Tendon-Driven Continuum Robot with Modular Stiffness and In-Situ Self Pose Estimation

Continuum robots have gained attention for their compliance and adaptability compared to rigid-link robots, enabling safe interaction and smooth continuous deformation inspired by biological structures such as octopus arms and elephant trunks [1]. These traits support dexterous manipulation and navigation in confined or cluttered spaces, making them suitable for search [2], minimally invasive surgery [3], inspection [4], and human–robot interaction [5]. Most existing continuum robots are still task-specific and built in an ad hoc way, with fixed geometry, actuation, and mechanical properties that are hard to reconfigure. As a result, adapting them to new tasks often requires major redesign and refabrication, limiting scalability, and broader adoption. This lack of modularity stands in contrast to the versatility that continuum robots are theoretically capable of providing. Other major barriers to the practical deployment of continuum robots are sensoring and state estimation. Unlike rigid robots with a small set of joint variables, continuum robots effectively have infinite degrees of freedom, making shape and pose estimation difficult. Many approaches therefore rely on external infrastructure (e.g., motion capture, cameras, or tracking rigs) [6], [7], which works in the lab but increases cost and complexity and limits use outside controlled environments. To address these limitations, we present a self-contained modular continuum robotic platform that combines mechanical reconfigurability with embedded proprioceptive sensing.

The proposed system is designed around modular continuum joints that can be rapidly assembled, replaced, or reconfigured to meet different task requirements without sacrificing the fundamental advantages of continuum robots, namely their smooth shape morphing and compliance. The mechanical behavior of each joint is programmable through interchangeable elements with precomputed stiffness characteristics, enabling users to specify desired robot shapes and deformation profiles in a principled manner. Beyond mechanical modularity, the platform enables self-pose estimation with onboard sensing, avoiding external tracking. We use magnetic sensing with machine learning to infer the configuration of each joint, and keep the learning pipeline modular by training one model on the movements of a single joint that can be reused across different robot joints. This reduces training effort and supports scalable reconfiguration while maintaining reliable pose estimates when trained on sufficiently diverse coil data.

The main contributions of this paper are as follows:

1. The design of a modular, reconfigurable continuum robotic platform.

2. A method for programming robot shape through interchangeable joints with stiffness properties that enhance performance for priority tasks while maintaining broad functionality.

3. A self-contained pose estimation approach using magnetic sensing and modular machine learning models trained at the joint level.

4. Experimental validation of the proposed system in a set of traversal and grasping tasks.

The platform is constructed from identical “rigid” core segments, which are 3D printed carbonfiber reinforced PET, that are connected by compliant 3D printed TPU joints. Each core segment houses the electronics for pose estimation and includes dovetail features for joint attachment, as well as outer slots for interchangeable PLA rings whose diameter and geometry can be customized for specific applications. This architecture supports an arbitrary number of segments, limited primarily by tendon actuation. The bending stiffness of the joints connecting the modular rigid segments was estimated using finite element studies performed in ANSYS/Mechanical (ANSYS, Inc.). For neck diameters ranging from 3 - 7 mm, we analyzed a single segment of the robot with a three-dimensional, static and geometrically nonlinear model subjected to a 30 degree bend angle on one end and a fixed boundary condition on the other. The reaction moment (M) at the fixed end was determined at every three degrees of bend angle (θ). The bending/rotational stiffness of the joint (k) was determined by fitting a linear relationship between the reaction moment (M) and bend angle (θ) according to M = kθ.

In each segment, a custom designed printed circuit board (PCB) is embedded to sense the magnetic field of the coil of the nearby segments. Each PCB includes a microcontroller (ATTiny3224), a 9 DoF IMU that includes magnetometer readings (ICM-20948), a linear drop out regulator (TPS72118DBVT), a logic shifter (SN74AXC4T774), as well as a diode and a N Channel MOSFET (NMOS) to control the coil. The microcontroller on the PCB acts as a bridge that transfers SPI communication to the actual IMU and I2C communication with an Arduino MEGA to relay the sensor data as well as to activate and deactivate the coil during sensing. Each PCB is daisy chained with JST XH 5 Pin connectors that includes the power lines for the coil and onboard components as well as I2C communication lines between each component and the Arduino MEGA that interfaces with the Raspberry Pi for pose inference.

Our pose estimation maps magnetometer measurements in an actively generated magnetic field to relative position via machine learning. We assume: (i) joints maintain a fixed arc length, with bending as the dominant deformation mode and (ii) joints pivot about their geometric center during bending. Together, these assumptions constrain inter-segment motion to a dome-like (hemispherical) manifold. Under these constraints and assumptions, we adopt a data-driven approach to determine position based on magnetic field measurements whereby we collect magnetic field data over the reachable relative workspace via randomized “wobble” motions and train a neural network to learn the mapping from field measurements to position. Because all segments use identical manufacturer-matched electromagnetic coils, we train a single joint-level network over one joint’s reachable workspace and reuse it across all joints, yielding a modular sensing pipeline.

The training data for the magnetic to position mapping were collected using a two-segment setup. During data collection, we execute the following sampling sequence: The procedure begins by moving all four motors I = [M1, M2, M3, M4] to their center positions [ci]. For each waypoint k, two values (xk, yk) are sampled uniformly at random from the square [-1, 1] × [-1, 1]. These values are scaled by an amplitude A and used to generate target motor commands as symmetric offsets about the motor centers; the resulting commands are clipped to the valid range [0, 4095]. The motors are then moved to these target positions. Once the robot reaches the waypoint, the coil is kept off and the ambient magnetic bias bk is estimated by averaging 10 sensor samples, with a short delay ∆t between samples. Next, the coil is turned on and a short burst of sensor readings [sk, j] (11 samples total) is recorded, again with delay ∆t between samples. Finally, the system logs the waypoint index k, the sampled inputs (xk, yk), the motor state (e.g., q), the bias bk, and the full coil-on burst sk∗, and then proceeds to the next waypoint. After every 500 waypoints, the system pauses for a 10-minute cooldown period. We use an accelerometer measurements recorded concurrently with the magnetometer data to recover the corresponding 3D ground-truth pose for each waypoint during measurement bursts, using gravity as a constraint since gravity is perpendicular to the ground plane. Since we assume that the movement space of a joint is part of a hemisphere, we find an azimuth angle θ and a colatitude angle λ to represent the position in spherical coordinate that corresponds to each magnetometer reading. For each sample, let the measured acceleration in the sensor/body frame be T ab = [ax,b, ay,b, az,b] and we use a world-aligned right-handed frame with axes (X,Y, Z) = (West, North, Out). We define the colatitude λ as arccos clip rz, −1, 1. Azimuth θ is defined as θ = atan2 rx, ry, where θ ∈ [−π, π). We then transform the two angles into a 3D direction vector for prediction: d(θ, λ) = [sin λ sin θ]. The input for the neural network has a dimension of 3, the normalized magnetometer, and the output of our network has a dimension of 3 as well, the normalized direction vector. For training the neural network, we used a cosine loss defined as: lcos = 1 − clip m̂⊤ d̂, −1, 1, where m̂i and d̂i represent the normalized predicted direction vector from the magnetometer readings and the ground truth direction vector, respectively. We optimized and trained the network using AdamW with best hyperparameters found via Bayesian optimization in Optuna.

In real life inference of the joint position, we deploy our trained model in ONNX format on a Raspberry Pi running a Docker container with ROS2 Humble. To minimize magnetic interference between the joints in our full length robot, we turn only one coil on and collect the magnetometer data of segment that neighbors the coil at any given time. For each joint, we spend 20 ms for the magnetic field in the activated coil to settle down. The process is repeated across all the segments along the length of the robot in a sequential fashion. At each cycle, all coils on N bridges are turned off. The bridge to sample is selected sequentially as j = (cycle mod N) + 1, so that j cycles through 1,..., N. With all coils off, the system waits an ambient settling time tamb, then reads the IMU from bridge j. From the returned packet s j (accelerometer/gyroscope/magnetometer), the magnetometer vector is extracted and used as the ambient magnetic bias b j for that bridge. After a short delay t∆, the coil associated with bridge (j − 1) is turned on, the system waits a coil-field settling time tcoil, and the IMU on bridge j is read again. From this second readout, the raw magnetometer vector mraw is extracted and debiased via j raw mdeb j = m j - b j. The resulting debiased measurement is stored in the sequential array as seq[j − 1] ← mdeb j. When j = N, the full array seq is transmitted for pose inference.

The experiments validated the effectiveness and generality of the proposed self-sensing modular continuum robotic platform in diverse manipulation tasks. In general, our experiments show that the proposed magnetic self-pose estimator captures robot shape across various conditions and, unlike other magnetic approaches [19], [24], requires no kinematic prior. Scenario 1 validates the method in an unobstructed setting. In Scenario 2 (planar obstruction), the estimator remains accurate under disturbances that cause complex unknown kinematics such as forced deformations of contact forces. Scenario 3 further removes any kinematic observability by not utilizing tendon actuation and instead moving the robot by hand. Despite motion-capture failure, our trained sensing scheme still produces qualitatively correct pose estimates, highlighting the value of in-situ sensing relative to visionbased methods. Quantitatively, our RMSE is comparable to accelerometer-based self-pose sensing reported in [27], indicating that magnetic sensing is a viable alternative for tendon-driven modular robots.

The limitations include: First, the coil for magnetic field generation is bulky (19 mm × 12 mm). Second, while the interferencemitigation sampling scheme improves signal quality, it makes the global sampling rate scale poorly with robot length. For our 10-linkage system, the maximum stable rate is ∼1.2 Hz, which is insufficient for tasks that requires fast movements such as pick and place in a factory setting. Parallelized or faster acquisition schemes are therefore an important direction for future work.

In summary, we present the design of a modular, tendon driven continuum robot with variable stiffness joints and embedded self pose sensing capabilities. We demonstrate how a mechanical analysis can be used to tailor a particular configuration of joints to produce a desired actuated shape, and the combined modularity and variable stiffness functionality was confirmed to be compatible with a novel self pose estimation scheme utilizing magnetic sensing and machine learning. This unique combination of modularity, variable stiffness, and in-situ pose estimation establishes our design as a promising platform for future research on the capabilities of continuum robot systems.

The experiments show that the proposed magnetic selfpose estimator captures robot shape across various conditions and, unlike other magnetic approaches [19], requires no kinematic prior. Scenario 1 validates the method in an unobstructed setting. In Scenario 2 (planar obstruction), the estimator remains accurate under disturbances that cause complex unknown kinematics such as forced deformations of contact forces. Scenario 3 further removes any kinematic observability by not utilizing tendon actuation and instead moving the robot by hand. Despite motion-capture failure, the system still produces qualitatively correct pose estimates, highlighting the value of in-situ sensing relative to visionbased methods. Quantitatively, our RMSE is comparable to accelerometer-based self-pose sensing reported in [27], indicating that magnetic sensing is a viable alternative for tendon-driven modular robots.

The experiments also demonstrate the reconfigurability of the platform and its ability to adapt to different task requirements: First, as shown in Fig. 6(a), we combined limbs with different effective lengths and flexible joints with different bending stiffness to promote a non-uniform curvature under tendon actuation. The choice of effective lengths and stiffness enabled the robot to bend around an obstacle and reach a target position for a switch. Second, we performed a spherical grasping task using TPU stopper modules coated with Eco-Flex 00-30 (Smooth-On, Inc), as shown in Fig. 6(b). This configuration provided both compliant contact and increased surface friction, allowing the robot to stably envelope and grasp a sphere with a diameter of approximately 12 cm. Third, we used square limb modules together with red stopper inserts designed to provide larger gripping force for cloth grasping, as shown in Fig. 6(c). Compared with the previous configuration, this setup generated stronger local contact forces and was better suited for handling deformable objects. In all three tasks, we used the proposed pose estimation method to predict the robot shape during operation. The results show that the method can effectively estimate both the relative positions between neighboring segments and the overall body configuration while maintaining good performance during real contact-rich interactions.

In conclusion, we present the design of a modular, tendon driven continuum robot with variable stiffness joints and embedded self pose sensing capabilities. We demonstrate how a mechanical analysis can be used to tailor a particular configuration of joints to produce a desired actuated shape, and the combined modularity and variable stiffness functionality was confirmed to be compatible with a novel self pose estimation scheme utilizing magnetic sensing and machine learning. This unique combination of modularity, variable stiffness, and in-situ pose estimation establishes our design as a promising platform for future research on the capabilities of continuum robot systems.

In conclusion, we present the design of a modular, tendon driven continuum robot with variable stiffness joints and embedded self pose sensing capabilities. We demonstrate how

Here are specific improvements for AI systems derived from this scientific paper:

1. Improve real-world deployment robustness by integrating a modular, self-sensing perception layer into existing robotic platforms.

2. Enhance manipulation and navigation capabilities of continuum robots in unstructured or confined environments through the implementation of variable stiffness control and closed-loop pose estimation.

3. Develop autonomous grasping systems for soft, deformable objects by leveraging magnetic sensing for in-situ shape estimation, eliminating reliance on external vision systems during contact tasks.

4. Create scalable and rapidly adaptable robotic architectures where new functional configurations (shapes/stiffness profiles) can be programmed via modular hardware swaps, significantly reducing the need for complete robot redesigns.

5. Design learning frameworks that utilize joint-level models trained on single-joint dynamics, allowing for efficient knowledge transfer and rapid reconfiguration across different continuum robot segments without retraining the entire system from scratch.