发表机构
Purdue University（普渡大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
针对经典 Hill--Clohessy--Wiltshire 方程无法处理持续推力参考轨迹的问题，提出基于 $\mathrm{SE}_2(3)$ 李群的相对运动框架，通过线性化失配并推导双脉冲规划器，实现推力参考下的成功交会。
AI 中文摘要
经典的 Hill--Clohessy--Wiltshire 方程假设参考轨迹为无外力作用的开普勒轨迹，而这一假设在需要持续推力的任务中不再成立。为解决这一局限，我们在 $\mathrm{SE}_2(3)$ 李群上构建了一个相对运动框架，将位置、速度和姿态统一编码在一个几何状态中。为了计算上的可处理性，我们对两航天器之间的重力失配和体坐标系控制失配进行了线性化，并在每种情况下推导了被忽略的高阶项的紧致解析上界。在圆形滑行开普勒假设下，该框架精确地还原了 Hill--Clohessy--Wiltshire 方程，从而将经典交会理论确立为一个特例而非独立的线性化方法。对于推力参考轨迹，状态转移矩阵获得了经典公式中不存在的姿态-平移耦合非对角块，而吸收控制失配则将线性化中心重新定位到两航天器输入的平均值。直接从状态转移矩阵推导出的双脉冲交会规划器同时考虑了这两种效应。数值模拟证实了经典方程恢复到机器精度，证明了在经典规划器失效的推力参考轨迹下交会能够成功，并在整个转移过程中验证了重力和控制失配线性化界的有效性。
英文摘要
The classical Hill--Clohessy--Wiltshire equations assume an unforced Keplerian reference trajectory, an assumption that is violated by missions requiring continuous thrust. We address this limitation with a relative motion framework on the $\mathrm{SE}_2(3)$ Lie group that encodes position, velocity, and attitude in a unified geometric state. For computational tractability we linearize both the gravity mismatch and the body-frame control mismatch between the two vehicles, deriving tight analytic upper bounds on the neglected higher-order terms in each case. Under circular coasting Keplerian assumptions the framework recovers the Hill--Clohessy--Wiltshire equations exactly, establishing classical rendezvous theory as a special case rather than an independent linearization. For thrusting reference trajectories the state transition matrix acquires off-diagonal attitude--translation coupling blocks absent from classical formulations, and absorbing the control mismatch re-centers the linearization at the mean of the two vehicles' inputs. A two-impulse rendezvous planner derived directly from the state transition matrix accounts for both effects. Numerical simulations confirm recovery of the classical equations to machine precision, demonstrate successful rendezvous about a thrusting reference where classical planners fail, and validate the gravity and control mismatch linearization bounds throughout the transfer.
