(a) Obstacles detected and filled (b) Positive obstacles
Figure 4.16: Figure 4.16a shows the output after the obstacles have been detected and filled. Black regions are obstacle-free while the light regions contain obstacles. Figure 4.16b shows only the positive obstacles. Almost 1,500 distinct obstacle groups were detected, of which approximately 25% of them were positive obstacles.
manifold using any of a number of triangulation algorithms. This analysis used the triangulation method put forth by de Berg et al. in [29] and summarized in Section 4.5.4, although Chazelle’s algorithm [27], for example, would also be a valid alternative. The full BTM is presented in Figure 4.17 and close-details of the traingulation are shown in Figure 4.18.
With a start and goal location separated by over 300 positive obstacles, there are many different sleeves in the associated BTM. Most of these sleeves, however, do not represent realistic homotopy classes for a rover path. For example, any sleeve that has the rover traversing directly across the plane fromx= 0 km tox= 1 km would be difficult to realize in practice. Given an anchor point ρa = [550, 1100]T and a goal location g= [500, 4700]T, a few candidate sleeves are presented for the reader in Figure 4.19.
Next, the sleeve is discretized every 1 m along the x and y-coordinates, the SHP from each point in the sleeve to the anchor location is computed, and a static controllability test is implemented according to Equation 4.41, with µ = .3 approximating the lunar friction coefficient. Using the
“central path” sleeve of Figure 4.19a, the anchor reachable sets from the goal location to the anchor are computed and represented in Figure 4.20. The red region represents the union of all the anchor reachable sets, the green and black diamonds represent the anchor and goal locations, respectively, and the yellow “holes” in the red region are the negative obstacles.
Anchor reachable sets corresponding to smaller and larger values of the coefficient of friction,µ, can be seen in Figure 4.21. When the rover has less traction (smallerµ), the anchor reachable sets become narrower (Figure 4.21a), while more traction allows Axel to access a larger subset of the sleeve (Figure 4.21b). These computations show a simple controllability calculation using the simple Coulomb model, however one could easily apply an arbitrarily complex controllability model based on the rover’s dynamics and available terrain data.
Using the discretized reachable configurations, it is now straightforward to compute a round-trip path (anchor to the goal and back) using any preferred optimization criteria. Staying within the anchor reachable sets (red region of Figure 4.20), the ascent is guaranteed to be homotopic to the descent path, precluding the possibility of tether ensnarement around an obstacle. Furthermore, by avoiding the edges of the anchor reachable sets, the rover can minimize lateral tether forces and reduce the risk of tip-over. Such a computation would have greatly benefited the Dante II rover during its ascent out of the Mt. Spurr volcano.
Figure 4.17: The 1 km × 4 km region of positive obstacles from Figure 4.16b as a boundary triangulated manifold
Figure 4.18: Details of the BTM presented in Figure 4.17. Blue region is background, red lines form the triangulation, and the white regions are obstacles.
(a) Central path (b) Veering right (c) Veering left
Figure 4.19: A few example sleeves for the BTM in Figure 4.17. The sleeve is given by the yellow fill. The anchor and goal triangles are presented in green and light blue, respectively.
Figure 4.20: The “central path” sleeve from Figure 4.19a is shown here. The red section indicates the region within the sleeve where Axel would remain controllable using µ = .3 for Coulomb’s law. Yellow “holes” in the red region demarcate the negative obstacles, which are not part of the anchor reachable sets. The green and light blue diamonds indicate the anchor and goal locations, respectively.
(a)µ=.2 (b)µ=.4
Figure 4.21: Computations of the anchor reachable sets with different levels of traction modeled by varying the coefficient of friction under the Coulomb model. The red section indicates the safe/controllable subset of the sleeve, and the green and light blue diamonds indicate the anchor and goal locations, respectively.
Chapter 5
Tether Tension Prediction
In addition to its wheel design and on-board intelligence,Axel’s mobility on extreme terrain is the result of its ability to actively control its tether. By reeling and unreeling the tether, it is able to both ascend and descend steep slopes, and it can even raise itself while free-hanging.
Paramount to extreme terrain mobility is ensuring thatAxel’s tether never severs due to piercing, abrasion, or overstress. Without the ability to recharge its batteries through power connections in the tether, Axel could still potentially function passively on one end of the DuAxel (see Chapter 6), but at the bottom of a crater the rover would be left with limited mobility and a diminishing battery supply.
In January of 1993, Carnegie Mellon University, with the support of NASA funding, took their eight-legged rappelling rover,Dante (the predecessor toDante II), on an expedition to Antarctica.
The target for their field test was Mt. Erebus, the southernmost active volcano on Earth. The rover’s tether, which carried both power and communication signals, would serve asDante’s lifeline during the expedition. Upon reaching the lip of the volcano,Dante began its descent. Unfortunately, after taking only a few steps, the tether snapped and the entire expedition was brought to an abrupt halt [45].
Understanding the tether forces while maneuvering on sloped terrain is therefore essential to ensuring the safety of the rover. When computed beforehand, configurations that lead to high tether stresses can be avoided, and Axel can navigate safely within the cord’s tolerances. This chapter models different rover configurations and maneuvers in order to predict the forces experienced by the tether. Doing so provides insight into this key component ofAxel’s mobility and increases the rover’s robustness on extreme terrain.