Proprioceptive Shape Estimation of Tensegrity Manipulators Using Energy Minimisation
Authors: Tufail Ahmad Bhat, Shuhei Ikemoto
Organizations: Graduate School of Life Science and Systems Engineering, Kyushu Institute of Technology, 2-4 Hibikino, Wakamatsu, Kitakyushu, Fukuoka, Japan
Shape estimation is fundamental for controlling continuously bending tensegrity manipulators, yet achieving it remains a challenge. Although using exteroceptive sensors makes the implementation straightforward, it is costly and limited to specific environments. Proprioceptive approaches, by contrast, do not suffer from these limitations. So far, several methods have been proposed; however, to our knowledge, there are no proven examples of large-scale tensegrity structures used as manipulators. This paper demonstrates that shape estimation of the entire tensegrity manipulator can be achieved using only the inclination angle information relative to gravity for each strut. Inclination angle information is intrinsic sensory data that can be obtained simply by attaching an inertial measurement unit (IMU) to each strut. Experiments conducted on a five-layer tensegrity manipulator with 20 struts and a total length of 1160 mm demonstrate that the proposed method can estimate the shape with an accuracy of 2.1 % of the total manipulator length, from arbitrary initial conditions under both static conditions and maintains stable shape estimation under external disturbances.
Figures & tables
Fig. 1: Tensegrity manipulator with self-shape awareness a) The tensegrity manipulator provides a balance between rigidity and flexibility. b) TM40 : a highly redundant five-layer tensegrity manipulator. c) Schematic of the three-dimensional TM40 tensegrity structure with the proprioceptive sensors, used for the energy minimisation algorithm. d) Estimated nodal positions. e) Full-scale estimated shape of the tensegrity manipulator determined from the spatial nodal positions.
Fig. 2: Schematic representation of the tensegrity manipulator. a) Geometric representation of the nodal positions, centre positions, and orientations of the strut elements. b) Transformation of the strut orientation from spherical to Cartesian coordinates.
Fig. 3: The figure shows the visualisation of the optimisation process in RViz2. Given the inclination angles of the strut elements, the method estimates the spatial nodal positions of the structure by minimising the energy function.
Fig. 4: Five-Layer tensegrity manipulator: Each module is stacked with alternating twist direction to form a five-layer tensegrity manipulator. The structure consists of 80 active and passive tension elements and 20 rigid struts. Each pneumatic cylinder is controlled independently, and the system has a ROS2-compatible embedded AD/DA module to actuate the 40 pneumatic cylinders.
Fig. 5: Sensor module for inclination angles: a) The manipulator consists of 20 struts. b) Each is equipped with a 6-axis IMU sensor. c) The inclination angles are measured from the longitudinal axis direction of each strut element and the gravitational vector.
Fig. 6: The plot shows the energy of the structure in the collapsed (green) and expanded (blue) states. The yellow-shaded region represents the cumulative energy of all cable elements for the first 20 steps of optimisation, while the light blue-shaded region shows the decrement in total energy over the optimisation steps.
Fig. 7: The heatmaps in the left and middle panels show the gradients of the energy function with respect to the mean of x , y , and z centre positions during the collapsed and expanded states, respectively. The right panel shows gradients with respect to the yaw angles of the struts. Each row corresponds to a strut element. In all the heatmaps, the gradients gradually vanish as the optimisation steps increase, which indicates convergence.
Fig. 8: The left and right sides show estimations starting from collapsed and expanded states, respectively. In both cases, the estimated shape gradually converges toward the actual shape as the optimisation steps increase. By the 1000th step, both cases reach a minimal energy state.
Fig. 9: The estimated shape initialised from random unknown parameters. The figure shows that the optimisation problem consistently converges to the same optimal solution across all 10 experiments.
Node Group
Avg. x
Avg. y
Avg. z
d
Error
% Error
Top ( n1 – n4 )
4.82
4.24
655.82
1135.5
2.11
24.5
Base ( n37 – n40 )
-4.26
-5.70
-479.59
–
–
–
TABLE I: Estimated manipulator length in mm.
Fig. 10: The left panel shows the estimated shape of the tensegrity manipulator before convergence. The right panel shows the converged shape, and then the manual external perturbations are applied at the tip of the arm, which induce bending to the left and right. The shape estimator approximately estimated these deformations, and the corresponding shapes of the TM-40 are visualised in RViz2.
Unlike conventional rigid-link robots defined by discrete joints, continuum robots pose a fundamental challenge for expressing their complex continuous bending configurations for closed-loop control. Several modelling approaches have been proposed for conventional continuum robots, but tensegrity-based continuum robots remain largely open. Moreover, many of these approaches assume a continuous elastic backbone and are therefore not directly applicable to tensegrity manipulators, whose bodies are networks of rigid struts and tensioned cables. This work presents a reduced-order model for shape and posture control of a tensegrity-based continuum manipulator. The manipulator is modelled as a serially connected parallel-link mechanism. The proposed method is formulated as an optimization problem that uses geometric constraints of the tensegrity structure together with information from the Inertial Measurement Unit (IMU) sensors embedded in the strut elements. To the best of our knowledge, this work presents the first experimental demonstration of a real-time IMU-based shape estimation method on a full-scale tensegrity manipulator and demonstrates posture control using a simple Proportional-Integral (PI) controller. The results show that the proposed method can estimate the shape of both single-module tensegrity structures and multi-module tensegrity manipulators from arbitrary static configurations and achieve desired postures.
This paper presents a scalable, open-source visuotactile sensing system for tensegrity robots that enables six-axis wrench estimation and contact detection. The proposed endcap sensor integrates an elastomeric shell, a 3D-printed thermoplastic polyurethane (TPU) interface, and a rigid base housing an embedded camera and LED illumination ring. A novel gyroid-infill bonding technique is introduced to form a durable elastomer-TPU interface without adhesives, yielding a lightweight and modular design compatible with large-scale tensegrity structures. A tactile-to-wrench neural network maps shear vector fields to six-dimensional force and torque measurements. Experimental results demonstrate accurate and stable wrench estimation with a mean squared error (MSE) of 0.1531 on static validation data and out-of-domain generalization under dynamic motion. Furthermore, full-system integration on a 12 kg tensegrity robot confirms the sensor's ability to reliably identify ground contacts. The system substantially improves the practicality of tactile feedback for tensegrity robots, offering a low-cost, reproducible, and physically interpretable pathway toward contact-aware proprioception and state estimation. Open source files are available at github.com/Jonathan-Twz/tensegrity-gelfoot
Wenzhe Tong, Jonathan Mi, Xili Yi +2
Robotics Department, University of Michigan, Ann Arbor, MI, 48109, USA
Tensegrity robots possess lightweight and resilient structures but present significant challenges for state estimation due to compliant and distributed ground contacts. This paper introduces a symmetry-aware heterogeneous graph neural network (Sym-HGNN) that infers contact states directly from proprioceptive measurements, including IMU and cable-length histories, without dedicated contact sensors. The network incorporates the robot's dihedral symmetry D3 into the message-passing process to enhance sample efficiency and generalization. The predicted contacts are integrated into a state-of-the-art contact-aided invariant extended Kalman filter (InEKF) for improved pose estimation. Simulation results demonstrate that the proposed method achieves up to 15% higher accuracy and 5% higher F1-score using only 20% of the training data compared to the CNN and MI-HGNN baselines, while maintaining low-drift and physically consistent state estimation results comparable to ground truth contacts. This work highlights the potential of fully proprioceptive sensing for accurate and robust state estimation in tensegrity robots. Code available at: https://github.com/Jonathan-Twz/Tensegrity-Sym-HGNN
Wenzhe Tong, Yicheng Jiang, Chi Zhang +2
Robotics Department, University of Michigan, Ann Arbor, MI 48109 USA