Design optimization of a cable actuated parallel ankle rehabilitation robot: A fuzzy based multi-objective evolutionary approach

1.1 Convergence versus diversity issue with MOEA

A complete Pareto front should consist of solutions in two major regions (A&B) of the objective space as shown in the Fig. 1. The first region (A) has solutions which are obtained as a result of convergence whereas the second region of interest (B) is the extremities of Pareto front which exhibits diversity maintaining ability of an optimization process. While region (A) has equitable design solutions, the extremities (region B) provide design solutions biased towards an individual objectives. As discussed before, end users are often interested in finding trade-off solutions which are biased and favour selected objective at the expense of others. Apparently, abilities of existing EAs are often compromised in exploring extremities of the PF [18–20, 23]. Therefore, it is required to revisit the conventional sorting and selection approaches in order to devise alternatives. A new method, suggesting fuzzy based sorting and selection of competing design solutions in order to explore the extremities of PF is proposed in this paper for ankle robot design optimization. Optimization results from the proposed method are further compared with the results obtained through NSGA-II. The fuzzy based sorting and selection mechanisms are further explained in the following section.

2.1 Fuzzification of objectives

First of all, an initial population of robot design solutions is randomly initialized and their respective objective function values are computed in a similar manner as is carried out in other EAs. During fuzzification stage, these objective function values are converted into fuzzy variables using fuzzy sets. Gaussian activation functions (AF) shown in Fig. 2 are chosen to define fuzzy objectives due to their smooth transition between activation functions. Later, a decision is also made on the number of activation functions, their shapes and their standard deviations. An illustration describing fuzzification of two objective functions is provided in Fig. 2. Here both the objectives (a & b) are normalized on a scale of 0 to10 and defined using four Gaussian AFs namely, AF1, AF2, AF3 and AF4, shown with subscripts ‘a’ and ‘b’ respectively. The number of AFs to be associated with an objective function can be varied and it is left to the user to define. Once the objective function values for the initial population of solutions are available, their extreme values are used (1–3) to further calculate the parameters of AFs, namely, the minimum fuzziness positions (shown by A, B, C & D points in Fig. 3) and the standard deviation (σ). A i = 0.75 * min ( f i ) ; B i = A i + R i ( M i - 1 ) ; C i = B i + R i ( M i - 1 ) ; D i = 1.25 * max ( f i ) ;(1)σ i = R i / ( 2 M i - 1 )(2)R i = ( 1.25 * max ( f i ) - 0.75 * min ( f i ) )(3)

Here R _i stands for the range of objective functions (f _i ), standard deviation (σ _i ) is calculated in a manner that the entire range of the objective function is covered. The total number of activation functions for ith objective function is given by M _i . The universe of discourse (or the range) of the objectives is dynamically updated in each iteration and the fuzzy parameters are also altered accordingly.

It is important to note that during EA process, competing design solutions are compared in terms of their objective values and the better or dominating solutions are selected for further evolution. The concept or the definition of dominance is more qualitative than quantitative and hence we propose to use fuzzy dominance concept in place of conventional crisp non-dominance criterion [16]. While comparing two solutions, normally a solution providing ‘better’ or ‘not worse’ objectives is selected. Since ‘better’ and ‘worse’ are qualitative variables, it is not possible to decide upon the extent of these qualitative variables (how much better or worse) using crisp numbers.

The argument being stressed here is that when the extent of solution accuracy is not available (which is true in the real life applications), a qualitative comparison of the objective function values is more justifiable than the numerical comparison. Depending on the goals of optimization, namely, convergence or diversity, there can be two variants of fuzzy based sorting. When the aim of the optimization is to achieve convergence, an fuzzy sorting genetic algorithm (FSGA) can be used. Readers are encouraged to refer author’s previous work [10] wherein FSGA was explained in the pretext of ankle robot design optimization. However, when the aim of optimization is to obtain diversity in the competing solution, a biased fuzzy sorting genetic algorithm (BFSGA) can be used which is being proposed in the present research. The proposed BFSGA approach tries to find design solutions which excel in at least one objective, thereby encouraging solutions in the extremities of PF. It can be easily illustrated that combining the populations from the two approaches, it is possible to obtain a complete PF which fulfils convergence as well as divergence requirements equally. The methodology and implementation of the BFSGA are further explained below.

Following fuzzification of the objectives, it is possible to represent numerical values of the objectives by a collection of linguistic terms in some proportion. Later, with the application of fuzzy inference mechanism through a set of rule-base on these fuzzy objectives, it is possible to rank individual candidate solutions based on their objective values [25].

To begin with, AFs are assigned unique scores as shown in Table 1. Here zero activation score (AS) is given to the first AF and is augmented by unity for subsequent higher AFs. In general the activation score for mth activation function can be given by following relation. AS ( m ) = m – 1(4)

Later in the process, an overall activation score (OAS) for each candidate solution is computed based on the collective fuzzy index of all its objective functions values. In order to compute this score, inference mechanism of fuzzy systems is used which works through its rule-base. In fuzzy systems rule-base is a collection of if and then statements connecting the antecedent (input) and consequent variables (output). General structure of a rule-base is as below. If f 1 is AF i 1 and , … … … and f N is AF im then AS i is y i

Here f₁ … f _N are the objectives treated as antecedents to the fuzzy system, AF_i1 … AF _im are the AFs corresponding to the objectives, AS _i is the activation score for ith rule and numerical value (y _i ) of this activation score can also be termed as the consequent of the rule. The consequent of an individual rule can be calculated using (5) which is influenced by the objective function with the lowest activation score. y i = 1 + min j | AS ( m ij ) |(5)

Here i represent the rule index and m _ij is the activation function for objective function j in rule i. Total number of rules N _R is derived from the number of AFs and the number of antecedent variables which is given by following relation [4]. N R = ∏ j = 1 N M j(6)

Here j is the index for the objective function, N stands for the total number of objective functions and M _j is the total number of activation functions for objective function j. Thus, when two objectives are represented using four activation functions each (as shown in Fig. 2), a total of 4² i.e. 16 rules shall be formed. These rules are the combination of all possible arrangements of AFs of the two objectives. Numerical values of objectives shall find varying degree of fulfillment in all the rules and each rule shall give an output correspondingly. The OAS of a candidate solution is the weighted average of all the rule outputs as further explained below.

The weighting of objectives (8) for each rule can be found by considering the product of all the applicable activation function values (7). Here f _j is the input objective value whereas f ¯ ij (mean of AFs) and σ _ij (standard deviations of AFs) depend on the activation function m _ij and are updated during successive iterations based on the limiting values (f_min, f_max) of objectives (1–3). The constant ‘a’ in the relation (7) is given by ( 1 σ 2 π ). AF ij ( f j , f ¯ ij , σ ij ) = ae - ( f j - f ¯ ij ) 2 2 σ ij(7)w i = ∏ j = 1 N AF ij(8)

The final crisp output of the fuzzy inference or the OAS of a candidate solution is the weighted average of all the individual rule consequents (y _i ) for given set of objective values. OAS = ∑ i = 1 N R ( w i y i ) ∑ i = 1 N R w i(9)

The fuzzy dominant front number Y^* for a solution can then be obtained from this crisp output using (11), where floor (.) is used to represent the function which returns the integer which is less than or equal to the argument. Y * = floor ( OAS )(10)

Later, the objective space is divided into fuzzy dominant fronts using the activation scores of the corresponding objectives. The total number of fuzzy dominating fronts is same as the number of activation functions used to define objectives. This further means that, for a MOP with three objectives defined by four activation functions each, there will be four fuzzy fronts. Increasing number of AFs, the number of fronts dividing the objective space can also be increased which will further enhance the discrimination capability for selection of better solutions amongst the good ones. Furthermore, since the number of fronts for a MOP is definite, the end user can determine the quality of solutions obtained after each epoch. The algorithm can be terminated as soon as the solutions belonging to the first fuzzy front are obtained.

2.4 Termination criterion

The proposed BFSGA algorithm shall terminate when either the solutions in the desired front are achieved or when the front index remains unchanged over successive iterations. It is important to note here that to carry out front index calculation for termination criterion it is required that the extreme values of objectives are known and available. Extreme objective values can be obtained either using a single objective optimization approach or by generating a random population of solutions and objectives thereafter.

2.5 Final solution from the Pareto front

As a result of the optimization, a fuzzy PF is obtained consisting of solutions which are all equally good, as they have same activation score as one. However, the end user is always interested in a singular solution which should be the best amongst the good ones. In such case, normally, the user makes a decision based on his/her experience and selects the final solution. This approach is quite subjective and requires experienced and skilled decision makers. Therefore, in order to help users in selecting a final solution, it is proposed to make use of the OAS index obtained from Equation (9).

The OAS calculated using (9) (before using floor (.) function) is a real number which is different for all the solutions in the fuzzy Pareto front. Further, the solution with minimum OAS provides the best compromised combination of all the objectives (for a minimization goal) and the same can be selected as a final solution.

3.1 Problem formulation

Following BFSGA, all the six objectives (F₁ … F₆) were first converted into fuzzy functions (Fig. 8) and were also transformed into minimization goals. The inherent mechanism of EAs can only minimize an objective and therefore it is essential to invert the objectives which are subjected to maximization. Four Gaussian functions are chosen for their activation functions and their individual universe of discourse (range) is determined through simulation. Later, while the algorithm progresses, the universe of discourse of the objectives is dynamically updated based on their limiting values ( f min k , f max k ) over successive iterations (k). Thus, even if a particular solution does not change between iterations, it may not have the same fuzzy front index in succeeding epochs.

3.3 Parameter selection

Performance of a parallel robot is very sensitive to its geometry [22], therefore eight geometrical parameters (q₁, …,  q₈) were identified and varied under defined constraints to obtain different ankle robot configurations or designs. These parameters are the polar coordinates of eight actuators connection points on both the platforms as shown in Fig. 7c. The limits on geometrical constraints given by (21) are in radians and millimeters. π 12 ≤ q 1 , q 3 ≤ π 3 and π 12 ≤ q 5 , q 7 ≤ 4 π 9 where80 ≤ q 2 , q 6 , q 8 ≤ 120 and 120 ≤ q 4 ≤ 160(21)

Here, q₁,  q₃,  q₅  &  q₇ are angle measures whereas q₂,  q₄,  q₆  &  q₈ are radial measures of the actuator connections points as shown in Fig. 7c.

Using random numbers (Knuth’s random number generator [37]) for geometric parameters, initial population of 500 design solutions is generated and various steps required for BFSGA approach (Fig. 6) and NSGA-II are followed in two different experiments. During these experiments, only the selection mechanisms used was different (fuzzy sorting for BFSGA and non-dominated sorting for NSGA-II) whereas all other essential parameters were kept same [38] such as, population size: 500; crossover probability: 0.95; real-parameter mutation probability: 0.05; distribution index for crossover: 10; distribution index for mutation: 50.

Experiments continued until design solutions in the first PF are obtained. Unexpectedly, during experiments with NSGA-II, all the 500 solutions became non-dominated and were assigned to the first PF after close to 10 iterations (in different experiments with NSGA-II). After a thorough analysis of the results, it is observed that, evolutionary algorithms, such as NSGA-II, normally fail to establish a clear discrimination between solutions while optimizing many objectives simultaneously. Consequently, all the competing solutions become non-dominated in few iterations, resulting a pseudo Pareto front. Previous studies have also revealed similar problems using non-dominance approach [39, 40]. On the other hand, following BFSGA, first PF solutions are obtained after 25 iterations (solutions normally converged to the first PF after almost 25∼30 iterations in different experiments). Out of 500 solutions, 118 solutions are placed in the first PF, whereas the remaining solutions are placed in subsequent fronts. In order to select a final design solution from 118 PF solutions, once again the fuzzy approach discussed in Section II B is used.

Overall activation scores (9), for all these 118 design solutions, are unique real numbers which can be easily used for comparison. Since the objective functions are transformed into minimization goals, a design solution with minimum OAS shall give best values for all the design objectives. Subsequent to BFSGA experiments, a design solution with minimum OAS as 1.63 is selected amongst 118 final PF solution. Design solutions obtained from both the approaches (BFSGA & NSGA-II) exhibiting minimum OAS are displayed in Table 3 for a quick comparison. Final population of 500 solutions, after BFSGA experiment are also shown in Fig. 9. Owing to the difficulty of plotting combinations of all the six objectives, three important objectives, namely condition number, norm of actuator forces & stiffness index, are plotted against each of them. A plot (Fig. 10), showing results from NSGA-II for stiffness indices and norm of actuator forces is also provided for further comparison and analysis.