# Transition Analysis and Its Application to Global Path Determination for a Biped Climbing Robot
## 1 Introduction  
Trusses typically comprise a number of triangular units constructed with straight rigid members whose ends are connected through joints. In modern architecture, spatial trusses are widely used in the construction of roofs, towers, bridges, and the like. Celebrated buildings include, for example, the Bird's Nest Stadium in Beijing, the Eiffel Tower in Paris, and the Auckland Harbor Bridge. Besides, scaffolds on which workers process the exterior of buildings are also typical spatial trusses. At present, truss-associated routine tasks such as construction, painting, inspection, maintenance, and so on, rely highly on manual labor, signifying a great risk to workers' safety. Robots are ideal assistants or substitutes for human workers carrying out these high-rise and high-intensity tasks. In the past decades, a number of robots have been developed for climbing on trusses or truss members, including, SM\({}^{2}\)[1], ROMA [2], the brachiating robot [3], TREPA [4; 5], WOODY [6; 7], Shady3D [8], RiSE [9; 10], UT-PCR [11], 3 DCLIMBER [12], Treebot [13], the tendril-based bio-inspired robot [14], Climbot [15; 16], and the Snake Robot [17]. Configuring different locomotion and attaching mechanisms, these robots differ significantly in mobility and flexibility. In-depth discussions on this topic could be found in [18; 19]. Among these robots, SM\({}^{2}\), ROMA, Shady3D, 3 DCLIMBER, Treebot and Climbot, having the characteristic of bipedal climbing, are considered to be dominant. These biped climbing robots (BiCRs) generally comprise of an arm-like serial body for locomotion and grippers at both ends for attachment. Benefiting from the bipedal climbing patternsinspired by arboreal primates [20], BiCRs have the flexibility to imitate transition between branches, as illustrated in Figure 1. This transition capability is paramount for robots switching objects to be climbed upon, especially for executing tasks on large-scale spatial complex structures. Another distinct advantage of BiCRs is the combination of manipulation and mobility. Hence, BiCRs are also known as mobile manipulators [1; 15]. As a featured example, SM2 was designed to work on the truss and other exterior surfaces of the Space Station Freedom for performing routine tasks. Besides, the two underactuated miniature climbing robots in [21], MATS [22], Frambot [23] and W-Climbot [24] are also typical BiCRs. Although coupled climbing robots have been developed in the past, most research focused on the system prototype and experimental verification of climbing patterns on simple structures [25; 26]. Few attention in the literature has been paid to the perception and planning of this type of robots when traveling in truss environments. In this paper, we study the BiCR climbing path planning problem. Routine truss-associated tasks usually involve the transition from one member to another. Therefore, the transition issue must be handled well before looking into the climbing path planning problem. BiCR transit was qualitatively described in [2; 27; 28], but neither quantitative analysis nor executable output can be found in detail. In [29], a BiCR with five degrees of freedom (DoF) was proven capable of transiting between two cylindrical members at any relative orientation. Unfortunately, the distance of the members was not considered in this paper, and it was assumed that the robot could always reach its target member at any time. This is not the case in real situations. Moreover, the issue of where and how to transit was not dealt with, preventing the transition from practice. In [30; 31], approaches for determining the graspable region in a climbing cycle were presented. The resulting graspable region, however, corresponds to a fixed grip only, i.e., one end attached at a known point. Theoretically, the entire operational region for transition could be numerically obtained by sampling the gripping point on both members alternatively, solving the graspable region for each and then merging all regions.