Continuum robots are flexible, thin manipulators capable of navigating confined or delicate environments making them well suited for surgical applications. Previous approaches to continuum robot state estimation typically rely on simplified, deterministic actuation models. In contrast, our method jointly estimates robot shape, external loads, internal stresses, and actuation inputs. We adopt a discrete Cosserat rod formulation and show that, when paired with a midpoint integration rule, it achieves high numerical accuracy with relatively few state nodes. This discretization naturally induces a factor-graph structure for sparse nonlinear optimization on SE(3). We extend the formulation with actuation factors for tendon-driven robots and combine multiple rod graphs for parallel continuum robots with closed-loop topologies. By explicitly including actuation variables in the state, the linearized system can be reused to extract manipulator Jacobians, which we leverage in performing trajectory tracking. Finally, we validate the approach experimentally on a surgical concentric tube robot. Overall, our approach enables principled, real-time estimation across multiple continuum robot architectures, accounting for actuation uncertainty and providing direct access to manipulator Jacobians.

翻译：连续体机器人是一种柔性细长机械臂，能够在受限或精密环境中导航，因此非常适合手术应用。以往连续体机器人状态估计方法通常依赖于简化的确定性驱动模型。相比之下，我们的方法联合估计机器人形状、外部载荷、内部应力及驱动输入。我们采用离散Cosserat杆模型，并证明当结合中点积分规则时，该方法能以较少的状态节点实现较高的数值精度。该离散化过程自然诱导出SE(3)上稀疏非线性优化的因子图结构。我们通过为腱驱动机器人添加驱动因子扩展了该模型，并为具有闭环拓扑结构的并联连续体机器人组合了多杆图。通过将驱动变量显式纳入状态，线性化系统可被复用以提取机械臂雅可比矩阵，我们在轨迹跟踪中利用了此特性。最后，我们在手术同心管机器人上通过实验验证了该方法。总体而言，我们的方法实现了跨多种连续体机器人架构的规范化实时估计，同时考虑了驱动不确定性并提供了机械臂雅可比矩阵的直接访问途径。