We consider the scenario where there are two classes of objects present in a flat enclosed region,W. For convenience we will refer to the two classes as ‘blue’ and ‘red’. Without loss of generality, one of these classes of objects will be considered to be of interest (i.e., those need to be manipulated and transported), while the other consists of obstacles or objects that are not of interest. LetO=R1∪R2∪ · · · ∪Rr∪Br+1∪ Br+2∪ · · · ∪Br+b ⊆ W, whereR1, R2,· · ·, Rrarercounts of red objects, andBr+1, Br+2,· · · , Br+b
arebcounts of blue objects. Each object,RiorBj, is assumed to be connected and arbitrarily shaped.
A flexible cable is attached, at its two ends, to two robots that are capable of navigating on the flat surface. Given an initial configuration of the cable and the robots (Figure 6.1(a)), we need to first make the robots follow paths to the boundary of the enclosed region,∂W, such that the final cable configuration ‘separates’ the blue objects from the red, which we call theseparating configuration(Figure 6.2(b)). Once that is achieved, the robots can move along ∂W to enclose one type of objects and “pull” them out, thus separating and transporting those objects.
and ∂W2 as in Figure 6.2(a). It is clear that the robot paths and cable configurations that describe the
problem and achieve the desired objective are sufficiently described up to homotopy. That is, ifC1andC2
are two cable configurations that are in the same homotopy class [15], then,“C1separates the two types of objects”⇐⇒“C2separates the two types of objects”(Figure 6.3(a)). Likewise, if a particular set of robot
paths,{τ1, τ2}, carry the cable from the initial configuration to the desired separating configuration (up to
homotopy), another set of paths,{τ10, τ20}, that are homotopic to the first set (i.e. τ10 ∼τ1andτ20 ∼τ2) will
achieve the same objective.
In addition to this, it should also be noted that the homotopy class of the cable configuration that achieves the separation of the two types of objects is not unique either. For example, in Figure 6.3(a), the configuration
C3 is in a different homotopy class from C1 or C2, but still separates the two types of objects. C0 in
Figure 6.1(b) is another example. Furthermore, for a given desired separating configuration of the cable (up to homotopy), the homotopy classes of the robot paths that can carry the cable from its initial configuration to the separating configuration, are not unique either (Figure 6.3(b)).
Thus, it is useful to develop a notion of optimality to more precisely define the problem objectives. It is natural to use length of the robot paths to the optimization criteria.
For the theoretical foundation and for setting up the optimization problem, we will make the following assumptions:
i. The objects are assumed to be stationary rigid bodies – that is, the cable cannot ‘pass through’ any of the objects, and that on contact of the cable with the objects the objects do not move. In the implementation (Section 6.5.2) we will however relax the conditions that the objects need to be stationary.
ii. The cable is flexible, and there is no restriction on the length of the cable (i.e.the cable will not fall short and tug on the robots). We assume that the cable can either be spooled out as required from a cable reel residing on the robots, or may stretch as in an elastic band.
One simple and intuitive strategy to solve this problem is to drive all the robot on one (left) boundary of the workspace and fix one robot. Then the other robot travels from this (left) boundary of the workspace to the opposite (right) boundary while passing one (red) object above it and the other (blue) object under it as shown in Figure 6.4(a). It is a simple and intuitive algorithm to find and drive to a separating configuration. However, this algorithm requires that all the obejcts should have different value ofX coordinates. And this algorithm does not guarantee the shortest traveling distance. In the same environment, the cable configura- tion or the path of a robot shown in Figure 6.4(b) is shorter than the one in Figure 6.4(a). The path can be shorter if we allow the robot to gobackwardto minimize the overall traveling distance. Also, we should allow the robot to go backward if the objects are not size less points like this case. For example, we cannot find a path with this strategy in the environment of Figure 6.1(b). Also, this algorithm does not guarantee the shortest path of robot from the initial configuration and we need another procedure to drive the robot to a specific part of the boundary of workspace.
In this work, we divide and solve this manipulation and transportation problem into two steps. The first step is to find a separating configurations, which works as initial configuration of the simple controller for transportation, which will be discussed in Section 6.3. And the second problem is to navigate the robots to a separating configurations.
e
s C
(a) A separating configuration achieved by intuitive planning.
e
s C'
(b) A separating configuration with less travel distance.
Figure 6.4: An example of separating configurations achieve by intuition when considering point objects.
Figure 6.5: An example of separating configurations which requires smart controller for transporting.