Sensitivity Analysis, Synthesis and Gait Classification of Reconfigurable Klann Legged Mechanism
Abstract
:1. Introduction
 Sensitivity analysis: Conduct a detailed analysis of the sensitivity of the reconfigurable Klann legged mechanism to variations in the geometric parameters using Sobol’s method. Towards this end, the kinematics of the KLM is also presented.
 Reconfiguration strategy: Studying the sensitivity of the Klann mechanism model makes it possible to reconfigure using a minimal number of actuators, i.e, using only two rotary actuators, five types of gait are achieved.
 Gait classification: Implement machine learning approaches, including K Nearest Neighbors (KNNs), Decision Tree, and Random Forest algorithms, for the classification of generated gaits based on altered positions of the lower rocker and the angle of the end effector.
 Experimental validation: Validate the proposed reconfigurable Klann legged mechanism through experimental assembly, ensuring practical applicability and performance verification.
2. Kinematics and Gaits of KLM
2.1. Position Analysis
2.2. Continuous and Discontinuous Gaits
3. Sensitivity Analysis for Reconfigurable KLM
3.1. Parametric Sensitivity Analysis
3.2. Computation of Total Sensitivity Index Using Monte Carlo Technique
Algorithm 1 Algorithm to find the Total Sensitivity Index (TSI) of parameters. 

3.3. Illustration with the Reconfigurable KLM
 Changing link lengths: This straightforward approach involves adjusting the lengths of the links, except the crank and end effector, using linear actuators. This method was studied in previous works and resulted in six useful gait patterns.
 Changing coupler link angles: The coupler is a key component that connects the input and output links. Adjusting its angle alters the foot’s trajectory.
 Changing grounded joints hinge positions: This novel approach proposes changing the position of the revolute joint connecting the crank, lower, and upper rocker to the fixed base, affecting the coupler joint’s range of motion and the end effector’s overall movement.
 Crank length (${a}_{1}$): The standard crank length considered is 11 cm, and it is varied from 3 to 14 cm. The range in which the crank length can be varied is ideally from ${a}_{1}\to 0$ to 14, so with an interval of 14 cm. The variation observed in the gait trajectory is that of scale, where both the stride length and height are increasing with the increase in the crank length as shown in Figure 5a.
 Lower rocker length (${a}_{2}$): The standard rocker length is considered 13 cm, and it is varied from 12 to 23 cm with a step of 1 cm as shown in Figure 5b. An increase in the length of ${a}_{2}$ leads to an increase in the height and a decrease in stride and vice versa. The step heights remain approximately the same. The slope of the return stroke also changes considerably.
 Upper rocker length (${a}_{6}$): The standard upper rocker length is kept as 17, and the length is varied from 14 to 35 cm. The decrease in length below 14 results in a discontinuous gait while the increase in length is allowed to vary. Figure 5c shows the variation due to the change in the length of ${a}_{6}$.
 Coupler length (${a}_{3},{a}_{7}$): The effect of the change in the length parameter of CD and DL or effectively CL is that the stride height effectively remains the same, whereas the stride length increases. The variation is also allowed and checked with the increase and decrease in the length by 10%.
 Base length (${a}_{4}$): The standard length of ${a}_{4}$ is considered 29. An increase in its dimension leads to an elongation of the stride length and a reduction in step height and vice versa. This suggests that the changes approximately maintain the area under the curve by the generated gaits. The range to vary this length is also lesser, i.e., −28 to 30.4 as shown in Figure 5e.
 Leg angle (${\u03f5}_{68}$): The angle in between for the leg, i.e., angle ELM, is varied and the effect is a change in the orientation of the gait, which as a result leads to a decrease in the effective length of the contact point with the ground. The standard value is kept as 160 degrees, and it is varied between 140 and 170 degrees.
 Select the number of input geometric parameters for sensitivity analysis.
 Define the range for perturbation values associated with these parameters. Gaussian distribution was utilized to allocate perturbations to each parameter. Intervals were specified as (${b}_{i}\pm \delta b$) and (${\varphi}_{i}\pm \delta \varphi $) for length parameters and angular parameters.
 Sensitivity analyses were conducted for the selected parameters, focusing on their impact on the robot’s outputs, specifically its position and orientation.
 The sensitivity indices for these parameters were subsequently determined using Algorithm 1.
3.4. Discussion on Aspects of Sobol’s Method
4. Classification of Gaits Using Machine Learning Approaches
5. Experimental Studies with the Proposed Design of Reconfigurable KLM
5.1. Single Reconfigurable KLM Testbed
5.2. Identification of the Geometric Parameters Using Computer Vision
 Image preprocessing: converts the image to grayscale, captured using the camera as shown in the experimental setup in Figure 9a.
 Joint identification: applying masking to identify joints based on their color. Red Mask with (a) circle in the quadrant (${Q}_{2}$ or ${Q}_{3}$) is Joint 2 (j2), (b) circle in the quadrant (${Q}_{1}$ or ${Q}_{4}$) with the least Xcoordinate is Joint 4 (J4), (c) circle in the quadrant (${Q}_{1}$) with the least Ycoordinate is Joint 8 (J8), while the remaining one is Joint 7 as in Figure 9b. Using a green mask, (a) if the circle is in quadrant (${Q}_{1}$), then it is Joint 5; (b) if the circle is in quadrant (${Q}_{2}$), it is Joint 1 (J1), and if the circle is in quadrant (${Q}_{4}$), then it is Joint 3 (J3). The gait trace, i.e., the end of the leg movement, is traced using the blue mask. Following joint identification, the algorithm calculates the Euclidean distances between points and normalizes them based on the crank length ratio. The experiment, conducted with a physical model by varying the crank rotation angles, hinge position of joint J3 and $\u03f5$ variations, yields the results presented in Table 2. The code for machine vision estimation is also shared in the online folder with the link (Link for the codes and data set used: https://drive.google.com/drive/folders/1U8lTBJXoNhxRnVc7QtoswXQwm1mRSVy4?usp=sharing, accessed on 25 January 2024).
5.3. CAD Assembly and Scaled Reconfigurable KLM
6. Conclusions
Author Contributions
Funding
Institutional Review Board Statement
Informed Consent Statement
Data Availability Statement
Conflicts of Interest
Correction Statement
Appendix A
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Class  Precision  Recall  F1Score  

KNN  DT  RF  KNN  DT  RF  KNN  DT  RF  
Hammering  0.90  0.90  0.92  0.96  0.78  0.92  0.93  0.84  0.92 
Digging  1.00  0.74  0.97  0.95  1.00  0.95  0.97  0.85  0.96 
Jam Avoidance  0.87  0.89  0.86  0.92  0.82  0.92  0.89  0.86  0.89 
Standard  0.85  0.90  0.90  0.79  0.58  0.85  0.81  0.70  0.88 
Step Climbing  0.98  0.93  0.97  0.97  0.99  0.97  0.97  0.96  0.97 
Symbolic Ratio  Actual Ratio  Average Ratio  Standard Deviation 

${a}_{5}/{a}_{1}$  1.54  1.55  0.07 
${a}_{6}/{a}_{1}$  2.36  2.38  0.18 
${a}_{3}/{a}_{1}$  2.54  2.55  0.18 
${a}_{5}/{a}_{1}$  2.54  2.55  0.11 
${a}_{7}/{a}_{1}$  2.09  2.06  0.16 
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Hayat, A.A.; Megalingam, R.K.; Kumar, D.V.; Rudravaram, G.; Nansai, S.; Elara, M.R. Sensitivity Analysis, Synthesis and Gait Classification of Reconfigurable Klann Legged Mechanism. Mathematics 2024, 12, 431. https://doi.org/10.3390/math12030431
Hayat AA, Megalingam RK, Kumar DV, Rudravaram G, Nansai S, Elara MR. Sensitivity Analysis, Synthesis and Gait Classification of Reconfigurable Klann Legged Mechanism. Mathematics. 2024; 12(3):431. https://doi.org/10.3390/math12030431
Chicago/Turabian StyleHayat, Abdullah Aamir, Rajesh Kannan Megalingam, Devisetty Vijay Kumar, Gaurav Rudravaram, Shunsuke Nansai, and Mohan Rajesh Elara. 2024. "Sensitivity Analysis, Synthesis and Gait Classification of Reconfigurable Klann Legged Mechanism" Mathematics 12, no. 3: 431. https://doi.org/10.3390/math12030431