# Motion Planning of Differentially Flat Planar Underactuated Robots

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Mathematical Model of the Underactuated Robot

- The last link center of mass (n) lies on the axis of the n-th joint;
- The overall center of mass of the links n and $n-1$ lies on the axis of the $(n-1)$-th joint.

- There are neither gravitational nor Coriolis-centrifugal torques on the last joints because the last two links are fully balanced (the last two elements of vectors $\mathit{b}(\mathit{q},\dot{\mathit{q}})$ and $\mathit{g}\left(\mathit{q}\right)$ are null);
- The last element of $\tau $ is null because there is no motor on the last joint (i.e., a passive joint);
- Matrices ${\mathit{C}}_{n}$ and ${\mathit{K}}_{n}$ are entirely null except the last bottom-right element, in which there are the torsional stiffness (${k}_{n}$) and damping coefficient (${c}_{n}$) of the passive joint; this happens because neither elastic nor dissipative phenomena are included in the actuated joints.

## 3. Trajectory Planning

#### 3.1. Undamped Robot (${c}_{n}=0$)

- Initial point: Initial values of flat variable ${y}_{1}$ are based on the initial values of joint variables, and flat variable derivatives are set to zero:$$\begin{array}{c}{y}_{11}\left(0\right)={\displaystyle \sum _{i=1}^{n}}{q}_{i,0}\\ {\dot{y}}_{11}\left(0\right)={\ddot{y}}_{11}\left(0\right)={y}_{11}^{\left(3\right)}\left(0\right)={y}_{11}^{\left(4\right)}\left(0\right)=0\end{array}$$
- Via point: The value of flat variable ${y}_{1}$ is based on joint values at the via point, with continuity of the flat variable derivatives and Equation (10) with ${c}_{n}=0$:$$\begin{array}{c}{y}_{1j}\left(1\right)={\displaystyle \sum _{i=1}^{n}}{q}_{i,via}\\ {y}_{1j}\left(1\right)={y}_{1,j+1}\left(0\right)\phantom{\rule{0.277778em}{0ex}},\phantom{\rule{0.277778em}{0ex}}{\dot{y}}_{1j}\left(1\right)={\dot{y}}_{1,j+1}\left(0\right)\phantom{\rule{0.277778em}{0ex}},\phantom{\rule{0.277778em}{0ex}}{\ddot{y}}_{1j}\left(1\right)={\ddot{y}}_{1,j+1}\left(0\right)\\ {y}_{1j}^{\left(3\right)}\left(1\right)={y}_{1,j+1}^{\left(3\right)}\left(0\right)\phantom{\rule{0.277778em}{0ex}},\phantom{\rule{0.277778em}{0ex}}{y}_{1j}^{\left(4\right)}\left(1\right)={y}_{1,j+1}^{\left(4\right)}\left(0\right)\end{array}$$$${\ddot{y}}_{1j}\left(1\right)=-\frac{{q}_{n,via}\left(t\right){k}_{n}}{{I}_{n}^{*}}$$
- Final point: The final joint values of flat variable ${y}_{1}$ are based on the final values of joint variables, and flat variable derivatives are set to zero:$$\begin{array}{c}{y}_{1f}\left(1\right)={\displaystyle \sum _{i=1}^{n}}{q}_{i,f}\\ {\dot{y}}_{1f}\left(1\right)={\ddot{y}}_{1f}\left(1\right)={y}_{1f}^{\left(3\right)}\left(1\right)={y}_{1f}^{\left(4\right)}\left(1\right)=0\end{array}$$

#### 3.2. Damped Robot (${c}_{n}\ne 0$)

- Initial point: Initial values of flat variable ${y}_{1}$ are based on the initial values of joint variables, and flat variable derivatives are set to zero up to the fifth derivative:$$\begin{array}{c}{y}_{11}\left(0\right)={\displaystyle \sum _{i=1}^{n}}{q}_{i,0}\\ {\dot{y}}_{11}\left(0\right)={\ddot{y}}_{11}\left(0\right)={y}_{11}^{\left(3\right)}\left(0\right)={y}_{11}^{\left(4\right)}\left(0\right)={y}_{11}^{\left(5\right)}\left(0\right)=0\end{array}$$
- Via point: The value of flat variable ${y}_{1}$ is based on joint values at the via point, with continuity of the flat variable derivatives and Equation (10):$$\begin{array}{c}{y}_{1j}\left(1\right)={\displaystyle \sum _{i=1}^{n}}{q}_{i,via}\\ {y}_{1j}\left(1\right)={y}_{1,j+1}\left(0\right)\phantom{\rule{0.277778em}{0ex}},\phantom{\rule{0.277778em}{0ex}}{\dot{y}}_{1j}\left(1\right)={\dot{y}}_{1,j+1}\left(0\right)\phantom{\rule{0.277778em}{0ex}},\phantom{\rule{0.277778em}{0ex}}{\ddot{y}}_{1j}\left(1\right)={\ddot{y}}_{1,j+1}\left(0\right)\\ {y}_{1j}^{\left(3\right)}\left(1\right)={y}_{1,j+1}^{\left(3\right)}\left(0\right)\phantom{\rule{0.277778em}{0ex}},\phantom{\rule{0.277778em}{0ex}}{y}_{1j}^{\left(4\right)}\left(1\right)={y}_{1,j+1}^{\left(4\right)}\left(0\right)\phantom{\rule{0.277778em}{0ex}},\phantom{\rule{0.277778em}{0ex}}{y}_{1j}^{\left(5\right)}\left(1\right)={y}_{1,j+1}^{\left(5\right)}\left(0\right)\end{array}$$$${q}_{n,via}=-\frac{{I}_{n}^{*}}{{k}_{n}}\frac{{\ddot{y}}_{1j}\left(1\right)}{{T}_{j}^{2}}+\frac{{I}_{n}^{*}{c}_{n}}{{k}_{n}^{2}}\frac{{y}_{1j}^{\left(3\right)}\left(1\right)}{{T}_{j}^{3}}$$
- Final point: The final joint values of flat variable ${y}_{1}$ are based on the final values of joint variables, and flat variable derivatives are set to zero up to the fifth derivative:$$\begin{array}{c}{y}_{1f}\left(1\right)={\displaystyle \sum _{i=1}^{n}}{q}_{i,f}\\ {\dot{y}}_{1f}\left(1\right)={\ddot{y}}_{1f}\left(1\right)={y}_{1f}^{\left(3\right)}\left(1\right)={y}_{1f}^{\left(4\right)}\left(1\right)={y}_{1f}^{\left(5\right)}\left(1\right)=0\end{array}$$

## 4. Numerical Results

- ${c}_{n}=0$ both for planning and in the model. This is the case of an ideal robot;
- ${c}_{n}=0$ for planning and ${c}_{n}\ne 0$ in the model. This test shows the effect of neglecting the damping during trajectory planning, whereas the actual robot has relevant damping phenomena in the passive joint;
- ${c}_{n}\ne 0$ both for planning and in the model. This test is the most realistic case. Because of the presence of damping, the polynomial orders increase [15].

## 5. Experimental Validation

## 6. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## References

- Deng, M.; Kubota, S. Nonlinear Control System Design of an Underactuated Robot Based on Operator Theory and Isomorphism Scheme. Axioms
**2021**, 10, 62. [Google Scholar] [CrossRef] - Zhang, T.; Zhang, D.; Zhang, W. Task-based configuration synthesis of an underactuated resilient robot. Robotics
**2023**, 12, 121. [Google Scholar] [CrossRef] - Firouzeh, A.; Salehian, S.S.M.; Billard, A.; Paik, J. An under actuated robotic arm with adjustable stiffness shape memory polymer joints. In Proceedings of the 2015 IEEE International Conference on Robotics and Automation (ICRA), Seattle, WA, USA, 26–30 May 2015; pp. 2536–2543. [Google Scholar]
- Qin, G.; Ji, A.; Cheng, Y.; Zhao, W.; Pan, H.; Shi, S.; Song, Y. Design and motion control of an under-actuated snake arm maintainer. Robotica
**2022**, 40, 1763–1782. [Google Scholar] [CrossRef] - Quaglia, G.; Tagliavini, L.; Colucci, G.; Vorfi, A.; Botta, A.; Baglieri, L. Design and Prototyping of an Interchangeable and Underactuated Tool for Automatic Harvesting. Robotics
**2022**, 11, 145. [Google Scholar] [CrossRef] - Gupta, S.; Kumar, A. A brief review of dynamics and control of underactuated biped robots. Adv. Robot.
**2017**, 31, 607–623. [Google Scholar] [CrossRef] - He, B.; Wang, S.; Liu, Y. Underactuated robotics: A review. Int. J. Adv. Robot. Syst.
**2019**, 16, 1729881419862164. [Google Scholar] [CrossRef] - Franch, J.; Agrawal, S.K.; Sangwan, V. Differential Flatness of a Class of n-DOF Planar Manipulators Driven by 1 or 2 Actuators. IEEE Trans. Autom. Control
**2010**, 55, 548–554. [Google Scholar] [CrossRef] - Tonan, M.; Doria, A.; Bottin, M.; Rosati, G. Influence of Joint Stiffness and Motion Time on the Trajectories of Underactuated Robots. Appl. Sci.
**2023**, 13, 6939. [Google Scholar] [CrossRef] - Oriolo, G.; Nakamura, Y. Control of mechanical systems with second-order nonholonomic constraints: Underactuated manipulators. In Proceedings of the 30th IEEE Conference on Decision and Control, Brighton, UK, 11–13 December 1991; Volume 3, pp. 2398–2403. [Google Scholar]
- De Luca, A.; Oriolo, G. Trajectory planning and control for planar robots with passive last joint. Int. J. Robot. Res.
**2002**, 21, 575–590. [Google Scholar] [CrossRef] - Agrawal, S.; Sangwan, V. Differentially flat designs of underactuated open-chain planar robots. IEEE Trans. Robot.
**2008**, 24, 1445–1451. [Google Scholar] [CrossRef] - Sangwan, V.; Kuebler, H.; Agrawal, S.K. Differentially flat design of under-actuated planar robots: Experimental results. In Proceedings of the 2008 IEEE International Conference on Robotics and Automation, Pasadena, CA, USA, 19–23 May 2008; pp. 2423–2428. [Google Scholar]
- Bottin, M.; Rosati, G. Comparison of Under-Actuated and Fully Actuated Serial Robotic Arms: A Case Study. J. Mech. Robot.
**2022**, 14, 034503. [Google Scholar] [CrossRef] - Tonan, M.; Doria, A.; Bottin, M.; Rosati, G. Oscillation-free point-to-point motions of planar differentially flat under-actuated robots: A Laplace transform method. Robotica
**2024**, 1–19. [Google Scholar] [CrossRef] - Franch, J.; Reyes, À.; Agrawal, S.K. Differential flatness of a class of n-DOF planar manipulators driven by an arbitrary number of actuators. In Proceedings of the 2013 European Control Conference (ECC), Zurich, Switzerland, 17–19 July 2013; pp. 161–166. [Google Scholar]
- Sangwan, V.; Agrawal, S.K. Robustness of a flatness based controller against parametric uncertainties for a class of under-actuated planar manipulators. In Proceedings of the 2017 American Control Conference (ACC), Seattle, WA, USA, 24–26 May 2017; pp. 3735–3740. [Google Scholar]
- Chettibi, T. Smooth point-to-point trajectory planning for robot manipulators by using radial basis functions. Robotica
**2019**, 37, 539–559. [Google Scholar] [CrossRef] - Gasparetto, A.; Boscariol, P.; Lanzutti, A.; Vidoni, R. Path Planning and Trajectory Planning Algorithms: A General Overview. In Motion and Operation Planning of Robotic Systems: Background and Practical Approaches; Springer International Publishing: Cham, Switzerland, 2015; pp. 3–27. [Google Scholar]
- Sangwan, V.; Agrawal, S. Effects of viscous damping on differential flatness-based control for a class of under-actuated planar manipulators. IEEE Control Syst. Lett.
**2018**, 2, 67–72. [Google Scholar] [CrossRef] - Wang, C.; Savkin, A.V.; Garratt, M. Collision free navigation of flying robots among moving obstacles. In Proceedings of the 2016 35th Chinese Control Conference (CCC), Chengdu, China, 27–29 July 2016; pp. 4545–4549. [Google Scholar]
- Wang, J.; Wang, J.; Han, Q.L. Receding-Horizon Trajectory Planning for Under-Actuated Autonomous Vehicles Based on Collaborative Neurodynamic Optimization. IEEE/CAA J. Autom. Sin.
**2022**, 9, 1909–1923. [Google Scholar] [CrossRef] - Agrawal, S.K.; Sangwan, V. Design of under-actuated open-chain planar robots for repetitive cyclic motions. In Proceedings of the International Design Engineering Technical Conferences and Computers and Information in Engineering Conference, Philadelphia, PA, USA, 10–13 September 2006; Volume 42568, pp. 1057–1066. [Google Scholar]
- Chen, W.; Xiong, C.; Chen, W.; Yue, S. Mechanical adaptability analysis of underactuated mechanisms. Robot. Comput. Integr. Manuf.
**2018**, 49, 436–447. [Google Scholar] [CrossRef] - Narikiyo, T.; Sahashi, J.; Misao, K. Control of a class of underactuated mechanical systems. Nonlinear Anal. Hybrid Syst.
**2008**, 2, 231–241. [Google Scholar] [CrossRef] - Biagiotti, L.; Melchiorri, C. Trajectory Planning for Automatic Machines and Robots; Springer Science & Business Media: Berlin/Heidelberg, Germany, 2008. [Google Scholar]
- Trinh, M.; Schwiedernoch, R.; Gründel, L.; Storms, S.; Brecher, C. Friction Modeling for Structured Learning of Robot Dynamics. In Proceedings of the Congress of the German Academic Association for Production Technology; Springer: Berlin/Heidelberg, Germany, 2022; pp. 396–406. [Google Scholar]
- Tonan, M.; Bottin, M.; Doria, A.; Rosati, G. A Modal Approach for the Identification of Joint and Link Compliance of an Industrial Manipulator. Mech. Mach. Sci.
**2022**, 122, 628–636. [Google Scholar]

**Figure 2.**Simulated trajectories. with one via point. The white points are the two extremities of the underactuated link, and the filled points represent the passive joint.

**Figure 3.**Simulated flat variables with one via point. The different colors represent the different polynomial functions joining the subsequent pairs of points.

**Figure 5.**Simulated trajectories with two via points. The white points are the two extremities of the underactuated link, and the filled points represent the passive joint.

**Figure 6.**Simulated flat variables with two via points. The different colors represent the different polynomial functions joining the subsequent pairs of points.

**Figure 8.**Two different trajectories with two different via points but the same obstacle within the workspace.

**Figure 9.**(

**a**) Prototype of the two-DOF robot used for the experimental validation. (

**b**) The two obstacles positioned in the working area.

**Figure 10.**Experimental results with a two-DOF robot: Cartesian trajectories (

**a**) and joint values (

**b**).

Parameters | Units | Link 1 | Link 2 |
---|---|---|---|

${m}_{i}$ | kg | $3.0\times {10}^{-2}$ | $1.2\times {10}^{-2}$ |

${m}_{ci}$ | kg | 0 | $1.2\times {10}^{-2}$ |

${I}_{Gi}$ | kg · m^{2} | $5.8\times {10}^{-5}$ | $4.9\times {10}^{-5}$ |

${a}_{i}$ | m | $1.3\times {10}^{-1}$ | $8.5\times {10}^{-2}$ |

${a}_{Gi}$ | m | $7.1\times {10}^{-2}$ | 0 |

${a}_{ci}$ | m | 0 | $8.5\times {10}^{-2}$ |

Test | ${\mathit{n}}_{\mathit{via}}$ | ${\mathit{c}}_{\mathit{n}}$ in Planning | ${\mathit{c}}_{\mathit{n}}$ in Model | Order of Polynomials |
---|---|---|---|---|

1 | 1 | 0 | 0 | 8-7 |

2 | 1 | 0 | ≠0 | 8-7 |

3 | 1 | ≠0 | ≠0 | 9-9 |

4 | 2 | 0 | 0 | 8-6-7 |

5 | 2 | 0 | ≠0 | 8-6-7 |

6 | 2 | ≠0 | ≠0 | 9-7-9 |

**Table 3.**Numerical tests motion parameters. Each couple of ${\mathit{q}}_{via}$ and each value of ${t}_{via}$ are related to one via point.

Test | ${\mathit{q}}_{\mathit{via}}$${(}^{\circ})$ | ${\mathit{q}}_{\mathit{f}}$${(}^{\circ})$ | ${\mathit{t}}_{\mathit{via}}$ (s) | ${\mathit{t}}_{\mathit{f}}$ (s) |
---|---|---|---|---|

1, 2, 3 | $[95\phantom{\rule{0.277778em}{0ex}},-87]$ | $[180\phantom{\rule{0.277778em}{0ex}},0]$ | 0.17 | 0.64 |

4, 5, 6 | $[104\phantom{\rule{0.277778em}{0ex}},-92]$, $[80\phantom{\rule{0.277778em}{0ex}},88]$ | $[180\phantom{\rule{0.277778em}{0ex}},0]$ | 0.27, 0.52 | 0.7 |

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**MDPI and ACS Style**

Tonan, M.; Bottin, M.; Doria, A.; Rosati, G.
Motion Planning of Differentially Flat Planar Underactuated Robots. *Robotics* **2024**, *13*, 57.
https://doi.org/10.3390/robotics13040057

**AMA Style**

Tonan M, Bottin M, Doria A, Rosati G.
Motion Planning of Differentially Flat Planar Underactuated Robots. *Robotics*. 2024; 13(4):57.
https://doi.org/10.3390/robotics13040057

**Chicago/Turabian Style**

Tonan, Michele, Matteo Bottin, Alberto Doria, and Giulio Rosati.
2024. "Motion Planning of Differentially Flat Planar Underactuated Robots" *Robotics* 13, no. 4: 57.
https://doi.org/10.3390/robotics13040057