Estimating Infinite-Dimensional Continuum Robot States From the Tip

Abstract

Knowing the state of a robot is critical for many problems, such as feedback control. For continuum robots, state estimation is an incredible challenge. First, the motion of a continuum robot involves many kinematic states, including poses, strains, and velocities. Second, all these states are infinite-dimensional due to the robot's flexible property. It has remained unclear whether these infinite-dimensional states are observable at all using existing sensing techniques. Recently, we presented a solution to this challenge. It was a mechanics-based dynamic state estimation algorithm, called a Cosserat theoretic boundary observer, which could recover all the infinite-dimensional robot states by only measuring the velocity twist of the tip. In this work, we generalize the algorithm to incorporate tip pose measurements for more tuning freedom. We also validate this algorithm offline using experimental data recorded from a tendon-driven continuum robot. We feed the recorded tendon force and tip measurements into a numerical solver of the Cosserat rod model based on our robot. It is observed that, even with purposely deviated initialization, the state estimates by our algorithm quickly converge to the recorded ground truth states and closely follow the robot's actual motion.

Figures (8)

• Figure 1: Overview of our Cosserat theoretic boundary observer algorithm and experimental results. This algorithm predicts the robot state based on the Cosserat rod model, actuator force measurements, and injected correction based on tip pose and velocity measurements. Experimental results show that the state estimates for the poses and velocities along the robot closely follow the ground truth states.
• Figure 2: Our continuum robot model. Using Cosserat rod theory, our robot is modeled as a continuous set of rigid cross-sections stacked along a centerline parametrized by $s$ from the base ($s=0$) to the tip ($s=\ell$) where $\ell$ is the total length. $g(s,t)\in SE(3)$ is the pose (homogeneous transform matrix) of the cross-section at location $s$ and time $t$. Each cross-section defines a body frame. The robot undergoes internal wrenches (due to elastic deformation and actuation) and external wrenches (such as gravity).
• Figure 3: Planar view of a single disk, 10 of which are attached along our robot backbone. The ridge at the central hole extends 1 mm from the surface of the disk and is located on the side of the disk facing the robot base. The small holes are 25 mm from the center. Each disk's thickness is 2 mm.
• Figure 4: Setup for experiments. We placed one LED tracking marker at the center of each of the ten disks to measure their ground truth position. We placed one IMU on each of the base and tip disks to measure the tip orientation relative to a known reference orientation. We used a force sensor to measure the tension of a tendon routed through the holes on top of the 10 disks. The body frame of the base disk was defined by the red axes attached to the base disk, which was also chosen to be the spatial frame. The body frames attached to all other cross-sections were similarly defined.
• Figure 5: Snapshots of the experiment and offline validation results of the algorithm. In the top row, the robot exhibited planar motions under a 50 g tip load and the time-varying tendon tension. In the bottom row, the black dots are the ground truth positions of the markers on the backbone, and the purple curve is the estimated position of the backbone by our algorithm. We purposely initialized our algorithm with a deviated configuration. The estimated positions quickly converged to the ground truth positions and exhibited close tracking of the robot's actual motion.