A novel structure of fuzzy logic controller is presented for trajectory tracking and vibration control of a flexible joint manipulator. The rule base of fuzzy controller is divided into two sections. Each section includes two variables. The variables of first section are the error of tip angular position and the error of deflection angle, while the variables of second section are derivatives of mentioned errors. Using these structures, it would be possible to reduce the number of rules. Advantages of proposed fuzzy logic are low computational complexity, high interpretability of rules, and convenience in fuzzy controller. Implementing of the fuzzy logic controller on Quanser flexible joint reveals efficiency of proposed controller. To show the efficiency of this method, the results are compared with LQR method. In this paper, experimental validation of proposed method is presented.

1. Introduction

The trajectory tracking control of robotic manipulators with joint flexibility has received considerable attention, owing to the complexity of the problem. Many robots incorporate harmonic drives for speed reduction, and it is known that such drives introduce tensional elasticity into the joints [1]. Industrial robots generally have elastic elements in the transmission systems, which may result in the occurrence of tensional vibrations when a fast response is required. For many manipulators, joint elasticity may arise from several sources, such as elasticity in gears, belts, tendons, bearings, hydraulic lines, and so forth, and may limit the speed and dynamic accuracy achievable by control algorithms designed assuming perfect rigidity at joints. A proper choice of mathematical model for a control system design is a crucial stage in the development of control strategies for any system. This is particularly true for robotic manipulators due to their complicated dynamics [2].

Experimental evidence suggests that joint flexibility should be taken into account in both modelling and control of manipulators if high performance is to be achieved. To model this elastic behaviour in the joints, the link is considered as connected to rotor through a tensional spring of stiffness 𝐾. The introduction of joint flexibility in the robot model considerably complicates the equations of motion. In particular, the order of the related dynamics becomes twice that of the rigid robots, and the number of degrees of freedom is larger than the number of inputs, making the control task difficult.

Research on the dynamic modelling and control of flexible robots has received increased attention in the last decades. A first step towards designing an efficient control strategy for manipulators with flexible joints must be aimed at developing dynamic models that can characterize the flexibility of the joints accurately. The controller design that minimizes the effects of the flexible displacements in lightweight robots is highly demanded in many industrial and space applications that require accurate trajectory control. In control applications of robot, manipulators with flexible arms are targeted either to reach a target position or to follow a prescribed trajectory. In the first case to reach a target position, a short settling time is desired while a large robot arm displacement is planned in the second case to follow a prescribed trajectory. In both cases, strong control actions are applied to the robot arm, and, as a result, undesired behaviours could appear if vibrations induced in the robot arm are not considered [3].

The control issue of the flexible joint is to design the controller so that link of robot can reach a desired position or track a prescribed trajectory precisely with minimum vibration to the link. In order to achieve these objectives, various methods using different technique have been proposed such as follow: linear quadratic regulation (LQR) control [4], adaptive output-feedback controller based on a backstepping design [5–7], nonlinear control based on the feedback linearization technique and the integral manifold technique [8, 9], robust control based on PD control [10] and robust H∞ control [11], fuzzy control, PD fuzzy and neural network [12–15], optimal control [16, 17], and so forth [2, 18–23].

In this paper, a novel structure of fuzzy logic controller is designed to control trajectory tracking and vibration of a flexible joint manipulator system. The rule base of fuzzy controller is divided into two sections. Each section includes two variables. The variables of the first section are the error of tip angular position and the error of deflection angle, while the variables of the second section are derivatives of mentioned errors. Using these structures, it would be possible to reduce the number of rules. The proposed fuzzy logic controller is implemented on the Quanser flexible joint to show efficiency of proposed controller.

The paper is organized as follows. The flexible joint manipulators are described in Section 2. Section 3 explains modelling of the flexible joint manipulator. Section 4 is focused on introductory to fuzzy logic controller. The proposed fuzzy logic controller are described in Section 5. Section 6 illustrates the implementation results. Finally, the conclusion is presented in Section 7.

2. The Flexible Joint Manipulator System

The flexible joint manipulator system considered in this work is shown in Figure 1, where πœƒ is the tip angular position and 𝛼 is the deflection angle of the flexible link. The base of the flexible joint manipulator which determines the tip angular position of the flexible link is driven by servomotor, while the flexible link will response based on base movement. The deflection of link will be determined by the flexibility of the spring as their intrinsic physical characteristics [24].

3. Modelling of the Flexible Joint Manipulator

This section provides a brief description on the modeling of the flexible joint manipulator system, as a basis of a simulation environment for development and assessment of the nonlinear control. The Euler-Lagrange formulation is considered in characterizing the dynamic behavior of the system. Considering the motion of the flexible joint system on a two-dimensional plane, the potential energy of the spring can be formulated as [24]1𝑉=2𝐾stiff𝛼2,(1) where 𝐾stiff is the joint stiffness. The kinetic energies in the system arise from the moving hub and flexible link can be formulated as1𝑇=2𝐽eqΜ‡πœƒ2+12𝐽Armξ€·Μ‡ξ€Έπœƒ+̇𝛼2,(2) where 𝐽eq and 𝐽Arm are the equivalent inertia and total link inertia, respectively. To obtain a closed-form dynamic model of the flexible joint, the energy expressions in (1) and (2) are applied to formulate the Lagrangian; that is,1𝐿=π‘‡βˆ’π‘‰=2𝐽eqΜ‡πœƒ2+12𝐽Armξ€·Μ‡ξ€Έπœƒ+̇𝛼2βˆ’12𝐾stiff𝛼2.(3)

Two generalized coordinates are πœƒ and 𝛼. Let the generalized torque corresponding to the generalized tip angle be 𝑇outputβˆ’π΅eqΜ‡πœƒ. Use Lagrangian’s equation as follows:πœ•ξ‚΅πœ•π‘‘πœ•πΏπœ•Μ‡πœƒξ‚Άβˆ’πœ•πΏπœ•πœƒ=𝑇outputβˆ’π΅eqΜ‡πœ•πœƒ,ξ‚€πœ•π‘‘πœ•πΏξ‚βˆ’πœ•Μ‡π›Όπœ•πΏπœ•π›Ό=0.(4)

The equation of motion is obtained as below:𝐽eqΜˆπœƒ+𝐽Armξ€·Μˆξ€Έπœƒ+Μˆπ›Ό=𝑇outputβˆ’π΅eqΜ‡π½πœƒ,Armξ€·Μˆξ€Έπœƒ+Μˆπ›Ό+𝐾stiff𝛼=0,(5) where 𝐡eq is the equivalent viscous damping and 𝑇output is the output torque on the load from the motor, defined as𝑇output=πœ‚π‘šπœ‚π‘”πΎπ‘‘πΎπ‘”ξ€·π‘‰π‘šβˆ’πΎπ‘”πΎπ‘šΜ‡πœƒξ€Έπ‘…π‘š,(6) where πœ‚π‘š is the motor efficiency, πœ‚π‘” is the gearbox efficiency, 𝐾𝑑 is the motor torque constant, 𝐾𝑔 is the high gear ratio, πΎπ‘š is the motor back-EMF constant, and π‘…π‘š is the armature resistance. The linear model of the uncontrolled system can be represented in a state-space form as shown in (7); that is,Μ‡π‘₯1=π‘₯3,Μ‡π‘₯2=π‘₯4,Μ‡π‘₯3=π‘Žπ‘₯2+𝑏π‘₯3+𝑐𝑒,Μ‡π‘₯4=𝑑π‘₯2+𝑓π‘₯3βˆ’π‘π‘’,(7) where Μ‡π‘₯=[πœƒπ›ΌπœƒΜ‡π›Ό]𝑇 and π‘Ž, 𝑏, 𝑐, 𝑑, and 𝑓 are given asπΎπ‘Ž=stiff𝐽eq,𝑏=βˆ’πœ‚π‘šπœ‚π‘”π‘˜π‘‘π‘˜π‘šπΎ2𝑔+𝐡eqπ‘…π‘šπ½eqπ‘…π‘š,πœ‚π‘=π‘šπœ‚π‘”πΎπ‘‘πΎπ‘”π½eqπ‘…π‘š,𝐾𝑑=βˆ’stiff𝐽eq+𝐽Arm𝐽eq𝐽Arm,πœ‚π‘“=π‘šπœ‚π‘”π‘˜π‘‘π‘˜π‘šπΎ2𝑔+𝐡eqπ‘…π‘šπ½eqπ‘…π‘š.(8) In (7), the input 𝑒 is the input voltage of the servomotor, π‘‰π‘š which determines the flexible joint manipulator base movement. In this study, the values of the parameters are defined as Table 1. Directions of torque to reduce the deflection angle when link moves anticlockwise and clockwise are shown in Figures 2(a) and 2(b), respectively.

4. Fuzzy Logic Controller

The first studies in the field of fuzzy systems and control have been making a big progress motivated by the practical success achieved in industrial process control applications. Fuzzy systems can be used either as open-loop controllers or as closed-loop controllers, as shown in Figure 3. When used as a closed-loop controller, the fuzzy system measures the outputs of the process and takes control actions on the process continuously. Applications of fuzzy systems in industrial processes belong to this category. The fuzzy controller uses a form of quantification of imprecise information (input fuzzy sets) to generate by an inference scheme, which is based on a knowledge base of control force to be applied on the system.

Two of the difficulties in the design of fuzzy control systems are to generate the membership functions and choice of the fuzzy rules. In fact, the decision-making logic is the way in which the controller output is generated. It uses the input fuzzy sets, and the decision is taken according to the values of the inputs. Moreover, the knowledge base comprises knowledge of application domain and the attendant control goals.

5. Proposed Fuzzy Logic Controller

The proposed fuzzy logic controller including four inputs; the error of tip angular position β€œπ‘’1,” the error of deflection angle β€œπ‘’2,” and their derivatives β€œπ‘‘π‘’1” and β€œπ‘‘π‘’2”. The input voltage of the servomotor is considered as a output of fuzzy controller. For each input and output variable, three membership functions in the names of β€œpositive,” β€œnegative,” and β€œZero” are defined. Figure 4 shows the structure of designed fuzzy logic controller.

As Figure 4 shows the rule base of fuzzy controller is divided to two small sections, each section contains 9 rules. Using these sections reduced the number of rules from 81 rules to 18 rules. So, not only computational complexity decreases but also the interpretability of rule base increases. Another advantage of proposed rule base is that the implementation of fuzzy controller is more convenient.

Figure 5 demonstrates membership function of four inputs and one output of the suggested controller. Because the model of system is nonminimum phase, the effect of tip angular position’s error and deflection angle’s error as two inputs are reverse of each other. Therefore, when tip angular position’s error and deflection angle’s error are in the same sign, the input voltage of the servomotor should be zero, and, when they are in different sign, the input voltage, depending on conditions, should be positive or negative. Table 1 shows rule base of the proposed fuzzy logic controller. The two mentioned sections are observable in Table 2.

6. Implementation Results

In this investigation, the designed fuzzy controller is applied to the flexible joint manipulator of the Quanser experimental set in order to testify the trajectory tracking capability of it. The input is applied at the tip angular of flexible joint manipulator. The tip angular position of the flexible joint manipulator is required to follow a trajectory of square pulse with amplitude of 30 degrees and the frequency of 0.1 Hz. As Figure 6(a) shows, the flexible joint manipulator tracks the desire trajectory. Figure 6(b) demonstrates deflection angle. It is observable that deflection angle amplitude’s range is satisfactory and it has a suitable damping ratio. The summation of two signals, 𝛾, is shown in Figure 6(c). It has very important concept. It shows the behaviour of flexible joint with respect to fix-point of system. As the figure demonstrates, there is not any unacceptable overshoot and undershoot in 𝛾. So, the designed controller with 18 rules can satisfy desired specifications of system.

To show the efficiency of the designed controller, sinusoidal trajectory is considered as another desired trajectory which the tip angular position of the flexible joint manipulator should track. Figures 7(a) and 7(b) demonstrate that tip angular position follows the desired trajectory satisfactorily and deflection angle is in the acceptable range, respectively. As Figure 7(c) shows, the summation of two mentioned signals, 𝛾, is suitable.

The LQR controller is applied to the system for comparison with the proposed fuzzy controller. The results for the first input are illustrated in Figures 8(a), 8(b), and 8(c).

As Figure 8(b) shows, there is a significant overshoot in the deflection angle which is not desirable. Figure 7(c) demonstrated the flexible joint manipulator did not track exactly and there is tracking error in the designed LQR controller.

Figures 9(a), 9(b), and 9(c) illustrated the results of applying sinusoidal trajectory as the second desired trajectory to the LQR controller.

As results show, there is a significant tracking error in flexible joint manipulator trajectory which is not desirable at all. Comparison of the proposed fuzzy controller with LQR controller reveals the efficiency of proposed controller.

The proposed reduced rule-based fuzzy control is effective and applicable than the conventional fuzzy control. Reducing fuzzy rules is helpful to decrease the computational time and computational resources.

7. Conclusion

In this paper, a novel structure of fuzzy logic controller was presented for trajectory tracking and vibration control of a flexible joint manipulator. Dividing rule base into two sections, such that each contains 9 rules, reduced the number of rules from 81 rules to 18 rules. The proposed fuzzy logic controller decreased computational complexity, and it increased interpretability of rules. The designed fuzzy logic controller implemented to the Quanser experimental set. Comparison of the proposed method with the designed LQR controller reveals the efficiency of it. The suggested fuzzy controller not only decreased tracking error significantly but also overshoot in the deflection angle is more acceptable in comparison to LQR controller. Result shows efficiency of proposed controller in trajectory tracking of desired trajectories.


Authors are much grateful to Dr. Faezeh Farivar, a member of the ISLAB of the K. N. Toosi University, for her guidance and enthusiasm. Also, they are indebted to the Intelligent System Laboratory (ISLAB) of the K. N. Toosi University for the support provided during this study.