4.5 Algorithm
4.5.3 Ascent Path Planning
platform, and so it is assumed that the coordinates of the mother ship are given. The tether is fixed to the anchor, but its geometry is otherwise governed by its interaction with the terrain and with the robot. Likewise, the coordinates of a goal location,g, are given.
Definition 4.5 (Extreme Terrain Motion Planning). Given a terrain model consisting of tether- free and tether-demand intersecting planes, a set of obstacles, an initial tethered robot configuration and tether anchor point, the solution to theextreme terrain motion planning problem is a feasible round-trip path from the anchor point to the goal configuration and back.
In some cases, more than one feasible path may exist, and thus an optimal path could be chosen based on different criteria, such as “safest” or “shortest.” Here, a feasible path is one where, to the resolution of the available terrain model and surface characteristics,Axel is controllable at all times during descent and ascent.
Figure 4.7: Top view of intersecting tether-demand and tether-free planes with obstacles and in- termediate anchor points. The dotted sections represent the edges of theanchor reachable set (see Definition 4.8).
shortest homotopic path (SHP) from the anchor to the robot’s configuration. In other words, this is the shortest path that is smoothly deformable to the tether’s geometry (without passing through an obstacle). An algorithm for computing the SHP in a 2D plane with obstacles can be found in [31]. Aspects of this algorithm’s construction will be summarized in Section 4.5.4.
Once the SHP from the anchor to a given robot configuration has been computed, one next identifies the intermediate anchor points, which are needed to determine whether or not a given ascent is feasible.
Definition 4.6. Anintermediate anchor point is a point at which the taut tether contacts one or more of the obstacles,O1, ..., ON.
Starting with the anchor point, a0, the intermediate anchor points, a1, ...,ak, are indexed in increasing order along the tether from the anchor point (Figure 4.7). For a given configuration and SHP, it is necessary to determine whether or not the steep terrain rover can navigate around the sequence of intermediate anchor points and ascend back to the tether-free plane, which motivates the following definition.
Definition 4.7. (Passability) An intermediate anchor point,aj, ispassablefrom robot configuration q if, given q and an SHP with anchor points a0, ...,aj, the robot can reach a configuration which removesaj from the SHP and makesaj−1the most immediate anchor point.
With Definitions 4.6 and 4.7, the ascent path planning problem simplifies to finding a taut tether configuration containing a sequence of passable anchor points betweenganda0.
Figure 4.8: Pseudo-code of the steep terrain tethered robot planning algorithm
Finding the set of all feasible ascents can now be accomplished with two steps. First, one computes all of the SHPs connecting the anchor, a0, to the goal, g. Algorithms for finding the shortest path of a given homotopy type already exist [31, 56] (see Section 4.5.4 for a summary of one such algorithm). This problem is simplified by restricting the SHP search to only consider shortest paths which do not wind around an obstacle, since it is generally hazardous to encircle terrain features with the rover’s tether. Let{Si(a0,g)}, i= 1, ..., NS, denote this set of taut tether paths connectinga0 tog. Assuming a finite number of obstacles,N, in the region of interest,NS is finite. Likewise, there are a finite number of intermediate anchor points,Ai, in path Si(a0,g).
The second step is then to determine whether or not the intermediate anchors of Si(a0,g) are passable, which can be accomplished with the help of one more definition.
Definition 4.8(Anchor Reachable Set). Given thejth anchor point of theith SHP,ai,j, theanchor reachable set,Ci,j(q), is the set of points that are reachable from the robot configuration,q, while it is tied to anchorai,j.
Generally, the anchor reachable sets will depend upon the SHP, the terrain angle, the terrain traction model, and the robot’s dynamic capabilities.
Anchor reachable sets have three types of edges: 1) an edge which, when crossed, changes the list of anchor points in the SHP (SHP edge), 2) an edge which is the limit of reachable configurations (reachable edge), and 3) an edge which may be both (1) and (2). In Figure 4.7, the SHP edge
diate anchor point. Once an intermediate anchor point is found to be passable, the crossed SHP edge is used to calculate a starting configuration, qfor the analysis of the following reachable set.
Optionally,qbecomes set-valued as the starting point for the computation of Ci,j−1(q).
If all intermediate anchor points ofSi(a0,g) are passable, a kinodynamic motion planning algo- rithm [41] can be used to search for feasible or optimal paths fromgto a0 within the space of the associated reachable sets.
The descent path planning problem is similar to the ascent planning problem except that it is further constrained to consider only paths which are homotopic to the feasible ascent paths, {Sif eas(a0,g)}. As described in Section 4.5.4, the sleeve framework of Hershberger and Snoeyink [31] is used to search for descent paths within the feasible ascent homotopy class.
In summary, the basic tethered robot steep terrain motion planning algorithm can be summarized with the pseudo-code presented in Figure 4.8. The next sections will provide additional technical details regarding the key steps of the planning algorithm.