# Precise Position Control of Holonomic Inchworm Robot Using Four Optical Encoders

^{*}

^{†}

## Abstract

**:**

## 1. Introduction

## 2. Holonomic Inchworm Robot

#### 2.1. Structure

#### 2.2. Principle

#### 2.3. Dynamic Model

_{L}and k

_{S}are the spring constants of the six PAs in the compression and shear deformations, respectively. The d

_{F}is the enforced displacement of PZT-F, and d

_{B}, d

_{R}

_{1}, d

_{R}

_{2}, d

_{L}

_{1}, and d

_{L}

_{2}are similar.

_{1}, P

_{2}, and P

_{3}. O

_{1}, O

_{2}, and O

_{3}are the initial positions of P

_{1}, P

_{2}, and P

_{3}, respectively. Furthermore, x

_{1}and y

_{1}are the coordinates of P

_{1}; (x

_{2}, y

_{2}) and (x

_{3}, y

_{3}) similarly define P

_{2}and P

_{3}, respectively. P is the center of gravity of EM-1. The position of P is represented by the orthogonal coordinate system used for x and y. O is the origin of P. We assume that EMs are rigid bodies.

_{1}, P

_{2}, and P

_{3}. Therefore, the PAs move to the free leg; m is the mass of the EMs, I is the moment of inertia of the gravity centers of the EMs, and r is the distance between the center and end of the EMs. From Figure 1, Newton’s equations of motion for EM-1 and the related parameters are represented by Equations (1)–(7).

#### 2.4. Input Voltages

_{F}–V

_{R}

_{2}are the input voltages to the PAs. V

_{F}–V

_{R}

_{2}was obtained by solving Equation (1) using the approximation of harmonic oscillations with no damping.

_{p-p}. The offset voltage was determined to be V

_{0}= 60 V, which is half the maximum voltage of 120 V. K

_{1}and K

_{2}are given by Equation (10).

## 3. XYθ Position Sensor

#### 3.1. Structure

#### 3.2. Signal Processing

**V**), as shown in Equation (9). The minimum measurement cycle of the FPGA was 0.35 μs, and the maximum calculated speed was 2800 mm/s. The FPGA measured the displacement of a sinusoidal vibration up to 1273 Hz with a displacement amplitude of 30 µm.

#### 3.3. Measurement Principle

_{1}′ is the position after moving $\Delta \overrightarrow{L}$ from its initial position. $\Delta \overrightarrow{L}$ is defined as a vector component of the translational movement as follows:

_{1}is the initial position of encoder-1, which is defined as (X

_{10}, Y

_{10}) in the XY coordinate system. E

_{2}, E

_{3}, and E

_{4}are defined similarly. R is the geometrical center of E

_{1}, E

_{2}, E

_{3}, and E

_{4.}

_{1}″ is the position of encoder-1 after moving $\Delta \overrightarrow{{R}_{1}}$ from E

_{1}′. $\Delta \overrightarrow{{R}_{1}}$ is defined as the vector component of rotational movement. We define (X

_{1}, Y

_{1}) as the XY coordinates of E

_{1}″, and (X

_{1}′, Y

_{1}′) as the X′Y′ coordinates of E

_{1}. The other parameters are defined similarly. We obtain the following coordinate transformation of the initial positions of E

_{k}from Equations (11) and (12) as follows:

_{1}′, X

_{2}′, Y

_{3}′, and X

_{4}′ using the four encoders, which are represented by Equation (14) as follows:

#### 3.4. Experimental Results

## 4. Sequence Control of Multiple Step Motions

#### 4.1. Control Sequence

_{T}is the target position, and e

_{T}

_{1}is the threshold of the distance error e for coarse motion; e

_{T}

_{2}is the fine motion, δθ

_{T}is the threshold of the orientation error δθ, and W

_{1}and W

_{2}are the step lengths of the coarse and fine motions, respectively. We determined e

_{T}

_{1}= 80 μm, e

_{T}

_{2}= 15 μm, δθ

_{T}= 0.06°, W

_{1}= 60.0 μm, and W

_{2}= 5.0 μm. The robot moved to an inchworm frequency of 100 Hz.

#### 4.2. Experimental Results

_{T}

_{1}with coarse movement, (1) Coarse1. It switched to fine movement and moved near the center of the range of e

_{T}

_{2}, (2) Fine1. If the center was within the range of e

_{T}

_{2}, the orientation angle θ was measured. If it was out of the range of δθ

_{T}, as shown in this case, the orientation was corrected by rotation, (3) Rotation1. If it moved outside the range of e

_{T}

_{2}, it approached the corner again using fine movements, (4) Fine2. If both the center and orientation were within the corresponding ranges of e

_{T}

_{2}and δθ

_{T}, respectively, the robot changed to a coarse movement toward the next corner, (5) Coarse2.

## 5. 3-Axis PID Control of One-Step Motion

#### 5.1. Transfer Function

**y**(t) as

**Y**(s). The Laplace transforms of the other parameters are defined similarly.

**d**(t) and

_{m}**w**(t) as vectors composed of the modeling error and uncertainties of the measured displacement, respectively. Here,

**k**,

_{I}**k**, and

_{P}**k**are vectors composed of integral, proportional, and derivative gains along the x-, y-, and θ-axes, respectively;

_{D}**r**(t) is the target position, and

**e**(t) is the deviation between

**r**(t) and

**y**(t). We applied a first-order Butterworth filter with a cutoff filter of 50 Hz to

**E**(

**s**) before derivative (D) control to minimize the time delay of the primary experiments.

#### 5.2. Experimental Results

**k**,

_{I}**k**, and

_{P}**k**for each experimental condition by using a heuristic method to obtain a no-overshoot trajectory. We conducted experiments five times for each condition, and similar results were obtained. We determined the target travel lengths r

_{D}_{i}as 1, 5, and 10 μm and the target moving directions φ

_{i}as 0°, 30°, 60°, and 90° for translational movements. We also determined the target rotational displacement of θ

_{i}as 0 and 14.8 millidegrees in the θ-axis. In addition, we determined the thresholds of the X-, Y-, and θ-axes as X

_{T}= ±0.14 μm, Y

_{T}= ±0.14 μm, and θ

_{T}= ±0.4 millidegrees around their corresponding targets.

_{S}, Y

_{S}, and θ

_{S}were similarly defined. As shown in Figure 10a for the XY trajectories, we succeeded in precisely controlling the position of the free leg, although non-negligible oscillations in the X- and Y-axes were generated for the directions of 60° and 90°. Figure 10b,c compares the plots of X, X

_{S}, Y, Y

_{S}, θ, and θ

_{S}vs. time for the 0 °and 60 °directions, and their settling times were approximately 65 and 78 ms, respectively. We assumed that the oscillation was attributed to electrical noise from a commercial 50 Hz AC power supply, as observed in Figure 10b,c,e,f.

_{P}, Y

_{P}, and θ

_{P}. Figure 10d shows the XY trajectories. Figure 10e,f shows plots of X, X

_{P}, Y, Y

_{P}, θ, and θ

_{P}vs. time for the 0° and 60° directions, and their settling times were approximately 122 and 130 ms, respectively. The oscillations decreased to ±0.5 μm in the X and Y axes, as shown in Figure 10.

## 6. Conclusions and Future Prospects

^{2}and −25–25°. We demonstrated the sequential positioning control of the multiple-step motion from the centimeter to the micrometer range. We have also demonstrated PID control of the one-step motion from the micrometer to the sub-micrometer range.

## Supplementary Materials

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

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**Figure 1.**Comparison of typical micromanipulation robots and XY stage [22,24,27]. Scores are evaluated based on the reference [27] (refer to Table S1 of the Supplementary Materials for the quantitative comparison of the performances).

**Figure 2.**Holonomic inchworm robot: (

**a**) structure (top view); (

**b**) overview; (

**c**) principle of motion (rightward movement); (

**d**) motion patterns; (

**e**) dynamic model.

**Figure 3.**XYθ position sensor: (

**a**) magnified view of the measurement area; (

**b**) assembly drawing; (

**c**) encoder installation plate; (

**d**) integrated 2-DoF scale.

**Figure 6.**Measuring errors of XYθ-position sensor and variations in the ΔX, ΔY, and Δθ outputs after calibration: (

**a**) Experimental setup for evaluation of the measuring performance of XYθ-position sensor; (

**b**) Plots of XYθ-axes errors vs. X-axis displacement before calibration; (

**c**) Plots of XYθ-axes errors vs. Y-axis displacement before calibration; (

**d**) Plots of XYθ-axes errors vs. θ-axis displacement before calibration; (

**e**) Plots of XYθ-axes errors vs. X-axis displacement after calibration; (

**f**) Plots of XYθ-axes errors vs. Y-axis displacement after calibration; (

**g**) Plots of XYθ-axes errors vs. θ-axis displacement after calibration.

**Figure 8.**Star-shaped trajectories with five target points: (

**a**) XY trajectory; (

**b**) X, Y vs. time; (

**c**) θ vs. time.

**Figure 10.**PID control of one step motion (left: step reference, right: parabolic reference with rise time of 100 ms): (

**a**) plots of XY trajectories along ${\varphi}_{i}=0,30,60,\mathrm{and}90\xb0$; (

**b**) plots of XYθ vs. time for $\left({r}_{i},{\varphi}_{i},{\theta}_{i}\right)=\left(10\mathsf{\mu}\mathrm{m},0\xb0,0\mathrm{m}\xb0\right)$; (

**c**) plots of XYθ vs. time for $\left({r}_{i},{\varphi}_{i},{\theta}_{i}\right)=\left(10\mathsf{\mu}\mathrm{m},60\xb0,14.8\mathrm{m}\xb0\right)$; (

**d**) plots of XY trajectories along ${\varphi}_{i}=0\mathrm{and}60\xb0$; (

**e**) plots of XYθ vs. time for $\left({r}_{i},{\varphi}_{i},{\theta}_{i}\right)=\left(10\mathsf{\mu}\mathrm{m},0\xb0,0\mathrm{m}\xb0\right)$; (

**f**) plots of XYθ vs. time for $\left({r}_{i},{\varphi}_{i},{\theta}_{i}\right)=\left(10\mathsf{\mu}\mathrm{m},60\xb0,14.8\mathrm{m}\xb0\right)$.

Characteristic Value | Quantity |
---|---|

Step length (120 V) | ~65 μm |

Resolution (15–25 °C, less than 50% rH) | Less than 10 nm |

DoF | X, Y, θ |

Natural Frequency (blocked free) | X: 413, Y: 418, θ: 476 Hz |

Maximum Velocity [frequency] | ~6.5 mm/s [100 Hz] |

Repeatability (CV; ratio of SD of final points to a path length with 10 mm path) [frequency] | ~3% [100 Hz] |

Maximum payload | <1000 g |

Dimension | 86 × 86 × 15 mm |

Weight | 100 g |

Characteristic Value | Quantity |
---|---|

Displacement (100 V) | 95.5 ± 5 μm |

Generative Force (100 V) | 18.0 N |

Spring constant | 115,000 N/m |

Capacitance | 1.04 μF |

Resolution (15–25 °C, less than 50% rH) | 1.52 nm |

Natural Frequency (blocked free) | 1.45 kHz |

Dimension | 12.9 × 6.4 × 9.2 mm |

Weight | 4 g |

Characteristic Value | Quantity |
---|---|

Resolution [μm/count] | 0.1 |

Maximum measurement speed [mm/s] | 800 |

Dimension [mm] | 15 × 10 × 1.5 |

Characteristic Value | Quantity |
---|---|

Measurement range X × Y [mm], θ [°] | 16 × 16, ±25 |

Measurement resolution in X (Y) [μm], θ [millidegrees] | 0.1, 0.3 |

Uncertainty in static state in X (Y) [μm], θ [millidegrees] | ±0.2, ±0.6 |

Measurement frequency [MHz] | 2.86 |

Maximum measurable speed [mm/s] | 800 |

Principle of measurement | Incremental |

Measurement accuracy in X and Y (−8~8mm) [%] | 0.08–0.18 |

Measurement accuracy in θ (−25~25°) [%] | 0.06–0.19 |

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## Share and Cite

**MDPI and ACS Style**

Tanabe, K.; Shiota, M.; Kusui, E.; Iida, Y.; Kusama, H.; Kinoshita, R.; Tsukui, Y.; Minegishi, R.; Sunohara, Y.; Fuchiwaki, O.
Precise Position Control of Holonomic Inchworm Robot Using Four Optical Encoders. *Micromachines* **2023**, *14*, 375.
https://doi.org/10.3390/mi14020375

**AMA Style**

Tanabe K, Shiota M, Kusui E, Iida Y, Kusama H, Kinoshita R, Tsukui Y, Minegishi R, Sunohara Y, Fuchiwaki O.
Precise Position Control of Holonomic Inchworm Robot Using Four Optical Encoders. *Micromachines*. 2023; 14(2):375.
https://doi.org/10.3390/mi14020375

**Chicago/Turabian Style**

Tanabe, Kengo, Masato Shiota, Eiji Kusui, Yohei Iida, Hazumu Kusama, Ryosuke Kinoshita, Yohei Tsukui, Rintaro Minegishi, Yuta Sunohara, and Ohmi Fuchiwaki.
2023. "Precise Position Control of Holonomic Inchworm Robot Using Four Optical Encoders" *Micromachines* 14, no. 2: 375.
https://doi.org/10.3390/mi14020375