Modeling Kinematic Uncertainty of Tendon-Driven Continuum Robots via Mixture Density Networks

TL;DR

The paper addresses the challenge of uncertain and computationally expensive kinematics for tendon-driven continuum robots by introducing a learned Gaussian mixture kinematic model produced by a mixture density network. This MDN outputs a distribution over workspace geometries conditioned on tendon displacements, enabling explicit reasoning about geometric uncertainty and faster inference than traditional Cosserat rod models. Empirical results show the MDN achieves about $0.33$ ms per evaluation (vs $0.39$ ms for the Cosserat rod) and can reduce collision probability in a simulated chest cavity from $15\%$ to $<7\%$ via Bayesian optimization. These findings demonstrate uncertainty-aware planning for safer tendon-driven continuum manipulators and suggest broader applicability to other continuum robot types.

Abstract

Tendon-driven continuum robot kinematic models are frequently computationally expensive, inaccurate due to unmodeled effects, or both. In particular, unmodeled effects produce uncertainties that arise during the robot's operation that lead to variability in the resulting geometry. We propose a novel solution to these issues through the development of a Gaussian mixture kinematic model. We train a mixture density network to output a Gaussian mixture model representation of the robot geometry given the current tendon displacements. This model computes a probability distribution that is more representative of the true distribution of geometries at a given configuration than a model that outputs a single geometry, while also reducing the computation time. We demonstrate one use of this model through a trajectory optimization method that explicitly reasons about the workspace uncertainty to minimize the probability of collision.

Figures (7)

• Figure 1: A Mixture Density Network (MDN) is trained to produce a Gaussian mixture model representation of the robot's geometry given the tendon displacements. The negative log-likelihood is computed between the MDN's output and ground-truth point clouds collected from the physical robot. The MDN is then trained to minimize the negative log-likelihood.
• Figure 2: An example mixture density network architecture that takes the list of tendon displacements $d=(d_1,d_2,...,d_n)$ as input and outputs $\boldsymbol{\mu}_i$, $\mathrm{U}_i$, and $w_i$ independently for each component of the resulting Gaussian mixture model.
• Figure 3: An example configuration in the training data set. (Left) One particular instance of the configuration. (Right) The concatenated distribution of all point clouds gathered during data collection illustrating the inherent uncertainty associated with the geometry of the robot at a given configuration.
• Figure 4: Loss function values at different numbers of mixture components on the test data set. The results suggest that 5 mixture components are the minimum number of components necessary to sufficiently minimize the negative log-likelihood.
• Figure 5: An example set of outputs from the mixture density network after training with five mixture components compared with the ground-truth geometric distribution. The red points are sampled from the Gaussian mixture model output from the mixture density network. The blue points are the points collected from the robot during data collection. The sampled points demonstrate how the learned Gaussian mixture model is capable of representing the full distribution of geometries as a result of the kinematic uncertainty.