The SGR body is essentially an inverted membrane tubing. Releasing the inner inverted membrane material via a central tendon (tendon C) and increasing the air pressure of the SGR would enable the robot to grow at the tip. Inside the SGR body, there is a pivot joint mechanism which has an embedded, pressure-tunable bladder. The pivot joint mechanism is composed of a bladder holder and a rotor driven by two actuating tendons tendon A and tendon B (Figures 1A,B). The rotor is stacked on top of the bladder holder, and they are connected via a rotation shaft, allowing the rotor to freely rotate around the shaft. The rotor is of a boat-anchor shape with two actuating tendons attached on its top. These tendons can control the rotation of the rotor with respect to the bladder holder, and thereby, engage with the robot body’s outer membrane to bend the robot body locally (Figure 1A). The bladder holder is a hollow tube embedded with a pressure-tunable bladder, and the inner inverted membrane material of the robot body goes through the bladder holder as well (Figures 1B,C). As such, inflating or deflating the bladder can change the friction between the bladder holder and the inner membrane material, enabling them to move together (coupled, when the bladder is inflated) or separately (decoupled, when deflated).

The detailed steering process depicted in Figures 1D–I is systematically described as follows:

Stage 1: Initial State (Figure 1D): The robot initiates linear eversion with pneumatic pressure P o . At this stage, tendons A , B remain relaxed, retraction tendon C slightly tensioned to prevent excessive robot eversion.

Stage 2: Steering (Figures 1E,F): When encountering an obstacle, to avoid and passing through, the bladder pressure ( P B ) is increased, enhancing friction bond to securely anchor the pivot joint mechanism with the inner inverted material of the robot; this ensures that the pivot joint does not translate under the tendon tension during the bending of the robot. Then, one steering tendon, e.g., tendon B , is tensioned to firmly engage the pivot mechanism against the robot’s internal membrane with the rotor and to bend the robot body at the location of the pivot joint.

Stage 3: Decoupled Growth for Obstacle Avoidance (Figure 1G): The robot grows by increasing pneumatic pressure P O and releasing the retraction tendon C . The bladder is deflated so that pivot joint does not move forward together with the inner membrane material (decoupled); in the meantime, the pivot joint keeps stationary because the rotor is anchored to the membrane wall of the robot body, which maintains the bending of the robot so that it grows to the steered direction to avoid the obstacle.

Stage 4: Coupled Growth (Figures 1H,I): After successfully passing the obstacle, all tendon tensions are released. The bladder is re-inflated to restore frictional coupling between the pivot joint and the inner membrane. Pressure P O is increased to resume the pivot joint’s forward movement together with the everting body along the newly established trajectory. The robot body then relies on the surrounding environment to preserve the bending configuration, allowing it to continue growth or repeat the steering cycle as needed.

To capture the characteristic of the soft growing robot equipped with a pivot joint mechanism, a kinematic model was theoretically derived. The model maps the actuation space (tendon displacement) to the configuration space (robot spinning angle and pivot joint position) and ultimately to the task space (tip poses). The model was then experimentally verified.

As shown in the robot schematic diagram (Figure 2), the robot can be treated as two sections connected by a pivot joint. The pivot joint is controlled by one tendon on each side, with its position coupled with the releasing of the central retraction tendon when the bladder is inflated during steering stage. The total length of the robot L is L 1 + L 2 , where L 1 is the length of the first section and L 2 represents the second. The growth of the whole robot is controlled by the release of the central retraction tendon L = d c / 2 , where d c is the release length of the central retraction tendon.

To bend the robot body, the rotor must first contact with the body inner membrane (Figure 2A). Further pulling the tendon would bend the robot. Here, we built the relationship between the tendon input displacement and the robot bending angle based on the geometry of a triangle (see the dashed red lines in Figure 2C) formed by the width of the robot (SGR diameter) D , the rotor contact length ( l r ) , and the projection length l p of rotor along the bended section of the robot. Pulling the tendon would shorten l p and thus increase θ . Here, the robot bending angle is θ 1 = θ − 90 ° . For this triangle, we have Equation 6 l r 2 = D 2 + l p − d 2 − 2 D l p − d cos ⁡ θ (6)

We can get the angle of the pivot joint with the robot outer membrane (Equation 7): θ = arccos l p − d 2 + D 2 − l r 2 2 l p − d D (7)

As the rotor contact the membrane, the initial l p (Figure 2A) can be define as Equation 8: l p = l r 2 − D 2 (8)

The pivot joint can produce a controllable bend of up to 90 ° . Further tendon pulling displaces the pivot away from the wrinkle, after which the tube buckles rather than steers; therefore, curvature beyond 90 ° is not controllable and was excluded from our operational envelope.

As the robot steering angle θ 1 = θ − 90 ° , the end effector position can be determined. By giving the position of the pivot joint, the whole robot can be simplified as an two link robot, and the tip point can be calculated as Equation 9: x = L 1 + L 2 ⁡ cos θ 1 y = L 2 ⁡ sin θ 1 . (9)

The kinematic model was validated by comparing its simulations against experimental deformation profiles under both bending and unbending conditions. The predicted robot end effector position (Equation 9) were compared against experimentally recorded deformation profiles under both bending (Figure 4A) and unbending motions (Figure 4B). As shown in Figure 4, the simulated end effector position closely matched the experimental observations, with average relative errors of 0.46 % during bending and 0.40 % during unbending, respectively. The relative error for each end effector is calculated as the average distance from the position of the end effector in simulation to the corresponding end effector in the experiment, divided by the robot length. These results confirm that the proposed model accurately captures the bending configuration of the robot and provides a reliable basis for motion planning and control.

The robot exhibited a clear buckling behavior in response to tendon actuation. In the initial stage from 0 ° to approximately 8 ° (Figure 5B), the required tension increased rapidly with the bending angle. Once the bending exceeded this critical buckling angle (Figure 5C), the additional tension needed for further bending tended to stabilize (Figure 5A). This critical point corresponds to the condition where the internal compressive force offsets the axial tensile stress, leading to the onset of local wrinkling. The critical tension required to initiate buckling increased with pressure, following the relationship defined in Equation 3. To experimentally determine the parameter K , we repeated the bending tests without internal pressure (i.e., P = 0 p s i ). During these tests, we measured the bending forces exerted by the tendon to form and maintain the wrinkles. The results indicated a consistent tendon force value of approximately 0.39   N , which represents the inherent stiffness of the pivot joint structure in the absence of internal pressurization. Based on these new measurements, the parameter K was set to 0.39   N and subsequently utilized for the data fitting presented through Equation 3. Once buckling occurred, the tension required to further enlarge the deformation remained nearly unchanged. In this work, we choose a low air pressure (0.3 p s i ) during steering to keep required steering force low.

In addition to the internal pressure used for tip eversion and growth, the bladder pressure, which controls the anchoring of the pivot joint, also plays a critical role in the robot’s functionality. As previously described, higher inflation pressures generate larger frictional forces between the bladder and the inverted inner tube membrane, effectively securing the pivot joint to the robot body and enabling co-growth. Conversely, reducing the bladder pressure releases internal constraints, allowing the robot to extend independently of the joint.

The test measured the axial frictional resistance provided by the bladder–tube interface at different pressure levels (Figure 5D). Results showed that when the bladder pressure reached 1 p s i , the resulting frictional force exceeded 2 N , which was sufficient to reliably anchor the pivot joint during growth. These findings provide a practical guideline for selecting actuation pressures that ensure sufficient friction to anchor the pivot joint during co-growth, while still allowing for its controlled release when further extension is required.

The robot can actively steer and grow via tip eversion. However, after bending, it lacks dedicated mechanisms to hold its shape. Instead, posture is passively maintained through contact with the environment—a form of morphology stabilization enabled by external constraints.

To quantify the role of environmental contact in maintaining robot posture, we measured the interaction force between the robot body and external supports under different internal pressures and contact positions, corresponding to different steering angles. As shown in Figure 6A, the contact force exhibited a transient peak upon initial contact, with values not exceeding 0.15 N , followed by a steady-state phase where the force stabilized around 0.06 N . Furthermore, the measured contact force decreased with increasing distance from the pivot point (Figures 6B,C),in agreement with the mechanical principle that torque is proportional to force times distance. The required contact force also increased with larger bending angles, indicating that greater deformation demands higher external force to maintain the resulting posture. These findings indicate that environment-assisted stabilization relies on relatively low contact forces, sufficient to retain the robot’s posture without exerting excessive mechanical load on its surroundings.

Overall, while the robot relies on passive environmental constraints to maintain its postural configuration after deformation, the contact force involved remains low. This suggests that the stabilization mechanism does not significantly burden or damage the environment, nor does it require strong or specially structured external supports. The low-contact nature of this stabilization strategy enhances the system’s adaptability and ensures compatibility with a wide range of operating conditions without requiring structural modifications to the environment.

The navigation capabilities of the soft growing robot equipped with pivot joints were evaluated in structured maze-like environments. As shown in Figures 7A,B, the robot performed an initial turning rotor to cross the first corner of a maze. Figure 7C demonstrates its ability to continue tip eversion and growth after steering. Subsequently, in Figures 7D–F, the robot executed a secondary bending motion, successfully navigating toward the end of a bifurcated path.

These sequential deformations highlight the robot’s capacity for complex navigation and repeated directional change. In Figures 7G,H, under manual control, the robot was steered to explore an alternative branch within the same maze, demonstrating its capability for operator-guided decision-making in unstructured settings. Finally, Figures 7I–L ppresents the robot operating within an altered maze configuration constructed from the same modular components. In this scenario, the robot continues to exhibit consistent exploratory behavior, with Figures 7I,J depicting the robot choosing the upper branch at the first Y-junction and subsequently executing another 90° turn in the same direction. Conversely, Figures 7K,L demonstrates the robot initially choosing the alternate path at the first Y-junction and performing a further bend along this route. The repositioning of panels was intentionally chosen to clearly illustrate the robot’s adaptability across different environmental layouts.

During steering, the pivot joint rotor must first establish firm contact with the robot’s inner wall before tendon displacement effectively induces bending of the leading segment. This required sequence inherently introduces some delay, which may slightly impact real-time manual control responsiveness.

After repeated testing, we observed that multiple bending and inflation cycles could potentially cause material degradation in the robot’s root membrane, increasing susceptibility to unintended bending. Additionally, repeated bladder inflation-deflation cycles occasionally caused partial extrusion of the bladder from its holder, potentially leading to punctures and subsequent air leakage. Such damage could impair the reliability of the locking mechanism.

These results confirm that the robot’s steering and growth are not limited to preprogrammed trajectories. Instead, the system can be manually controlled in real time to explore unfamiliar, unstructured environments, reflecting strong potential for use in navigation tasks where environmental layouts are unknown or dynamic.