Conventional mobile robots have difficulty navigating highly unstructured spaces such as caves and forests. In these environments, a highly extendable limb could be useful for deploying hooks to climb over terrain, or for reaching hard-to-access sites for sample collection. This article details a new form of a multimodal mobile robot that utilizes a novel tape spring limb named EEMMMa (elastic extending mechanism for mobility and manipulation). Its innovative U-shaped tape structure allows it to handle loads in tension as well as compression. It can also bend using mechanical multiplexing for a lightweight and compact design that is well suited for mobile robots. For mobility, the limb can extend prismatically to deploy grappling hook anchors to suspend and transport the main body, or even serve as legs. For manipulation, the limb can morph its shape to bend around or over obstacles, allowing it to retrieve distant objects or position cameras around corners. The EEMMMa-1 prototype detailed in this article successfully demonstrates climbing ladders and shelves in 1.5 body lengths per second, and can bend up to 100 deg. A simplified model of the bending kinematics is developed and analyzed. This article concludes by detailing future EEMMMa applications and theories to strengthen the model in future studies.
Many environments are still difficult for modern mobile robots to traverse, such as cave systems and forest canopies. These have highly irregular features in all directions like stalagmites, cliffs, tunnels, branches, and vines that can make navigation difficult for wheeled or flying vehicles. To safely move through these environments, legged climbing is one method that can offer the required terrain adaptability and safety. Spider monkeys are a good example of a legged creature that is well adapted to movement in complex environments. They suspend themselves in forest canopies using not just their limbs but also their tails, as depicted in Fig. 1. This gives them an additional anchoring point for safety and allows them to climb swiftly and reach for food with better stability .
Many current climbing robots are designed with applications such as surveillance or inspection in mind. There are a wide variety of mechanical approaches to climbing. The simplest type is wheeled climbing robots [3–5] that can drive vertically but only along flat surfaces. Legged systems adhere to walls using microspines [6–8], gecko adhesives , and even suction cups .
These examples reveal important attributes for successful climbing. First, most climbing robots are preferably small and lightweight, since they must be able to support their own weight while navigating terrain. Additionally, redundancy is essential for safety, with multiple limbs or anchors to prevent the system from falling if an anchor fails. Finally, legged systems offer much greater adaptability to surface variations, but result in bulky and sluggish systems with greater complexity. The mobility of legged climbers is also limited by the length of their limbs, which confines their reachable workspace to a relatively small area. This prevents their effective use in environments where anchor points are sparse. These systems could be significantly enhanced by utilizing a long-reach extendable limb, which would allow a greater workspace of anchor points to be reached and could greatly simplify the planning process by climbing long distances in a single step.
Tape springs can be used to offer this greatly enhanced reach in a lightweight and compact package thanks to their ability to be spooled. These thin curved strips exhibit natural directional stiffness that allows them to serve as structural members and exhibit different properties while folded. By utilizing an extendable tape spring limb, grappling hooks can be placed at precise locations from long distances, with a workspace that extends many times past the robot's body length.
As initially proposed in our previous work , this article details a novel form of multimodal mobile robot that uses extendable tape spring limbs for both mobility and manipulation tasks named EEMMMa (Elastic Extending Mechanism for Mobility and Manipulation). EEMMMa's tape spring structure allows it to serve as a versatile, lightweight, long-reach limb. The limb utilizes an innovative closed-loop structure that grants it the ability to handle both tensile and compressive loads and also enables controlled bending.
Figure 2(a) shows the EEMMMa-1 prototype, a lightweight single degree-of-freedom (DOF) robot that uses an extendable tape spring mechanism to demonstrate promising mobility and manipulation abilities. EEMMMa-1 can climb shelves and ladders using compliant hooks and can ascend rough vertical walls when equipped with microspines. The extendable limb can also morph its shape with a form of mechanical multiplexing. As seen in Fig. 2(b), the limb can bend using a simple braking function to change the limb's kinematics. This allows a second rotational DOF to be effectively controlled using only a single primary motor. As a result, the system is very lightweight, since it can achieve more complex motions without needing additional large actuators at the distal regions. A summary video of its capabilities is presented online.2
This innovative combination of features makes the EEMMMa limb very well suited for future mobile robot platforms for use in rough terrain. With the ability to extend long distances, the limb could be used to deploy grappling hook anchors at distant points to suspend and transport the main body. This can enable safe and fast mobility inside cave tunnels or through forest canopies as depicted in Fig. 3(a). Since the limb can also support significant loads in compression, EEMMMa could be used for legged locomotion as well, serving as highly extendable legs to elevate the body as depicted in Fig. 3(b). The ability to step over obstacles rather than traveling around them would be highly advantageous in environments that feature wide-spanning hazards such as water and mud in swamps and would make path planning much easier and safer.
EEMMMa's ability to bend also grants it great potential as a manipulator, as it can bend around or over obstacles and extend to reach into tight spaces. This could allow it to retrieve distant or submerged objects, position cameras around corners, or place grappling anchors above overhangs such as cliffs. Additionally, the tape's elastic properties enable it to self-correct from perturbations for tasks that require alignment. The limb is also relatively safe to use for manipulation tasks since it simply elastically deforms during a collision or if a target is missed. These concepts are showcased in this article with successful EEMMMa-1 single-limb demonstrations of climbing, standing, and bending to place a target on an overhanging surface. More complex manipulation tasks such as target retrieval in tight spaces will be the subject of future studies.
Section 2 outlines the overall design and operation of the EEMMMa-1 prototype. Section 3 shows demonstrations of mobility and manipulation tasks. Section 4 provides a preliminary analysis of the system's behavior during bending, outlines a simplified kinematic model, and details experiments performed to compare with the model. Section 5 summarizes the future implications of this system and outlines additional work needed to refine the model.
2 Design and Implementation
EEMMMa-1's design will be broken down into five main parts: (1) background information, (2) an overview of operations, (3) the tape spring that forms the main structure of the extendable limb, (4) the main body containing the motorized spool, tension management subsystems, and electronics, and (5) the end effector that serves as the end of the limb and has a braking function to initiate bending.
EEMMMa's tape spring-based design offers several advantages over existing extendable or bendable robotic limb designs. Conventional extendable limbs that utilize telescoping  or collapsible scissor structures [13,14] exhibit good loading capabilities and rigidity. However, they have high mechanical complexity, relatively poor compactness, and only expand the available robot's available workspace by a few times its body span. Additionally, many bending robotic limbs have recently been developed, with a variety of soft robots and continuum arms that offer excellent adaptability to arbitrary geometries that could be useful for manipulation tasks [15–18]. However, these also suffer from high complexity, requiring complex cable-driven drives with multiple actuators, or accompanying air/liquid systems for pressure-based actuation schemes. This often results in manipulators that are bulky or heavy for their size and poorly suited for use on a lightweight mobile robot. These issues can be remedied with tape springs, which offer far better deployable length with low weight and excellent compactness thanks to their ability to be tightly spooled.
Tape springs are curved thin shells of material that have the ability to elastically deform and transition between a straight configuration and a folded configuration as depicted in Fig. 4(a). In this way, localized “folds” generally serve as revolute joints, while unfolded straight segments can serve as links that can withstand significant forces in tension, as well as limited compression forces and bending moments. The rigidity exhibited by unfolded segments can be attributed to their transverse curvature, which increases the energetic cost of bending longitudinally [21,22].
Tape springs exhibit several useful features for serving as flexible or structural members. When a moment is applied, tape spring segments will not fold until a peak moment is reached. As depicted in Fig. 4(a), this localized fold exhibits zero transverse curvature and a uniform longitudinal curvature. If the peak moment is exceeded, the tape spring will exhibit a snap-through buckling behavior with a sudden change in stiffness properties at the fold as it is formed. Additionally, because of the directionality of the tape's curvature, the value of the peak moment changes depending on the direction of the applied moment. As depicted in Fig. 4(b), tapes subjected to “equal-sense” bending will fold much more easily than for “opposite-sensed” bending , which requires a significantly higher applied moment to form a coherent fold. It should also be noted that the development and propagation of folds are highly dependent on the loading and boundary conditions present at the end sections of the tape spring . Tape springs also exhibit a level of self-actuation due to their spring properties, which will cause the tape to elastically return to its unfolded neutral state when bent or twisted .
The benefits of tape spring mechanisms have been previously explored in a variety of fields, but there has been little prior work on utilizing the long range of such mechanisms for mobility or manipulation. Tape springs are used in deployable space structures, including extendable booms , automatically deploying solar reflectors [27,28], and large closed-loop structures , which take advantage of tape spring's compactness and ability to self-deploy from a folded position to achieve large structures with little to no active actuation.
One notable example is a planar 3-DOF manipulator for unmaned aerial vehicles (UAV), which utilizes the long reach of the tape to deploy an end effector below a UAV. This design utilizes a mechanical node that travels along the tape's length to “pinch” and induce a fold to control the bend location and angle up to 55 deg, and can extend up to 2 m . This system depends on a bulky traveling node that increases the system weight and limits its ability to be deployed in directions not aligned with gravity.
Another example is ReachBot , which utilizes a mobility scheme similar to our previously proposed concept. ReachBot uses single tapes for its limbs that extend prismatically and are designed to be loaded in tension for suspending the main body. In comparison, EEMMMa's unique U-shaped tape spring design and mechanical multiplexing result in a more versatile system in a lightweight package. It can handle significant compressive loads to apply the contact forces required for certain manipulation tasks and can bend to reorient its end effectors to reach certain geometries or around obstacles.
2.2 Overview of Operations.
Designed primarily to demonstrate climbing on shelves and ladders, EEMMMa-1 is equipped with two sets of compliant hooks to climb successive levels as seen in Fig. 2(c). The first set of hooks is attached to the end effector. The second set of hooks is located on the main body and serves to anchor the body at the current shelf or ladder rung.
The hook-engaging sequence can be seen in Fig. 5. As the tape extends vertically, the end effector hooks gently deform to allow them to pass above the next level. The hooks then spring back to their original positions after they clear the level, resulting in a one-way locking effect. The main body then retracts the spool to reel itself upward. When the body approaches the next level, the second set of sloped hooks on the body passively guide the shelf or ladder rung onto the load-bearing back portion of the hook, establishing a new anchor point. Once the body hooks pass the next level and establish stable contact, this grappling and anchoring sequence can be repeated.
The overall weight of the system is 685 g, consisting of a 640 g main body, 35 g end effector assembly equipped with hooks, and 10 g of steel tape.
2.3 Tape Spring Limb.
The tape provides the main structure for the limb and is stored in a spool in the main body. It is folded into a U-shape as seen in Fig. 6(b). In the U-shaped bend, an idler pulley passively follows the traveling fold at the “end” of the limb, allowing the end effector to move up and down. The idler and the U-shaped tape form a pulley system that provides mechanical advantage, halving the torque required to lift the main body while climbing.
The U-shaped tape has one end connected to the spool and the other end fixed to the main body to form a closed loop. This layout essentially creates three tape regions: two unfolded segments placed back to back and one folded segment connecting the two that makes up the bend of the U. When tape springs are placed back to back, the overall structure exhibits significantly improved stiffness, since one of the tapes is subjected to opposite-sense bending regardless of the bend direction [31–33]. EEMMMa-1 utilizes this advantageous property while only requiring actuation of a single continuous tape. Additionally, the elastic spring properties of the tape assist with maintaining the orientation of the hooks during extension and engagement, resulting in robust climbing capabilities.
EEMMMa-1 can initiate shape morphing using a form of mechanical multiplexing to bend the limb, depicted in Fig. 7(a). By manually activating a braking function at the end effector, the system can switch its kinematic mode. The brake is activated by rotating a small screw, which presses a rubber pad against the tape's surface and idler pulley. This locks the end effector relative to the tape's surface, creating a local “fixed” boundary condition. Since one segment is “fixed,” it will not change length, while the other actuated segment will decrease in length, depicted in Fig. 7(b). The disparity in lengths causes the upper half of the assembly to rotate until folds are generated, which functionally serve as revolute joints. This simple on–off braking function allows EEMMMa-1 to bend with minimal added weight, effectively granting the arm 2-DOF capabilities with only a single main motor, although it cannot actuate both DOF simultaneously. While the braking function requires manual activation for this proof-of-concept prototype, future versions will have this feature fully automated.
It should be noted that bending was found to be much easier to initiate when the spool was retracted rather than extended. This is because retracting pulls the tape in tension, allowing it to transmit loads much better than compressive loads. Additionally, when the limb is bent, one tape segment experiences equal-sense bending and folds more easily, while the other tape segment experiences opposite-sense bending and serves as the main structure for supporting loads due to its greater rigidity. The tape's U-shape was chosen to be oriented with transverse curvatures facing outward, so the fixed length is loaded in opposite sense while bending. This allows it to serve as a better structure for reliably producing a coherent L-shaped limb.
The material used is a uniform segment of prestressed steel tape cut from a Pittsburgh brand 12 ft. × ½ in. tape measure, with a 0.006 in. thickness. This tape width of ½ in. is relatively small when compared to 1 − ¼ in. used in other projects [20,34]. This results in reduced bending stiffness and limb rigidity, but also easier shape morphing since the peak moment required to induce folds is lower. Since this prototype was designed for climbing, the reduced bending stiffness is inconsequential since the tape is almost always loaded in tension. The tape's total length is 1 m, allowing the limb to extend 50 cm away from the main body. This distance was chosen specifically for climbing ladders, which commonly have rungs with ∼4 cm diameter and spaced ∼30 cm apart. Both ends of the tape have 3 mm holes drilled in their centers to rigidly connect to the spool or frame.
2.4 Main Body.
The main body forms the primary structure that houses the motorized spool, tension management subsystem, and electronics. These components are placed in specific locations to make the center of mass close to vertically aligned with the tape axis. This is to reduce pitching moments that can disrupt hook alignment while climbing or cause the system to fall . The frame is primarily composed of 3D-printed polylactic acid (PLA) and steel fasteners and measures 130 × 140 × 95 mm.
The 40 mm diameter spool is surrounded by an outer casing with a small exit hole to confine the coiled tape loops, which will unwind themselves or push out of plane due to instabilities while coiled . The inside of the spool casing is lined with a strip of nylon to reduce friction.
After leaving the spool, the tape enters the tension management subsystem. An output roller covered with nonslip rubberized surface (neoprene) grips the tape as it exits to maintain a tension force inside of the tape that keeps the tape properly coiled around the spool. The spool and roller have the same diameter and are geared 1:1 to ensure the tape deploys smoothly. While this is sufficient for shorter lengths, longer lengths of spooled tape will result in the spool diameter decreasing as more tape is deployed, which may cause tensioning issues for future prototypes.
Finally, the tape exits the main body at the output hole, which has a curved shape that follows the tape's transverse curvature. This is to ensure the tape deploys in an unfolded state for maximum rigidity. When there are disturbances from forces at the end effector, the output hole's curved surfaces reduce the likelihood of a stress concentration, which may result in an unwanted fold.
2.5 End Effector.
The end effector assembly provides a structure at the end of the limb for mounting any additional mechanisms that interact with the environment such as hooks or cameras. Inside the end effector is the idler pulley, which is wrapped in rubberized tape and is 29 mm in diameter. Folded tapes exhibit a characteristic longitudinal curvature based on the tape's manufacture, which is 29 mm for this tape.
This is also the mounting distance between the fixed end of the tape and the spool output, which was chosen to be 29 mm for simplicity to make the tape segments parallel while extending vertically. Changing this mounting distance in future studies may result in different desirable limb properties. The slight angle between the segments should allow the limb to handle horizontal loads better and can increase the effects of the applied bending torque from the motor, allowing folds to be generated more easily.
An outer casing surrounds the idler and tape. The inside of the casing contains a small nylon pad at the tip of the fold, which allows the end effector to transfer loads to the tape in both tension and compression with minimal friction. The casing also prevents the traveling fold from splitting into two separate folds when loads are applied.
In the following demonstrations, EEMMMa-1 shows its effectiveness as a multimodal platform for both mobility and manipulation in a lightweight, compact package. By leveraging the tape spring's unique properties, EEMMMa-1 can demonstrate climbing, bending, pushing, and pulling. These tasks require only two parameters to be controlled: the length change of the tape, and the on/off of the end effector brake. All demonstrations were performed with simple open-loop control and manual input.
To verify the limb's ability to handle loads in tension, EEMMMa-1 was subjected to climbing trials in three scenarios: climbing a shelf, a ladder, and a rough vertical wall. Trials were first conducted on wire-frame shelves made of 5 mm diameter wire, seen in Fig. 8. The shelves had a thickness of 22 mm and were spaced 254 mm apart for a total of 276 mm to ascend per level. EEMMMa-1 can ascend at 19 cm/s, traversing a level at top speed in about 2 s. This is about 1.5 body lengths per second, which matches that of the fastest wall climbing robots .
For ladder climbing trials, straight vertical ladders were used, which are commonly seen in industrial or mechanical environments such as factories, buildings, and ships. The ladder chosen had cylindrical rungs with 2 cm diameter, spaced 28 cm apart for a total of 30 cm to ascend per level. The larger diameter of the ladder rungs caused more significant perturbations from the compliant hooks. When climbing at the top speed, these perturbations caused hook alignment issues during multiple climbs in succession, although trials at slower speeds were successful.
For the wall scaling trials, the end effector and body compliant hooks were swapped with small microspine arrays, seen in Fig. 9. The microspine array is composed of four small steel hooks, with sharp points that engage in asperities on the rough surface. Outer housings and rubber bands provide rotational and translational compliance for engagement and load sharing. EEMMMa-1 could successfully cling to the wall and ascend small distances. However, this prototype was unable to perform multiple grappling and anchoring sequences in succession due to two main effects. First, the main body's center of mass being slightly off center caused the body to pitch and the limb to extend at an angle. At large extensions, this caused the end effector microspines to be too far away from the wall to engage the surface properly. Additionally, the microspine array footprint was small enough to be approximated as a point contact, which sometimes caused the anchor to twist off the surface during perturbations. A future body redesign and microspine array upgrade could alleviate these issues.
For all climbing trials, the elastic spring properties of the tape proved to be beneficial for resisting unwanted forces or moments at the end effector. Hooks require directional engagement, and they must approach the grappling features at a specific orientation to be effective. This is especially important for microspines, which can peel away or fail to engage if they are not properly aligned with the gripping surface. Because correctional forces are passively supplied by the tape's spring properties, the climbing sequence is robust and simple to control. However, future designs with larger extension lengths and heavier end effectors may experience additional difficulties since the end effector may oscillate over long periods without added damping. These trials demonstrate EEMMMA-1's ability to pull loads against gravity, which is vital for suspending the main body in midair from above, or for retrieving samples from below.
Robotic manipulation tasks commonly require reaching a target in space, so the next set of tests was devised to demonstrate EEMMMa-1's ability to reach a desired location on a plane using controlled bending. Since it is trivial to reach any single point along the 1-DOF linear path, trials involved reaching two points on the plane, seen in Fig. 7(a).
For the first set of trials, the limb was pointed vertically, extending in the direction opposite gravity. This was selected as the most relevant scenario for this system, since climbing actions generally involve vertical movement, and bending would be advantageous for reaching above tables or steps. For the first test, the limb was first extended to point A located 20 cm above the main body at coordinates (0,20) cm. Next, the end effector braking system was engaged, allowing the motor to initiate limb bending. Actuation was applied slowly until a fold was created in both tape segments, essentially serving as a new revolute joint. After rotating 90 deg, the end effector successfully reached point B located at (10,10) cm. Video footage was taken on a gridded background to verify repeatability.
Subsequent bending trials to other positions revealed additional phenomena. Angling the revolute joint more than 100 deg resulted in the folds suddenly migrating toward the main body, causing the limb to “collapse.” Limb collapse also occurred when extending the spool to reach most points in the X direction (see Fig. 7(b) for axes). This is due to the fixed segment being loaded in equal-sense bending rather than opposite-sense bending, which will cause buckling under a much lower peak moment. It was also found that the location of the fold could be controlled through dynamic inputs, which will be discussed in Sec. 4.
As a test application for bending, EEMMMa-1 was placed on a stone staircase to demonstrate anchoring to the top surface of steps with microspines, as shown in Fig. 10. Although EEMMMa-1 was not designed to fully climb stairs, these trials still showcase the limb's mechanical multiplexing abilities to reach a target position and orientation that is otherwise inaccessible with a simple prismatic joint. The limb was first extended vertically above the next step, and a fold was induced to angle the end effector downward. The limb was intentionally actuated to generate the fold close to the base instead of the midpoint to allow the hooks to contact the ground. The tape was then retracted slowly, allowing the spines to fully engage and pull the body in. Test footage showed that the tape's natural spring properties assist with maintaining proper spine orientation during approach, engagement, and retraction.
These preliminary tests demonstrate the potential for EEMMMa to serve as a flexible, long-reach manipulator arm. Future versions of EEMMMa will use these abilities to bend around or over obstacles to position grippers, cameras, or other instruments in difficult-to-reach places.
A simple “standing” test was performed as a demonstration of the limb's ability to handle loads in compression by serving as an extendable prismatic leg, as seen in Fig. 11. First, the end effector was placed in a vice with the main body carefully positioned directly above. The limb was extended, and the main body was released, with its weight creating a compressive load on the limb. These static loading tests were successful up to 20 cm of limb extension. Beyond this point, the limb's rigidity was insufficient to prevent small perturbations from causing the body's center of mass to shift, which resulted in a collapse. Dynamically extending the limb also caused collapses since reaction moments at the main body resulted in the center of mass shifting and would cause the system to topple over.
These tests demonstrate the end effector's effectiveness at allowing the limb to handle compressive loads. Because the tape is a single continuous U-shape, loads can be transferred evenly between the two opposing segments. The end effector housing effectively confines the U-shaped tape fold without it splitting or propagating, which would result in a greatly reduced ability to handle loads as a manipulator. The ability to successfully handle compressive loads is promising for future EEMMMa experimental prototypes, which will feature multiple limbs and not exhibit the same collapsing issues due to load sharing between the limbs.
As an additional test for mobility, EEMMMa-1 was equipped with passive wheels on the underside of the main body to demonstrate 1-DOF crawling along the floor, depicted in Fig. 12. The end effector was equipped with the microspine attachment, and the limb was extended. When outstretched, gravity causes the limb to sag slightly, allowing the end effector microspines to contact the floor. The tape was retracted slowly to engage the microspines, and then a fast retraction pulled the body forward. The main body would then coast linearly on the floor, carried by momentum. As the body approaches the end effector, the microspines would naturally disengage due to the changing angle of engagement. This demonstrates EEMMMa's ability to be used with other mobility schemes and operate with them in combinations for various effects. For example, this could be useful if a rover lost power to its wheels, but could still operate using its EEMMMa manipulator arm for mobility.
4 Analysis and Discussion
4.1 Bending Behavior.
This section details preliminary, mostly qualitative and geometric studies of the limb's overall bending behavior to highlight certain phenomena while extending and bending. These observations will be used to form the basis and assumptions for a simplified kinematic model of the system. Fully characterizing the tape's complex nonlinear behavior while bending is highly challenging and is outside the scope of this initial investigation. Future studies will focus on forming a complete model of the tape's moment–rotation relationships and accurately predicting the end effector's position during all bending stages.
Snapshots from the bending trials are shown in Fig. 13 that displays three stages of behavior that the limb exhibits as it bends from 0 deg to 90 deg. Under normal 1-DOF operation, the tape operates as a single continuous piece of material, moving the end effector as the idler pulley rolls along the tape's length. Activating the brake at the end effector functionally separates the tape into two segments, S1 and S2. The actuated segment S1 on the right is attached to the spool and has a variable length that can be changed by actuating the motor. The segment S2 on the left now has a fixed length due to its end attachment to the main body. By retracting the tape, S1's length is reduced, and the end effector begins to rotate due to the disparity in lengths. As the bending angle increases, the limb goes through three stages of distinct bending behavior:
Stage 1: Both S1 and S2 are unfolded and behave like beams.
Stage 2: S2 has a fold, and S1 is unfolded but bends and twists.
Stage 3: Both S1 and S2 have folds that each behave like a revolute joint.
In stage 1, depicted in Fig. 13(a), both tape segments experience a small horizontal displacement. S2 is subjected to opposite-sense bending, and it can be treated as a single flexible beam until buckling occurs. However, due to the tape's very thin cross section, this can be considered a large deflection, which exhibits geometric and material nonlinearities that will be studied in the future.
In stage 2 (Fig. 13(b)), the fixed segment S2 has folded, while the actuated segment S1 has not, but continues to experience combined bending and twisting. When folded, the curve flattens and the cross section becomes rectangular. Folds will generally occur at the midpoint of segments, since the loads at the ends are equal and opposite. Due to this symmetry, the fold should not propagate or travel along the tape's length, assuming minimal effects from external forces.
In this stage, S2 behaves as two rigid segments connected by a virtual “revolute joint,” which approximates the rotation between the two segments as a point hinge despite the fact that the fold is actually a finite length of tape with zero curvature. An analytical planar rod model outlined in Seffen and Pellegrino's  characterizes the folded configuration in this way, with two rigid bars and a nonlinear rotational spring that accounts for the bending stiffness of the fold. Meanwhile, S1 experiences combined flexural-torsional deformation that can be potentially characterized using Mansfield's equations  in future analyses.
In stage 3 (Fig. 13(c)), both segments have folded at their centers, essentially creating a combined virtual revolute joint for the limb to form the L-shape. The limb can easily be returned to the straight prismatic configuration by equalizing the lengths. This is partially assisted by the tape itself, which will attempt to straighten in order to relieve the accumulated strain energy from bending .
4.2 Two-Dimensional Bending Kinematics.
4.3 Experimental Comparison.
To analyze the accuracy of the formulated bending kinematic equations, a series of tests were conducted to compare the actual end effector position with calculated theoretical values during bending. This was also done to identify what factors might contribute most to deviations from the simplified model.
To collect position data, the end effector was outfitted with a colored marker that was tracked real time via webcam. The collected position dataset was then mapped from pixel space to the actual X-Y position in meters. For each experiment, the tape spring limb was extended to a set initial length (S1(0) and S2(0) set to some value). Bending mode was initiated, and the spool was commanded to slowly retract a set distance (q′(t) < 0) until the limb reached 90 deg, with the end effector moving along a curved path through the first quadrant of the X-Y plane.
Figure 15 shows a selection of data points from multiple bending trials at two different initial lengths. Predicted values are overlaid as solid lines, which were calculated using the kinematics equations outlined in Sec. 4.2. For the first trial, the limb was extended to 27 cm before bending. This distance is half the total extension length and is a good representation of the limb's average behavior. In the second trial, the limb was extended to 14 cm before bending to show behavior at smaller extensions. In addition, the limb was tested at greater extensions (above 40 cm), but the limb would frequently collapse after bending more than −45 deg due to the greater sensitivity at long extensions. Slight vibrations would cause uncontrolled fold migration and limb collapse, leading to inconsistent results, so that they are omitted in this study.
For these randomly selected data points, there is a small amount of variance in the measured positions between trials. This is likely due to a combination of minor factors, including slight disturbances from vibrations, tracking errors from the camera, and slightly different starting conditions caused by backlash and friction in the spool.
Figure 15 also shows the fourth-order polynomial trendlines for the collected data compared to the predicted values. These graphs show that the model tracks well from 0 deg < θ < −70 deg. For a given angle, the model deviates from the actual X-Y position by no more than 3% for both trials. Bending beyond this region causes the actual data to deviate significantly from the model's prediction.
These deviations can possibly be explained by the effects of gravity and the rotational spring forces from folding. While the model accounts for the fold's geometric properties by simplifying it as a point hinge, it does not account for moments at the hinge or the effect of external torques that can increase the applied bending moment and affect the rotation angle. The weight of the end effector generates an additional torque on the tape, causing the end effector to sag slightly in -Y just below the position predicted by the simple kinematic model. The model also does not account for the rotational spring force in the fold, which partially counteracts this weight.
The fold location also appears to migrate slightly away from the midpoint when bending at angles beyond −70 deg. Since the kinematic equations were formulated with the assumption that the fold is always at the tape segments’ midpoints, the end effector's actual position begins to deviate significantly from the model. It should be noted from Fig. 15 that the fold migration appears to be predictable, since the end effector traveled along the same approximate path between separate trials. This fold migration may be caused during the physical transition from stage 2 to stage 3. The sudden formation of the fold causes a significant disturbance to the system as the tape snaps through to its new configuration.
To better observe these effects, Fig. 16 shows the error in the Y position between the predicted path and the trendline for the measured data for the 27 cm trials. As the end effector moves further in the + X direction, we can see the effects of the three stages of bending previously shown in Fig. 13. The vertical dashed lines at X = 4.2 cm and X = 10.8 cm separate the graph into the three bending stages as observed from captured test footage.
In stage 1 (0 < X < 4.2), neither tape segment has a coherent fold. The end effector's weight is still supported by the mostly vertical tape segments, so it does not cause much sagging and the error remains relatively low. It is interesting to note that although the kinematic model does not account for the nonlinearities associated with the tape segments’ large displacements, the error is low in this region.
In stage 2 (4.2 < X < 10.8), one of the tapes has folded. The newly generated fold causes the folded segment to have reduced rotational stiffness. The weight from the end effector contributes a growing amount of external torque since its moment arm increases as it moves further in +X, which causes the error to increase.
In stage 3 (X > 10.8), both segments have formed folds, but the fold migration appears to take effect. The fold migration changes the kinematics of the system and causes the Y position to drop much faster than expected. After intersecting with the predicted path, the error increases rapidly as the end effector follows a new path. Additionally, the sagging effect from the end effector weight is much more pronounced due to its large +X distance, as well as from the decreased rotational stiffness of both tape segments.
While outside the scope of this article, these tests reveal that a purely geometric study is limited and that future models should better capture the fold's moment characteristics. This includes more complex behaviors like fold formation and migration experienced between stage 2 and stage 3. The moment analysis should also take into account external torques from both the motor and the weight of the end effector, and possibly the weight of the tape itself at longer extensions.
4.4 3D Bending and Dynamic Input Investigations.
In the bending experiments outlined earlier, all end effector movement occurs in the 2D X-Y plane only. This is because the tape is deployed vertically. Actuating the spool in this configuration applies a bending torque about the Z-axis, and the weight of the end effector generates a torque that is also about the Z-axis. Thus, all rotations and forces cause movement in X-Y only, and the system remains in a single plane.
However, when the tape is deployed horizontally as depicted in Fig. 17, inducing a fold can cause the limb to bend out of plane, resulting in a 3D fold. This is because actuating the spool applies a torque about the Y-axis, but the moment arm from the weight of the end effector causes a rotation about the Z-axis. In this case, the fold will exhibit both bending and twisting in θ and ϕ. Since ϕ is a function of the Z-axis moment arm, it can be controlled by simply extending the tape. This method allows effective control of the end effector's out-of-plane displacement until the peak moment is reached and the limb collapses. Future versions of EEMMMa can potentially use this phenomenon to grant the limb 3-DOF without another actuator, taking advantage of gravity to achieve out-of-plane movement.
Additional tests show that the limb can be manipulated in even more ways if dynamic inputs are used. In preliminary tests, the tape was retracted in short bursts, resulting in oscillations at the end effector. If the tape was retracted again before the oscillations settled, the fold location could be altered depending on the state of the end effector, depicted in Fig. 18. The timing of these actuation bursts can also potentially be used to minimize oscillations at the end effector by canceling out vibrations with well-timed retractions. This phenomenon has been observed in previous tape spring studies where the fold travels due to impulse–momentum interactions , where the tape is modeled as a traveling hinge with hinge position and rotation angle as two independent degrees-of-freedom.
While analyzing and controlling these additional degrees-of-freedom are outside the scope of this article, these phenomena are important to note for their potential to enhance EEMMMa's available workspace and manipulation capabilities in the future
5 Conclusion and Future Work
This paper detailed our novel concept of utilizing tape spring mechanisms for robotic limbs that can serve dual mobility and manipulation tasks. The unique closed-loop U-shape of EEMMMa's tape spring grants several advantages over a single-tape structure, including an enhanced ability to handle both tensile and compressive loads, as well as the ability to bend with mechanical multiplexing. The innovative combination of features results in a simple, compact, and lightweight system that is well suited for future mobile robot platforms. The EEMMMa-1 prototype presented in this article demonstrates the versatility of a single EEMMMa limb, providing key functions such as climbing and bending with only one primary actuator.
Future versions of platforms that utilize multiple EEMMMa limbs are currently in development, including a 3-DOF system capable of vertical movement between two walls. The final goal envisioned for this project is a three- or four-limbed robot capable of moving freely in 3D space using its extendable limbs. This can be a suspended robot as previously depicted in Fig. 3(a) or a legged robot as depicted in Fig. 3(b).
The principles explored in EEMMMa could also be used to improve platforms that focus purely on either mobility or manipulation. EEMMMa's morphability could be used for a closed-loop continuous tape as a tank tread for movement, utilizing the pulley-brake mechanism to morph the shape of the treads to move over obstacles, as seen in Fig. 19(a). EEMMMa could also serve as fingers for a compliant gripper that morphs its shape to conform around objects, as seen in Fig. 19(b). The tape's steel construction could be favorable for conforming to shapes with sharp corners that could damage other soft robotic shape-morphing manipulators.
The next stage of research for this project will be forming a more complete model of the limb's tape spring segments and accurately predicting the end effector's movement during all three stages of bending. Existing works can help analytically or numerically characterize the tape's initial large deflection using large-deflection curved beam theories, especially for thin-walled structures [39–41]. For future real-time control schemes, analytical methods would be preferred for a lighter computational load.
Additionally, the folded bending stiffness and rotation behavior should be characterized. The useful planar rod model for folded tapes developed by Seffen and Pellegrino  assumes a point hinge with two rigid bars, but is limited in that it does not account for the creation or migration of folds. A semi-analytical method used by Brougeois and Guinot [24,42] builds upon this by accounting for the changes in the tape's cross-sectional shape. This method can be used to model the creation, splitting, and migration of folds. It should be noted that many of these existing theories deal primarily with opposite-sense bending, with S1's combined equal-sense bending and twisting in stage 2 proving difficult to model.
When bending at larger extensions, it may be necessary to consider the tape's own weight as a distributed load since this will cause the tape to curve and invalidate the planar rod model's assumption that the unfolded segments are rigid. In this case, it might be necessary to adjust each segment's geometry and end conditions to account for the curvature of the member depending on the amount of extension.
Finally, it will be important to characterize the tape's hinge when subjected to 3D folds while bending in a horizontal orientation. Most of the previously mentioned theories apply only to the planar 2D case. However, Walker and Aglietti  detail an analytical model that can be used to determine the hinge moment for 3D bending in a skewed tape spring system. This model is slightly limited in that it is only valid for a hinge located at the midpoint of the tape segment.
Examining these theories and integrating them into a full model of the limb will form the basis for EEMMMa's control schemes and will help push EEMMMa's potential to enable a variety of long-reach manipulators and locomotion schemes for future mobile robots.
Conflict of Interest
There are no conflicts of interest.
Data Availability Statement
The datasets generated and supporting the findings of this article are obtainable from the corresponding author upon reasonable request.