Real-Time Shape Estimation of Tensegrity Structures Using Strut Inclination Angles

TL;DR

The paper tackles real-time shape estimation for tensegrity robots that lack traditional joints by formulating an energy-based optimization using only strut inclination angles from onboard IMUs. It introduces a gradient-descent framework that minimizes the elastic energy to recover strut centers and orientations, leveraging a known connectivity matrix and cable stiffness while circumventing magnetometer reliance due to interference. Experimental validation on a Class 1 four-strut tensegrity demonstrates real-time performance (≈0.52 ms per step) with MAEs on the order of a few millimeters for strut centers and tens of millimeters for node positions, across static and dynamic deformations. The approach reduces sensor complexity and shows potential for broader deployment in tensegrity-based robotics, with future work aimed at scaling to larger structures and accommodating variable stiffness.

Abstract

Tensegrity structures are becoming widely used in robotics, such as continuously bending soft manipulators and mobile robots to explore unknown and uneven environments dynamically. Estimating their shape, which is the foundation of their state, is essential for establishing control. However, on-board sensor-based shape estimation remains difficult despite its importance, because tensegrity structures lack well-defined joints, which makes it challenging to use conventional angle sensors such as potentiometers or encoders for shape estimation. To our knowledge, no existing work has successfully achieved shape estimation using only onboard sensors such as Inertial Measurement Units (IMUs). This study addresses this issue by proposing a novel approach that uses energy minimization to estimate the shape. We validated our method through experiments on a simple Class 1 tensegrity structure, and the results show that the proposed algorithm can estimate the real-time shape of the structure using onboard sensors, even in the presence of external disturbances.

Figures (9)

• Figure 1: Left TM40: Tensegrity manipulator consisting of 20 struts, 40 pneumatic actuators, and 40 cables. Middle: Schematic of TM40. Unlike articulated robots with well-defined joints, this manipulator has no joints. Right: The estimated shape of a simple Class 1 tensegrity.
• Figure 2: Geometric Representation of a Tensegrity Structure (a) Overview of the Class 1 tensegrity structure. (b) Geometric representation of $i$th strut element in the $xyz$-coordinate frame and the relevant angles, such as the inclination angle $\phi_i$ and the rotational angle $\theta_i$ around the gravity vector or yaw angle. (c) Schematic representation of $k$th cable element, connecting nodes $\mathbf{n}_{ik}$ and $\mathbf{n}_{jk}$ in a two-dimensional plane.
• Figure 3: The experimental setup is based on a Class 1 tensegrity structure consisting of four struts and twelve cables. Each strut is equipped with an onboard M5stick device that includes an integrated IMU to measure the inclination angles $\boldsymbol{\phi}$. We also affixed three markers to each strut to obtain the ground truth.
• Figure 4: A high-level overview of the system architecture. The MoCap nodes are responsible for the ground truth (red) data published to RViz2, whereas the inclination angles $\boldsymbol{\phi}$ are used in the optimization process obtained from the M5-Stick. The optimized shape (black) has also been published to RViz2. For further details, refer to Section III.
• Figure 5: The figure shows the comparison of the actual shape (red) and the estimated shape (black) of the tensegrity structure at the initial step (Step 1) and after optimization (Step 300). The results show estimated shape improves and aligns more closely with the actual configuration as the optimization steps increase.