Dynamic analysis of an isoglide3-T3 parallel robot.

Rat, Nadia-Ramona ; Neagoe, Mircea ; Gogu, Grigore 等

1. INTRODUCTION

Parallel robots with uncoupled motions are a novel robot class recently approached and presented in the literature (Gogu, 2004; Gogu, 2008) due to their performances: robustness, high speed and precision--specific to parallel robots, and simple kinematic models--specific to serial robots.

Isoglide3-T3 (Figure 1) is a representative structure of 3DOF parallel maximally regular robots (Gogu, 2008), the most studied in the last period by the robotic researchers under different denominations: Orthogonal Tripteron (Gosselin et al., 2004), CPM (Kim &Tsai, 2002).

Few studies concerning the dynamic modeling of the parallel robots with uncoupled motions are presented in the literature. Different methods can be applied to dynamic modeling of this robot type in both hypotheses of rigid and flexible links. In this paper, the Lagrange method with multipliers is used for the closed form dynamic modeling in the hypothesis of rigid links. These results are then compared with CAD model implemented in Adams. The modeling with flexible links hypothesis is made with Adams Autoflex module (Rat et al., 2005).

2. DYNAMIC MODELLING AND SIMULATION OF ISOGLIDE3-T3 PARALLEL ROBOT

Isoglide3-T3 (Figure 1) is a maximally regular parallel robot composed by a mobile platform 5 connected to the robot base through three independent PRRR type kinematic chains (1 Prismatic + 3 Revolute joints) (Gogu, 2004).

[FIGURE 1 OMITTED]

The following types of parallel manipulators (PMs) have been identified in the literature (Gogu, 2008): (i) maximally regular PMs if the Jacobean J is an identity matrix, (ii) fully-isotropic PMs, if the Jacobean J is a diagonal matrix with identical diagonal elements throughout the entire workspace, (iii) PMs with uncoupled motions if J is a diagonal matrix with different diagonal elements, (iv) PMs with decoupled motions, if J is a triangular matrix and (v) PMs with coupled motions if J is neither a triangular nor a diagonal matrix. Maximally regular and fully-isotropic PMs give a one-to-one mapping between the actuated joint velocity space and the external velocity space.

2.1 Dynamic modeling of ISOGLIDE3-T3 with the hypothesis of the rigid links

The Lagrange method with multipliers was succefully applied to derive the dynamic closed form model of Isoglode3T3 robot, in the premises:

* rigid elements with distributed masses;

* gravity is oriented in negative sense of the Z axis;

* no external load.

Starting on the Lagrange equations with multipliers, a relative complex closed form dynamic model was obtained using the Maple software:

[k.summation over (i=1)] [partial derivative][[GAMMA].sub.i]/[partial derivative][q.sub.j] = d/dt ([partial derivative]L/[partial derivative][q.sub.j]--[[??].sub.j], (1)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (2)

where: [[lambda].sub.i] are the Lagrange multipliers; L--system

Lagrangean; [q.sub.j]--displacement of actuated joint j; [[??].sub.j] -generalized external force; and [[GAMMA].sub.i]--the Lagrange multipliers. For example, the expression of the force F[q.sub.1] is:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (3)

For numerical testing of this dynamic model, a given linear trajectory between two points in Cartesian space and a 5th degree polynomial low of movement were used. Even the dynamic model is complex (see eq. 4, where kinematic parameters from actuated and passive joints are included), the obtained driving forces on the trajectory (Figure 2, continuous lines) have a similar profile with the displacement variation. We can see that [Fq.sub.3] is the biggest driving force, due to the gravity effect.

2.2 Dynamic modeling of ISOGLIDE3-T3 with the hypothesis of flexible links

The dynamic modeling in the flexible links hypothesis was derived in Adams AutoFlex module, by considering the following assumptions:

1. The mobile platform 5 and the links 2ABC (see Figure 1) were considered as rigid elements (compact small size elements);

2. The first three natural link frequencies have been taken into consideration.

3. A simplified geometry of the Isoglide3-T3 structure was used (cylindrical type elements), maintaining the same inertial properties as in the model with rigid links.

4. The active joint movements (the same 5th degree polynomial interpolation) are considered as inputs in the simulation process.

A significant jump at the beginning of the trajectory can be observed while driving forces are obtained in the rigid links hypothesis (Figure 2--continuous lines); the jump is less important in the case of flexible links hypothesis (Figure 2--dashed lines) due to the elasticity of the robot links.

[FIGURE 2 OMITTED]

These results emphasize the effects of the link elasticity on the driving forces and on the precision of the pass planning.

[FIGURE 3 OMITTED]

The difference between the driving forces in the actuated prismatic joints obtained the flexible--rigid link hypothesis are represented in Figure 3 and the associated positioning errors in Figure 4.

[FIGURE 4 OMITTED]

3. CONCLUSIONS

A dynamic modeling of maximally regular parallel robots, with application to the ISOGLIDE3-T3 topology has been presented in this paper. A closed form dynamic model has been set up in the rigid link hypothesis, and the dynamic behavior for flexible link hypothesis has been obtained by using Adams Autoflex module.

The numerical simulations on a given trajectory emphasize the influence of link flexibility on the driving forces and the positioning errors.

With obtained analytical dynamic model a control model can be add to completing the efectueted study.

4. REFERENCES

Gogu G., (2004). Structural synthesis of fully-isotropic translational parallel robots via theory of linear transformations. European Journal of Mechanics / A Solids, Vol. 23, pp 1021-1039

Gogu, G. (2008). Structural synthesis of parallel robots, Part 1: Methodology, Springer Verlag

Gosselin, C.M., Kong, X., Foucault S. & Bonev, I.A. (2004). A fully-decoupled 3-dof translational parallel mechanism, In: Parallel Kinematic Machines in Research and Practice, 4th Chemnitz Parallel Kinematics Seminar, pp 595-610

Kim, S.H. & Tsai, L-W. (2002). Evaluation of a cartesian parallel manipulator, In: Advances in robot kinematics, Lenarcic J., Thomas F. (eds), Kluwer Academic Publishers, pp 21-28

Rat, N., Rizk, R., Gogu, G. & Neagoe, M. (2005). Comportement des robots paralleles a mouvements decouples et corps deformable, Proceeding of 17eme Congres Francais de Mecanique, Troyes, Sept. 2005