On the Inverse Dynamic Problem of a 3-P RRR Parallel Manipulator, Tripteron Iman Yahyapour * , Milad Hasanvand † , Mehdi Tale Masouleh * , Mojtaba Yazdani * ,Siavash Tavakoli † * Faculty of New Sciences and Technologies {iman.yahyapour, m.t.masouleh}@ut.ac.ir, mojtaba.yazdani@hotmail.com † K.N. Toosi University of Technology {hasanvand.milad, siavash.tavakoli.1}@gmail.com Abstract—This paper presents a new approach for the dy- namic modeling of a 3-degree-of-freedom translational parallel manipulator, called Tripteron. The reasoning used in this paper is based on a judicious concept in detaching the whole mecha- nism into several subsystems and consecutive synergies between kinematic analysis, Lagrangian and Newtonian approaches. In this regard, Tripteron is made equivalent to four subsystems and the equations of kinematic constraints are derived for all the constituting subsystems. Thus upon resorting to Lagrangian approach and blending it with the latter kinematic relations, the dynamic model of each limb in task space is obtained. The dynamic model of the end-effector, as the constraint to the problem, is written in virtue of Newton-Euler’s approach where yields to three differential equations. Finally, the problem leads to a system of three independent equations. For the sake of comparison, the results are put into contrast by a dynamic analyser software. The results obtained by both approaches are coherent which affirms the correctness of the proposed algorithm. I. I NTRODUCTION The primary studies of robotic mechanical systems, Parallel Mechanisms (PMs) among others, leads inevitably to the study of its kinematics and dynamic properties. The main reason for which PMs reveal a challenges in the kinematic and dynamic analysis, is due to the fact that they have not anthropomorphic character, like the serial manipulators, which is mainly caused by the presence of passive joints and having multi-link limbs. The complexity in the dynamic analysis of overconstraint PMs, which is the central subject of this paper, increases considerably where the redundant constraints imposed by each limb should be take into account. In the context of PMs, a few works have been conducted on their dynamic analysis. Thanh et al. [1] presented a gen- eral approach for the inverse dynamic problem of redundant and reduced-DOF PMs. Li and Xu [2] designed a modified DELTA parallel robot [3] with applications in cardiopulmonary resuscitation and analyzed the dynamic properties. Dasgupta and Mruthyunjaya [4] resorted to Newtonian approach to solve the inverse dynamic problem of the 6-DOF Gough- Stewart platform. They separated each link of Gough-Stewart platform and applied Newton-Euler’s concept to solve the inverse dynamic problem of this manipulator. Zhao and Li [5] investigated the inverse dynamic analysis of a 3-RRRT joint manipulator. They combined screw theory and principle of virtual work approach to solve the inverse dynamic problem of this manipulator. In [6], the dynamic model of Tripteron, the mechanism under study in this paper, is explored using Newton-Euler approach. In the latter study, the angular veloc- ity, the angular acceleration and the acceleration of the centre of the mass of each link are computed individually. Then the mechanism was dissembled into seven subsystems consisting six links plus the EE (end-effector) and the free body dia- gram of each link was established by substituting the latter kinematic values into the links of the mechanism. Detaching the mechanism to seven subsystems yields to 42 unknowns and 42 equations (six equations for each subsystem). Upon an appropriate solution strategy, the dynamic model is claimed to be found computationally inexpensively. This paper aims at obtaining the dynamic model of Tripteron as an overconstraint PM, which belongs to the so-called Multipteron family introduced in [7] with Quadrupteron and Pentapteron [8] as the 4-DOF and 5-DOF members of this family, respectively. The points that make this approach more efficient than the one presented in [6] is the method of detaching the mechanism into different subsystems, which leads to a fewest set of differential equations to be solved. In the approach proposed in this paper, Tripteron is sep- arated from specific joints which leads to four subsystems, consisting three P RR subsystems plus the EE. Then the kinematic property of each P RR subsystem is derived and a Euler-Lagrange equation in task space is written for each of the latter subsystems, which results in 9 equations for all limbs. Finally, in order to couple the latter 9 equations with the dynamic equations of the EE, as constraints to the problem, the dynamic model of the EE is written by resorting to Newton- Euler approach, which are three differential equations along each axis of the Cartesian coordinates. The solution of the above 12 equations, 9 from the dynamic model of the limbs plus 3 equations from the dynamic model of the EE, results in the actuation forces, i.e., the inverse dynamic problem of the Tripteron. The remainder of this paper is organized as follows. First the architecture of Tripteron is broadly reviewed. Then the essential to establish a general framework for obtaining the dynamic model of PMs is laid down by touching upon some preliminary concepts in dynamic of robotic mechanical sys- tems. The proposed general approach is applied to Tripteron. Finally the results obtained by the latter approach are put into