Lifting and stabilizing of two-wheeled wheelchair system using interval type-2 fuzzy logic control based spiral dynamic algorithm

ABSTRACT

: Change of error for the orientation angle of Link2

INTRODUCTION
Based on their functions and capabilities, wheelchairs have practically been used by the elderly and disabled people alike as transportation and mobile tools. Often, limited flexibility has been seen within a standard four-wheeled wheelchair due to its bulky exterior and limited transformational capability. Such limitation is overcomed by the development of a transformational wheelchair (four-wheeled -> twowheeled). The later system which mimicked the concept of an inverted pendulum has enabled greater satisfactions to users' mobility requirements; yet, it has also uncovered new challenges in stability, often due to its complexity and non-linearity [1]- [13].
In seek of resolution to the fore-mentioned challenge, the current study focuses on the control strategy to lift and stabilize a transformational two-wheeled wheelchair, with the transformation mechanism of four-wheeled to two-wheeled wheelchair. Herewith, two links mechanism have been incorporated within the studied two-wheeled wheelchair with additional dynamics and disturbances as same as a double link inverted pendulum scenario. Thus, this signifies the need for a powerful control strategy towards maintaining system's stability via reduced vibrations, thereby ensuring user's well-placed security. A number of controllers have been involved in fulfilling such requirements [14]- [26]. Compared to other controllers, linear quadratic regulator (LQR) [27] as well as proportional integral derivative (PID) [28] were the most commonly adopted controllers in balancing two-wheeled wheelchairs with a transformation mechanism. Whereas, some have adopted the applications of intelligent control algorithms, like fuzzy logic and neural network, in achieving system's stability-fuzzy logic type-1 has often been the basic fuzzy logic system found within other published research [29], [30]. Fuzzy logic is well-suited for approximating the nonlinearities [31]. Studies have shown that fuzzy strategies are an effective and relatively simple tool to use for nonlinear problems [32], [33]. Apart from the nonlinearity of the systems, the significant other consideration is that it imposes difficulties to the solution of control problem. An excellent controller for nonlinear systems and complex control problems is the fuzzy logic control (FLC) that has been considered as an ideal controller for mitigating complexity in the control system. Interval type-2 fuzzy logic control (IT2FLC) can take into consideration the uncertainities in a two-wheeled wheelchair system, such as whether it is currently moving or not, and respond effectively on nonlinear complex systems such as type-1 FLC wheelchair system [34]- [37]. Therefore, intelligent control approaches following nature-inspired algorithms (e.g., optimized fuzzy logic), despite being minimally explored within previous literatures, have hereby been highlighted in catering the inherent nonlinearity of a transformational two-wheeled system. Following the discussions above, an integration of IT2FLC with spiral dynamics algorithm (SDA) optimization, has been adopted within this study as its intelligent control approach. As introduced by Tamura and Yasuda [38], SDA optimization follows a logarithmic spiral algorithm which possesses a converging spiral motion towards the center. It hereby fulfills the diversification and search strategies [39]. Not to mention, such optimization method has been applied towards optimizing the scaling factors including output and input gain of a stair climbing wheelchair [40], which proven its applicability within a fuzzy logic control algorithm (fuzzy logic type-1), as being currently investigated.
The main focus of this research is placed on developing an IT2FLC-based control system, through the application of SDA optimization, in reducing vibrations and handling disturbances, further achieving stability during the transformation process of a transformational two-wheeled wheelchair from its standard four-wheeled configuration. To evaluate the system's performance, the proposed model has been investigated in the Simwise 4D environment for the purpose of visualization, and then integrated with MATLAB/SIMULINK. Herewith, the remainder of the paper is arranged as follows: The transformational two-wheeled wheelchair system modeling methodology is presented in section 2. Section 3 contains the results along with a brief discussion, while the conclusion is represented in section 4.

System model and parameters
Based on a double links IP system, wheelchair with transformational mechanism between four and two wheels was designed using SimWise 4D software. Similar techniques were employed on the previous research that utilized type-1 fuzzy logic controller (T1FLC) as its movement controller [18]. However, such system was being improved within the current research, by integrating IT2FLC as its controller. Such improvements were further verified by making performance comparison between both the former and current controllers.
This model's design had been inspired by previous works [18]- [22]. Basically, three independent actuators were involved, which were located on each of the two wheels in generating torque-left (τL) and right (τR); with the third one being situated in between Link1 and Link2 as the second torque (τ2). The wheelchair's design was largely consisted of two wheels driven by the left and right motors, individually. Whereas, another two caster wheels (front wheels), classified as the second group wheels, will be lifted during the transformation process. In particular, the wheelchair's mechanism actually consists of two links, the first of which was the rod that attached the axle to the left and right motors; and the second of which was the seat of the wheelchair that was connected to the first link. Herewith, tilt angle for Link1 and Link2 were called δ1 (θ1) and δ2 (θ2), respectively. The schematic diagrams for the system in normal position (Figure 1 (a)), as well as when the wheelchair is transformed (Figure 1 (b)), are hereby illustrated in Figure 1. Figure 1 also figured out the turbulence for 360-degree rotation for the wheelchair in real simulation with the human load [41].
The system used SimWise 4D for designing the model as a software module, as it is a powerful tool in which 3D multibody dynamic motion simulation was performed by analyzing and optimizing of 3D finite element, yet manageable simulative domain (Motion+FEA=4D) [41]. The integration of MATLAB/SIMULINK2015b considered as the second software to develop the system's controller towards controlling the wheelchair's performance, it worked as representative of the mechanical system in SimWise 4D where the whole system, inclusive of hydraulics, electronics, and controls, could be tested. As such, the full design of the four to two-wheeled wheelchair model with load had been previously shown in [41], which covered basic dimensions and specifications of the system.

Interval type-2 fuzzy logic controller
There are five different components that are utilized in type-2 FLC including rule base, fuzzifier, fuzzy inference engine, type-reducer, and defuzzifier. The inputs as well as outputs of the system can be presented by interval the type-2 fuzzy sets, towards developing type-2 FLC within the control of a mobile robot [42], [43]. In general, the process for type-2 FLC starts off with the crisp inputs from input sensors being fuzzified into type-2 input fuzzy sets. This first step only considers singleton fuzzification, which would then activate the fuzzy inference engine and rule base towards producing type-2 output fuzzy sets. Following this, the type-2 output fuzzy sets will be processed by type-reducer through undertaking centroid calculation. Then they are combined with the gained type-1 fuzzy sets to further generate the type-reduced sets. Lastly, defuzzifier takes on the role to defuzzify the type-1 reduced fuzzy outputs, in attaining the crisp outputs to be fed to the actuators [44]. In view of clarification, Figure 2 depicts the block diagram of the entirely of the process undertaken for a type-2 FLC system.

Fuzzifier
A fuzzifier gives a crisp input vector with ρ inputs, into the input fuzzy sets, which can then be generalized as the type-2 fuzzy sets (Ãχ).
In view of singleton fuzzification's ability in enabling swift computing, and its ability to be applied on robots in real-time application, it has been heavily recommended for the current research. One notable point of non-zero membership is that there is only a single point of non-zero membership in the input fuzzy for the Singleton fuzzification; Ãχ (type-2 singleton) with a coefficient, μ if [44] μÃχ ( ) = 1/1 for = ' (2) and,

Rule base
By considering a type-2 FLC robot have ρ inputs, and c outputs; The ith rule in the multiple-input-multiple-output (MIMO) FLC can be written as: R MIMO : IF 1 is Ṽ 1 and … … χ is Ṽ Here, the number of rules in the rule base is denoted by M [38]. Type-2 FLC has a rule base that is very similar to type-1 FLC, as shown by [45]. Therefore, by satisfying this condition, R i MIMO can be classified as a group of multiple-input-multiple-output (MISO) R i kMISO rules, whereas R i kMISO is a rule relating multiple ρ inputs and the kth single output with k=1, ..., c [44].

Fuzzy inference engine
Mappings for the output type-2 sets are obtained from the type-2 input by combining the fuzzy inference engine rules. The different antecedents are connected inside the fuzzy inference engine by using the Meet operation. The next step is to execute a combination operation known as Join. In more detail, the input and output membership grades are combined through the use of the extended sup-star composition. The MISO fuzzy rule base comprised of M rule contains ρ inputs for each rule, 1 ϵ X 1 , . . . . . , ϵ X ρ (8) and one output, can be written as, According to [39], R i k MISO is described by the membership function, and, can be written as, A type-2 fuzzy input set, Aχ comprises of a single element x' in the Singleton fuzzification. Each of μ̃ ( ) is non-zero solely at a single point, which is = , . The meet under t-norm is employed in the interval type-2 FLC. The outcomes of the input along with the antecedents operations in an interval type-1 set are represented by, where, and In the firing set, the input and the operations are included that can be expressed as [38],

Type reduction
In the type-2 FLC system, two stages of type reduction are applied. The first stage would be to calculate centroid of the rule consequents. For any output of k (t=1, ..., T, where T=number of output fuzzy sets), Type-2 interval consequent set of ith rule, Ẽ i k will be one of the outputs of type-2 interval fuzzy sets Ẽ t k.
Calculation of the consequent centroid, while not being part of the control cycle, would be done once prior to the robot's primary movements. In this case, the two end points, ylk and yrk, would need to be computed for any output k in the main equation in order to obtain the type-reduced sets. Following this, the type-reduced sets shall be defuzzified, in delivering crisp outputs to the actuators.

Defuzzification
As previously described in [44], the utmost left and right points, ylk and yrk will be determined following the type reduction step. Herewith, the interval set based upon average between both ylk and yrk is defuzzified, which then generates the crisp output for each output k. In (18) describes the crisp output defuzzification as: In the proposed system, there are three inputs and two outputs that are observed. These include the torque for Link1 at the left and right wheels being the first and second input, the torque for Link2 being the third input; whereas, the angle of body orientation for Link1 (left and right wheels) as a whole, and Link2 being the two outputs, respectively. The first subsystem includes controller inputs for the orientation angle of Link1 which are error (E1) and change of error (ΔE1). Similarly, the second subsystem includes the controller inputs for the orientation angle of Link2, which are error (E2) and change of error (E2).
In general, there are 5 separate levels that have been included in the membership function including negative big (NB), positive big (PB), positive small (PS), negative small (NS) and zero (Z). By following the IF-THEN rule, then it generates 25 rules (5x5) altogether. Table 1 illustrates these 25 rules by using the inputs E and ΔE in IT2FLC. Thus, Figure 3 describes the entirety of integration for SimWise 4D, alongside the controller in MATLAB/SIMULINK.  Figure 3. Integration between SimWise 4D and MATLAB/SIMULINK

Spiral dynamic algorithm
As understood from Tamura and Yasuda, the SDA approach is based on the metaheuristic optimization, which draws inspiration from natural spiral patterns including galaxy, hurricanes and tornado [38]. As a basic criterion, SDA optimization would establish a balance in combining both the exploration and exploitation strategies. Basically, the search agents for this spiral-inspired algorithm would move from the outer part area of a spiral, slowly towards its centre in a twirling manner, following the gradual increase in the number of iterations. Also known as the intensification phase, such occurrence is taken place within the spiral, with the centre as its destination. Thus, the equation of mathematical modeling for SDA is represented by, Here, θ is defined as the angle of rotation, which rotates with a variation from 0 to 2π. The center of the spiral is denoted by χ*, whereas the number of iterations is denoted by k. In is a matrix. The radius of the spiral is denoted by r, ranging from 0 to 1. The number of points and the maximum point is denoted by i and m, respectively [39]. The implementation flowchart of SDA has been illustrated in Figure 4 [38].
In the proposed work, for the spiral dynamic trajectory; radius, r=0.95 and Ө=π/4 were employed since they provide the optimum performance [38]. The number of search agents was applied as 30, while the number of iterations employed in this work was measured at 30. This optimization was sought to minimise the error in the angular position while mitigating external disturbances. Most importantly, the total of the 14 inputs and output gains as a whole, as well as δ and σ, were applied to determine the size of membership function within IT2FLC.

Spiral dynamic algorithm optimization
The research objectives have hereby been achieved, following the completed results obtained from the trial and error method, as demonstrated in the previous paper [41]. All of the values obtained were then used for the purpose of optimization towards improving the controller. The SDA approach has been selected for optimization within the current study as it has been tested to be one of the best optimization methods for this type of system. Overall, the Simulink block diagram with SDA for the system's optimization is shown in Figure 5. Figure 5 also describes throughout the optimization stage when SimWise 4D was chosen as the replication of the real application, all the parameters involved were taken into consideration in order to design a 4-to-2 wheelchair. To gain the best 14 parameters, SDA MATLAB/SIMULINK and visual representation were conducted at the same time during optimization process. The specifications include length, weight, mass, gravity, friction, angles, distance, velocity, joints, motors and uncertainties have been considered in this work. K1 until K14 were hereby optimized to achieve a stable performance. Noted that K1 until K12 were the input and output gains; whereas, K13 and K14 were known as the corresponding delta and sigma, which were controller used to control the system's parameters. Following optimization being implemented within the investigated system, the data was recorded, similar to that of the trial and error method. The vibrations were reduced by the IT2FLC and remain stable as the system was able to achieve zero-degree tilt angle in the upright position.

Comparison with other controllers
In order to demonstrate the excellence of IT2FLC-SDA as a controller during the transformational process of a two-wheeled wheelchair, SDA optimization results were then verified and compared against other controllers-FLC Type-1 by Chotikunnan [29], and FLC Type-1 and PID by Ghani [16], [17]. Specifically, three sorts of external disturbances were employed to the system during the lifting and stabilizing tasks, towards assessing the robustness of the controllers; which consisted of positive, negative and positive-negative disturbances. The results, alongside condition without any disturbance, is presented as:

Without disturbances signals
Under the absence of external disturbance, the obtained data for the system implemented with IT2FLC-SDA, and other tested controllers have been presented in Table 2. As seen, IT2FLC-SDA has prevailed to be the second best option when it comes to traveling distance following the simulation, with a recorded travelling distance of 0.6 meters which bested that of FLC type-1 and PID by Ghani [16], [17]; yet, felt short behind FKC Type-1 by Chotikunnan [29] at a traveling distance of merely 0.36 meters. However, such shortfall has been overshadowed by the shortest settling time recorded for IT2FLC-SDA, at a mere 1.5 seconds. Despite its slightly further traveling distance, the settling time of the proposed controller has outshined that of Chotikunnan [29] as recorded at 4.6 seconds; thus, supporting IT2FLC-SDA as the best controller, among others, under the absence of external disturbance.

Positive-negative disturbance signals
As for the condition of positive-negative disturbance, external disturbances were concurrently applied from both directions as per being previously tested independently in testing the system's compatibility to uncertain situations. Unlike disturbances from a single direction, the current simulation started off with impulses injected from the 3 rd second mark, which then consecutively applied within a 10 seconds interval (13 th , 23 rd , 33 rd second marks). Accounting for both positive and negative directions, positive disturbances with a magnitude of 1260 N were applied at the 3 rd and the 23 rd second marks; whereas, negative disturbances with a magnitude of -470 N were applied at the 13 th and 33 rd second marks, as illustrated in Figure 6. Following the simulation, the input torques for Link1 and Link2 are separately presented in Figure 7 and Figure 8; while, Figure 9 and Figure 10 represent the output angles of the wheelchair for both Link1 and Link2 respectively. Comparatively, IT2FLC-SDA has yielded a massively closer wheelchair position to the initial point, following the influence of external stimulusa difference in travel distance of 9.899 meters. This has also been the case for settling time, where IT2FLC-SDA settled 3.4 seconds faster than that of Type-1 FLC; a settling time of less than 2 seconds indicating higher stability. Besides, smaller values have been recorded for both maximum and minimum torques on both positive and negative disturbances as shown in Table 3.

CONCLUSION
In this paper, IT2FLC with SDA optimization has been implemented as the control algorithm for tuning both the controller's input and output gains, as well as its control parameters within a transformational two-wheeled wheelchair system. Following this, comprehensive comparisons have been conducted in term of tilt angle and stability performance under several disturbance conditions. The results hereby indicated that IT2FLC based SDA has managed to improve both the wheelchair's settling time and traveling distance when undertaking its transformation system from four-wheeled to two-wheeled. As such, the proposed algorithm has also demonstrated its superiority in performance during the case of independent positive and negative disturbances, with 50% greater robustness in both traveling distance and settling time, as compared to the former controllers. This has further demonstrated the proposed controller's ability to withstanding greater disturbances from both directions. However, as effective as it is, future work with regards to IT2FLC-SDA can place emphasis on additional features, with the like of vertically extendable seat on the wheelchair. Since this work is the first stage of the research, SimWise 4D acts as the replicator of the real application and almost likely to be real because the software includes the variation of needed for a real situation such as type of material used, gravity, and friction. Therefore, the second stage of this work will be including the real hardware application and predictably to be in success rate in the future work.