# Development and Improvement of a Piezoelectrically Driven Miniature Robot

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## Abstract

**:**

## 1. Introduction

## 2. Overall Design

## 3. Piezoelectric Actuator Manufacturing

## 4. Transmission Parameters Design

_{a}in parallel with a spring k

_{a}and a damper b

_{a}and has a certain mass m

_{a}to describe the dynamic performance [30]. Since the lift and swing DOFs of the transmission are fully decoupled, the lift DOF including a crank-slider mechanism can be analyzed separately. The flexure hinge in the transmission is equivalent to a pseudo-rigid body model that includes a rigid pin joint and a torsional spring [20]. The illustration of the lift powertrain model is shown in Figure 5.

_{a}is the actuator block force generated when the end displacement is 0, which is a constant value under a determined drive voltage; δ

_{a}is the bending displacement of the actuator, which is variable; k

_{a}is the actuator stiffness; b

_{a}is the actuator damping; k

_{i}and ϕ

_{i}(i = 1, 2, 3) is the bending stiffness and the rotation angles of the flexure hinges, respectively; F

_{leg}is the force experienced by the leg; δ

_{leg-y}is the lift displacement of the leg; m

_{a}is the actuator mass; m

_{i}and L

_{i}(i = 0, 1, 2) are the masses and lengths of the links in the crank-slider mechanism, respectively; Δv

_{i}and Δω

_{i}are the velocity changes and angular velocity changes of the links from the beginning to the end of the motion, respectively; similarly, Δv

_{leg}and Δω

_{leg}are the velocity changes and angular velocity changes of the leg, respectively; and J

_{i}and J

_{leg}are the moments of inertia of the links and leg, respectively.

_{a}of the actuator. Therefore, the output characteristic curves of F

_{a}and δ

_{a}can be obtained by linearly fitting the experimental results of the output force and output displacement with the least squares method. The experimental data and fitting results will be provided in Section 6.1. The kinematics of the crank-slider mechanism can be given by the following closed equation:

_{leg}of the leg can be given as:

_{a}= 195 mN and the actuator stiffness k

_{a}= 491 N/m (as we will describe in Section 5). Meanwhile, to reduce the impact of the robot sagging under load, the leg output displacement δ

_{leg-y}is specified as 1.5 mm. For the flexure hinges, we designed their lengths and widths to be 120 μm and 1.6 mm, respectively, for ease of manufacturing. Commercially available polyimide films for miniature robots are typically 20 μm, 30 μm, 40 μm, and 50 μm in thickness. Therefore, the stiffness of the flexure hinges can be calculated [25]. To sum up, with the link lengths (L

_{1}, L

_{2}, L

_{leg-x}, and L

_{leg-y}) as the design variable and the leg output force as the objective function, the optimization problem of the robot’s payload capacity is determined in this regard. We limit the length, width, and height dimensions of the whole robot to 5 cm × 5 cm × 3 cm, and, considering the assembly relationship between the various components of the robot, as well as the inherent geometrical constraints of the crank-slider mechanism, the constraints of this optimization problem can be given by the following equations:

_{1}, L

_{2}, L

_{leg-x}, and L

_{leg-y}] of the links are shown in Figure 6a. The horizontal axis refers to the data points formed by different combinations of link lengths. The row vectors in Figure 6a represent the lengths (unit: mm) of various links corresponding to the maximum output force of the leg. It can be seen from Figure 6a that, for any thickness of the hinge, there exists an optimal linkage combination that maximizes the output force of the robot leg, and the output force converges with the number of iterations. It is clear that the robot leg has a maximum output force of 0.0236 N when the thickness of the flexure hinge is 20 μm. Based on the optimization results in Figure 6a, the relationship curve between the hinge thickness and the maximum leg force corresponding to the optimal linkage combination under this hinge thickness was plotted, as shown in Figure 6b. Obviously, the leg output force increases as the hinge thickness decreases. However, the robot body with 20 μm-thick flexure hinges has a sag of close to 2 mm in the robot locomotion test, so it can not produce effective movement. Therefore, we choose hinges with a thickness of 30 μm for the transmission. Correspondingly, the lengths of the links in the crank-slider mechanism are L

_{1}= 8 mm, L

_{2}= 1.2 mm, L

_{leg-x}= 7.4 mm, and L

_{leg-y}= 13.4 mm, respectively.

_{1}, L

_{2}, L

_{leg-x}, and L

_{leg-y}) specified in Equation (6) is divided at 1 mm intervals, that is to say, L

_{1}= [3 4… 8], L

_{2}= [0.5 1.5… 4.5], L

_{leg-x}= [1.5 2.5… 14.5], and L

_{leg-y}= [4 5… 15]. L

_{1}, L

_{2}, L

_{leg-x}, and L

_{leg-y}, respectively, contain 6, 5, 14, and 12 elements. Then, four four-dimensional matrices containing 6 × 5 × 14 × 12 elements are generated using MATLAB’s ‘ndgrid’ function, and the corresponding elements of the four matrices at the same position constitute 5040 co-ordinates (i.e., the combination of lengths of each link). Taking two two-dimensional matrices M and N as an example, the corresponding elements of M and N at the same position refer to the two elements in the i-th row and j-th column of these two matrices. According to Equation (5), an image of the F

_{leg}= f (L

_{1}, L

_{2}, L

_{leg-x}, L

_{leg-y}) is plotted as shown in Figure 7. Obviously, different co-ordinate points correspond to different leg output forces, and there exists a co-ordinate point (combination of link lengths) that maximizes the leg output force.

## 5. Experiments

#### 5.1. Force and Displacement Experiments of Piezoelectric Actuators

#### 5.2. The Leg’s Quasi-Static Force Experiments

_{leg-y}) is adjusted by changing the height of the positioning stage. The positioning accuracy of the positioning stage is 0.02 mm. During the experiment, the distance between the leg and the positioning stage is changed at intervals of 0.2 mm until the leg cannot contact the sensor under the actuator drive.

#### 5.3. Dynamic Model Identification Experiments of the Powertrain

#### 5.4. Locomotion Test of the Robot

## 6. Results and Discussion

#### 6.1. Forces and Displacements of Piezoelectric Actuators

_{out}corresponding to this bending displacement δ

_{a}driven at a 210 V DC voltage. The results are shown in Figure 13c, the slope of the curve is the stiffness k

_{a}of the actuator, and the force F

_{a}is the output force generated when the end displacement of the actuator is 0, that is, k

_{a}= 491 N/m, and F

_{a}= 195 mN.

#### 6.2. The Leg’s Quasi-Static Force Results

_{i}was determined. The corresponding optimal linkage combination (L

_{1}, L

_{2}, L

_{leg-x}, and L

_{leg-y}) has also been determined. The piezoelectric force F

_{a}of the actuator and the stiffness k

_{a}are also analyzed in Section 6.1. Therefore, except for the parameters of leg force and displacement, all other parameters in Equation (5) are known, and the output force of the robot leg is only determined by the leg displacement. Therefore, the relation of leg force F

_{leg}versus lift displacement δ

_{leg-y}can be obtained through MATLAB simulation. The experimental results of the robot’s leg force are lower than the simulation results, because the exoskeleton of the robot is not rigid and the flexure hinges will buckle when subjected to load, and the bending displacement of the actuator is already greater than the theoretical displacement.

#### 6.3. Dynamic Model Identification Results of Powertrain

_{n}is the natural frequency, k is the amplification coefficient, and ξ is the damping ratio. Based on this second-order oscillation model, the experimental results are fitted by MATLAB’s fittype and fit functions. The damping ratios for the swing and lift DOFs of the front left leg are 0.0685 and 0.0866, respectively; the natural frequencies are 61.88 Hz and 72.17 Hz, respectively; and the amplification coefficients are 3.499 and 3.428, respectively. The results indicate that the front left powertrain is under-damped and there is a 90° phase offset between the leg output and input signals when the drive frequency is close to the natural frequency.

#### 6.4. The Locomotion Performance of the Robot

## 7. Conclusions

## Supplementary Materials

## Author Contributions

## Funding

## Institutional Review Board Statement

## Data Availability Statement

## Conflicts of Interest

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**Figure 1.**(

**a**) A CAD model of the miniature robot. (

**b**) A schematic diagram of the powertrain. (

**c**) The prototype of the miniature robot contrasted with a coin.

**Figure 2.**(

**a**) Explosion view of the actuator. (

**b**) The CAD model of the actuator. (

**c**) The physical picture of the actuators.

**Figure 3.**(

**a**) The stacking order of the actuator components. (

**b**) The positional relationship of the alumina, PZT, and FR-4 jig in the same plane. (

**c**) The top view schematic diagram of the stacked laminate.

**Figure 4.**(

**a**) The stacked laminate using discrete PZT pieces and its cutting path. (

**b**) The stacked laminate using a whole piece of PZT and its cutting path.

**Figure 6.**(

**a**) The optimization results of the leg output force on different combination lengths [L

_{1}, L

_{2}, L

_{leg-x}, and L

_{leg-y}] of the links for flexure hinges of different thicknesses. (

**b**) Relationship between different hinge thicknesses and the corresponding maximum leg output forces.

**Figure 7.**The relationship between the output force of the legs and the co-ordinate points formed by different combinations of link lengths.

**Figure 8.**(

**a**) Experimental setup for measuring actuator forces. (

**b**) Experimental setup for measuring actuator displacements.

**Figure 10.**Experimental setup for frequency response of the swing (

**left**) and lift (

**right**) DOFs of the leg.

**Figure 11.**(

**a**) Layout diagram of the actuators below the circuit board. (

**b**) The drive signals of eight actuators for the robot to move forward in a trot gait. (

**c**) The motion trajectory of two robot legs on the same side under the drive signal of trot gait. (

**d**) Footfall patterns of the trot gait.

**Figure 13.**(

**a**) Experimental results of the block force (peak-to-peak value) of three different actuator versions. (

**b**) Experimental results of the displacement (peak-to-peak value) of three different actuator versions. (

**c**) Force–displacement curves for ‘AC’ actuators driven at 210 V. Each error bar is the standard deviation acquired from five actuators with the same design parameters.

**Figure 14.**The experimental and simulation results of the leg force at different displacements. Each error bar of the experimental results is the standard deviation of the five repeated experiments.

**Figure 15.**Frequency responses and second-order oscillation model fitting of the powertrain of the front left leg.

**Figure 16.**(

**a**) The speed of the robot with no payload under different drive frequencies. (

**b**) The speed of the robot with different payloads. Each error bar is the standard deviation of five repeated experiments.

**Figure 17.**Representative frames captured by camera when the miniature robot reaches a speed of 48.66 cm/s.

Robots | Length (cm) | Mass (g) | Maximum Speed (cm/s) | Highlights | Refs |
---|---|---|---|---|---|

SMR-O | 4.6 | 1.8 | 48.66 | Greater payload capacity of 5.5 g compared to HAMR; exoskeletons, legs, and hip joints monolithically integrated and manufactured. | This paper |

HAMR-VP | 4.4 | 1.27 | 37 | Improved manufacturing and assembly speed of robots through the pop-up process. Increased payload capacity to 1.35 g compared to previous versions of robots. | [14] |

DASH | 10 | 16.2 | 150 | The highest speed among miniature robots. | [17] |

MinRAR | 5.5 | 16 | 52 | A fast speed driven at the resonance frequency. | [26] |

RoACH | 3 | 2.4 | 3 | The first robot to use smart composite microstructure technology. | [9] |

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## Share and Cite

**MDPI and ACS Style**

Wu, G.; Wang, Z.; Wu, Y.; Zhao, J.; Cui, F.; Zhang, Y.; Chen, W.
Development and Improvement of a Piezoelectrically Driven Miniature Robot. *Biomimetics* **2024**, *9*, 226.
https://doi.org/10.3390/biomimetics9040226

**AMA Style**

Wu G, Wang Z, Wu Y, Zhao J, Cui F, Zhang Y, Chen W.
Development and Improvement of a Piezoelectrically Driven Miniature Robot. *Biomimetics*. 2024; 9(4):226.
https://doi.org/10.3390/biomimetics9040226

**Chicago/Turabian Style**

Wu, Guangping, Ziyang Wang, Yuting Wu, Jiaxin Zhao, Feng Cui, Yichen Zhang, and Wenyuan Chen.
2024. "Development and Improvement of a Piezoelectrically Driven Miniature Robot" *Biomimetics* 9, no. 4: 226.
https://doi.org/10.3390/biomimetics9040226