#
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

- Gosselin, C.M.; Sefrioui, J.; Richard, M.J. Solutions polynomiales au problème de la cinématique directe des manipulateurs parallèles plans à trois degrés de liberté. Mech. Mach. Theory
**1992**, 27, 107–119. [Google Scholar] [CrossRef] - Peisach, E.E. Determination of the position of the member of three-joint and two-joint four member Assur groups with rotational pairs. Machinowedenie
**1985**, 5, 55–61. (In Russian) [Google Scholar] - Pennock, G.R.; Kassner, D.J. Kinematic analysis of a planar eight-bar linkage: Application to a platform-type robot. J. Mech. Des.
**1992**, 114, 87–95. [Google Scholar] [CrossRef] - Wohlhart, K. Direct kinematic solution of the general planar Stewart platform. In Proceedings of the 3rd International Conference on Computer Integrated Manufacturing, Troy, NY, USA, 20–22 May 1992; pp. 403–411. [Google Scholar]
- Gosselin, C.; Merlet, J.P. The direct kinematics of planar parallel manipulators: Special architectures and number of solutions. Mech. Mach. Theory
**1994**, 29, 1083–1097. [Google Scholar] [CrossRef] - Kong, X.; Gosselin, C. Forward displacement analysis of third-class analytic 3-RPR planar parallel manipulators. Mech. Mach. Theory
**2001**, 39, 1009–1018. [Google Scholar] [CrossRef] - Collins, C.L. Forward kinematics of planar parallel manipulators in the Clifford algebra of P
^{2}. Mech. Mach. Theory**2002**, 37, 799–813. [Google Scholar] [CrossRef] - Rojas, N.; Thomas, F. The forward kinematics of 3-RPR planar robots: A review and a distance-based formulation. IEEE Trans. Robot.
**2011**, 27, 143–150. [Google Scholar] [CrossRef] - Mimura, N.; Funahashi, Y. A new analytical system applying 6 DOF parallel link manipulator for evaluating motion sensation. In Proceedings of the 1995 IEEE International Conference on Robotics and Automation, Nagoya, Japan, 21–27 May 1995; pp. 227–233. [Google Scholar] [CrossRef]
- Gosselin, C.M. Parallel computational algorithms for the kinematics and dynamics of planar and spatial parallel manipulators. ASME J. Dyn. Syst. Meas. Control
**1996**, 118, 22–28. [Google Scholar] [CrossRef] - McAree, P.R.; Daniel, R.W. A fast, robust solution to the Stewart platform forward kinematics. J. Robot. Syst.
**1996**, 13, 407–427. [Google Scholar] [CrossRef] - Šika, Z.; Kočandrle, V.; Stejskal, V. An investigation of properties of the forward displacement analysis of the generalized Stewart platform by means of general optimization methods. Mech. Mach. Theory
**1998**, 33, 245–253. [Google Scholar] [CrossRef] - Der-Ming, K. Direct displacement analysis of a Stewart platform mechanism. Mech. Mach. Theory
**1999**, 34, 453–465. [Google Scholar] [CrossRef] - Dhingra, A.K.; Almadi, A.N.; Kohli, D. A Groebner-Sylvester hybrid method for closed-form displacement analysis of mechanisms. ASME J. Mech. Des.
**2000**, 122, 431–438. [Google Scholar] [CrossRef] - Merlet, J.P. Solving the forward kinematics of a Gough-type parallel manipulator with interval analysis. Int. J. Robot. Res.
**2004**, 23, 221–235. [Google Scholar] [CrossRef] - Shi, X.; Fenton, R.G. Forward kinematic solution of a general 6 DOF Stewart platform based on three point position data. In Proceedings of the Eighth World Congress on the Theory of Machines and Mechanism, Prague, Czechoslovakia, 26–31 August 1991; pp. 1015–1018. [Google Scholar]
- Stoughton, R.; Arai, T. Optimal sensor placement for forward kinematics evaluation of a 6-DOF parallel link manipulator. In Proceedings of the IEEE/RSJ International Workshop on Intelligent Robots and Systems (IROS), Osaka, Japan, 3–5 November 1991; pp. 785–790. [Google Scholar] [CrossRef]
- Cheok, K.C.; Overholt, J.L.; Beck, R.R. Exact methods for determining the kinematics of a Stewart platform using additional displacement sensors. J. Robot. Syst.
**1993**, 10, 689–707. [Google Scholar] [CrossRef] - Merlet, J.P. Closed-form resolution of the direct kinematics of parallel manipulators using extra sensors data. In Proceedings of the IEEE International Conference on Robotics and Automation (ICRA), Atlanta, GA, USA, 2–6 May 1993; pp. 200–204. [Google Scholar] [CrossRef]
- Han, K.; Chung, W.; Youm, Y. New resolution scheme of the forward kinematics of parallel manipulators using extra sensor data. ASME J. Mech. Des.
**1996**, 118, 214–219. [Google Scholar] [CrossRef] - Parenti-Castelli, V.; Gregorio, R.D. Real-time computation of the actual posture of the general geometry 6-6 fully-parallel mechanism using two extra rotary sensors. J. Mech. Des.
**1998**, 120, 549–554. [Google Scholar] [CrossRef] - Bonev, I.A.; Ryu, J. A new method for solving the direct kinematics of general 6-6 Stewart platforms using three linear extra sensors. Mech. Mach. Theory
**2000**, 35, 423–436. [Google Scholar] [CrossRef] - Vertechy, R.; Parenti-Castelli, V. Accurate and fast body pose estimation by three point position data. Mech. Mach. Theory
**2007**, 42, 1170–1183. [Google Scholar] [CrossRef] - Schulz, S.; Seibel, A.; Schreiber, D.; Schlattmann, J. Sensor concept for solving the direct kinematics problem of the Stewart-Gough platform. In Proceedings of the IEEE/RSJ International Conference on Intelligent Robots and Systems (IROS), Vancouver, BC, Canada, 24–28 September 2017; pp. 1959–1964. [Google Scholar] [CrossRef]
- Schulz, S.; Seibel, A.; Schlattmann, J. Closed-form solution for the direct kinematics problem of planar 3-RPR parallel mechanisms. In Proceedings of the IEEE International Conference on Robotics and Automation (ICRA), Brisbane, Australia, 21–25 May 2018; pp. 968–973. [Google Scholar] [CrossRef]
- Seibel, A.; Schulz, S.; Schlattmann, J. On the direct kinematics problem of parallel mechanisms. J. Robot.
**2018**, 2018, 2412608. [Google Scholar] [CrossRef] - Schulz, S.; Seibel, A.; Schlattmann, J. Assembly modes of general planar 3-RPR parallel mechanisms when using the linear actuators’ orientations. In Advances in Mechanism and Machine Science, Proceedings of the 15th IFToMM World Congress on Mechanism and Machine Science, Krakow, Poland, 30 June–4 July 2019; Uhl, T., Ed.; Springer: Cham, Switzerland, 2019; Volume 73, pp. 279–288. [Google Scholar] [CrossRef]
- Merlet, J.P. Parallel Robots; Springer: Dordrecht, The Netherlands, 2006. [Google Scholar] [CrossRef]
- Merlet, J.P. Singular configurations of parallel manipulators and Grassmann geometry. Int. J. Robot. Res.
**1992**, 8, 45–56. [Google Scholar] [CrossRef] - Li, H.; Gosselin, C.M.; Richard, M.J. Determination of maximal singularity-free zones in the workspace of planar three-degree-of-freedom parallel mechanisms. Mech. Mach. Theory
**2006**, 41, 1157–1167. [Google Scholar] [CrossRef] - Zein, M.; Wenger, P.; Chablat, D. Non-singular assembly-mode changing motions for 3-RPR parallel manipulators. Mech. Mach. Theory
**2008**, 43, 480–490. [Google Scholar] [CrossRef] - Wenger, P.; Chablat, D. Kinematic analysis of a class of analytic planar 3-RPR parallel manipulators. In Computational Kinematics; Kecskeméthy, A., Müller, A., Eds.; Springer: Berlin, Germany, 2009; pp. 43–50. [Google Scholar] [CrossRef]
- Wenger, P.; Chablat, D.; Zein, M. Degeneracy study of the forward kinematics of planar 3-RPR parallel manipulators. ASME J. Mech. Des.
**2006**, 129, 1265–1268. [Google Scholar] [CrossRef] - Briot, S.; Arakelian, V.; Bonev, I.A.; Chablat, D.; Wenger, P. Self-motions of general 3-RPR planar parallel robots. Int. J. Robot. Res.
**2008**, 27, 855–866. [Google Scholar] [CrossRef] - Caro, S.; Binaud, N.; Wenger, P. Sensitivity analysis of 3-RPR planar parallel manipulators. ASME J. Mech. Des.
**2009**, 129, 121005. [Google Scholar] [CrossRef] - Staicu, S. Power requirement comparison in the 3-RPR planar parallel robot dynamics. Mech. Mach. Theory
**2009**, 44, 1045–1057. [Google Scholar] [CrossRef] - Chablat, D.; Jha, R.; Caro, S. A framework for the control of a parallel manipulator with several actuation modes. In Proceedings of the IEEE International Conference on Industrial Informatics (INDIN), Poitiers, France, 19–21 July 2016; pp. 190–195. [Google Scholar] [CrossRef]
- Moezi, S.A.; Rafeeyan, M.; Zakeri, E.; Zare, A. Simulation and experimental control of a 3-RPR parallel robot using optimal fuzzy controller and fast on/off solenoid valves based on the PWM wave. ISA Trans.
**2016**, 61, 265–286. [Google Scholar] [CrossRef] - Boudreau, R.; Turkkan, N. Solving the forward kinematics of parallel manipulators with a genetic algorithm. J. Robot. Syst.
**1996**, 13, 111–125. [Google Scholar] [CrossRef] - Sheng, L.; Wan-Long, L.; Yan-chun, D.; Liang, F. Forward kinematics of the Stewart platform using hybrid immune genetic algorithm. In Proceedings of the IEEE International Conference on Mechatronics and Automation, Luoyang, China, 25–28 June 2006; pp. 2330–2335. [Google Scholar] [CrossRef]
- Rolland, L.; Chandra, R. Forward kinematics of the 6-6 general parallel manipulator using real coded genetic algorithms. In Proceedings of the IEEE/ASME Conference on Advanced Intelligent Mechatronics (AIM), Singapore, 14–17 July 2009; pp. 1637–1642. [Google Scholar] [CrossRef]
- Rolland, L.; Chandra, R. The forward kinematics of the 6-6 parallel manipulator using an evolutionary algorithm based on generalized generation gap with parent-centric crossover. Robotica
**2016**, 34, 1–22. [Google Scholar] [CrossRef] - Yee, C.S.; Lim, K.B. Forward kinematics solution of Stewart platform using neural networks. Neurocomputing
**1997**, 16, 333–349. [Google Scholar] [CrossRef] - Parikh, P.J.; Lam, S.S.Y. A hybrid strategy to solve the forward kinematics problem in parallel manipulators. IEEE Trans. Robot.
**2005**, 21, 18–25. [Google Scholar] [CrossRef] - Didrit, O.; Petitot, M.; Walter, E. Guaranteed solution of direct kinematic problems for general configurations of parallel manipulator. IEEE Trans. Robot. Autom.
**1998**, 14, 259–266. [Google Scholar] [CrossRef] - Dieudonne, J.E.; Parrish, R.V.; Bardusch, R.E. An Actuator Extension Transformation for a Motion Simulator and Inverse Transformation Applying Newton-Raphson’s Method; NASA Technical Report TN D-7067; NASA Langley Research Center: Hampton, VA, USA, 1972.
- Nguyen, C.C.; Zhou, Z.L.; Antrazi, S.S.; Campbell, C.E. Efficient computation of forward kinematics and Jacobian matrix of a Stewart platform-based manipulator. In Proceedings of the IEEE Proceedings of the SOUTHEASTCON ’91, Williamsburg, VA, USA, 7–10 April 1991; pp. 869–874. [Google Scholar] [CrossRef]
- Merlet, J.P. Direct kinematics of parallel manipulator. IEEE Trans. Robot. Autom.
**1993**, 9, 842–846. [Google Scholar] [CrossRef] - Liu, K.; Fitzgerald, J.M.; Lewis, F.L. Kinematic analysis of a Stewart platform manipulator. IEEE Trans. Ind. Electron.
**1993**, 40, 282–293. [Google Scholar] [CrossRef] - Yang, C.; Zheng, S.; Jin, J.; Zhu, S.; Han, J. Forward kinematics analysis of parallel manipulator using modified global Newton-Raphson method. J. Cent. South Univ. Technol.
**1996**, 17, 1264–1270. [Google Scholar] [CrossRef] - Vertechy, R.; Parenti-Castelli, V. Robust, fast and accurate solution of the direct position analysis of parallel manipulators by using extra-sensors. In Parallel Manipulators, towards New Applications; Wu, H., Ed.; I-Tech Education and Publishing: Vienna, Austria, 2008; pp. 133–154. [Google Scholar]
- Zhuang, H. Self calibration of parallel mechanisms with a case study on Stewart platform. IEEE Trans. Robot. Autom.
**1997**, 13. [Google Scholar] [CrossRef] - Chiu, Y.J. Forward kinematics of a general fully parallel manipulator with auxiliary sensors. Int. J. Robot. Res.
**2001**, 20, 401–414. [Google Scholar] [CrossRef] - Arai, T.; Cleary, K.; Nakamura, T. Design, Analysis and Construction of a Prototype Parallel Link Manipulator. In Proceedings of the IEEE/RSJ International Workshop on Intelligent Robots and Systems (IROS), Ibaraki, Japan, 3–6 July 1990; pp. 205–212. [Google Scholar] [CrossRef]
- Baron, L.; Angeles, J. The direct kinematics of parallel manipulators under redundant sensors. IEEE Trans. Robot. Autom.
**2000**, 16, 12–19. [Google Scholar] [CrossRef] - Baron, L.; Angeles, J. The kinematic decoupling of parallel manipulators using joint-sensor redundancy. IEEE Trans. Robot. Autom.
**2000**, 16, 644–651. [Google Scholar] [CrossRef] - Bonev, I.A.; Ryu, J.; Kim, N.J.; Lee, S.K. A simple new closed-form solution of the direct kinematics of parallel manipulators using three linear extra sensors. In Proceedings of the IEEE/ASME International Conference on Advanced Intelligent Mechatronics (AIM), Atlanta, GA, USA, 19–23 September 1999; pp. 526–530. [Google Scholar] [CrossRef]
- Bonev, I.A.; Ryu, J.; Kim, N.J.; Lee, S.K. A closed-form solution to the direct kinematics of nearly general parallel manipulators with optimally located three linear extra sensors. IEEE Trans. Robot. Autom.
**2001**, 17, 148–156. [Google Scholar] [CrossRef] - Etemadi-Zanganeh, K.; Angeles, J. Real-time direct kinematics of general six-degree-of-freedom parallel manipulators with minimum-sensor data. J. Robot. Syst.
**1995**, 12, 833–844. [Google Scholar] [CrossRef] - Han, K.; Chung, W.; Youm, Y. Local Structurization for the Forward Kinematics of Parallel Manipulators Using Extra Sensor Data. In Proceedings of the IEEE International Conference on Robotics and Automation (ICRA), Nagoya, Japan, 21–27 May 1995; pp. 514–520. [Google Scholar] [CrossRef]
- Nair, R.; Maddocks, J.H. On the Forward Kinematics of Parallel Manipulators. Int. J. Robot. Res.
**1994**, 13, 171–188. [Google Scholar] [CrossRef] - Innocenti, C. Closed-Form Determination of the Location of a Rigid Body by Seven In-Parallel Linear Transducers. J. Mech. Des.
**1998**, 120, 293–298. [Google Scholar] [CrossRef] - Parenti-Castelli, V.; Gregorio, R.D. Determination of the Actual Configuration of the General Stewart Platform Using Only One Additional Sensor. J. Mech. Des.
**1999**, 121, 21–25. [Google Scholar] [CrossRef] - Parenti-Castelli, V.; Gregorio, R.D. A New Algorithm Based on Two Extra-Sensors for Real-Time Computation of the Actual Configuration of the Generalized Stewart-Gough Manipulator. J. Mech. Des.
**2000**, 122, 294–298. [Google Scholar] [CrossRef] - Tancredi, L.; Merlet, J.P. Extra sensors data for solving the forward kinematics problem of parallel manipulators. In Proceedings of the 9th World Congress on the Theory of Machines and Mechanisms, Milan, Italy, 29 August–2 September 1995; pp. 2122–2126. [Google Scholar]
- Tancredi, L.; Teillaud, M.; Merlet, J.P. Forward Kinematics of a Parallel Manipulator with Additional Rotary Sensors Measuring the Position of Platform Joints. In Computational Kinematics; Merlet, J.P., Ravani, B., Eds.; Solid Mechanics and Its Applications; Kluwer Academic Publishers: Dordrecht, The Netherlands, 1995; Volume 40, pp. 261–270. [Google Scholar] [CrossRef]
- Vertechy, R.; Dunlop, G.R.; Parenti-Castelli, V. An accurate algorithm for the real-time solution of the direct kinematics of 6-3 Stewart platform manipulators. In Advances in Robot Kinematics; Lenarčič, J., Thomas, F., Eds.; Solid Mechanics and Its Applications; Kluwer Academic Publishers: Dordrecht, The Netherlands, 2002; Volume 40, pp. 369–378. [Google Scholar] [CrossRef]
- Jin, Y. Exact solution for the forward kinematics of the general Stewart platform using two additional displacement sensors. In Proceedings of the 23th ASME Biennial Mechanism Conference, Minneapolis, MN, USA, 11–14 September 1994; pp. 491–945. [Google Scholar]
- Baron, L.; Angeles, J. A linear algebraic solution of the direct kinematics of parallel manipulators using a camera. In Proceedings of the 9th World Congress on the Theory of Machines and Mechanisms, Milano, Italy, 29 August–2 September 1995; pp. 1925–1929. [Google Scholar]
- Hesselbach, J.; Bier, C.; Pietsch, I.; Plitea, N.; Buttenbach, S.; Wogersien, A.; Guttler, J. Passive joint-sensor applications for parallel robots. In Proceedings of the IEEE/RJS International Conference on Intelligent Robots and Systems (IROS), Sendai, Japan, 28 September–2 October 2004; pp. 3507–3512. [Google Scholar] [CrossRef]
- Besnard, S.; Khalil, W. Calibration of parallel robot using two inclinometers. In Proceedings of the IEEE International Conference on Robotics and Automation (ICRA), Detroit, MI, USA, 10–15 May 1999; pp. 1758–1763. [Google Scholar] [CrossRef]
- Yun, X.; Lizarraga, M.; Bachmann, E.R.; McGhee, R.B. An improved quaternion-based Kalman filter for real-time tracking of rigid body orientation. In Proceedings of the IEEE/RSJ International Conference on Intelligent Robots and Systems (IROS), Las Vegas, NV, USA, 27–31 October 2003; pp. 1074–1079. [Google Scholar] [CrossRef]
- Mahony, R.; Hamel, T.; Pflimlin, J.M. Nonlinear complementary filters on the special orthogonal group. IEEE Trans. Autom. Control
**2008**, 53, 1203–1218. [Google Scholar] [CrossRef] - Madgwick, S.O.H.; Harrison, A.J.L.; Vaidyanathan, R. Estimation of IMU and MARG orientation using a gradient descent algorithm. In Proceedings of the IEEE International Conference on Rehabilitation Robotics (ICORR), Zurich, Switzerland, 29 June–1 July 2011; pp. 179–185. [Google Scholar] [CrossRef]
- Valenti, R.G.; Dryanovski, I.; Xiao, J. Keeping a good attitude: A quaternion-based orientation filter for IMUs and MARGs. Sensors
**2015**, 15, 19302–19330. [Google Scholar] [CrossRef] - Briot, S.; Martinet, P.; Rosenzveig, V. The hidden robot: An efficient concept contributing to the analysis of the controllability of parallel robots in advanced visual servoing techniques. IEEE Trans. Robot.
**2015**, 31, 1337–1352. [Google Scholar] [CrossRef]

**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 ${\left[10\phantom{\rule{0.166667em}{0ex}}\mathrm{mm}\phantom{\rule{1.em}{0ex}}50\phantom{\rule{0.166667em}{0ex}}\mathrm{mm}\phantom{\rule{1.em}{0ex}}{0}^{\circ}\right]}^{\top}$ (blue), for ${\left[50\phantom{\rule{0.166667em}{0ex}}\mathrm{mm}\phantom{\rule{1.em}{0ex}}20\phantom{\rule{0.166667em}{0ex}}\mathrm{mm}\phantom{\rule{1.em}{0ex}}{20}^{\circ}\right]}^{\top}$ (red) and for ${\left[0\phantom{\rule{0.166667em}{0ex}}\mathrm{mm}\phantom{\rule{1.em}{0ex}}0\phantom{\rule{0.166667em}{0ex}}\mathrm{mm}\phantom{\rule{1.em}{0ex}}{0}^{\circ}\right]}^{\top}$ (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 ${\left[{\sigma}^{2}\left(x\right)\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.166667em}{0ex}}{\sigma}^{2}\left(y\right)\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.166667em}{0ex}}{\sigma}^{2}\left(\gamma \right)\right]}^{\top}$. 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