Multi-Contact Force Estimation for Continuum Robots via Gaussian-Parameterized Factor Graphs

Reconstructing the shape of continuum manipulators from sparse, noisy sensor data is a challenging task, owing to the infinite-dimensional nature of such systems. Existing approaches broadly trade off between parametric methods that yield compact state representations but lack probabilistic structure, and Cosserat rod inference on factor graphs, which provides principled uncertainty quantification at the cost of a state dimension that grows with the spatial discretization. This letter combines the strength of both paradigms by estimating the coefficients of a low-dimensional Geometric Variable Strain (GVS) parameterization within a factor graph framework. A novel kinematic factor, derived from the Magnus expansion of the strain field, encodes the closed-form rod geometry as a prior constraint linking the GVS strain coefficients to the backbone pose variables. The resulting formulation yields a compact state vector directly amenable to model-based control, while retaining the modularity, probabilistic treatment and computational efficiency of factor graph inference. The proposed method is evaluated in simulation on a 0.4 m long tendon-driven continuum robot under three measurement configurations, achieving mean position errors below 2 mm for all three scenarios and demonstrating a sixfold reduction in orientation error compared to a Gaussian process regression baseline when only position measurements are available.

Recent factor graph approaches to continuum robot state estimation have been successful for quasi-static applications and spatiotemporal estimation using white-noise kinematic motion priors. However, when inertial effects are significant, these approximations may fail to capture the underlying physics, limiting accuracy during dynamic motions. In contrast, our approach approximates the Cosserat rod dynamics of continuum robots. We write inertia and damping as equivalent applied loads, so that the dynamic balance retains the algebraic form of the static one from prior work with quasi-static robots. Without backbone observations, the framework reduces to a stochastic forward simulation of the robot's motion. Given observations, it jointly refines kinematic and dynamic states and infers external loads, among other states. We validate the approach through simulation and experiments, demonstrating stochastic forward simulation as well as state estimation on tendon-driven continuum robots.

Continuum robots often operate in uncertain environments, where accurate state estimation is essential for safe interactions. Estimate uncertainty is inherently spatially non-uniform: confidence varies depending on where measurements are available. Global estimation accuracy is not always the top priority, but rather achieving sufficient confidence at task-relevant locations along the robot. This extended abstract introduces mechanically reconfigurable sensing enabling uncertainty-shaping in state estimation for continuum robots. We present a concept hardware design demonstrating the feasibility of longitudinal translation of a sensor within a continuum robot. We demonstrate that state estimation confidence can be reconfigured by varying the sensor location, and show a reduction of full-body shape estimation errors when sliding the sensor back and forth over time, compared to a single fixed tip sensor.