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Simulation Modelling Practice and Theory
journal homepage: www.elsevier.com/locate/simpat
Singularity-free dynamic equations of vehicle–manipulator systems Pål J. From a,*, Vincent Duindam b, Kristin Y. Pettersen a, Jan T. Gravdahl a, Shankar Sastry b a b
Department of Engineering Cybernetics, Norwegian University of Science and Technology, Norway Department of EECS, University of California, 253 Cory Hall, Berkeley, CA 94720-1770, USA
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In this paper we derive the singularity-free dynamic equations of vehicle–manipulator systems using a minimal representation. These systems are normally modeled using Euler angles, which leads to singularities, or Euler parameters, which is not a minimal representation and thus not suited for Lagrange’s equations. We circumvent these issues by introducing quasi-coordinates which allows us to derive the dynamics using minimal and globally valid non-Euclidean conﬁguration coordinates. This is a great advantage as the conﬁguration space of the vehicle in general is non-Euclidean. We thus obtain a computationally efﬁcient and singularity-free formulation of the dynamic equations with the same complexity as the conventional Lagrangian approach. The closed form formulation makes the proposed approach well suited for system analysis and model-based control. This paper focuses on the dynamic properties of vehicle–manipulator systems and we present the explicit matrices needed for implementation together with several mathematical relations that can be used to speed up the algorithms. We also show how to calculate the inertia and Coriolis matrices and present these for several different vehicle–manipulator systems in such a way that this can be implemented for simulation and control purposes without extensive knowledge of the mathematical background. By presenting the explicit equations needed for implementation, the approach presented becomes more accessible and should reach a wider audience, including engineers and programmers. Ó 2010 Elsevier B.V. All rights reserved.
Article history: Received 9 October 2009 Received in revised form 17 January 2010 Accepted 18 January 2010 Available online 25 January 2010 Keywords: Robot modeling Vehicle–manipulator dynamics Singularities Quasi-coordinates
1. Introduction A good understanding of the dynamics of a robotic manipulator mounted on a moving vehicle is important in a wide range of applications. Especially, the use of robots in harsh and remote areas has increased the need for vehicle–robot solutions. A robotic manipulator mounted on a moving vehicle is a ﬂexible and versatile solution well suited for these applications and will play an important role in the operation and surveillance of remotely located plants in the very near future. Recreating realistic models of for example space or deep-sea conditions is thus important. Both for simulation and for model-based control the explicit dynamic equations of vehicle–manipulator systems need to be implemented in a robust and computationally efﬁcient way to guarantee safe testing and operation of these systems. One example of such a system is spacecraft–manipulator systems [1–5] which are emerging as an alternative to human operation in space. Operations include assembling, repair, refuelling, maintenance, and operations of satellites and space stations. Due to the enormous risks and costs involved with launching humans into space, robotic solutions evolve as the most cost-efﬁcient and reliable solution. However, space manipulation involves quite a few challenges. In this paper we focus on
* Corresponding author. Tel.: +1 4790690932. E-mail address: email@example.com (P.J. From). 1569-190X/$ - see front matter Ó 2010 Elsevier B.V. All rights reserved. doi:10.1016/j.simpat.2010.01.012
P.J. From et al. / Simulation Modelling Practice and...