# Transition Analysis and Its Application to Global Path Determination for a Biped Climbing Robot
## 1 Introduction
Despite probabilistic completeness, this is intensive in terms of time and computational resources. Lacking a close-formed solution for transition analysis, concrete transiting motion is always ignored in studies of path planning on trusses. Hierarchical control structure and multi-phase control strategy were discussed in [32], where only the truss junctions were considered attachable by SM2. Additionally, the initial and final states of the robot's two feet for each climbing cycle were artificially designated. In other words, neither path planning nor member-to-member transition were required by SM2. In [2], the path planning of ROMA, a robot designed for inspection applications in 3D environments, was modeled as a classical traveling salesman problem (TSP), taking a climbing step as the smallest unit for the consideration of energy cost. The TSP was finally solved by optimizing the energy consumption. However, only the middle and both ends of a member are allowed to attach for the robot. Similar research was done in [28], with a stricter calculation of the energy consumed in terms of a specific climbing gait. One obvious but important issue in both [2; 28] is that the ability  
Figure 1: The inspiration for biped climbing robots. of a gripper to grip a member on the expected point was neglected. In the real world, the spatial relationships of gripper fingers and members must be properly considered. Otherwise, the robot may fail in attachment, and thereby fall down. This issue was well handled in [27] by defining a node with its position, direction, and face on a truss. Truss members were dispersed into a number of limited nodes, each representing a discrete gripping point (The term \"gripping point\" used in this paper consists of the position vector and the orientation matrix to locate a grip in 3D space.). The best path to a destination on the truss was then determined by optimizing the path length and the cost of difficult maneuvers. However, owing to the limitation in locomotion flexibility, Shady3D has only one type of climbing gait, and performs simple transition within a plane cooperating with another unit. Consequently, the path planning of Shady3D robots is simpler than that of a more flexible BiCR with greater DoF. To assemble a truss with multiple Shady3D robots, the truss navigation problem was investigated in [33], where each Shady3D was treated as a single movable point. The trunks and branches of trees form natural truss structures upon which the path planning problem was studied in [34]. A grid map with bottom-up rings comprised by discrete points was used to model the surfaces of the trunk and branch. A sequence of gripping points was afterwards arranged by considering the motion cost, gravity, robot orientation, and its reachability. This method is applicable to climbing robots with nonenclosure gripping, i.e., Treebot, as stated in the paper, but not to BiCRs, such as SM\({}^{2}\) and Climbot, which use enclosure gripping and can flip over the head and tail in climbing. In a word, to the best of our knowledge, few efforts have been made in the field of BiCR path planning on three-dimensional (3D) complex trusses, while complete transition analysis has not yet been reported. Our Climbot, originally developed in 2007 [15], is a robot designed to carry out high-rise routine tasks. Figure 2 shows the latest version of Climbot and its kinematic diagram. Compared with other BiCRs, Climbot has more climbing gaits and a stronger ability to negotiate obstacles and transit between structural members owing to its agile body and control, transplanted from powerful and light-weight industrial robots. To address the path planning problem of BiCRs in 3D complex truss environments, firstly, we propose a theoretical analysis of the transition between any two given members. The principle of transition analysis was previously presented in [35], and will be further improved by considering the grip accessibility in this paper.