Risley prisms encounter a control singularity in the center region of the field of regard (FOR) during target tracking. When the emerging beam or the line of sight (LOS) of the system tracks a target near the system rotation axis, the prisms need to rotate at an extremely high speed and even make an instantaneous 180° flip. Due to the limited maximum speeds of driving motors, the control singularity problem challenges the drive and control of the prisms when tracking a continuous and smooth path close to or passing through the system rotation axis, restricting the system,s real-time target tracking capability. Although three-element Risley prisms can eliminate these singularities, adding a third prism not only enlarges the size and increases the cost but also demands complex control algorithms. To maintain simplicity, two prisms seem a reasonable choice. However, to relieve control difficulties, it is beneficial to discuss the characteristics and sources of the control singularity problem, which can assist in guiding the control system design and exploring solutions to the singularity problem. In our current study, based on our previous research on the nonlinearity problem in Risley-prism-based target tracking, we aim to analyze the inverse solutions of prism orientations for targets in the center region of the FOR. Then, the characteristics and root causes of the control singularity problem are disclosed in continuous and discrete time domains. Moreover, the internal mechanism and implementation effect of the optimal-solution method for resolving the singularity problem are further investigated.

Focusing on the center of the FOR, the inverse solutions of prism orientations are obtained using the two-step method, and their singularities are then analyzed. For the targets passing through the center or moving near the center [Fig. 2(a)], the ratios of the rotational speed of the prisms to the slewing rate of the beams, denoted as the M values, are calculated in continuous time domains. By analyzing the M value, the origin of the singularity is uncovered, and the performance characteristics of the singularity are discussed. For target tracking in the discrete time domain (Fig. 3), the required rotation angles of the prisms for tracking the target from one point to other points in the center zone are calculated (Fig. 4) and the average M values are derived (Fig. 5). Based on these results, the characteristics and root causes of the singularity in discrete time domains are studied (Figs. 6 and 7). The principle basis is revealed to explain why the optimal-solution method can mitigate the control singularity problem. The requirements of target tracking for driving and controlling prism rotation and the angular region of tracking blind zone in the center region are estimated (Figs. 6 and 8).

For the center of the FOR, the singularity of the inverse solutions of prism orientations results from the uncertainty of the azimuth angle. For the targets moving near the center [Fig. 2(a)], the tangential movement leads to large variation in azimuth, yielding the maximum M value, denoted as M_m [Fig. 2(b)]. As the altitude angle approaches zero, M_m increases sharply and becomes infinite [Fig. 2(c)]. For the targets passing through the center, the optimal-solution method can solve the singularity problem. In the discrete time domain (Fig. 3), if the same set of solutions is used to track a target when the target moves from one side of the center to the other side, the average M value increases significantly (Fig. 5). This singularity problem can be alleviated by switching the solutions, that is, adopting the optimal-solution method (Fig. 6). However, a strong control ability to drive prism rotation is still necessary for tracking the target within a certain angle range near the center. For a given rotational double prism system and tracking application, a tracking blind zone with a certain angle range exists in the center region of the FOR (Fig. 8).

Based on the inverse solutions of prism orientations obtained using the two-step method, the ratios of the rotational speed of the prisms to the slewing rate for the beams are calculated. The characteristics and root causes of the control singularity problem are analyzed when the system tracks a target in the center region of the FOR in continuous-time and discrete-time domains respectively. It is discovered that for target tracking in the center region, the control singularity problem stems from the large variation in target azimuth, caused by the tangential movement of the target. The closer the target is to the center of the FOR, the more prominent the control singularity problem is. The optimal-solution method can relieve the control difficulties of prism rotation resulting from the azimuth jump of the target crossing the center. There is still a tracking blind zone in the center region of the FOR, and its angle range is determined by the driving and control ability of the system to prism rotation and the target tracking requirements. The proposed analysis methods and results can provide a foundation for the design of the prism drive control scheme and the evaluation of the system tracking performance.