#
Performance Evaluation of a Sensor Concept for Solving the Direct Kinematics Problem of General Planar 3-R__P__R Parallel Mechanisms by Using Solely the Linear Actuators’ Orientations

## Abstract

**:**

## 1. Introduction

## 2. Review of Classical Solutions for the Direct Kinematics Problem

#### 2.1. Analytical Solution

#### 2.2. Numerical Solution

#### 2.3. Additional Sensor Solution

## 3. Assembly Modes when Using the Linear Actuators’ Orientations

## 4. Cramér-Rao Lower Bound

## 5. Experiments

#### 5.1. Experimental Device

#### 5.2. Dynamic Orientation Measurement

#### 5.3. Accuracy of the Orientation Measurements

#### 5.4. Accuracy of Static Pose Detections

#### 5.5. Comparing Analytic Orientation-Based Results with Iterative Length-Based Results for Static Pose Detections

#### 5.6. Accuracy of Dynamic Pose Detections

## 6. Conclusions

## Supplementary Materials

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**General planar 3-RPR parallel mechanism with the three base platform joints A, B and C and the three manipulator platform joints D, E and F. The pose of the manipulator platform is given by the position of joint D and the platform’s orientation $\gamma $ with respect to the shown coordinate system.

**Figure 2.**Assembly modes (shown in blue, red, green, orange, yellow and brown) for the manipulator platform of the general planar 3-RPR parallel mechanism when using the linear actuators’ lengths ${\rho}_{1}$, ${\rho}_{2}$ and ${\rho}_{3}$ from Equation (14).

**Figure 3.**Solutions for the general planar 3-RPR parallel mechanism when using a Newton-Raphson algorithm with the linear actuators’ lengths: solution for ${\left(\right)}^{10}$ (blue), for ${\left(\right)}^{50}$ (red) and for ${\left(\right)}^{0}$ (green) as initial pose estimates.

**Figure 4.**General planar 3-RPR parallel mechanism with two additional passive linear actuators with the base platform joints G and H. The active linear actuators are shown in black and the supplementary passive linear actuators are shown in blue.

**Figure 5.**Actual solution (blue) for the general planar 3-RPR parallel mechanism when using two additional lengths in addition to the linear actuators’ lengths. The second solution is shown in red.

**Figure 6.**The two assembly modes (shown in blue and red) for the manipulator platform of the general planar 3-RPR parallel mechanism when using the linear actuators’ orientations: (

**a**) results for ${\phi}_{1}={82.8750}^{\circ}$, ${\phi}_{2}={96.0453}^{\circ}$ and ${\phi}_{3}={106.5502}^{\circ}$ and (

**b**) results for ${\phi}_{1}={82.8750}^{\circ}$, ${\phi}_{2}={94.7360}^{\circ}$ and ${\phi}_{3}={101.0877}^{\circ}$.

**Figure 7.**Experimental prototype of the general planar 3-RPR parallel mechanism with inertial measurement units (IMUs) mounted on the linear actuators and an Arduino Mega with a display integrated in the base to calculate and show the two assembly modes of the manipulator platform.

**Figure 8.**Experimental test bench to investigate IMUs on the dependency of the orientation angles’ variances on the orientation angle (

**a**). Experimental results of the InvenSense MPU-9250: (

**b**) variances of the raw angle and suitable fifth-order polynomial fit and (

**c**) variances of the filtered orientation angle and suitable fifth-order polynomial fit.

**Figure 9.**Results for the ten investigated static poses with 500 repetitions obtained experimentally from the raw accelerometer values (red) and the filtered orientation angles (blue). The errors in each axis, $\Delta x$, $\Delta y$ and $\Delta \gamma $, are displayed in a boxplot. Dimensions are in mm and ${}^{\circ}$. 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.**Results for the first five investigated static poses with 500 repetitions obtained experimentally from the filtered orientation angles (blue) and by simulation using the corresponding Cramér-Rao lower bound (CRLB) (purple). The errors in each axis, $\Delta x$, $\Delta y$ and $\Delta \gamma $, are displayed in a boxplot. Dimensions are in mm and ${}^{\circ}$. 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 11.**Trajectories of the first (blue), second (red) and third (green) manipulator platform joint during the dynamic experiment. The trajectories were recorded by a camera with 30 fps and the joints’ positions were analysed using image processing.

**Figure 12.**Pose of the manipulator platform during the dynamic experiment calculated from the raw (red) and the filtered (blue) linear actuators’ orientations: (

**a**) x-position, (

**b**) y-position and (

**c**) orientation angle $\gamma $. As reference (black), the positions and orientations calculated from the optically analysed manipulator platform joints are used.

**Figure 13.**Boxplots of the position and orientation errors of the manipulator platform’s pose during the experiment calculated with the raw orientation angles (red) and the complementary filtered orientation angles (blue). 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 14.**Conventional (

**a**) and proposed control concept (

**b**) for controlling the manipulator platform’s pose of a parallel mechanism. The conventional control concept uses the linear actuators’ lengths, whereas the proposed control concept uses the linear actuators’ orientations. In contrast to the conventional control concept, the proposed control concept can guarantee an analytic solution of the direct kinematics problem.

**Table 1.**Constants of the fifth-order polynomial for describing the orientation angle’s variances of the raw and the filtered orientation angle as a function of the orientation angle itself.

${\mathit{a}}_{0}$ | ${\mathit{a}}_{1}$ | ${\mathit{a}}_{2}$ | ${\mathit{a}}_{3}$ | ${\mathit{a}}_{4}$ | ${\mathit{a}}_{5}$ | |
---|---|---|---|---|---|---|

${\phi}_{\mathrm{acc}}$ | 3.8553$\phantom{\rule{0.166667em}{0ex}}\times \phantom{\rule{0.166667em}{0ex}}{10}^{-2}$ | $\phantom{-}$2.7241$\phantom{\rule{0.166667em}{0ex}}\times \phantom{\rule{0.166667em}{0ex}}{10}^{-1}$ | −2.7631$\phantom{\rule{0.166667em}{0ex}}\times \phantom{\rule{0.166667em}{0ex}}{10}^{-2}$ | 1.0431$\phantom{\rule{0.166667em}{0ex}}\times \phantom{\rule{0.166667em}{0ex}}{10}^{-3}$ | −1.5467$\phantom{\rule{0.166667em}{0ex}}\times \phantom{\rule{0.166667em}{0ex}}{10}^{-5}$ | 7.8730$\phantom{\rule{0.166667em}{0ex}}\times \phantom{\rule{0.166667em}{0ex}}{10}^{-8}$ |

${\phi}_{\mathrm{com}}$ | 1.9579$\phantom{\rule{0.166667em}{0ex}}\times \phantom{\rule{0.166667em}{0ex}}{10}^{-3}$ | −1.6773$\phantom{\rule{0.166667em}{0ex}}\times \phantom{\rule{0.166667em}{0ex}}{10}^{-5}$ | −5.1628$\phantom{\rule{0.166667em}{0ex}}\times \phantom{\rule{0.166667em}{0ex}}{10}^{-7}$ | 5.0914$\phantom{\rule{0.166667em}{0ex}}\times \phantom{\rule{0.166667em}{0ex}}{10}^{-8}$ | −8.2501$\phantom{\rule{0.166667em}{0ex}}\times \phantom{\rule{0.166667em}{0ex}}{10}^{-10}$ | 4.2277$\phantom{\rule{0.166667em}{0ex}}\times \phantom{\rule{0.166667em}{0ex}}{10}^{-12}$ |

**Table 2.**Investigated static poses and mean values of the calculated poses (solution I and solution II) after 500 measurements obtained from the raw orientation angles. Dimensions are in mm and ${}^{\circ}$.

Pose | Actual Pose | Solution I | Solution II |
---|---|---|---|

${\left[\begin{array}{ccc}\mathit{x}& \mathit{y}& \mathit{\gamma}\end{array}\right]}^{\top}$ | ${\left[\begin{array}{ccc}{\mathit{x}}_{\mathbf{I}}& {\mathit{y}}_{\mathbf{I}}& {\mathit{\gamma}}_{\mathbf{I}}\end{array}\right]}^{\top}$ | ${\left[\begin{array}{ccc}{\mathit{x}}_{\mathbf{II}}& {\mathit{y}}_{\mathbf{II}}& {\mathit{\gamma}}_{\mathbf{II}}\end{array}\right]}^{\top}$ | |

1 | ${\left[\begin{array}{ccc}146.76& 190.46& \phantom{-}14.01\end{array}\right]}^{\top}$ | ${\left[\begin{array}{ccc}148.10& 190.89& \phantom{-}14.63\end{array}\right]}^{\top}$ | ${\left[\begin{array}{ccc}309.13& 398.44& -146.85\end{array}\right]}^{\top}$ |

2 | ${\left[\begin{array}{ccc}\phantom{1}90.71& 212.00& -20.38\end{array}\right]}^{\top}$ | ${\left[\begin{array}{ccc}\phantom{1}94.96& 220.66& -24.55\end{array}\right]}^{\top}$ | ${\left[\begin{array}{ccc}146.43& 340.08& -\phantom{1}84.27\end{array}\right]}^{\top}$ |

3 | ${\left[\begin{array}{ccc}137.55& 206.21& -\phantom{1}7.71\end{array}\right]}^{\top}$ | ${\left[\begin{array}{ccc}137.68& 206.42& -\phantom{1}6.67\end{array}\right]}^{\top}$ | ${\left[\begin{array}{ccc}250.39& 375.34& -107.89\end{array}\right]}^{\top}$ |

4 | ${\left[\begin{array}{ccc}155.25& 191.61& \phantom{-}15.72\end{array}\right]}^{\top}$ | ${\left[\begin{array}{ccc}155.41& 190.95& \phantom{-}17.04\end{array}\right]}^{\top}$ | ${\left[\begin{array}{ccc}322.82& 396.65& -151.26\end{array}\right]}^{\top}$ |

5 | ${\left[\begin{array}{ccc}123.65& 211.69& -11.96\end{array}\right]}^{\top}$ | ${\left[\begin{array}{ccc}124.50& 211.93& -11.72\end{array}\right]}^{\top}$ | ${\left[\begin{array}{ccc}217.26& 369.73& -\phantom{1}99.65\end{array}\right]}^{\top}$ |

6 | ${\left[\begin{array}{ccc}\phantom{1}69.22& 215.68& -12.16\end{array}\right]}^{\top}$ | ${\left[\begin{array}{ccc}\phantom{1}74.53& 228.82& -18.83\end{array}\right]}^{\top}$ | ${\left[\begin{array}{ccc}124.70& 382.65& -\phantom{1}90.18\end{array}\right]}^{\top}$ |

7 | ${\left[\begin{array}{ccc}107.01& 190.51& \phantom{-1}0.71\end{array}\right]}^{\top}$ | ${\left[\begin{array}{ccc}107.98& 192.76& -\phantom{1}0.73\end{array}\right]}^{\top}$ | ${\left[\begin{array}{ccc}219.71& 392.17& -120.73\end{array}\right]}^{\top}$ |

8 | ${\left[\begin{array}{ccc}\phantom{1}64.62& 186.05& \phantom{-}15.84\end{array}\right]}^{\top}$ | ${\left[\begin{array}{ccc}\phantom{1}66.82& 191.85& \phantom{-}10.75\end{array}\right]}^{\top}$ | ${\left[\begin{array}{ccc}156.08& 448.07& -144.02\end{array}\right]}^{\top}$ |

9 | ${\left[\begin{array}{ccc}125.21& 161.73& \phantom{-}13.32\end{array}\right]}^{\top}$ | ${\left[\begin{array}{ccc}125.31& 162.46& \phantom{-}13.50\end{array}\right]}^{\top}$ | ${\left[\begin{array}{ccc}276.07& 357.91& -153.10\end{array}\right]}^{\top}$ |

10 | ${\left[\begin{array}{ccc}132.37& 157.14& \phantom{-1}8.30\end{array}\right]}^{\top}$ | ${\left[\begin{array}{ccc}132.71& 158.64& \phantom{-1}7.45\end{array}\right]}^{\top}$ | ${\left[\begin{array}{ccc}284.36& 339.89& -142.95\end{array}\right]}^{\top}$ |

**Table 3.**Variances and results for the Cramér-Rao lower bound for the first five static poses when using raw orientation angles and when using filtered orientation angles. The variances are displayed as ${\left(\right)}^{{\sigma}^{2}}$. Dimensions are in mm${}^{2}$ and ${}^{\circ 2}$.

Pose | Variances for Raw Orientation Angles | Variances for Filtered Orientation Angles | ||
---|---|---|---|---|

Experiments | CRLB | Experiments | CRLB | |

1 | $\left[\begin{array}{c}\phantom{1}1.8895\\ \phantom{1}4.4329\\ \phantom{1}6.9274\end{array}\right]$ | $\left[\begin{array}{c}\phantom{1}1.6150\\ \phantom{1}4.1013\\ \phantom{1}7.6160\end{array}\right]$ | $\left[\begin{array}{c}\phantom{1}0.0337\\ \phantom{1}0.2244\\ \phantom{1}0.2335\end{array}\right]$ | $\left[\begin{array}{c}\phantom{1}0.0715\\ \phantom{1}0.1697\\ \phantom{1}0.3470\end{array}\right]$ |

2 | $\left[\begin{array}{c}\phantom{1}7.3331\\ 65.1472\\ 20.3617\end{array}\right]$ | $\left[\begin{array}{c}\phantom{1}7.7918\\ 57.1507\\ 23.6220\end{array}\right]$ | $\left[\begin{array}{c}\phantom{1}0.1493\\ \phantom{1}2.4503\\ \phantom{1}0.6109\end{array}\right]$ | $\left[\begin{array}{c}\phantom{1}0.2968\\ \phantom{1}2.2853\\ \phantom{1}0.9335\end{array}\right]$ |

3 | $\left[\begin{array}{c}\phantom{1}0.6376\\ \phantom{1}7.9631\\ \phantom{1}6.0536\end{array}\right]$ | $\left[\begin{array}{c}\phantom{1}1.2437\\ \phantom{1}9.4596\\ \phantom{1}9.7948\end{array}\right]$ | $\left[\begin{array}{c}\phantom{1}0.0767\\ \phantom{1}0.4116\\ \phantom{1}0.3189\end{array}\right]$ | $\left[\begin{array}{c}\phantom{1}0.0582\\ \phantom{1}0.3787\\ \phantom{1}0.4445\end{array}\right]$ |

4 | $\left[\begin{array}{c}\phantom{1}1.5456\\ \phantom{1}3.5633\\ \phantom{1}5.6761\end{array}\right]$ | $\left[\begin{array}{c}\phantom{1}2.5668\\ \phantom{1}4.9758\\ \phantom{1}7.7996\end{array}\right]$ | $\left[\begin{array}{c}\phantom{1}0.0594\\ \phantom{1}0.1653\\ \phantom{1}0.2352\end{array}\right]$ | $\left[\begin{array}{c}\phantom{1}0.1070\\ \phantom{1}0.2006\\ \phantom{1}0.3483\end{array}\right]$ |

5 | $\left[\begin{array}{c}\phantom{1}1.6415\\ 14.4296\\ \phantom{1}8.1143\end{array}\right]$ | $\left[\begin{array}{c}\phantom{1}3.0573\\ 18.2156\\ 13.9312\end{array}\right]$ | $\left[\begin{array}{c}\phantom{1}0.0593\\ \phantom{1}0.4560\\ \phantom{1}0.3463\end{array}\right]$ | $\left[\begin{array}{c}\phantom{1}0.1263\\ \phantom{1}0.7334\\ \phantom{1}0.5923\end{array}\right]$ |

**Table 4.**Investigated static poses and mean offset errors $\Delta x$, $\Delta y$ and $\Delta \gamma $ of the analytic, orientation-based formulation and the iterative length-based solution (Newton-Raphson algorithm). Dimensions are in mm and ${}^{\circ}$. Poses, where the algorithm fails to converge are indicated by a $--$.

Pose | Actual Pose | Offset Error Solution I | Offset Error Newton-Raphson Algorithm |
---|---|---|---|

${\left[\begin{array}{ccc}\mathit{x}& \mathit{y}& \mathit{\gamma}\end{array}\right]}^{\top}$ | ${\left[\begin{array}{ccc}\mathsf{\Delta}\mathit{x}& \mathsf{\Delta}\mathit{y}& \mathsf{\Delta}\mathit{\gamma}\end{array}\right]}^{\top}$ | ${\left[\begin{array}{ccc}\mathsf{\Delta}\mathit{x}& \mathsf{\Delta}\mathit{y}& \mathsf{\Delta}\mathit{\gamma}\end{array}\right]}^{\top}$ | |

1 | ${\left[\begin{array}{ccc}146.76& 190.46& \phantom{-}14.01\end{array}\right]}^{\top}$ | ${\left[\begin{array}{ccc}-1.25& -\phantom{1}0.23& -0.66\end{array}\right]}^{\top}$ | ${\left[\begin{array}{ccc}\phantom{-}\phantom{1}8.18& -\phantom{1}9.22& \phantom{-}4.09\end{array}\right]}^{\top}$ |

2 | ${\left[\begin{array}{ccc}\phantom{1}90.71& 212.00& -20.38\end{array}\right]}^{\top}$ | ${\left[\begin{array}{ccc}-3.99& -\phantom{1}7.83& \phantom{-}3.88\end{array}\right]}^{\top}$ | ${\left[\begin{array}{ccc}-\phantom{1}1.85& -2.77& \phantom{-}\phantom{1}1.58\end{array}\right]}^{\top}$ |

3 | ${\left[\begin{array}{ccc}137.55& 206.21& -\phantom{1}7.71\end{array}\right]}^{\top}$ | ${\left[\begin{array}{ccc}\phantom{-}0.11& \phantom{-}\phantom{1}0.39& -1.35\end{array}\right]}^{\top}$ | ${\left[\begin{array}{ccc}-\phantom{1}0.42& -3.79& \phantom{-}\phantom{1}1.38\end{array}\right]}^{\top}$ |

4 | ${\left[\begin{array}{ccc}155.25& 191.61& \phantom{-}15.72\end{array}\right]}^{\top}$ | ${\left[\begin{array}{ccc}-0.05& \phantom{-}\phantom{1}1.03& -1.48\end{array}\right]}^{\top}$ | ${\left[\begin{array}{ccc}\phantom{-}\phantom{1}2.77& -\phantom{1}4.41& \phantom{-}0.78\end{array}\right]}^{\top}$ |

5 | ${\left[\begin{array}{ccc}123.65& 211.69& -11.96\end{array}\right]}^{\top}$ | ${\left[\begin{array}{ccc}-0.59& \phantom{-}\phantom{1}0.51& -0.60\end{array}\right]}^{\top}$ | ${\left[\begin{array}{ccc}-\phantom{1}0.82& -2.92& \phantom{-}\phantom{1}1.36\end{array}\right]}^{\top}$ |

6 | ${\left[\begin{array}{ccc}\phantom{1}69.22& 215.68& -12.16\end{array}\right]}^{\top}$ | ${\left[\begin{array}{ccc}-5.16& -12.22& \phantom{-}6.42\end{array}\right]}^{\top}$ | ${\left[\begin{array}{ccc}\phantom{-}\phantom{1}1.32& -\phantom{1}3.69& \phantom{-}2.25\end{array}\right]}^{\top}$ |

7 | ${\left[\begin{array}{ccc}107.01& 190.51& \phantom{-1}0.71\end{array}\right]}^{\top}$ | ${\left[\begin{array}{ccc}-0.90& -\phantom{1}1.85& \phantom{-}1.26\end{array}\right]}^{\top}$ | ${\left[\begin{array}{ccc}-\phantom{1}3.12& \phantom{-}0.21& -\phantom{1}1.08\end{array}\right]}^{\top}$ |

8 | ${\left[\begin{array}{ccc}\phantom{1}64.62& 186.05& \phantom{-}15.84\end{array}\right]}^{\top}$ | ${\left[\begin{array}{ccc}-2.21& -\phantom{1}5.55& \phantom{-}5.16\end{array}\right]}^{\top}$ | ${\left[\begin{array}{ccc}-32.48& \phantom{-}5.86& -11.14\end{array}\right]}^{\top}$ |

9 | ${\left[\begin{array}{ccc}125.21& 161.73& \phantom{-}13.32\end{array}\right]}^{\top}$ | ${\left[\begin{array}{ccc}-0.02& -\phantom{1}0.41& -0.15\end{array}\right]}^{\top}$ | ${\left[\begin{array}{ccc}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.166667em}{0ex}}--& \phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.166667em}{0ex}}--& \phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.166667em}{0ex}}--\end{array}\right]}^{\top}$ |

10 | ${\left[\begin{array}{ccc}132.37& 157.14& \phantom{-1}8.30\end{array}\right]}^{\top}$ | ${\left[\begin{array}{ccc}-0.25& -\phantom{1}1.26& \phantom{-}0.82\end{array}\right]}^{\top}$ | ${\left[\begin{array}{ccc}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.166667em}{0ex}}--& \phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.166667em}{0ex}}--& \phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.166667em}{0ex}}--\end{array}\right]}^{\top}$ |

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## Share and Cite

**MDPI and ACS Style**

Schulz, S.
Performance Evaluation of a Sensor Concept for Solving the Direct Kinematics Problem of General Planar 3-R__P__R Parallel Mechanisms by Using Solely the Linear Actuators’ Orientations. *Robotics* **2019**, *8*, 72.
https://doi.org/10.3390/robotics8030072

**AMA Style**

Schulz S.
Performance Evaluation of a Sensor Concept for Solving the Direct Kinematics Problem of General Planar 3-R__P__R Parallel Mechanisms by Using Solely the Linear Actuators’ Orientations. *Robotics*. 2019; 8(3):72.
https://doi.org/10.3390/robotics8030072

**Chicago/Turabian Style**

Schulz, Stefan.
2019. "Performance Evaluation of a Sensor Concept for Solving the Direct Kinematics Problem of General Planar 3-R__P__R Parallel Mechanisms by Using Solely the Linear Actuators’ Orientations" *Robotics* 8, no. 3: 72.
https://doi.org/10.3390/robotics8030072