SoPrA: Soft Continuum Proprioceptive Arm

• SoPrA is a soft continuum robotic arm featuring pneumatically actuated chambers combined with embedded flex sensors for closed-loop proprioception.
• It employs an analytical dynamic model that integrates nonlinear elasticity and real-time sensor fusion to estimate shape and force without external tracking.
• Its modular design facilitates scalability and effective dynamic manipulation in environments lacking reliable exteroceptive feedback.

A Soft Continuum Proprioceptive Arm (SoPrA) is a pneumatically actuated, compliant robotic manipulator designed for dynamic interaction, manipulation, and force estimation in unstructured or obstructed environments. Distinguished by its full-body proprioceptive sensing—achieved through embedded flex sensors and an analytical dynamic model—SoPrA advances beyond single-component soft actuators by integrating actuation, self-sensing, and model-based estimation into a unified physical and computational system. This allows for accurate shape and force reconstruction without dependence on external motion capture, rendering the system especially effective when exteroceptive measurements are impractical.

1. Physical Architecture and Fabrication Principles

SoPrA’s structure comprises multiple fiber-reinforced soft pneumatic chambers, each directly cast from silicone elastomer and reinforced to optimize compliance and load-bearing capability. Chambers are bundled into segments, each containing three inflatable actuators arranged around a central, inextensible core. This core serves two critical functions: routing of pneumatic lines and hosting of lightweight, integrated two-axis capacitive flex sensors along the backbone. Such integration facilitates distributed, internal measurement of segment bending, a cornerstone for closed-loop proprioceptive estimation.

Scalability is enabled via 3D-printed connectors, which allow addition or removal of segments and seamless hardware integration (e.g., a pleated gripper at the distal tip). Segment geometry is tapered, assigning larger cross-sectional area and thus higher potential actuation torque to proximal segments, which is necessary given the cascaded transmission of bending torques along the trunk. All hardware routing, including pneumatic lines and sensor wiring, is internal, preserving a continuous and robust envelope against external disturbance.

2. Analytical Dynamic Modeling

The core of SoPrA’s estimation and control system is an analytical dynamic model that captures nonlinear elasticity, inertia, and damping properties of the complete soft structure. The dynamic equations are formulated as:

$$A\mathbf{p} + J^\top \mathbf{f} = B(\mathbf{q})\,\ddot{\mathbf{q}} + \mathbf{c}(\mathbf{q},\dot{\mathbf{q}}) + \mathbf{g}(\mathbf{q}) + K\mathbf{q} + D\dot{\mathbf{q}}$$

where $\mathbf{p}$ is the vector of chamber pressures, $A$ is the actuation-force mapping, $J$ is the task-to-joint-space Jacobian, $\mathbf{f}$ is the tip force, $\mathbf{q}$ the configuration vector (e.g., bending angles of piecewise constant curvature [PCC] elements), $B(\mathbf{q})$ the state-dependent inertia matrix, $\mathbf{c}(\cdot)$ the Coriolis/centrifugal term, $\mathbf{g}(\cdot)$ gravity, $K$ and $D$ the elastic stiffness and damping matrices. The elastic moment for a PCC segment is calculated as:

$M_e = \frac{4\mu I}{l}\theta$

with $\mu$ the silicone shear modulus, $I$ second moment of area, $l$ the arc length, and $\theta$ the bending angle.

This model is constructed to admit real-time evaluation, such that both open-loop simulation and closed-loop estimation (with data correction from proprioceptive sensors) are computationally efficient and practically deployable.

3. Integrated Proprioception and Sensor Fusion

Proprioception in SoPrA is achieved through the combination of analytical modeling and embedded two-axis digital flex sensors. Each sensor measures the net bending along its host segment, outputting a signal $s = S\mathbf{q}$, where $S$ is a matrix aggregating the contribution of each internal shape parameter (e.g., segmental bending angle). When there are more configuration variables than sensor channels, this mapping is not directly invertible, but provides a strong linear constraint.

Sensor data is fused with the model-based estimate using an optimization procedure:

$$\text{minimize} \quad \| A\mathbf{p} - g(\mathbf{q}_0) - K\mathbf{q} \|^2 \quad \text{subject to} \quad s = S\mathbf{q}$$

This procedure refines the configuration $\mathbf{q}$, correcting model errors using sensor data and ensuring internal consistency across segments. When estimating external tip forces, the optimization is extended:

$$\min_{\mathbf{q}, \mathbf{f}_{\text{ext}}} \ \| A\mathbf{p} - g(\mathbf{q}_0) + J^\top(\mathbf{q}_0)\mathbf{f}_{\text{ext}} - K\mathbf{q} + \mathbf{f}_0 \|^2 + r\| \mathbf{f}_{\text{ext}} \|^2$$

where $r$ is a regularizer and $\mathbf{f}_0$ a tare offset.

4. Estimation of External Forces and Model Refinement

Force sensing in SoPrA is accomplished without exteroceptive instrumentation. After actuation, the enhanced model—trained and corrected via proprioceptive signals—can extract at each time point both the shape and the external load applied at the tip (e.g., during object grasping). Experimental results demonstrate a robust linear relationship between estimated and actual loads (17 g–117 g range), independent of trunk configuration, although absolute accuracy is limited by modeling error and sensor noise. The system thus satisfies real-world requirements for tasks demanding force estimation or manipulation of irregular/fragile objects.

5. Experimental Performance Validation

SoPrA’s model and sensor suite were validated under both dynamic actuation and loaded conditions. Under a regime of open-loop sinusoidal actuation, the model (with three PCC elements per segment) predicted tip position with mean errors reported in the range of 1.3–2.0 cm—substantially improved over single-segment models. In static grasp trials, the enhanced estimation method reliably mapped flex sensor data to external force magnitude.

All tasks were performed without motion capture feedback, confirming the feasibility and stability of the sensor-model integration. This performance demonstrates the system’s ability to generalize across actuation conditions and external loads.

6. Functional Capabilities and Deployment Contexts

Because SoPrA operates with true proprioception—combining model-based estimation and embedded flex sensor feedback—it is suitable for deployment in visually or physically obstructed settings, such as within debris fields, dark/confined spaces, or other environments where LIDAR or camera-based tracking cannot function. The platform supports complex manipulation, including picking objects, force-controlled grasping, and potentially compliant human-interactive behaviors.

The modular and scalable design, enabled by 3D-printed connectors and soft hardware, implies extensibility to greater segment counts, specialized end effectors, or alternative sensor integration, while maintaining computational tractability.

7. Limitations and Further Directions

Although SoPrA’s proprioceptive scheme provides substantial autonomy and flexibility, certain limitations were noted:

• The inversion of the sensor-to-shape mapping is only approximate for multi-segment/multi-sensor systems; errors grow with geometric complexity.
• External force estimation, while linearly correlated, did not perfectly reproduce actual mass or load vectors, particularly with heavier objects or large configuration deviations.
• Real-world durability, sensor drift, or unforeseen nonlinearities in the silicone soft body may require additional calibration or adaptive refinement over time.

Future research directions include improving the joint estimation of shape and force under dynamic contact, integrating learning-based model correction to compensate for unmodeled effects, and developing application-specific gripper designs. The aim is to enhance performance for tasks such as force-sensitive object handling or operation in cluttered, uninstrumented environments.

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SoPrA’s architecture unites soft, scalable pneumatic actuation, analytical model-based dynamics, and integrated flex sensor proprioception to enable self-contained estimation of configuration and external forces, supporting compliant manipulation tasks in settings where exteroceptive feedback is absent or unreliable (Toshimitsu et al., 2021).