Geometric operator for simulation and control of tele-operated robots.

Iacob, Robert ; Aurite, Traian ; Popa, Razvan 等

1. INTRODUCTION

During the last decades the use of robots in modern manufacturing industries was intensively increased. Robot technology has been applied in many fields such as industry, agriculture, for domestical and medical purposes/operations etc. Tele-operated robots represent a special category, frequently used for operations in radioactive or dangerous environments. Generally involved in hazardous operations, these robots require monitoring in continuous time in order to avoid any kind of collisions or abnormal operations which can represent a considerable risk.

Industrial standard simulation software, like RobotStudio (ABB.com), proposed by the manufacturers of the robots does not offer enough flexibility for real-time monitoring and there is no possibility to control a tele-operated robot with a haptic device. In order to develop a complete control and simulation platform there is a strong need to represent the valid movements. Using this data in a virtual simulation and control software, perturbations due to collision detections can be avoided. More than that, having a graphical representation is very usefull to enlarge the user perception of the real environment.

In this context, the aim of this work is to present the concepts of a geometric operator able to represent at any moment all the valid movements of the end-effector of a tele-operated robot.

2. GEOMETRIC OPERATOR

2.1 Unit sphere concept

In order to have a data representation of the valid trajectories at a time, a geometric operator is needed. Among the current approaches, the unit sphere concept, or Gauss sphere, is interesting. Woo (Woo, 1993) proposed a spherical operator, for translation directions representation, used in five axis machining algorithm. It allows determining the possible translation displacements of a component according to its planar contacts with its neighbouring components.

[FIGURE 1 OMITTED]

The principle is that each planar contact between two surfaces divides the unit sphere into two hemispheres. Indeed, the goal is to build the space containing all the possible directions of movement, called Non-Directional Blocking Graph (NDBG), starting from a polyhedral representation of the environment and an analysis of the planar contacts between the polyhedrons. The unit sphere defines a point P on a spherical surface such that the direction defined by the origin of the sphere and P defines the translation (or the rotation) movement.

Another field of application is the analysis and simulation of the assembly/disassembly process. Wilson and Latombe (Wilson & Latombe, 1994) have developed the concept by adding a second sphere to the valid rotation movements representation. Some examples showing the hemispheres of Translations, and Rotations are presented in Fig. 1. The combined translation-rotation operator was used by Romney (Romney & Godard, 1995) to create a framework for assembly sequence generation.

2.2 Unit ball concept

The unit sphere concept is a powerful tool for describing translation and rotation movements but it is limited because there is no representation for the helical movements. In order to have a complete geometric description of all possible movements we propose the unit ball concept (Fig. 2). Compared to the unit sphere, the unit ball defines a volume and a point P in that volume which defines a helical movement as follows. The direction defined by the origin of the ball and P defines the direction of the helical movement and the distance of P to the origin defines its pitch. The pitch varies between 0, i.e. a rotation, when P coincides with the origin to [infinity], i.e. a translation, when P lies on the surface of the unit ball.

The concept is based on Chasles's theorem (Kumar, 2000): "Any general movement in 3D space reduces to a helical movement". In the scope of the screw theory, this movement is the central axis of a screw; this one is a combination of two elementary motions: a translation and a rotation. It should be noted that in order to get a helical movement in a given direction [??], there must be one rotation axis co-linear to [??], the two motions being dependant from each other.

[FIGURE 2 OMITTED]

2.3 Real-time control

Real-time control is one of the first categories of application needing models of component mobility. (Rakic, 2007) presented a robot control software which allows the user to give commands in real-time but without a description of the environment and without a representation of the valid movements.

Having the ability to model and to represent translations, rotations and helical movements is mandatory to re-produce realistic impressions. In the context of real-time control linked with an immersive framework, the mobilities between components represent a mean to set up kinematic constraints for haptic devices, thus reducing the complexity of collision detection algorithms. Indeed, kinematic constraints reduce the number of free degrees of freedom, hence reducing the diversity of interferences as well as the computation time. In addition, immersive simulations require a capability to switch, in a transparent manner, from the kinematically constrained mode to the free mode so that the user's immersion is of high quality. To this end, input from position and force sensors available in a haptic environment must be used to identify constraints that need to stay consistent with the sensors, i.e. kinematic constraints must be transparently activated or deactivated in accordance with the user's movement and the diversity of kinematic constraints must be able to cope with the whole range of sensor data to avoid unrealistic changes between modes.

As a consequence, a general operator for describing and combining the mobilities of the end-effector is needed. Missing representation may lead to movements which are not allowed or may discard solutions appearing as obvious from a user's point of view. In addition, having the ability to model the mobilities in a transparent way is also critical to avoid un-realistic movements. Also, interfacing the mobility model with path finding algorithms is a mean to provide more realistic boundary conditions than just trajectory extreme points.

2.4 Combination operator

As mentioned before, the new ball concept is a powerful tool for the geometric representation of the three categories of movements: translation, rotation and helical ones. There are two areas of interest: active domain [D.sub.1] and restrictive domain [D.sub.2]. The active domain is the unit ball of all the valid movements for the end-effector--i.e. a complete ball when there is no object in the space around. The restrictive domain is a hemisphere which represents the invalid movements--i.e. a half of a ball for a planar surface.

The combination operator is based on the resulting trajectories between two families of trajectories associated to [D.sub.1] and [D.sub.2] and it must be able to describe any type of mobility. In order to obtain the compatible trajectories between two domains [D.sub.1] and [D.sub.2] i.e. rotation, translation, helical movements, one must combine the geometric representations of their families of trajectories.

[FIGURE 3 OMITTED]

Translations are essentially characterized by their directions, i.e. their amplitude must be identical. Hence, the unit sphere of translations ([S.sub.T]) is well suited to define the geometric domain ([G.sub.T]) of valid translation directions. Similarly, rotations are mainly characterized by their directions and the boundary angles of finite rotations are not taken into account, hence a geometric domain ([G.sub.R]) over the unit sphere of rotations ([S.sub.R]) is also well suited to describe a set of rotation axes. Finally, helical movements must be characterized by their direction and their pitch, whose interval is [0, [infinity]]. The combination of these parameters can be used to describe a volume domain [G.sub.H], such that one face, [F.sub.1H] lies on ST that corresponds to an infinite pitch. The helices with 0 pitch are located at the origin of [S.sub.T], hence the lateral face of [G.sub.H], [F.sub.2H], is a set of vectors defining the directions of the helical movements and located at the boundary of [G.sub.R]. Let us also define [F.sup.P.sub.2H], the central projection of [F.sub.2H] on [S.sub.R]. Based on the Chasles's theorem, [G.sub.H] has the following properties:

[F.sub.1H] [subset] [G.sub.T] [subset] [S.sub.T] and [F.sup.P.sub.2H] [subset] [G.sub.R] [subset] [S.sub.R],

which ensures the validity of a family of trajectories, whatever the configuration considered. It must be noted that [S.sub.T], [S.sub.R] shares the same origin.

Based on these definitions, the representation of a combination of geometric descriptors of the mobilities for a planar surface is given in Fig. 3. The middle plane characterizes the planar surface; the allowable movements are represented by the upper hemisphere and the invalid movements by the under hemisphere; the thick line depicts the helical movements whose pitches range from 0 to [infinity]. To distinguish more easily [S.sub.T] and [S.sub.R], [S.sub.R] is homothetic to [S.sub.T] with respect to the origin of [S.sub.R] with a ratio of 1/2. The use of the proposed operator is suitable especially for real time simulations because it is able to describe wide diversity movements for the components, as they can be generated in real time by the user.

3. CONCLUSION

Further work will address the implementation of the proposed combination operator in robot control software. As part of this implementation, the set of geometric constraints needed to characterize the respective positions of surfaces will be added the geometry data. In addition to this perspective, the connection between sensor data available in real time simulations and the description of families of trajectories will be addressed to be able to generate a kinematically constrained haptic behaviour.

4. REFERENCES

Kumar, V. (2000). About Robotics, (MEAM 520), University of Pennsylvania Press, 2000

Rakic, S. (2007). Robot control platform, Proceedings of the 18th International DAAAM Symposium, Katalinic, B. (Ed.), pp 419-421, ISBN 3-901509-58-5, Croatia

Romney, B., Godard, C. (1995). An efficient system for geometric assembly sequence generation, Proceedings of the ASME Conference, pp 699-712, Atlanta, USA

Wilson, R.H. & Latombe, J.C. (1994). Geometric reasoning about mechanical assembly, Journal of Artificial Inteligence, vol. 71(2), pp 361-396

Woo, T.C. (1993). Visibility maps and spherical algorithms, Journal of Computer-Aided Design, vol. 26(1), pp 6-16 *** http://www.abb.com--ABB RobotStudio