**1. Introduction**

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Obstacle avoidance and robot path planning problems have gained sufficient research attention due to its indispensable application demand in manufacturing vis-à-vis material handling sector, such as picking-and- placing an object, loading / unloading a component to /from a machine or storage bins. Visibility map in the configuration space (*c-space*) has become reasonably instrumental towards solving robot path planning problems and it certainly edges out other techniques widely used in the field of motion planning of robots (e.g. Voronoi Diagram, Potential Field, Cellular Automata) for *unstructured environment*. The c-space mapping algorithms, referred in the paper, are discussed with logic behind their formulation and their effectiveness in solving path planning problems under various conditions imposed a-priori. The visibility graph (*v-graph*) based path planning algorithm generates the equations to obtain the desired joint parameter values of the robot corresponding to the ith intermediate location of the end - effector in the collision - free path. The developed c-space models have been verified by considering first a congested workspace in 2D and subsequently with the real spatial manifolds, cluttered with different objects. New lemma has been proposed for generating c-space maps for higher dimensional robots, e.g. having degrees-of-freedom more than three. A test case has been analyzed wherein a seven degrees-of-freedom revolute robot is used for articulation, followed by a case-study with a five degrees-of-freedom articulated manipulator (RHINO XR-3) amidst an in-door environment. Both the studies essentially involve new c-space mapping thematic in higher dimensions.

Tomas Lozano Perez' postulated the fundamentals of Configuration Space approach and proved those successfully in spatial path planning of robotic manipulators in an environment congested with polyhedral obstacles using an explicit representation of the manipulator configurations that would bring about a collision eventually [Perez', 1983]. However, his method suffers problem when applied to manipulators with revolute joints. In contrast to rectilinear objects, as tried by Perez', collision-avoidance algorithm in 2D for an articulated two-link planar manipulator with circular obstacles have been reported also [Keerthi & Selvaraj, 1989]. The paradigm of *automatic transformation* of obstacles in the cspace and thereby path planning is examined with finer details [De Pedro & Rosa, 1992], such as *friction* between the obstacles [Erdmann, 1994]. Novel c-space computation algorithm for convex planar algebraic objects has been reported [Kohler & Spreng, 1995],

Spatial Path Planning of Static Robots Using Configuration Space Metrics 419

The paper has been organized in six sections. The facets of our proposition towards configuration space maps in 3D are discussed in the next section. Section 3 delineates the c-space mapping algorithms, with the logistics and analytical models. The features of the path planning metrics and the algorithm in particular, using the concept of visibility graph, have been reported in section 4. Both 2D and spatial workspaces have been postulated, with an insight towards the analytical modeling, in respective cases alongwith test results. Section 5 presents the case study of robot path planning. Finally section 6

The robotic environment is modeled through *discretization* of the 3D space into a number of 2D planes (Cartesian workspace), corresponding to a finite range of *waist* /base rotation of the robot2. Thus, modeling has been attempted with the sectional view of the obstacles in 2D

The obstacles are considered to be *regular*, i.e. having finite shape and size with standard geometrical features with known vertices in Cartesian co-ordinates. For example, obstacles with shapes such as cube, rectangular paralleopiped, trapezoid, sphere, right circular cylinder, right pyramid etc. have been selected (as primitives) for modeling the environment. A complex obstacle has been modeled as a Boolean combination of these primitives, to have polygonal convex shape preferably. However, concave obstacles can also be used in the algorithms by approximating those to the nearest convex shapes, after considering their 'convex hulls' (polytones). Irregular-shaped obstacles have also been modeled by considering their envelopes to be of convex shapes. Circular obstacles have been approximated to the nearest squares circumscribing the original circles, thereby

Features of the developed technique, namely, "Slicing Method", are: i] alongwith shoulder, elbow and wrist (pitch only) rotations, waist rotation of the robot is considered, which is guided by a finite range vis-à-vis a finite resolution; ii] the entire 3D workspace is divided into a number of 2D planes, according to the number of 'segments' of the waist rotation; iii] for every fixed angle of rotation of the waist, a 2D plane is to be constructed, where all other variables like shoulder, elbow and wrist pitch movements are possible; iv] corresponding to each of the 2D slices of the workspace, either one obstacle entirely or a part of it will be

Corresponding to each slice, one c-space map (considering only two joint variables at a time) is to be developed and likewise, several maps will be obtained for all the remaining slices. The final combination of the colliding joint variable values will be the union of all those sets of the values for each slice. Nevertheless, the process of computation can be simplified by taking some finite number of slices, e.g. four to five planes. Figure 1 pictorially illustrates the above-mentioned postulation, wherein the robotic workspace consists of various categories of obstacles, like regular geometry (e.g. obstacle 'A', 'B' & 'C') and integrated geometry3 (e.g.

2 Base rotation is earmarked for articulated robots, whereas suitable angular divisions of the entire

generated, depending on the value of the resolution chosen for waist rotation.

planar area, i.e. 3600 are considered for robots with non-revolute joints (e.g. prismatic).

3 Boolean combination of regular geometries is considered.

**2. Configuration space map in 3D space: Our proposition** 

concludes the paper.

possessing *pseudo-vertices*.

obstacle 'D' & 'E').

plane.

**2.1 Modeling of robot workspace** 

while *slicing* approach for the same is tried for curvilinear objects [Sacks & Bajaj, 1998] & [Sacks, 1999]. Nonetheless, various intricacies of the global c-space mapping techniques for a robot under static environment have been surveyed to a good extent [Wise & Bowyer, 2000]. Although the theoretical paradigms of c-space technique for solving *find-path* problem have been largely addressed in the above literature vis-à-vis a few more [Brooks, 1983], [Red & Truong-Cao, 1985], [Perez', 1987], [Hasegawa & Terasaki, 1988], [Curto & Monero, 1997], the bulging question of tackling collision detection under a typical manufacturing scenario, cluttered with real-life multi-featured obstacles remains largely unattended.

Survey reports on motion planning of robots in general, have been presented, with special reference to path planning problems of lower dimensionality [Schwartz & Sharir, 1988] & [Hwang & Ahuja, 1992]. The find-path problem under sufficiently cluttered environment has been studied with several customized models, such as using distance function [Gilbert & Johnson, 1985], probabilistic function [Jun & Shin, 1988], time-optimized function [Slotine & Yang, 1989], shape alteration paradigms [Lumelsky & Sun, 1990a] and sensorized stochastic method [Acar et al, 2003]. Even, novel *path transform function* for guiding the search for find-path in 2D is reported [Zelinsky, 1994], while the same for manipulators with higher degrees-of-freedom is also described [Ralli & Hirzinger, 1996]. All these treatises are appreciated from the context of theoretical estimation, but lacks in simulating all kinds of polyhedral obstacles.

Based on the c-space mapping, algorithmic path planning in 2D using *visibility* principle is studied [Fu & Liu, 1990], followed by exhaustive theoretical analysis on visibility maps [Campbell & Higgins, 1991]. However, issues regarding computational complexity involved in developing a typical visibility graph, which is O (n2), 'n' being the total number of vertices in the map, is analyzed earlier [Welzl, 1985]. The concept of *M-line*1 and its uniqueness in generating near-optimal solutions against heuristic-based search algorithms has also been examined [Lumelsky & Sun, 1990b].

Several researchers have reviewed the facets of path planning problem in a typical spatial manifold. A majority of these models are nothing but extrapolation of proven 2D techniques in 3D space [Khouri & Stelson, 1989], [Yu & Gupta, 2004] & [Sachs et al, 2004]. However, new methods for the generation of c-space in such cases (i.e. spatial) have been exploited too [Brost, 1989], [Bajaj & Kim, 1990] & [Verwer, 1990]. Customized solution for rapid computation of c-space obstacles has been addressed [Branicky & Newman, 1990], using geometric properties of collision detection between *known* static obstacles and the manipulator body, while sub-space method is being utilized in this regard [Red et al, 1987]. The usefulness of several new algorithms using v-graph technique has been demonstrated in spatial robotic workspace [Roy, 2005].

It may be mentioned at this juncture with reference to the citations above, that, although celebrated, a distinct methodology of using c-space mapping for higher dimensional robots as well as in spatial workspace is yet to be tuned. Our approach essentially calls on this lacuna of the earlier researches. We proclaim our novelty in adding new facets to the problem in a generic way, like: a] *rationalizing* configuration space mapping for *higher dimensional* (e.g. 7 or 8 degrees-of-freedom) *robots*; b] *preferential selection* of joint-variables for configuration space plots in 2D; c] extension of 2D path planning algorithm in 3D through *slicing technique (*creation, validation & assimilation of c-space slices) and d] *searching*  collision-free path in 3D, using novel *visibility map-based algorithm*.

<sup>1</sup> Mean Line, as referred in the literature concerning the visibility graph-based path planning of robots.

The paper has been organized in six sections. The facets of our proposition towards configuration space maps in 3D are discussed in the next section. Section 3 delineates the c-space mapping algorithms, with the logistics and analytical models. The features of the path planning metrics and the algorithm in particular, using the concept of visibility graph, have been reported in section 4. Both 2D and spatial workspaces have been postulated, with an insight towards the analytical modeling, in respective cases alongwith test results. Section 5 presents the case study of robot path planning. Finally section 6 concludes the paper.
