Neuronal ConstraintHandling Technique for the Optimal Synthesis of ClosedChain Mechanisms in Lower Limb Rehabilitation
Abstract
:1. Introduction
1.1. Contributions
1.2. Paper Organization
2. ConstraintHandling Technique Based on a Neural Network for the Differential Evolution Algorithm
2.1. Statement of the Mechanism Synthesis Problem
2.2. General Overview of the Differential Evolution Algorithm
2.3. Neuronal ConstraintHandling Technique
Algorithm 1 Pseudocode of the DE/RAND/1/BIN with the inclusion of the NCH technique (R1BNCH). 

3. Study Cases
3.1. Case 1: Four–Bar Linkage Mechanism
3.2. Case 2: Cam–Linkage Mechanism
4. Results
4.1. Experiment Conditions: Algorithm Parameter Tuning and Neuronal Constraint Handling Training Process
4.2. Algorithm Performance Analysis
4.2.1. Descriptive Statistics
4.2.2. Confidence Intervals and Inferential Statistics
4.2.3. Overall Evaluation of the Proposed NCH through Study Cases
5. Conclusions
Author Contributions
Funding
Institutional Review Board Statement
Informed Consent Statement
Data Availability Statement
Acknowledgments
Conflicts of Interest
Abbreviations and Nomenclature
Abbreviations
NCH  Neuronal ConstraintHandling 
DE  Differential Evolution 
DE/RAND/1/BIN  DE variant with random mutation an binomial crossover 
SQP  Sequential Quadratic Programming 
GA  Genetic Algorithm 
PSO  Particle Swarm Optimization 
MUMSA  Malaga University Mechanism Synthesis Algorithm 
POEMA  Pareto Optimum Evolutionary multiobjective Algorithm 
GAFL  GA–Fuzzy Logic 
AG  AntGradient 
CS  Cuckoo Search 
ICA  Imperialist Competitive Algorithm 
TLBO  TeachingLearningBased Optimization 
HLIDE  Hybrid Lagrange Interpolation DE 
CMDE  CombinedMutation DE 
CHT  ConstraintHandling Techniques 
Nomenclature
$\overline{J}$  Weighted objective function 
${J}_{i}$  ith objective function 
x  Design variable vector 
${g}_{j}$  jth inequality constraint 
${h}_{k}$  kth equality constraint 
${x}_{min}$ & ${x}_{max}$  Upper and lower design variable vector bounds 
${\mathbf{X}}^{G}$  Population of individuals in a G generation 
${X}_{p}^{G}$  pth individual in the ${\mathbf{X}}^{G}$ population 
${\mathbf{U}}^{G}$  Offspring individuals in a G generation 
$\varphi $  Constraint distance 
$f\left(\mu \right)$  Sigmoid function 
${a}_{r}^{s}$  rth neuron in sth layer in NCH technique 
${\widehat{w}}_{t,r}^{s}$  tth wight for ${a}_{r}^{s}$ neuron 
${a}_{r}^{s}$  rth bias in sth layer for ${a}_{r}^{s}$ neuron 
${w}_{i}$  ith weight in the objective function 
$[{\overline{x}}_{p}^{i},{\overline{y}}_{p}^{i}]$  Desired path points 
$[{x}_{p}^{i},{y}_{p}^{i}]$  Mechanism path points 
${r}_{i}$  ith length link 
${\theta}_{2}^{i}$  ith crank angle 
$[{x}_{0},{y}_{0}]$  Ground link origin 
$[{r}_{{c}_{x}},{r}_{{c}_{y}}]$  Lengths in the coupler link 
$\beta ,\gamma $ & $\eta $  Link angles 
e  Slider displacement 
$\alpha $  Slider angular position 
${R}_{0}$  Cam base radius 
${\Theta}_{ref}$  Normalization angle parameter 
${R}_{ref}$  Normalization radius parameter 
${d}_{ij}$  Distance between i to j point 
$CR$  Crossover factor 
${F}_{min}$ & ${F}_{max}$  Maximum and minimum scale factor limit 
$MR$  Mutation rate 
$Pf$  Probabilistic factor in SR 
$cp$  Control the relaxation of constraints 
$Tc$  Maximum iterations to relax constraints 
$RG$  Number of attempts to improve a solution 
$Pg$  Probabilistic factor to improve a solution 
Appendix A. Trajectories for Rehabilitation in the Study Cases
Trajectory 1  Trajectory 2  

x [m]  y [m]  x [m]  y [m] 
0.7429  0.188  0.7429  0.188 
0.6551  0.1573  0.6551  0.1573 
0.6014  0.1388  0.6014  0.1388 
0.5189  0.1149  0.5189  0.1249 
0.4159  0.1012  0.4159  0.1212 
0.3001  0.1074  0.3001  0.1174 
0.1964  0.1375  0.1964  0.1375 
0.1639  0.1662  0.1639  0.1662 
0.1605  0.2003  0.1605  0.2003 
0.1934  0.2256  0.1934  0.2256 
0.2619  0.2251  0.2619  0.2251 
0.4201  0.1808  0.4201  0.2108 
0.6474  0.1607  0.6474  0.1907 
0.7429  0.188  0.7429  0.188 
x [mm]  y [mm]  x [mm]  y [mm]  x [mm]  y [mm] 

−315.8706  −725.914  102.286  −784.1517  249.9019  −624.0724 
−305.426  −729.4993  113.9412  −782.7289  232.1635  −626.9287 
−295.562  −732.633  126.2456  −780.9599  214.5033  −631.2082 
−284.9638  −735.7146  137.8298  −779.0587  195.4565  −636.575 
−272.9513  −738.8459  150.0497  −776.7977  175.5689  −643.126 
−259.5651  −742.091  162.2326  −774.345  154.0295  −650.5493 
−245.4816  −745.2649  175.691  −771.2814  132.9875  −658.2092 
−230.7078  −748.3559  188.404  −768.0873  110.6898  −666.7843 
−217.2474  −750.9006  201.7067  −764.4419  88.04307  −675.2332 
−201.8657  −754.024  214.2527  −760.6806  62.68539  −684.4142 
−188.5033  −756.6797  228.0001  −756.1684  37.99372  −693.0726 
−173.8345  −759.6436  240.3332  −751.8006  13.98859  −701.2471 
−161.1907  −762.2906  253.8222  −746.6116  −13.02161  −709.5296 
−147.8624  −764.9876  265.2127  −741.7514  −38.76389  −716.7951 
−134.5299  −767.7142  277.7069  −736.0049  −65.7052  −723.5015 
−122.5271  −770.2517  289.3726  −730.1277  −93.18724  −729.2118 
−111.1662  −772.6216  300.7921  −723.7715  −119.8744  −733.994 
−98.40761  −774.9819  310.6746  −717.4391  −147.5638  −737.2401 
−86.97769  −777.2058  320.2523  −710.5785  −172.2932  −739.7084 
−74.81197  −779.1819  328.2496  −703.7132  −197.8191  −740.6221 
−63.29198  −781.0032  335.1887  −696.3953  −222.7483  −740.2338 
−51.73248  −782.6248  339.9644  −689.448  −244.9169  −738.7976 
−40.13729  −784.046  343.6156  −681.9446  −266.9157  −735.9746 
−28.51026  −785.2659  345.5832  −674.264  −284.0916  −733.2587 
−16.16956  −786.2028  346.533  −666.2336  −299.6447  −729.774 
−4.491148  −787.0124  344.1531  −659.0253  −313.0036  −726.1448 
7.891441  −787.5247  340.7879  −651.3812  −321.6559  −723.6518 
19.6055  −787.922  335.573  −644.3309  −328.838  −721.1686 
30.63857  −788.0627  327.8983  −637.91  −332.1062  −720.0498 
43.05355  −787.9534  318.5382  −632.043  −332.155  −720.179 
54.77622  −787.6019  307.4652  −627.8585  −329.6154  −721.2705 
67.19192  −787.0725  295.572  −624.2872  −323.2063  −723.8421 
78.90697  −786.3287  281.0296  −622.9766  −316.7307  −726.2375 
91.2885  −785.2526  266.4392  −622.3635 
x [mm]  y [mm]  x [mm]  y [mm]  x [mm]  y [mm] 

−315.8706  −725.914  102.286  −777.1517  249.9019  −624.0724 
−305.426  −729.4993  113.9412  −776.7289  232.1635  −626.9287 
−295.562  −732.633  126.2456  −776.9599  214.5033  −631.2082 
−284.9638  −735.7146  137.8298  −776.0587  195.4565  −636.575 
−272.9513  −738.8459  150.0497  −774.7977  175.5689  −643.126 
−259.5651  −742.091  162.2326  −773.345  154.0295  −650.5493 
−245.4816  −745.2649  175.691  −771.2814  132.9875  −658.2092 
−230.7078  −748.3559  188.404  −768.0873  110.6898  −666.7843 
−217.2474  −750.9006  201.7067  −764.4419  88.04307  −675.2332 
−201.8657  −754.024  214.2527  −760.6806  62.68539  −680.4142 
−188.5033  −756.6797  228.0001  −756.1684  37.99372  −685.0726 
−173.8345  −759.6436  240.3332  −751.8006  13.98859  −687.2471 
−161.1907  −762.2906  253.8222  −746.6116  −13.02161  −692.5296 
−147.8624  −764.9876  265.2127  −741.7514  −38.76389  −697.7951 
−134.5299  −767.7142  277.7069  −736.0049  −65.7052  −701.5015 
−122.5271  −770.2517  289.3726  −730.1277  −93.18724  −702.2118 
−111.1662  −772.6216  300.7921  −723.7715  −119.8744  −701.994 
−98.40761  −774.9819  310.6746  −717.4391  −147.5638  −703.2401 
−86.97769  −777.2058  320.2523  −710.5785  −172.2932  −703.7084 
−74.81197  −777.1819  328.2496  −703.7132  −197.8191  −704.6221 
−63.29198  −777.0032  335.1887  −696.3953  −222.7483  −702.2338 
−51.73248  −776.6248  339.9644  −689.448  −244.9169  −700.7976 
−40.13729  −777.046  343.6156  −681.9446  −266.9157  −697.9746 
−28.51026  −778.2659  345.5832  −674.264  −284.0916  −698.2587 
−16.16956  −778.2028  346.533  −666.2336  −299.6447  −699.774 
−4.491148  −778.0124  344.1531  −659.0253  −313.0036  −701.1448 
7.891441  −778.5247  340.7879  −651.3812  −321.6559  −703.6518 
19.6055  −777.922  335.573  −644.3309  −328.838  −705.1686 
30.63857  −778.0627  327.8983  −637.91  −332.1062  −708.0498 
43.05355  −777.9534  318.5382  −632.043  −332.155  −710.179 
54.77622  −777.6019  307.4652  −627.8585  −329.6154  −713.2705 
67.19192  −777.0725  295.572  −624.2872  −323.2063  −718.8421 
78.90697  −777.3287  281.0296  −622.9766  −316.7307  −721.2375 
91.2885  −777.2526  266.4392  −622.3635 
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Study  Mechanisms  Metaheuristic Algorithms  ConstraintHandling Technique 

[5]  four–bar mechanism  Genetic Algorithm (GA)  Penalty Function (PF) 
[41]  Hand robot mechanism  Pareto Optimum Evolutionary multiobjective Algorithm (POEMA)  Feasibility Rules (FR) 
[31]  four–bar mechanism  Differential Evolution (DE)  PF 
[32]  Sixbar mechanism  DE  PF 
[33]  four–bar mechanism  DE  PF 
[34]  four–bar mechanism  GA–fuzzy logic (GAFL)  PF 
[17]  four–bar and Sixbar mechanisms  Málaga University Mechanism Synthesis Algorithm (MUMSA)  PF 
[35]  four–bar mechanism  GA, DE, Particle Swarm Optimization (PSO)  PF 
[36]  four–bar mechanism  Antgradient (AG)  PF 
[45]  four–bar mechanism  GA–DE  PF 
[6]  Sixbar mechanism  Cuckoo Search (CS)  PF 
[37]  four–bar mechanism  Imperialist Competitive Algorithm (ICA), GA, DE, PSO  PF 
[8]  four–bar mechanism  Modified Krill Herd  PF 
[46]  four–bar mechanism  TeachingLearningBased Optimization (TLBO), GA, PSO  PF 
[38]  four–bar mechanism  Hybrid Lagrange Interpolation DE (HLIDE)  PF 
[39]  four–bar and Sixbar mechanisms  Hybridization DE with Generalized Reduced Gradient  PF 
[47]  four–bar mechanism  DE  FR 
[48]  four–bar mechanisms  CS, TLBO, DE, MUMSA autoadaptive modified DE, combinedmutation DE (CMDE)   
Study  Mechanism in Rehabilitation  Metaheuristic Algorithms  CHT 
[21]  Sixbar mechanism in finger rehabilitation  MUMSA  FR 
[22]  cam–linkage mechanism in gait rehabilitation  GA  PF 
[7]  four–bar mechanism in gait rehabilitation  DE  FR 
[23]  four–bar mechanism in gait rehabilitation and orthotic devices  PSO, TLBO   
[3]  Eightbar mechanism in lower limb rehabilitation  DE  FR 
[29]  Eightbar, four–bar and cam–linkage mechanisms in lower limb rehabilitation  DE, PSO, MUMSA, GA  FR, PF, StochasticRanking (SR), $\u03f5$Constraint ($\u03f5$C) 
Algorithm  CHT  Parameters 

DE/RAND/1/BIN  FR  $CR=0.95$, ${F}_{min}=0.13$, ${F}_{max}=0.95$ 
SR  $CR=0.91$, ${F}_{min}=0.23$, ${F}_{max}=0.96$, $Pf=0.15$  
EC  $CR=0.89$, ${F}_{min}=0.20$, ${F}_{max}=0.81$, $Pg=0.07$, $Tc=590$, $Rg=5$, $cp=6$  
PF  $CR=0.95$, ${F}_{min}=0.17$, ${F}_{max}=0.91$, ${\varpi}_{k}=\mathrm{10,000}$  
GA  FR  $CR=0.80$, $MR=0.06$ 
SR  $CR=0.23$, $Pf=0.20$, $MR=0.05$  
EC  $CR=1$, $Pg=0.05$, $Tc=610$, $Rg=4$, $cp=8$, $MR=0.08$  
PF  $CR=0.83$, $MR=0.11$, ${\varpi}_{k}=\mathrm{10,000}$  
PSO  FR  ${v}_{min}=0.0$, ${v}_{max}=0.01$, $C1=1.29$, $C2=2.04$ 
SR  $Pf=0.69$, ${v}_{min}=0.0$, ${v}_{max}=0.01$, $C1=2.22$, $C2=1.06$  
EC  $Pg=0.05$, $Tc=610$, $Rg=4$, $cp=8$, ${v}_{min}=0.0$, ${v}_{max}=0.01$, $C1=1.10$, $C2=1.79$  
PF  ${v}_{min}=0.05$, ${v}_{max}=0.17$, $C1=2.04$, $C2=1.06$, ${\varpi}_{k}=\mathrm{10,000}$  
MUMSA  FR  $CR=0.73$, ${F}_{min}=0.34$, ${F}_{max}=0.78$, $R=0.87$, $MR=0.05$ 
SR  $CR=0.81$, ${F}_{min}=0.69$, ${F}_{max}=0.79$, $R=0.73$, $MR=0.08$, $Pf=0.31$  
EC  $CR=0.93$, ${F}_{min}=0.71$, ${F}_{max}=0.99$, $R=0.75$, $MR=0.03$, $Pg=0.08$, $Tc=257$, $Rg=3$, $cp=9$  
PF  $CR=0.02$, ${F}_{min}=0.13$, ${F}_{max}=0.78$, $R=0.04$, $MR=0.02$, ${\varpi}_{k}=\mathrm{10,000}$ 
Algorithm  CHT  Parameters 

DE/RAND/1/BIN  FR  $CR=0.87$, ${F}_{min}=0.51$, ${F}_{max}=0.53$ 
SR  $CR=0.91$, ${F}_{min}=0.46$, ${F}_{max}=0.70$, $Pf=0.52$  
EC  $CR=0.92$, ${F}_{min}=0.53$, ${F}_{max}=0.56$, $Pg=0.04$, $Tc=430$, $Rg=5$, $cp=9$  
PF  $CR=0.93$, ${F}_{min}=0.52$, ${F}_{max}=0.63$  
GA  FR  $CR=0.98$, $MR=0.11$ 
SR  $CR=0.57$, $Pf=0.12$, $MR=0.17$  
EC  $CR=1$, $Pg=0.08$, $Tc=640$, $Rg=4$, $cp=6$, $MR=0.14$  
PF  $CR=0.02$, $Pg=0.21$  
PSO  FR  ${v}_{min}=0.05$, ${v}_{max}=0.24$, $C1=0.29$, $C2=3.12$ 
SR  $Pf=0.48$, ${v}_{min}=0.04$, ${v}_{max}=0.23$, $C1=1.60$, $C2=1.07$  
EC  $Pg=0.06$, $Tc=600$, $Rg=2$, $cp=2$, ${v}_{min}=0.04$, ${v}_{max}=0.26$, $C1=1.90$, $C2=0.86$  
PF  ${v}_{min}=0.12$, ${v}_{max}=0.34$, $C1=2.47$, $C2=0.34$  
MUMSA  FR  $CR=0.99$, ${F}_{min}=0.78$, ${F}_{max}=0.86$, $R=0.31$, $MR=0.03$ 
SR  $CR=0.88$, ${F}_{min}=0.71$, ${F}_{max}=0.74$, $R=0.89$, $MR=0.03$, $Pf=0.41$  
EC  $CR=0.91$, ${F}_{min}=0.82$, ${F}_{max}=0.85$, $R=0.83$, $MR=0.03$, $Pg=0.03$, $Tc=340$, $Rg=2$, $cp=5$  
PF  $CR=0.98$, ${F}_{min}=0.66$, ${F}_{max}=0.73$, $R=0.89$, $MR=0.12$ 
Parameter  Four–Bar  Cam–Linkage  Parameter  Four–Bar  Cam–Linkage 

${w}_{1,1}^{1}$  $5.1507$  $9.7041$  ${w}_{3,2}^{3}$  $4.5326$  $1.7223$ 
${w}_{2,1}^{1}$  $5.7785$  $4.6228$  ${w}_{1,1}^{4}$  $6.4775$  $9.5093$ 
${w}_{3,1}^{1}$  $7.5795$  $9.6850$  ${w}_{2,1}^{4}$  $0.7554$  $3.7442$ 
${w}_{4,1}^{1}$  $5.2518$  $3.5547$  ${w}_{3,1}^{4}$  $0.8913$  $0.0632$ 
${w}_{1,2}^{1}$  $8.4485$  $1.6000$  ${w}_{1,2}^{4}$  $5.6703$  $6.2279$ 
${w}_{2,2}^{1}$  $4.8087$  $5.5538$  ${w}_{2,2}^{4}$  $5.7473$  $5.4144$ 
${w}_{3,2}^{1}$  $6.4018$  $4.7216$  ${w}_{3,2}^{4}$  $2.1010$  $1.4408$ 
${w}_{4,2}^{1}$  $0.0997$  $4.5662$  ${w}_{1,1}^{5}$  $5.0501$  $5.8695$ 
${w}_{1,3}^{1}$  $6.4759$  $9.3248$  ${w}_{2,1}^{5}$  $2.7255$  $4.0260$ 
${w}_{2,3}^{1}$  $6.9908$  $0.6948$  ${w}_{3,1}^{5}$  $5.471$  $0.5826$ 
${w}_{3,3}^{1}$  $6.9994$  $9.1374$  ${u}_{1}^{1}$  $5.0333$  $5.8695$ 
${w}_{4,3}^{1}$  $0.9477$  $5.2051$  ${u}_{2}^{1}$  $2.7255$  $4.0260$ 
${w}_{1,1}^{2}$  $6.4335$  $7.9333$  ${u}_{3}^{1}$  $5.471$  $0.5826$ 
${w}_{2,1}^{2}$  $6.7337$  $0.4223$  ${u}_{1}^{2}$  $5.0333$  $7.6293$ 
${w}_{3,1}^{2}$  $9.1879$  $3.7733$  ${u}_{2}^{2}$  $3.2977$  $7.3464$ 
${w}_{1,2}^{2}$  $2.2342$  $7.7584$  ${u}_{3}^{2}$  $0.5276$  $0.2524$ 
${w}_{2,2}^{2}$  $6.4004$  $8.9817$  ${u}_{1}^{3}$  $4.4916$  $5.0776$ 
${w}_{3,2}^{2}$  $0.2861$  $6.4264$  ${u}_{2}^{3}$  $6.1729$  $2.9905$ 
${w}_{1,1}^{3}$  $7.0605$  $7.6129$  ${u}_{3}^{3}$  $9.4461$  $9.8085$ 
${w}_{2,1}^{3}$  $3.3614$  $8.5456$  ${u}_{1}^{4}$  $3.4358f$  $6.9378$ 
${w}_{3,1}^{3}$  $3.2812$  $3.3421$  ${u}_{2}^{4}$  $0.8191$  $4.6794$ 
${w}_{1,2}^{3}$  $4.3912$  $2.1478$  ${u}_{3}^{4}$  $1.1605$  $4.4777$ 
${w}_{2,2}^{3}$  $8.4263$  $7.2631$  ${u}_{1}^{5}$  $1.7526$  $2.6718$ 
${F}_{max}$  $0.95$  $0.53$  ${F}_{min}$  $0.13$  $0.51$ 
$CR$  $0.95$  $0.87$ 
AlgorithmCHT  Mean  Std.  Median  Min  Max  NFS 

R1BFR  $0.02619$  $0.01033$  $0.03075$  $0.003966$  $0.04198$  0 
GAFR  $0.03709$  $0.006465$  $0.03808$  $0.02067$  $0.05163$  0 
PSOFR  $0.03626$  $0.01057$  $0.03566$  $0.02025$  $0.05834$  0 
MUMSAFR  $0.03233$  $0.007363$  $0.03338$  $0.01265$  $0.0421$  0 
R1BSR  $0.03088$  $0.008759$  $0.03394$  $0.002988$  $0.04118$  0 
GASR  $0.03578$  $0.01002$  $0.03773$  $0.01142$  $0.05438$  0 
PSOSR  $0.02846$  $0.007408$  $\mathbf{0.02765}$  $0.004439$  $0.04094$  0 
MUMSASR  $0.03104$  $0.00883$  $0.0341$  $0.01653$  $0.04237$  0 
R1B$\u03f5$C  $0.03395$  $\mathbf{0.004956}$  $0.03378$  $0.02432$  $0.04756$  0 
GA$\u03f5$C  $0.04107$  $0.009803$  $0.04155$  $0.01859$  $0.0544$  0 
PSO$\u03f5$C  $0.03441$  $0.007179$  $0.03341$  $0.02315$  $0.0567$  0 
MUMSA$\u03f5$C  $0.02708$  $0.01299$  $0.0295$  $0.003726$  $0.04752$  0 
R1BPF  $0.02486$  $0.01167$  $0.03135$  $0.003076$  $\mathbf{0.03391}$  0 
GAPF  $0.0397$  $0.008355$  $0.038$  $0.02922$  $0.05966$  0 
PSOPF  $0.03371$  $0.01283$  $0.03478$  $0.005304$  $0.06198$  0 
MUMSAPF  $0.02407$  $0.009266$  $0.02789$  $0.009579$  $0.03866$  0 
R1BNCH  $\mathbf{0.02255}$  $0.01324$  $0.02978$  $\mathbf{0.002167}$  $0.03585$  0 
AlgorithmCHT  Mean  Std.  Median  Min  Max  NFS 

R1BFR  $0.02357$  $0.01494$  $0.03304$  $0.002539$  $0.04297$  0 
GAFR  $0.03521$  $0.01041$  $0.03816$  $0.01467$  $0.05109$  0 
PSOFR  $0.03724$  $\mathbf{0.00713}$  $0.03643$  $0.02442$  $0.05452$  0 
MUMSAFR  $0.03205$  $0.009535$  $0.03412$  $0.01266$  $0.048$  0 
R1BSR  $0.02685$  $0.01408$  $0.03331$  $0.002685$  $0.04368$  0 
GASR  $0.04522$  $0.01005$  $0.04495$  $0.02564$  $0.06512$  0 
PSOSR  $0.0353$  $0.01015$  $0.03663$  $0.003119$  $0.05145$  0 
MUMSASR  $0.03064$  $0.01027$  $0.03104$  $0.01039$  $0.05104$  0 
R1B$\u03f5$C  $0.02956$  $0.009226$  $0.03264$  $0.01179$  $0.04592$  0 
GA$\u03f5$C  $0.03973$  $0.009862$  $0.03818$  $0.02489$  $0.05877$  0 
PSO$\u03f5$C  $0.03608$  $0.009744$  $0.03674$  $0.01091$  $0.05446$  0 
MUMSA$\u03f5$C  $0.02675$  $0.01264$  $0.03091$  $\mathbf{0.002507}$  $0.05102$  0 
R1BPF  $0.02509$  $0.01224$  $\mathbf{0.02461}$  $0.002746$  $0.04512$  0 
GAPF  $0.03458$  $0.01353$  $0.04058$  $0.005144$  $0.05076$  0 
PSOPF  $0.03452$  $0.01104$  $0.03651$  $0.01113$  $0.05355$  0 
MUMSAPF  $0.02689$  $0.007722$  $0.02891$  $0.00876$  $\mathbf{0.03767}$  0 
R1BNCH  $\mathbf{0.02336}$  $0.01577$  $0.03441$  $0.002572$  $0.04166$  0 
AlgorithmCHT  Mean  Std.  Median  Min  Max  NFS 

R1BFR  $0.7046$  $0.05785$  $0.6858$  $0.6778$  $0.9428$  0 
GAFR  $19.72$  $34.7$  $7.387$  $2.206$  $145.5$  0 
PSOFR  $5.092$  $2.945$  $4.244$  $2.232$  $13.56$  1 
MUMSAFR  $1.641$  $0.8234$  $1.158$  $0.6581$  $3.145$  0 
R1BSR  $0.7533$  −  $0.7533$  $0.7533$  $\mathbf{0.7533}$  29 
GASR  $3.123$  $1.743$  $2.455$  $1.892$  $8.379$  0 
PSOSR  $3.407$  $4.932$  $2.441$  $1.334$  $28.89$  0 
MUMSASR  $1.779$  $1.324$  $1.206$  $0.6633$  $6.056$  0 
R1B$\u03f5$C  $0.6886$  $\mathbf{0.04365}$  $0.6751$  $0.6625$  $0.8178$  0 
GA$\u03f5$C  $5.154$  $2.55$  $3.98$  $2.339$  $11.08$  0 
PSO$\u03f5$C  $2.336$  $1.067$  $2.17$  $0.983$  $6.829$  0 
MUMSA$\u03f5$C  $1.797$  $1.694$  $0.8433$  $0.6595$  $5.484$  0 
R1BPF  $0.7961$  $0.3161$  $0.7118$  $0.6722$  $2.422$  0 
GAPF  $2.714$  $0.4742$  $2.506$  $2.077$  $3.639$  0 
PSOPF  $2.91$  $1.138$  $2.899$  $1.388$  $7.702$  0 
MUMSAPF  $2.901$  $2.317$  $2.341$  $0.7097$  $10.47$  0 
R1BNCH  $\mathbf{0.6823}$  $0.05252$  $\mathbf{0.6632}$  $\mathbf{0.658}$  $0.8157$  0 
AlgorithmCHT  Mean  Std.  Median  Min  Max  NFS 

R1BFR  $0.7011$  $0.08787$  $0.6694$  $0.6592$  $0.9423$  0 
GAFR  $6.38$  $6.634$  $3.33$  $2.098$  $35.07$  0 
PSOFR  $3.705$  $2.125$  $3.331$  $1.706$  $13.79$  2 
MUMSAFR  $1.601$  $1.468$  $0.8563$  $\mathbf{0.6418}$  $5.802$  0 
R1BSR  −  −  −  −  −  30 
GASR  $3.77$  $2.143$  $2.737$  $1.889$  $7.992$  0 
PSOSR  $5.722$  $11.51$  $2.966$  $1.03$  $62.69$  0 
MUMSASR  $1.681$  $1.233$  $1.042$  $0.649$  $4.779$  0 
R1B$\u03f5$C  $0.686$  $0.06462$  $0.6591$  $0.6556$  $0.8979$  0 
GA$\u03f5$C  $5.369$  $2.458$  $4.001$  $1.8$  $8.895$  0 
PSO$\u03f5$C  $2.596$  $1.027$  $2.506$  $1.394$  $7.299$  1 
MUMSA$\u03f5$C  $1.755$  $1.594$  $0.9007$  $0.6536$  $5.441$  0 
R1BPF  $0.7394$  $0.1097$  $0.691$  $0.6796$  $1.056$  0 
GAPF  $2.836$  $0.8672$  $2.561$  $2.065$  $6.465$  0 
PSOPF  $2.983$  $1.476$  $2.779$  $1.151$  $7.049$  0 
MUMSAPF  $3.183$  $5.02$  $2.204$  $0.6996$  $28.75$  0 
R1BNCH  $\mathbf{0.6588}$  $\mathbf{0.03913}$  $\mathbf{0.6487}$  $0.6462$  $\mathbf{0.8027}$  0 
Trajectory 1  

Limits  R1BFR  R1BSR  R1B$\mathbf{\u03f5}$C  R1BPF  R1BNCH 
Low  0.0223  0.0276  0.0321  0.0205  0.0176 
Up  0.0300  0.0341  0.0358  0.0292  0.0275 
GAFR  GASR  GA$\mathbf{\u03f5}$C  GAPF  
Low  0.0347  0.0320  0.0374  0.0366  
Up  0.0395  0.0395  0.0447  0.0428  
PSOFR  PSOSR  PSO$\mathbf{\u03f5}$C  PSOPF  
Low  0.0323  0.0257  0.0317  0.0289  
Up  0.0402  0.0312  0.0371  0.0385  
MUMSAFR  MUMSASR  MUMSA$\mathbf{\u03f5}$C  MUMSAPF  
Low  0.0296  0.0277  0.0222  0.0206  
Up  0.0351  0.0343  0.0319  0.0275  
Trajectory 2  
Limits  R1BFR  R1BSR  R1B$\mathbf{\u03f5}$C  R1BPF  R1BNCH 
Low  0.0180  0.0216  0.0261  0.0205  0.0175 
Up  0.0291  0.0321  0.0330  0.0297  0.0292 
GAFR  GASR  GA$\mathbf{\u03f5}$C  GAPF  
Low  0.0313  0.0415  0.0360  0.0295  
Up  0.0391  0.0490  0.0434  0.0396  
PSOFR  PSOSR  PSO$\mathbf{\u03f5}$C  PSOPF  
Low  0.0346  0.0315  0.0324  0.0304  
Up  0.0399  0.0391  0.0397  0.0386  
MUMSAFR  MUMSASR  MUMSA$\mathbf{\u03f5}$C  MUMSAPF  
Low  0.0285  0.0268  0.0220  0.0240  
Up  0.0356  0.0345  0.0315  0.0298 
Trajectory 1  

Hypotesis  Bonferroni 
MUMSAPF vs. R1BNCH  1 
MUMSAPF vs. R1BPF  1 
R1BNCH vs. R1BPF  1 
Trajectory 2  
Hypotesis  Bonferroni 
R1BFR vs. R1BNCH  1 
R1BFR vs. R1BPF  $0.59012$ 
R1BNCH vs. R1BPF  $0.90510$ 
Trajectory 1  

AlgorithmCHT  Set 1  Set 2  Set 3  Set 4 
R1BNCH  0.00219  0.00305  0.00216  0.00214 
R1BPF  0.00234  0.00848  0.00401  0.00215 
MUMSAPF  0.01340  0.01142  0.01166  0.00976 
Trajectory 2  
AlgorithmCHT  Set 1  Set 2  Set 3  Set 4 
R1BNCH  0.00255  0.00257  0.00266  0.00263 
R1BPF  0.00257  0.00307  0.00341  0.00354 
R1BFR  0.00270  0.00260  0.00271  0.00353 
Trajectory 1  

Limits  R1BFR  R1BSR  R1B$\mathbf{\u03f5}$C  R1BPF  R1BNC 
Low  0.6830    0.6723  0.6780  0.6627 
Up  0.7262    0.7049  0.9141  0.7019 
GAFR  GASR  GA$\mathbf{\u03f5}$C  GAPF  
Low  6.7614  2.4716  4.2015  2.5364  
Up  32.6767  3.7737  6.1055  2.8906  
PSOFR  PSOSR  PSO$\mathbf{\u03f5}$C  PSOPF  
Low    1.5653  1.9376  2.4854  
Up    5.2484  2.7343  3.3350  
MUMSAFR  MUMSASR  MUMSA$\mathbf{\u03f5}$C  MUMSAPF  
Low  1.3340  1.2845  1.1647  2.0357  
Up  1.9489  2.2735  2.4295  3.7658  
Trajectory 2  
Limits  R1BFR  R1BSR  R1B$\mathbf{\u03f5}$C  R1BPF  R1BNC 
Low  0.6683    0.6618  0.6984  0.6442 
Up  0.7339    0.7101  0.7803  0.6735 
GAFR  GASR  GA$\mathbf{\u03f5}$C  GAPF  
Low  3.9029  2.9697  4.4512  2.5120  
Up  8.8576  4.5703  6.2872  3.1597  
PSOFR  PSOSR  PSO$\mathbf{\u03f5}$C  PSOPF  
Low    1.4245  2.2053  2.4316  
Up    10.0193  2.9865  3.5342  
MUMSAFR  MUMSASR  MUMSA$\mathbf{\u03f5}$C  MUMSAPF  
Low  1.0532  1.2205  1.1600  1.3086  
Up  2.1492  2.1417  2.3506  5.0576 
Trajectory 1  

Hypotesis  Bonferroni 
R1BEC vs. R1BFR  $\mathbf{9.0179}\times {\mathbf{10}}^{\mathbf{4}}$ 
R1BEC vs. R1BNCH  $\mathbf{4.2514}\times {\mathbf{10}}^{\mathbf{2}}$ 
R1BFR vs. R1BNCH  $\mathbf{3.8933}\times {\mathbf{10}}^{\mathbf{9}}$ 
Trajectory 2  
Hypotesis  Bonferroni 
R1BEC vs. R1BFR  $8.4557\times {10}^{2}$ 
R1BEC vs. R1BNCH  $\mathbf{1.8685}\times {\mathbf{10}}^{\mathbf{5}}$ 
R1BFR vs. R1BNCH  $\mathbf{5.7132}\times {\mathbf{10}}^{\mathbf{11}}$ 
Trajectory 1  

AlgorithmCHT  Set 1  Set 2  Set 3  Set 4 
R1BNCH  0.65626  0.65628  0.65606  0.65637 
R1BFR  0.67261  0.67827  0.67514  0.67639 
R1B$\u03f5$C  0.67012  0.66972  0.66852  0.66230 
Trajectory 2  
AlgorithmCHT  Set 1  Set 2  Set 3  Set 4 
R1BNCH  0.64382  0.64224  0.64307  0.64441 
R1BFR  0.64948  0.65921  0.65997  0.65742 
R1B$\u03f5$C  0.65648  0.65463  0.65531  0.64569 
AlgorithmCHT  Win  Draw  Total 

R1BNCH  4  4  8 
R1BPF  0  4  4 
R1BFR  0  3  3 
R1BEC  1  1  2 
MUMSAPF  0  2  2 
Design variable  ${x}_{1}$ [m]  ${x}_{2}$ [m]  ${x}_{3}$ [m]  ${x}_{4}$ [m]  ${x}_{5}$ [rad]  ${x}_{6}$ [m]  ${x}_{7}$ [m]  ${x}_{8}$ [m] 
Value  0.5993  0.3056  0.5188  0.4359  0.5584  −0.0820  0.3773  0.5959 
Design variable  ${x}_{9}$ [m]  ${x}_{10}$ [rad]  ${x}_{11}$ [rad]  ${x}_{12}$ [rad]  ${x}_{13}$ [rad]  ${x}_{14}$ [rad]  ${x}_{15}$ [rad]  ${x}_{16}$ [rad] 
Value  −0.1457  −0.8068  −1.1432  0.6441  0.7786  0.9475  1.1921  1.5975 
Design variable  ${x}_{17}$ [rad]  ${x}_{18}$ [rad]  ${x}_{19}$ [rad]  ${x}_{20}$ [rad]  ${x}_{21}$ [rad]  ${x}_{22}$ [rad]  ${x}_{23}$ [rad]  
Value  2.0538  2.5685  3.0521  −2.6381  −1.9459  −1.1755  −0.8090 
Design variable  ${x}_{1}$ [mm]  ${x}_{2}$ [mm]  ${x}_{3}$ [mm]  ${x}_{4}$ [mm]  ${x}_{5}$ [mm] 
Value  222.2690  592.4821  895.9455  686.9534  400.3920 
Design variable  ${x}_{6}$ [mm]  ${x}_{7}$ [mm]  ${x}_{8}$ [mm]  ${x}_{9}$ [rad]  ${x}_{10}$ [rad] 
Value  506.1880  509.3375  683.1163  0.4579  −1.0462 
Design variable  ${x}_{11}$ [rad]  ${x}_{12}$ [rad]  ${x}_{13}$ [mm]  ${x}_{14}$ [mm]  ${x}_{15}$ [mm] 
Value  −0.1572  −0.4755  −295.5667  14.9513  33.3667 
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MuñozReina, J.S.; VillarrealCervantes, M.G.; CoronaRamírez, L.G.; ValenciaSegura, L.E. Neuronal ConstraintHandling Technique for the Optimal Synthesis of ClosedChain Mechanisms in Lower Limb Rehabilitation. Appl. Sci. 2022, 12, 2396. https://doi.org/10.3390/app12052396
MuñozReina JS, VillarrealCervantes MG, CoronaRamírez LG, ValenciaSegura LE. Neuronal ConstraintHandling Technique for the Optimal Synthesis of ClosedChain Mechanisms in Lower Limb Rehabilitation. Applied Sciences. 2022; 12(5):2396. https://doi.org/10.3390/app12052396
Chicago/Turabian StyleMuñozReina, José Saúl, Miguel Gabriel VillarrealCervantes, Leonel Germán CoronaRamírez, and Luis Ernesto ValenciaSegura. 2022. "Neuronal ConstraintHandling Technique for the Optimal Synthesis of ClosedChain Mechanisms in Lower Limb Rehabilitation" Applied Sciences 12, no. 5: 2396. https://doi.org/10.3390/app12052396