3.6 Stability analysis of the mechanism
3.6.2 Modification of design parameters
For fixed design parameters, it was found that the mechanism remained unstable under natural conditions. In order to determine stable configurations, it is necessary to study the derivatives of the total potential energy against the pose variables. As the mechanism has dependencies on the two pose variables, the following steps are carried out:
• Generation of the Hessian matrix or the mechanism stiffness
Stability analysis of the mechanism 101
• Estimation of the determinant value of the Hessian matrix
• Estimation of the value of the second-order derivative of total potential energy with respect to one of the tilt angles
The Hessian matrix and its determinant are given by:
Hi=
∂2Uj
∂η2
∂2Uj
∂η∂ϕ
∂2Uj
∂η∂ϕ
∂2Uj
∂ϕ2
=
f11 f12
f21 f22
(3.55)
det(Hi) =f11f22−f122 (3.56)
where fori= [1,2], j = [3-SPS-U , 4-SPS-U]
As the total potential energy is a function of the two pose variables, the following conditions are possible [Daw03] :
1. The total potential energy has a relative maximum when det(Hi) > 0 and f11 < 0 2. The total potential energy has a relative minimum when det(Hi) > 0 and f11 > 0
From Figure 3.20, the first condition could be observed for both mechanisms as there exists a relative maximum withf11< 0. As the mechanism will be integrated along with the bio-inspired robot of Chapter-2, the value of rf is retained as 11 mm [VCB19]. When the mechanism is coupled with the bio-inspired robot, there exists a preload on the mechanism due to the presence of motor modules and its cables. The existing prototype weighs around 6.5 N [CVB18].
However, a design modification will be done on the robot to arrive at a compact design with reduced motor sizing for overcoming pipe bends in Chapter-5. Thus, a preloading of 2 N and 1.5 N along each spring for the 3-SPS-U and 4-SPS-U mechanisms are considered for the analysis.
For determining optimal values of h and spring stiffness k under the presence of a preload, a simplified optimization problem using Genetic algorithm is carried out in MATLAB [CF95]. A stochastic approach is being followed for extracting multiple solutions that could be distributed over a domain. The optimization problem is stated as:
Maximize: f11(x)
subject to constraints: g1: det(H) ≥0 g2: f11(x) ≥0 wherex= [h, k]T
The objective function aims to maximize the derivative f11 such that it remains positive along with the determinant of the Hessian matrix throughout the optimization process. The equations for the objective function and the determinant value of the Hessian matrix for both mechanisms are given by:
f11(i) = 11j(11k−22h2k−Flh)
2 (3.57)
det(Hi) = 12j2(Flh+ 22h2−11k)2
4 (3.58)
where for i= [1,2] , j = [3,4] for 3-SPS-U & 4-SPS-U also for i= 1, F1=F2 =F3=Fl= 2 N for 3-SPS-U
and for i= 2, F1=F2 =F3=F4 =Fl= 1.5 N for 4-SPS-U
The lower and upper bounds for h are set between 0.2 to 0.8 while for the spring these values are set between 0.1 to 1. By using Eqs. (3.57) to (3.58) and the bounds, the optimization problem is solved using the function gain MATLAB [CF95]. The results of solutions obtained from the optimization problem is represented below in Figure 3.21.
0.2 0.4 0.6 0.8 1
h 0.1
0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
k (N/mm)
0.2 0.4 0.6 0.8 1
h 0.1
0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
k (N/mm)
(a) (b)
Figure 3.21 – Solutions for optimization problem obtained by Genetic Algorithm for the (a) 3-SPS-U and (b) 4-SPS-U mechanisms
In Figures 3.21a and 3.21b, the solutions obtained by Genetic algorithm in MATLAB are depicted as black colored scatter points over the contour of the objective function. The values of h andk for Figures 3.21a and 3.21b are provided in Tables 3.6 and 3.7. It could be observed that optimum values of h are distributed between 0.6 to 0.7. For each value of h, an optimum value for spring stiffness k is obtained. From the results of Figure 3.21 and Tables 3.6 and 3.7, the value of k that matches with springs available at LS2N is 0.75 N/mm. The rows having values of k closer to 0.75 N/mm are highlighted in green in Tables 3.6 and 3.7. For this value of k, Eq. (3.57) is differentiated with respect to h and equated to zero under external forces of 2 N and 1.5 N for the 3-SPS-U and 4-SPS-U mechanisms respectively. The plot of parameterh versus f11 for both mechanisms atk=0.75 N/mm is represented in Figure 3.22.
Stabilityanalysisofthemechanism103 Parameterh Spring stiffnessk(N/mm) (N.mm/rad)f11 det(H)
0.628 0.531 0.366 0.134
0.612 0.433 0.477 0.227
0.549 0.245 0.455 0.207
0.655 0.820 0.489 0.239
0.613 0.438 0.477 0.227
0.636 0.589 0.569 0.323
0.632 0.559 0.447 0.200
0.650 0.742 0.576 0.331
0.546 0.239 0.503 0.253
0.643 0.656 0.609 0.371
0.627 0.520 0.518 0.268
0.636 0.589 0.569 0.323
0.657 0.862 0.294 0.086
0.654 0.804 0.486 0.236
0.646 0.698 0.368 0.135
0.567 0.282 0.437 0.191
0.654 0.806 0.433 0.188
0.641 0.635 0.611 0.373
0.645 0.685 0.404 0.163
0.603 0.393 0.442 0.195
Table 3.6 – Optimum values ofk andh for the 3-SPS-U mechanism from Figure 3.21a
Parameterh Spring stiffnessk(N/mm) (N.mm/rad)f11 det(H)
0.664 0.749 0.486 0.236
0.663 0.737 0.323 0.104
0.621 0.360 0.567 0.322
0.663 0.735 0.381 0.145
0.669 0.859 0.275 0.076
0.618 0.348 0.506 0.256
0.646 0.524 0.348 0.121
0.663 0.724 0.703 0.494
0.649 0.550 0.441 0.194
0.656 0.629 0.440 0.193
0.657 0.639 0.542 0.293
0.660 0.679 0.665 0.379
0.665 0.761 0.293 0.442
0.663 0.738 0.323 0.086
0.665 0.773 0.363 0.109
0.641 0.482 0.245 0.131
0.667 0.816 0.560 0.060
0.658 0.652 0.556 0.313
0.654 0.601 0.533 0.309
0.635 0.436 0.496 0.284
Table 3.7 – Optimum values ofkand h for the 4-SPS-U mechanism from Figure 3.21b
0 0.2 0.4 0.6 0.8 1 h
-200 -150 -100 -50 0 50 100 150
f 11 (N.mm/rad)
f11 vs h h= 0.649 at f
11 = 0
0 0.2 0.4 0.6 0.8 1
h -250
-200 -150 -100 -50 0 50 100 150 200
f 11 (N.mm/rad)
f11 vs h h= 0.663 at f
11 = 0
(a) (b)
F1 = F
2 = F
3 = 2 N F
1 = F
2 = F
3 = F
4 = 1.5 N
Figure 3.22 – Optimum value ofh at spring stiffness of 0.75 N/mm for the (a) 3-SPS-U and (b) 4-SPS-U mechanisms under the presence of external forces
From Figures 3.22a and 3.22b, a value of 0.649 and 0.663 for h is obtained for the 3-SPS-U and 4-SPS-U mechanisms wheref11is zero. For ease of calculations and prototyping, a common value ofh = 0.6 is taken for both the mechanisms. The stability plot for both the mechanisms with the modified design parameters under the presence of a preload is represented below in Figure 3.23.
Figure 3.23 – Plot of total potential energy versus the tilt angles η and ϕ for h = 0.6 at the home-pose condition for the (a) 3-SPS-U and (b) 4-SPS-U mechanisms
Stability analysis of the mechanism 105 It could be observed from Figures 3.23a and 3.23b that a stable configuration has been obtained for both mechanisms under the presence of a preload. Using standard parts available at LS2N, the 3-SPS-U tensegrity mechanism is realized. A standard universal joint of 68 mm length and springs of 0.75 N/mm stiffness are employed to assemble the tensegrity mechanism.
The base and end-effector are realized by 3D printing. Two prototypes with h= 1 and h= 0.6 are realized where the corresponding values of rf are 34 mm and 56.7 mm. The prototypes are represented below in Figure 3.24.
(a) (b)
Figure 3.24 – Prototypes of the 3-SPS-U tensegrity mechanism realized at LS2N with (a)h = 1 and (b)h=0.6
From Figure 3.24a, it could be observed that under natural conditions, the mechanism re-mains unstable when h= 1. Upon the design parameter modification of h from 1 to 0.6, a stable configuration could be observed in Figure 3.24b. The stability analysis was carried out by fixing the free lengths as 0 mm. However, upon considering a value of 35 mm as free length for the spring employed in the prototypes depicted in Figure 3.24, the mechanism still remained unstable at h=1 upon numerical analysis, which is not presented here. At the home-pose and under the absence of any external forces, the stability of the mechanism purely depends on the parameterhand the stiffness of the tension spring. The influence of parameterhplays an essen-tial role to determine the overall stability of the mechanism. Since the design parameter h has been modified, the singularity analysis is carried out again to identify the maximum tilt limits for the mechanism. Experimental validation of the mechanism represented in Figure 3.24b will be carried out in Chapter-4 to demonstrate the actuation strategies.