Most of the wheelchair users are paraplegics, who are not able to move on their own due to permanent injury in their lower extremities. These wheelchairs are four-wheeled and have certain limitations due to design and control mechanism. For example, the wheelchairs cannot move to a higher level, lift the front wheel and stay in an upright position. As a result, wheelchair users cannot reach certain heights to pick and place things on the shelves, and cupboards, etc. without any assistant and also cannot have eye-to-eye conversation with normal people effectively. On the other hand, a two-wheeled wheelchair has a unique characteristic that may help disabled and elderly people who use the wheelchair as the main means of transport and can also use the wheelchair for these added advantages. Now the idea is to transform the standard four-wheeled wheelchair into a two-wheeled upright wheelchair to facilitate such manuoverability. The front wheels (casters) can be lifted up and stabilized as an inverted pendulum, thus increasing the level of height achievable while in the upright position. Similarly, when this upright position is no longer needed it may be transformed back into its normal four-wheeled position. The schematic diagram of the twowheeled wheelchair is shown in Figure 1. The transformation will result in a highly nonlinear and complex system. Since a human has quite significant mass sitting on the wheelchair, the two-wheeled wheelchair can be modeled with double links that mimic double inverted pendulum scenario that need a clever control strategy.

Most of the classical control design methodologies such as Nyquist, Bode, state-space, optimal control, root locus, *H* , and -analysis are based on assumptions that the process is linear and stationary and hence is represented by a finite dimensional constant coefficient linear model. These methods do not suit complex systems well because few of those represent uncertainty and incompleteness in system knowledge or complexity in design. But the fact is the real world is too complex. As the complexity of a system increases, quantitative analysis and precision become difficult. The increasing complexity of dynamical systems such as this coupled with stringent performance criteria, which are sometimes subject to human satisfaction, necessitates the use of more sophiticated control approaches. However, many processes that are nonlinear, uncertain, incomplete or non-stationary have subtle and

Modular Fuzzy Logic Controller for Two-Wheeled Wheelchair 39

described by expert knowledge, normally possessed by human. A fuzzy logic controller

Input Output

Defuzzification

Mechanism

Fuzzification is a process of transforming an observed input space to fuzzy sets within a universe of discourse. This process consists of associating to each fuzzy set a membership function (MF). These functions can be thought of as maps from the real numbers to the interval *I* 0,1 . If there are *n* fuzzy sets associated with a given quantity *x R* , such *n* maps : , 1, , *FR I i n i* are defined. They determine to what extent the linguistic label associated with fuzzy set *i* characterizes the current value of *x*. There are different kinds of MFs used in designing fuzzy controllers. The most common choices are triangular, trapezoidal, Gaussian and bell shaped MFs. There is no exact method for choosing an MF,

Inference is used to describe the process of formulating a nonlinear mapping from a given input space to an output space. The mapping then provides a basis from which decisions can be taken. The process of fuzzy inference involves the MFs, fuzzy logic operators and rule-base. Generally there are three types of commonly used fuzzy inference. They differ mainly in the consequent part of their fuzzy rules, aggregations and defuzzification procedures. Thus selecting a different fuzzy inference will result in different computational time. The three common fuzzy inferences are: Mamdani fuzzy inference, Sugeno fuzzy inference and Tsukamoto fuzzy inference. The choice of a particular inference mechanism is eventually problem dependent and availability of information

Mamdani type fuzzy modeling was proposed as the first attempt to control a steam engine and boiler by a set of linguistic control rules by (Mamdani 1974). In this type of inference, Max-min is the most common rule of composition used. In this composition rule, the inferred output of each rule is a fuzzy set chosen from the minimum firing strength. On the

Rule-base

(FLC) has the basic configuration illustrated in Figure 2.

Fig. 2. Fuzzy logic control

ii. Inference mechanism

about the system in question.

i. Fuzzification

iii. Rule-base iv. Defuzzification

Fuzzification Inference

Generally, a fuzzy logic controller consists of the following components:

and the designer mainly relies upon an expert knowledge or use heuristic rule.

interactive exchanges with the operating environment and are controlled by skilled human operators successfully. Rather than mathematically model the process, the human operator models the process in a heuristic or experiential manner. It is evident that human knowledge is becoming more and more important in control systems design. This experiential perspective in controller design requires the acquisition of heuristic and qualitative, rather than quantitative, knowledge or expertise from the human operator. During the past several decades, fuzzy control has emerged as one of the most active and powerful areas for research in the application of such complex and real world systems using fuzzy set theory (Zadeh, 1965).

Fig. 1. Schematic diagram of wheelchair with three under actuated joints

Due to many significant advantages of wheelchair usage, this research presents findings of the research carried out on the implementation of new architecture of modular intelligent control strategies on the two-wheeled wheelchair model. The multi-objective control involves lifting and stabilizing of Link1 and Link2 of double-inverted pendulum like twowheeled wheelchair, wheelchair backward and forward motion control as well as position. It is hoped that the proposed model, mechanisms and control could be of benefit to a wheelchair user, thus enhancing wheelchair technology for paraplegics and elderly.

#### **2. Intelligent control approach**

Intelligent control systems have evolved from existing controllers in a natural way competing demanding challenges of the time and are not defined in terms of specific algorithms. They employ techniques that can sense and reason without much *a priori* knowledge about the environment and produce control actions in a flexible, adaptive and robust manner (Harris, 1994). In general, by intelligent control approaches, it is mainly meant the methodologies of fuzzy logic, neural networks, and genetic algorithms. These methodologies have shown to be effective in controlling complex nonlinear systems. The control of complex nonlinear systems has been approached over the last few decades using fuzzy logic techniques due to the fact that fuzziness itself is easy to implement and can be described by expert knowledge, normally possessed by human. A fuzzy logic controller (FLC) has the basic configuration illustrated in Figure 2.

Fig. 2. Fuzzy logic control

Generally, a fuzzy logic controller consists of the following components:

38 Fuzzy Logic – Controls, Concepts, Theories and Applications

interactive exchanges with the operating environment and are controlled by skilled human operators successfully. Rather than mathematically model the process, the human operator models the process in a heuristic or experiential manner. It is evident that human knowledge is becoming more and more important in control systems design. This experiential perspective in controller design requires the acquisition of heuristic and qualitative, rather than quantitative, knowledge or expertise from the human operator. During the past several decades, fuzzy control has emerged as one of the most active and powerful areas for research in the application of such complex and real world systems using fuzzy set theory (Zadeh, 1965).

Link2

Link1

δ1

2

Fig. 1. Schematic diagram of wheelchair with three under actuated joints

Due to many significant advantages of wheelchair usage, this research presents findings of the research carried out on the implementation of new architecture of modular intelligent control strategies on the two-wheeled wheelchair model. The multi-objective control involves lifting and stabilizing of Link1 and Link2 of double-inverted pendulum like twowheeled wheelchair, wheelchair backward and forward motion control as well as position. It is hoped that the proposed model, mechanisms and control could be of benefit to a

Intelligent control systems have evolved from existing controllers in a natural way competing demanding challenges of the time and are not defined in terms of specific algorithms. They employ techniques that can sense and reason without much *a priori* knowledge about the environment and produce control actions in a flexible, adaptive and robust manner (Harris, 1994). In general, by intelligent control approaches, it is mainly meant the methodologies of fuzzy logic, neural networks, and genetic algorithms. These methodologies have shown to be effective in controlling complex nonlinear systems. The control of complex nonlinear systems has been approached over the last few decades using fuzzy logic techniques due to the fact that fuzziness itself is easy to implement and can be

wheelchair user, thus enhancing wheelchair technology for paraplegics and elderly.

δ2

*R*

**2. Intelligent control approach**

 and *L*

on each wheel

iv. Defuzzification

Fuzzification is a process of transforming an observed input space to fuzzy sets within a universe of discourse. This process consists of associating to each fuzzy set a membership function (MF). These functions can be thought of as maps from the real numbers to the interval *I* 0,1 . If there are *n* fuzzy sets associated with a given quantity *x R* , such *n* maps : , 1, , *FR I i n i* are defined. They determine to what extent the linguistic label associated with fuzzy set *i* characterizes the current value of *x*. There are different kinds of MFs used in designing fuzzy controllers. The most common choices are triangular, trapezoidal, Gaussian and bell shaped MFs. There is no exact method for choosing an MF, and the designer mainly relies upon an expert knowledge or use heuristic rule.

Inference is used to describe the process of formulating a nonlinear mapping from a given input space to an output space. The mapping then provides a basis from which decisions can be taken. The process of fuzzy inference involves the MFs, fuzzy logic operators and rule-base. Generally there are three types of commonly used fuzzy inference. They differ mainly in the consequent part of their fuzzy rules, aggregations and defuzzification procedures. Thus selecting a different fuzzy inference will result in different computational time. The three common fuzzy inferences are: Mamdani fuzzy inference, Sugeno fuzzy inference and Tsukamoto fuzzy inference. The choice of a particular inference mechanism is eventually problem dependent and availability of information about the system in question.

Mamdani type fuzzy modeling was proposed as the first attempt to control a steam engine and boiler by a set of linguistic control rules by (Mamdani 1974). In this type of inference, Max-min is the most common rule of composition used. In this composition rule, the inferred output of each rule is a fuzzy set chosen from the minimum firing strength. On the

Modular Fuzzy Logic Controller for Two-Wheeled Wheelchair 41

geometric progression in the number of required rules for general purpose tracking and control situations. Moreover, it should be achieved without compromising the robustness

A generic problem with an FLC is that the number of rules grow exponentially with the number of input-output variables and linguistic terms for each variable. For a complete rulebase with input variables *Xi n i*| 1, , with linguistic terms *A j ij*| 1, , *mi* and output variables *Yk l k*| 1, , with linguistic terms *Bkj*| 1, , *j p k* , the number of rules will be

> 1 *n i i R m*

If (X1 is A11) and ... and (Xn is Anm) Then (Y1 is B11) and ... and (Yl is Blp) This large number of rules complicates the design of an FLC, because for each of the *R* different premises the expert must provide a combination of term sets for the output variables, which is nearly impossible for a human expert to guess. It is possible to omit a set of rules if it could be guranteed that a certain combination of input-output variables will never occur during control of the dynamic system. A modular structure of FLCs with

For large scale and complex systems, the reduction in computation and design complexity remains a challenge of intelligent control systems. Hierarchical and modular methodology have gained wide popularity because of its simplicity in design and robustness. There are several approaches in decomposing a system into modules such as decentralized approach, time-scale decomposition, hierarchical system, and workspace decomposition (Siljak, 1991). For control problems with multiple objectives of different priority, sub-controller with a subset of input-output variables can be designed for each objective. Furthermore, each antecedent can be decomposed into single input modules. Each fuzzy module is designed to

minimum number of input-output variables can reduce the number of rules *R*.

(1)

and capability of the complete system.

The rules have the form

**3. Modular fuzzy control**

Fig. 3. MIMO FLC for lifting and stabilizing the wheelchair

other hand, in max-product rule of composition the inferred output of each rule is a fuzzy set scaled down by its firing strength via algebraic product.

A fuzzy system is characterized by a set of linguistic statements based on expert knowledge. The expert knowledge is usually in the form of if-then rules, which are easily implemented by fuzzy conditional statements in fuzzy logic. The collection of fuzzy rules that are expressed as fuzzy conditional statements forms the rule base or the rule set of an FLC. A rule consists of two parts, antecedent and consequent. For example, typical rule in Mamdani-type fuzzy model with four-inputs and three-outputs FLC can be expressed by the following linguistic conditional statement.

If (*X1* is *Ai*) and (*X2* is B*j*) and (*X3* is C*k*) and (*X4* is D*l*)

then (Y1 *is* U*p*) and (Y*2* is V*q*) and (Y*3* is W*r*)

where {X1, X2, X3, X4} are the inputs with linguistic terms { *Ai,* Bj, C*k*, Dl} and {Y1, Y2, Y3} are the outputs with linguistic terms {*Up,* Vq, W*r*}.

Defuzzification is basically a mapping from a space of fuzzy control actions defined over an output universe of discourse into a space of nonfuzzy (crisp) control actions. In a sense this is the inverse of the fuzzification even though mathematically the maps need not be inverses of one another. In general, defuzzification can be viewed as a function : *n DF I R* , mapping a fuzzy vector *F x* with *n* fuzzy sets to a real number. There are different methods of defuzzification. However, simple methods are available to use depending on the application, among them Centre of Gravity Method (COG), and Weighted Average Method are widely used in Mamdani-type FLC and Sugeno-type FLC. Each method is problem dependent, but the experts should know that these methods are available and should try to see which works best for the application.

The two-wheeled wheelchair model involves lifting and stabilizing the two links (Link1 and Link2) similar to a double-inverted pendulum and hence is a multi-objective control problem. Considering the complexity and non-linearity of the wheelchair, the controller has to be designed in such a way to produce the required torques, namely *R* , *L* and 2 , for acting at three different locations on the wheelchair for lifting the casters/chair and stabilizing the system. The torque *R* and *L* represent the input torque to the right and left wheels respectively. 2 represents the torque between Link1 and Link2 to cater for the whole weight of the human body. Angular positions of Link1 and Link2, 1 and 2 respectively, are measured using sensors attached to the wheelchair. This characterizes the system as a highly nonlinear multi-input multi-output (MIMO) system. Fuzzy logic control is therefore very appropriate to use in this case. To achieve upright position for the two links, they need to be lifted and stabilized to zero degree (relative to vertical axis) upright position. This may be realised with a single controller. However, this will lead to a huge fuzzy rule-base. A conventional fuzzy controller with 4 inputs *e ee e* 1 12 2 , ,, and 3 outputs *R L* , , 2 (inputs-outputs are shown in Figure 3) has significant drawback in terms of computational complexity, which increases with the dimension of the system variables; the number of rules increases exponentially as the number of system variables increases. A strategy is sought to simplify the development process and reduce the

other hand, in max-product rule of composition the inferred output of each rule is a fuzzy

A fuzzy system is characterized by a set of linguistic statements based on expert knowledge. The expert knowledge is usually in the form of if-then rules, which are easily implemented by fuzzy conditional statements in fuzzy logic. The collection of fuzzy rules that are expressed as fuzzy conditional statements forms the rule base or the rule set of an FLC. A rule consists of two parts, antecedent and consequent. For example, typical rule in Mamdani-type fuzzy model with four-inputs and three-outputs FLC can be expressed by

If (*X1* is *Ai*) and (*X2* is B*j*) and (*X3* is C*k*) and (*X4* is D*l*)

then (Y1 *is* U*p*) and (Y*2* is V*q*) and (Y*3* is W*r*) where {X1, X2, X3, X4} are the inputs with linguistic terms { *Ai,* Bj, C*k*, Dl} and {Y1, Y2, Y3} are

Defuzzification is basically a mapping from a space of fuzzy control actions defined over an output universe of discourse into a space of nonfuzzy (crisp) control actions. In a sense this is the inverse of the fuzzification even though mathematically the maps need not be inverses of one another. In general, defuzzification can be viewed as a function : *n DF I R* , mapping a fuzzy vector *F x* with *n* fuzzy sets to a real number. There are different methods of defuzzification. However, simple methods are available to use depending on the application, among them Centre of Gravity Method (COG), and Weighted Average Method are widely used in Mamdani-type FLC and Sugeno-type FLC. Each method is problem dependent, but the experts should know that these methods are available and should try to

The two-wheeled wheelchair model involves lifting and stabilizing the two links (Link1 and Link2) similar to a double-inverted pendulum and hence is a multi-objective control problem. Considering the complexity and non-linearity of the wheelchair, the controller has

acting at three different locations on the wheelchair for lifting the casters/chair and

respectively, are measured using sensors attached to the wheelchair. This characterizes the system as a highly nonlinear multi-input multi-output (MIMO) system. Fuzzy logic control is therefore very appropriate to use in this case. To achieve upright position for the two links, they need to be lifted and stabilized to zero degree (relative to vertical axis) upright position. This may be realised with a single controller. However, this will lead to a huge

terms of computational complexity, which increases with the dimension of the system variables; the number of rules increases exponentially as the number of system variables increases. A strategy is sought to simplify the development process and reduce the

 , *L*

represent the input torque to the right and left

represents the torque between Link1 and Link2 to cater for the

2 (inputs-outputs are shown in Figure 3) has significant drawback in

 and 2 , for

> and 2

 1 12 2 , ,, and 3

to be designed in such a way to produce the required torques, namely *R*

 and *L*

whole weight of the human body. Angular positions of Link1 and Link2, 1

fuzzy rule-base. A conventional fuzzy controller with 4 inputs *e ee e*

set scaled down by its firing strength via algebraic product.

the following linguistic conditional statement.

the outputs with linguistic terms {*Up,* Vq, W*r*}.

see which works best for the application.

stabilizing the system. The torque *R*

wheels respectively. 2

outputs

 *R L* , ,  geometric progression in the number of required rules for general purpose tracking and control situations. Moreover, it should be achieved without compromising the robustness and capability of the complete system.

Fig. 3. MIMO FLC for lifting and stabilizing the wheelchair

A generic problem with an FLC is that the number of rules grow exponentially with the number of input-output variables and linguistic terms for each variable. For a complete rulebase with input variables *Xi n i*| 1, , with linguistic terms *A j ij*| 1, , *mi* and output variables *Yk l k*| 1, , with linguistic terms *Bkj*| 1, , *j p k* , the number of rules will be

$$R = \prod\_{i=1}^{n} m\_i \tag{1}$$

The rules have the form

$$\text{If } (\mathsf{X}\_1 \text{ is } \mathsf{A}\_{11}) \text{ and } \dots \text{ and } (\mathsf{X}\_{\mathsf{n}} \text{ is } \mathsf{A}\_{\mathsf{nm}}) \text{ Then } (\mathsf{Y}\_1 \text{ is } \mathsf{B}\_{11}) \text{ and } \dots \text{ and } (\mathsf{Y}\_{\mathsf{l}} \text{ is } \mathsf{B}\_{\mathsf{p}}).$$

This large number of rules complicates the design of an FLC, because for each of the *R* different premises the expert must provide a combination of term sets for the output variables, which is nearly impossible for a human expert to guess. It is possible to omit a set of rules if it could be guranteed that a certain combination of input-output variables will never occur during control of the dynamic system. A modular structure of FLCs with minimum number of input-output variables can reduce the number of rules *R*.

#### **3. Modular fuzzy control**

For large scale and complex systems, the reduction in computation and design complexity remains a challenge of intelligent control systems. Hierarchical and modular methodology have gained wide popularity because of its simplicity in design and robustness. There are several approaches in decomposing a system into modules such as decentralized approach, time-scale decomposition, hierarchical system, and workspace decomposition (Siljak, 1991). For control problems with multiple objectives of different priority, sub-controller with a subset of input-output variables can be designed for each objective. Furthermore, each antecedent can be decomposed into single input modules. Each fuzzy module is designed to

Modular Fuzzy Logic Controller for Two-Wheeled Wheelchair 43

of switches that gathers information from the subsystems and sends supervisory (threshold condition) instructions back to the subsystems. The supervisor in this case is the condition, (if the angular position error of Link1 and Link2, -5° < e < 5°, then Link1-stabilizing and Link2-stabilizing are activated). In this case, the switches coordinate the condition fulfilment of all the criteria for the activation of actuator to work accordingly. The reference position for lifting and stabilizing of both links is 0 degree at the upright position. The parameters 'a' and 'b' in the figure show the fuzzy input scaling factors (input gain) such that if the stabilizing subsystem of Link1 or Link2 is activated, the sensitivity of the fuzzy inputs is increased by giving higher gain (about 10 times) of a and b. The outputs from the system that are fed back to the controller are the angular position of Link1 (δ1) and the angular position of Link2 (δ2). The control approach using this modular strategy is believed to work well with the independently allocated tasks. In the figure, eδ1 shows the angular position error of Link1, ∆eδ1 represents the change of angular position error of Link1. The effect of Link2 onto Link1 is taken into account by using the angular position error of Link2, eδ2 as the fuzzy input for FLC1 and FLC2. In these two controllers, eδ2 represents the angular position error of Link2, while ∆eδ2 represents the change of angular position error of Link2. Similarly the effect of Link1 onto Link2 is taken into account by using the angular position

2

0 δ2

Coordinator

Switch

Switch

The MFC is also known as hierarchical fuzzy control (HFC), and the two terms are used interchangeably. It is discussed in detail for two-wheeled application in (Ahmad et al. 2011).

R, L δ1

error of Link1, eδ1 as the input for FLC3 and FLC4.

Stabilizing

FLC 2

FLC 1

Lifting

Lifting

FLC 3

Stabilizing

FLC 4

Subsystems

Fig. 5. Modular FLC for Two-wheeled wheelchair.

**4. MFC for two-wheeled wheelchair**

a

b

0

Supervisor

+

handle one specific input affiliated with one of the decoupled antecedents *Xi n i*| 1, , and produces a crisp action *Yk l k*| 1, , where *k l* 1, , . Such a generic modular architecture is shown in Figure 4.

Fig. 4. Modular FLC

A typical fuzzy rule issues an appropriate output action by evaluating the related inputs from the measurement data. In the conventional IF-THEN fuzzy inference formulation, all of the system's input parameters are suggested as antecedents in the fuzzy rule. The total possible number of fuzzy rules that can be generated for the rule base is *Lk* where *k* is the number of inputs and *L* is the number of fuzzy linguistic terms or MFs. As compared to the modular FLC design, each input represents one fuzzy control module. The total number of rules for each module is determined by the number of MFs *L*. Thus, the total number of fuzzy rules for all *k* modules is *kL*. This clearly shows a significant reduction in the number of fuzzy rules from *Lk* to *k* as well as savings in computation.

The mathematical model of the two-wheeled wheelchair incorporates three independent actuators; derived from Figure 1, corresponding to control output to be fed into the system. The angular position of Link1 and Link2, denoted as δ1 and δ2 respectively, are the controlled variables that will determine the system performance. The control challenge relates to the fact that there is more than one mechanism acted upon with the same actuator. For example, to transform the wheelchair into an upright two-wheeled wheelchair, the torques determined by fuzzy control are located at both right and left wheels. At the same time, if linear motion is considered, the same actuator needs to provide enough torque such that the wheelchair will still move forward or backward while in the upright position. Lifting and stabilizing consist of two system output parameters to be considered, namely angular position of Link1, δ1 and angular position of Link2, δ2. Therefore a modular fuzzy logic control (MFC) is adopted to realize this multi-function two-wheeled wheelchair.

The MIMO system with an objective of achieving zero degree upright position is decomposed into small and simpler subsystems: Link1-lifting, Link1-stabilizing, Link2 lifting, and Link2-stabilizing. The structure of the modular FLC for the wheelchair is illustrated in the block diagram in Figure 5. Accordingly, this type of FLC can deal with, for example, N subsystems located at different levels, where each subsystem manages its own control strategy and communicates with the coordinator. The coordinator comprises a pair

handle one specific input affiliated with one of the decoupled antecedents *Xi n i*| 1, , and produces a crisp action *Yk l k*| 1, , where *k l* 1, , . Such a generic modular

A typical fuzzy rule issues an appropriate output action by evaluating the related inputs from the measurement data. In the conventional IF-THEN fuzzy inference formulation, all of the system's input parameters are suggested as antecedents in the fuzzy rule. The total possible number of fuzzy rules that can be generated for the rule base is *Lk* where *k* is the number of inputs and *L* is the number of fuzzy linguistic terms or MFs. As compared to the modular FLC design, each input represents one fuzzy control module. The total number of rules for each module is determined by the number of MFs *L*. Thus, the total number of fuzzy rules for all *k* modules is *kL*. This clearly shows a significant reduction in the number

The mathematical model of the two-wheeled wheelchair incorporates three independent actuators; derived from Figure 1, corresponding to control output to be fed into the system. The angular position of Link1 and Link2, denoted as δ1 and δ2 respectively, are the controlled variables that will determine the system performance. The control challenge relates to the fact that there is more than one mechanism acted upon with the same actuator. For example, to transform the wheelchair into an upright two-wheeled wheelchair, the torques determined by fuzzy control are located at both right and left wheels. At the same time, if linear motion is considered, the same actuator needs to provide enough torque such that the wheelchair will still move forward or backward while in the upright position. Lifting and stabilizing consist of two system output parameters to be considered, namely angular position of Link1, δ1 and angular position of Link2, δ2. Therefore a modular fuzzy logic control (MFC) is adopted to realize this multi-function two-wheeled wheelchair.

The MIMO system with an objective of achieving zero degree upright position is decomposed into small and simpler subsystems: Link1-lifting, Link1-stabilizing, Link2 lifting, and Link2-stabilizing. The structure of the modular FLC for the wheelchair is illustrated in the block diagram in Figure 5. Accordingly, this type of FLC can deal with, for example, N subsystems located at different levels, where each subsystem manages its own control strategy and communicates with the coordinator. The coordinator comprises a pair

of fuzzy rules from *Lk* to *k* as well as savings in computation.

architecture is shown in Figure 4.

Fig. 4. Modular FLC

of switches that gathers information from the subsystems and sends supervisory (threshold condition) instructions back to the subsystems. The supervisor in this case is the condition, (if the angular position error of Link1 and Link2, -5° < e < 5°, then Link1-stabilizing and Link2-stabilizing are activated). In this case, the switches coordinate the condition fulfilment of all the criteria for the activation of actuator to work accordingly. The reference position for lifting and stabilizing of both links is 0 degree at the upright position. The parameters 'a' and 'b' in the figure show the fuzzy input scaling factors (input gain) such that if the stabilizing subsystem of Link1 or Link2 is activated, the sensitivity of the fuzzy inputs is increased by giving higher gain (about 10 times) of a and b. The outputs from the system that are fed back to the controller are the angular position of Link1 (δ1) and the angular position of Link2 (δ2). The control approach using this modular strategy is believed to work well with the independently allocated tasks. In the figure, eδ1 shows the angular position error of Link1, ∆eδ1 represents the change of angular position error of Link1. The effect of Link2 onto Link1 is taken into account by using the angular position error of Link2, eδ2 as the fuzzy input for FLC1 and FLC2. In these two controllers, eδ2 represents the angular position error of Link2, while ∆eδ2 represents the change of angular position error of Link2. Similarly the effect of Link1 onto Link2 is taken into account by using the angular position error of Link1, eδ1 as the input for FLC3 and FLC4.

Subsystems

#### **4. MFC for two-wheeled wheelchair**

The MFC is also known as hierarchical fuzzy control (HFC), and the two terms are used interchangeably. It is discussed in detail for two-wheeled application in (Ahmad et al. 2011).

Modular Fuzzy Logic Controller for Two-Wheeled Wheelchair 45

The system was tested with both configurations and the performances, with and without coupling were comparably similar, see Figure 6. Therefore, as seen the second configuration performed well with fewer fuzzy rules fired, and this configuration is used in implementing

0

100

Angular position

Torque between

Fig. 6. System performance comparison between coupled fuzzy inputs and decoupled fuzzy

The MFC is thus adopted for the two-wheeled wheelchair mechanisms, and the

The MFC can be divided into two significant categories, primary and secondary (Bessacini and Pinkos 1995). The controller is categorized according to different objectives. The control structure for achieving an upright two-wheeled maneuverable wheelchair is depicted in Figure 7. The general function of MFC is to minimize the errors in system responses considered. The primary goal unit caters for the upright control, which consists of lifting and stabilizing to the upright position and the transformation back to normal four-wheeled position of Link1 and Link2. These controllers are active most of the time even during maneuver. The secondary unit is activated by the coordinator (switch), with certain condition pre-set for output activation. It consists of different unique objectives involving linear motion control, steering control, additional chair height extension control. Each

objective in the secondary goal unit is discussed in detail in the following sections.

Link1 and Link2 (Nm)

of Link2 (degree)

0

10

0 10 20 30

coupled decoupled

0 10 20 30

coupled decoupled

Time (s)

Time (s)

the motion control for two-wheeled wheelchair.

inputs in terms of δ1, δ2, R, L and 2.

corresponding research objectives are:

Linear motion control (forward or backward)

Lifting and stabilizing control

Steering motion control

0

100

Wheel Torques (Nm)

200

0

50

Angular position

of Link1 (degree)

100

0 10 20 30

Right torque coupled Left torque coupled Right torque decoupled Left torque decoupled

coupled decoupled

0 10 20 30

Time (s)

Time (s)

The goal of the controller is to produce the required torques, namely *R* , *L* and 2 , for acting at three different locations on the wheelchair for lifting and stabilizing. The torques *R* and *L* represent the input torque to the right and left wheels respectively. On the other hand, 2 represents the torque between Link1 and Link2 to be used to cater for the whole weight of the human body. Angular positions of Link1 and Link2, 1 and 2 respectively, are measured using sensors attached to the wheelchair in Visual Nastran (VN). To achieve upright position of the two links, they need to be lifted and stabilized at zero degree upright position. The goal may be treated as a single objective control that is having Link1 and Link2 at the 0 degree upright position with one controller. This will increase significantly the computational complexity, which increases with the number of system variables; the number of rules increases exponentially as the number of system variables increases.

## **4.1 Rules reduction strategy for general purpose tracking and control situations**

The strategy is sought without compromising the robustness and capability of the system. Such a strategy relies mainly on three concepts, (Ahmad et al. 2011).

To assess the effect of coupling in the fuzzy control, the system is tested with two different configurations, which mainly differ at the input side of the controller, as shown in Table 1.

Table 1. Different input configurations of modular fuzzy logic controller

acting at three different locations on the wheelchair for lifting and stabilizing. The

 respectively, are measured using sensors attached to the wheelchair in Visual Nastran (VN). To achieve upright position of the two links, they need to be lifted and stabilized at zero degree upright position. The goal may be treated as a single objective control that is having Link1 and Link2 at the 0 degree upright position with one controller. This will increase significantly the computational complexity, which increases with the number of system variables; the number of rules increases exponentially as the number of system

whole weight of the human body. Angular positions of Link1 and Link2, 1

**4.1 Rules reduction strategy for general purpose tracking and control situations**

Such a strategy relies mainly on three concepts, (Ahmad et al. 2011).

error of Link1, ∆eδ1 - Angular position error of

error of Link2, ∆eδ2 - Angular position error of

Link1, eδ1

Link2, eδ2

Link2, eδ2

Link1, eδ1

The strategy is sought without compromising the robustness and capability of the system.

To assess the effect of coupling in the fuzzy control, the system is tested with two different configurations, which mainly differ at the input side of the controller, as shown in Table 1.

**With coupling effect Without coupling effect**

5 x 5 x 3 = 75 rules 5 x 5 = 25 rules

Table 1. Different input configurations of modular fuzzy logic controller

Link1, ∆eδ1

Link2, ∆eδ2

represent the input torque to the right and left wheels respectively. On

represents the torque between Link1 and Link2 to be used to cater for the

 , *L*

 and 2 , for

> and

The goal of the controller is to produce the required torques, namely *R*

torques *R*

2

the other hand, 2

variables increases.

Independence

**Link1 (Lifting & Stabilizing)**

**Link2 (Lifting & Stabilizing)**

**Rules of each lifting**

**and stabilizing**

 Functional Relationship Command Manipulation

 and *L*

The system was tested with both configurations and the performances, with and without coupling were comparably similar, see Figure 6. Therefore, as seen the second configuration performed well with fewer fuzzy rules fired, and this configuration is used in implementing the motion control for two-wheeled wheelchair.

Fig. 6. System performance comparison between coupled fuzzy inputs and decoupled fuzzy inputs in terms of δ1, δ2, R, L and 2.

The MFC is thus adopted for the two-wheeled wheelchair mechanisms, and the corresponding research objectives are:

The MFC can be divided into two significant categories, primary and secondary (Bessacini and Pinkos 1995). The controller is categorized according to different objectives. The control structure for achieving an upright two-wheeled maneuverable wheelchair is depicted in Figure 7. The general function of MFC is to minimize the errors in system responses considered. The primary goal unit caters for the upright control, which consists of lifting and stabilizing to the upright position and the transformation back to normal four-wheeled position of Link1 and Link2. These controllers are active most of the time even during maneuver. The secondary unit is activated by the coordinator (switch), with certain condition pre-set for output activation. It consists of different unique objectives involving linear motion control, steering control, additional chair height extension control. Each objective in the secondary goal unit is discussed in detail in the following sections.

Modular Fuzzy Logic Controller for Two-Wheeled Wheelchair 47

Fig. 8. Block diagram of two-wheeled wheelchair system motion

Fig. 9. Block diagram for linear motion control

shown in Figure 9.

This confined space is normally found in the domestic environment (home, office and library). Within such environment, linear motion is executed alone before steering is done and vice versa. The block diagram for linear motion control of two-wheeled wheelchair is

The FLC for linear motion (FLC3) consists of two inputs and two outputs. The controller inputs are the position error, *e* and the change of position error, ∆*e*, while the controller

Fig. 7. Adapted modular intercepts fuzzy logic system (Bessacini and Pinkos 1995)

#### **4.2 Simulation based performance analysis**

The overall motion control for two-wheeled wheelchair is represented in Figure 8.

#### **a. FLC for linear motion**

The linear motion control generally consists of forward and backward (reverse) motion control. They are both characterized as secondary systems (Bessacini and Pinkos 1995) since the system needs to fulfill the primary target to achieve the upright position for both links. Therefore MFC as discussed in Section 4 is very appropriate to implement.

Similar structure of FLC used for lifting and stabilizing is adopted for linear motion control. The controls differ in terms of input and output scaling factors due to different reference points executed. The control strategy designed in Matlab/Simulink was integrated with wheelchair model, which was developed in VN software environment as a plant. The motion (forward, backward or steering) takes place after lifting and stabilizing has been achieved. Results show that the MFC strategy designed works very well and gives good system performance.

In the current studies of wheelchair mobility, much research has been conducted on wheelchair mobility in large spaces (outdoor mobility) (Vries et al. 1999; Wong et al. 2007). In those researches, the distance and angle are considered at the same time to give output torque of the wheels. On the other hand, note that the two-wheeled wheelchair is designed for use in confined spaces, such that the linear motion and the steering motion are independent.

Fig. 7. Adapted modular intercepts fuzzy logic system (Bessacini and Pinkos 1995)

The overall motion control for two-wheeled wheelchair is represented in Figure 8.

Therefore MFC as discussed in Section 4 is very appropriate to implement.

The linear motion control generally consists of forward and backward (reverse) motion control. They are both characterized as secondary systems (Bessacini and Pinkos 1995) since the system needs to fulfill the primary target to achieve the upright position for both links.

Similar structure of FLC used for lifting and stabilizing is adopted for linear motion control. The controls differ in terms of input and output scaling factors due to different reference points executed. The control strategy designed in Matlab/Simulink was integrated with wheelchair model, which was developed in VN software environment as a plant. The motion (forward, backward or steering) takes place after lifting and stabilizing has been achieved. Results show that the MFC strategy designed works very well and gives good

In the current studies of wheelchair mobility, much research has been conducted on wheelchair mobility in large spaces (outdoor mobility) (Vries et al. 1999; Wong et al. 2007). In those researches, the distance and angle are considered at the same time to give output torque of the wheels. On the other hand, note that the two-wheeled wheelchair is designed for use in confined spaces, such that the linear motion and the steering motion are independent.

**4.2 Simulation based performance analysis**

**a. FLC for linear motion**

system performance.

Fig. 8. Block diagram of two-wheeled wheelchair system motion

This confined space is normally found in the domestic environment (home, office and library). Within such environment, linear motion is executed alone before steering is done and vice versa. The block diagram for linear motion control of two-wheeled wheelchair is shown in Figure 9.

Fig. 9. Block diagram for linear motion control

The FLC for linear motion (FLC3) consists of two inputs and two outputs. The controller inputs are the position error, *e* and the change of position error, ∆*e*, while the controller

Modular Fuzzy Logic Controller for Two-Wheeled Wheelchair 49

(Z), Positive Small (PS) and Positive Big (PB). The membership function for inputs and outputs of FLC3 is shown in Figure 10. Table 2 shows the implemented fuzzy rules for FLC3 controller. The two consecutive rows in the output part represent two fuzzy outputs, τR (first row) and τL (second row). The rules developed are predetermined using expert knowledge available such that all the errors should be brought back to the reference point immediately.

The system was commanded to move forward after 4s, where at this time the two links had been stabilized at the upright position. Figure 11 shows the final position of forward mechanism execution while Figure 12 to Figure 18 show the results over 15s of simulation time for forward movement of the two-wheeled wheelchair. The wheelchair was set to move 1.5m forward from its initial position. The results show that the FLC approach worked very well with the wheelchair system on two wheels. Figure 11 shows the final wheelchair position when it was set to move forward to 1.5m from the origin. It is noted from Figures 12 and 13 that both links settled after 4s from starting time of linear motion, which can be considered quite good performance for the initial attempts of parameters setting. Link1 tilted with a positive angle from the 0° upright position. This configuration was automatically adjusted to initiate the forward motion. The corresponding wheelchair position is shown in Figure 14. It is noted that as much as 0.1m of the steady state error appeared when it settled. Figure 15 shows the wheel torques (τR and τL) from the lifting and stabilizing controller of Link1 (FLC1), and the wheel torques from the linear motion control is shown in Figure 16. The torques vary from +40Nm to -40Nm during the forward motion with positive slope during initial phase of travel. The resultant wheel torques contributed by the lifting and stabilizing control as well as the linear motion control are shown in Figure 17.

The torque between Link1 and Link2 (τ2) given by (FLC2) is shown in Figure 18.

Fig. 11. Final position of 1.5m forward motion

*Forward Motion*

outputs are the torques, τR and τL. The fuzzy inputs are normalized so that they can be generalized and then processed using the fuzzy rules. Moreover, the input normalization is done due to the complexity of predetermining the range of change of position error, ∆*e*. Gaussian (bell shaped) type membership functions with default parameters given by Matlab/Simulink are used for all inputs and outputs. The membership levels for each input and outputs are five in total. These comprise Negative Big (NB), Negative Small (NS), Zero

Fig. 10. Membership functions for inputs and outputs for FLC3 of linear motion control

Table 2. Fuzzy rules for linear motion

(Z), Positive Small (PS) and Positive Big (PB). The membership function for inputs and outputs of FLC3 is shown in Figure 10. Table 2 shows the implemented fuzzy rules for FLC3 controller. The two consecutive rows in the output part represent two fuzzy outputs, τR (first row) and τL (second row). The rules developed are predetermined using expert knowledge available such that all the errors should be brought back to the reference point immediately.

#### *Forward Motion*

48 Fuzzy Logic – Controls, Concepts, Theories and Applications

outputs are the torques, τR and τL. The fuzzy inputs are normalized so that they can be generalized and then processed using the fuzzy rules. Moreover, the input normalization is done due to the complexity of predetermining the range of change of position error, ∆*e*. Gaussian (bell shaped) type membership functions with default parameters given by Matlab/Simulink are used for all inputs and outputs. The membership levels for each input and outputs are five in total. These comprise Negative Big (NB), Negative Small (NS), Zero

0

1

0

**NB NS Z PS PB**

PB PB PB PS Z PB PB PB PS Z

PB PB PS Z NS PB PB PS Z NS

PB PS Z NS NB PB PS Z NS NB

PS Z NS NB NB PS Z NS NB NB

Z NS NB NB NB Z NS NB NB NB

0.5

Degree of membership

Fig. 10. Membership functions for inputs and outputs for FLC3 of linear motion control

0.5

Degree of membership

1

NB NS Z PS PB

ChangeOfError

NB NS Z PS PB

Left torque

NB NS Z PS PB

Error

NB NS Z PS PB

Right torque

**∆e**

**NB**

**NS**

**Z**

**PS**

**PB**

Table 2. Fuzzy rules for linear motion

0

1

0

**e**

0.5

Degree of membership

0.5

Degree of membership

1

The system was commanded to move forward after 4s, where at this time the two links had been stabilized at the upright position. Figure 11 shows the final position of forward mechanism execution while Figure 12 to Figure 18 show the results over 15s of simulation time for forward movement of the two-wheeled wheelchair. The wheelchair was set to move 1.5m forward from its initial position. The results show that the FLC approach worked very well with the wheelchair system on two wheels. Figure 11 shows the final wheelchair position when it was set to move forward to 1.5m from the origin. It is noted from Figures 12 and 13 that both links settled after 4s from starting time of linear motion, which can be considered quite good performance for the initial attempts of parameters setting. Link1 tilted with a positive angle from the 0° upright position. This configuration was automatically adjusted to initiate the forward motion. The corresponding wheelchair position is shown in Figure 14. It is noted that as much as 0.1m of the steady state error appeared when it settled. Figure 15 shows the wheel torques (τR and τL) from the lifting and stabilizing controller of Link1 (FLC1), and the wheel torques from the linear motion control is shown in Figure 16. The torques vary from +40Nm to -40Nm during the forward motion with positive slope during initial phase of travel. The resultant wheel torques contributed by the lifting and stabilizing control as well as the linear motion control are shown in Figure 17. The torque between Link1 and Link2 (τ2) given by (FLC2) is shown in Figure 18.

Fig. 11. Final position of 1.5m forward motion

Modular Fuzzy Logic Controller for Two-Wheeled Wheelchair 51

0 5 10 15

0 5 10 15

Time (s)

0 5 10 15

Time (s)

Right torque Left torque

Right torque Left torque

Right torque Left torque

Time (s)

Fig. 15. Wheel torques, τR and τL due to lifting and stabilizing control, FLC1 (Nm)

Fig. 16. Wheel torques, τR and τL due to linear motion control, FLC3 (Nm)

Fig. 17. Resultant wheel torques due to FLC1 and FLC3 (Nm)

0

50

Resultant torques (Nm)

100

150

Wheel torques from

linear motion control (Nm)

0

50

100

0

50

Wheel torques from lifting

and stabilizing control (Nm)

100

150

Fig. 12. Angular position of Link1, δ1 (degree)

Fig. 13. Angular position of Link2, δ2 (degree)

Fig. 14. Wheelchair position, x (m)

0 5 10 15

Time (s)

0 5 10 15

Time (s)

0 5 10 15

Time (s)

0

Fig. 14. Wheelchair position, x (m)

0.5

1

Wheelchair position (m)

1.5

2

Fig. 13. Angular position of Link2, δ2 (degree)

Angular position of Link2 (degree)

0

10

Fig. 12. Angular position of Link1, δ1 (degree)

0

20

40

Angular position of Link1 (degree)

60

80

100

Fig. 15. Wheel torques, τR and τL due to lifting and stabilizing control, FLC1 (Nm)

Fig. 16. Wheel torques, τR and τL due to linear motion control, FLC3 (Nm)

Fig. 17. Resultant wheel torques due to FLC1 and FLC3 (Nm)

Modular Fuzzy Logic Controller for Two-Wheeled Wheelchair 53

This new capacity increased the number of DOF of the two-wheeled wheelchair. Thus, it is noticeable challenge to control the two-wheeled wheelchair where limited actuators are available for different functions. Therefore, suitable controllers are needed, and FLC is adopted. Results show that the FLC strategy works well and gives good system

Two inputs and two outputs FLC is developed to control the steering motion. The membership functions used are shown in Figure 20. The membership levels for each input

0

1

0

0.5

Degree of membership

0.5

Degree of membership

1

NB NS Z PS PB

ChangeofError

NB NS Z PS PB

Left torque

performance.

Fig. 19. Block diagram for steering motion control

NB NS Z PS PB

WCrotError

NB NS Z PS PB

Right torque

Fig. 20. Membership levels for inputs and outputs of steering control

0

1

0

0.5

Degree of membership

0.5

Degree of membership

1

Fig. 18. Torque between Link1 and Link2, τ2, FLC2 (Nm)

#### **b. FLC for steering motion**

A steering motion is needed when the two-wheeled wheelchair needs to change its direction. The two-wheeled wheelchair can rotate to the right or to the left depending on which direction it is commanded. There are two different approaches where steering could be realized (Tanimoto et al., 2009). Similar direction of wheel rotation with different magnitudes could lead to steering motion (moving both wheels forward with different magnitudes). The first approach causes bigger turning radius as compared to the second approach. The second approach to realize steering motion is by giving different direction of wheel rotation (moving right wheel forward and left backward). The output torques in this work given by the FLC used for steering motion covers both approaches according to the steering error and the change of the steering error. In contrast to normal steering for mobile robots, steering motion for the two-wheeled wheelchair is executed after the upright position has been achieved; Link1 and Link2 at the 0° upright position. Therefore the complexity in this configuration is higher than the steering motion using four wheels, since other motion controls are active at the same time.

A block diagram for steering motion control of two-wheeled wheelchair is shown in Figure 19. As discussed earlier, for reasons of simplicity, the torques applied to the two wheels are the same in magnitudes (one output torque from the controller) so as to move the wheelchair only forward or backward. Then each right and left wheel torque is made independent to realize the steering motion. The weight here represents the human body weight, for which an average 70kg human is used. Sensors are attached at the respective reference bodies for control and measurement. The control signals applied to the wheelchair model comprise the right torque, τR (Nm), left torque, τL (Nm) and torque between Link1 and Link2, τ2 (Nm). The measured outputs from the wheelchair system that consist of the angular position of Link1, δ1, (degree), angular position of Link2, δ2, (degree) and wheelchair rotation angle about the vertical axis, ψ (degree) are compared with the target references.

The wheelchair system modeled in VN software environment was used as a plant and controlled with the developed FLC in the Matlab/Simulink environment. The steering motion introduced takes place after the lifting and stabilizing mechanism has been achieved.

0 5 10 15

Time (s)

A steering motion is needed when the two-wheeled wheelchair needs to change its direction. The two-wheeled wheelchair can rotate to the right or to the left depending on which direction it is commanded. There are two different approaches where steering could be realized (Tanimoto et al., 2009). Similar direction of wheel rotation with different magnitudes could lead to steering motion (moving both wheels forward with different magnitudes). The first approach causes bigger turning radius as compared to the second approach. The second approach to realize steering motion is by giving different direction of wheel rotation (moving right wheel forward and left backward). The output torques in this work given by the FLC used for steering motion covers both approaches according to the steering error and the change of the steering error. In contrast to normal steering for mobile robots, steering motion for the two-wheeled wheelchair is executed after the upright position has been achieved; Link1 and Link2 at the 0° upright position. Therefore the complexity in this configuration is higher than the steering motion using four wheels, since

A block diagram for steering motion control of two-wheeled wheelchair is shown in Figure 19. As discussed earlier, for reasons of simplicity, the torques applied to the two wheels are the same in magnitudes (one output torque from the controller) so as to move the wheelchair only forward or backward. Then each right and left wheel torque is made independent to realize the steering motion. The weight here represents the human body weight, for which an average 70kg human is used. Sensors are attached at the respective reference bodies for control and measurement. The control signals applied to the wheelchair model comprise the right torque, τR (Nm), left torque, τL (Nm) and torque between Link1 and Link2, τ2 (Nm). The measured outputs from the wheelchair system that consist of the angular position of Link1, δ1, (degree), angular position of Link2, δ2, (degree) and wheelchair rotation angle about the vertical axis, ψ

The wheelchair system modeled in VN software environment was used as a plant and controlled with the developed FLC in the Matlab/Simulink environment. The steering motion introduced takes place after the lifting and stabilizing mechanism has been achieved.

**b. FLC for steering motion**

Fig. 18. Torque between Link1 and Link2, τ2, FLC2 (Nm)

other motion controls are active at the same time.

(degree) are compared with the target references.

Torque between

Link1 and Link2 (Nm)

0

50

100

This new capacity increased the number of DOF of the two-wheeled wheelchair. Thus, it is noticeable challenge to control the two-wheeled wheelchair where limited actuators are available for different functions. Therefore, suitable controllers are needed, and FLC is adopted. Results show that the FLC strategy works well and gives good system performance.

Fig. 19. Block diagram for steering motion control

Two inputs and two outputs FLC is developed to control the steering motion. The membership functions used are shown in Figure 20. The membership levels for each input

Fig. 20. Membership levels for inputs and outputs of steering control

Modular Fuzzy Logic Controller for Two-Wheeled Wheelchair 55

0 5 10 15

Time (s)

0 5 10 15

Time (s)

Fig. 21. Final steering position for 30° reference point

Fig. 23. Angular position of Link2, δ2 (degree)

Angular position of Link2 (degree)

0

10

Fig. 22. Angular position of Link1, δ1 (degree)

0

20

40

Angular position of Link1 (degree)

60

80

100

and output comprise Negative Big (NB), Negative Small (NS), Zero (Z), Positive Small (PS) and Positive Big (PB). The two inputs used were the wheelchair rotation error (eψ) and the change of wheelchair rotation error, ∆eψ. The controller outputs are the right and left wheel torques, τR and τL. All membership functions of input and output parameters are normalized for ease of control. Table 3 shows the implemented fuzzy rules for steering motion control (FLC4), where the first row relates to right-wheel torque and the second (shaded) row relates to the left-wheel torque.

Table 3. Fuzzy rules for steering motion

#### *Steering to 30°*

The final position of the wheelchair system can be seen in Figure 21. Figures 22 and Figure 23 show the angular positions of Link1, δ1 and Link2, δ2 respectively when the system was set to steer at 30° causing the two-wheeled wheelchair to rotate to the left from its initial position. Both links settled with small steady state error after the steering settlement was achieved. As noted, they settled in less than 4s. As noted in Figure 24, the wheelchair rotated very near to 30°, with < 0.1° of the steady state error. The output torques from each lifting and stabilizing control of Link1 as well as the steering control are shown in Figures 25 and 26 respectively. Note that the output torques from FLC1 had the same magnitude and direction for both right and left wheels. On the other hand, the output torque from FLC4 had the same magnitude but different in direction representing the fuzzy rules output for steering motion. The torque between Link1 and Link2 can be seen in Figure 27. As noted, it changed between +30Nm and -20Nm to maintain the upright stability of Link2 with human payload during the steering motion. The resultant torques for both fuzzy controllers (FLC1+FLC4) is shown in Figure 28. The system was then tested to rotate at a different angle (negative angle leading to rotation to the right).

and output comprise Negative Big (NB), Negative Small (NS), Zero (Z), Positive Small (PS) and Positive Big (PB). The two inputs used were the wheelchair rotation error (eψ) and the change of wheelchair rotation error, ∆eψ. The controller outputs are the right and left wheel torques, τR and τL. All membership functions of input and output parameters are normalized for ease of control. Table 3 shows the implemented fuzzy rules for steering motion control (FLC4), where the first row relates to right-wheel torque and the second

**NB NS Z PS PB**

NB NB NB NS Z PB PB PB PS Z

NB NB NS Z PS PB PB PS Z NS

NB NS Z PS PB PB PS Z NS NB

NS Z PS PB PB PS Z NS NB NB

Z PS PB PB PB Z NS NB NB NB

The final position of the wheelchair system can be seen in Figure 21. Figures 22 and Figure 23 show the angular positions of Link1, δ1 and Link2, δ2 respectively when the system was set to steer at 30° causing the two-wheeled wheelchair to rotate to the left from its initial position. Both links settled with small steady state error after the steering settlement was achieved. As noted, they settled in less than 4s. As noted in Figure 24, the wheelchair rotated very near to 30°, with < 0.1° of the steady state error. The output torques from each lifting and stabilizing control of Link1 as well as the steering control are shown in Figures 25 and 26 respectively. Note that the output torques from FLC1 had the same magnitude and direction for both right and left wheels. On the other hand, the output torque from FLC4 had the same magnitude but different in direction representing the fuzzy rules output for steering motion. The torque between Link1 and Link2 can be seen in Figure 27. As noted, it changed between +30Nm and -20Nm to maintain the upright stability of Link2 with human payload during the steering motion. The resultant torques for both fuzzy controllers (FLC1+FLC4) is shown in Figure 28. The system was then tested to rotate at a different angle

(shaded) row relates to the left-wheel torque.

**NB**

**NS**

**Z**

**PS**

**PB**

(negative angle leading to rotation to the right).

Table 3. Fuzzy rules for steering motion

*Steering to 30°*

**eψ**

**∆eψ**

Fig. 21. Final steering position for 30° reference point

Fig. 22. Angular position of Link1, δ1 (degree)

Fig. 23. Angular position of Link2, δ2 (degree)

Modular Fuzzy Logic Controller for Two-Wheeled Wheelchair 57

0 5 10 15

Right torque Left torque

Time (s)

0 5 10 15

Time (s)

Fuzzy logic is one of the control techniques that is very close to human feelings and expressions. It can be easily understood and implemented although the knowledge about classical or conventional control system is not much identified. Nevertheless the general knowledge of the system involved must be generally known otherwise it is difficult to formulate a fuzzy controller for such system. If the system involved is known to be linear, and simple thus it is more worth to start with conventional Proportional-Integral-Differential (PID) controller. Otherwise if the system is known to be very complex, nonlinear and ill-defined type of system, then it is suggested to use one of the computational approaches such as fuzzy logic. This method was successfully implemented in the twowheeled wheelchair system where a modular fuzzy control (MFC) was developed and implemented for controlling lifting and stabilizing mechanism, linear and steering motion control. Note that since a wheelchair is a main means of transport for disabled and elderly people, this two-wheeled wheelchair system would allow the user to achieve a higher level of height without assistance and hence independence. The wheelchair has been modeled as a double inverted pendulum. The integrated two-wheeled wheelchair with a human model

Fig. 28. Resultant wheel torques (Nm)

0

50

Resultant wheel torques (Nm)

**5. Conclusion**

100

150

Fig. 27. Torque between Link1 and Link2, τ2 (Nm)

Torque between

Link1 and Link2 (Nm)

0

50

Fig. 24. Wheelchair rotation, ψ (degree)

Fig. 25. Wheel torques (τR and τL) from FLC1 (Nm)

Fig. 26. Wheel torques (τR and τL) from steering motion control, FLC4 (Nm)

Fig. 27. Torque between Link1 and Link2, τ2 (Nm)

Fig. 28. Resultant wheel torques (Nm)

#### **5. Conclusion**

56 Fuzzy Logic – Controls, Concepts, Theories and Applications

0 5 10 15

Time (s)

0 5 10 15

Right torque Left torque

Right torque Left torque

Time (s)

0 5 10 15

Time (s)

Fig. 26. Wheel torques (τR and τL) from steering motion control, FLC4 (Nm)

Fig. 24. Wheelchair rotation, ψ (degree)

0

Wheel torques from

steering control (Nm)

50

100

Fig. 25. Wheel torques (τR and τL) from FLC1 (Nm)

0

50

Wheel torques from lifting

and stabilizing of Link1 (Nm)

100

150

0

10

Wheelchair rotation (degree)

20

30

40

Fuzzy logic is one of the control techniques that is very close to human feelings and expressions. It can be easily understood and implemented although the knowledge about classical or conventional control system is not much identified. Nevertheless the general knowledge of the system involved must be generally known otherwise it is difficult to formulate a fuzzy controller for such system. If the system involved is known to be linear, and simple thus it is more worth to start with conventional Proportional-Integral-Differential (PID) controller. Otherwise if the system is known to be very complex, nonlinear and ill-defined type of system, then it is suggested to use one of the computational approaches such as fuzzy logic. This method was successfully implemented in the twowheeled wheelchair system where a modular fuzzy control (MFC) was developed and implemented for controlling lifting and stabilizing mechanism, linear and steering motion control. Note that since a wheelchair is a main means of transport for disabled and elderly people, this two-wheeled wheelchair system would allow the user to achieve a higher level of height without assistance and hence independence. The wheelchair has been modeled as a double inverted pendulum. The integrated two-wheeled wheelchair with a human model

**4**

Bin Zi1,2

*China*

**Fuzzy Control System Design and**

 **Cable-Driven Manipulators**

*1School of Mechanical and Electrical Engineering,*

 *Zhejiang University, Hangzhou, Zhejiang*

**Analysis for Completely Restrained**

 *China University of Mining and Technology, Xuzhou, Jiangsu, 2The State Key Laboratory of Fluid Power and Mechatronic Systems,*

Cable-driven manipulators, referred to as the overhead crane and rotary crane are widely used in the manufacturing and construction industries in order to move heavy objects as illustrated in Fig. 1. Cable-driven manipulators are relatively simple form, with multiple cables attached to a mobile platform or end-effector. The end-effector may be equipped with various attachments, including hooks, cameras, robotic grippers and so on.  For the preceding reasons, cable-driven manipulators have received attention and have been recently studied since 1980s (Behzadipour & Khajepour, 2005; Ghasemi et al., 2008; Motoji,

**1. Introduction**

2004; Oh & Agrawal, 2005; Pham et al., 2006).

Fig. 1. Crane-type cable manipulator.

has been imported as the plant into Matlab/Simulink environment for control and evaluation purposes. Therefore, fuzzy logic techniques have been found suitable for control of the two-wheeled wheelchair.

A Modular Fuzzy logic Control (MFC) approach has been adopted, where the control tasks are divided into primary and secondary tasks (subsystems), and FLC modules have been designed and executed for the various control tasks accordingly. Among the control tasks, lifting and stabilizing in the upright position are considered as the primary control system task. Secondary system tasks include linear motion and steering motion. The MFC strategy developed is based on a hierarchical approach whereby the primary subsystem must be executed followed by selection of secondary subsystems. Both linear and steering motions have been successfully controlled independently using a two-input two-output PD-type FLC.

The proposed MFC has been successfully implemented and tested within simulated exercises for two-wheeled wheelchair application. The results presented proved that the MFC approach works very well in controlling highly nonlinear systems such as a wheelchair on two wheels and significantly reduces the number of rules.

## **6. Acknowledgment**

The authors would like to express their appreciation to the International Islamic University Malaysia (IIUM) for sponsoring the publication of this book chapter.

## **7. References**

## **Fuzzy Control System Design and Analysis for Completely Restrained Cable-Driven Manipulators**

Bin Zi1,2

*1School of Mechanical and Electrical Engineering, China University of Mining and Technology, Xuzhou, Jiangsu, 2The State Key Laboratory of Fluid Power and Mechatronic Systems, Zhejiang University, Hangzhou, Zhejiang China*

## **1. Introduction**

58 Fuzzy Logic – Controls, Concepts, Theories and Applications

has been imported as the plant into Matlab/Simulink environment for control and evaluation purposes. Therefore, fuzzy logic techniques have been found suitable for control

A Modular Fuzzy logic Control (MFC) approach has been adopted, where the control tasks are divided into primary and secondary tasks (subsystems), and FLC modules have been designed and executed for the various control tasks accordingly. Among the control tasks, lifting and stabilizing in the upright position are considered as the primary control system task. Secondary system tasks include linear motion and steering motion. The MFC strategy developed is based on a hierarchical approach whereby the primary subsystem must be executed followed by selection of secondary subsystems. Both linear and steering motions have been successfully controlled independently using a two-input two-output PD-type FLC. The proposed MFC has been successfully implemented and tested within simulated exercises for two-wheeled wheelchair application. The results presented proved that the MFC approach works very well in controlling highly nonlinear systems such as a

The authors would like to express their appreciation to the International Islamic University

Ahmad, S., N. H. Siddique and M. O. Tokhi (2011). A modular fuzzy control approach for

Bessacini, A. F. and R. F. Pinkos (1995). A hierarchical fuzzy controller for intercept

Engelbrecht, A. P. (2007). *Computational intelligence: An introduction*. Chichester, England,

Mamdani, E. H. (1974). Application of fuzzy algorithms for control of a simple dynamic

Vries, T. J. A. D., C. V. Heteran and L. Huttenhuis (1999). Modeling and control of a fast

Wong, C.-C., H.-Y. Wang, S.-A. Li and C.-T. Cheng (2007). Fuzzy controller designed by GA for two-wheeled mobile robots. *International Journal of Fuzzy Systems* 9(1).

moving, highly maneuverable wheelchair. *International Biomechatronics Workshop*,

two-wheeled wheelchair. *Journal of Intelligent and Robotic Systems* Springer Journal,

guidance with a forbidden zone. Newport, Naval Undersea Warfare Center

wheelchair on two wheels and significantly reduces the number of rules.

Malaysia (IIUM) for sponsoring the publication of this book chapter.

process. *Proceeding of IEEE* 121(12): 1585-1588.

Siljak, D. (1991) *Decentralized Control* of *Complex Systems,* Academic Press**.**

Zadeh, L. A. (1965). Fuzzy sets. *Information and Control* 8(3): 338 - 353.

Reznik, L. (1997). *Fuzzy Controllers*. Oxford, Newnes.

of the two-wheeled wheelchair.

**6. Acknowledgment**

Division.

Vol. 64(3-4), pp. 401-426.

John Wiley & Sons.

Enschede, Netherlands.

**7. References**

Cable-driven manipulators, referred to as the overhead crane and rotary crane are widely used in the manufacturing and construction industries in order to move heavy objects as illustrated in Fig. 1. Cable-driven manipulators are relatively simple form, with multiple cables attached to a mobile platform or end-effector. The end-effector may be equipped with various attachments, including hooks, cameras, robotic grippers and so on.  For the preceding reasons, cable-driven manipulators have received attention and have been recently studied since 1980s (Behzadipour & Khajepour, 2005; Ghasemi et al., 2008; Motoji, 2004; Oh & Agrawal, 2005; Pham et al., 2006).

Fig. 1. Crane-type cable manipulator.

Fuzzy Control System Design and Analysis for Completely Restrained Cable-Driven Manipulators 61

design. Results and discussions are presented in Section 5. Finally, concluding remarks are

The CRCM suspends an end-effector (clog) by four cables and restrains all motion degrees of freedom for the object using the cables and gravitational force when the end-effector moves within the workspace. In the design, of each cable in the CRCM, one end is connected to the end-effector, the other end rolls through a pulley fixed on the top of the relative pillar and then is fed into a servo mechanism, with which cable length can be altered. The design of CRCM follows a built-up modular system, as illustrated in Fig.2. The system comprises several components: servo motor, belt pulley drive mechanism, speed reducer, girder,

Reliability, long-distance transmission, high speed and precision are paramount for the CRCM design. The structure of the CRCM is shown in Fig.3, and the end-effector is driven by four sets of servomechanism. Belts are looped over pulleys. In a two pulley system, the belt can either drive the pulleys in the same direction. As a source of motion, a conveyor belt is one application where the belt is adapted to continually carry a load between two distant

A simple schematic of the CRCM representing the coordinate systems is shown in Fig. 4. With the bottom of the pillar corresponding to the point *B*3 as the origin, a Cartesian coordinate system is established. The end-effector is predigested as a particle whose location coordinates are *Axyz* (,,) , and the distance between each pulley center whose coordinates are (,,) *Bxy z iiii* and the end-effector is ( 1,2,3,4) *il i* . Four pillars have the same height and are arrayed in a rectangular on the ground, whose deformation in movement is ignored. In order to simplify the calculation, the cable is treated as a kind of massless rigid body, which

points. Typically, gears and elastic drive belts are applied to transmit motion.

provided in Section 6.

**2. Mechanical design**

Fig. 2. Model of the CRCM.

**3. Modeling and analysis**

has no deformation, and can only sustain tension.

windlass, cable pillar, cable, end-effector, and so on.

Cable-driven manipulators can be classified as either incompletely restrained or completely restrained (Bosscher & Ebert-Uphoff, 2006). Cable-driven manipulators are underconstrained if it relies on gravity to determine the pose (position and orientation) of the end-effector, while they are completely restrained if the pose of the end-effector is completely determined by the lengths of the cables. As you know, dynamics is a huge field of study devoted to studying the forces required to cause motion. In order to accelerate the robot from rest, glide at a constant end-effector velocity, and finally decelerate to a stop, a complex set of torque functions must be applied by the joint actuators (Craig, 2005). The motivation for this paper comes directly from the design, mechanics analysis, and control of completely restrained cable-driven manipulators (CRCM) with 3 Degrees of Freedom (DOF). As demonstrated in (Anupoju et al., 2005), servomechanism dynamics constitute an important component of the complete robotic dynamics. Therefore, the dynamics of the servomotors and its gears must be modeled for further control design. However, the literature on the CRCM system including the actuator dynamics is sparse.

CRCM systems are multivariable in nature. The control of the multivariable systems is a complicated problem due to the coupling that exists between the control inputs and the outputs, and the multivariable systems are nonlinear and uncertain, therefore, their control problem becomes more challenging (Chien, 2008; Yousef et al., 2009). In order to achieve a high-precision performance, the controller of the CRCM must effectively and accurately manipulate the motion trajectory. It is well known that up until now, a conventional proportional-integral-derivative (PID) controller has been widely used in industry due to its simple control structure, ease of design, and inexpensive cost (Reznik et al., 2000; Visioli, 2001). However, the CRCM is a multivariable nonlinear coupling dynamic system which suffers from structured and unstructured uncertainties such as payload variation, external disturbances, etc. As a result, the PID controller cannot yield a good control performance for this type of control system. For dealing with nonlinear effects, various control algorithms have been proposed. Among them, adaptive control and fuzzy logic system algorithm draw much attention due to the applicability for typically highly nonlinear systems (Chang, 2000; Soyguder & Alli, 2010; Su & Stepanenko, 1994). The idea of fuzzy set and fuzzy control is introduced by Zadeh in an attempt to control systems that are structurally difficult to model (Feng, 2006; Zadeh, 1965). Fuzzy controllers have been well accepted in control engineering practice. The major advantages in all these fuzzy-based control schemes are that the developed controllers can be employed to deal with increasingly complex systems to implement the controller without any precise knowledge of the structure of entire dynamic model. As a knowledge-based approach, the fuzzy controller usually depends on linguistics-based reasoning in design. However, even though a system is well defined mathematically, the fuzzy controller is still preferred by control engineers since it is relatively more understandable whereas expert knowledge can be incorporated conveniently. Recently, the fuzzy controller of nonlinear systems was studied by many authors and has also been extensively adopted in adaptive control of robot manipulators (Chen et al., 1996; Labiod et al., 2005; Purwar et al., 2005; Yoo & Ham, 1998). It has been proven that adaptive fuzzy control is a powerful technique and being increasingly applied in the discipline of systems control, especially when the controlled system has uncertainties and highly nonlinearities (Yu et al., 2011).

This chapter is organized as follows. First, the mechanical system is designed in Section 2. Then, modeling and analysis of the cable-driven manipulator are described in Section 3. Section 4 presents the developed systematic approach for the adaptive fuzzy controller design. Results and discussions are presented in Section 5. Finally, concluding remarks are provided in Section 6.

## **2. Mechanical design**

60 Fuzzy Logic – Controls, Concepts, Theories and Applications

Cable-driven manipulators can be classified as either incompletely restrained or completely restrained (Bosscher & Ebert-Uphoff, 2006). Cable-driven manipulators are underconstrained if it relies on gravity to determine the pose (position and orientation) of the end-effector, while they are completely restrained if the pose of the end-effector is completely determined by the lengths of the cables. As you know, dynamics is a huge field of study devoted to studying the forces required to cause motion. In order to accelerate the robot from rest, glide at a constant end-effector velocity, and finally decelerate to a stop, a complex set of torque functions must be applied by the joint actuators (Craig, 2005). The motivation for this paper comes directly from the design, mechanics analysis, and control of completely restrained cable-driven manipulators (CRCM) with 3 Degrees of Freedom (DOF). As demonstrated in (Anupoju et al., 2005), servomechanism dynamics constitute an important component of the complete robotic dynamics. Therefore, the dynamics of the servomotors and its gears must be modeled for further control design. However, the

CRCM systems are multivariable in nature. The control of the multivariable systems is a complicated problem due to the coupling that exists between the control inputs and the outputs, and the multivariable systems are nonlinear and uncertain, therefore, their control problem becomes more challenging (Chien, 2008; Yousef et al., 2009). In order to achieve a high-precision performance, the controller of the CRCM must effectively and accurately manipulate the motion trajectory. It is well known that up until now, a conventional proportional-integral-derivative (PID) controller has been widely used in industry due to its simple control structure, ease of design, and inexpensive cost (Reznik et al., 2000; Visioli, 2001). However, the CRCM is a multivariable nonlinear coupling dynamic system which suffers from structured and unstructured uncertainties such as payload variation, external disturbances, etc. As a result, the PID controller cannot yield a good control performance for this type of control system. For dealing with nonlinear effects, various control algorithms have been proposed. Among them, adaptive control and fuzzy logic system algorithm draw much attention due to the applicability for typically highly nonlinear systems (Chang, 2000; Soyguder & Alli, 2010; Su & Stepanenko, 1994). The idea of fuzzy set and fuzzy control is introduced by Zadeh in an attempt to control systems that are structurally difficult to model (Feng, 2006; Zadeh, 1965). Fuzzy controllers have been well accepted in control engineering practice. The major advantages in all these fuzzy-based control schemes are that the developed controllers can be employed to deal with increasingly complex systems to implement the controller without any precise knowledge of the structure of entire dynamic model. As a knowledge-based approach, the fuzzy controller usually depends on linguistics-based reasoning in design. However, even though a system is well defined mathematically, the fuzzy controller is still preferred by control engineers since it is relatively more understandable whereas expert knowledge can be incorporated conveniently. Recently, the fuzzy controller of nonlinear systems was studied by many authors and has also been extensively adopted in adaptive control of robot manipulators (Chen et al., 1996; Labiod et al., 2005; Purwar et al., 2005; Yoo & Ham, 1998). It has been proven that adaptive fuzzy control is a powerful technique and being increasingly applied in the discipline of systems control, especially when the controlled system has uncertainties

This chapter is organized as follows. First, the mechanical system is designed in Section 2. Then, modeling and analysis of the cable-driven manipulator are described in Section 3. Section 4 presents the developed systematic approach for the adaptive fuzzy controller

literature on the CRCM system including the actuator dynamics is sparse.

and highly nonlinearities (Yu et al., 2011).

The CRCM suspends an end-effector (clog) by four cables and restrains all motion degrees of freedom for the object using the cables and gravitational force when the end-effector moves within the workspace. In the design, of each cable in the CRCM, one end is connected to the end-effector, the other end rolls through a pulley fixed on the top of the relative pillar and then is fed into a servo mechanism, with which cable length can be altered. The design of CRCM follows a built-up modular system, as illustrated in Fig.2. The system comprises several components: servo motor, belt pulley drive mechanism, speed reducer, girder, windlass, cable pillar, cable, end-effector, and so on.

Reliability, long-distance transmission, high speed and precision are paramount for the CRCM design. The structure of the CRCM is shown in Fig.3, and the end-effector is driven by four sets of servomechanism. Belts are looped over pulleys. In a two pulley system, the belt can either drive the pulleys in the same direction. As a source of motion, a conveyor belt is one application where the belt is adapted to continually carry a load between two distant points. Typically, gears and elastic drive belts are applied to transmit motion.

Fig. 2. Model of the CRCM.

## **3. Modeling and analysis**

A simple schematic of the CRCM representing the coordinate systems is shown in Fig. 4. With the bottom of the pillar corresponding to the point *B*3 as the origin, a Cartesian coordinate system is established. The end-effector is predigested as a particle whose location coordinates are *Axyz* (,,) , and the distance between each pulley center whose coordinates are (,,) *Bxy z iiii* and the end-effector is ( 1,2,3,4) *il i* . Four pillars have the same height and are arrayed in a rectangular on the ground, whose deformation in movement is ignored. In order to simplify the calculation, the cable is treated as a kind of massless rigid body, which has no deformation, and can only sustain tension.

Fuzzy Control System Design and Analysis for Completely Restrained Cable-Driven Manipulators 63

11 11

*l xx yy zz l xx yy zz l xx yy zz l xx yy zz*

22 2 2

As the pillars are arrayed in a rectangular on the base, according to the geometric

2222

The general dynamic equations of motion can be derived from the Lagrangian method. In the model, the end-effector is assumed to act as a point mass and the cable is treated as a kind of massless rigid body. As a result, the kinetic energy, *K* , and the potential energy, *P* ,

cable in or let it out, therefore, it is desirable to regard the cable lengths as the variables. By substituting the forward kinematic equations (2) into the kinetic energy and potential energy shown in Eqs. (4) and (5), respectively, the Lagrange's Equation can be written in the

> ( ) *i i ii dK K P Q dt q q q*

where and the variables, *i q* (for *i* 1,2,3,4 ), the generalized forces, *Qi* , respectively, can be

22 2 2

( )( )( ) ( )( )( ) ( )( )( ) ( )( )( )

22 22

33 33

44 44

The forward kinematic equations can be found by solving (1) for (,,) *xyz* , which results in

1 2 2 1 21

( ) / 2( ) ( ) / 2( ) ( )( ) *i i ii*

*xl l x x xx y l l y y yy z l xx y y z*

2 3 3 2 32 222

2 22

2 22

2 22

2 22

*l* and the end-effector location *Axyz* (,,) , can be

4213 *llll* (3)

1 2 22 ( ) 2 *K mx y z* (4)

*l* , 2*l* , 3*l* and 4*l* , are directly controlled by rotating the winch to reel the

*P mgz* (5)

(6)

(1)

(2)

The relationship between the cable length *i*

of the end-effector can be written in Cartesian coordinates as

where *m* is mass of end-effector, *g* is acceleration of gravity.

relationship, the geometric constraint of the cable length is calculated as

easily obtained as follows:

the following:

The cable lengths, 1

following form

expressed as

Fig. 3. Mechanical structure of the CRCM

Fig. 4. Structure model of the CRCM

Pulley Cable Cable pillar

Girder

Servomotor

Servomotor

Fig. 3. Mechanical structure of the CRCM

Fig. 4. Structure model of the CRCM

Belt-pulley drive mechanism

Speed reducer

End-effector (clog)

Windlass

End-effector (clog)

Girder

Cable

The relationship between the cable length *i l* and the end-effector location *Axyz* (,,) , can be easily obtained as follows:

$$\begin{cases} l\_1 = \sqrt{(\mathbf{x} - \mathbf{x}\_1)^2 + \left(y - y\_1\right)^2 + \left(z - z\_1\right)^2} \\ l\_2 = \sqrt{\left(\mathbf{x} - \mathbf{x}\_2\right)^2 + \left(y - y\_2\right)^2 + \left(z - z\_2\right)^2} \\ l\_3 = \sqrt{\left(\mathbf{x} - \mathbf{x}\_3\right)^2 + \left(y - y\_3\right)^2 + \left(z - z\_3\right)^2} \\ l\_4 = \sqrt{\left(\mathbf{x} - \mathbf{x}\_4\right)^2 + \left(y - y\_4\right)^2 + \left(z - z\_4\right)^2} \end{cases} \tag{1}$$

The forward kinematic equations can be found by solving (1) for (,,) *xyz* , which results in the following:

$$\begin{cases} \mathbf{x} = \left(l\_1^{-2} - l\_2^{-2} + \mathbf{x}\_2^{-2} - \mathbf{x}\_1^{-2}\right) / \, 2(\mathbf{x}\_2 - \mathbf{x}\_1) \\ y = \left(l\_2^{-2} - l\_3^{-2} + y\_3^{-2} - y\_2^{-2}\right) / \, 2(y\_3 - y\_2) \\ z = -\sqrt{l\_i^2 - (\mathbf{x} - \mathbf{x}\_i)^2 - (y - y\_i)^2} + z\_i \end{cases} \tag{2}$$

As the pillars are arrayed in a rectangular on the base, according to the geometric relationship, the geometric constraint of the cable length is calculated as

$$l\_4^2 + l\_2^2 = l\_1^2 + l\_3^2\tag{3}$$

The general dynamic equations of motion can be derived from the Lagrangian method. In the model, the end-effector is assumed to act as a point mass and the cable is treated as a kind of massless rigid body. As a result, the kinetic energy, *K* , and the potential energy, *P* , of the end-effector can be written in Cartesian coordinates as

$$K = \frac{1}{2}m(\dot{\mathbf{x}}^2 + \dot{\mathbf{y}}^2 + \dot{\mathbf{z}}^2) \tag{4}$$

$$P = \text{mgz} \tag{5}$$

where *m* is mass of end-effector, *g* is acceleration of gravity.

The cable lengths, 1 *l* , 2*l* , 3*l* and 4*l* , are directly controlled by rotating the winch to reel the cable in or let it out, therefore, it is desirable to regard the cable lengths as the variables. By substituting the forward kinematic equations (2) into the kinetic energy and potential energy shown in Eqs. (4) and (5), respectively, the Lagrange's Equation can be written in the following form

$$\frac{d}{dt}(\frac{\partial K}{\partial \dot{q}\_i}) - \frac{\partial K}{\partial q\_i} + \frac{\partial P}{\partial q\_i} = Q\_i \tag{6}$$

where and the variables, *i q* (for *i* 1,2,3,4 ), the generalized forces, *Qi* , respectively, can be expressed as

Fuzzy Control System Design and Analysis for Completely Restrained Cable-Driven Manipulators 65

In which, 1234 [ ]*T T T T T T* ; *ir* is the radius of the windlass, (for 1,2,3,4 *i* ). For more

The nominal model of CRCM including servomechanism dynamics is described by the

2 ( )

*d d nKU K J dt dt*

It is also well known that there is a dual relation between externally applied wrench on the end-effector and the cable tensions required to keep the system in equilibrium. The above dynamic model is valid only for 0 *Ti* , i.e., the cables are in tension. Clearly, the equation (12) is a non-homogeneous linear quaternary equations. The solution of the equations will be multiple. For the sake of this, the suitable solution is found through MATLAB software

In general, a fuzzy logic system consists of four parts: the knowledge base, the fuzzifier, the fuzzy inference engine, and the defuzzifier. There are many different choices for the design of fuzzy system if the mapping is static. In this study, we consider a MIMO fuzzy logic system (Liu, 2008; Yoo & Ham, 2000). Supposing the fuzzy logic system performs a mapping

fuzzy knowledge base consists of a collection of fuzzy IF–THEN rules in the following form

*T*

fuzzy system, respectively, *l Fi* and *l Cj* (for 1,2, , *l M* ) are linguistic variables, and *M* is the number of fuzzy rules. Based on the fuzzy inference engine working on fuzzy rules, the

The output of the fuzzy control system with singleton fuzzifier, product inference engine,

1 1

*M n l*

*l i j M n*

*l i*

1 1

*m VV V R m* , *V R j* (for *j m* 1,2, , ). For a MIMO system, the

*l F* and...and *n x* is *l Fn*

( )

( )

*x*

*l i*

*F i*

*l i*

*i j F*

*y x*

*d bi*

*mi mi*

2

(12)

*n UU U R n* , *U R i* (for

(14)

*l C* and...and *ym* is *l Cm* (13)

*yy y* , *m V* are the input and output vectors of the

*mi mi*

details on the specification of the drive transmission system, refer to (Zi et al., 2009).

2 2

*d d KU J K dt dt*

*bi i U C ni*

() (,)

*Dqq Cqq*

from fuzzy sets in *n U R* to fuzzy sets in *m V R* , where 1

( )*l R* : IF 1 *x* is 1

center average defuzzifier is in the following form (Yoo & Ham, 2000)

*y*

*U C mi*

based on the pseudo-inverse method.

THEN 1 *y* is 1

*n xx x U* , and 1

defuzzifier maps fuzzy sets in *U* to a crisp point in *V* .

*T*

**4. Adaptive fuzzy control**

*i n* 1,2, , ), 1

where 1

following formulation

$$q\_i = \begin{bmatrix} l\_1 \ l\_2 \ l\_3 \ l\_4 \end{bmatrix}^T \tag{7}$$

$$Q\_i = \left[ -\tau\_1 \left\langle r\_1 - \tau\_2 \left\langle r\_2 - \tau\_3 \right\rangle r\_3 - \tau\_4 \left\langle r\_4 \right\rangle \right]^\dagger \tag{8}$$

The windlass torques, 1 , 2 , 3 and 4 , are the control inputs 1*r* , 2*r* , 3*r* and 4*r* , is the radius of the windlass.

Given the equations of motion shown above, using the assumptions along with various substitutions and algebraic manipulations of the CRCM derived, the dynamic equation of the CRCM can be expressed as

$$D(q)\ddot{q} + C(q, \dot{q}) + \tau\_d = \sigma \tag{9}$$

where 4 4 *D q*( ) is the inertia matrix which is symmetric positive define, 4 *Cqq* (,) is a nonlinear Coriolis/centripetal/gravity vector terms, 4 *d* represents the disturbance which is bounded, and 4 is the input torque vector with 1234 *T* . The 4x4 matrix *D q*( ) and the 4x1 vector *Cqq* (,) will be referred to as *D* and *C* respectively. The details of these expressions will be omitted for the sake of brevity.

The dynamic model is presented in two parts: one is directed to the structural model (CRCM) above and the other is related to the actuator dynamics (servo mechanism). We have already developed mechanics equations of the drive transmission system (Zi et al., 2009), and briefly outline here. The extendable actuator of each subsystem of the CRCM system is comprised of an alternating current (AC) servomotor & drive unit, belt pulley drive mechanism, two-level cycloid-gear speed reducer, and windlass. To simplify matters, here without regard to the belt pulley drive mechanism, the next step servomechanism model is developed. Without going into details, the servomechanism dynamic model is briefly described by the following formulation,

$$\begin{cases} \mathcal{K}\_{lI} \mathcal{U}\_{\mathcal{C}} = \mathcal{J}\_{mi} \frac{d^2 \theta\_{mi}}{dt^2} + \mathcal{K}\_{oo} \frac{d \theta\_{mi}}{dt} \\\\ \tau\_{hi} = n\_i (\mathcal{K}\_{lI} \mathcal{U}\_{\mathcal{C}} - \mathcal{K}\_{oo} \frac{d \theta\_{mi}}{dt} - \mathcal{J}\_{mi} \frac{d^2 \theta\_{mi}}{dt^2}) \end{cases} \tag{10}$$

where *bi* (for 1,2,3,4 *i* ) is the torque of the windlass; *ni* is the gear ratio; *UC* is control voltage ; *UC* and *Kw* are positive constant, respectively; *mi J* denotes the moment of inertia of the motor; *ni J* is the equivalent moment of inertia including motor, speed reducer, flywheel and windlass, and *mi* is the rotor angular position.

The driving force of cable *Ti* (for 1,2,3,4 *i* ) can be expressed as

$$\begin{cases} K\_{lI} \mathbf{U}\_{\gets} = \mathbf{J}\_{mi} \frac{d^2 \theta\_{mi}}{dt^2} + \mathbf{K}\_{oo} \frac{d \theta\_{mi}}{dt} \\\\ T\_i = \frac{\mathbf{r}\_{bi}}{r\_i} = \frac{n\_i}{r\_i} (\mathbf{K}\_{lI} \mathbf{U}\_{\gets} - \mathbf{K}\_{oo} \frac{d \theta\_{mi}}{dt} - \mathbf{J}\_{mi} \frac{d^2 \theta\_{mi}}{dt^2}) \end{cases} \tag{11}$$

In which, 1234 [ ]*T T T T T T* ; *ir* is the radius of the windlass, (for 1,2,3,4 *i* ). For more details on the specification of the drive transmission system, refer to (Zi et al., 2009).

The nominal model of CRCM including servomechanism dynamics is described by the following formulation

$$\begin{cases} \mathcal{K}\_{\rm l} \mathcal{U}\_{\rm C} = J\_{mi} \frac{d^2 \theta\_{mi}}{dt^2} + \mathcal{K}\_{oo} \frac{d \theta\_{mi}}{dt} \\\\ \boldsymbol{\tau}\_{hi} = n\_i (\mathcal{K}\_{\rm l} \mathcal{U}\_{\rm C} - \mathcal{K}\_{oo} \frac{d \theta\_{mi}}{dt} - \mathcal{J}\_{mi} \frac{d^2 \theta\_{mi}}{dt^2}) \\\\ D(q)\ddot{q} + \mathcal{C}(q\_{\prime} \dot{q}) + \boldsymbol{\tau}\_{d} = \boldsymbol{\tau}\_{hi} \end{cases} \tag{12}$$

It is also well known that there is a dual relation between externally applied wrench on the end-effector and the cable tensions required to keep the system in equilibrium. The above dynamic model is valid only for 0 *Ti* , i.e., the cables are in tension. Clearly, the equation (12) is a non-homogeneous linear quaternary equations. The solution of the equations will be multiple. For the sake of this, the suitable solution is found through MATLAB software based on the pseudo-inverse method.

#### **4. Adaptive fuzzy control**

64 Fuzzy Logic – Controls, Concepts, Theories and Applications

1234

 11 22 33 44 *T Q rrrr i*

Given the equations of motion shown above, using the assumptions along with various substitutions and algebraic manipulations of the CRCM derived, the dynamic equation of

() (,) *Dqq Cqq d*

where 4 4 *D q*( ) is the inertia matrix which is symmetric positive define, 4 *Cqq* (,) is a

matrix *D q*( ) and the 4x1 vector *Cqq* (,) will be referred to as *D* and *C* respectively. The

The dynamic model is presented in two parts: one is directed to the structural model (CRCM) above and the other is related to the actuator dynamics (servo mechanism). We have already developed mechanics equations of the drive transmission system (Zi et al., 2009), and briefly outline here. The extendable actuator of each subsystem of the CRCM system is comprised of an alternating current (AC) servomotor & drive unit, belt pulley drive mechanism, two-level cycloid-gear speed reducer, and windlass. To simplify matters, here without regard to the belt pulley drive mechanism, the next step servomechanism model is developed. Without going into details, the servomechanism dynamic model is

> 2 2

*d d KU J K dt dt*

2 2

*d d KU J K dt dt*

*bi i U C ni*

*U C mi*

2 ( )

*d d nKU K J dt dt*

where *bi* (for 1,2,3,4 *i* ) is the torque of the windlass; *ni* is the gear ratio; *UC* is control voltage ; *UC* and *Kw* are positive constant, respectively; *mi J* denotes the moment of inertia of the motor; *ni J* is the equivalent moment of inertia including motor, speed reducer,

*mi* is the rotor angular position.

*i U C ni*

*n dd T KU K J r r dt dt*

*mi mi*

*bi i mi mi*

*mi mi*

2 ( )

which is bounded, and 4 is the input torque vector with 1234

*d*

2

2

*mi mi*

The windlass torques, 1

the CRCM can be expressed as

of the windlass.

 , 2 , 3 and 4

briefly described by the following formulation,

flywheel and windlass, and

The driving force of cable *Ti* (for 1,2,3,4 *i* ) can be expressed as

*U C mi*

*i i*

nonlinear Coriolis/centripetal/gravity vector terms, 4

details of these expressions will be omitted for the sake of brevity.

*T*

*i q llll* (7)

, are the control inputs 1*r* , 2*r* , 3*r* and 4*r* , is the radius

(8)

(9)

represents the disturbance

 *T*

(10)

(11)

. The 4x4

In general, a fuzzy logic system consists of four parts: the knowledge base, the fuzzifier, the fuzzy inference engine, and the defuzzifier. There are many different choices for the design of fuzzy system if the mapping is static. In this study, we consider a MIMO fuzzy logic system (Liu, 2008; Yoo & Ham, 2000). Supposing the fuzzy logic system performs a mapping from fuzzy sets in *n U R* to fuzzy sets in *m V R* , where 1 *n UU U R n* , *U R i* (for *i n* 1,2, , ), 1 *m VV V R m* , *V R j* (for *j m* 1,2, , ). For a MIMO system, the fuzzy knowledge base consists of a collection of fuzzy IF–THEN rules in the following form

$$\mathcal{R}^{(l)}: \text{IF } \mathbf{x}\_1 \text{ is } F\_1^l \text{ and...and } \mathbf{x}\_n \text{ is } F\_n^l$$

$$\text{THEN } y\_1 \text{ is } \mathbb{C}\_1^{(l)} \text{ and...and } y\_m \text{ is } \mathbb{C}\_m^{(l)} \tag{13}$$

where 1 *T n xx x U* , and 1 *T yy y* , *m V* are the input and output vectors of the fuzzy system, respectively, *l Fi* and *l Cj* (for 1,2, , *l M* ) are linguistic variables, and *M* is the number of fuzzy rules. Based on the fuzzy inference engine working on fuzzy rules, the defuzzifier maps fuzzy sets in *U* to a crisp point in *V* .

The output of the fuzzy control system with singleton fuzzifier, product inference engine, center average defuzzifier is in the following form (Yoo & Ham, 2000)

$$\begin{aligned} \boldsymbol{y}\_{j} &= \frac{\sum\_{l=1}^{M} \overline{\boldsymbol{y}}\_{j}^{l} \left( \prod\_{i=1}^{n} \mu\_{\boldsymbol{F}\_{i}^{l}} (\boldsymbol{x}\_{i}) \right)}{\sum\_{l=1}^{M} \left( \prod\_{i=1}^{n} \mu\_{\boldsymbol{F}\_{i}^{l}} (\boldsymbol{x}\_{i}) \right)} \end{aligned} \tag{14}$$

Fuzzy Control System Design and Analysis for Completely Restrained Cable-Driven Manipulators 67

1

*i ii*

4

*r d i ii i*

1

 (22)

4

*r i ii i*

1

(24)

*D q q C q q Fi D q q K s* (25)

1

*T T T T*

4

*Fqq*, can be written as

(26)

 (23)

*n T T*

*i*

where \* *iii* , *i* (for *i*=1, 2, 3, 4) is the parameter vector, \* *i* is the ideal parameter,

*T T*

is nonlinear function. Since the disturbance is related to the position and velocity

*T T*

It is considered that the fuzzy logic compensation control is to approach just for the external

 ˆ , , *T F i qq qq*

ˆ () (,) ,

1

4

ˆ , , ˆ , ˆ , ,

*F qq q q Fqq*

2 2 3 3

ˆ , ,

*F qq q q*

, ˆ ,

*F qq q q q q F qq*

disturbance, and the fuzzy logic system *Fqq* , for the CRCM system is defined as

To prove the negative definition of ( ) *V t* , the time derivative of (21) is given as follows

can be written in the form of *Fqq* (,) . Hence, Eq. (22) can be rewritten as

*V t s Dq C F*

*q q*, is fuzzy basis function (for *i*=1,2,3,4).

From the previous results, the control law is given as follows

where *K diag K D i* ( ) , 0 *Ki* (for *i* 1,2,3,4 ), and ˆ

*V t s Dq C*

*r sqq* (20)

(21)

Then Eq. (18) can be rewritten as

where *d*

signal, *d*

where

Let us consider the Lyapunov function candidate

and *i* is a positive definite diagonal matrix.

1 2

*V t s Ds*

where *l j y R* (for *j m* 1,2, , ) is a criop value at which the membership function *l C* for output fuzzy set reaches its maximum,and *l* ( ) *i F i x* is the membership function of the linguistic variable *i x* , defined as

$$\mu\_{F\_i^l}(\mathbf{x}\_i) = \exp\left[-\frac{(\mathbf{x}\_i - \overline{\mathbf{x}\_i^l})^2}{\sigma^2}\right] \tag{15}$$

where *l i x* and are respectively, the mean and the deviation of the Gaussian membership function. The fuzzy control system inputs are composed of the five linguistic terms: NB (Negative Big), NO (Negative Medium), SS (Zero), PO (Positive Medium), and PB (Positive Big).

As the fixed nonlinear mapping in the hidden layer, ( ) *x* is defined as

$$\sigma\_l(\mathbf{x}) = \frac{\prod\_{i=1}^n \mu\_{\mathbf{f}\_i^l}(\mathbf{x}\_i)}{\sum\_{l=1}^M \left(\prod\_{i=1}^n \mu\_{\mathbf{f}\_i^l}(\mathbf{x}\_i)\right)} \tag{16}$$

In order to maintain the consistent performance of the fuzzy control system in situations where there is uncertainty variation, the fuzzy control system should be adaptive. Therefore, (14) can be rewritten as

$$y\_j = \Theta\_j^T \varepsilon(\mathbf{x}) \tag{17}$$

where 1 ( ) ( ), , ( ) *T M x x xR M* is the fuzzy antecedent function vector, and 1 , , *T M M j j j yy R* is the center of the fuzzy subset *Cj* .

In the following analysis, it will be assumed that the dynamic model of the robot manipulator to be controlled is well known, and all the state variables can be measurable. The control system requirements for the CRCM are similar to those of almost all manipulators. In order to follow the desired continuously differentiable and uniformly bounded trajectory *d q* and keep the tracking error ( ) *d et q q* approach zero, a sliding surface, *s*, is defined in the stable state space (Liu, 2008). The most common sliding surface is chosen as follows

$$s = \dot{e} + \mathcal{A}e\tag{18}$$

where is a positive definite design parameter matrix.

Now introduce the variable *r q* , and define

$$
\dot{q}\_r(t) = \dot{q}\_d(t) - \lambda e(t) \tag{19}
$$

Then Eq. (18) can be rewritten as

66 Fuzzy Logic – Controls, Concepts, Theories and Applications

*j y R* (for *j m* 1,2, , ) is a criop value at which the membership function *l C*

*l* ( ) exp

*i*

*x*

*F i*

( )

*x*

*i F i*

*x x*

function. The fuzzy control system inputs are composed of the five linguistic terms: NB (Negative Big), NO (Negative Medium), SS (Zero), PO (Positive Medium), and PB (Positive

1

*i l M n*

*n*

1 1

In order to maintain the consistent performance of the fuzzy control system in situations where there is uncertainty variation, the fuzzy control system should be adaptive. Therefore,

*l i*

*j j y*

In the following analysis, it will be assumed that the dynamic model of the robot manipulator to be controlled is well known, and all the state variables can be measurable. The control system requirements for the CRCM are similar to those of almost all manipulators. In order to follow the desired continuously differentiable and uniformly bounded trajectory *d q* and keep the tracking error ( ) *d et q q* approach zero, a sliding surface, *s*, is defined in the stable state space (Liu, 2008). The most common sliding surface

> *se e*

*q t q t et r d*

2 2 ( )

*l i i*

are respectively, the mean and the deviation of the Gaussian membership

( )

*x*

*l i*

( ) *T*

*x x xR M* is the fuzzy antecedent function vector, and

*F i*

( )

*x*

*l i*

*F i*

( ) *x* is defined as

output fuzzy set reaches its maximum,and *l* ( )

As the fixed nonlinear mapping in the hidden layer,

linguistic variable *i x* , defined as

*x* is the membership function of the

(15)

(16)

*x* (17)

(18)

(19)

for

where *l*

where *l*

Big).

*i x* and

(14) can be rewritten as

1 , ,

is chosen as follows

where

where 1 ( ) ( ), , ( ) *T M*

*T M M*

Now introduce the variable *r q* , and define

*j j j yy R* is the center of the fuzzy subset *Cj* .

is a positive definite design parameter matrix.

$$s = \dot{q} - \dot{q}\_r \tag{20}$$

Let us consider the Lyapunov function candidate

$$V(t) = \frac{1}{2} \left( \mathbf{s}^T D \mathbf{s} + \sum\_{i=1}^n \tilde{\Theta}\_i^T \Gamma\_i \tilde{\Theta}\_i \right) \tag{21}$$

where \* *iii* , *i* (for *i*=1, 2, 3, 4) is the parameter vector, \* *i* is the ideal parameter, and *i* is a positive definite diagonal matrix.

To prove the negative definition of ( ) *V t* , the time derivative of (21) is given as follows

$$\dot{V}\left(t\right) = -\mathbf{s}^{T}\left(D\ddot{\boldsymbol{q}}\_{r} + \mathbf{C} + \boldsymbol{\tau}\_{d} - \boldsymbol{\tau}\right) + \sum\_{i=1}^{4} \tilde{\Theta}\_{i}^{T} \Gamma\_{i} \dot{\tilde{\Theta}}\_{i} \tag{22}$$

where *d* is nonlinear function. Since the disturbance is related to the position and velocity signal, *d* can be written in the form of *Fqq* (,) . Hence, Eq. (22) can be rewritten as

$$\dot{V}\left(t\right) = -\mathbf{s}^{T}\left(D\ddot{\boldsymbol{\eta}}\_{r} + \mathbf{C} + F - \tau\right) + \sum\_{i=1}^{4} \tilde{\Theta}\_{i}^{T} \Gamma\_{i} \dot{\tilde{\Theta}}\_{i} \tag{23}$$

It is considered that the fuzzy logic compensation control is to approach just for the external disturbance, and the fuzzy logic system *Fqq* , for the CRCM system is defined as

$$\hat{F}\left(q,\dot{q}\middle|\tilde{\Theta}\right) = \Theta\_i^T \varepsilon\left(q,\dot{q}\right) \tag{24}$$

where *q q*, is fuzzy basis function (for *i*=1,2,3,4).

From the previous results, the control law is given as follows

$$\tau = D(q)\ddot{q} + \mathcal{C}(q, \dot{q})\hat{F}\_i\left(q, \dot{q} \middle| \tilde{\Theta}\right) - K\_D s \tag{25}$$

where *K diag K D i* ( ) , 0 *Ki* (for *i* 1,2,3,4 ), and ˆ *Fqq*, can be written as

$$
\hat{F}(\boldsymbol{q},\dot{\boldsymbol{q}}|\boldsymbol{\Theta}) = \begin{bmatrix}
\hat{F}\_1(\boldsymbol{q},\dot{\boldsymbol{q}}|\boldsymbol{\Theta}) \\
\hat{F}\_2(\boldsymbol{q},\dot{\boldsymbol{q}}|\boldsymbol{\Theta}) \\
\hat{F}\_3(\boldsymbol{q},\dot{\boldsymbol{q}}|\boldsymbol{\Theta}) \\
\hat{F}\_4(\boldsymbol{q},\dot{\boldsymbol{q}}|\boldsymbol{\Theta})
\end{bmatrix} = \begin{bmatrix}
\Theta\_1^T \mathcal{E}(\boldsymbol{q},\dot{\boldsymbol{q}}) \\
\Theta\_2^T \mathcal{E}(\boldsymbol{q},\dot{\boldsymbol{q}}) \\
\Theta\_3^T \mathcal{E}(\boldsymbol{q},\dot{\boldsymbol{q}}) \\
\Theta\_4^T \mathcal{E}(\boldsymbol{q},\dot{\boldsymbol{q}})
\end{bmatrix} \tag{26}
$$

Fuzzy Control System Design and Analysis for Completely Restrained Cable-Driven Manipulators 69

*x t y t*

Fig. 5 displays the workspace of the end-effector of the CRCM. The spatial helical following trajectory and the spatial circle following trajectory of the end-effector are shown in Fig. 6

Fig. 7, Fig. 8, Fig. 10 and Fig. 11 show the changes in length and the tension of the cables in the two different trajectories tracking, respectively. As can be seen in Figs. 6-11, the above

0.2

0.2

Y coordinates (m) X coordinates (m)

X coordinates (m) Y coordinates (m)

0.4

0.6

0.8 1 0.2 0.4 0.6 0.8 1 1.2

0.4

0.6

0.8

1

0.2 0.05

*z t*

and Fig. 9, respectively.

Fig. 5. Workspace of the end-effector.

Z coordinates (m) Z coordinates (m)

Fig. 6. Following trajectory of the end-effector.

0.5 0.3cos(0.1 ) 0.5 0.3sin(0.1 )

(32)

The fuzzy approximation error is defined as

$$w = F(q, \dot{q}) - \hat{F}(q, \dot{q} \vert \tilde{\Theta}) \tag{27}$$

Substituting Eqs. (25)-(27) into Eq. (23), the following equation can be derived

$$\begin{split} \dot{V}\left(t\right) &= -\boldsymbol{s}^{T} \left(D\ddot{\boldsymbol{q}}\_{r} + \boldsymbol{C} + \boldsymbol{F} - \boldsymbol{\tau}\right) + \sum\_{i=1}^{4} \tilde{\Theta}\_{i}^{T} \Gamma\_{i} \dot{\tilde{\Theta}}\_{i} \\ &= -\boldsymbol{s}^{T} \left(\Theta\_{i}^{T} \boldsymbol{\varepsilon}\left(\boldsymbol{q}, \dot{\boldsymbol{q}}\right) + \boldsymbol{w} + \boldsymbol{K}\_{D} \boldsymbol{\dot{s}}\right) + \sum\_{i=1}^{4} \tilde{\Theta}\_{i}^{T} \Gamma\_{i} \dot{\tilde{\Theta}}\_{i} \\ &= -\boldsymbol{s}^{T} \boldsymbol{K}\_{D} \boldsymbol{s} - \boldsymbol{s}^{T} \boldsymbol{w} + \sum\_{i=1}^{4} \left(\tilde{\Theta}\_{i}^{T} \Gamma\_{i} \dot{\tilde{\Theta}}\_{i} - \boldsymbol{s}\_{i} \Theta\_{i}^{T} \boldsymbol{\varepsilon}\left(\boldsymbol{q}, \dot{\boldsymbol{q}}\right)\right) \end{split} \tag{28}$$

Then, the adaptive law is defined as

$$\dot{\Theta}\_i = -\Gamma\_i^{-1} s\_i \varepsilon(\boldsymbol{q}\_\prime \dot{\boldsymbol{q}}) \tag{29}$$

Since the minimum approximation error, *w* , can be sufficiently small through designing the fuzzzy logic system with enough rules, and satisfying 4 1 , *T T i ii ii i s qq* =0. In addition, *KD* 0 . Consequently, we get

$$\dot{V}\{t\} - \mathbf{s}^T \mathbf{K}\_D \mathbf{s} - \mathbf{s}^T w < \mathbf{0} \tag{30}$$

Based on Lyapunov stability theory, and the result of Eq. (30), it is shown that the closedloop system is asymptotically stable, and the scheduled control object can be realized.

#### **5. Results and analysis**

In order to justify the dynamic modeling the CRCM, we performed a series of simulations. This section presents two motion cases of the end-effector for dynamic simulation. A simulation for the dynamic model of the CRCM was carried out by Matlab 7.0 software. Some parameters of the CRCM are given as follows: the height of the pillar is 2 m, Pillars *B*1~ *B*4 are distributed evenly on the vertices of a square, with the side length of 2 m, and the quality of the end-effector is 5 kg. The acceleration of gravity *g* is 9.8 2 *m s* / .

The spatial circle trajectory can be expressed as

$$\begin{cases} x = 1 + 0.3 \times \cos(0.2\pi t) \\ y = 1.5 + 0.3 \times \sin(0.2\pi t) \\ z = 1 \end{cases} \tag{31}$$

And the spatial helical trajectory is as follows

$$\begin{cases} x = 0.5 + 0.3 \cos(0.1 \pi t) \\ y = 0.5 + 0.3 \sin(0.1 \pi t) \\ z = 0.2 + 0.05t \end{cases} \tag{32}$$

Fig. 5 displays the workspace of the end-effector of the CRCM. The spatial helical following trajectory and the spatial circle following trajectory of the end-effector are shown in Fig. 6 and Fig. 9, respectively.

Fig. 5. Workspace of the end-effector.

68 Fuzzy Logic – Controls, Concepts, Theories and Applications

Substituting Eqs. (25)-(27) into Eq. (23), the following equation can be derived

*V t s Dq C F*

,

*s qq w K s*

4

*T T T i D i ii*

*T T*

1

 1 , *i iis qq*

Since the minimum approximation error, *w* , can be sufficiently small through designing the

Based on Lyapunov stability theory, and the result of Eq. (30), it is shown that the closedloop system is asymptotically stable, and the scheduled control object can be realized.

In order to justify the dynamic modeling the CRCM, we performed a series of simulations. This section presents two motion cases of the end-effector for dynamic simulation. A simulation for the dynamic model of the CRCM was carried out by Matlab 7.0 software. Some parameters of the CRCM are given as follows: the height of the pillar is 2 m, Pillars *B*1~ *B*4 are distributed evenly on the vertices of a square, with the side length of 2 m, and the

> 1 0.3 cos(0.2 ) 1.5 0.3 sin(0.2 )

*x t y t*

quality of the end-effector is 5 kg. The acceleration of gravity *g* is 9.8 2 *m s* / .

1

*z*

fuzzzy logic system with enough rules, and satisfying 4

*TT T T D i ii ii i*

*sKs sw s qq*

ˆ *w Fqq Fqq* , , (27)

,

(29)

, *T T i ii ii*

*s qq*

=0. In

(31)

1

( ) 0 *T T Vt sK s s w D* (30)

*i*

(28)

1

*i*

4

*r i ii i*

1 4

The fuzzy approximation error is defined as

Then, the adaptive law is defined as

addition, *KD* 0 . Consequently, we get

The spatial circle trajectory can be expressed as

And the spatial helical trajectory is as follows

**5. Results and analysis**

Fig. 6. Following trajectory of the end-effector.

Fig. 7, Fig. 8, Fig. 10 and Fig. 11 show the changes in length and the tension of the cables in the two different trajectories tracking, respectively. As can be seen in Figs. 6-11, the above

Fuzzy Control System Design and Analysis for Completely Restrained Cable-Driven Manipulators 71

0.8

0 2 4 6 8 10 12 14 16 18 20

Time (s)

1.3 1.4 1.5 1.6 1.7

Y coordinates (m) X coordinates (m)

1

1.2

Fig. 9. Following trajectory of the end-effector.

0.4 0.6 0.8 1 1.2 1.4 1.6

Z coordinates (m)

Fig. 10. Changes in tension of cable for the circle motion

5

10

15

20

Tension (N)

25

30

35

formulation tracks the planned trajectory relatively well. From the above simulation results, it can be concluded that the dynamic modeling is justified.

Fig. 7. Changes in length of cable for the helical motion.

Fig. 8. Changes in tension of cable for the helical motion.

Fig. 9. Following trajectory of the end-effector.

formulation tracks the planned trajectory relatively well. From the above simulation results,

0 2 4 6 8 10 12 14 16 18 20

0 2 4 6 8 10 12 14 16 18 20

Time (s)

Time (s)

it can be concluded that the dynamic modeling is justified.

Fig. 7. Changes in length of cable for the helical motion.

1

1.2

1.4

1.6

1.8

Length (m)

2

2.2

2.4

2.6

2.8

Fig. 8. Changes in tension of cable for the helical motion.

0

5

10

15

20

Tension (N)

25

30

35

40

Fig. 10. Changes in tension of cable for the circle motion

Fuzzy Control System Design and Analysis for Completely Restrained Cable-Driven Manipulators 73

0 0.5 1 1.5 2 2.5 3 3.5 4

*q* and *q* (m)

Fig. 12. Membership function of input variables.

0

0.2

0.4

Degree of membership

0.6

0.8

1

Fig. 13. Simulink model of the CRCM.

Fig. 11. Changes in length of cable for the circle motion.

In order to assess the performance of the adaptive fuzzy control system of the CRCM, simulations in spatial circle trajectory motion have been performed. The initial length configuration of the cables of the CRCM is given as *q*(0) [1.32 1.71 2.22 1.93]T, and the other consequent parameters areinitialized to zero. The nonlinearity *Fqq* (,) is estimated by using five Gaussian fuzzy sets for *q* and *q* , which is constructed, as shown in Fig 12. The disturbance vector is 15sin(20 ) 10sin(20 ) 10sin(20 ) 15sin(20 ) *T d tttt* . The design parameters of the controller are determined as 10 , 0.001 , *K I D* 250 , and *I* is a 4 4 matrix. The resulting fuzzy set must be converted to a signal that can be sent to the process as a control input. Based on S-Function, the Simulink model of the CRCM is shown in Fig 13.

Figs. 14 and 15 display the trajectory tracking of the end-effector of the CRCM, respectively. From Fig. 14, the above formulation tracks the planned trajectory relatively well. Figs. 16 and 17 illustrate the position trajectory and the position errors of the end-effector in x, y, z directions, respectively. The changes in length and the length trajectory tracking errors of the cables 1234 *llll* ,,, are shown in Fig. 18 and Fig. 19, respectively. In Figs. 16 and 18, the desired trajectory is indicated in a red solid line, and the actual output is in a blue dash line, and from Fig. 16 and Fig. 18, it can be seen that the actual and desired trajectories almost overlap each other.

Fig. 12. Membership function of input variables.

In order to assess the performance of the adaptive fuzzy control system of the CRCM, simulations in spatial circle trajectory motion have been performed. The initial length configuration of the cables of the CRCM is given as *q*(0) [1.32 1.71 2.22 1.93]T, and the other consequent parameters areinitialized to zero. The nonlinearity *Fqq* (,) is estimated by using five Gaussian fuzzy sets for *q* and *q* , which is constructed, as shown in Fig 12. The

0 2 4 6 8 10 12 14 16 18 20

Time (s)

4 4 matrix. The resulting fuzzy set must be converted to a signal that can be sent to the process as a control input. Based on S-Function, the Simulink model of the CRCM is shown

Figs. 14 and 15 display the trajectory tracking of the end-effector of the CRCM, respectively. From Fig. 14, the above formulation tracks the planned trajectory relatively well. Figs. 16 and 17 illustrate the position trajectory and the position errors of the end-effector in x, y, z directions, respectively. The changes in length and the length trajectory tracking errors of

desired trajectory is indicated in a red solid line, and the actual output is in a blue dash line, and from Fig. 16 and Fig. 18, it can be seen that the actual and desired trajectories almost

*llll* ,,, are shown in Fig. 18 and Fig. 19, respectively. In Figs. 16 and 18, the

*d tttt* . The design

, 0.001 , *K I D* 250 , and *I* is a

disturbance vector is 15sin(20 ) 10sin(20 ) 10sin(20 ) 15sin(20 ) *T*

Fig. 11. Changes in length of cable for the circle motion.

1.4

1.6

1.8

Length (m)

2

2.2

2.4

2.6

parameters of the controller are determined as 10

in Fig 13.

the cables 1234

overlap each other.

Fig. 13. Simulink model of the CRCM.

Fuzzy Control System Design and Analysis for Completely Restrained Cable-Driven Manipulators 75

0 2 4 6 8 10

0 2 4 6 8 10

0 2 4 6 8 10

0 2 4 6 8 10

0 2 4 6 8 10

0 2 4 6 8 10

Time (s)

Time (s)

Fig. 16. Position trajectory of the end-effector in x, y and z directions

0.5

1

1

0.95

0.01

0.01

0

0

0.01

0

1.05

1.5

Position trajectory (m)

Position errors (m)

2

1

1.5

Fig. 17. Position errors of the end-effector in x, y and z directions.

robustness.

Fig. 20 displays the disturbance *d* and its compensator, and the control input torques of the windlass are shown in Fig. 21. From the simulation results, it may be concluded that the adaptive fuzzy control strategy can achieve a favourable control performance and has high

Fig. 14. Following trajectory of the end-effector.

Fig. 15. Following trajectory of the end-effector

0.8

1.2

Y coordinates (m) X coordinates (m)

X coordinates (m)

0.7 0.8 0.9 1 1.1 1.2 1.3 1.4

1.4

1.6

1

1.2

1.4

Fig. 14. Following trajectory of the end-effector.

1.8 0.9

0.95

Z coordinates (m)

1

1.05

1.1

Fig. 15. Following trajectory of the end-effector

1.1

1.2

1.3

1.4

1.5

Y coordinates (m)

1.6

1.7

1.8

1.9

Fig. 16. Position trajectory of the end-effector in x, y and z directions

Fig. 20 displays the disturbance *d* and its compensator, and the control input torques of the windlass are shown in Fig. 21. From the simulation results, it may be concluded that the adaptive fuzzy control strategy can achieve a favourable control performance and has high robustness.

Fig. 17. Position errors of the end-effector in x, y and z directions.

Fuzzy Control System Design and Analysis for Completely Restrained Cable-Driven Manipulators 77

0 1 2 3 4 5 6 7 8 9 10

0 1 2 3 4 5 6 7 8 9 10

0 1 2 3 4 5 6 7 8 9 10

0 1 2 3 4 5 6 7 8 9 10

0 1 2 3 4 5 6 7 8 9 10

0 1 2 3 4 5 6 7 8 9 10

0 1 2 3 4 5 6 7 8 9 10

Time (s)

Fig. 20. The disturbance *d* and its compensator.

Disturbance and compe

nsator

(Nm)

Control input torques

(Nm)

Fig. 21. The control input torques of the windlass 1234 ,,,

Cable parallel manipulators are a class of robotic mechanisms whose simplicity of design, light weight and ability to support large loads make them useful in many industrial and military settings. This chapter presented in detail a 3-DOF, 4-cable CRCM for its adaptive

Time (s)

0 1 2 3 4 5 6 7 8 9 10

**6. Conclusion**

Fig. 18. Length tracking of the cables 1234 *llll* ,,,

Fig. 19. Length tracking errors of the cables 1234 *llll* ,,,

Fig. 20. The disturbance *d* and its compensator.

Fig. 21. The control input torques of the windlass 1234 ,,,

## **6. Conclusion**

76 Fuzzy Logic – Controls, Concepts, Theories and Applications

0 1 2 3 4 5 6 7 8 9 10

0 1 2 3 4 5 6 7 8 9 10

0 1 2 3 4 5 6 7 8 9 10

0 1 2 3 4 5 6 7 8 9 10

Time (s)

0 1 2 3 4 5 6 7 8 9 10

0 1 2 3 4 5 6 7 8 9 10

0 1 2 3 4 5 6 7 8 9 10

0 1 2 3 4 5 6 7 8 9 10

Time (s)

*llll* ,,,

*llll* ,,,

Fig. 18. Length tracking of the cables 1234

Length tracking errors

(m)

1 1.5 2

1 1.5 2

Length tracking (m)

1.5 2 2.5

1.5 2 2.5

Fig. 19. Length tracking errors of the cables 1234

Cable parallel manipulators are a class of robotic mechanisms whose simplicity of design, light weight and ability to support large loads make them useful in many industrial and military settings. This chapter presented in detail a 3-DOF, 4-cable CRCM for its adaptive

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