Acta Astronautica Vol.51, No.12, pp.831–842, 2002 ? 2002 Elsevier Science Ltd. All rights reserved Printed in Great Britain www.elsevier.com/locate/actaastro PII: S0094-5765(02)00032-2 0094-5765/02/$-see front matter ATWO-DIMENSIONALAPPROACHTOMULTIBODYFREEDYNAMICS INSPACEENVIRONMENT P. GASBARRI† Dipartimento Aerospaziale, Universit a di Roma La Sapienza, Via Eudossiana 18—00184 Roma, Italy (Received 8 August 2000; revised version received 12 December 2001) Abstract—The equation of motion of a multibody system, described here as a chain of rigid bars and revolute joints orbiting around the Earth, is derived. For each bar two translational and one rotational equilibrium equations are written. The forces acting on each body are the gravitational forces and the reaction forces (unknown) acting on it’s end joints. The complete set of equilibrium equations consists of NX dierential equations, where NX is the order of the state vector. The total number of unknowns is NX + NR where NR =2NJ and NJ is the number of joints. The NR additional equations, to make the system determinate, are provided by the nondierential compatibility equations. The resulting system is a set of dierential algebraic equations (DAE) for which the well-known method of reducing the system to ordinary dierential equations (ODE) is applied. Since the internal forces are associated with the relative displacements between the bodies, which are small fractions of the distance of the multibody spacecraft from the center of the Earth, the task of obtaining these forces from inertial coordinates, from a numerical viewpoint, could be impossible.Sotheproblemisreformulatedinsuchawaythattheequationofmotionofthesystem, containsglobalquantitieswherenointernalforcesappear,andlocalequationswhereinternalforces do appear. In the latter one, only quantities of the same order of the spacecraft dimensions are present. Numerical results complete the work. ? 2002 Elsevier Science Ltd. All rights reserved 1. INTRODUCTION Robots working in space have been studied exten- sively in recent years [1,2]. The basic function of a robot is to move payloads from one position to another. In general, a space robot diers from an industrialrobotfortwomainreasons:(a)industrial robotsaremountedonaxedbase,whereasspace robotsaremountedonorbitalspaceplatformscapa- ble of translations and rotations (this circumstance leads to problems not encountered in xed robots, e.g., an unwanted motion of the satellite body due to the motion of the manipulator arms); (b) space robots must be very light, and hence very exible whereas industrial ones are characterized by their bulky and sti arms. The extreme exibility of the arms may cause undamped elastic vibrations, which tend to aect adversely the performance of the manipulator it- self [3]. Other important research in the eld of space-based robot with free-ying characteristics †Corresponding author. E-mail address: gasbarri@pcsantini2.ing.uniroma1.it (P. Gasbarri). has been carried out by Meirovitch [4,5], Rey- hanoglu [6], Yamada [7] in recent years; the main scope of these studies is the possibility of control- ling the spacecraft system by merely maneuvering the robot arms. Yamada et al. [8,9] and Modi [10] have shown that either the desired arm trajectory and the wanted base attitude can be achieved si- multaneously, or that the desired joint angles and the desired base attitude can be obtained simulta- neously. As far as the problem of modeling multibody dynamic systems is concerned, many approaches have been presented in the last decades [11,12]. Amongthesewecanndthe“directpathmethod” proposed by Ho [13], “Kane’s approach”, Kane [14], the “perturbation technique” proposed by Meirovitch [15], the “Lagrangian approach” used by Modi et al. [16]. In the case of the Lagrangian approach, we can say that, the kinetic energy ex- pression becomes extremely complicated as the number of bodies increases making it dicult to manage it to obtain the equilibrium equations. On the contrary this approach automatically satises holonomic constraints. 831