Abstract
Aerial manipulators (AM) exhibit particularly challenging, non-linear dynamics; the UAV and its manipulator form a tightly coupled dynamic system, mutually impacting each other. The mathematical model describing these dynamics forms the core of many solutions in non-linear control and deep reinforcement learning. Traditionally, the formulation of the dynamics involves Euler angle parametrization in the Lagrangian framework or quaternion parametrization in the Newton-Euler framework. The former has the disadvantage of giving birth to singularities and the latter being algorithmically complex. This work presents a hybrid solution, combining the benefits of both, namely a quaternion approach leveraging the Lagrangian framework, connecting the singularity-free parameterization with the algorithmic simplicity of the Lagrangian approach. We do so by offering detailed insights into the kinematic modeling process and the formulation of the dynamics of a general aerial manipulator. The obtained dynamics model is validated experimentally against a real-time physics engine. A practical application of the obtained dynamics model is shown in the context of a computed torque feedback controller (feedback linearization), where we analyze its real-time capability with increasingly complex AM models.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Bartelds, T., Capra, A., Hamaza, S., Stramigioli, S., Fumagalli, M.: Compliant aerial manipulators: toward a new generation of aerial robotic workers. IEEE Robot. Autom. Lett. 1(1), 477–483 (2016). https://doi.org/10.1109/LRA.2016.2519948
Ollero, A.: Aerial robotic manipulators. In: Encyclopedia of Robotics. https://doi.org/10.1007/978-3-642-41610-1_78-1, pp 1–8. Springer, Berlin (2019)
Mohiuddin, A., Tarek, T., Zweiri, Y., Gan, D.: A Survey of Single and Multi-UAV Aerial Manipulation. Unmanned Systems 8(2), 119–147 (2020). https://doi.org/10.1142/S2301385020500089
Ikeda, T., Yasui, S., Fujihara, M., Ohara, K., Ashizawa, S., Ichikawa, A., Okino, A., Oomichi, T., Fukuda, T.: Wall contact by octo-rotor UAV with one DoF manipulator for bridge inspection. IEEE Int. Conf. Intell. Robots Syst., pp. 5122–5127. https://doi.org/10.1109/IROS.2017.8206398 (2017)
Korpela, C., Orsag, M., Oh, P.: Towards valve turning using a dual-arm aerial manipulator, IEEE Int. Conf. Intell. Robots Syst., pp. 3411–3416. https://doi.org/10.1109/IROS.2014.6943037 (2014)
Jimenez-Cano, A.E., Martin, J., Heredia, G., Ollero, A., Cano, R.: Control of an aerial robot with multi-link arm for assembly tasks,. In: Proceedings - IEEE International Conference on Robotics and Automation, pp. 4916–4921. https://doi.org/10.1109/ICRA.2013.6631279 (2013)
Heredia, G., Jimenez-Cano, A., Sanchez, I., Llorente, D., Vega, V., Braga, J., Acosta, J., Ollero, A.: Control of a multirotor outdoor aerial manipulator. In: IEEE/RSJ International Conference on Intelligent Robots and Systems, no. Iros, 2014, pp. 3417–3422, IEEE. https://doi.org/10.1109/IROS.2014.6943038 (2014)
Garimella, G., Kobilarov, M.: Towards model-predictive control for aerial pick-and-place, Proceedings - IEEE International Conference on Robotics and Automation, pp. 4692–4697. https://doi.org/10.1109/ICRA.2015.7139850(2015)
Bernard, M., Kondak, K.: Generic slung load transportation system using small size helicopters, Proceedings - IEEE International Conference on Robotics and Automation, pp. 3258–3264. https://doi.org/10.1109/ROBOT.2009.5152382 (2009)
Michael, N., Fink, J., Kumar, V.: Cooperative manipulation and transportation with aerial robots. Auton. Robot. 30(1), 73–86 (2011). https://doi.org/10.1007/s10514-010-9205-0
Ding, C., Lu, L., Wang, C., Ding, C.: Design, sensing, and control of a novel uav platform for aerial drilling and screwing. IEEE Robot. Autom. Lett. 6(2), 3176–3183 (2021). https://doi.org/10.1109/LRA.2021.3062305
Sanchez-Lopez, J.L., Castillo-Lopez, M., Voos, H.: Semantic situation awareness of ellipse shapes via deep learning for multirotor aerial robots with a 2D LIDAR, 2020 Int. Conf. Unmanned Aircr. Syst., ICUAS 2020, pp. 1014–1023. https://doi.org/10.1109/ICUAS48674.2020.9214063 (2020)
Sanchez-Lopez, J.L., Molina, M., Bavle, H., Sampedro, C., Suárez Fernández, R.A., Campoy, P.: A Multi-Layered Component-Based Approach for the Development of Aerial Robotic Systems: The Aerostack Framework. J. Intell. Robot. Syst.: Theory Appl. 88(2-4), 683–709 (2017). https://doi.org/10.1007/s10846-017-0551-4
Mellinger, D., Lindsey, Q., Shomin, M., Kumar, V.: Design, modeling, estimation and control for aerial grasping and manipulation. IEEE Int. Conf. Intell. Robots Syst., 2668–2673. https://doi.org/10.1109/IROS.2011.6048556 (2011)
Lippiello, V., Ruggiero, F.: Exploiting redundancy in Cartesian impedance control of UAVs equipped with a robotic arm. In: IEEE/RSJ International Conference on Intelligent Robots and Systems, pp. 3768–3773, IEEE. https://doi.org/10.1109/IROS.2012.6386021 (2012)
Suarez, A., Jimenez-Cano, A.E., Vega, V.M., Heredia, G., Rodriguez-Castano, A., Ollero, A.: Lightweight and human-size dual arm aerial manipulator. 2017 Int. Conf. Unmanned Aircr. Syst. ICUAS 2017, pp. 1778–1784. https://doi.org/10.1109/ICUAS.2017.7991357 (2017)
Orsag, M., Korpela, C., Oh, P.: Modeling and control of MM-UAV: Mobile manipulating unmanned aerial vehicle. J. Intell. Robot. Syst.: Theory Appl. 69(1-4), 227–240 (2013). https://doi.org/10.1007/s10846-012-9723-4
Manukyan, A., Olivares-Mendez, M.A., Geist, M., Voos, H.: Deep reinforcement learning-based continuous control for multicopter systems. In: 2019 6th International Conference on Control, Decision and Information Technologies. pp. 1876–1881. https://doi.org/10.1109/CoDIT.2019.8820368 (2019)
Bruno Siciliano, O.K.: Springer Handbook of Robotics. Springer, Heidelberg (2008). https://doi.org/10.1007/978-3-540-30301-5
Kim, S., Choi, S., Kim, H.J.: Aerial manipulation using a quadrotor with a two DOF robotic arm, IEEE Int. Conf. Intell. Robots Syst., pp. 4990–4995. https://doi.org/10.1109/IROS.2013.6697077 (2013)
Jiao, R., Dong, M., Ding, R., Chou, W.: Control of quadrotor equipped with a two dof robotic arm, ICARM 2018 - 2018 3rd Int. Conf. Adv. Robot. Mechatron., pp. 437–442, https://doi.org/10.1109/ICARM.2018.8610770 (2019)
Abaunza, H., Castillo, P., Lozano, R., Victorino, A.: Quadrotor aerial manipulator based on dual quaternions, 2016 Int. Conf. Unmanned Aircr. Syst., ICUAS. pp. 152–161, https://doi.org/10.1109/ICUAS.2016.7502589 (2016)
Abaunza, H., Castillo, P., Victorino, A., Lozano, R.: Dual quaternion modeling and control of a quad-rotor aerial manipulator. J. Intell. Robot. Syst. Theory Appl. 88(2-4), 267–283 (2017). https://doi.org/10.1007/s10846-017-0519-4
Hemingway, E.G., O’Reilly, O.M.: Perspectives on Euler angle singularities, gimbal lock, and the orthogonality of applied forces and applied moments. Multibody Syst. Dyn. 44(1), 31–56 (2018). https://doi.org/10.1007/s11044-018-9620-0
Tassora, A.: An optimized Lagrangian multiplier approach for interactive multibody simulation in kinematic and dynamical digital prototyping, VII ISCSB, Ed. CLUP. pp. 3–12 (2001)
Emami, S.A. , Banazadeh, A.: Simultaneous trajectory tracking and aerial manipulation using a multi-stage model predictive control. Aerosp. Sci. Technol. 112, 106573 (2021). https://doi.org/10.1016/j.ast.2021.106573
Fanni, M., Khalifa, A.: A new 6-DOF quadrotor manipulation system: Design, kinematics, dynamics, and control. IEEE/ASME Trans. Mech. 22(3), 1315–1326 (2017). https://doi.org/10.1109/TMECH.2017.2681179
Wang, T., Umemoto, K., Endo, T., Matsuno, F.: Dynamic hybrid position/force control for the quadrotor with a multi-degree-of-freedom manipulator. Artificial Life and Robotics 24(3), 378–389 (2019). https://doi.org/10.1007/s10015-019-00534-0
Nascimento, T.P., Saska, M.: Position and attitude control of multi-rotor aerial vehicles: a survey. Annu. Rev. Control 48, 129–146 (2019). https://doi.org/10.1016/j.arcontrol.2019.08.004
Acosta, J., Sanchez, M., Ollero, A.: Robust control of underactuated aerial manipulators via IDA-PBC. In: 53rd IEEE Conference on Decision and Control, Vol. 2015-Febru, IEEE, pp. 673–678. https://doi.org/10.1109/CDC.2014.7039459http://ieeexplore.ieee.org/document/7039459/ (2014)
Ali, Z.A., Li, X.: Controlling of an under-actuated Quadrotor UAV equipped with a manipulator. IEEE Access 8, 34664–34674 (2020). https://doi.org/10.1109/ACCESS.2020.2974581
Liang, J., Chen, Y., Lai, N., He, B., Miao, Z., Wang, Y.: Low-complexity prescribed performance control for unmanned aerial manipulator robot system under model uncertainty and unknown disturbances, IEEE Trans. Ind. inform. PP (c) 1. https://doi.org/10.1109/TII.2021.3117262 (2021)
ROS: URDF XML Specifications. http://wiki.ros.org/urdf/XML (2013)
Murray, R.M., Li, Z., Sastry, S.S.: A Mathematical Introduction to Robotic Manipulation, 1st edn. CRC Press, Boca Raton (1994). https://doi.org/10.1201/9781315136370, https://www.taylorfrancis.com/books/9781351469791
Craig, J.: Introduction to Robotics: Mechanics and Control, 3rd edn. Pearson Education, London (2005)
Siciliano, B., Sciavicco, L., Villani, L., Oriolo, G.: Robotics, Advanced Textbooks in Control and Signal Processing. Springer, London (2009). https://doi.org/10.1007/978-1-84628-642-1
Baumgarte, J.: Stabilization of constraints and integrals of motion in dynamical systems. Comput. Methods Appl. Mech. Eng. 1(1), 1–16 (1972). https://doi.org/10.1016/0045-7825(72)90018-7
Möller, M., Glocker, C.: Rigid body dynamics with a scalable body, quaternions and perfect constraints. Multibody Syst. Dyn. 27(4), 437–454 (2012). https://doi.org/10.1007/s11044-011-9276-5
Udwadia, F.E, Schutte, A.D.: An Alternative Derivation of the Quaternion Equations of Motion for Rigid-Body Rotational Dynamics. J Appl Mechan 77(4), 0–4 (2010). https://doi.org/10.1115/1.4000917
Schutte, A., Udwadia, F.: New approach to the modeling of complex multibody dynamical systems. J. Appl. Mech., Transactions ASME, vol 78 (2) https://doi.org/10.1115/1.4002329 (2011)
Bjerkeng, M., Pettersen, K.Y.: A new Coriolis matrix factorization. In: Proceedings - IEEE International Conference on Robotics and Automation. pp. 4974–4979. https://doi.org/10.1109/ICRA.2012.6224820 (2012)
Graf, B.: Quaternions and dynamics, dynamical systems. arXiv:0811.2889 (2008)
Kotarski, D., Krznar, M., Piljek, P., Simunic, N.: Experimental Identification and Characterization of Multirotor UAV Propulsion. J. Phys.: Conf. Ser. 870(1), 012003 (2017). https://doi.org/10.1088/1742-6596/870/1/012003
Yoon, M.: Experimental identification of thrust dynamics for a multi-rotor helicopter. Int. J. Eng. Res. Technol. (IJERT) 4,(11) (2015)
Yoon, S., Howe, R.M., Greenwood, D.T.: Constraint violation stabilization using gradient feedback in constrained dynamics simulation. J. Guid. Control. Dyn. 15(6), 1467–1474 (1992). https://doi.org/10.2514/3.11410
Braun, D.J., Goldfarb, M.: Eliminating constraint drift in the numerical simulation of constrained dynamical systems. Comput. Methods Appl. Mech. Eng. 198(37-40), 3151–3160 (2009). https://doi.org/10.1016/j.cma.2009.05.013
Sherif, K., Nachbagauer, K., Steiner, W.: On the rotational equations of motion in rigid body dynamics when using Euler parameters. Nonlinear Dyn. 81(1-2), 343–352 (2015). https://doi.org/10.1007/s11071-015-1995-3
Xu, J., Halse, K.H.: Dual quaternion variational integrator for rigid body dynamic simulation. arXiv:1611.00616
Erwin, B., Coumans, Y.: PyBullet, a Python module for physics simulation for games, robotics and machine learning. http://pybullet.org (2020)
Solà, J.: Quaternion kinematics for the error-state Kalman filter. arXiv:1711.02508
Solà, J., Deray, J., Atchuthan, D.: A micro Lie theory for state estimation in robotics. pp. 1–17, arXiv:1812.01537 (2018)
Funding
This project was partially supported by the European Commission Horizon2020 research and innovation programme, under the grant agreement No 101017258 (SESAME); by the Luxembourg National Research Fund (FNR) 5G-SKY project (ref. C19/IS/113713801); and by a partnership between the Interdisciplinary Center for Security Reliability and Trust (SnT) of the University of Luxembourg and the Army of Luxembourg.
Author information
Authors and Affiliations
Contributions
Conceptualization: H. Voos, J. L. Sanchez-Lopez, P. Kremer; Methodology: P. Kremer, J. L. Sanchez-Lopez; Formal analysis and investigation: P. Kremer; Writing original draft preparation: P. Kremer; Writing - review and editing: P. Kremer, J. L. Sanchez-Lopez; Software: P. Kremer; Funding acquisition: H. Voos; Resources: P. Kremer, J. L Sanchez-Lopez, H. Voos; Supervision: H. Voos, J. L. Sanchez-Lopez.
Corresponding author
Additional information
Publisher’s Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Appendix A: Quaternion Fundamentals
Appendix A: Quaternion Fundamentals
This appendix briefly introduces quaternions with the focus on unit quaternions, which play a fundamental role in representing singularity-free orientations in three-dimensional space. Its most relevant aspects are summarized here. For further details the reader is referred to [38, 42, 50] and [51].
A quaternion is an expression defined by a set of four coefficients \(q_{w}, q_{x}, q_{y}, q_{z} \in \mathbb {R}\), and three symbols i,j,k,
which can be put in the more convenient form of
where \(q_{w}\in \mathbb {R}\) is the scalar part and \(\bar {\mathbf {q}}_{v}=\left (q_{i}, q_{j}, q_{k}\right )\in \mathbb {R}^{3}\) is the vector part containing the three imaginary coefficients.
The quaternion group \(\mathbb {H}\) is endowed with the non-commutative quaternion product defined as
The norm of a quaternion is defined as
In the context of mechanics, quaternions describe both a (uniform) scaling and a rotation [38]. A quaternion q with \(\left \Vert \mathbf {q}\right \Vert =1\), element of S3, is called a unit quaternion, parameterizing a 4-dimensional unit-sphere, and, contrary to a general quaternion in \(\mathbb {H}\), solemnly describe the orientation of a rigid body. S3 is a double cover of SO(3) meaning that q and −q characterize the same orientation.
The inverse of a quaternion is defined as
The conjugate of a quaternion is obtained by negating its vector part
and is equal to the inverse in case \(\left \Vert {\mathbf {q}}\right \Vert =1\).
The quaternion identity rotation is defined as
Unit quaternions can directly be obtained from axis-angle notation (the equivalent of the Euler notation in quaternion space) s.t.:
where \(\bar {\mathbf {\mathbf {u}}}\) represents the (normalized) axis of rotation and 𝜃 the angle of rotation.
A quaternion is called pure, if its scalar part is zero:
A vector \(\bar {\mathbf {\mathbf {u}}} \in \mathbb {R}^{3}\) defined in body frame can be rotated into the world frame by q ∈S3 using the double quaternion product:
The vectors \(\bar {\mathbf {\mathbf {u}}}\) and \(\bar {\mathbf {\mathbf {u}}}'\) have the same magnitude if and only if \(\left \Vert {\mathbf {q}}\right \Vert =1\). Otherwise, the length of \(\bar {\mathbf {\mathbf {u}}}\) is scaled by the norm of q.
Composition of two quaternions \(\mathbf {q}_{1}, \mathbf {q}_{2} \in \mathit {S}^{3}\) is similar to the composition of rotation matrices:
The quaternion product is linear and as such it can be expressed as a matrix-vector product:
with the operators
and
wherein \(\left [.\right ]_{\times }: \mathbb {R}^{3} \rightarrow \mathbb {R}^{3\times 3}\) is the cross product operator
Applying Eqs. A.15 and A.16 to A.10 yields the rotation of a vector in matrix notation:
The matrix equivalent of Eq. A.8 is given by the Rodrigues rotation formula \(R\left(\theta ,\bar{\mathbf {u}}\right): \mathbb{R} \times \mathbb{R}^{3} \rightarrow \mathit{SO(3)}\), which represents a rotation of an angle 𝜃 around an arbitrary axis ∥u and is define as
Within this work, the quaternion is always assumed to be unit length.
Rights and permissions
Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article's Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article's Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/.
About this article
Cite this article
Kremer, P., Sanchez-Lopez, J.L. & Voos, H. A Hybrid Modelling Approach for Aerial Manipulators. J Intell Robot Syst 105, 74 (2022). https://doi.org/10.1007/s10846-022-01640-1
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/s10846-022-01640-1