Optimal manipulator parameter selection using evolutionary optimization technique

Figures & data

Figure 1. Flow diagram for the proposed evolutionary approach.

Figure 2. A 2-DOF RR planar manipulator and its parameters.

Table 1. Manipulator task specifications for the deterministic case.

Trajectory	Coordinates of start point (xim,yim)	Coordinates of destination point (xfm,yfm)	Time to travel (s)
Cubic	(0.65, 0)	(0.4, 0.3)	2
Quintic	(0.65, 0)	(0.4, 0.3)	2

Figure 3. Function history of deterministic case for cubic trajectory.

Figure 4. Function history of deterministic case for quintic trajectory.

Table 2. Manipulator task specifications for case with effect of noise.

Case	Coordinates of start point (xim,yim)	Coordinates of destination point (xfm,yfm)	Time to travel (s)
(i)	(0.65, 0)	(0.4, 0.3)	2
(ii)	(0.65, 0.05)	(0.4, 0.3)	2
(iii)	(0.65, 0.1)	(0.4, 0.3)	2
(iv)	(0.65, 0.05)	(−0.4, 0.3)	2
(v)	(0.65, 0.1)	(−0.4, 0.3)	2
(vi)	(0.4, 0.3)	(0.65, 0)	2

Table 3. Parameter constraints for manipulator.

Link 1 length l1 range (m)	Link 2 length l2 range (m)	Link 1 mass m1 range (kg)	Link 2 mass m2 range (kg)

Table 4. Optimal parameters of manipulator for the deterministic case.

Trajectory	Link 1 length l1 (m)	Link 2 length l2 (m)	Link 1 mass m1 (kg)	Link 2 mass m2 (kg)	Function value × 10^− 2(m)
Cubic	0.4502313	0.44993491	6.255106	7.985928	0.092029
Quintic	0.54994198	0.41548853	9.975794	4.014191	0.010061

Table 5. L₈ Orthogonal array.

Experiment no.	Column number
1	2	3	4	5	6	7
1	1	1	1	1	1	1	1
2	1	1	1	2	2	2	2
3	1	2	2	1	1	2	2
4	1	2	2	2	2	1	1
5	2	1	2	1	2	1	2
6	2	1	2	2	1	2	1
7	2	2	1	1	2	2	1
8	2	2	1	2	1	1	2

Table 6. Noise for design and process parameters.

Parameters	Tolerances
Lengths of link l1 and l2 (m)	±0.0003
Mass of link m1 and m2 (kg)	±0.015
Torque at joint τ1 and τ2 (Nm)	±0.05

Table 7. Assumed tolerances for control factors.

Parameters	Tolerances
Link lengths l1, l2 (m)	±0.0003
Link masses m1, m2 (kg)	±0.015

Table 8. Assumed population member values sent by DE.

l1 (m)	l2 (m)	m1 (kg)	m2 (kg)
0.425	0.375	6	4

Table 9. Combinations created from the population member values using OA.

Experiment no.	Parameters
l1 (m)	l2 (m)	m1 (kg)	m2 (kg)
1	0.4247	0.3747	5.985	3.985
2	0.4247	0.3747	5.985	4.015
3	0.4247	0.3753	6.015	3.985
4	0.4247	0.3753	6.015	4.015
5	0.4253	0.3747	6.015	4.985
6	0.4253	0.3747	6.015	4.015
7	0.4253	0.3753	5.985	3.985
8	0.4253	0.3753	5.985	4.015

Table 10. Optimal parameters for cubic trajectory with DE.

Case	Link 1 length l1 (m)	Link 2 length l2 (m)	Link 1 mass m1 (kg)	Link 2 mass m2 (kg)	Mean positional error × 10^− 2 (m)	Number of function evaluations	Time for 50 generations (s)
(i)	0.45115423	0.44025207	6.585616	7.886889	0.130762	698	146580
(ii)	0.4516324	0.44804167	7.843923	7.99053	0.120789	796	167160
(iii)	0.45044698	0.35094645	6.191169	7.998867	0.231646	812	170520
(iv)	0.45021022	0.35036626	6.257418	7.95036	0.236336	728	152880
(v)	0.45713918	0.44867156	6.727582	7.969148	0.111177	764	160440
(vi)	0.45083597	0.44943789	6.037671	7.714803	0.125251	735	154350

Table 11. Optimal parameters for quintic trajectory with DE.

Case	Link 1 length l1 (m)	Link 2 length l2 (m)	Link 1 mass m1 (kg)	Link 2 mass m2 (kg)	Mean positional error × 10^− 2 (m)	Number of function evaluations	Time for 50 generations (s)
(i)	0.53526522	0.44429454	9.728534	7.9974	0.120093	694	145740
(ii)	0.53052684	0.44841063	9.185676	7.976033	0.101264	812	170520
(iii)	0.52195604	0.44669770	9.498453	7.947742	0.097292	812	170520
(iv)	0.45139675	0.44965546	9.898766	7.609616	0.252178	726	152460
(v)	0.45054925	0.44976276	9.920583	7.85658	0.218845	754	158340
(vi)	0.45159362	0.44882879	9.955927	7.763131	0.054702	755	158550

Figure 5. Function history for cubic trajectory with DE.

Figure 6. Function history for quintic trajectory with DE.

Figure 7. Function history for cubic trajectory with GA.

Table 12. Optimal parameters for cubic trajectory from GA.

Case	Link 1 length l1 (m)	Link 2 length l2 (m)	Link 1 mass m1 (kg)	Link 2 mass m2 (kg)	Mean positional error × 10^− 2 (m)	Number of function Evaluations	Time for 50 generations (s)
(i)	0.50090856	0.44894164	7.706868	7.627576	0.298985	3030	636300
(ii)	0.47413563	0.44741994	9.953008	7.720614	0.299325	3030	636300
(iii)	0.54346479	0.43160096	6.322149	7.602735	0.309823	3030	636300
(iv)	0.45234867	0.36078641	7.03435	7.697754	0.249708	3030	636300
(v)	0.53186493	0.43093319	9.415876	7.70108	0.303738	3030	636300
(vi)	0.52985308	0.42991547	7.672331	4.650942	0.273784	3030	636300

Table 13. Optimal parameters for quintic trajectory from GA.

Case	Link 1 length l1 (m)	Link 2 length l2 (m)	Link 1 mass m1 (kg)	Link 2 mass m2 (kg)	Mean positional error × 10^− 2 (m)	Number of function evaluations	Time for 50 generations (s)
(i)	0.53524039	0.42170688	7.675625	7.908822	0.268773	3030	636300
(ii)	0.50382344	0.41220353	7.517503	4.441751	0.277632	3030	636300
(iii)	0.54346479	0.43160096	6.322149	7.602735	0.289652	3030	636300
(iv)	0.53005154	0.42992842	6.672026	6.639223	0.27548	3030	636300
(v)	0.52977983	0.35500796	7.660979	4.705873	0.258589	3030	636300
(vi)	0.51232633	0.38042798	7.277755	5.522419	0.269944	3030	636300

Figure 8. Function history for quintic trajectory with GA.