# Design and Testing of a 3-DOF Robot for Studying the Human Response to Vibration

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

^{−3}and 1.4 × 10

^{−2}m/s

^{2}in the frequency range between 1 and 50 Hz, proving the good outcome of the design process.

## 1. Introduction

#### 1.1. Effects of Vibrations on the Human Body

^{2}Root Mean Square (RMS), generated by an electrohydraulic vibrator. Several papers have focused on the effect of the vibration direction on the response of the human body. For example, Paddan et al. [11] investigated the effect of floor vibration along three axes (vertical, fore- and aft-, and horizontal) on standing subjects; however, the vibration along the different directions was not provided simultaneously. Tarabini et al. [19] described a system for the measurement of the mass matrix of a standing human subject when simultaneously excited with vertical and horizontal vibration. The system able to provide multi-axial acceleration consists of two single-axis electrodynamic shakers (one for the vertical and one for the horizontal direction) connected to compliant plates, which allow movement in one horizontal direction, while being rigid in the vertical and along the other horizontal direction. Subashi et al. [20] proposed a lumped parameter model of the human body that could mimic the response of the standing human body when exposed to only vertical WBV; however, the effect of the cross talk between the axis is so relevant that the authors had to include horizontal and rotational degrees of freedom to model it. Therefore, it is possible to state that an excitation of the human body along one axis also generates relevant effects along the other two axes. Moreover, to the best of the authors’ knowledge, there are no studies in which the effect of simultaneous three-axial vibration on the standing human body has been investigated. Therefore, the purpose of this paper is to propose a machine able to generate simultaneous translational vibration along the three mutually perpendicular spatial axes, which can be used to generate the signal required to improve the measurement of the human response to vibration.

#### 1.2. Linear Delta Robot Selection

#### 1.3. Functional Requirements

^{2}along the three axes. Finally, the maximum weight carried by the shaker is defined as 200 kg, which corresponds to the apparent mass in the vertical direction of a standing subject weighing 100 kg subjected to vertical vibrations at the resonance frequency of 5–6 Hz [14].

## 2. Materials and Methods

#### 2.1. Design of the Delta Shaker

#### 2.1.1. Kinematic Optimization

_{i}, and the length of the links, i.e., the modulus of the vector l

_{i}. The radius of the platform, i.e., the modulus of the vector b

_{i}, was set to 250 mm, given the fact that is must be wide enough to support a standing person.

_{j}is the component of the position vector of the center of the moving platform along the j direction and D is the difference between the radius of the base and the radius of the moving platform. Appendix A includes all the analytical steps necessary to obtain such an equation.

^{®}, 1 Apple Hill Drive, Natick, MA, USA) has been used to process all the computations.

#### 2.1.2. Actuator Selection

^{2}divided by the number of components and the signal along the vertical axis was increased by 50%. The signal duration was 2 s. Given the random components, 25 different signals were created to explore the different possibilities.

_{max}is the maximum torque that the motor can provide, p is the pitch of the ball screw, J

_{s}is the total inertia of the rotating parts, g is the acceleration of gravity, and M is the total translating mass divided by three. The inertia of the rotating part is the sum of the inertia of the motor; the screw, which is modeled as a cylinder; and the inertia of the joint between the two. The joint that was selected is the model smartflex

^{®}1/932.333 manufactured by Mayr, which has a mass moment of inertia of 1.04 × 10

^{−4}kg·m

^{2}. The total translating mass is the sum of the mass of the payload and the mass of the translating parts of the shaker, which was set to 32 kg, plus the mass of the translating part of the screw.

#### 2.1.3. FEM Modal Analysis

^{3}kg/m

^{3}, E = 69 GPa, ν = 0.33) of all the parts, except the base, for which steel was assigned (ρ = 7.86 × 10

^{3}kg/m

^{3}, E = 210 GPa, ν = 0.33). The effect of the bearings was modeled using the coupling and tie interaction property. For each bearing, the outer surface of the rotating ring and the inner surface of the hole were coupled with a separate reference point at the center. The two reference points were then constrained to only rotate along the desired direction. Following this, the two surfaces were tied together to avoid interpenetration.

#### 2.1.4. Experimental Modal Analysis

#### 2.2. Performance Tests

^{2}. The frequencies tested were from 5 to 20 Hz, with a step of 5 Hz. For each frequency, the acquisition time was 10 s and the acquisition frequency was 2048 Hz. The signals were acquired through six single-axis piezoelectric accelerometers 4508B (PCB Piezotronics, Inc. 3425 Walden Avenue, Depew, NY, USA), which were connected to two NI 9234 acquisition boards. Both cards were connected to the NI cDAQ9174 (National Instruments Corporation, 11500 N MoPac Expwy, Austin, TX, USA). The data were collected and analyzed through Labview-based software on a PC. The position of the accelerometers on the platform was changed for each direction of motion, as shown in Figure 3. Figure 3a shows the setup for the test conducted along the X direction. Three accelerometers were oriented to measure along the X direction: one at the center of the platform, one on the left side, and one on the right side, with the latter two representing the positive and negative side of the Y axis, respectively. Two accelerometers were oriented to measure along the vertical direction, on the top and bottom of the moving platform, i.e., along the positive and negative side of the X axis, respectively. Figure 3b shows the position and the orientation of the accelerometers during the tests along the Y direction. The setup is equivalent to the one previously explained, but rotated by 90° along the vertical axes. Figure 3c shows the setup for the tests executed to evaluate the performance along the vertical direction. For this test, all the accelerometers were rotated to measure along the vertical direction. One accelerometer was placed at the center of the platform, three accelerometers were placed at each short side of the moving platform, and the last accelerometer was placed on the long side of the moving platform at the negative side of the X axis. The RMS value of the signal acquired by the different accelerometers was then compared.

^{2}. Moreover, the actual signal imposed on the moving platform was modified to compensate for the frequency response function of the shaker, thanks to a specific feature of the control system. The total duration of the test was 120 s. The acceleration of the end effector was measured as was explained for the previous test, but using only one three-axis piezoelectric accelerometer 356A22, placed at the center of the platform. The measured acceleration was divided into 120 segments with a duration of 1 s, and the average modulus of all the segments, multiplied by the Hanning window, was computed. The moduli of the resulting signals were compared with a ±3 dB tolerance band applied to the reference signal. Moreover, the mean quadratic error between the moduli of the actual signals and the reference signal was computed in the frequency band of interest (i.e., 1–50 Hz). To characterize the background noise, the root mean squared values of the components higher than 50 Hz were computed.

## 3. Results

#### 3.1. Experimental Modal Analysis

#### 3.2. Measured Performances

^{2}, except in the Y direction when a person is standing on the platform. Table 5 shows the RMS value of the components with a frequency higher than the maximum imposed value, as a measure of the background noise. All the values are lower than 10

^{−2}m/s

^{2}, and the noise is higher along the Z direction both with and without load.

## 4. Discussion

^{2}between 1 and 50 Hz along the vertical direction, the RMS of the acceleration along the two other axes was 0.17 m/s

^{2}. When the same signal was imposed on a commercial single-axis shaker (model LDS V830), the RMS of the acceleration along the X and Y axes was 0.07 and 0.05 m/s

^{2}, respectively. This drawback is explained by the fact that the proposed machine is designed to simultaneously work along the three axes.

## 5. Conclusions

^{2}when imposing pseudo-random noise with an RMS value of 1 m/s

^{2}and a flat spectrum in the frequency range of 1–50 Hz, along the three axes simultaneously. Given these results, within the limitations explained in the previous section, the machine presented here is capable of meeting the design requirements, and is able to generate the signal necessary for measuring the full apparent mass matrix of standing subjects.

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Appendix A

- Reference coordinate system: O
_{xyz}; - Unit vectors of the reference coordinate system O
_{xyz}: $\left(\begin{array}{ccc}{\widehat{x}}_{o}& {\widehat{y}}_{o}& {\widehat{z}}_{o}\end{array}\right)$; - Reference system integral with the moving platform: TCP
_{xyz}. Due to the linear delta kinematic constraints, this reference system is always parallel to O_{xyz}; - Unit vectors of the moving platform coordinate system TCP
_{xyz}: $\left(\begin{array}{ccc}{\widehat{x}}_{TCP}& {\widehat{y}}_{TCP}& {\widehat{z}}_{TCP}\end{array}\right)$; - Position vector of TCP
_{xyz}: p; - Components of p along the reference coordinates system O
_{xyz}:$$\left\{\begin{array}{c}{p}_{x}\\ {p}_{y}\\ {p}_{z}\end{array}\right\};$$ - Linear actuator and joint index: i;
- Distance between O
_{xyz}and the projection of the linear actuators along the plane defined by $\left(\begin{array}{cc}{\widehat{x}}_{o}& {\widehat{y}}_{o}\end{array}\right)$: s; - Position vector of the projection of the ith linear actuator along the plane $\left(\begin{array}{cc}{\widehat{x}}_{o}& {\widehat{y}}_{o}\end{array}\right)$: s
_{i}. Given the radial symmetry of the linear delta, its components are$$\left\{\begin{array}{c}s\mathrm{cos}\left(\frac{2\pi}{3}\left(\mathrm{i}-1\right)\right)\\ s\mathrm{sin}\left(\frac{2\pi}{3}\left(\mathrm{i}-1\right)\right)\\ 0\end{array}\right\};$$ - Distance between TCP
_{xyz}and the joints attached to the platform: b; - Position vector of the ith joint attached to the platform in the reference system TCP
_{xyz}: b_{i}. As stated before, its components can be immediately computed:$$\left\{\begin{array}{c}b\mathrm{cos}\left(\frac{2\pi}{3}\left(\mathrm{i}-1\right)\right)\\ b\mathrm{sin}\left(\frac{2\pi}{3}\left(\mathrm{i}-1\right)\right)\\ 0\end{array}\right\};$$ - Vector of the distance between the position of the ith linear actuator end and its projection along the plane $\left(\begin{array}{cc}{\widehat{x}}_{o}& {\widehat{y}}_{o}\end{array}\right):$ q
_{i}. Its components are defined as$$\left\{\begin{array}{c}0\\ 0\\ {q}_{i}\end{array}\right\};$$ - Length of the linear delta links: l;
- Vector connecting the joint of the ith linear actuator to the ith joint of the moving platform: l
_{i}; - Unit vector of l
_{i}: ${\widehat{v}}_{i};$ Vector connecting the projection of the ith linear actuator on the plane $\left(\begin{array}{cc}{\widehat{x}}_{o}& {\widehat{y}}_{o}\end{array}\right)$ and the ith joint of the moving platform: d_{i};With reference to Figure 1, it is possible to write the following equations:$${\underset{\_}{l}}_{i}={\underset{\_}{d}}_{i}-{\underset{\_}{q}}_{i}$$$${\underset{\_}{d}}_{i}=\underset{\_}{p}+{\underset{\_}{b}}_{i}-{\underset{\_}{s}}_{i}.$$

_{i}and l

_{i}, it is possible to obtain the following quadratic equation with the unknown q

_{i}:

_{i}as a function of d

_{i}. Given the position of the platform with respect of the links of the delta, a negative solution is taken.

_{i}with the terms provided by (A2). The relationship can be further simplified given the fact that the vectors b

_{i}and s

_{i}are defined without components along ${\widehat{z}}_{o}$. In this way, it is possible to compute each q

_{i}as a function of p:

_{i}with the terms provided by (A2):

_{i}, b

_{i}, and s

_{i}are known and equal for each i, hence it is possible to write

_{i}and s

_{i}, the following equation can be computed:

_{x}and p

_{z}, and between p

_{y}and p

_{z}. Therefore, by substituting p

_{x}and p

_{y}in (A10) computed for i = 1, with the functions of p

_{z}defined before, it is possible to write a quadratic equation with the unknown of p

_{z}. Such an equation has the following parameters:

_{z}. Given the geometry of the delta, only the positive solution is relevant. Then, from (A11) and (A12), it is possible to compute the other two components of the vector p.

_{i}, according to (A1), while ${\underset{\_}{\dot{d}}}_{i}$ is equal to $\dot{\underset{\_}{p}},$ according to (A2), because the other terms are constant in time. Therefore, by dividing both terms by the length of the links of the delta, the following relationship can be established:

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**Figure 3.**Position and sensing direction of the accelerometers on the moving platform of the delta shaker for the performance tests along X direction (

**a**), Y direction (

**b**) and Z direction (

**c**).

**Figure 5.**Comparison of the modal analysis results obtained through the Finite Element Method (FEM) and experimentally.

**Figure 6.**Signal acquired by the reference accelerometer placed at the center of the platform when imposing a 15 Hz sinusoidal signal along the X direction, in both the time and frequency domain.

**Figure 7.**Comparison of the imposed acceleration signal and the acceleration of the moving platform, free and loaded with a person standing on it, along the three directions.

Parameter | Lower Bound (mm) | Step (mm) | Upper Bound (mm) |
---|---|---|---|

Base radius | 250 | 10 | 500 |

Link length | 150 | 5 | 250 |

**Table 2.**Maximum forces and moments at the joints, computed through the multibody dynamic simulation.

Fx (N) | Fy (N) | Fz (N) | Mx (Nm) | My (Nm) | Mz (Nm) |
---|---|---|---|---|---|

1916 | 1908 | 282 | 148 | 181 | 320 |

C10 | C20 | C30 | D1 | D2 | D3 |
---|---|---|---|---|---|

0.9636 | −0.6213 | 0.3265 | 0 | 0 | 0 |

**Table 4.**Mean quadratic errors of the moduli of the acceleration signals measured on the moving platform at frequencies between 1 and 50 Hz.

a_{x} (mm/s^{2}) | a_{y} (mm/s^{2}) | a_{z} (mm/s^{2}) | |
---|---|---|---|

No load | 5.7 | 8.2 | 6.3 |

Loaded | 9.6 | 14.0 | 9.8 |

a_{x} (mm/s^{2}) | a_{y} (mm/s^{2}) | a_{z} (mm/s^{2}) | |
---|---|---|---|

No load | 4.9 | 6.5 | 8.2 |

Loaded | 4.6 | 6.0 | 7.8 |

© 2019 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Marzaroli, P.; Albanetti, A.; Negri, E.; Giberti, H.; Tarabini, M. Design and Testing of a 3-DOF Robot for Studying the Human Response to Vibration. *Machines* **2019**, *7*, 67.
https://doi.org/10.3390/machines7040067

**AMA Style**

Marzaroli P, Albanetti A, Negri E, Giberti H, Tarabini M. Design and Testing of a 3-DOF Robot for Studying the Human Response to Vibration. *Machines*. 2019; 7(4):67.
https://doi.org/10.3390/machines7040067

**Chicago/Turabian Style**

Marzaroli, Pietro, Alessandro Albanetti, Edoardo Negri, Hermes Giberti, and Marco Tarabini. 2019. "Design and Testing of a 3-DOF Robot for Studying the Human Response to Vibration" *Machines* 7, no. 4: 67.
https://doi.org/10.3390/machines7040067