The trajectories of the $\mathrm{O}(1)$-Kepler problem at level $n\ge 2$ are completely determined. It is found in particular that a non-colliding trajectory is an ellipse, a parabola or a branch of hyperbola according as the total energy is negative, zero or positive. Moreover, it is shown that the group $\mathrm{GL}(n, \mathbb R)/\mathrm{O}(1)$ acts transitively on both the set of oriented elliptic trajectories and the set of oriented parabolic trajectories. The method employed here is similar to the one used by Levi-Civita in the study of planar Kepler problem in 1920.

In situ tissue biopsy with an endoluminal catheter is an efficient approach for disease diagnosis, featuring low invasiveness and few complications. However, the endoluminal catheter struggles to adjust the biopsy direction by distal endoscope bending or proximal twisting for tissue sampling within the tortuous luminal organs, due to friction-induced hysteresis and narrow spaces. Here, we propose a pneumatically-driven robotic catheter enabling the adjustment of the sampling direction without twisting the catheter for an accurate in situ omnidirectional biopsy. The distal end of the robotic catheter consists of a pneumatic bending actuator for the catheter's deployment in torturous luminal organs and a pneumatic rotatable biopsy mechanism (PRBM). By hierarchical airflow control, the PRBM can adjust the biopsy direction under low airflow and deploy the biopsy needle with higher airflow, allowing for rapid omnidirectional sampling of tissue in situ. This paper describes the design, modeling, and characterization of the proposed robotic catheter, including repeated deployment assessments of the biopsy needle, puncture force measurement, and validation via phantom tests. The PRBM prototype has six sampling directions evenly distributed across 360 degrees when actuated by a positive pressure of 0.3 MPa. The pneumatically-driven robotic catheter provides a novel biopsy strategy, potentially facilitating in situ multidirectional biopsies in tortuous luminal organs with minimum invasiveness.