzzy IF-THEN rules.The output of a multi-input and multi-output fuzzy logic system with centre-average defuzzifier, product inference, and singleton fuzzifier takes the following form: where l j y is the point in j V at which fuzzy membership function The fuzzy basis function can be defined as Thus, Eq. ( 4) can be rewritten as follows: where is the fuzzy basis function vector, and Then, the overall output of a multi-input and multioutput fuzzy logic system can be rewritten as where jth column of the matrix  .

Adaptive control schemes
In this section, two types of adaptive control schemes combining computed-torque controller and fuzzy compensators are designed to compensate the uncertainties.The configuration of the adaptive control scheme based on a feed-back fuzzy compensator (FBC) is shown in Fig. 1.The controller takes the actual commands as the input variables of fuzzy rules, and the error is used to tune parameters of the fuzzy compensator.The robot joint torques in the two types of controller are depicted as follows.
  where ff  and fb  are the joint torques in the FBC-based and FFC-based adaptive controllers, respectively; , d d q q  are the desired position and velocity; , q q  are the actual position and velocity.For the sake of simplicity, the following torque equation is used to represent both schemes ( 8) and ( 9): Thus, the corresponding closed loop tracking error dynamic equation is (under the assumption Since the control objective is to generate joint torques  to reduce the error signal  in Eq. ( 11), clearly minimizing the error signal  by the use of fuzzy compensator allows us to achieve ideal computed-torque control directly.

Fuzzy compensator design
In order to make the joint motions of the robotic system follow the desired trajectories, the computed-torque controller 0  in Figs. 1 and 2 is connected to the compensator f  to generate a controller signal  for the robot manipulator.That is, the control law is given by where 0  is the output torque of the computed-torque controller, defined as Eq.( 2), and f  is the output torque of the fuzzy compensator.
The  in Eq. ( 11) is defined by an ideal fuzzy logic system as where   x  is reconstruction error of the fuzzy logic system; , , , is the fuzzy basis function vector, and is the input vectors of the fuzzy system.
  in Eq.( 13) is an optimal weight matrix and is defined as follows:   arg min sup where , x    denote the sets of suitable bounds on  and x , respectively, and x   , which can be defined as follows: Thus, the fuzzy compensator control law f  in Eq. ( 12) can be defined as follows: From Eq. ( 12) and Eq. ( 16), the tracking error dynamic Eq. ( 11) can be rewritten as where 0 On the basis of the above discussions, the following result can be established by Lyapunov stability theory.
Theorem 2. If we select the following control law based on the adaptive fuzzy compensator in control torque Eq. ( 11) where P is a unique n n  positive definite symmetrical matrix, which satisfies the following Lyapunov equation: Then, the closed-loop system is asymptotically stable.
Proof: Let us define a Lyapunov function candidate as follows:   The derivative of V with respect to time is given by From the adaptive control law Eq.( 19) and Lyapunov Eq. ( 20), we can get Therefore, based on Rayleigh-Ritz theorem 1, we have the following inequality: The V  is negative as long as the term in the brace is positive, which implies The negative semi-definiteness of V  outside the compact set implies the boundedness of , V X and  .Integrating Eq. ( 24) with respect to time From Eq. ( 26), we obtain 2 X L  .In addition, the boundedness of X implies X L   .From Eq. ( 17) and the boundedness of , X  , we can get is concluded by Barbalat's lemma [21].Therefore, the stability of the closed-loop system is guaranteed using the control law Eq.( 19).

Simulation results and discussions
In this section, we examine the performance of the proposed adaptive fuzzy computed-torque controller through simulations on a three-link rotary robot manipulator gripping an unknown load, shown in Fig. 3.Moreover, we also make a comparison between the proposed controllers and conventional controllers.As shown in Fig. 3, let   , ,

T
x y z denote the position of the end-effector, and the manipulator is modelled as three rigid links (of lengths 1 L , 2 L and 3 L ) with point masses at the distal ends of the links ( 1M , 2 M and 3 M ).Using the conventional Euler-Lagrangian approach, we derive the robot dynamic equation of the three-link rotary robot manipulator as follows: 2 sin( ) S q q   , and 23 2 3 cos( ) C q q   .
. Moreover, a 40kg uncertainty payload is attached to the third link, and coulomb friction and viscous friction forces .In the simulation, the initial configuration of the three-link rotary robot manipulator is set as . To simulate these control schemes of the three-link rotary robot manipulator, the kinematics and dynamic models obtained from Eq. ( 27) are implemented within the MATLAB environment, and Matlab/Simulink software is used as the simulation tool.

Comparison studies between adaptive controller and conventional controller
For the purposes of comparison, simulation studies in three cases were conducted, and the tracking performances are described in Figs. 4, 5 and 6.As can be seen from Fig. 4 (a-c), the computed-torque controller based on precise dynamical knowledge of robotic manipulators without uncertainties indeed yields convergence of the tracking errors.Furthermore, as can be observed from the input torques applied to the three joints in Fig. 4 (d), the computed-torque controller with precise dynamics shows smooth control performances.However, uncertain dynamics will make the computed-torque control approach unattractive to industry.The computed-torque control scheme is designed according to nominal parameters instead of actual parameters.Furthermore, model uncertainties attached to the third link and friction forces added to each joint are also taken into account in the dynamic model.Compared with the first case, it can be easily seen from Fig. 5 (a-c) that the tracking performances in the second case are much worse than in the first case.The evaluation of input torques for the three joints in Fig. 5 (d) displays relatively large oscillations, which might go beyond the limits of actuators.Thus, the conclusion can be drawn that the computed-torque controller with uncertain dynamics will become unstable.
Case 3: In order to decrease the tracking errors caused by the uncertain dynamics in the second case, the proposed adaptive computed-torque controller combined with the fuzzy compensator in Eq. ( 19) is used to control robotic manipulators with uncertain dynamics.
in Eq. ( 11) due to uncertainties and external disturbances.The input vector of the fuzzy compensator is defined as . The universe of discourse of each fuzzy input is divided into five fuzzy labels, i.e., NB, NS, ZO, PS, PB.The five membership functions are defined as , where i C are -1, -0.5, 0, 0.5, and 1 as i A are NB, … , PB, respectively, and i  is equal to 0.2124.The initial values of weight parameters  in adaptive Eq. ( 19) are specified as zero matrices.In addition, the positive definite matrix Q in Lyapunov Eq. ( 20) and the gain matrix  in the adaptive Eq. , respectively.The uncertainties and external disturbances in the dynamics equation are the same as in the second case.As can be seen from Fig. 6, the oscillations of the tracking errors are remarkably reduced compared with the second case, and the input torques for the three joints have smooth control actions.Therefore, these simulation results demonstrate that the computed-torque controller combined with a fuzzy compensator can achieve better control performances with uncertainties and external disturbances in the dynamics equation than the computedtorque controller without a fuzzy compensator.

Comparison studies between FBC-based and FFC-based adaptive controllers
In order to extensively investigate the control performances of the FBC-based and FFC-based adaptive controllers, simulation studies are performed on the three-link rotary robot manipulator gripping an unknown load, shown in Fig. 3   where p E is the position tracking error; v E is the velocity tracking error; N is the number of elements of the position vectors; , di i q q are the desired and actual trajectories, respectively.As listed in Table 1, there are very clear differences between the three control schemes.The two adaptive controllers demonstrate extremely good control performances compared with the uncompensated case.Moreover, the FFC-based adaptive controller performs slightly better in the circular trajectory than the FBCbased adaptive controllers.The trajectories of the FFCbased adaptive controller and the uncompensated case for one cycle are plotted together in Fig. 7, where the label "A" denotes the trajectory of the FFC-based adaptive controller and the label "B" represents the trajectory of the uncompensated case.As can be seen from Fig. 7, the trajectory of the FFC-based adaptive controller is more close to the desired trajectory.

Conclusions
This paper discusses two types of adaptive control schemes, where feed-forward and feed-back fuzzy compensators are used to compensate modelling uncertainties and external disturbances.Comparisons of their performances with conventional computed-torque controllers under the condition of these uncertainties are carried out.Comparative results demonstrate that the two types of adaptive control schemes are effective in improving control performances in terms of modelling uncertainties and external disturbances.The convergence and stability of the two control schemes are proved by using the Lyapunov method.Computer simulation of a three-link rotary robot manipulator is carried out.Simulation results show that the trajectory tracking errors of the robotic manipulator can be significantly reduced.Further research should be carried out on the more general reduction algorithm of the number of fuzzy rules.


achieves its maximum value, which is assumed to be 1; l i A and l j B are the linguistic variables of the fuzzy sets in the subspace i U and j V , described by their membership functions the number of fuzzy subset.

Figure 1 .
Figure 1.Configuration of FBC-based adaptive controller.The configuration of the adaptive control scheme based on a feed-forward fuzzy compensator (FFC) is illustrated in Fig. 2. At first glance, the FBC-based and FFC-based adaptive controllers have a common adaptive learning

Case 1 :
Figure 4  shows the tracking performances of three joints where the computed-torque controller takes control action without uncertainties.These control parameters are chosen as

Figure 4 .Case 2 :
Figure 4. Tracking performances of three joints where the computed-torque controller takes control action without uncertainties: (a) the first joint; (b) the second joint; (c) the third joint; (d) input torques for three joints.Case 2: Figure 5 illustrates a set of simulation results of the tracking performances when the computed-torque controller is applied to the robot manipulator with uncertain dynamics with the same control parameters as in the first case.

Fig. 5 .
Fig. 5. Tracking performances of the three joints where the computed-torque controller takes control action with uncertainties: (a) the first joint; (b) the second joint; (c) the third joint; (d) input torques for all three joints.

Fig. 6 .
Fig. 6.Tracking performances of the three joints where the computed-torque controller plus fuzzy compensator takes control action with uncertainties: (a) the first joint; (b) the second joint; (c) the third joint; (d) input torques for all three joints.Fig.6 depicts the tracking performances of the three joints in this case.The fuzzy compensator is used to compensate the uncertain parts (19) are chosen as   50,50,50,50,50,50 Q diag  and   100,100,100 diag   . The control performances are tested by tracking a circular movement in the horizontal plane following the desired trajectory 0.3 0.1sin( ) measured by the following equations of tracking error over one training cycle of a trajectory.

Fig. 7 .
Fig. 7. End point tracking of a circular trajectory for uncompensated and FFC-based control schemes.

Table 1
summarizes the tracking errors after convergence of the two controllers.

Table 1 .
Tracking errors after convergence.