A general spatial multi-loop linkage optimization model for motion generation with static loading

Figures & data

Figure 1. RSSR-SS linkage (left) with joint descriptions and (right) with displacement variables.

Figure 2. RSSR-SS (a) $\theta -{\mathbf{u}}_{{\mathbf{a}}_0}$, (b) $\omega -\mathbf{w}$ and (c) $\gamma -{\mathbf{u}}_{\mathbf{a}}$ coupler rotations.

Figure 3. (a) Four-bar linkage assembly configurations and (b) branch defect.

Figure 4. Order difference with four precision points (orders (a) 1-2-3-4 and (b) 1-2-4-3).

Figure 5. Four-bar linkage with circuit defect.

Table 1. Equation, Gradient and Hessian File Sizes (in MB for $N=2$ coupler positions) in Matlab.

Eq. No.	$f\left(\mathbf{X}\right)$	$\mathrm{\nabla }f$	$\mathbf{H}\left(f\right)$
15 (for 2 pos.)	2.564	847.304	*18954
16	0.001	0.001	0.004
17	0.001	0.001	0.004
18	0.001	0.015	0.073
19	0.001	0.017	0.080
20	0.001	0.031	0.404
21	0.001	0.034	0.440
22	0.002	0.054	1.208
23	0.855	257.754	*5766
24	0.854	257.725	*5765
25	0.855	257.725	*5765
26	0.003	0.113	1.901
27	1.172	542.304	*9123
28	37.007	*12229	*205728
sum	43.318	*14390	*251103

*estimated minimum file size

Figure 6. Precision positions and precision position force.

Table 2. Precision position coordinates.

Pos. #	${\mathbf{p}}^{\ast }$ [m]	${\mathbf{q}}^{\ast }$ [m]	${\mathbf{r}}^{\ast }$ [m]
1	−0.7500,−0.7500,−0.5000	0.2159,−1.0088,−0.5000	−0.5154, 0.1254, −0.0774
2	0.3750, 0.3750, 0.2500	1.3664, 0.5055, 0.2500	0.2476, 1.3429, 0.0336
3	1.5000, 1.5000, 1.0000	2.3660, 2.0000, 1.0000	1.2132, 1.9967, 0.1808

Table 3. Initial and calculated RSSR-SS linkage variable values.

variable	initial values	calculated values
a0	−3, −3, 3	−1.0140. −1.6574, 3.3824
a1	−3, −3, 3	−1.1595, −1.3124, 0.3007
$\mathbf{u}{}_{{\mathbf{a}}_0}$	−0.5, 0.5, 0.5	−0.8087, 0.5792, 0.1030
b0	5, −5, 5	2.2400, −1.6886, 2.3063
b1	1, −1, 1	0.7723, −1.8300, 0.3007
$\mathbf{u}{}_{{\mathbf{b}}_0}$	−0.5, 0.5, 0.5	−0.7956, 0.2126, 0.5672
c0	1, 5, 5	0.3312, 1.5657, 3.7417
c1	−1, 1, 1	−0.6903, 0.4384, 1.1459
θ1–2, φ1–2, γ1–2	−50°, −50°, 100°	−43.4329°,−71.9654°, 75.5734°
Number of Iterations: 239 Elapsed Time: 9262 s

Table 4. Coupler position coordinates achieved by the synthesized RSSR-SS linkage.

Pos. #	p [m]	q [m]	r [m]
1	−0.7500,−0.7500,−0.5000	0.2159, −1.0088,−0.5000	−0.5154, 0.1254, −0.0774
2	0.3777, 0.3781, 0.2484	1.3693, 0.5079, 0.2500	0.2514, 1.3464, 0.0330

Figure 7. Synthesized RSSR-SS linkage at (top) initial and (bottom) final achieved coupler positions.

Figure 8. Driven link displacement angles (versus driving link rotation) for synthesized linkage.

Figure 9. Synthesized RSSR-SS linkage at (top) initial and (bottom) final achieved coupler positions.

Figure 10. Driven link displacement angles (versus driving link rotation) for synthesized linkage.

Table 5. Initial and calculated RSSR-SS linkage variable values.

variable	initial values	calculated values
a0	−3, −3, 3	3.6571, −1.8056, 6.2335
a1	4, −4, 1	0.7723, −1.8300, 0.3007
$\mathbf{u}{}_{{\mathbf{a}}_0}$	0.5, 0.5, 0.5	0.8781, −0.2178, −0.4261
b0	−2, 0.5, 2	−3.3603, 0.3239, 5.6743
b1	−5, −5, 0.5	−1.1595, −1.3124, 0.3007
$\mathbf{u}{}_{{\mathbf{b}}_0}$	0.5, 0.5, 0.5	0.6435, −0.6179, 0.4517
c0	−2, 2, 2	−0.0911,3.7265,5.7921
c1	−4, 4, 1	−0.6903, 0.4384, 1.1459
θ1–2, φ1–2, γ1–2	30°, 50°, 70°	47.9849°, 40.3923°, 127.6369°
Number of Iterations: 261 Elapsed Time: 10121 s

Table 6. Coupler position coordinates achieved by the synthesized RSSR-SS linkage.

Pos. #	p [m]	q [m]	r [m]
1	−0.7500,−0.7500,−0.5000	0.2159,−1.0088,−0.5000	−0.5154, 0.1254, −0.0774
3	1.5009, 1.5007, 1.0032	2.3671, 2.0004, 1.0028	1.2147, 1.9961, 0.1830