Dynamic simulations regarding the influence of the load-rigidity correlation on the working accuracy of a medical Triglide parallel robot

N. R. Rat*, M. Neagoe**, D. Diaconescu***, S. D. Stan**** *Transilvania University of Brasov, B-dul Eroilor No 29, 500036 Romania, E-mail: ncretescu@unitbv.ro **Transilvania University of Brasov, B-dul Eroilor No 29, 500036 Romania, E-mail: mneagoe@unitbv.ro ***Transilvania University of Brasov, B-dul Eroilor No 29, 500036 Romania, E-mail: dvdiaconescu@unitbv.ro ****Technical University of Cluj-Napoca, Bdul Muncii, 400641 Romania, E-mail: sergiustan@ieee.ro


Introduction
In the last three decades, parallel robots have been increasingly studied and developed from both theoretical viewpoints and practical applications, as the medical one.
Parallel robots are the structures with closed kinematic chains, composed by an end-effector (the mobile platform) with n DOF (degrees of freedom), connected to a base by two or more kinematical chains called legs (that can be a simple or a complex kinematic chains) [1][2][3].
For a like 3 DOF structure (Fig. 1), that is also the object of this paper, Stan et al. approached relevant kinematic and control aspects in [4,5].
In this paper, starting from another dynamic applications [6][7][8] and using ADAMS simulations, the influence of the load-rigidity correlation on the dynamic response of the considered medical Triglide robot is highlighted; this approach compares two robot models (with rigid and with elastic connecting rods), taking into account adjustable load and adjustable rigidity of the rods AB (Fig. 1).The resulting conclusions are useful for the accuracy of the medical robot applications.

Inverse kinematical model
The 3 DOF Triglide parallel robot (Fig. 1) is composed by a base N connected at a mobile platform P 2 by 3 glide actuators NA i ; i = 1, 2, 3 and 3 parallelogram legs whit spherical joints (A i1 , A i2 , B i1 , B i2 ).
The inverse kinematics of this structure supposes to establish the joint variable vector q(q 1 , q 2 , q 3 ) of the Fig. 1 Kinematic scheme of a branch from the considered medical parallel robot [9] actuators as against of the end-effector variable vector P 2 (x p , y p , z p ) of the mobile platform center against of the system Nxyz.The actuators' vector can be obtained from equations of the type (Fig. 1) which α i and β i = 0º, 120º, 240º.This kinematical model, elaborated and simulated in [9,10], was now extended to the rigid robot dynamic modeling.

Dynamic simulation using ADAMS software
For the considered medical Triglide robot (Fig. 1) [9], the influence of the load-rigidity correlation on the dynamic response is highlighted further; using ADAMS software, this approach compares two robot models (rigid robot and robot with elastic rods), with adjustable load and adjustable rigidity of the connecting rods AB.
The next prerequisites were taken into consideration: • kinematic and dynamic modelling is made in a fixed system Oxyz whose origin O is described in the system Nxyz by vector NO (0.14 m, 0.13 m, 1.11 m); • robot task: to describe a circle with a radius of 50 mm in a tilted plane; • the first model of rigid robot, whose mass features are given in Table 1, is taken as reference, and the second model with elastic rods with adjustable load and adjustable rod elasticity (rods with ∅10 mm and rods with ∅20 mm) is analysed; adjustable roads' elasticity (that has the main influence) was obtained using ADAMS AutoFlex module, taking into account only the natural frequency smaller than 200 Hz for ∅10 mm and than 500 Hz for ∅20 mm; • the material used for the elements was aluminium; • the adjustable load is achieved by adjustable external mass: 0.5, 1.5 and 2.5 kg.The simulations, in which the medical robot has rods with ∅10 mm and external mass of 0.5 and 1.5 kg, show that the difference between dynamic response of the rigid robot model and the response of the robot model with elastic connecting rods can be neglected; the relevant differences interfere for the mass of 2.5 kg (Figs. 2 -7).The dynamic ADAMS simulations, for the case of 2.5 kg, are graphically illustrated in Figs.3-8 and highlight the following properties.
• The displacement of the point P 2 (the centre of the mobile platform), in the case of ∅10 mm elastic roads, has a notable difference as against of rigid roads' case; this difference induces undesirable effects for speed and acceleration (Figs. 4, 5 and 6).
In case of rods with ∅20 mm, the both models (rigid and elastic) give similar displacements (Fig. 4, b), with relative errors (Fig. 4, c) closed to zero (neglected errors).• Because the responses of motors are similar, only the M3 motor response is illustrated in Figs.7 and  8.
All displacements of the motor's are close variations (Fig. 7); this closeness can be observed also for the motor forces (Fig. 8, c), but not in the case of the elastic rods with 10 mm diameter (Fig. 8, a), in which the differences are very large; unlike the 10 mm diameter case, the 20 mm case assures a proper elasticity, for the given system, and implicitly a better response that is close to rigid case response.In the last case (with external mass of 2.5 kg), although the weight of moving elements (6 roads, 3 driving motors and the mobil platform) totalizes 3.11 kg for 20 mm roads' diameter (Table ), unlike 1.16 kg for the 10 mm roads'diameter, the dynamic responses of the robot with elastical elements highlight a much better working behaviour in the case with 20 mm roads' diameter (Fig. 3,  a and b).The large elastic deformations from Fig. 2, a and Fig. 3, a 1.This paper presents the dynamic analysis by ADAMS simulations (displacement, speeds, accelerations, forces and rods' deformation), considering three values for external load (of 0.5, 1.5 and 2.5 kg) and two values for rods' diameter (of 10 and 20 mm).
2. Te results of dynamic simulations highlight that the differences between dynamic responses of the parallel robot with rigid rods and the robot with elastic roads are negligible for the external load of 0.5 and 1.5 kg.The relevant differences interfere, the external load of 2.5 kg, be-tween elastic model with 10 mm roads' diameter and with 20 mm roads' diameter.
3. While dynamic response of the robot with elastic roads of 20 mm diameter (and external mass of 2.5 kg) is almost the same with the rigid robot response (the proper elasticity assure a good system stability), the response of the robot with elastic roads of 10 mm diameter (and the same external mass) shows that the structure has control difficulties.
4. In accordance with the previous conclusion, a parallel robot of Triglide type can achieve a precise trajectory in medical applications, if a good correlation between roads' rigidity, external loads and robot mass is accomplished.

LYGIAGRETAUS MEDICININIO TRIJŲ SLYDIMO
3D model of Triglide robot with rigid (a) and elastic (b) rods of ∅10 mm and external mass of 2.5 kg 3D model of Triglide robot with elastic rods of ∅10 mm (a) and ∅20 mm (b) and external mass of 2.5 kg

Fig. 4 Fig. 5 Fig. 6 Fig. 7
Fig. 4 The platform center polar-ray variation for rods with ∅10 mm (a) and with ∅20 mm (b) in the premise of rigid rods (continuous line) and elastic rods (dashed line); the difference between the polar rays afferent to the elastic model and rigid model with ∅20 mm roads (c)

Table
Mass and inertial property of all elements in movement