Robotic leaping enhanced by thrust-induced hypogravity, achieving precise, predictable, and extended jumps

Domain: neuroscience
Published: 2026-02-09
Editorial outcome: major_revision

Abstract
Robotic jumping research advances engineering and biomimicry frontiers, prioritizing range, precision, and predictability to navigate unstructured environments. Earth’s gravity necessitates powerful actuators and lightweight bodies in robotic designs for maximal jump height. While many robots excel in statical environments, precise, predictable jumps in dynamic settings remain challenging. We realized this with a bipedal robot leveraging thrust-induced hypogravity, alongside dual regulation of aerial attitude and parabolic trajectory via thrust vectoring. Hypogravity multiplies leap range (max: 6.9 m) despite leg force saturation, enabling the robot to clear multi-level stairs, a 2.35-m-high wall, and 3-m-wide stream. Parabolic trajectory regulation allows leap distance precision/consistency surpassing existing thrust-assisted hybrids and leg-only jumpers. It enables pre-jump prediction of aerial/landing positions and timing, facilitating leaps in dynamic scenarios: through fast-moving windows (3.8 m/s), onto shifting, confined targets, and against wind disturbance. This research establishes extended range, precise, and predictable jumping through self-generated hypogravity and parabolic trajectory regulation.

Introduction
Jumping locomotion, a biomechanically efficient strategy for surmounting large obstacles and enabling rapid traversal across unstructured terrains, is extensively adopted in nature. Jumping specialists in animal kingdom can achieve superior jumping ranges, cope with dynamic environmental uncertainties via predictive jumping, and stably land onto confined target areas. However, for robotics, the pursuit of extended leap range also leads to an inherent compromise of leap precision, predictability, landing stability and safety. Conquering this trade-off to reach animal-level jumping performance with trajectory predictability, landing precision and stability, and ultra-broad range remains a key challenge. Paradigms for maximizing jumping elevation frequently involve enhancing the power-to-weight ratio, entailing advancements in jumping actuation systems. A prevalent approach is the use of catapult mechanisms, such as loading springs with cam mechanisms1–5, winching mechanisms6–15, and mini-scale jumpers with bistable structures adopting torque reversal mechanisms16–21 (also commonly adopted by robotic grippers22–24), soft buckling beam25,26, or shell27–30 structures. Nonetheless, catapult-based jumping’s energy-storing phases, often much longer than the leap aerial phases, hinder agility and continuity. Its reliance on elastic energy poses difficulties for precise feedback control of trajectories or landing positions, most such jumpers are manually, or open loop controlled. In contrast, actuator direct-drive methods (e.g., motor31–33, electrohydraulic actuation34,35) enable agile, continuous jumping with better controllability. Still, their limited instantaneous power restricts ultra-high leaps (e.g., 3m high), with the highest record (1.25m) achieved by Salto-1P36.

Previous works have employed fixed wings as aerial assistance for elongated aerial phases7,37–40. While these systems cover enhanced leap distances, the added wing structures also increase overall system mass and vertical air drag, impairing jumping elevation. Meanwhile, the aerodynamic forces generated by wings also interfere with ballistic leap trajectories, causing difficulties in aerial path and landing position prediction. Truong et al.41. used vertical flapping wings combined with catapult launch mechanism, enhancing jump height by about 30%. However, its maximum height remains restrained under 1 meter, and its cyclical flapping forces distort the trajectory’s parabolic feature, introducing both trajectory and landing position unpredictability.

The vast majority of robotic jumping or hopping research is under the ubiquitous constraint of Earth’s gravity1–21,25–40,42–46. There is a lack of research unveiling the potential of thrust-induced hypogravity (using air thrust to offset Earth’s gravity) for jumping motions. Our rationale for studying hypogravity jumping is multifold: (1) Leap range enhancement. Linear gravity reduction shows a superior enhancement effect (hyperbolic) on jumping elevation compared to linear launch velocity increase (quadratic). Theoretical height can achieve infinity with the gravity fully offset; (2) Landing safety. Reduced landing speed (close to take-off speed) ensures deducted landing impact; (3) Energy efficiency. Compared to direct flying, legged propulsion force instantly establishes a high initial speed, only requiring a constant thrust below robot’s weight, while flight demands constant upward acceleration (thrust ≥ weight). (4) Alternative ways for parabolic trajectory modulation. Jumping height gain (h=v^2/2g*) can be tuned via the gravity constant g*.

In literature, the precise control of parabolic leap trajectory and landing position is a challenging issue. For leap trajectory control, current state-of-the-art robotic jumpers mainly utilize the delicate control of launch speed and angle (e.g., Salto-1P47 (2020), Moobot19), showcasing effective tracking of individual parabolas, or regulate touchdown attitude (e.g., Salto-1P48 (2018), Hopcopter49, flapping-wing passive hopper50) for overall hopping path control. All these works lack parabolic trajectory regulation during mid-flight. In real-world applications, terrain surface variations could easily cause the deviation of launch or touchdown angle, leading to large trajectory errors (e.g., In a chair-desk jumping task, Salto-1P48 has a strict 0.7-degree touchdown angle error tolerance). Also, extreme cases such as wind disturbances could distort a trajectory’s ballistic pattern, rendering such methods insufficient. For robustness and predictability of leap patterns, we investigate a previously underexplored locomotion pattern: parabolic trajectory regulation under thrust-induced hypogravity condition. It involves studying the discrete-point errors in every timestamp of a full leap parabola, under the guidance from a reference parabolic target with coupled time and positions (time-coupled parabolic trajectories, TCPT). Previous fixed-thrust-based jumping hybrids (thrust direction fixed to airframe) employed active thrust propulsion for touchdown attitude and height regulation49,50, non-parabolic position control51, or horizontal velocity and height control52, to initiate rebound-based motions. These systems exhibit limited precision in regulating individual parabolic trajectories, not only because of their control objectives but the inherent fixed-thrust configuration limitations. Specifically, attitude adjustments require airframe rotation that induces extra translations, while positional control rigidly requires airframe rotation, restricting the control flexibility of inclination. Another technical contribution of this work lies in decoupling the rigidity between the regulation of inclination and parabolic trajectory and enables simultaneous control of the two components.

Following the mid-flight regulation of leap trajectory, balanced landing without secondary bounces is critical for tasks involving confined targets. Many jumpers show imbalance upon landing, hence using self-righting mechanism to recover from instability5,6,8–10,53–55. Soft-bodied jumpers27,29,30,34,56–61 can partly reduce secondary bounces with inherent compliance. Yet, balanced and bounce-free landing (stop at the first land point) is insufficient in the literature.

To tackle the aforementioned challenges, we devise a 370g bipedal robot called GravOff (Gravity offset) robot (Fig. 1A) that first investigates robotic jumping under thrust-induced hypogravity while regulating the aerial-phase parabolic trajectories. The component of hypogravity endow the robot with range-tunable projectile motions, extending leap ranges multifold despite the saturation of leg joint torque. This enables the robot to achieve a height jump up to 46 times body height (6.9 m), and facilitates large, rapid spatial transitions in field tests: leaping over multi-level stairs, a 2.35 m-high wall, a 3 m-wide stream, etc. An energy comparison study between hypogravity jumping and trajectory-controlled flight modes reveal that the total energy cost of hypogravity jumping modes is less than 10% of that of the flight mode when overcoming the same medium-scale obstacle of ~5 body heights. The legs cooperate with thrusters to control directional jumping motions, enabling both longitudinal and lateral leaps. During aerial descent, the legs act as an active deceleration mechanism, dissipating 50% of the impact momentum compared to direct landings, thereby facilitating bounce-free landings within confined targets. Legs’ propulsion and active deceleration are coordinated with vector-thrust and attitude control to enable seamless consecutive leaps in challenging scenarios, including jumps across inclination-varied targets and across area-confined, dynamic targets. With directed takeoff, the robot leaps under thrust-induced hypogravity while regulating the projectile motion for aerial-phase and landing-point precision and predictability. We use synchronous thrust vectoring to enable dual regulation of inclination and parabolic trajectory. The system reveals effectiveness in targeted jumping and shows an order of jumping distance (JD) precision and consistency improvement than our fixed-thrust-based regulation case and previous precise leg-only jumpers, which could achieve landings within a robot-projection-sized target region (0.25 m by 0.25 m) with sub-centimeter distance error and standard deviation. Under interference, such as large takeoff angle deviations (up to 15°) and wind disturbance (3.5 m/s), the robot can remain targeted landing with small variations. Owing to the time-coupled feature of the reference parabola (i.e., the TCPT), the timing for all positions in the subsequent parabolic curve, including the landing point, can be estimated. This allows the robot to execute well-timed leaps in response to dynamic environmental changes. The robot demonstrates this capability by conquering rapid-moving obstacles, such as leaping through a pendular window at 3.8 m/s and a rotating window at 100 deg/s. Combined with jumping, the robot also conducts versatile motions in field tests, including legged crawling (on grassland, sand, cobbles, etc.) and air-propelled wheeling (reaching 24 body length/s).

Results
Prototyping the GravOff bipedal robot
Bush crickets (Tettigoniidae) have long and thin legs, but can jump in tens of milliseconds. Their elongated limbs facilitate an extended acceleration phase while allowing for reduced muscle force demand to achieve high take-off velocities. They power their jumps by muscle direct actuation that exert forces on long levers and showcase superior quickness than spring-actuated jumpers (such as locust) for escape movements62. The GravOff robot also utilizes direct actuation method, with a pair of 3-Degree-of-Freedom (DOF) long legs (leg extension ratio reaches 340%) to initiate jumps resembling bush crickets (Fig. 1B, C), experiencing balanced crouching state, synchronous leg extension, and the airborne phase. For the proposed hypogravity effect, the four thrusters are controlled at a constant rotational speed while balancing the robot simultaneously (by a cascaded Proportional-Integral-Derivative (PID) attitude controller, see Fig. S2 and Table S2). The thrusters are arranged with a relatively wide span (220 mm by 190 mm) for long moment arms to facilitate balancing stability and aerial attitude control.

The 2-DOF jumping unit (Fig. 1D(i)) features a parallelogram mechanism and is reinforced with two additional bars for increased lateral stiffness. A brushless DC motor (Fig. 1D(ii)) serves as the primary power source, driving the upper link of the mechanism. The second DOF, controlled by servo 1, manages the overall rotation of the jumping unit. We adopt this integrated design for the jumping unit to reduce the moment of inertia and alleviate torque demand for the leg actuator. To mitigate landing impact, a 3D-printed TPU shock absorber is placed between the upper and lower links of the four-bar mechanism. A third DOF is adopted for hip rotation to enable legged assistive crawling and locomotion mode transformation (to wheeling). A lockable wheeling structure featuring a micro screw-slider mechanism (Fig. 1D(iii)) is applied at the foot for both the jumping mode (wheels locked) and a high-speed wheeling mode for flat terrain. A thrust vectoring (TV) unit (see Fig. 1D(iv)) works in conjunction with the thrusters to change the initial launch direction of a jump. It eliminates the horizontal thrust component and prevents foot sliding when the robot starts jumping in the tilted balancing pose. All components used by the robot are listed in Table S1 and the electronics architecture is discussed in the supplementary section S1 and Fig. S1.

Leg extension trajectory generation
During jumps, the leg extension trajectory, shown in Fig. 2A, is generated by the coordinated motion of the leg motor and leg servo 1 (In the context, the leg motor refers to the brushless motor in the leg actuator in Fig. 1D(ii)). We applied a slightly convex profile for the foot trajectory by linearly mapping the angle of the leg motor (controlled at 100 deg/s) to leg servo 1, with the two actuators rotating in opposite directions (see supplementary methods in Fig. S3 and section S2). Results show that the recorded trajectories (10 trials) align well with the reference trajectory, with slight deviations from linkage oscillations (see also Supplementary Movie 1). The leg motor is controlled by a bi-directional electronic speed controller (ESC; the control signals and proper settings are discussed in supplementary sections S1, S4 and Table S4). The leg motor’s position and speed control are based on two fine-tuned clamped P controllers (see supplementary section S3, Fig. S5).

There exists a difference in torque demand for the two DOF in the jumping unit, as illustrated by Fig. 2B(ii). While the legs exert forces on the ground, the reactive force produces a back driving torque (counterclockwise) on the actuators. This torque aligns with the direction of the servo 1 input torque and opposes that of the leg motor. This design reduces the torque demand on servo 1, requiring it only to regulate the leg trajectory while the leg motor handles the primary torque generation for jumping. Result also reveals that this leg mechanism can provide sufficient torque demand for vertical jumps more than five times of the robot’s body heights, without the gravity reduction (thrusters’ actuation), as presented in supplementary Fig. S10, section S8 and Supplementary Movie 2 (clip 1). Still, the leap range was shown to be markedly enhanced with constant gravity reduction in comparison to the case of leg-only jumping (see Fig. S11 and clip 2 in Supplementary Movie 2).

Achieving leap extreme via hypogravity
We tested the peak leap height by reducing gravity to microgravity level. As shown in Fig. 2B and Supplementary Movie 3, the GravOff robot attained an experimental peak leap height of 6.9 meters (46 times body height) near gPluto (0.620 m/s2), a record surpassing most jumpers working under Earth gravity (Fig. 2C). The robot achieved this with an estimated takeoff velocity of 2.85 m/s (derived by g*ΔT^2 = 2). By comparison, achieving equivalent height under Earth gravity requires 11.62 m/s—a velocity exceeding typical power limitations of legged propulsion systems. From a landing safety perspective, the reduced takeoff velocity in constant hypogravity lowers landing impact momentum to just 25% of an equivalent Earth-gravity jump, thereby reducing damage susceptibility. Overall, the robot demonstrates a broad achievable hypogravity and height range, as depicted in the red shaded area in Fig. 2C, with g* ranges from 0.062 to 0.822g (Here, 0.822g corresponds to 10% threshold throttle Earth ensuring stable balancing), and h ranges from 1 to 46 times body height. In the test, the equivalent gravity g* was set by referring to the quasi-linear gravity-throttle relationship presented in supplementary section S7 and Fig. S9.

Energetics study of the hypogravity jumping
Under thrust-induced hypogravity, jumping elevation is determined by the combined contribution of the legged propulsion and gravity reduction. While hypogravity reduces the energy demand on the leg mechanism, generating the thrust itself requires energy expenditure. To investigate the energetics of hypogravity jumping, we included a statistical test with control variate method to study the total energy cost required to achieve a fixed jump height (0.8m) under varying levels of equivalent gravity g*. Eight groups of tests were conducted for the energy analysis, with g* ranging from 0.1g to 0.8g. For each group, the target value of equivalent gravity (g*) was set using the thrust-throttle mapping in Fig. S9, and leg extension speed was adjusted to achieve the target apex height (0.8m; as the controlled variable) within ±0.05m, as validated in Fig. 3A. The results demonstrate a significant reduction in total energy consumption with increasing hypogravity (g*), decreasing from 242.68J at g*=0.1 to 49.52J at g*=0.8. Conversely, gravity reduction substantially lowers the leg energy demand (calculated by Table S5) from 27.09J at g*=0.8 to 5.37J at g*=0.1. The power is calculated by the product of smoothed voltage and current data (measured by the voltage and current sensors presented in Fig. S1). Detailed data for all trials are provided in Fig. S6 and Table S5-S8.

Our study further demonstrates that hypogravity jumping constitutes a more energy-efficient strategy than flight for clearing medium-scale obstacles (2-8 BH). This was further validated through direct energy comparisons between trajectory-controlled flight and hypogravity leaps, when clearing a 0.73m table, as shown in Fig. 3C–E and Supplementary Movie 4. The flight was commanded with a reference parabola with a nominal takeoff condition of v0=0.6 m/s, ϕ0= -13°, g=0.018g, controlled by the fixed-rotor (FR) based MPC (see details in Methods), while gravity-offset leaps employed open-loop pre-tilt launches with ϕ0 = -6°. The comparison reveals that hypogravity jumps require a total energy demand <10% and average power demand around 40% (56.28W versus 139.14W) of the flight mode. Even using the lighter, legless configuration (193.8g, Table S1) for a fair comparison, flight still consumes more energy (minimum 266.16J) than jumping (56.52J). This energy cost is calculated by multiplying the critical lift power (interpolated from Table S5) by the average flight duration in following the reference flight path (Fig. 3D). For optimal efficiency, the energy feature of hypogravity jumping can guide the strategy of tuning equivalent gravity according to obstacles’ sizes. For small-to-medium obstacles (<8 body heights, see illustration in Fig. 3F), leg dominant jumps can be utilized to ensure efficiency (with g* set to 0.6g–0.8g). For large obstacles where legs propulsion forces saturate, gravity level is reduced accordingly to extend parabolic trajectory range (as exemplified by Fig. 2B).

Longitudinal jumping with the pre-tilt method
To achieve more consistent takeoff angles and velocities, the robot employs the pre-tilt method for longitudinal jumps (see Fig. 4A and Fig. 1D(iv)). During the pre-tilt phase, a 0.6-s ramp-up angle command ϕ0 is sent from the Teensy controller to the low-level PID attitude controller while the tilt servos (Fig. 1D(iv)) simultaneously rotate to a negative angle -ϕ0. Once the pre-tilt stance is achieved and stabilized, the robot applies synchronous extension commands to both legs. This method aligns the ground reaction force closely with the leg extension line, resulting in a consistently directed initial velocity. Owing to the thrust vectoring unit, which eliminates horizontal thrust components, the robot can maintain stability during the pre-tilt phase under oscillatory angle commands, as demonstrated in Supplementary Movie 5. Beyond this pre-tilt-based longitudinal jumping method, the GravOff robot can also initiate lateral jumps by creating phase differences between its leg motors during extension, as shown in Supplementary Section S6, Fig. S8, and Supplementary Movie 6.

Leap trajectory regulation with decoupled inclination control
The precision of parabolic jumping trajectories is regulated after the robot establishes its initial state (x0 comprising three-dimensional position and velocity) in the jumping-specific coordinate frame QJ (Fig. 4B; transformed from ground coordinate QG, detailed in Methods) via legged propulsion. To address initial state deviations and aerial perturbations, a reference TCPT guides the robot’s parabolic motion to ensure trajectory precision and robustness. This TCPT reference includes both trajectory positions and velocities, governed by three tunable parameters: nominal takeoff velocity v*, inclination angle ϕ0 (here also termed pitch), and equivalent gravity g*:

xref = [v0 sin(ϕ0)t + x0, v0 cos(ϕ0)t - g*t^2/2 + z0, v0 sin(ϕ0), v0 cos(ϕ0) - g*t]^T (1)

In each time frame of the aerial phase, the trajectory error ei = xi - xrefi is passed as state input to a time-varying model-based predictive controller (MPC) that regulates motions in the x-y plane and z-direction (see the “Methods” section for details).

While standard FR-based dynamics provide sufficient control degrees of freedom for 3D trajectory regulation (roll θ for x-direction, pitch ϕ for y-direction, total thrust ΣT for z-direction), this approach becomes inadequate in jumping scenarios involving terrain surface variations, where the robot must dynamically adjust its body inclination alongside parabolic trajectory regulation to ensure proper contact and stability. However, FR systems require attitude changes to modulate trajectories, i.e., by tilting the entire body to generate horizontal acceleration, which fundamentally couples trajectory control with body inclination.

To decouple inclination control from trajectory regulation, we utilized synchronous thrust vectoring to govern longitudinal trajectory (see comparison in Fig. 4C) and established a thrust-vectoring-based MPC for the discussed jumping scenario (see Methods). The thrust-vectoring-based method assigns controller commands as linear rotor tilt angle εr, roll θ, and total thrust ΣT, while the inclination ϕcmd being freely tuned. These commands are passed as reference signals to the low-level attitude controller and tilt servos, as shown in the control workflow (Fig. 4D). Here, the measured attitude angles θmea, ϕmea, along with the previous εr,k-1 (absent in FR mode), Tj are used for state-space model update. Both TV-based MPC and FR-based MPC can regulate the TCPT and can be interchanged during distinct mid-air phases. Concurrently, the leg unit is controlled using current height information in preparation for terrain interaction (active and passive damping, detailed in supplementary section S9).

Leaping onto confined targets
To showcase the performance of the TV-based trajectory regulation, we design a task where the robot leaps onto its projection-sized target region (0.25m by 0.25m) under dynamically varying inclination commands, as shown in Fig. 5A and Supplementary Movie 7. The GravOff robot initiated the jump in a start region with a pre-tilt angle of -15° and a desired speed set of 2.0 m/s (by referring to the statistical relationship between motor speed and takeoff speed, see Fig. S7B and section S5 for details), followed by the ascending phase where the inclination was maintained, and the descending phase where a step change δϕ=30° lasting around 0.27s occurs, before landing onto the target region. Figure 5B shows the trajectory reference, result and the TV-based MPC predictions at each time frame, and Fig. 5C displays the robot’s attitude angles, rotor tilt command, and resultant thrust command. The results reveal that the thrust vectoring command εr smoothly adapts to the variation of robot’s inclination (Fig. 5C(i), (ii)), maintaining the resultant thrust quasi-vertically upward while adjusting longitudinal trajectories, as εr takes -ϕ as the its reference angle (see details in “Methods” section). The predictions are updated at each time frame, projecting trajectories ten steps ahead. Figure 5D details the absolute prediction errors for each step in every time frame 10 steps ahead of the end time (for the jump presented in Fig. 5A).

To statistically evaluate how well the predictive controller captures the dominant dynamics of the robotic system and the precision of the parabolic path regulation, we conducted repeatability tests for three cases: TV mode with varying ϕ, TV mode with fixed ϕ, and FR mode. Each case was tested in the targeted leaping task (Fig. 5A) across five repeated trials. We assessed the statistical performance of the predictions in three directions using the prediction mean absolute error (PMAE), calculated by:

PMAE(k) = 1/pstps * Σ [x_i(k+j|k) - x_i(k+j)] (2)

Results in Fig. 5E reveal that the experimental trajectories align well with the parabolic references, with the mean absolute prediction error constrained below 0.05m in all three dimensions. Overall, the mean prediction error exhibited decreasing trends and converged to zero gradually.

The repeatability tests also reveal that the TV-based parabolic trajectory regulation shows better consistency and preciseness of jumping distance, compared to the FR configuration. Among the three groups, TV mode with fixed ϕ demonstrated the best result in target landing (see Fig. 5G(i)), attaining a mean distance error of 3.8mm with minimal variation. The jumping distance standard deviation (JDSD) reached 3.901mm, an order of improvement compared to the FR case (4.589cm), and almost a quarter of the record achieved by previous precise baselines such as Salto-1P (best record: 1.6cm47) and Moobot (best record: 1.3cm19), despite our jumper’s considerably larger size.

Robust tests were conducted to analyze the TV-based controller’s performance under takeoff condition deviations (see Fig. 5G(iv)). We included four perturbed groups: three with takeoff angle deviations (-10°, +10°, and +15°) and one with lateral speed deviation (~0.2 m/s). All four groups employed the TV-based regulation with varying ϕ (launch with the corresponding pre-tilt angle, and land with a back tilt of +15°). Each group was tested over five trials, and all 20 tests confined the landing points within the target region (see Fig. 5F) with an average jumping distance error of 1cm, and a JDSD of 1.77cm. The controller commands for the jumps under nominal takeoff conditions are presented in Fig. S20, and robustness-test commands are in Figs. S21 and S22. The comparisons between MPC model predictions and experimental trajectories for each trial in Fig. 5G are shown in Figs. S23 and S24.

Bounce-free landings
Following a well-controlled parabolic trajectory, a stable robotic landing is the critical final step. The GravOff bipedal robot employed two landing methods according to targets’ dimension and impact energy: passive damping (PD) and active damping (AD). The PD method utilizes the TPU shock absorber (see Fig. 1D) for jumps with high impact momentum, while AD is adopted for missions requiring a precise and bounce-free landing within confined targets.

We compared the proposed landing methods in two vertical jumping tasks, as illustrated in Fig. 6A, B. With AD, the robot maintained the legs extended in the aerial phase and employed the legs’ synchronous contraction near the ground contact point as a soft deceleration mechanism (see Fig. 6A(i) and Supplementary Movie 8). In this process, the leg contraction speed was set at apex leap height and timing for flexion was based on a height threshold method (see supplementary section S9). The contraction movement of AD method extended ground contact duration (0.338s) twofold compared to PD method (0.17s), while reducing the average contact force (FAD = maAD = 0.367 × 5.23 = 1.92N; Fig. 6A(ii)) to a half of body weight and only a quarter of PD group (FPD = maPD = 0.367 × 20.6 = 7.56N; Fig. 6B(ii)). As a result, AD produced an impulse that almost dissipates the landing momentum: IAD = FAD ΔT = 0.648 N·S ≈ mvland = 0.69 N·S, enabling the robot to remain stationary at the first contact point. The PD groups saw an impulse (IPD = FPD ΔT = 20.6 × 0.367 × 0.17 = 1.285 N·s) that was 1.6 times its landing momentum, resulting in residual bounces deviating the robot 0.55m away from the initial landing point. To initiate vertical jumps, both the AD and PD groups applied a leg motor rotational speed command of 800 deg/s with the attitude controlled horizontal.

In a jumping task onto target placed 1.35 meters away, the active damping method enabled the bounce-free landing without deviating out of the bounds (Fig. 6C, Supplementary Movie 8). The PD method, however, saw a deviation of about 0.2m (see Supplementary Movie 8). After TCPT tracking, a back tilt identical to the launch angle was initiated at a height of 0.6 meters for the preparation of active damping. This ensured that the leg contact force and trajectory tangent at the landing point were aligned, preventing unstable flipping and falling upon contact.

Leap against wind disturbances
Robustness to disturbances is a critical factor in tracking problems, yet it is rarely explored in the context of jumping, especially during the aerial phase when disturbances like wind can affect landing position. In a targeted jumping task, we investigated our robot’s response to wind disturbances generated by an electric fan, measured at 3.5 m/s using an anemometer (model RA110, Dongguan SANLIANG Electronics Co., Ltd, China) placed 1 meter away. Tests were conducted under three scenarios: TV-based jumping with inclination control, FR-based jumping, and jumping with open-loop control (attitude maintains level), as shown in Fig. 7A and Supplementary Movie 9. Results indicate that wind disturbances significantly affected all groups near the apex of the trajectory. However, the controller-enabled groups compensated for displacement loss in the y-direction by decreasing the inclination angle ϕ (FR-based case) or the tilt angle εr (TV-based case), as demonstrated in Fig. 7B. Despite imperfect tracking of the TCPT during descent, both controllers allowed the robot to reach the 0.25 × 0.25-meter target region (landing points correspond to trajectory endpoints in Fig. 7C; data presented as mean ± SD; all landing positions are presented in Table S15). Although both methods enabled the robot to reach the target region, FR-based MPC caused deviation beyond the region after first contact due to necessary forward inclination demands compensating for trajectory errors; in contrast, TV-based MPC maintained final position proximity to target with proper inclination control, as compared in Supplementary Movie 9.

Continuous leaps across varying inclined targets
To exemplify the advantages of the TV configuration compared to the FR configuration, we challenged the GravOff robot with a demanding continuous jumping scenario featuring targets of varying surface angles (see Fig. 7D, Supplementary Movie 10). The inclined target platforms increased in height progressively from 0.6m (first platform: 20° inclination, 0.3m × 0.3m) to 1.04m (second platform: -25° inclination, 0.4m × 0.4m). In this task, the robot leaped and regulated three consecutive parabolas, traversed inclined targets A, B, before reaching the final target C. The parabolas have the nominal takeoff conditions set by [ϕ* v* g*] and values set to [-15°, 2.0, 0.278g], [20°, 2.2, 0.278g], and [-35°, 2.1, 0.278g], respectively. Once the robot landed on one inclined platform with legged assistive damping, the takeoff reference position was updated, with the previous first contact point as the start of the next reference parabola, as demonstrated in Fig. 7E. This task features the coordinated control of the inclination ϕ, tilt rotor angle εr, and the legs’ damping and propulsion, in both the aerial and stance phases. While regulating the TCPT, the inclination angle was simultaneously controlled, ensuring adaptation to the inclined target upon the stance phase (controlled perpendicular to the target surface). The Corresponding control signals for the three components are shown in Fig. 7F. The damping signals were generated before the stance phase, with the motor speed set according to an estimate of the landing speed (see details in section S9). And the leg propulsion signals were generated once the legs reached full contraction with the speed set by equation (S23).

Predictive leaps through static/dynamic aerial obstacles
Since the tracking of parabolic trajectories is rigidly time-coupled, it indicates the timings for subsequent aerial positions ahead of a leap. Utilizing this feature, our robot showcased predictive jumps to tackle dynamic/static aerial obstacles. As shown in Fig. 8 and Supplementary Movie 11 (clip 1), we designed three scenarios where the robot was required to jump through a stationary or shifting window (0.51 by 0.35m) with an error tolerance of 5cm (lateral) and 10cm (vertical). In the task of crossing the stationary window, the robot was required to follow the TCPT, adjust its leg pose to pass through the window without contact, and realize a soft landing (see Fig. 8A). The robot’s legs experienced four phases: extension for jumping, contraction, second extension for damping preparation, and active damping. Figure 8B shows the controlled leap curve aligns closely with the reference, regulating the cross-point error to 1.4cm (lateral) and 2.6cm (vertical) for safe passage. The robot attained a 6.56g peak acceleration and experienced a 0.278g hypogravity, and softly landed with an average acceleration below g (Fig. 8C). The aerial phase attitude angle and corresponding controller commands are presented in Fig. 8D.

The second jumping task involves a continuously rotating window that hinders the robot from the targeted landing (see Fig. 8E, F and Supplementary Movie 11, clip 2). In this case, the robot must cross it. To ensure success, its Center of Mass (CoM) was expected to cross the window plane at the time when the window plane was perpendicular to the positive y-axis (expected crossing position). With prior knowledge of the leap climbing time, including the time consumed in TCPT (0.56s), acceleration (~0.1s, estimated from statistical results, see Table S16), wireless communication and leg motor startup (~0.2s; see Table S17), the robot made a predictive jump before the window reached the expected crossing position. The jump was initiated in advance at a phase angle of -84.3 degrees (0.86s ahead), as shown in Fig. 8G, H. The actual CP (crossing position) deviated from the expected CP by 2.7 degrees. The exact time consumed in the climbing phase was 0.833s, slightly smaller than the planned time of 0.86s.

The third task exhibited jumping through the window in rapid pendular motion, as shown in Fig. 8I, J and Supplementary Movie 11 (clip 3). The GravOff robot jumped toward the direction in which the pendular window swung and crossed through it before landing on the targeted area. The motion of the window was determined by a releasing height of 1.9m (geometrical center height), resulting in a peak speed of 3.8 m/s at the crossing position. As the second group, the trigger timing was also determined by the climbing time of TCPT and the motion pattern of the aerial barrier. Specifically, the jumping signal was generated by the time when ϕwin = 60.3° (Fig. 8K, L), which was 0.814s before the expected CP. This task highlights an increased precision requirement for leap timing and TCPT tracking, where a mere 0.1s delay (leading to a deviation of around 20 degrees) from the expected CP can lead to a collision or failure to pass through the window, as revealed by Fig. 8K. In the experiment, a 3.9-degree deviation (captured by the indoor positioning system) between the actual and expected CP.

Consecutive jumping missions involving moving, confined targets
To showcase the predictive feature of future position in TCPT-tracking-based jumping, we included another type of jumping experiment in which the landing target position varied with time. Specifically, we introduced a rotational motion on a square landing region (a 0.33-by-0.3-meter platform driven by a speed control servo) slightly broader than the robot’s foot span. Figure 9A demonstrates this scenario. The robot initiated a jump, followed the desired TCPT, employed the active-damping technique to land on the moving target, and seamlessly performed a second jump to land on a safety net, reaching apex heights of 0.95m and 1.44m in the two jumps. The jumping tasks experienced three acceleration spikes and two aerial hypogravity phases (g*=0.264g), shown in Fig. 9B. To land on the moving target, trigger timing was calculated according to the elapsed time in the reference TCPT (1.302s), acceleration (~0.1s), signal transfer and motor start-up (~0.2s), and the rotational speed of the target region. We implemented this task with three rotational speeds for the platform: 33°/s, 76°/s and 93°/s (Supplementary Movie 12, clip 1). A similar time-related representation as Fig. 8 has been used to demonstrate the two ballistic curves of this jumping task (see Fig. 9C).

In the group with ωplatform = 93°/s, the trigger setpoint corresponded to the position where ψplatform = 1.602 × 93° ≈ 149° (0° corresponds to the position where the radius of the platform is perpendicular to the y-axis of QJ).

We conducted a repeatability test including ten trials using the group with ωplatform = 76°/s, as shown in Fig. 9D. Seven out of ten trials successfully landed and initiated a second jump on the rotating platform, and the remaining three failed trials revealed leg slippage due to late landing or bad tracking of TCPT (see Supplementary Movie 12, clip 2). The angular position of the platform ψplatform is presented in Fig. 9E, and the trigger setpoints were chosen by counting a predictive time of 1.602s ahead of the expected landing point (0°). The landing points (where the z-direction acceleration turned positive from a negative value in the aerial phase) were statistically studied, with the average x and y location ± SD, as shown in Fig. 9F. Upon contact, the result reveal that the mean value of the angular position (-2.7°) was close to the expected value (0°), shown in Fig. 9G and Table S18.

Versatile locomotion in field tests
In addition to indoor TCPT tracking-based jumping, we evaluated the GravOff bipedal robot in field settings by integrating its three locomotion modes: air-propelled passive wheeling, synchronous legged crawling, and jumping (Fig. 10). Figure 10A depicts transitions between these modes. The legged crawling posture serves as an intermediate state for jumping and wheeling (see supplementary section S12 for mode transformation and crawling control) and proved effective as an alternative to jumping or hopping on terrain with unpredictable surface contact angles, such as grassland (Fig. 10B, Supplementary Movie 13), sand (Fig. 10C), cobbles (Fig. 10D), and grooved roads (Fig. 10E). The crawling motion enables minor turns and incremental steps, facilitating precise launch positioning and orientation adjustments during field-based jumping tasks.

For speed and efficiency, the air-propelled passive wheeling was adopted for locomotion on flat terrain. A speed test in (Fig. 10F, Supplementary Movie 14) revealed that the robot could attain a peak speed of 4.8 m/s (around 24 body lengths per second), with a 50% throttle command.

The hypogravity jumping strategy facilitates rapid traversal of large outdoor barriers. As shown in Fig. 10G, the robot successfully leaped over a 2.35m-high wall under quasi-moon gravity, achieving a 3.3-meter apex height. To reach target locations, the robot executed locomotion mode transformations to navigate different types of terrains. This was demonstrated in scenarios involving targets above a staircase (crawling-jumping transition in Fig. 10H; wheeling-jumping transition in Fig. 10I), atop a 1.6m wall (Fig. 10J, crawling-jumping transition), and across a 3m-wide stream (Fig. 10K, crawling-jumping transition), as shown in Supplementary Movie 15.

Methods
Definition of coordinates
The dynamic modeling of the proposed jumper considers a total of four coordinate frames, including the ground coordinate frame QG, the jumping coordinate Q, the robot’s body coordinate QB, and the local rotor coordinate Qr fixed to the front and rear rotor pairs (the two rotor pairs use the same coordinate as they have synchronized rotation angle), as demonstrated in Fig. 4B, C. The ground coordinate system QG is established in the motion capture system’s calibration process. The jumping coordinate frame (QJ) is defined at the robot’s jump initiation position and orientation, with the zJ axis aligned with zG axis and the yJ axis oriented toward the robot’s current forward direction at the moment of jump. The body coordinate QB is fixed at the robot’s center of mass (COM) and is aligned with the robot’s main body.

Ground contact phase dynamics
The ground contact phase dynamics constitute both the robot’s acceleration phase and landing phase (including the active damping and passive damping). Before the launch of jumping, the front and rear rotor pairs have synchronized rotation angles and help to set at an arbitrary pre-tilt angle without slippage. In the acceleration phase, we assumed that the legs have a small linkage deformation in the lateral direction (i.e., along yB, refer to Fig. 4A) and the extension trajectory and ground reactive forces are generated in the xB-zB plane. The ground contact phase dynamics can be described by Newton-Euler equations, with the translational and rotational dynamics described in QJ:

mp€J = JBR(BrRTsum + fleg) - mge3 - fdrag (3)

Aerial-phase dynamics
Once the jump is initiated, the robot encounters the aerial phase resembling a projectile motion under a gravity lower than the Earth’s, as the resultant thrust force is maintained quasi-constant. And to ensure the compensated part of gravity is nearly constant, the rotation angles of the two rotor pairs are also synchronized (at the same εr). By eliminating the leg force and torque terms, the aerial-phase dynamics can be simplified to:

mp€J = JRTsum - mge3 - fdrag,z (9)
JΘ€B = -Θ_B × JΘ_B + BrRτthr - MB εr (10)

Coordinate transformation from QG to QJ
Since trajectory regulation is conducted in the jumping coordinate frame Q. To obtain the position and velocity in Q, a transformation from ground coordinate QG (where motion capture system data are obtained) to local jumping coordinate QJ is performed:

pJ = JGRpG - LG (11)
vJ = JRvG (12)

Parabolic trajectory regulation with decoupled inclination control
To preserve ballistic characteristics in the jumping trajectory despite the tilt thrust, a method is required to maintain the resultant thrust vector as vertically as possible within the jump coordinate frame. Simultaneously, the tilt mechanism must smoothly modulate the trajectory in the forward direction. This can be realized by synchronizing the rotation angle of the tilt rotor (as the acceleration phase) and establishing a control reference that has εr(t) + ϕ(t) = 0. To achieve smooth regulation of jumping trajectories rather than rigid tracking of TCPT at one time, we employed model-based predictive control (MPC) that could provide the anticipation of future states and allow for overall control of trajectory precision and smoothness via weights tuning.

The translational dynamics in Eq. (10) is utilized to establish the state-space models for the MPC. To address nonlinearities in (10) and for computational simplicity, the model is decomposed into planar (x-y) and vertical (z) dynamics:

[x_J y_J u_J v_J]^T = [v_J u_J ΣT( -sεr(cψsθsϕ - sψcϕ) + cεr(cϕsθcψ + sϕsψ))/m - fdrag,x/m, ΣT(-sεr(sψsθsϕ + cϕcψ) + cεr(cϕsθsψ - sϕcψ))/m - fdrag,y/m]^T (13)

w_J = ΣT(-sεr cθsϕ + cεr cθcϕ)/m - fdrag,z/m - g (14)

Here [uJ vJ wJ]^T corresponds to the velocity states. We assume the air drag is negligible in the planar (x-y) direction. The vertical drag fdrag,z, however, was found to be a noticeable disturbance term as the robot encounters much more vertical speed facing the thrusters (this is identical to the case where thrusters are under relative air flow), which alters the actual thrust generated. Here, fdrag,z is modelled as a velocity-related damping term (fdrag,z = cd wJ, cd = -0.0625 based on the test results in Fig. S9).

Robot fabrication
The leg actuator uses a high KV brushless DC motor (TMotor F1408 3950KV, Tiger Motor International Co., Ltd, China) reduced by a four-stage steel gear set (teeth numbers: 11; 25/10; 35/8; 36/14; 44, reduction ratio 112.5) manufactured using powder metallurgy (compressing steel powder in a mold and sintered between 1120 and 1250°C). The gears are sandwiched between two 1.5mm-thick CNC-machined (using a carbon fiber board, Wuxi Aike Technology Co., Ltd) spacers. The gears use axial stoppers (3D-printed with ESUN hard tough resin, Shenzhen Esun Industrial Co. Ltd, using Elegoo Saturn 2, Shenzhen Elegoo Technology Co., Ltd) to ensure engagement. To read the angle position using the magnetic encoder (As5600, Ams-OSRAM AG), a radial ring magnet (D7-d3-h3) is mounted to the other side of the output shaft (Fig. 1D(ii)), and rotated to the calibrated position (180 degrees) before being super-glued (using ergo 5210 adhesive, Kisling AG). A steel-gear servo motor (DS1906B, GDW Digital Servo Manufacturer Co., Ltd, China) is mounted beside (in opposite direction) the brushless motor by a 3D-printed fixer (see Fig. 1D(ii)). The output shaft of the servo and a coaxial steel shaft (a two-head threaded shaft in 1.5mm, with one end inserted into the gearbox) fix the whole leg actuator to a 3D-printed hip fixer (polyethylene terephthalate glycol (PETG), printed with Bambu Lab A1 mini, Shenzhen Tuo Zhu Technology C0., Ltd, China). The leg upper links (1.5mm thick, Yangzhou Lihong Carbon Fiber Materials Co., Ltd) were also CNC machined and were mounted with flanged ball bearings (F681X, NSK Ltd.) at the joints, and linked by a 3D-printed two-axis connector (Fig. 1D(i)) to the lower link (a D5-d4mm hollow carbon fiber tube). The TPU impact absorber was 3D-printed (TPU95A-HF, FusRock Co., Ltd) and mounted between the two linkages of the upper leg.

The brake mechanism is driven by a coreless motor (RIC-615D, Ric Motor Co., Ltd, China) reduced by a gear reduction set (two stages: 33/8-33/9) and a small screw slider mechanism. The final-stage gear drives the rotation of a screw rod (M1.6×0.35) and a hex nut (also M1.6×0.35) mounted into the 3D-printed slider (3D-printed with UV resin) transfer rotation into linear motion. A slice of 400-grit sandpaper was glued to the brake for increased friction. The thrust vectoring unit uses a pair of 1:1 angle gear (14 teeth, 3D-printed UV resin), with the output gear fixed to a hollow carbon fiber tube (D5-d4) using a tapping screw. At the two ends of the tube, the thrusters (GTS V2 1202.5 6000KV, RCinpower Co., Ltd, China) were mounted in the same horizontal plane.

Indoor positioning setup
The robot tracking was conducted in a 3.4m × 3.2m × 2.2m arena, with six evenly spaced motion capture cameras (SWIFT 30, LUSTER LightTech Co., Ltd., China) mounted on a cubic aluminum frame. The angle of the cameras was adjusted to ensure the experimental space was in the cameras’ field of view (83° × 66° for each camera). Before conducting TCPT tracking test, the cameras were calibrated using a 250mm wand and the world coordinate frame and levelness of ground were set and tuned using a calibration triangle with an embedded level vial.

Procedures for leaping in dynamic environments
Infrared markers were attached to the robot (7 markers), the window (5 markers), and the rotational target platform (5 markers) to construct rigid bodies. The robot’s CoM was calculated by SolidWorks (version 2019, Dassault Systèmes) and was set in the motion capture software Moca 2.3.0.8. Their position and attitude data were streamed to Matlab (2019b, MathWorks, Inc.) for real-time leap control. Before leaping, the robot was placed at the launch point by referring to the initial point of the tracked trajectory. The leap tasks calculated the exact trigger points. During implementation, since the streamed data are discrete, the experimental trigger timings were set to be the first data points surpassing the calculated timing (Figs. 7H, L, 8E). In the window-crossing tasks, the contraction and extension of the legs in the aerial phase were controlled according to the current elapsed time in TCPT, which reflects the robot’s spatial position. For tasks in Fig. 7, considering the motor startup time and signal transfer (~200ms) and contraction process (~160ms), the contraction signals were generated 400ms before reaching the window to ensure safe passage. The leg extension signal (only for task in Fig. 7H) was generated 70ms (~5 MPC loops) after window crossing. For the task in Fig. 9, the legs first adopted a slow contraction signal before reaching the height threshold for damping to eliminate preparation time for motor start-up (see supplementary section S9 for details). The leg command signals, thrust, pitch and roll angle commands were all generated in Matlab and transmitted wirelessly to the development board Teensy (Teensy® 4.0) via two HC-12 modules (transmitter and receiver, Guangzhou Huicheng Information Technology Co., Ltd, China) to control leg movement and attitude.

Peer review rounds
[{"round": 1, "reviewer_comments": {"reviewer_1": "This work presents the design of a legged-aerial robot in which the legs are recruited to enhance jumping capabilities.\nThe proposed design incorporates two two-link legs with parallelogram structures on each side and four vectorable thrusters\nattached to the body. The legs are equipped with actuated wheels at the end, which assist during the acceleration phase of\njumping.\nThe authors demonstrate experimental validation of the platform across several scenarios, including walking on rough\nterrain, jumping, and wheeled locomotion.\nHowever, there are significant concerns regarding the control framework and the actual contributions of the mechanical\ndesign:\nThere is minimal evidence supporting the implementation and effectiveness of MPC. For instance, it is unclear how well the\nprediction model captures the dominant dynamic behavior of the actual system. The authors do not present any comparison\nbetween the model's predictions and experimental results—an essential step to justify the use of MPC. Without this\ncomparison, it is unclear why MPC was considered appropriate in the first place.\nA key technical question remains unanswered: What is the actual contribution of the legs to the jumping performance? Are\nthe legs actively contributing to jump thrust, or are they merely synchronized with body motion during lift-off without\nmeaningful mechanical impact? In other words, is this a legged robot that jumps, or is it essentially a drone that lifts off while\nextending its legs? The manuscript provides little evidence to clarify this distinction.\nThe paper is saturated with jargon, starting with the title. Phrases such as 'regulating aerial-phase dynamic precision' are\nvague and difficult to interpret.\nSeveral pieces of data appear to be inaccurate. For example, Table 1 incorrectly states that Salto has no aerial regulation\ncapability, which is not correct. Salto regulates its roll dynamics using thrusters.\nMathematical equations are difficult to follow due to missing symbols and unconventional notations. For example, in\nEquation 20, variables are defined with arrows, which is not a standard.\nThe figures are excessively dense and cluttered, making it hard to extract clear insights. Moreover, the manuscript includes\ntoo many figures, which dilutes the clarity of the key messages.\nOverall, the paper does not provide sufficient evidence to evaluate whether the design enables meaningful collaboration\nbetween the legs and thrusters to enhance jumping performance. The main contributions of the work are unclear and get lost\nin the volume of information presented.\nClarification and focused validation -- particularly around the role of the legs and the justification for using MPC -- are\nessential.", "reviewer_2": "This manuscript presents a control method for robotic leaping in a hybrid robot that integrates quadrotors with jumping legs.\nThe core contribution lies in the design of a model predictive controller (MPC) for generating Time-Coupled Parabolic\nTrajectories (TCPTs) during the aerial phase of motion. The manuscript includes a variety of experimental demonstrations,\nincluding extremely high vertical leaps, lateral jumps with directional control, and traversal of moving obstacles. These\nresults are presented clearly and effectively.\nThe paper is generally well written and easy to follow. The experimental section is particularly strong, with clear descriptions\nand visualizations that effectively communicate the robot’s dynamic capabilities. The explanation of the controller design is\nalso easy to understand, making the manuscript accessible to a broad robotics audience.\nThat said, the novelty of the work appears limited in both the hardware design and control methodology. From a hardware\nstandpoint, the concept of combining legged systems with thrusters is not new. Several prior works have proposed and\ndemonstrated similar hybrid locomotion platforms. Notable examples include:\n• Songnan Bai et al., An agile monopedal hopping quadcopter with synergistic hybrid locomotion, Science Robotics, 2024\n• Kyunam Kim et al., A bipedal walking robot that can fly, slackline, and skateboard, Science Robotics, 2021\nThese works already established the feasibility and benefits of integrating aerial propulsion with legged mobility. Therefore,\nthe hardware presented in this manuscript does not constitute a significant advancement beyond existing designs.\nWith regard to control, while the authors highlight their approach for enabling TCPTs as a key contribution, the\nimplementation essentially follows standard model predictive control techniques that track pre-planned reference trajectories\nover time. The dynamics considered during the aerial phase are fundamentally those of a conventional quadrotor, and the\ncontrol problem is equivalent to trajectory tracking in existing aerial robotics literature—a problem that is well studied and\nconsidered mature. Thus, the manuscript does not present a novel control formulation or fundamental advancement in\ncontrol theory.\nMoreover, while the authors claim that their controller enables reduced impact forces, higher jump heights, and more\npredictable trajectories, these outcomes are largely expected when employing quadrotor control during flight. The\nexperimental maneuvers—although visually impressive—could likely be executed with quadrotor control alone, without\nrequiring the use of the jumping leg.\nIn summary, while the manuscript offers a set of compelling experimental demonstrations with a hybrid aerial-leg robot, it\nlacks a clear technical contribution in either hardware innovation or control methodology. For this reason, the reviewer\nbelieves that the work does not meet the threshold for publication in Nature Communications, where novelty and\nfundamental insight are expected.\nMinor Comment: In the vicinity of Equations (3), (4), and (5), the definitions of variables such as I, v, and Fb appear to omit\nR(real numbers), which should be clarified for correctness and completeness.", "reviewer_3": "Summary\nThe paper introduces GravOff, a bipedal jumping robot that combines leg actuation with quadrotor thrust assistance to\nsimulate hypogravity conditions. It proposes time-coupled parabolic trajectory (TCPT) tracking for aerial regulation and\nclaims significant improvements in leap height and accuracy. While the work presents an ambitious hardware system and\ndemonstrates several high-performance jumping tasks, the experimental design and claims raise significant concerns. In its\ncurrent form, the work lacks rigorous comparative justification and overstates the contribution of the legs in what is effectively\na drone-aided aerial robot.\nStrengths\n- Interesting mechanical design: Inspired by bush crickets, the direct-drive leg mechanism is thoughtfully implemented with a\nhigh leg extension ratio and a compact jumping mechanism.\n- Impressive leap feats: The robot demonstrates high jumps (up to 6.9 m), predictive leaping through moving windows, and\nrobust aerial tracking.\n- A comprehensive set of experiments includes wind disturbances, confined landings, and dynamic targets.\nWeaknesses and Major Concerns\n1. Unjustified Energy Comparison (Fig. 2C, 2D)\nThe robot achieves high leaps by combining leg actuation with constant vertical thrust from four rotors. Yet it compares its\n“leg energy” with other jumpers that use only leg power.\nThis is misleading: the constant lift from rotors (simulating hypogravity) drastically reduces the energy required from legs.\nThere is no reporting of total energy consumption, which is crucial to assess the full system's energy efficiency.\n2. Is this really a legged jumper or a drone with legs?\nDuring the aerial phase, the robot behaves essentially as a quadrotor with minor leg involvement. It tracks a trajectory using\nMPC and controls attitude with PID, similar to drones.\nThe legs are largely dispensable after takeoff. This raises the question: what is the role of legs in overall motion beyond\nlaunching?\nThe introduction and discussion should clarify the necessity and contribution of legs—especially since many tasks could be\nachieved with a tilt-controlled quadrotor.\n3. Core claims about hypogravity need deeper physical justification\nThe paper claims that reduced gravity reduces the take-off speed and hence energy demand. While kinematically true (same\nheight at lower (cid:1) = 2 (cid:2)h), this ignores the constant energy drawn by rotors to offset gravity.\nSince the robot must apply vertical thrust over the full duration of the aerial phase, this introduces substantial energy draw,\nwhich is not accounted for.\n4. No comparison to existing quadrotor-jumper hybrids or multi-modal drones\nPrior work on drone-leg hybrid systems (e.g., flapping-wing assisted jumpers, or quadruped drones like FLaSH) should be\ndiscussed.\nIt's unclear how this robot advances beyond those in terms of efficiency, precision, or novelty.\n5. Overclaiming precision and control performance\nThe TCPT-MPC tracking is interesting, but the claim of “full predictability of all future positions and timing” is too strong—\nespecially in real-world conditions where disturbances or modeling errors exist.\nThere is no noise or uncertainty modeling in MPC nor any robustness bounds in the presence of actuation delay or ground\ncontact variability.\nSuggestions for Improvement\n- Include full power analysis (leg + rotors).\n- Add baseline comparisons to (a) drone-only trajectory control, and (b) leg-only jumpers, if available."}, "author_response": "Dear EditorandReviewers,\nWesincerely expressour gratitudefor consideringourmanuscript (NCOMMS-25-19757)andfor your valuable\ntime and patience. Your critical comments and insightful analysis have provided us with meaningful reflections\nand inspiration to explorenew methods to address theissues raised.Your recognition istruly encouraging and\nmotivates us to further strengthen this work. To address all concerns well, we make extensively revisions to the\nmanuscript.Thetextscoloredinbluecorrespondtonewinformation,andthetextshighlightedinyellowarethe\nrevisedinformation. Specifically,we wouldlike highlight themajor revisionsand additions:\n1. Afterrevision,wehaveestablisheda methodbydecouplingtherigiditybetweeninclinationcontroland\nparabolic trajectory regulation that is an inherent constraint for fixed-thrust or fix-rotor configuration\n(predominantly used by existing prominent thrust-assisted jumpers and our previous case). The method is\nenabledbysynchronous thrustvectoring duringparabolic trajectoryregulation,wherethetilt-rotorpairs adapt\nto the freely-assigned inclinationangle while it regulates thelongitudinal paths (y-direction). We revised Fig. 4\nwith the illustration of the method. We reorganize the Methods section with the inclusion of ground contact\nphase dynamics that additionally considers the leg’s reaction force and vector thrust forces, as well as the part\nfor the parabolic trajectory regulation with the altered dynamics and different control outputs. Some\nunconventional mathematical symbols and notations are revised for clarity (with the definitions presented in\nnew TableS16).\n2. We include the modelling of uncertainty and noise existed in the hypogravity jumping case. Previous\nmodelling did not considers the existence of thrust uncertainty when there are relative speed between the\nrobot and the air. This change of actual thrust is obvious both ascent and descent phase (please see Fig. S24\nand Section S12). The uncertainty is modeled as a first-order speed-related damping term, estimated by a\nverticaljumpingtestwithaconstantthrustcommand(seealsoSectionS12).Consideringthevibrationofrotors,\nnoisealsoexist intheactualthrust (asalsorevealedby Fig.S9C).Thethrustnoiseismodeled asarandomwalk\nnoise (equation(24)). The noise and modelling are added to discrete state-space model (equation (25)), in\nwhichtheMPCsolver makes predictions andcontrol actions.\n3. The contrition and necessity of the legs are clarified. We have investigated the energy characteristics of\nhypogravity jumping under different contribution levels of legged propulsion and gravity reduction, and\ncompared it to drone-only trajectory control case (please check the new result section: Energetics study of\nhypogravity jumping and Fig. 3, Fig. S6, Table S5-S7). The result highlight the necessity of the legs as an\nenergy-efficient component for hypogravity jumping. An obstacle clearance task (onto a 0.73 m table) reveals\nthe large gap of energy cost betweenhypogravity jumping and trajectory-controlled flight (please see Fig. 3D,\nE, and new video 3). The detailed discussion regarding the contribution of the legs are presented in boht\nIntroduction(paragraph6,coloredin blue)andDiscussionsection(parargraph2).\n4. Largely improved jumping distance precision and consistency. The Result section “Leap onto confined\ntargets” isrevisedbyincludingtheVT-based parabolictrajectoryregulationfor thejumpingtaskonto arobot’s\nprojection-sized (0.25-by-0.25 m) region, along with demonstrations in the new Fig. 5 and video 6. The result\nshows vector thrust can realize parabolic trajectory regulation and targeted landing with the inclination (𝜙) \ncontrolledto beboth fixed (maintaining0 degrees)andvaried (by an step change of30 degrees), as shown by\nthetrajectories(Figs.5B,5Eand5G),landingpoints(Fig.5FandTableS11),andcontrollercommands(Fig.5C,\nFig. S18). The result (Fig. 5G) reveals that the jumping distance precision and consistency see an order of\nimprovement compared to our previous fixed-rotor-based regulation method, as well as existing precise\nbaselinesof leg-only jumpers,whichis comparedinTable1.\n5. Reduced delay in the parabolic trajectory regulation (longitudinal direction). The previous\nfixed-rotor-based trajectory regulation relies on attitude inclination to produce horizontal thrust, which\nencountersdelays(>0.2s forrotating20degrees)aswellasoscillationsandsteady-statedeviations.By contrast,\nthe vector-thrust-based regulation has better responsiveness (the vectoring mechanism take around\n0.015-0.02s to rotate 20 degrees). Its oscillations and steady-state angle deviations are also negligible (with\nproper priorcalibration).\n6. Tovalidatetheeffectivenessofmodel-basedpredictivecontrolandaccuracyofpredictions,theResults\nsection includes further comparisons between the predicted trajectory segments and actual trajectory (please\nsee Figs. 5B, 5D). To statistically evaluate prediction performance, the Prediction Mean Absolute Error (PMAE;\ncalculated by equation (2) ) is presented to for both the VT and FR configurations along three dimensions,\npresentedinFig. 5E.\n7. Further validation tests demonstrating the advantage of the new method over the fixed-thrust case\narepresented.(1)Tohighlight thedualcontrollabilityofinclinationandparabolicmotion,andthecollaboration\nbetween the legs and thrusters, the robot is tackled with inclined targets, in a continuous leap mission, where\nthe inclination must be adjusted in the aerial phase to adapt to the platform’s inclination while regulating the\nparabolic trajectories (please see section Figs.7D-7F, supplementary video 9). (2) the targeted jumping task\nagainst wind disturbance (measured 3.5 m/s) is updated, with the inclusion of the VT case. Both the VT and FR\ncases are able to ensure the robot to reach the targets. The FR case requires a large forward inclination for\ntrajectory compensation, resulting in unwanted landing contact angle (deviating robot out of bound). The VT\nbased case enables robot to stay proximity to the region with active legged damping, owing to the actively\ncontrolled inclination (please see Figs. 7A-7C, Video 8; here, active legged damping was not applied in\nFR-casetoavoidcrashes).\n8. Table 1 is revised and include four more comparison items: “Aerial-attitude control”, “Aerial inclination and\nparabola dual control”, “jumping distance precision”,” jumping distance consistency”. Previous “regulating\nparabolicdynamicprecision” isrevised to“Parabolic positionalregulation”,to avoid ambiguity.\n9. Thefiguresaretransformed intovector format after revision.\nTable of contents for response:\nResponseto reviewer1:pages3-11.\nResponseto reviewer2:pages12-18.\nResponseto reviewer3:pages19-26."}, {"round": 2, "reviewer_comments": {"reviewer_1": "The readers of this work expects to see two substantiated results, as I also stated in my first review: (1) if MPC is used, what\nis the accuracy of the prediction model? and (2) what is the contribution of the legs (i.e., what makes this robot different from\na quadcopter).\nI specifically looked for these results throughout the draft but was unable to locate them.\nMy first concern relates to the prediction model. Fig 5-b shows reference (commanded) versus predicted trajectories during\nthe flight phase, while Fig 5-g shows reference and trials but not prediction model versus trials. The prediction model error\nhas to be directly compared with the trials. The point of this comparison is to demonstrate MPC effectiveness, since MPC\nrelies on the prediction model to control the actual system. A straightforward way to establish this is to record open-loop\ntrajectories of the physical robot and compare the robot states with the prediction model states.\nMy second concern relates to the role of the legs. Fig 3 shows energy-based analyses of leg and thruster actuators, but this\nanalysis alone cannot reveal meaningful insight. For example, leg actuators can be high energy in terms of being high\ntorque at low speed (as is typical of off-the-shelf servos) while still being incapable of producing the high-torque, high-speed\nmovements needed for dynamic jumping maneuvers. Future readers of this paper will expect to see clear evidence that this\nrobot is not merely a quadcopter. At present, no technical results demonstrate whether the system is capable of jumping\nwithout thrusters.\nA simple experimental setup could address this: remove the propellers (not the brushless motors), constrain the robot’s\nmotion in all directions except Z using a simple jig, then perform trials with leg activation only and record the\nresulting jump trajectories. If the legs contribute meaningfully, the robot should achieve some measurable jump. Leg\nversus thruster contributions could then be quantified, for example, by comparing the body-length height achieved with legs alone versus\nlegs plus thrusters.\nMy other concerns remain as well too. The terminology needs careful reconsideration. Phrases such as “vector thrust MPC”\nor “vector-thrust-based jump” are not standard. Perhaps the intended terms are “thrust-vector-based” or “thrust-vectoring-\nbased.” There are several other examples throughout the manuscript where terms are unclear or unconventional, and these\nshould be revised.", "reviewer_2": "The authors have made substantial revisions, including hardware modifications, additional control design, and new\nexperimental demonstrations to address the reviewers’ comments. This reviewer is generally satisfied with the revisions and\nhas no further comments.", "reviewer_3": "While the authors have added an extensive energy analysis, the fundamental issues remain unresolved. The claim that\nE_leg is less than 10% of the total energy and thus the legs are \"efficient\" is misleading. This overlooks the indirect impact of\nthe legs on the system.\nThe legs significantly increase system mass, aerodynamic drag, and inertia. These factors directly increase the energy\nconsumption of the rotors (E_rotors), which are required to generate and maintain hypogravity. Therefore, E_leg cannot be\ninterpreted in isolation. A small direct energy input from the leg actuators does not imply a small contribution to overall\nsystem energy consumption.\nMoreover, the LP/GP formulation (where LP + GP = 1) is not a valid partition of energy. LP is derived from normalized takeoff\nvelocity squared, and GP is defined as normalized gravity reduction. These are not additive energy terms and cannot be\nused to infer energy tradeoffs in the way presented.\nThe energy comparisons with drone-only flight are also not properly controlled. The authors compare against drone modes\nwith different masses and thrust configurations. A fair comparison would require same-mass, same-hardware baselines with\nand without legs.\nFinally, the justification for the specific leg design remains weak. There is no evidence that this complex 3-DoF architecture\nprovides benefits beyond simpler alternatives. Most control and trajectory tracking are performed by the rotors, and the role\nof the legs beyond takeoff is limited.\nIn summary, the energy analysis is conceptually flawed and does not support the claimed efficiency or necessity of the leg\ndesign. A valid system-level energy breakdown, with controlled comparisons, is still missing."}, "author_response": "Dear Editor and referees,\nWe appreciate your careful and rigorous review of our manuscript entitled “Ultra-range, precise, and predictable\nrobotic leaping via parabolic trajectory regulation under thrust-induced hypogravity” (NCOMMS-25-19757A).\nAccordingtoyourcommentsandsuggestions,wehaveconductedadditionalexperimentalstudyandrevisionstothe\nmanuscript,andmakedetailedclarificationregardingthepointsthatappearedconfusing.Inthefollowingpages,we\nmake point-by-point reply to these remarks and your careful review would be highly appreciated.\nColorcodinginthemanuscript:thetextinbluecorrespondstonewcontentsandthosehighlightedinyellowarethe\nrevisions. Red texts are highlighted for clarification.\nTable of contents for responses:\nResponse to referee #1: Page 1\nResponse to referee #2: Page 8\nResponse to referee #3: Page 8"}, {"round": 3, "reviewer_comments": {"reviewer_3": "My concerns are not resolved. The effect of additional mass of the leg should be considered to claim energy efficiency. At\nthe end, legless drone v.s. Proposed system should bw compared for fairness. Legged drone during flight v.s. Legged drone\njumping does not justify the additional legs.\nThe revision is mostly rewriting. Some words are clearer, but fundamental analysis and reasoning is still not persuasive.\nI would recommend this work for a conference paper for the introduction of the concept. Not enough through evaluation and\nscientific findings for a journal work."}, "author_response": "Dear reviewer #3,\nWe appreciate your critical comments regarding the energy efficiency of the hypogravity jumping\nmode compared to the flight mode, when overcoming the same obstacle (the 0.73-m-high table\npresented in Fig, 3C and video 4). And we agree that considering the additional mass of leg when\ncomparingenergycost betweenjumpingandflightmode, isfairer,sincetheflightmodehasthelegs\nas unused mass. Following your point, we compare the energy cost of the robot’s flight mode in\nleglesscasev.s.thecompleterobotinjumpingmode.Anditturnsoutthat,fortheobstacleclearance\ntaskinFig.3C,hypogravityjumpingrequireslessenergyexpenditurecomparedtotheleglessdrone\ncase.\nWhenremovingthelegs,therobotweighs193.8g(Total:367g,thetwolegs,includingtwolegunits\nand hip rotation structures, weigh 173.2g, according to the robot’s component mass listed in Table\nS1,for convenience,seeTable R1in thisresponse letter).Theminimumpower requirementforthe\nflight of the legless configuration is to have the total thrust force be equal to the weight of legless\nconfiguration. To generate a thrust that lifts the legless robot’s weight (identical to generating a\nhypogravityof 0.528 g_earth), power requirement can be derivedfrom Table S5 (please see Table\nR2 in this response letter) by interpolating the mean values, yielding:\np  p(0.5g) 0.5280.5\nlegless   p 56.63 W (1)\np(0.6g) p(0.5g) 0.60.5 legless\nFollowing the same flight reference trajectory (defined by v* 0.6m/s, * 13,g* 0.018g) used\n0 0\nin Fig. 3C, where the average time elapsed in the five flight groups is 4.7 s (Fig. 3D). The minimal\nenergy cost of legless configuration’s flight mode would be:\nE  p ΔT 56.634.7266.16J\nlegless legless,min flight\nIn comparison, average cost of E is 56.56 J (the result is presented in Fig. 3E) is almost 1/5th of\njump\nthe energy cost of the optimal case of legless flight mode.\nHence, through this calculation, the hypogravity jumping mode (with the complete robot) turns out\nto be more energy-efficient than the flight mode when considering removing the legs’ mass.\nWe have also made revision to the main text by adding the statement regarding this fairer\ncomparison, please see paragraph 2 in the section “Energetics study of hypogravity jumping”. we\nalso present it here:\nEven using the lighter, legless configuration (193.8 g, Table S1) for a fair comparison, flight still\nconsumes more energy (minimum 266.16 J) than jumping (56.52 J). This energy cost is calculated\nby multiplying the critical lift power (interpolated from Table S5) by the average flight duration in\nfollowing the reference flight path (Fig. 3D).\nCoincidentally, the critical power requirement (56.63 W) for lifting the legless robot’s weight is very\nclosetotheaveragepowerrequirementofthehypogravityjumping(56.28W).However,differences\nbetween them are the climbing time between activation (at the bottom) and table-robot contact.\nLegged jumping ensures high vertical climbing rate, enabling the obstacle clearance task completed\n(i.e., upon robot-table contact, see Fig. 3D) within 1 seconds.\nEven if we assume that it is possible to lower the climbing time of the legless drone by assigning a\nfaster reference parabola (using higher v* and g*), and the legless robot can complete the same\n0\ntask using the same amount of time (1s), the power demand would also surpass that of the jumping\ncase (56.28 W) as increased thrust force is needed.\nWe truly appreciate your critical review and comments. If any further improvements are needed or\nquestions unanswered, we will consider them carefully and make revisions accordingly. Thank you!"}]

Decision letter