Type-2 fuzzy logic control for underactuated truss-like robotic finger with comparison of a type-1 case 1

Abstract

Underactuated mechanical systems have their own difficulties within the control criterion. As a particular and complex underactuated mechanical system, underactuated truss-like robotic finger(UTRF) is studied by establishing its dynamic model. The control problems include high nonlinearity, model inaccuracy and uncertainties. Type-2 fuzzy logic control method is supposed to be a proper way to solve these problems, because fuzzy logic control itself does not depend on an accurate model of the controlled object, and type-2 fuzzy logic control is able to handle uncertainties. Based on a brief introduction on type-2 fuzzy logic systems, an interval type-2 fuzzy logic controller is designed for UTRF to accomplish the goal of stabilization in its equilibrium point. As an extension of the type-1 fuzzy, the performances of the proposed controller are compared with the type-1 one case to show the advantages of the type-2 fuzzy. Simulation results show that the designed interval type-2 fuzzy logic controller is correct and effective and has better performances than that of type-1 fuzzy control.

Keywords

Underactuated robot nonlinear system fuzzy logic system type-2 fuzzy logic control

1 Introduction

Underactuated mechanical systems, such as inverted pendulum, anthropomorphic robotic hands and manipulators, have been worldwide concerns for the control community during recent years [1]. Different from fully-actuated systems, underactuated systems mean that they have more degrees of freedom than their number of independent actuators, leading to a series of benefits which may include slighter weight, lower cost, more compact structure [2], etc. Despite these benefits on the mechanisms, the lack of actuation along with model uncertainties significantly increases the difficulty of their control problems, among which the main one is that the lack of actuation makes the high nonlinearity impossible to be fully feedback linearized by feedback linearization method, which is commonly applied in nonlinear control systems. Therefore, other feasible control strategies should be pursued to accomplish thistask.

Type-2 fuzzy logic control method is supposed to be a proper way to accomplish this task, because fuzzy logic control itself does not depend on an accurate model of the controlled object, and type-2 fuzzy logic control is able to handle uncertainties. Proposed by Zadeh [3] in 1965, fuzzy sets and fuzzy logic are originally applied to handle incomplete information. While in real-world systems, there are a lot of uncertainties present. Zadeh then introduced Type-2 fuzzy sets (T2FSs) to give a new theory of fuzzy systems. As a result, the pre-existing fuzzy sets are referred to as Type-1 fuzzy sets (T1FSs). With the help of T2FSs, type-2 fuzzy logic systems (T2FLSs) can be built to effectively deal with control problems with uncertainties involved [4]. Type-2 fuzzy logic is able to handle uncertainties because it can model them and minimize their effects.

Since T1FS was firstly proposed, the type-1 fuzzy logic control(T1FC) approach has been widely applied in application to solve the control problem of complex nonlinear systems [5 –7]. With the development of T2FLSs, a variety of studies on the implementation of Type-1 and Type-2 fuzzy controllers has been made. Leticia Amador [8] made a comparison of the use of Type-1 and Type-2 fuzzy logic in the benchmark problem of the water tank, showing that T2FLSs manage better the uncertainty in real world problems, but with more computational expense. D. K. Sambariya [9] analyzed the performance of an interval type-2 fuzzy logic controller as a power system stabilizer, and the superiority of performance with interval type-2 fuzzy power system stabilizer over the corresponding type-1 case was validated. Yatak [10] implemented type-2 fuzzy controller for DC–DC boost converter fed by photovoltaic panels, showing that type-2 fuzzy controller had pointed out the best performance.

The research of type-2 fuzzy logic controllers seems to be thorough and applicable, however, when the complexity of the model and the controller increases, the operation becomes much more difficult, in which case the design procedure of its type-2 fuzzy logic controller tends to be more complicated. Therefore, design of type-2 fuzzy controllers for complex nonlinear systems still needs further research.

In this paper, the typical underactuated truss-like robotic finger is studied by establishing its dynamic model. Based on the dynamic model and the control problems in it, the basic concepts and feasibility of type-2 fuzzy logic control are analyzed. To reduce the computational complexity, an interval type-2 fuzzy logic controller is designed to make this robot stabilized in its equilibrium point. Then, the performances of the type-2 fuzzy logic controller and its corresponding type-1 controller are compared.

2 Dynamic model

The physical structure and dynamic model of UTRF(Underactuated Truss-like Robotic Finger) is introduced in this section.

2.1 Physical structure

The physical structure of UTRF is the foundation of its dynamic model. UTRF is composed of three parallelogram mechanisms, which is called deformable trusses structure when linked by cables and pulleys. Each cable is controlled by one motor, and it runs through the finger with the same direction of the diagonals, so one motor can control the finger to move toward a single direction. Considering the fact that the finger has more joints than the number of motors, this robot is underactuated. Combining several fingers together, a robotic paw can be formed to grab objects. The physical structure is shown in Fig. 1.

Fig. 1

Physical structure of UTRF.

The structure of UTRF is brand new while complex. The system is a kind of nonlinear system with strong dynamic couplings [11]. When put inverted in the gravity field, it can be regarded as a type of inverted pendulum, but much more complicated than a regular pendulum.

2.2 Dynamics of UTRF

Detailed derivation of establishing the dynamic model of UTRF is conducted in the author’s previous work [12]. Here shows the main idea of the modeling procedures.

We assume the bars which constitute the trusses to be uniform and with no elasticity involved. The weight and size of the pulleys are neglected and the cables are considered to be able to stretch toward its own direction with no deformation. The generalized model of UTRF is shown in Fig. 2.

Fig. 2

Schematic diagram of UTRF.

Based on Cartesian Coordinates [13], we define horizontal direction is x-axis and the vertical direction is y-axis. We put the root joint’s left point of UTRF to the zero point, i.e. origin of coordinates, and the angle between each joint and the vertical direction is defined as α₀, α₁, α₂ respectively. The line which passes through the zero point is cable 1, and the other is cable 2. Considering every parallelogram has the same form, each of them is consisted of two long bars and two short bars. The mass of every long bar is m₁, and length is l. The mass of short bar is m₂, and length is b.

Then we can use Lagrangian mechanics to write the dynamic equations of UTRF [14]. The dynamic model of UTRF is obtained by calculating the following Lagrange equation. $\frac{d}{dt} (\frac{\partial L}{\partial {\dot{α}}_{i}}) - \frac{\partial L}{\partial α_{i}} = Q_{i} (i = 0, 1, 2)$ (1)

Where L stands for the Lagrange operator: $L = T - V$ (2)

T stands for the kinetic energy, and V stands for the potential energy of the whole system. Q_i represents the equivalent torque which is imposed on angular coordinates [15].

The final dynamic model of UTRF is represented in the following form: $M (α) \ddot{α} + C (\dot{α}, α) + G (α) = QT$ (3)

Where ${\begin{matrix} M (α) = [\begin{matrix} M_{11} & M_{12} & M_{13} \\ M_{21} & M_{22} & M_{23} \\ M_{31} & M_{32} & M_{33} \end{matrix}] \\ C (\dot{α}, α) = [\begin{matrix} C_{1} \\ C_{2} \\ C_{3} \end{matrix}] \\ G (α) = [\begin{matrix} G_{1} \\ G_{2} \\ G_{3} \end{matrix}] \\ Q = [\begin{matrix} Q_{11} & Q_{12} \\ Q_{21} & Q_{22} \\ Q_{31} & Q_{32} \end{matrix}] \\ T = {[\begin{matrix} T_{1} & T_{2} \end{matrix}]}^{T} \end{matrix}$ (4)

In which $\begin{matrix} {\begin{matrix} M_{11} = (\frac{14}{3} m_{1} + 3 m_{2}) l^{2} \\ M_{12} = M_{21} = (3 m_{1} + 2 m_{2}) l^{2} cos (α_{0} - α_{1}) \\ M_{13} = M_{31} = (m_{1} + m_{2}) l^{2} cos (α_{0} - α_{2}) \\ M_{22} = (\frac{8}{3} m_{1} + 2 m_{2}) l^{2} \\ M_{23} = M_{32} = (m_{1} + m_{2}) l^{2} cos (α_{1} - α_{2}) \\ M_{33} = (\frac{2}{3} m_{1} + m_{2}) l^{2} \end{matrix} \\ {\begin{matrix} C_{1} = (3 m_{1} + 2 m_{2}) l^{2} sin (α_{0} - α_{1}) {\dot{α}}_{1}^{2} \\ + (m_{1} + m_{2}) l^{2} sin (α_{0} - α_{2}) {\dot{α}}_{2}^{2} \\ C_{2} = - (3 m_{1} + 2 m_{2}) l^{2} sin (α_{0} - α_{1}) {\dot{α}}_{0}^{2} \\ + (m_{1} + m_{2}) l^{2} sin (α_{1} - α_{2}) {\dot{α}}_{2}^{2} \\ C_{3} = - (m_{1} + m_{2}) l^{2} sin (α_{0} - α_{2}) {\dot{α}}_{0}^{2} \\ - (m_{1} + m_{2}) l^{2} sin (α_{1} - α_{2}) {\dot{α}}_{1}^{2} \end{matrix} \\ {\begin{matrix} G_{1} = - (5 m_{1} g + 3 m_{2} g) l sin α_{0} \\ G_{2} = - (3 m_{1} g + 2 m_{2} g) l sin α_{1} \\ G_{3} = - (m_{1} g + m_{2} g) l sin α_{2} \end{matrix} \\ {\begin{matrix} Q_{11} = - \frac{bl cos α_{0}}{\sqrt{l^{2} + b^{2} + 2 bl sin α_{0}}} \\ Q_{12} = \frac{bl cos α_{0}}{\sqrt{l^{2} + b^{2} - 2 bl sin α_{0}}} \\ Q_{21} = - \frac{bl cos α_{1}}{\sqrt{l^{2} + b^{2} + 2 bl sin α_{1}}} \\ Q_{22} = \frac{bl cos α_{1}}{\sqrt{l^{2} + b^{2} - 2 bl sin α_{1}}} \\ Q_{31} = - \frac{bl cos α_{2}}{\sqrt{l^{2} + b^{2} + 2 bl sin α_{2}}} \\ Q_{32} = \frac{bl cos α_{2}}{\sqrt{l^{2} + b^{2} - 2 bl sin α_{2}}} \end{matrix} \end{matrix}$ (5)

M (α) is called the positive inertial matrix [16]; C the Coriolis force matrix; G (α) the gravitational matrix; Q the coefficient matrix of input torque; T the input torque. represent the angle, angular velocity and angular acceleration [17, 18].

3 Type-2 fuzzy logic systems

3.1 Basic concepts

Zadeh first introduced Type-2 fuzzy sets (T2FSs) as an extension of the Type-1 fuzzy sets (T1FSs). The main difference between T1FS and T2FS lies in the membership functions (MFs) of their antecedent and consequent sets.

The above Fig. 3 shows the difference between a type-1 membership function and type-2 membership function. A type-1 fuzzy set is crisp, while a type-2 fuzzy set is itself fuzzy. As a consequence, a general type-2 membership grade could be any subset in the primary set [0, 1], while a type-1 membership grade is a crisp member in the primary set. Therefore, the subset of the primary set is then called the secondary membership.

Fig. 3

Type-1 and Type-2 fuzzy sets.

Figure 4 shows a type-2 Gaussian membership function, which is formed by blurring a type-1 Gaussian membership function towards right side. Then a shaded region is formed, which represents the uncertainty of the membership function, called footprint of uncertainty(FOU). The upper bound of FOU is called upper membership function and the lower bound is called lower membership function, denoted as UMF and LMF, respectively.

Fig. 4

Gaussian Type-2 membership function.

While in real applications, the fact that the secondary grades varies in an interval brings a heavy burden of calculating process. So the concept of interval type-2 fuzzy logic system (IT2FLS) is proposed by Mendel [19] and his students.

Figure 5 shows that the secondary grades of the type-2 membership function all equal 1, making all elements within the FOU have the same value of 1. Systems with this kind of membership functions are called IT2FLSs, which are typical cases of T2FLSs and have less calculating burden.

Fig. 5

Membership function of IT2FLS.

Considering the fuzzy logic control system, the rule base and inference engine of a type-2 fuzzy logic system (T2FLS) is extremely the same with that of a type-1 fuzzy logic system (T1FLS), while the output of inference engine is different because it is a type-2 fuzzy set. The structure of a T2FLS is shown in Fig. 6.

Fig. 6

Structure of a T2FLS.

The output of the inference engine is a type-2 fuzzy set, which is not possible to get a crisp output directly. Therefore, a type-reducer, which is not exited in a T1FLS, is needed to reduce the fuzzy set to a type-1 set. There are many type-reduction methods, among which the most popular one is center-of-set (COS) type-reduction method. Then by output processing, the crisp outputs could be obtained.

3.2 Feasibility for UTRF

The control goal for UTRF is to make it inverted in gravitational field and stabilized in this equilibrium point. The controllability and stabilizability of UTRF were analyzed in previous works [12], and a type-1 controller was designed to accomplish the control goal of stabilization. However, the previous works did not consider the uncertainties in the fuzzy system, which are indeed existed in real applications. Especially, in UTRF, the driving forces are conducted through cables, which will cause remarkable uncertainties in the control force of the system. Besides, UTRF’s dynamic model is highly nonlinear, while one advantage of fuzzy control method is that it does not depend on an accurate model of a system. Therefore, a type-2 control method, which is able to handle uncertainties and model inaccuracies, is feasible for UTRF in real applications.

4 Design of type-2 fuzzy logic controller

4.1 Structure of IT2FLS

To realize the goal of being stabilized in the inverted equilibrium point, human thinking method is imitated: drive the left cable to make UTRF move toward left and drive the right cable to make UTRF move toward right. The extent of the driving force depends on how much UTRF is deviated from the origin of the coordinate [20 –25]. Based on this principle, the design of a type-2 fuzzy logic controller is conducted in MATLAB/Simulink environment. The kind of interval type-2 fuzzy logic controller (IT2FLC) is chosen to reduce the calculation burden in the complex system.

There are three similar type-2 fuzzy controllers corresponding to three joints of UTRF, named it2f0, it2f1, it2f2. Each fuzzy controller has two inputs, which are the deviations of each joint’s angle and angular velocity. Each controller’s output is a control force obtained by type-2 fuzzy reasoning. The three outputs add up together as the control force to control this system. Since the three controllers have similar forms, we use the design of it2f0 to demonstrate the design process of interval type-2 fuzzy logic controllers.

4.2 Membership functions

By using the MATLAB toolbox proposed by Zuhtu Hakan Akpolat [26], the membership functions of IT2FLCs are designed. By investigating the controller it2f2, it is considered that the controller’s inputs and output should be described by linguistic variables first. The angle deviation of α₂ is denoted as E₂, and the angular velocity deviation of α₂ is denoted as EC₂. Depending on the angle’s deviation extent to the origin coordinate, five deviation levels are defined as Left Big(LB), Left Small(LS), Zero(O), Right Small(RS), and Right Big(RB), which can be used in the antecedents. The range of the levels is acquired by an open looped simulation of the UTRF’s dynamic model itself, so the maximum and minimum of the state variables with no controllers involved are obtained [–15, 15] in this case. According to the five levels and their relative ranges, the type-2 membership function of E₂ is designed, as is shown in Fig. 7.

Fig. 7

Type-2 membership function of E₂.

The angular velocity deviation of α₂ is denoted as EC₂, which is the differentiation of E₂. The five deviation levels of EC₂ are defined as Anticlockwise Big(AB), Anticlockwise Small(AS), Zero(O), Clockwise Small(CS), and Clockwise Big(CB). The range of these levels is [–6.5, 6.5]. The type-2 membership function of EC₂ is designed, as is shown in Fig. 8.

Fig. 8

Type-2 membership function of EC₂.

The output force of it2f2 is denoted as U₂. The five levels of U₂ are defined as Negative Big(NB), Negative Small(NS), Zero(O), Positive Small(PS), and Positive Big(PB). The range of it is determined by human thinking experience and simulation test, [–4380, 4380] in this case. The type-2 membership function of U₂ is designed, as is shown in Fig. 9.

Fig. 9

Type-2 membership function of U₂.

As comparisons, the corresponding type-1 fuzzy logic controllers’ membership functions are shown in Fig. 10.

Fig. 10

Type-1 membership function of E, EC, U₂.

Fuzzy controller it2f1 and it2f0 share the similar form of membership functions. Their relative parameters are shown in Table 1.

Table 1

Parameters of it2f1 and it2f0

Name	Meaning	Range
E₁	Angular Deviation of α₁	[–15,15]
EC₁	Angular Velocity Deviation of α₁	[–164,164]
U₁	Output Force of alpha1	[–682385,682395]
E₀	Angular Deviation of α₀	[–15,15]
EC₀	Angular Velocity Deviation of α₀	[–52,52]
U₀	Output Force of alpha0	[–1278360,1278360]

4.3 Fuzzy logic rules

The fuzzy logic rules are designed by imitating human thinking experiences. To illustrate the detailed fuzzy logic rules, we still use the fuzzy logic rules in it2f2 as a typical example, because the other two controllers share similar fuzzy logic rules. Here the angle deviation is denoted as E, and the angular velocity deviation is denoted as EC. The rules can be listed in a table as shown in Table 2.

Table 2
Fuzzy logic rules

EC AB AS O CS CB

E

LB PB PB PB PS O

LS PB PB PS O NS

O PB PS O NS NB

RS PS O NS NB NB

RB O NS NB NB NB

EC	AB	AS	O	CS	CB
E
LB	PB	PB	PB	PS	O
LS	PB	PB	PS	O	NS
O	PB	PS	O	NS	NB
RS	PS	O	NS	NB	NB
RB	O	NS	NB	NB	NB

Fuzzy logic rules are demonstrated by IF-THEN sentences. The part “IF” is the antecedent, and the part “THEN” is the consequent. In this case, every rule has two antecedents, i.e. E and EC, which are connected by AND method; every rule has one consequent. For instance, the first rule can be written as:

IF E is LB AND EC is AB,

THEN U is PB.

The first rule means if the angular deviation is Left Big(LB) and the angular velocity is Anticlockwise Big(AB), then we should give a Positive Big(PB) control force(U) to make UTRF move towards right. The remaining 24 rules can be written and understood similarly.

After editing all the fuzzy logic control rules to the FIS Rule Editor, the control rules’ surface is obtained, as is shown in Fig. 11.

Fig. 11

Fuzzy Rules Surface of it2f2.

The surface figures of it2f1 and it2f0 are similar in shape and only different in range values, so they are not shown here repetitiously.

5 Simulations

Simulations are conducted in MATLAB/Simulink environment [27]. The initial position states, i.e. three joints’ angles, of UTRF is set to be 0.02rads deviated from the origin coordinate. The control goal is expecting the three joints’ angles converge to zero so that the system is stabilized in the equilibrium point. Some basic parameters of UTRF are given in Table 3.

Table 3
Evaluation of variables

m ₁ 3 kg

m ₂ 2 kg

l 3 m

b 2 m

g 10 m/s²

ξ 0.3

The simulation result of interval type-2 fuzzy logic control of three angles is shown in Fig. 12. As a comparison, type-1 fuzzy logic control’s performance is shown in Fig. 13.

Fig. 12

IT2FLC result of angles.

Fig. 13

T1FC result of three angles.

These figures show that the initial state of each angle is 0.02rad, and the angles all converge to zero, which means both type-2 and type-1 fuzzy logic controllers are able to stabilize UTRF in the equilibrium point.

The simulation result of interval type-2 fuzzy logic control of three angular velocities is shown in Fig. 14. Also, type-1 fuzzy logic control’s performance of angular velocities is shown in Fig. 15 as a comparison.

Fig. 14

IT2FLC result of angular velocities.

Fig. 15

T1FC result of three angular velocities.

These figures show that the angular velocities of three angles converge to zero after adjustment procedure, which also means that UTRF is stabilized in the end.

The simulation result of the control force of interval type-2 fuzzy logic controller is shown in Fig. 16. Figure 17 shows the comparison.

Fig. 16

IT2FLC result of control force.

Fig. 17

T1FLC result of control force.

Theses figures show that the control force converges to constant, which is equal to the symmetrical force in the other rope, after the adjustment procedure, which means the forces in two ropes are equal in the end, making UTRF stabilized.

Both type-2 and type-1 fuzzy logic controllers are able to stabilize UTRF in the equilibrium point, but the performance of the interval type-2 fuzzy logic controller is much smoother and it needs less regulating times.

Since type-2 fuzzy logic is claimed to have better performance when uncertainties exist, simulations with uncertainties involved are conducted afterwards. We consider the mass and length of the short bar of UTRF are not constant, with the following uncertainties involved: ${\begin{matrix} m_{2} = 2 + Δ k_{1} sin (t) \\ b = 2 + Δ k_{2} sin (t) \\ 0.6 \leq Δ k_{1} = Δ k_{2} \leq 1.2 \end{matrix}$ (6)

Other parameters remain unchanged, then simulations are conducted to verify the control abilities of the IT2FLC and T1FLC.

The simulation results of three angles by IT2FLC and T1FLC with model uncertainties are shown in Figs. 18 and 19 respectively.

Fig. 18

Result of angles by IT2FLC with uncertainties.

Fig. 19

Result of angles by T1FLC with uncertainties.

The simulation results of three angular velocities by IT2FLC and T1FLC with model uncertainties are shown in Figs. 20 and 21 respectively.

Fig. 20

Result of velocities by IT2FLC with uncertainties.

Fig. 21

Result of velocities by T1FLC with uncertainties.

The simulation result of the control force(CF) by IT2FLC and T1FLC with model uncertainties are shown in Figs. 22 and 23 respectively.

Fig. 22

Result of CF by IT2FLC with uncertainties.

Fig. 23

Result of CF T1FLC with uncertainties.

From Figs. 18 to 23, it is obvious that when uncertainties involved, the performance of the interval type-2 fuzzy logic controller is influenced by the uncertainties, but it is still able to stabilize the system in about 4 seconds. The performance of type-1 fuzzy controller declines severely and even becomes unable to stabilize the system in one regulation period. The control force of type-1 fuzzy controller oscillates sharply, which means the controller acts toughly to control the system.

6 Conclusions

This paper mainly focuses on a particular underactuated mechanical system, called underactuated truss-like robotic finger(UTRF). The dynamic model of UTRF is established by using Lagrangian Mechanics. Aiming at solving the problem of model inaccuracy, high nonlinearity and uncertainties, type-2 fuzzy logic control method is analyzed by demonstrating the difference between type-2 and type-1 fuzzy systems and particularly introduces the type-2 fuzzy logic membership functions. Moreover, an interval type-2 fuzzy logic controller is designed in MATLAB/Simulink environment. Within the design procedure, the main principle which is relied on is human thinking strategy instead of accurate model. The uncertainties of the parameters are modeled in the fuzzy logic controller’s control force by using the type-2 fuzzy logic control method, thus handles the problem of uncertainties. Simulation results show that the designed IT2FL controller is feasible for UTRF to accomplish the goal of stabilization in equilibrium point, and type-2 fuzzy logic controller is smoother and it needs less regulating times. When model uncertainties involved, the performance of I2FLC triumphs over T1FLC obviously. These results show that a type-2 fuzzy logic controller is able to handle uncertainties and has a better performance than that of a type-1 case.

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