General Methods for Evaluating Collision Probability of Different Types of Theta-phi Positioners

TL;DR

This work addresses the collision risk in large arrays of theta-phi robotic fiber positioners by formulating a general static-collision probability model for RFPs, applicable to both equal- and unequal-arm configurations. It introduces a two-part collision calculation that accounts for overlap conflicts and eccentric-arm interactions, and validates the model against Monte Carlo simulations, leveraging CUDA to accelerate collision detection and enabling rapid iterative design. The study finds that employing a Poisson target distribution can reduce average collision probability by about $2.6\%$, and demonstrates strong agreement between the mathematical model and Monte Carlo results (R^2 ≈ 0.934 in large-scale tests). These results provide a practical framework for RFP design, tiling strategies, and target allocation to minimize collisions in next-generation multi-object spectroscopic facilities.

Abstract

In many modern astronomical facilities, multi-object telescopes are crucial instruments. Most of these telescopes have thousands of robotic fiber positioners(RFPs) installed on their focal plane, sharing an overlapping workspace. Collisions between RFPs during their movement can result in some targets becoming unreachable and cause structural damage. Therefore, it is necessary to reasonably assess and evaluate the collision probability of the RFPs. In this study, we propose a mathematical models of collision probability and validate its results using Monte Carlo simulations. In addition, a new collision calculation method is proposed for faster calculation(nearly 0.15% of original time). Simulation experiments have verified that our method can evaluate the collision probability between RFPs with both equal and unequal arm lengths. Additionally, we found that adopting a target distribution based on a Poisson distribution can reduce the collision probability by approximately 2.6% on average.

Figures (11)

• Figure 1: Schematic of equal-arm RFP. The dashed circle represents the patrol area of the RFP.
• Figure 2: Accessibility of equal-arm RFPs. Region a: only one RFP can reach, region b: two RFPs can reach, region c: three RFPs can reach.
• Figure 3: Accessibility of unequal-arm RFP in different regions. The shading in the figure indicates the coverage of each region by the RFPs. The depth of the color indicates the coverage of a single area by the RFP: the darker the color, the more RFPs can reach that area.
• Figure 4: Comparison of three different collision cases. (a) Type 1: Two RFP do not collide with each other. (b) Type 2: The RFP experiences collisions limited to its immediate neighboring ring of positioners. (c) Type 3: This type of collision involves the RFP's collision with RFP more than its neighbor positioners. In the images, the green region represents the area covered by a single RFP, the yellow region is covered by two RFPs, and the red region is covered by three RFPs.
• Figure 5: The diagram illustrating the calculation of collision probability. The green, blue and yellow areas represent the patrol area of one positioner, the potential collision zones of the eccentric arm, and the conflict area, respectively. The red areas represent the actual inner and outer diameters considering obstacle avoidance.