Previous on-manifold approaches to continuum robot state estimation have typically adopted simplified Cosserat rod models, which cannot directly account for actuation inputs or external loads. We introduce a general framework that incorporates uncertainty models for actuation (e.g., tendon tensions), applied forces and moments, process noise, boundary conditions, and arbitrary backbone measurements. By adding temporal priors across time steps, our method additionally performs joint estimation in both the spatial (arclength) and temporal domains, enabling full \textit{spacetime} state estimation. Discretizing the arclength domain yields a factor graph representation of the continuum robot model, which can be exploited for fast batch sparse nonlinear optimization in the style of SLAM. The framework is general and applies to a broad class of continuum robots; as illustrative cases, we show (i) tendon-driven robots in simulation, where we demonstrate real-time kinematics with uncertainty, tip force sensing from position feedback, and distributed load estimation from backbone strain, and (ii) a surgical concentric tube robot in experiment, where we validate accurate kinematics and tip force estimation, highlighting potential for surgical palpation.

翻译：以往基于流形的连续体机器人状态估计方法通常采用简化的 Cosserat 杆模型，这类模型无法直接考虑驱动输入或外部载荷。本文提出一个通用框架，该框架整合了驱动（如腱张力）、作用力与力矩、过程噪声、边界条件以及任意骨架测量的不确定性模型。通过引入跨时间步的时间先验，我们的方法进一步实现了空间（弧长）域与时间域的联合估计，从而能够进行完整的时空状态估计。通过对弧长域进行离散化，我们得到了连续体机器人模型的因子图表示，该表示可利用类似 SLAM 的快速批量稀疏非线性优化方法进行处理。本框架具有通用性，适用于广泛的连续体机器人类型；作为示例，我们展示了（i）仿真中的腱驱动机器人，我们演示了带不确定性的实时运动学、基于位置反馈的末端力感知，以及基于骨架应变的分布式载荷估计；（ii）实验中的外科同心管机器人，我们验证了精确的运动学与末端力估计，凸显了其在外科触诊中的应用潜力。