Modeling and Control of a Soft Robotic Arm Based on a Fractional Order Control Approach
<p>Soft robotic arm platform.</p> "> Figure 2
<p>Comparing panels (<b>a</b>,<b>b</b>), the maximum curvatures in the different directions of the soft arm show that angle <math display="inline"><semantics> <msub> <mi>θ</mi> <mn>2</mn> </msub> </semantics></math> is greater than angle <math display="inline"><semantics> <msub> <mi>θ</mi> <mn>1</mn> </msub> </semantics></math>. (<b>a</b>) Maximum curvature of the soft arm, angle <math display="inline"><semantics> <msub> <mi>θ</mi> <mn>1</mn> </msub> </semantics></math>, when bending in the direction of the vertices. (<b>b</b>) Maximum curvature of the soft arm, angle <math display="inline"><semantics> <msub> <mi>θ</mi> <mn>2</mn> </msub> </semantics></math>, when bending in the direction of the edges.</p> "> Figure 3
<p>Description of tendon distribution in the soft arm.</p> "> Figure 4
<p>The panels show the flexion of the soft robotic arm (<b>a</b>) from a lateral view in positive and negative pitch flexion and (<b>b</b>) from a top view in positive and negative yaw flexion. For a negative pitch bending, panel (<b>a</b>) shows how the vertices will touch, preventing further bending in that direction. For a positive pitch flexion, the edges need more bending for this contact to occur. The yaw flexion in panel (<b>b</b>) has symmetry in the positive and negative directions. The combination of pitch and yaw rotations allows movements with two degrees of freedom, and the limits of flexion vary according to the combined rotations.</p> "> Figure 5
<p>Identification data using motor position input and pitch output. Step references for <math display="inline"><semantics> <mi>α</mi> </semantics></math> within the range [−50,50] in steps of 10 degrees with a random <math display="inline"><semantics> <mrow> <mo>±</mo> <mn>1</mn> </mrow> </semantics></math> noise and with constant <math display="inline"><semantics> <mrow> <mi>β</mi> <mo>=</mo> <mn>0</mn> </mrow> </semantics></math> have been used. It can be seen that the pitch position reached in time is different for positive and negative bending.</p> "> Figure 6
<p>Identification data using motor velocity input and pitch (<b>left</b>) and yaw (<b>right</b>) output.</p> "> Figure 7
<p>Bode plots of the open loop systems with the resulting PI controllers. Control specifications: <math display="inline"><semantics> <mrow> <msub> <mi>ϕ</mi> <mi>m</mi> </msub> <mo>=</mo> <msup> <mn>60</mn> <mo>∘</mo> </msup> </mrow> </semantics></math> and <math display="inline"><semantics> <mrow> <msub> <mi>ω</mi> <mrow> <mi>g</mi> <mi>c</mi> </mrow> </msub> <mo>=</mo> <mn>1.5</mn> </mrow> </semantics></math> rad/s. (<b>a</b>) Bode plots of the open loop system with the PI controller corresponding to the pitch system. (<b>b</b>) Bode plots of the open loop system with the PI controller corresponding to the yaw system.</p> "> Figure 8
<p>Bode plots of the open loop systems with the resulting FOPI controllers. Control specifications: <math display="inline"><semantics> <mrow> <msub> <mi>ϕ</mi> <mi>m</mi> </msub> <mo>=</mo> <msup> <mn>60</mn> <mo>∘</mo> </msup> </mrow> </semantics></math> and <math display="inline"><semantics> <mrow> <msub> <mi>ω</mi> <mrow> <mi>g</mi> <mi>c</mi> </mrow> </msub> <mo>=</mo> <mn>1.5</mn> </mrow> </semantics></math> rad/s. (<b>a</b>) Bode plots of the open loop system with the FOPI controller corresponding to the pitch system. (<b>b</b>) Bode plots of the open loop system with the FOPI controller corresponding to the yaw system.</p> "> Figure 9
<p>System response with the PI controller for a pitch step reference of <math display="inline"><semantics> <mrow> <mi>α</mi> <mo>=</mo> <msup> <mn>40</mn> <mo>∘</mo> </msup> </mrow> </semantics></math> and a constant yaw angle of <math display="inline"><semantics> <mrow> <mi>β</mi> <mo>=</mo> <msup> <mn>0</mn> <mo>∘</mo> </msup> </mrow> </semantics></math>. Control specifications: <math display="inline"><semantics> <mrow> <msub> <mi>ϕ</mi> <mi>m</mi> </msub> <mo>=</mo> <msup> <mn>60</mn> <mo>∘</mo> </msup> </mrow> </semantics></math> and <math display="inline"><semantics> <mrow> <msub> <mi>ω</mi> <mrow> <mi>g</mi> <mi>c</mi> </mrow> </msub> <mo>=</mo> <mn>1.5</mn> </mrow> </semantics></math> rad/s.</p> "> Figure 10
<p>System response with the FOPI controller for a pitch step reference of <math display="inline"><semantics> <mrow> <mi>α</mi> <mo>=</mo> <msup> <mn>40</mn> <mo>∘</mo> </msup> </mrow> </semantics></math> and a constant yaw angle of <math display="inline"><semantics> <mrow> <mi>β</mi> <mo>=</mo> <msup> <mn>0</mn> <mo>∘</mo> </msup> </mrow> </semantics></math>. Control specifications: <math display="inline"><semantics> <mrow> <msub> <mi>ϕ</mi> <mi>m</mi> </msub> <mo>=</mo> <msup> <mn>60</mn> <mo>∘</mo> </msup> </mrow> </semantics></math> and <math display="inline"><semantics> <mrow> <msub> <mi>ω</mi> <mrow> <mi>g</mi> <mi>c</mi> </mrow> </msub> <mo>=</mo> <mn>1.5</mn> </mrow> </semantics></math> rad/s.</p> "> Figure 11
<p>Bode plots of the open loop systems with the resulting PI controllers. Control specifications: <math display="inline"><semantics> <mrow> <msub> <mi>ϕ</mi> <mi>m</mi> </msub> <mo>=</mo> <msup> <mn>60</mn> <mo>∘</mo> </msup> </mrow> </semantics></math> and <math display="inline"><semantics> <mrow> <msub> <mi>ω</mi> <mrow> <mi>g</mi> <mi>c</mi> </mrow> </msub> <mo>=</mo> <mn>5</mn> </mrow> </semantics></math> rad/s. (<b>a</b>) Bode plots of the open loop system with the PI controller corresponding to the pitch system. (<b>b</b>) Bode plots of the open loop system with the PI controller corresponding to the yaw system.</p> "> Figure 12
<p>Bode plots of the open loop systems with the resulting FOPI controllers. Control specifications: <math display="inline"><semantics> <mrow> <msub> <mi>ϕ</mi> <mi>m</mi> </msub> <mo>=</mo> <msup> <mn>60</mn> <mo>∘</mo> </msup> </mrow> </semantics></math> and <math display="inline"><semantics> <mrow> <msub> <mi>ω</mi> <mrow> <mi>g</mi> <mi>c</mi> </mrow> </msub> <mo>=</mo> <mn>5</mn> </mrow> </semantics></math> rad/s. (<b>a</b>) Bode plots of the open loop system with the FOPI controller corresponding to the pitch system. (<b>b</b>) Bode plots of the open loop system with the FOPI controller corresponding to the yaw system.</p> "> Figure 13
<p>PI and FOPI system response to a lemniscate of Bernoulli input reference. Control specifications: <math display="inline"><semantics> <mrow> <msub> <mi>ϕ</mi> <mi>m</mi> </msub> <mo>=</mo> <msup> <mn>60</mn> <mo>∘</mo> </msup> </mrow> </semantics></math> and <math display="inline"><semantics> <mrow> <msub> <mi>ω</mi> <mrow> <mi>g</mi> <mi>c</mi> </mrow> </msub> <mo>=</mo> <mn>5</mn> </mrow> </semantics></math> rad/s. Tip load: 0 g.</p> "> Figure 14
<p>PI and FOPI system response to a lemniscate of Bernoulli input reference. Control specifications: <math display="inline"><semantics> <mrow> <msub> <mi>ϕ</mi> <mi>m</mi> </msub> <mo>=</mo> <msup> <mn>60</mn> <mo>∘</mo> </msup> </mrow> </semantics></math> and <math display="inline"><semantics> <mrow> <msub> <mi>ω</mi> <mrow> <mi>g</mi> <mi>c</mi> </mrow> </msub> <mo>=</mo> <mn>5</mn> </mrow> </semantics></math> rad/s. Tip load: 500 g.</p> "> Figure 15
<p>FOPI and PI controller (<math display="inline"><semantics> <mrow> <msub> <mi>ϕ</mi> <mi>m</mi> </msub> <mo>=</mo> <msup> <mn>60</mn> <mo>∘</mo> </msup> </mrow> </semantics></math> and <math display="inline"><semantics> <mrow> <msub> <mi>ω</mi> <mrow> <mi>g</mi> <mi>c</mi> </mrow> </msub> <mo>=</mo> <mn>5</mn> </mrow> </semantics></math> rad/s) response to <math display="inline"><semantics> <mrow> <mi>α</mi> <mo>=</mo> <msup> <mn>40</mn> <mo>∘</mo> </msup> </mrow> </semantics></math> and <math display="inline"><semantics> <mrow> <mi>β</mi> <mo>=</mo> <msup> <mn>0</mn> <mo>∘</mo> </msup> </mrow> </semantics></math> with 500 g mass at the end.</p> ">
Abstract
:1. Introduction
2. Description of the Soft Arm Platform
3. Modeling Approach
4. System Identification
5. Control Specification and Tuning
5.1. The First Control Specifications: 60 and 1.5 rad/s
- Step 1.
- The system phase and phase slope found at are and , respectively.
- Step 2.
- The controller is required to contribute with a phase slope opposite to that of the system—that is, . Besides, in order to achieve the phase margin specification, the controller has to provide a phase at —that is, .
- Step 3.
- Based on these two values from Step 2, the fractional order resulting from the slopes graph available in [13] is .
- Step 4.
- Using these values, is computed (see [13] for more details), resulting .
- Step 5.
- Finally, the controller gain k is computed, resulting .
- Step 6.
- Therefore, according to the method, the controller parameters are , and .
5.2. The Second Control Specifications: 60 and 5 rad/s
6. Experimental Results
6.1. First Experiment: Using Control Specifications 60 and 1.5 rad/s
6.2. Second Experiment: Using Control Specifications 60 and 5 rad/s
7. Conclusions
Author Contributions
Funding
Data Availability Statement
Conflicts of Interest
References
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Plant | ||
---|---|---|
0.1937 | 0.1603 | |
0.1546 | 0.1246 |
Plant | |||
---|---|---|---|
0.0361 | 0.2205 | 0.3800 | |
0.0361 | 0.1682 | 0.3900 |
Plant | ||
---|---|---|
0.6689 | 1.5800 | |
0.5395 | 1.1740 |
Plant | |||
---|---|---|---|
0.3083 | 0.9967 | 0.4600 | |
0.3168 | 0.7401 | 0.5200 |
Data | Peak Value (deg) | Peak Time (s) | Overshoot (%) | RMSE |
---|---|---|---|---|
Simulation | 49.5 | 2.2 | 23.8% | - |
PI: 0 g | 50.4337 | 2.3 | 26.0842% | 0.9765 |
PI: 500 g | 52.8717 | 2.26 | 32.1794% | 2.9153 |
Data | Peak Value (deg) | Peak Time (s) | Overshoot (%) | RMSE |
---|---|---|---|---|
Simulation | 46 | 2.05 | 15.1% | - |
FOPI: 0 g | 46.4978 | 1.98 | 16.24% | 0.8020 |
FOPI: 500 g | 48.4998 | 1.96 | 21.24% | 2.8854 |
Trajectory Component | FOPI: 0 g | PI: 0 g | FOPI: 500 g | PI: 500 g |
---|---|---|---|---|
Pitch | 0.8576 | 0.7357 | 0.6749 | 11.5717 |
Yaw | 0.4862 | 0.4556 | 0.2271 | 0.6521 |
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Relaño, C.; Muñoz, J.; Monje, C.A.; Martínez, S.; González, D. Modeling and Control of a Soft Robotic Arm Based on a Fractional Order Control Approach. Fractal Fract. 2023, 7, 8. https://doi.org/10.3390/fractalfract7010008
Relaño C, Muñoz J, Monje CA, Martínez S, González D. Modeling and Control of a Soft Robotic Arm Based on a Fractional Order Control Approach. Fractal and Fractional. 2023; 7(1):8. https://doi.org/10.3390/fractalfract7010008
Chicago/Turabian StyleRelaño, Carlos, Jorge Muñoz, Concepción A. Monje, Santiago Martínez, and Daniel González. 2023. "Modeling and Control of a Soft Robotic Arm Based on a Fractional Order Control Approach" Fractal and Fractional 7, no. 1: 8. https://doi.org/10.3390/fractalfract7010008