The rigid tail was designed to be of similar shape to the robot’s legs (rectangular) but greater in length (90 mm in length, 3.5 mm thick, and 8 mm wide, Figure 3A). The top of the tail was connected to the rear servo motor. The bottom of the tail was filleted, with the removal of sharp corners key to ensuring the tail does not get stuck on obstacles as it moves. The primary goal of this design was to provide an extra contact point at the back section of the robot. Since the legs are coupled, only one of the legs of each segment is on the ground at each time step (Ozkan-Aydin and Goldman, 2021), and therefore a support triangle cannot be formed. Without a tail, the diagonal gait of the robot was interrupted and led to instability during locomotion (Ozkan-Aydin and Goldman, 2021), with the body of the tail striking the ground twice per gait cycle. This also led to foot slippage and insufficient ground clearance, which leads to problems when the robot attempts to climb and navigate obstacles.

The rigid tail has a passive role throughout the smooth surface experiments, serving as an additional contact point along the robot’s back. However, it is noteworthy that the lateral motion of the rigid tail (undulation of the tail from left to right like a lizard’s tail) could be actively controlled through the tail servo motor. This mechanism could be useful for climbing or walking on rough terrain. If the robot gets stuck on an obstacle, it can be pushed out of that position by repeatedly rotating its rigid tail from side to side.

The tail comprises five articulating vertebra-like segments of cuboidal shape (Figure 2). The fifth segment of the tail has a narrow, curved end, which is in contact with the ground. This segment includes a fillet on the interior face, allowing the tail to pass over obstacles and sharp ledges without causing the robot to get stuck. The tail was 3D printed using a Stratasys F170 printer with all parts intact. The links between each segment were designed to eliminate the need for assembly. Dimensions of the linkage component can be modified in order to alter the rigidity of the tail along with the range of motion.

Included in the design is a connection component, a box attached to the back of the robot. This component allowed the tail to attach and detach from the robot with ease. Furthermore, it elevates the attachment point of the tail significantly above the body. This is important as it allows a longer tail to be developed, leading to an increased range of motion and, thus, more effective robot locomotion. The connection component also includes small loops at the top and the side, which helped guide the cabling when wrapping around the reel.

To control the flexibility of the tail, we designed a mechanism that consists of a custom-made reel that is attached to the rear servo motor (Figures 2, 3B, Supplementary Video/0:08-0:52 s). Kevlar wire was then threaded through the front and rear channels of the tail and wrapped around the reel attached to the servo motor. When the servo motor was in its initial position, the tail was relaxed and very flexible (Figure 3B-left). The wires were wrapped around the reel in the same direction (counterclockwise (CCW)). The tension in the cables was carefully tuned so that when the reel rotated CCW 90°, the tail would be pulled into an approximate “s” shape (Figure 3B-right), similar to the shape of vertebrate tails observed in certain biological species such as the kangaroo (O’Connor et al., 2014). With an applied rotation of the tail servo motor of between 0 and 90° (CCW), the distance between the segments was reduced, increasing the rigidity (or decreasing the tail’s flexibility).

We conducted measurements of the horizontal pulling force against the displacement of the tip of the tail using a force gauge (see Figure 3C). As seen from Figures 3C, D, the relaxed tail can bend about 45° from its initial position with about zero pulling force (Supplementary Video). Furthermore, it is observed that the stiff tail configuration can exert approximately 1 N of normal force, while the relaxed tail configuration can only apply a significantly reduced force of 0.1 N (Supplementary Video/0:08-0:52 s). Two different implementations of this approach were tested, primarily the effect of periodically stiffening and relaxing the tail during key phases of the gait cycle and the effect of using a touch sensor to sense when the tail should stiffen and relax. More detail on the stiffening/relaxation timing can be found in Sections 3.3.3 and Section 3.3.4.

Due to the tail’s flexibility, the rigidity could be carefully controlled and altered via the actuation of the servo motor. Three unique tail states were used across the experiments: fully relaxed (or flexible), high rigidity, and periodic stiffening and relaxation.

The first two states were introduced at the end of Section 2.2.2 and in Figures 3B–D. The fully relaxed state involves no servo motor rotation and keeps it at its default position. As a result, there was no tension in the cables, allowing the tail to act passively and remain flexible. The high rigidity state was the inverse of the fully relaxed state, with the servo motor maintained at a constant rotation of 90° anticlockwise from its original position. The desired rigidity and tail shape can be achieved by choosing the appropriate degree of rotation within this a 0 to 90 degree range; however, in this study, we only used completely flexible and rigid states.

In the third state, we programmed the flexible tail of the robot to periodically stiffen and relax during pivotal moments of the gait cycle. We investigated two distinct gait-stiffening sequences for the purpose of our study Supplementary Video/0:53-1:08 s).

In the first sequence, tail stiffening occurred at 50% of the gait cycle (Figure 4), the midpoint of the swing phase of the front left leg, i.e., when it was reaching forward. Tail relaxation occurred at 100% of the gait cycle at the same instant during the swing phase of the front right leg. The stiff tail provided additional support to the robot, supplementing stability in conjunction with the front right leg and the back left leg during the flight phase. When the tail relaxed, all four legs of the robot were in contact with the ground, ensuring continuous stability. The stiff tail not only provided a counterbalance but also played a significant role in propelling the robot forward. On a smooth, flat surface, the robot with this configuration maintained a straight trajectory and covered an optimal distance in a cycle (Figure 5A).

In contrast, the second sequence presented a different outcome. In this sequence, the tail stiffened just before the front right leg touched the ground, with the body curved to the left. The tail was relaxed when the front left, and back right legs were in the air, leading to an unstable stance as the robot had to rely solely on the front right leg and the left back leg without any assistance from the tail. This imbalance led to the robot tilting backward. This configuration consistently deviated from a straight trajectory on a smooth, flat surface, veering to the right, indicative of a foot drop sequence. This gait-stiffening sequence resulted in covering less distance per cycle compared to the first sequence (Figure 5B).

Given these observations, we continued our experiments using the first sequence. It offered a clear advantage in stability and locomotion, making it a more efficient model for our robotic design. However, this does not rule out the potential that other gait-stiffening sequences might bring to the table. For future work, a more comprehensive study could explore other gait-stiffening sequences and their respective impacts on locomotion.

This experiment was carried out as described in section 3.1. In each of the five trials conducted, the robot could not meet the goal of traveling along a straight line while on level ground. The robot deviated 2.4° ± 0.71° per cycle from the straight path and traveled with a speed of 0.45 ± 0.03 BL/cycle (or 3.75 ± 0.25 cm/s), Figure 7B.

This test revealed many problems in the robot’s locomotion without a tail. Firstly, the robot’s body struck the ground twice during each gait cycle. The front of the robot pitched upwards (12.95° ± 6.6°) from the horizontal axis during the swing phase, causing the rear of the robot to strike the ground before foot placement could occur (Figure 6A; Figure 7A). Secondly, the rear legs of the robot did not have sufficient ground clearance during the swing phase, resulting in the rear swinging leg dragging along the ground. Finally, the front legs were found to have excessive ground clearance, leading to increased motion perpendicular to the ground, rather than the desired motion parallel to the ground. These three problems were inexplicably linked to the upward pitching of the robot and caused the robot’s diagonal gait to be asymmetrical and uneven. This impaired the robot’s balance, in turn hindering the robot’s ability to travel along a straight line.

Five replications of the previous experiment were conducted with identical protocols (Figure 6B). Quantitative analysis showed a substantial enhancement in locomotion speed (0.61 ± 0.09 BL/cycle or 5.1 ± 0.75 cm/s), a 34% increase compared to locomotion with no tail (Figure 7B). However, the robot still could not travel in a straight line and deviated from the path 3.13° ± 1.72° per cycle with a rigid tail.

The tail successfully raised the robot’s rear, eliminating any contact of the body with the ground. Upon close examination of the experiments, it was found that there was still an unpredictable and unwanted variance from the gait cycle. This occurred during the swinging phase of the front right leg. The front right leg made contact with the ground before the back left leg, interrupting the diagonal gait of the robot. This caused the front of the robot to pitch with 3.0° ± 1.91° and the tail to lift off the ground. There was further instability when the tail re-made contact with the ground.

Before the experiments were run on the flexible tail, the different tail states/configurations were evaluated. The tail could be configured in the following three ways: (1) a relaxed flexible tail where there was no tension being applied (Figure 3B-left), (2) a rigid tail where the servo motor was applying constant tension (Figure 3B-right), or (3) an alternating flexible to rigid tail, which the rotation of the servo motor would control.

It was found that the flexible tail with constant rigidity provided the best stability (less vertical oscillation, 2.08° ± 1.61°) to the robot while walking on flat, smooth terrain (Figure 6C) compared to other tail configurations (no tail = 12.95° ± 6.6°, rigid tail = 3.0° ± 1.91°, flexible tail-passive = 9.44° ± 5.13° and flexible tail-On/off = 3.67° ± 2.78°). A further improvement in the robot’s ability to travel in a straight line was observed in this test. The robot was capable of traveling along a straight path with 0.71° ± 0.98° rotation/cycle and 0.60 ± 0.04 BL/cycle (or 5 ± 0.33 cm/s) forward displacement, whereas the other tension settings, relaxed and alternating flexible/rigid tail, resulted in 1.48° ± 1.77° and 2.5° ± 1.77° rotations/cycle, respectively. The robot’s movements were smoother and more consistent due to the lack of sudden lurches or pitching. It was found that the side-to-side undulations of the tail, although passive and slight, helped maintain ground contact throughout all stages of the gait cycle. The front leg was still in contact with the ground just before the opposite back leg. However, this no longer caused instability. This is because the ground contact of the tail still provided sufficient support to stop rapid pitching.

Despite being in a rigid state, residual flexibility in the tail emerged as a crucial element in augmenting the robot’s stability and locomotion. This allowed the tail to make slight adjustments automatically when traversing the terrain. When foot strike occurred, the tail moved towards the body momentarily. During the flight phase, the tail quickly moved in the opposite direction away from the body until the tension of the reel limited it. The subtle movements of the tail played a pivotal role in ensuring that the tail remained in contact with the ground, preserving the overall stability of the robot with seamless transitions between gait phases. The robot’s stance position was now altered; the front of the robot was rotated slightly downwards. During the swing phase, the robot pitched marginally upwards until the body angle was approximately parallel to the ground. Then, when the foot strike occurred, the front of the robot pitched downwards to return to the stance position. The swing phase was then again initiated, and the cycle repeated. This resulted in a smooth, consistent gait without sudden and rapid pitching, which would cause the robot to change direction.

The constant stability experienced by the robot led to a further increase in locomotion speed. It traveled at a rate of 0.60 ± 0.04 BL/cycle, similar to the locomotion with the rigid tail.

With no tail, the robot could not travel along any incline successfully (Supplementary Video/1:43-1:52 s). It failed to make it to the end of the runway angled at 10° to the horizontal in five consecutive trials. During each of the trials, pitching occurred, resulting in the body striking the ground as well as low rear leg clearance. The cause of this rocking is identical to what is described in Section 3.1.1.

In the flat terrain tests, the pitching of the robot had only a minor effect on its locomotion capabilities. However, when traveling on an inclined surface, the instability caused by the pitching led to the constant slipping of the robot’s rear feet. This made the robot’s travel speed extremely slow (0.05 ± 0.02 BL/cycle). It also resulted in the robot eventually rotating 90° and walking off the runway, failing to meet the success criteria of the experiment.

The addition of a rigid tail allowed the robot to travel along a runway of inclines 10 and 15° with 0.45 ± 0.03 and 0.37 ± 0.04 BL/cycle, respectively (Supplementary Video/1:53-2:08 s). Notably, the robot experienced some degree of slippage during locomotion along these inclines. However, the robot’s hind legs exhibited adequate ground clearance, facilitating forward movement. When the angle was increased to 20°, the slippage of the feet was greater than the distance stepped forward; therefore, the robot traveled in the reverse direction.

The results obtained from our previous experiments have indicated that implementing a flexible tail with constant high tension (Figure 3D) yields the most optimal performance on flat terrain. We, therefore, first conducted our inclined experiments utilizing this tension setting exclusively (Supplementary Video/2:10-2:27 s). As the tail was maintained at a high rigidity throughout the experiment, the robot made continuous contact with the runway during each trial. When tested with this configuration, the robot successfully traveled along a 10, 15, and 20-degree runway with 0.40 ± 0.03, 0.37 ± 0.03, and 0.27 ± 0.06 BL/cycle, respectively. When the incline of the runway was increased to 25°, the robot could not travel along it with any tail type, as it would slide.

For inclines of 10 and 15°, the robot’s motion was only slightly inhibited by some minor slippage of the rear feet. It completed the tests efficiently while traveling along the straight line indicated on the runway. In contrast, the robot experienced more pronounced front and hind legs slippage when traversing inclines of 20°. Despite this, the robot could complete the test at a slightly reduced speed compared to the preceding trials. However, the robot’s performance was severely impaired when the angle of the incline was further increased to 25°. In this case, the robot could not make any net progress up the incline, as it slid back to its previous point after each step. It remained here indefinitely, not moving upwards along the track but also not moving downwards and falling off. This suggests that if a change was made to increase the feet’ grip, the robot could move along this incline successfully.

We also conducted experiments with two different flexible tail settings: a flexible tail in a relaxed setting and a flexible tail in a stiff/relaxed setting. In the relaxed setting (Supplementary Video/2:28-2:52 s), the robot successfully climbed both 10-degree and 15-degree inclines. However, some directional deviations were noted towards the end of the inclines. Specifically, the robot tended to deviate from its straight trajectory, shifting to the left or right. Numerical results indicated a locomotion speed of 0.29 ± 0.04 BL/cycle on a 10-degree incline, decreasing slightly to 0.24 ± 0.01 BL/cycle on a 15-degree incline. When attempting a 20-degree incline, the robot could not successfully climb, continually sliding down and sometimes veering to the sides of the board.

For the stiff/relaxed setting (Supplementary Video/2:55-3:15 s), similar results were obtained. The robot again successfully ascended the 10-degree and 15-degree inclined planes. However, it exhibited a similar pattern of directional deviation near the ends of the slopes, veering left or right after a few cycles. The recorded locomotion speeds were 0.27 ± 0.04 BL/cycle for the 10-degree incline and 0.23 ± 0.09 BL/cycle for the 15-degree incline. As with the relaxed setting, the robot failed to ascend the 20-degree incline, persistently sliding down and occasionally moving sideways.

Without a tail, the robot failed to climb any steps, meaning a result of 0/30, and rotated 30 ± 5 ^o per cycle (Supplementary Video/3:16-3:37 s). It was observed during the previous experiments that on flat terrain, the front of the robot pitched up during locomotion, and the rear legs’ ground clearance was extremely small. These two factors impaired the robot while climbing. During each trial, the robot placed its front legs on the first step. It moved forward on that step until the rear legs reached the edge of the step. When this happened, the rear legs could not lift enough to clear the ledge. The body was also making contact with the step. As a result, the robot was stuck indefinitely. The result of this experiment was significant, as it showcased the robot’s inability to climb or navigate over obstacles without getting stuck.

There was an improvement in the robot’s climbing ability when a rigid tail was added. The robot successfully climbed 8/30 steps with 0.2 ± 0.08 stairs/cycle over five trials (Figure 9A) with a 5.34 ± 1.75 ^o rotation/cycle (Supplementary Video/3:38-4:06 s). With the addition of the rigid tail, the robot could climb onto the first step in every trial. The robot no longer pitched upwards during locomotion, meaning the hind legs had sufficient ground clearance to reach the first step. The robot encountered difficulties moving forward on the second step and attempting to climb onto the next step (Figure 8B). This was due to the lack of contact between the tail and the ground, as the robot’s legs were on a higher elevation surface than the tail, which led to the loss of support and stabilization. Consequently, the robot’s gait reverted to its previous state without a tail, where the front end would incline upwards, and the rear legs would have limited to no ground clearance. When the tail approached the step’s edge, it became stuck, obstructing the robot’s progress. Despite being immobilized, the robot continued moving, but without any forward motion along the track. The front legs extended forward, but the tail constrained the robot in place.

In most trials, the robot continued to move forward until, due to slipping of the feet, it began to rotate to one side and ultimately move off the track. In two instances, sudden jerking forward caused by the robot’s instability allowed the robot’s tail to become free and climb to the next step and continue climbing, although this required numerous gait cycles and a considerable amount of time. It was observed that the robot had already started rotating before being freed, leading to the movement toward the track’s edge. Also, when the robot was attempting to climb while on the track, one of the hind legs made insufficient clearance to climb over the step, resulting in additional rotation and ultimately causing the robot to fall off the track.

It was evident in this test that the length of the robot’s tail is a critical factor in determining its climbing ability. It must be noted that although the robot can climb better with a rigid stick-type leg, this rigid leg can easily get caught on obstacles, not just when climbing. It also does not offer sufficient ground support because the contact area of the leg is small. Increasing the tail’s length increases the clearance height of the robot’s rear and provides more support during climbing, enhancing the robot’s stability and balance. However, longer tails are more likely to get caught on objects or ledges, leading to potential failures during locomotion. Meanwhile, shorter tails decrease the probability of getting stuck but also decrease the support provided during climbing and flat ground traversal, leading to reduced stability and insufficient ground clearance.

Making the tail longer and more flexible allows the robot to adapt the height of its backside to the environment and provide more support during climbing. For this experiment, the tail was configured to stiffen and relax periodically (Supplementary Video/4:07-4:37 s). This tail state is described in Section 2.2.3. In this test, the robot successfully climbed 30 steps out of a possible 30, with 0.45 ± 0.04 stairs/cycle and 0.66° ± 0.41° rotation/cycle, over 5 trials (Figure 9).

When the tail is stiffened, the robot can climb to a higher step by providing support in the form of constant ground contact, resulting in a large clearance distance between the legs and the ground. While the robot is on the higher step, the body remains parallel, resulting in smoother locomotion. However, due to the length of the tail, it can get stuck when it reaches the edge of a step, and the robot ceases to move forward.

To prevent this, the tail relaxes by releasing the tension on the cables. This allows the tail to move over the step and for the robot to continue climbing. Due to the periodic nature of the stiffening and relaxation of the tail, the tail may inhibit movement in some cases, and the robot may appear stuck momentarily. This occurs when the tail meets a step in its stiffened state. The robot then attempts to move forward for a brief period. During the forward swing, the tail is relaxed, and the robot accelerates rapidly forward. The feet of the robot slides along the surface, and as a result, the robot moves a further distance forward than in a typical cycle. The stiffening of the tail also thrusts the robot, pushing it upward and forward at a specific instant during the gait cycle to increase locomotion effectiveness.

An extra set of trials were conducted on the upward climb experiment, incorporating a touch sensor near the end of the tail to control the state of the tail (Supplementary Video/4:38-4:53 s). The robot climbed 26 out of 30 steps with 0.32 ± 0.04 stairs/cycle and 1.92° ± 1.25° rotation/cycle. During the tests, the tail remained rigid until it came into contact with a step. When the touch sensor recorded a force, it relaxed the tail until it was no longer in contact with the step. The tail then returned to its rigid state and continued traveling forward.

Using a sensor allowed the tail to be stiffened when attempting to climb, providing important stability and foot clearance to climb upwards successfully. It also allowed the tail to relax when encountering obstructions and prevented the robot from getting stuck. Although this tail configuration significantly improved the robot’s climbing ability compared to a completely rigid tail, it performed worse than the open-loop flexible tail. This was because the tail relaxation and stiffening timing were altered so that it was not occurring at the optimal stages during the gait cycle (Section 2.2.3). Additionally, our reliance on a single sensor located on the tail meant that the sensor could only detect stairs when the tail made contact from a specific point, resulting in occasional misses. To address this issue in the future, implementing a sensor array across the entire tail could provide more accurate data and control.

When the robot has no tail attached, it performs better at climbing downwards than upwards. The robot successfully traveled down 15 steps out of a possible 30 with 0.53 ± 0.06 stairs/cycle and 2.12 ± 0.47 ^o rotation/cycle over six trials. Previous experiments showed that when traveling along flat terrain, the robot pitches upwards, causing the rear to make contact with the ground. This still occurred when climbing downwards, but when the robot moved to a lower step, it violently pitched downwards, causing the front to hit the ground or runway. This caused further disturbance to the robot’s gait, leading to instability and inconsistencies in the robot’s direction of travel.

The robot successfully traveled down 21 steps with 0.62 ± 0.27 stairs/cycle and 0.92 ± 0.71 ^o rotation/cycle when a rigid tail was attached. The robot’s locomotion patterns were made more consistent by adding a tail. However, upon each foot placement, the robot pitched downwards, and the tail briefly lost contact with the supporting surface. This caused problems when moving onto a lower step. In those cases, the robot pitched to an even more extreme angle, resulting in the front of the robot making contact with the surface. Similar to the previous experiment, these collisions with the runway resulted in negative consequences such as misdirection and less efficient locomotion, albeit to a lesser degree.

With the addition of the flexible tail, the robot climbed down 30 out of a possible 30 steps with 0.71 ± 0.04 stairs/cycle and 1.3 ± 0.7 rotation/cycle. Different from the upward climbing, where the tail was periodically stiffened and relaxed, in these experiments, the tail was configured so that it was fully flexible, i.e., there was no tension in the cables and no periodic motion of the reel. This configuration of the tail made the robot more adaptable to its terrain. When stepping off each step, the tail extends to the point where it is almost entirely vertical. At this instant, the bottom of the tail is in direct contact with the ground (the narrow end piece of the tail). Simultaneously, the front leg springs bend, and the front of the robot makes contact with the ground. However, as the tail still supports the robot, there is little impact felt by it, and there are no deviations from the path. At all other stages, the tail is flexed away from the robot’s body. This allows the tail to continue to offer support when the legs of the robot have moved to a lower step.

Without a tail, the robot could traverse on mulch without being significantly affected by the softness of the surface (Figure 10A) compared to its locomotion on a smooth flat surface. However, when the robot encountered organic debris like leaves and fallen twigs, the legs of the robot would drag the debris along, reducing the robot’s velocity (Supplementary Video/5:23-5:34 s). When a flexible tail was added, which exhibited periodic stiffening and relaxation, the robot was more resilient to these obstacles and locomoted at 4.03 ± 0.15 cm/s (Supplementary Video/4:54-5:06 s). The tail’s flexibility allowed it to absorb the impacts of the obstacles and facilitate the robot’s smooth movement.

With no tail, the robot traveled extremely slowly (∼ zero speed) over pebbled surfaces. The robot’s legs repeatedly got stuck in gaps between stones due to the legs’ low ground clearance and the body’s pitching. This was compounded by the robot’s eventual entrapment, which resulted from one of its legs becoming firmly lodged between stones, rendering it immobile.

When the robot was tested with a flexible tail that exhibited periodic stiffening and relaxation, the locomotion velocity was increased to 2.04 ± 0.6 cm/s (Supplementary Video/5:07-5:21 s). The legs no longer got stuck due to the movement of the tail and the support it provided. Although the tail occasionally became stuck, the periodic relaxation of the tail allowed it to become dislodged quickly without impeding the robot’s overall performance (Figure 10B).

When traveling on sand, the robot’s rear legs did not lift off the surface as expected but instead were dragged along through the sand. This unexpected movement of the robot’s legs disrupted the gait cycle and subjected the rear legs to increased force and twisting, which the robot was not designed to handle (Supplementary Video/5:37-6:10 s). Continued use of the robot on sand would likely lead to fracturing of the legs or body, as well as damage to the rear motor due to the excessive forces being exerted.

In contrast to the issues encountered when the robot traversed a sandy terrain without a tail, adding a flexible tail proved to be an effective solution (Figure 10C). The tail’s ability to provide constant support and its side-to-side passive undulation was clearly visible in the tracks left by the robot (Figures 10C, D). Specifically, the tail enabled the robot to traverse small divots and uneven surfaces with a velocity of 3.45 ± 0.24 cm/s, which it previously could not navigate effectively (Supplementary Video/5:37-6:10 s).