# On Using Inertial Measurement Units for Solving the Direct Kinematics Problem of Parallel Mechanisms

## Abstract

**:**

## 1. Introduction

## 2. Direct Kinematics Solution for Planar 3-RPR Parallel Mechanisms

#### 2.1. Robust Orientation Measurements

#### 2.2. Robust Pose Calculations

#### 2.2.1. Linear Least-Squares Formulation

#### 2.2.2. Sensor Fusion

## 3. Experiment

#### 3.1. Achievable Sampling Rates

#### 3.2. Robust Orientation Measurements

#### 3.3. Robust Pose Calculations

## 4. Discussion

## 5. Conclusions

## Supplementary Materials

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**Planar 3-RPR parallel mechanism with three linear actuators, three base platform joints, and three manipulator platform joints.

**Figure 2.**Schematic diagram of the linear least-squares formulation for solving the direct kinematics problem of the planar 3-RPR parallel mechanism.

**Figure 3.**Specially designed prototype of a planar 3-RPR parallel mechanism. Actuonix L16-100-35-P linear actuators are used with a minimum length of 168 mm and a stroke length of 100 mm. As IMUs, InvenSense MPU-9250 sensors are chosen and mounted on the linear actuators as well as the manipulator platform. The device also consists of a power supply, a display, an Arduino Mega board with a data acquisition and pose calculation algorithm, and a motor shield for controlling the linear actuators’ lengths.

**Figure 4.**Trajectories of the first (blue), second (red), and third (green) manipulator platform joint during the experiments. The trajectory is recorded by a camera, and the joints’ positions are analyzed using image processing.

**Figure 5.**Orientation errors of the first linear actuator, the second linear actuator, the third linear actuator, and the manipulator platform. The orientation angles are obtained from the accelerometer values (blue), the complementary filter (red), and the Kalman filter (green). As reference, the orientations calculated from the optically analyzed manipulator platform joints are used.

**Figure 6.**Boxplots of the orientation errors of the first linear actuator, the third linear actuator, and the manipulator platform obtained from the accelerometer values, the complementary filter, and the Kalman filter, respectively. The box corresponds to the area in which the middle 50% of the errors lie while the whiskers indicate the area in which the middle 99.3% of the errors lie.

**Figure 7.**Lengths errors of the first, second, and third linear actuator. The lengths are calculated from the linear actuators’ potentiometer values and compared with the actual lengths that are calculated from the optically derived manipulator platform’s pose by using inverse kinematics.

**Figure 8.**Position and orientation errors of the manipulator platform’s pose calculated with the linear least-squares formulation and the filtered first and third linear actuators’ orientations. In addition, the position and orientation errors of the manipulator platform’s pose calculated with the linear actuators’ lengths and the Newton Raphson algorithm are shown. As ground truth, in both cases, the positions and orientations calculated from the optically analyzed manipulator platform joints are used.

**Figure 9.**Boxplots of the position and orientation errors of the manipulator platform’s pose during the experiment calculated with the linear least-squares formulation and the filtered first and third linear actuators’ orientations. In addition, boxplots of the position and orientation errors of the manipulator platform’s pose calculated with the linear actuators’ lengths and the Newton Raphson algorithm are shown. As ground truth, in both cases, the positions and orientations calculated from the optically analyzed manipulator platform joints are used. The box corresponds to the area in which the middle 50% of the errors lie while the whiskers indicate the area in which the middle 99.3% of the errors lie.

**Figure 10.**Position and orientation errors of the manipulator platform’s pose calculated with the linear least-squares formulation and the filtered first, second, and third linear actuators’ orientations. As ground truth, in both cases, the positions and orientations calculated from the optically analyzed manipulator platform joints are used.

Base Platform Joints | Manipulator Platform Joints | ||||
---|---|---|---|---|---|

${}^{1}{\mathit{p}}_{1{J}_{1,1}}$ | ${}^{1}{\mathit{p}}_{1{J}_{2,1}}$ | ${}^{1}{\mathit{p}}_{1{J}_{3,1}}$ | ${}^{2}{\mathit{p}}_{2{J}_{1,2}}$ | ${}^{2}{\mathit{p}}_{2{J}_{2,2}}$ | ${}^{2}{\mathit{p}}_{2{J}_{3,2}}$ |

$\left(\right)$ | $\left(\right)$ | $\left(\right)$ | $\left(\right)$ | $\left(\right)$ | $\left(\right)$ |

**Table 2.**Achievable sampling rates of the Arduino Mega depending on the calculations contained and the displayed data. The sampling rates are considered as the inverse of the mean time between two measurements evaluated over 1000 measurements.

No. | Included Calculation | Displayed Data | Sampling Rate |
---|---|---|---|

1 | length measurement (3 lengths) | 4 | 399.71 Hz |

2 | orientation measurement (4 IMUs) | 16 | 71.27 Hz |

3 | #1 + length control | 4 | 293.88 Hz |

4 | #1 + #2 + length control | 16 | 57.85 Hz |

5a | #4 + raw angles | 5 | 70.95 Hz |

5b | #4 + orientation filtering (complementary filter) | 5 | 60.41 Hz |

5c | #4 + orientation filtering (Kalman filter) | 5 | 53.02 Hz |

6a | #5a + linear least-squares formulation | 4 | 46.20 Hz |

6b | #5b + linear least-squares formulation | 4 | 44.82 Hz |

6c | #5c + linear least-squares formulation | 4 | 40.82 Hz |

7 | #3 + Newton Raphson algorithm | 4 | 3.82 Hz |

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**MDPI and ACS Style**

Schulz, S.
On Using Inertial Measurement Units for Solving the Direct Kinematics Problem of Parallel Mechanisms. *Robotics* **2019**, *8*, 99.
https://doi.org/10.3390/robotics8040099

**AMA Style**

Schulz S.
On Using Inertial Measurement Units for Solving the Direct Kinematics Problem of Parallel Mechanisms. *Robotics*. 2019; 8(4):99.
https://doi.org/10.3390/robotics8040099

**Chicago/Turabian Style**

Schulz, Stefan.
2019. "On Using Inertial Measurement Units for Solving the Direct Kinematics Problem of Parallel Mechanisms" *Robotics* 8, no. 4: 99.
https://doi.org/10.3390/robotics8040099