Kinematic and Dynamic Modeling of 3DOF Variable Stiffness Links Manipulator with Experimental Validation
Abstract
:1. Introduction
2. Mathematical Modelling
2.1. Kinematics
2.1.1. Forward Kinematics
2.1.2. Inverse Kinematics
2.2. Dynamics
3. Simulation
3.1. Simulation Environment
3.2. Workspace
4. Experimental Validation
4.1. Experimental Test Rig
4.2. Kinematic Model Experimental Results
4.3. Dynamic Model Experimental Results
5. Conclusions
Author Contributions
Funding
Institutional Review Board Statement
Informed Consent Statement
Data Availability Statement
Conflicts of Interest
Appendix A
References
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i | ||||
---|---|---|---|---|
1 | 90 | 0 | l1 | |
2 | 0 | l2 | 0 | |
3 | 0 | l3 | 0 |
Modeling Parameters | ||
---|---|---|
m1 = 0.77189 kg | Y1 = 0.04 m | I1yy = 1.83 × 10−3 |
m2 = 0.215 kg | Z1 = 0.0245 m | I1zz = 1.86 × 10−3 |
m3 = 0.14752 kg | X2 = 0.157 m | I2xx = 3.82 × 10−5 |
l1 = 0.04 m | Y2 = 0 | I2yy = 1.6 × 10−3 |
l2 = 0.29 m | Z3 = 0 | I2zz = 1.59 × 10−3 |
l3 = 0.240 m | X3 = 0.125 m | I3xx = 4.26 × 10−5 |
k1f = k2f = k3f = 1.66 | Y3 = 0 | I3yy = 7.45 × 10−4 |
c1f = c2f = c3f = 1.3 | Y4 = 0 | I3zz = 7.25 × 10−4 |
X1 = 0 | I1xx = 2.47 × 10−4 |
θ1 | θ2 | θ3 | Experimental (mm) | Theoretical (mm) | Error (mm) | |||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|
x | y | z | x | y | z | x | y | z | ||||
1 | 88.69 | 71.61 | −86.82 | - | 326.45 | 262.15 | - | 324.66 | 256.96 | - | 1.79 | 5.19 |
2 | 88.69 | 45.48 | −74.17 | - | 414.1 | 137.76 | - | 417.37 | 135.11 | - | 3.27 | 2.65 |
3 | 88.69 | 85.41 | −2.27 | - | 51.9 | 565.35 | - | 52.27 | 572.33 | - | 0.37 | 6.97 |
4 | 88.69 | 5.74 | −52.09 | - | 456.38 | −101.6 | - | 459.18 | −104.1 | - | 2.8 | 2.57 |
5 | 88.69 | 6.01 | −6.01 | - | 527.89 | 70.54 | - | 533.37 | 70.88 | - | 5.48 | 0.33 |
6 | 88.69 | 6.17 | −85.62 | - | 336.84 | −160.7 | - | 337.23 | −164.2 | - | 0.39 | 3.51 |
7 | 88.69 | 17.48 | −6.9 | - | 514.11 | 171.89 | - | 517.29 | 172.67 | - | 3.18 | 0.78 |
8 | 88.69 | 17.54 | −69.1 | - | 428.36 | −54.91 | - | 430.49 | −59.07 | - | 2.13 | 4.16 |
9 | 88.69 | 29.9 | −5.93 | - | 472.4 | 283.21 | - | 475.03 | 284.55 | - | 2.63 | 1.34 |
10 | 88.69 | 29.7 | −39.05 | - | 489.56 | 148.71 | - | 493.05 | 147.16 | - | 3.49 | 1.55 |
11 | 88.69 | 45 | −4.17 | - | 388.62 | 403.97 | - | 390.19 | 405.51 | - | 1.57 | 1.54 |
12 | 88.69 | 45.78 | −85.86 | - | 386.48 | 100.25 | - | 389.37 | 96.89 | - | 2.89 | 3.36 |
13 | 88.69 | 58.52 | −4.14 | - | 292.44 | 484.43 | - | 293.82 | 486.67 | - | 1.38 | 2.24 |
14 | 88.69 | 58.35 | −43.16 | - | 384.95 | 353.38 | - | 386.4 | 354 | - | 1.45 | 0.62 |
15 | 88.69 | 72.04 | −3.98 | - | 180.18 | 541.9 | - | 180.63 | 543.24 | - | 0.45 | 1.34 |
16 | 88.69 | 71.44 | −73.11 | - | 330.83 | 313.12 | - | 333.79 | 312.66 | - | 2.96 | 0.46 |
17 | 88.69 | 85.61 | −24.8 | - | 139.88 | 540.29 | - | 139.63 | 543.65 | - | 0.25 | 3.36 |
18 | 88.69 | 85.07 | −85.07 | - | 262.85 | 332.92 | - | 265.35 | 333.9 | - | 2.5 | 0.98 |
19 | 88.69 | 1.88 | 6.6 | 12.13 | 530.63 | - | 12.23 | 534.86 | - | 0.1 | 4.23 | - |
20 | 75.41 | 1.88 | 6.6 | 134.46 | 516.42 | - | 134.76 | 517.74 | - | 0.30 | 1.32 | - |
21 | 60.24 | 1.88 | 6.6 | 263.01 | 459.91 | - | 265.55 | 464.44 | - | 2.54 | 4.53 | - |
22 | 43.28 | 1.88 | 6.6 | 385.56 | 363.09 | - | 389.48 | 366.77 | - | 3.92 | 3.68 | - |
23 | 26.76 | 1.88 | 6.6 | 474.75 | 239.37 | - | 477.7 | 240.88 | - | 2.95 | 1.51 | - |
24 | 10.56 | 1.88 | 6.6 | 519.42 | 101.38 | - | 525.93 | 98.04 | - | 6.51 | 3.34 | - |
25 | −3.09 | 1.88 | 6.6 | 530.18 | −28.58 | - | 534.22 | −28.83 | - | 4.04 | 0.25 | - |
26 | 88.69 | 1.88 | 6.6 | - | 532.18 | 30.27 | 0 | 533.88 | 29.92 | 0 | 1.7 | 0.1225 |
Average error | 2.90 | 2.29 | 2.40 | |||||||||
Average percentage error | 0.81%% | 0.71% | 1.4% |
Time (sec) | Experimental (mm) | Theoretical (mm) | Error (mm) | |||||||
---|---|---|---|---|---|---|---|---|---|---|
X | y | z | x | y | z | x | y | z | ||
1 | 1.7 | 278.442 | 0 | 492.48 | 285.39 | 0 | 486.59 | 6.95 | 0 | 5.88 |
2 | 1.73 | 281.2061 | 0 | 491.55 | 295.12 | 0 | 480.22 | 13.91 | 0 | 11.33 |
3 | 1.76 | 305.3148 | 0 | 474.64 | 312.81 | 0 | 467.84 | 7.49 | 0 | 6.8 |
4 | 1.8 | 320.0337 | 0 | 464.22 | 323.09 | 0 | 460.13 | 3.06 | 0 | 4.09 |
5 | 1.83 | 330.6901 | 0 | 455.15 | 331.74 | 0 | 453.33 | 1.05 | 0 | 1.82 |
6 | 1.86 | 344.3517 | 0 | 445.08 | 348.3 | 0 | 439.47 | 3.95 | 0 | 5.61 |
7 | 1.9 | 354.0388 | 0 | 437.74 | 358.58 | 0 | 430.27 | 4.54 | 0 | 7.46 |
8 | 1.93 | 365.542 | 0 | 425.37 | 371.04 | 0 | 418.45 | 5.5 | 0 | 6.91 |
9 | 1.96 | 375.4381 | 0 | 416.44 | 376.71 | 0 | 412.8 | 1.27 | 0 | 3.63 |
10 | 2 | 386.3407 | 0 | 405.04 | 388.09 | 0 | 400.94 | 1.75 | 0 | 4.1 |
11 | 4 | 510.1 | 0 | 163.63 | 515.59 | 0 | 159.16 | 5.49 | 0 | 4.47 |
12 | 4.13 | 519.23 | 0 | 155.47 | 516.45 | 0 | 153.69 | 2.78 | 0 | 1.77 |
13 | 4.16 | 513.53 | 0 | 153.83 | 517.52 | 0 | 145.14 | 3.99 | 0 | 8.68 |
14 | 4.2 | 517.64 | 0 | 152.18 | 518.09 | 0 | 139.2 | 0.45 | 0 | 12.97 |
15 | 4.23 | 515.41 | 0 | 143.87 | 518.25 | 0 | 137.1 | 2.83 | 0 | 6.76 |
16 | 4.26 | 514.45 | 0 | 138.89 | 518.36 | 0 | 135.56 | 3.91 | 0 | 3.32 |
17 | 4.3 | 520.68 | 0 | 137.01 | 518.56 | 0 | 132.18 | 2.11 | 0 | 4.82 |
18 | 4.36 | 518.12 | 0 | 123.38 | 518.82 | 0 | 125.77 | 0.7 | 0 | 2.39 |
19 | 4.4 | 510.1 | 0 | 163.63 | 515.59 | 0 | 159.1 | 5.49 | 0 | 4.47 |
20 | 7.13 | 484.12 | 0 | 2.81 | 488.66 | 0 | 2.81 | 4.53 | 0 | 0 |
21 | 7.16 | 477.51 | −12 | 2.81 | 488.52 | −11.7 | 2.81 | 11 | 0.29 | 0 |
22 | 7.20 | 483.1 | −27 | 2.81 | 487.76 | −29.7 | 2.81 | 4.65 | 2.65 | 0 |
23 | 7.23 | 491 | −40.8 | 2.81 | 486.91 | −41.3 | 2.81 | 4.18 | 0.49 | 0 |
24 | 7.26 | 485.8 | −50.1 | 2.81 | 485.81 | −52.7 | 2.81 | 0.05 | 2.59 | 0 |
25 | 7.30 | 481.83 | −63.1 | 2.81 | 483.54 | −70.5 | 2.81 | 1.71 | 7.39 | 0 |
26 | 7.33 | 485.17 | −83.1 | 2.81 | 491.11 | −80.5 | 2.81 | 5.93 | 2.65 | 0 |
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Ramzy, M.R.; Hammad, S.; Maged, S.A. Kinematic and Dynamic Modeling of 3DOF Variable Stiffness Links Manipulator with Experimental Validation. Appl. Sci. 2024, 14, 5285. https://doi.org/10.3390/app14125285
Ramzy MR, Hammad S, Maged SA. Kinematic and Dynamic Modeling of 3DOF Variable Stiffness Links Manipulator with Experimental Validation. Applied Sciences. 2024; 14(12):5285. https://doi.org/10.3390/app14125285
Chicago/Turabian StyleRamzy, Mina R., Sherif Hammad, and Shady A. Maged. 2024. "Kinematic and Dynamic Modeling of 3DOF Variable Stiffness Links Manipulator with Experimental Validation" Applied Sciences 14, no. 12: 5285. https://doi.org/10.3390/app14125285