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

Abstract

Robotic devices can be potentially used to assist physical therapy treatments in order to restore musculoskeletal system malfunctions owing to neurological disorders. Cable actuated parallel robots, despite their obvious benefits such as enhanced workspace, light weight, and flexibility, are not popularly used in ankle rehabilitation treatments, due to their complex mechanism and cable actuation issues. In order to address these issues, it is recommended to carry out robot design optimization. However, design synthesis of the cable actuated parallel ankle robot calls for multi-objective optimization (MOO), since there are multiple and conflicting objectives to achieve. To acquire more choices between actuator forces, overall stiffness of robot (which is crucial for a cable based ankle robot) and other vital design objectives, it is required to explore the extreme ends of the Pareto Front (PF) more carefully. Existing multi-objective evolutionary algorithms (MOEAs) normally focus on the convergence and may not provide solutions at the extremities of PF. Capitalizing on this improvement opportunity, this paper presents a fuzzy based MOEA, namely, biased fuzzy sorting genetic algorithm (BFSGA) which encourages solutions in the extreme zones of the PF. It is shown in this paper that using proposed method, diversity in the populations is supported and in the process wider trade-off choices of objectives can be obtained. During ankle robot design optimization, crisp objectives are defined as fuzzy objectives and competing solutions are provided an overall activation score (OAS). Subsequently OAS is used to assign a fuzzy dominance ranking to the design solutions. It is found that the BFSGA approach performs well in exploring the extreme zones of the Pareto front, which are normally overlooked by other MOEA such as NSGA-II due to their inherent mechanism.

Keywords

Ankle robot parallel mechanism rehabilitation design optimization fuzzy methods

1 Introduction

Robot driven physical rehabilitation has been actively researched in the past two decades to help physiotherapists provide better treatment [1 –3]. However, existing designs of rehabilitation robots are extremely heavy and rigid and may not be suitable to work with human users. In order to advance the present state of robotic physiotherapy, newer designs of robots which are cable based and driven by pneumatic muscle actuators (PMA) are being investigated to provide compact, flexible and light weight solutions [4 –6]. Apart from being flexible, such robots provide wider workspace and are very light in weight since they do not employ heavy electric motors and associated mechanical components [7, 8]. A cable based parallel ankle rehabilitation robot (Fig. 7) had been proposed by authors for the ankle joint rehabilitation subsequent to neurological disorders [4]. The parallel ankle robot configuration has two platforms connected together by a parallel arrangement of links. While one of the platforms is kept stationary, the other platform, which accommodates the patient’s foot, moves through three rotational degrees of freedom (dof) keeping the position of the ankle joint unchanged. Cables connected in series with PMA can manoeuvre the moving platform and offer necessary flexibility and compliant motion. Nevertheless, despite obvious benefits, cable based parallel ankle robot has complex mechanisms and there are certain issues pertaining to the use of cables which daunt its wider practical use. Apparently, this robot can operate as long as the cables are maintained taut which further means that a single actuator can apply only unidirectional forces. Since cables and actuators (PMA) used are flexible, maintaining high stiffness is also an important issue. The cables cannot be subjected to very high forces which may cause breakage or undesired elongation affecting the positional accuracy. Normally, cable based parallel robots are redundantly actuated which means that the robot needs ‘(n+1)’ actuators to achieve ‘n-dof’ motion of the manipulator [8]. Eventually, design of a cable based ankle robot is a difficult task especially due to stiffness requirements, redundant actuation and force constraint on the cables. Since there are multiple and conflicting objectives to achieve, a multi-objective optimization was recommended and carried out [9, 10]. Multi-objective optimization had been also carried out while designing other instances of parallel robots [11 –14].

Multi-objective design optimization provides a population of Pareto optimal solutions, instead of a singular design solution [15]. Designers have freedom to select a suitable design from the set of Pareto optimal solutions. While other prevalent methods need multiple simulations to construct set of Pareto optimal solutions, multi-objective evolutionary algorithms (MOEA) have advantage of reaching to Pareto optimality in a single simulation effort [16]. Amongst different variants of existing evolutionary algorithms (EAs), NSGA-II is more popular which works on the basis of non-dominated sorting of prospective solutions during genetic based evolution [17]. However, some improvement opportunities were discovered in the prevailing EAs during the present research. Firstly, it was found that owing to their sorting mechanism they may not discriminate between competing solutions vividly [10 , 18–20] and give a pseudo PF eventually. Another constraint posed by EAs is the difficulty of maintaining diversity in the Pareto optimal solutions [21]. Author’s, in their previous work [10] had proposed a fuzzy dominance based selection approach, following which it is possible to establish a good discrimination between competing solutions and a better convergence as a consequence. However, it is found that maintaining diversity is rather more important in the present case of cable based robots which could not be achieved earlier. Since the subject ankle robot is actuated using cables and PMA, maintaining high overall stiffness is important. Altering geometry of the robot mechanism (in particular the actuator connection points at the two platforms of the parallel robot), its overall stiffness can be changed [22]. However, high stiffness also results in undesirable high actuator forces and therefore a selective trade-off is required to be established between the two goals. Analysing the Pareto optimal front obtained from our previous work [10], it was found that, the extreme-end solutions of the Pareto front could not be explored which could have provided more choices of trade-offs between actuator forces and overall robot stiffness. As a matter of fact, while one designer may select an equitable solution (design solution showing best convergence) from the set of Pareto optimal solutions, others may seek a biased solution (diversity in the Pareto front) favouring an important design objective. Therefore, we need an approach which can provide us better visibility in the extremities of the PF subsequent to the optimization process.

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 Fuzzy based sorting of design solutions

With the advent of fuzzy set theory it is possible to treat qualitative numbers vividly with conventional mathematical operators [24]. In other words, using fuzzy set theory, it is possible to manipulate qualitative variables without converting them into crisp numbers. Lately, fuzzy logic has become a popular heuristic approach to model real world phenomena which are non-linear, uncertain and ambiguous in nature [25].

Evolutionary optimization methods, such as NSGA-II, is discretely used to optimize fuzzy systems by optimally partitioning the universe of discourse of fuzzy variables and extracting an optimal set of fuzzy rule base [26 –29]. Similarly, fuzzy systems have also been used in past to improve overall performance of evolutionary optimization methods [30 –35]. However, the role of fuzzy mathematics in EAs has been limited either to the final selection of a suitable solution from the set of Pareto optimal solutions or for incorporating user preference to guide the convergence of EA in a desired direction. The present work attempts to address limitations of EAs mentioned in the previous section comprehensively using fuzzy logic based approach.

Steps involved in the implementation of the proposed approach are explained in the following sub-section. To begin with, the objective functions are defined as fuzzy variables and this process is termed as the fuzzification of objectives. Later, the construction of fuzzy dominant fronts is explained followed by an explanation on the fuzzy based sorting scheme. A simple hand calculation is also provided for clear manifestation of the algorithm.

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). $\begin{matrix} A_{i} & = & 0.75 * min (f_{i}); B_{i} = A_{i} + \frac{R_{i}}{(M_{i} - 1)}; \\ C_{i} & = & B_{i} + \frac{R_{i}}{(M_{i} - 1)}; D_{i} = 1.25 * max (f_{i}); \end{matrix}$ (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.

2.2 Overall Activation Score (OAS) calculation

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. $\begin{matrix} If f_{1} is {AF}_{i 1} and, \dots \dots \dots and \\ f_{N} is {AF}_{im} then {AS}_{i} is y_{i} \end{matrix}$

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]. $N_{R} = \prod_{j = 1}^{N} M_{j}$ (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 ${\bar{f}}_{ij}$ (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 $(\frac{1}{σ \sqrt{2 π}})$ . ${AF}_{ij} (f_{j}, {\bar{f}}_{ij}, σ_{ij}) = {ae}^{- \frac{{(f_{j} - {\bar{f}}_{ij})}^{2}}{2 σ_{ij}}}$ (7) $w_{i} = \prod_{j = 1}^{N} {AF}_{ij}$ (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. $OAS = \frac{\sum_{i = 1}^{N_{R}} (w_{i} y_{i})}{\sum_{i = 1}^{N_{R}} w_{i}}$ (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)

2.3 Fuzzy dominant fronts

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.

2.3.1 Hand calculation

An example problem is being solved now to demonstrate the evaluation of OAS for certain given objective values. Two objectives shown in Fig. 2 are being considered which have been defined using four AFs and their extreme values are assumed to be 0 and 10. The standard deviations for all the AFs are calculated using (2) and found to be (10/7) or 1.43 units. Positions or the minimum fuzziness points of all the AFs are shown in Fig. 2. Next, consider a candidate solution which gives out two objective function values (f₁ and f₂) as ‘4’ and ‘7’ units and it is required to find an OAS corresponding to this design solution. Outputs obtained at various steps of algorithm are displayed in Table 2 with proper reference to the equation numbers. There are sixteen rows in the table depicting sixteen rules. The first column contains outputs (y_i) from all the rules (6) which will be same for all the objective values. Column 2 and 3 show AF’s values for the given input objective values ‘4’& ‘7’ respectively. Apparently, objective value ‘4’ has complete activation (which is ‘1’) in AF2 whereas activation for objective value ‘7’ is equally shared (as 0.5) by AF3 & AF4. Column 4 displays 16 values obtained from (8) whereby column 2 is multiplied with the transpose of column 3. Columns 5 and 6 have numbers obtained after the necessary calculations carried out using (9). Finally in column 7 the floored value of the fuzzy front index is displayed as 2. Therefore a solution having set of objectives values which are (4 & 7) shall be placed in the second fuzzy front out of the total 4 fronts shown in Fig. 4. Shapes of activation functions and their placement in the objective space are also explained with illustrations in Fig. 5.

The overall activation score of a solution is termed as its fuzzy front index and the solution is accordingly placed in the appropriate front. While the first front is most desirable for a minimization goal and the fourth front remains target for a maximization problem. The floor function is applied on the crisp outputs (Y) in order to transform real numbers to integers and avoid giving a unique score to the competing solutions. The entire BFSGA process is also explained with a flowchart in Fig. 6.

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 Design optimization of the cable based parallel ankle rehabilitation robot

The proposed BFSGA approach is used to perform the design optimization of the cable based parallel ankle rehabilitation robot [36]. The ankle robot (Fig. 7a &b) works on a parallel mechanism comprising of a fixed and a moving platform. Four air muscle actuators arranged in parallel configuration, connect the two platforms and with their simultaneous actuation, drive the moving platform with 3-dof. Apart from the wearability and compactness requirements, use of cables and flexible PMA make robot design an exigent task. The moving platform of the robot has to accommodate patient’s foot of different sizes, thus the configuration of robot’s moving platform is also constrained (21). Ankle robot design is defined mainly by the coordinates of actuator connection points at the two parallel platforms (Fig. 7c). Six performance indices (PIs) are defined in order to evaluate competing designs. These PIs are considered as objectives and are further explained along with their mathematical formulation in the following section.

3.1 Problem formulation

3.1.1 Global condition number (F₁)

Condition number of ankle robot’s Jacobian matrix reflects sensitivity in the output Cartesian velocities for small changes in joint velocities. It is desired to obtain a robot design for which this condition number is close to unity in the entire workspace. A global condition number (GCN), which is subjected to minimization, has been formulated for ankle robot as below.

$\begin{matrix} Minimize \\ GCN = \frac{\int_{W} (k (J)) dw}{\int_{w} dw} \end{matrix}$ (11)

Here, k (J) is the condition number of robot’s Jacobian matrix for a given robotic orientation/pose in the workspace (w).

3.1.2 Robot workspace index (F₂)

The feasible workspace of the robot is desired to be maximized which is the conglomeration of the workspace points where the robot can reach with positive tension in the cables. At the same time robot should be prevented to fall in a singular configuration where the robot Jacobian matrix becomes rank deficient. The feasible workspace index (I) can be defined as below.

$\begin{matrix} Maximize \\ I = \frac{φ_{f}}{φ_{T}} \end{matrix}$ (12) $φ_{T} = (θ_{max} - θ_{min}) (φ_{max} - φ_{min}) (ψ_{max} - ψ_{min})$ (13)

Here φ_f is the feasible workspace and φ_T is the overall workspace with Euler angle limits as $θ_{- 25}^{+ 25}, φ_{- 40}^{+ 40} and ψ_{- 30}^{+ 30}$ .

3.1.3 Stiffness conditioning index (F₃)

Stiffness of individual link of the ankle robot contains stiffness of cable and PMA in series and therefore, the resultant stiffness, which is dominated by the less stiff PMA, can be given by following relation: $k = \frac{6 F}{[3 L - \frac{b^{2}}{L}]}$ (14)

Here L is the length of the PMA, b is the thread angle of the PMA mesh and F is the actuator force. It is desired to maximize the minimum stiffness amongst four links of the ankle robot. The overall stiffness matrix K of the ankle robot (containing four links) can be resolved in three matrices using singular value decomposition (SVD) as shown in (12): ${[K]}_{3 \times 3} = {[X^{T}]}_{3 \times 3} {[Σ]}_{3 \times 3} {[Y]}_{3 \times 3}$ (15) $\begin{matrix} Maximize \\ K_{\min} = \min (diag (Σ_{1}, Σ_{2}, Σ_{3})) \end{matrix}$ (16)

While, X and Y are orthogonal matrices, Σ is a diagonal matrix containing three singular values as (Σ₁, Σ₂, Σ₃). The minimum actuator stiffness or the minimum singular value (13) is being maximized in the present work.

3.1.4 Norm of actuator forces (F₄)

The ankle robot uses four parallel actuators and it was required to minimize the norm of all the actuator forces (|F|). Actuator force vector can be obtained from the desired moment (M_desired) and robot’s Jacobian(J). $M_{desired} = J^{T} F$ (17)

3.1.5 Error in achieving desired moments (F₅)

Owing to the redundant actuation, an extra dof is always available to alter the force vector and the robot design suitably. The aim is to find a robot design for which the error (ΔM_ext) between the desired and actual moments (obtained as a consequence of four actuator forces), is minimum. $M_{actual} = J^{T} F$ (18) $\begin{matrix} Minimize \\ Δ M_{ext} = M_{actual} - M_{desired} \end{matrix}$ (19)

3.1.6 Maximum actuator force (F₆)

Apart from minimizing the actuator force norm, it is also important to minimize the maximum actuator force in an individual actuator. $F = max (J {(J^{T} J)}^{- 1} M_{desired})$ (20)

3.2 Fuzzification of objectives

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. $\frac{π}{12} \leq q_{1}, q_{3} \leq \frac{π}{3} and \frac{π}{12} \leq q_{5}, q_{7} \leq \frac{4 π}{9}$ where $80 \leq q_{2}, q_{6}, q_{8} \leq 120 and 120 \leq q_{4} \leq 160$ (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.

3.4 Experimental results from BFSGA and NSGA-II

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.

4 Discussion on results

Robot design solutions obtained after BFSGA and NSGA-II are compared in terms of their objective values and PF shapes. Pareto front shapes from both the approaches are quite similar; however, BFSGA approach evidently exceled in finding extreme points which can be attributed to its selection mechanism whereby the extreme objective values are promoted.

This particular feature of BFSGA is more evident in the plot showing actuator forces and stiffnesses of solutions (Figs. 9 and 10). Comparing these results, it can be seen that BFSGA approach could successfully explore extreme end solutions whereas a clear PF is not visible in NSGA-II results.

Results obtained through optimization exhibit an expected trend for various objectives. It had been earlier established that minimizing the condition number it is possible to reduce the actuator force requirements [36], which is also evident from results displayed in Fig. 9. Further, stiffness index represents actuator stiffnesses which are calculated using Jacobian matrix of the robot. Therefore, stiffness should improve given that the Jacobian matrix is well conditioned which is again apparent from the results. There exists a positive correlation between actuator forces and robot stiffness. Stiffness and force from PMA are governed by the inside pressure of the actuator and therefore higher stiffness would mean higher actuator forces. Amidst this conflict, a trade-off is required to be established between the stiffness and the norm of actuator forces while maintaining the condition number of the robot as close tounity.

Apart from other benefits such as clear discrimination between solutions and qualitative presentation of objectives, the proposed approach has two major advantages. Firstly, it is able to provide wider choices of solutions compared to existing NSGA-II approach. Secondly, following BFSGA it is possible to select a final design solution from the hoard of PF solutions which otherwise poses a burden on the designers.

It is important to mention here that the final design solution, obtained through optimization, is found to abide by all the design constraints (21) and thus is a feasible design.

5 Conclusion

Evolutionary algorithms can be efficiently used to perform a comprehensive design optimization of cable actuated parallel robots whereby many design objectives are required to be simultaneously optimized. However, existing approaches, such as NSGA-II, may not establish a clear discrimination between solutions and during the course of optimization all the competing solutions quickly emerge out to be non-dominated. This results in a false PF wherein the entire population of solutions appears to be Pareto optimal, whereas it is not. The inherent mechanism of existing EAs focuses more on the convergence and therefore the extremities of PF remain unexplored. Finally, there is no mechanism by which a final design solution can be selected from equally good PF solutions.

In this paper, a biased fuzzy based sorting scheme is proposed which can effectively discriminate between good and better solutions and at the same time can provide a wider choice of solution by exploring extremities of the PF. It has also been shown in this research that the overall activation score, which is a unique number assigned to all the solutions and is a representative of their objective values, can be used to select the final design solution from the non-dominated PF solutions. Another interesting feature of the proposed fuzzy based optimization, which was discovered during present research, was its ability to incorporate user preference (for objectives) in the beginning of optimization. Normally, the user preference is considered posteriorly, whereby users are asked to select their preferred objectives from the final PF solutions after the conclusion of optimization. This approach is not desirable rather in order to save on time and resources; it is recommended to incorporate user preferences before commencing the optimization. It has been observed during the present research that by varying the standard deviation of Gaussian activation functions (used to define fuzzy objectives) it is possible to prioritize objectives. It is envisaged that in future, promising research work, shall be carried out in this direction.

Footnotes

Acknowledgments

This research is supported by the Start-up fund from the Faculty of Engineering and Information Sciences, University of Wollongong, NSW, Australia.

References

Hussain

, Xie

S.Q.

and Jamwal

P.K.

, Effect of cadence regulation on muscle activation patterns during robot assisted gait: A dynamic simulation study, IEEE Journal of Biomedical and Health Informatics17 (2013), 442–451.

Jamwal

P.K.

, Hussain

and Xie

S.Q.

, Review on design and control aspects of ankle rehabilitation robots, Disability and Rehabilitation: Assistive Technology10 (2015), 93–101.

Hussain

, State-of-the-art robotic gait rehabilitation orthoses: Design and control aspects, Neuro Rehabilitation35 (2014), 701–709.

Jamwal

P.K.

, Xie

S.Q.

, Hussain

and Parsons

J.G.

, An adaptive wearable parallel robot for the treatment of ankle injuries, IEEE/ASME Transactions on Mechatronics19 (2014), 64–75.

Hussain

, Xie

S.Q.

and Jamwal

P.K.

, An intrinsically compliant robotic orthosis for treadmill training, Medical Engineering and Physics34 (2012), 1448–1453.

Hussain

, Xie

S.Q.

and Jamwal

P.K.

, Robust nonlinear control of an intrinsically compliant robotic gait training orthosis, IEEE Transactions on Systems, Man, and Cybernetics: Systems43 (2013), 655–665.

Tsoi

Y.H.

and Xie

S.Q.

, Trajectory generation for ankle rehabilitation: A biomechanical model based approach, in IET Seminar Digest, 2010, pp. 1124–1129.

Pusey

, Fattah

, Agrawal

and Messina

, Design and workspace analysis of a 6-6 cable-suspended parallel robot, Mechanism and Machine Theory39 (2004), 761–778.

Jamwal

P.K.

, Hussain

and Xie

S.Q.

, Three-Stage Design Analysis and Multicriteria Optimization of a Parallel Ankle Rehabilitation Robot Using Genetic Algorithm, Automation Science and Engineering, IEEE Transactions on, 2014, pp. 1–14.

10.

Jamwal

P.K.

and Hussain

, Multi-criteria design optimization of a parallel ankle rehabilitation robot: Fuzzy dominated sorting evolutionary algorithm approach, IEEE Transactions on Systems, Man and Cybernetics: Systems46 (2016), 589–597.

11.

Bounab

, Multi-objective optimal design based kineto-elastostatic performance for the delta parallel mechanism, Robotica34(2) (2016), 258–273.

12.

Kelaiaia

, Company

, Zaatri

, Multiobjective optimization of parallel kinematic mechanisms by the genetic algorithms, Robotica30 (2012), 783–797.

13.

Kelaiaia

, Company

and Zaatri

, Multiobjective optimization of a linear Delta parallel robot, Mechanism and Machine Theory50 (2012), 159–178.

14.

Kelaiaia

, Zaatri

, Company

and Chikh

, Some investigations into the optimal dimensional synthesis of parallel robots, International Journal of Advanced Manufacturing Technology (2015).

15.

Coello

C.A.C.

and Lamont

G.B.

, Applications of multi-objective evolutionary algorithms, World Scientific (2004). ISBN 981-256-106-4Singapore.

16.

Deb

, Multi-Objective Optimization using Evolutionary Algorithms, John Wiley & Sons, Ltd, 2004.

17.

Deb

, Pratap

, Agarwal

and Meyarivan

, A fast and elitist multiobjective genetic algorithm: NSGA-II, IEEE Transactions on Evolutionary Computation6 (2002), 182–197.

18.

Bader

and Zitzler

, HypE: An algorithm for fast hypervolume-based many-objective optimization, Evolutionary Computation19 (2008), 45–76.

19.

Zitzler

, Two Decades Of Evolutionary Multi-Criterion Optimization: A Glance Back And A Look Ahead, in IEEE Symposium on Computational Intelligence in Multicriteria Decision Making, HI, USA, 2007, 318–318.

20.

Zou

, Chen

, Liu

and Kang

, A new evolutionary algorithm for solving many-objective optimization problems, IEEE Transactions on Systems, Man, and Cybernetics, Part B: Cybernetics38 (2008), 1402–1412.

21.

, Kwong

and Deb

, A dual-population paradigm for evolutionary multiobjective optimization, Information Sciences309 (2015), 50–72.

22.

Merlet

J.P.

, Determination of the optimal geometry of modular parallel robots, Proceedings of the 2003 IEEE International Conference on Robotics & Automation, Taipei, Taiwan, 2003, pp. 1197–1202.

23.

Rachmawati

and Srinivasan

, Incorporating the notion of relative importance of objectives in evolutionary multiobjective optimization, IEEE Transactions on Evolutionary Computation14(4) 2009, 530–546.

24.

Zadeh

L.A.

, Fuzzy sets, Information and Control8 (1965), 338–353.

25.

Pratihar

D.K.

and Hui

N.B.

, Evolution of fuzzy controllers and applications, in Studies in Computational Intelligence66 (2007), 47–69.

26.

AlcalÁ

, Ducange

, Herrera

, Lazzerini

and Marcelloni

, A multiobjective evolutionary approach to concurrently learn rule and data bases of linguistic fuzzy-rule-based systems, IEEE Transactions on Fuzzy Systems17 (2009), 1106–1122.

27.

Carmona

C.J.

, Gonzalez

, Jesus

M.J.D.

and Herrera

, NMEEF-SD: Non-dominated multiobjective evolutionary algorithm for extracting fuzzy rules in subgroup discovery, IEEE Transactions on Fuzzy Systems18 (2010), 958–970.

28.

Deng

, Chung

F.L.

and Wang

, Robust relief-feature weighting, margin maximization, and fuzzy optimization, IEEE Transactions on Fuzzy Systems18 (2010), 726–744.

29.

and Hu

, Two-step interactive satisfactory method for fuzzy multiple objective optimization with preemptive priorities, IEEE Transactions on Fuzzy Systems15 (2007), 417–425.

30.

Abbasnia

, Afshar

and Eshtehardian

, Time-cost trade-off problem in construction project management, based on fuzzy logic, Journal of Applied Sciences8 (2008), 4159–4165.

31.

Fernandez

, Cancela

and Olmedo

, Deriving a final ranking from fuzzy preferences: An approach compatible with the Principle of Correspondence, Mathematical and Computer Modelling47 (2008), 218–234.

32.

Fernandez

, Lopez

, Bernal

, Coello Coello

C.A.

and Navarro

, Evolutionary multiobjective optimization using an outranking-based dominance generalization, Computers & Operations Research37 (2010), 390–395.

33.

Liu

, Sun

and Wang

, The concept of approximation based on fuzzy dominance relation in decision-making, in Lecture Notes in Artificial Intelligence (Subseries of Lecture Notes in Computer Science), 2003, pp. 382–385.

34.

Reardon

B.J.

, Fuzzy logic versus niched Pareto multiobjective genetic algorithm optimization, Modelling and Simulation in Materials Science and Engineering6 (1998), 717–734.

35.

Wang

and Wu

, A new fuzzy dominance GA applied to solve many-objective optimization problem, in Second International Conference on Innovative Computing, Information and Control, ICICIC 2007, 2008.

36.

Jamwal

P.K.

, Xie

and Aw

K.C.

, Kinematic design optimization of a parallel ankle rehabilitation robot using modified genetic algorithm, Robotics and Autonomous Systems57 (2009), 1018–1027.

37.

Knuth

D.E.

, The art of computer programming, 02 (2000).

38.

Deb

and Tiwari

, Multi-objective optimization of a leg mechanism using genetic algorithms, Engineering Optimization37 (2005), 325–350.

39.

Khare

, Yao

and Deb

, Performance Scaling of Multi-Objective Evolutionary Algorithms, Springer, Berlin, 2003.

40.

Purshouse

R.C.

and Fleming

P.J.

, Evolutionary many-objective optimisation: An exploratory analysis, in Evolutionary Computation, 2003. CEC ’03. The 2003 Congression3 (2003), 2066–2073.

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

Abstract

Keywords

1 Introduction

1.1 Convergence versus diversity issue with MOEA

2 Fuzzy based sorting of design solutions

2.1 Fuzzification of objectives

2.3.1 Hand calculation

2.4 Termination criterion

2.5 Final solution from the Pareto front

3 Design optimization of the cable based parallel ankle rehabilitation robot

3.1 Problem formulation

3.1.1 Global condition number (F1)

3.3 Parameter selection

4 Discussion on results

5 Conclusion

Footnotes

Acknowledgments

References

3.1.1 Global condition number (F₁)