Many small animals, particularly insects, use power-amplification to generate rapid motions, such as jumping, that would otherwise be impossible given the standard power density of muscle. A common framework for understanding this power amplification is Latch-Mediated, Spring Actuated (LaMSA) jumping, in which a spring is slowly compressed, latched in its compressed state and the latch released to allow jumping. Motivated by the jumps of certain insect larvae, we consider an external latching mechanism via adhesion to a substrate that is quickly released for jumping. We show that the rate at which this adhesion is lost is crucial in determining the efficiency of jumping and, indeed, whether jumping occurs at all. As well as showing how release rate should be chosen to facilitate optimal jumping, our analysis underscores the importance of the interaction between latch-release dynamics and the elastic deformation of the jumper for power amplification, thereby providing new insight into the post-latch control of jumping.
Water condensation on superhydrophobic surfaces can generate spontaneous droplet jumping, enabling rapid condensate removal and improved thermal and mass transfer. Although this effect has been extensively demonstrated on densely packed nanostructures, the capability of microscale textures to support jumping condensation remains poorly understood. Here, we show that engineered microscale conical arrays can achieve efficient microdroplet jumping and reveal a previously unreported spacing-dependent critical transition between jumping and non-jumping regimes. In the jumping regime, by varying only the cone pitch, we identify a geometric threshold below which sub-10 micron droplets are rapidly removed, and above which jumping is suppressed, resulting in slower dynamics and larger departing droplets. In situ optical and environmental scanning electron microscopies reveal the mechanistic origin of this transition: dense arrays favour full Cassie droplets, which depart cleanly, while wider spacing favours partial Cassie droplets that retain a localized wet region initiating new nucleation. From these results, we construct a geometry-wetting design map linking microstructure spacing, dropl
Natural terrain complexity often necessitates agile movements like jumping in animals to improve traversal efficiency. To enable similar capabilities in quadruped robots, complex real-time jumping maneuvers are required. Current research does not adequately address the problem of online omnidirectional jumping and neglects the robot's kinodynamic constraints during trajectory generation. This paper proposes a general and complete cascade online optimization framework for omnidirectional jumping for quadruped robots. Our solution systematically encompasses jumping trajectory generation, a trajectory tracking controller, and a landing controller. It also incorporates environmental perception to navigate obstacles that standard locomotion cannot bypass, such as jumping from high platforms. We introduce a novel jumping plane to parameterize omnidirectional jumping motion and formulate a tightly coupled optimization problem accounting for the kinodynamic constraints, simultaneously optimizing CoM trajectory, Ground Reaction Forces (GRFs), and joint states. To meet the online requirements, we propose an accelerated evolutionary algorithm as the trajectory optimizer to address the complex
This paper presents a curriculum-based reinforcement learning framework for training precise and high-performance jumping policies for the robot `Olympus'. Separate policies are developed for vertical and horizontal jumps, leveraging a simple yet effective strategy. First, we densify the inherently sparse jumping reward using the laws of projectile motion. Next, a reference state initialization scheme is employed to accelerate the exploration of dynamic jumping behaviors without reliance on reference trajectories. We also present a walking policy that, when combined with the jumping policies, unlocks versatile and dynamic locomotion capabilities. Comprehensive testing validates walking on varied terrain surfaces and jumping performance that exceeds previous works, effectively crossing the Sim2Real gap. Experimental validation demonstrates horizontal jumps up to 1.25 m with centimeter accuracy and vertical jumps up to 1.0 m. Additionally, we show that with only minor modifications, the proposed method can be used to learn omnidirectional jumping.
Quadrupedal wheeled-legged robots combine the advantages of legged and wheeled locomotion to achieve superior mobility, but executing dynamic jumps remains a significant challenge due to the additional degrees of freedom introduced by wheeled legs. This paper develops a mini-sized wheeled-legged robot for agile motion and presents a novel motion control framework that integrates the Nonlinear Model Predictive Control (NMPC) for locomotion and the Differential Evolution (DE) based trajectory optimization for jumping in quadrupedal wheeled-legged robots. The proposed controller utilizes wheel motion and locomotion to enhance jumping performance, achieving versatile maneuvers such as vertical jumping, forward jumping, and backflips. Extensive simulations and real-world experiments validate the effectiveness of the framework, demonstrating a forward jump over a 0.12 m obstacle and a vertical jump reaching 0.5 m.
Achieving both target accuracy and robustness in dynamic maneuvers with long flight phases, such as high or long jumps, has been a significant challenge for legged robots. To address this challenge, we propose a novel learning-based control approach consisting of model learning and model predictive control (MPC) utilizing a variable-frequency scheme. Compared to existing MPC techniques, we learn a model directly from experiments, accounting not only for leg dynamics but also for modeling errors and unknown dynamics mismatch in hardware and during contact. Additionally, learning the model with variable-frequency allows us to cover the entire flight phase and final jumping target, enhancing the prediction accuracy of the jumping trajectory. Using the learned model, we also design variable-frequency to effectively leverage different jumping phases and track the target accurately. In a total of 92 jumps on Unitree A1 robot hardware, we verify that our approach outperforms other MPCs using fixed frequency or nominal model, reducing the jumping distance error 2 to 8 times. We also achieve jumping distance errors of less than 3 percent during continuous jumping on uneven terrain with rand
Dynamic and continuous jumping remains an open yet challenging problem in bipedal robot control. Real-time planning with full body dynamics over the entire jumping trajectory presents unsolved challenges in computation burden. In this paper, we propose a novel variable-model optimization approach, a unified framework of variable-model trajectory optimization (TO) and variable-frequency Model Predictive Control (MPC), to effectively realize continuous and robust jumping planning and control on HECTOR bipedal robot in real-time. The proposed TO fuses variable-fidelity dynamics modeling of bipedal jumping motion in different jumping phases to balance trajectory accuracy and real-time computation efficiency. In addition, conventional fixed-frequency control approaches suffer from unsynchronized sampling frequencies, leading to mismatched modeling resolutions. We address this by aligning the MPC sampling frequency with the variable-model TO trajectory resolutions across different phases. In hardware experiments, we have demonstrated robust and dynamic jumps covering a distance of up to 40 cm (57% of robot height). To verify the repeatability of this experiment, we run 53 jumping experim
Achieving precise target jumping with legged robots poses a significant challenge due to the long flight phase and the uncertainties inherent in contact dynamics and hardware. Forcefully attempting these agile motions on hardware could result in severe failures and potential damage. Motivated by these challenging problems, we propose an Iterative Learning Control (ILC) approach that aims to learn and refine jumping skills from easy to difficult, instead of directly learning these challenging tasks. We verify that learning from simplicity can enhance safety and target jumping accuracy over trials. Compared to other ILC approaches for legged locomotion, our method can tackle the problem of a long flight phase where control input is not available. In addition, our approach allows the robot to apply what it learns from a simple jumping task to accomplish more challenging tasks within a few trials directly in hardware, instead of learning from scratch. We validate the method via extensive experiments in the A1 model and hardware for various jumping tasks. Starting from a small jump (e.g., a forward leap of 40cm), our learning approach empowers the robot to accomplish a variety of challe
We focus on agile, continuous, and terrain-adaptive jumping of quadrupedal robots in discontinuous terrains such as stairs and stepping stones. Unlike single-step jumping, continuous jumping requires accurately executing highly dynamic motions over long horizons, which is challenging for existing approaches. To accomplish this task, we design a hierarchical learning and control framework, which consists of a learned heightmap predictor for robust terrain perception, a reinforcement-learning-based centroidal-level motion policy for versatile and terrain-adaptive planning, and a low-level model-based leg controller for accurate motion tracking. In addition, we minimize the sim-to-real gap by accurately modeling the hardware characteristics. Our framework enables a Unitree Go1 robot to perform agile and continuous jumps on human-sized stairs and sparse stepping stones, for the first time to the best of our knowledge. In particular, the robot can cross two stair steps in each jump and completes a 3.5m long, 2.8m high, 14-step staircase in 4.5 seconds. Moreover, the same policy outperforms baselines in various other parkour tasks, such as jumping over single horizontal or vertical disco
Jumping automata are finite automata that read their input in a non-sequential manner, by allowing a reading head to ``jump'' between positions on the input, consuming a permutation of the input word. We argue that allowing the head to jump should incur some cost. To this end, we propose four quantitative semantics for jumping automata, whereby the jumps of the head in an accepting run define the cost of the run. The four semantics correspond to different interpretations of jumps: the \emph{absolute distance} semantics counts the distance the head jumps, the \emph{reversal} semantics counts the number of times the head changes direction, the \emph{Hamming distance} measures the number of letter-swaps the run makes, and the \emph{maximum jump} semantics counts the maximal distance the head jumps in a single step, We study these measures, with the main focus being the \emph{boundedness problem}: given a jumping automaton, decide whether its (quantitative) language is bounded by some given number $k$. We establish the decidability and complexity for this problem under several variants.
In order to solve the problem of stable jumping of micro robot, we design a special mechanism: elastic passive joint (EPJ). EPJ can assist in achieving smooth jumping through the opening-closing process when the robot jumps. First, we introduce the composition and operation principle of EPJ, and perform a dynamic modeling of the robot's jumping process. Then, in order to verify the effectiveness of EPJ in controlling the robot's smooth jump, we design a simulation experiment based on MATLAB. Through comparative experiments, it was proved that EPJ can greatly adjust the angular velocity of the robot and increase the jump distance of the robot. Finally, we analyze each parameter in EPJ and performs parameter optimization. After optimization, EPJ achieves a completely flip-free jump of the robot, laying an important foundation for improving the mobility of micro-robot.
Legged robots with high locomotive performance have been extensively studied, and various leg structures have been proposed. Especially, a leg structure that can achieve both continuous and high jumps is advantageous for moving around in a three-dimensional environment. In this study, we propose a parallel wire-driven leg structure, which has one DoF of linear motion and two DoFs of rotation and is controlled by six wires, as a structure that can achieve both continuous jumping and high jumping. The proposed structure can simultaneously achieve high controllability on each DoF, long acceleration distance and high power required for jumping. In order to verify the jumping performance of the parallel wire-driven leg structure, we have developed a parallel wire-driven monopedal robot, RAMIEL. RAMIEL is equipped with quasi-direct drive, high power wire winding mechanisms and a lightweight leg, and can achieve a maximum jumping height of 1.6 m and a maximum of seven continuous jumps.
Jumping is essential for legged robots to traverse through difficult terrains. In this work, we propose a hierarchical framework that combines optimal control and reinforcement learning to learn continuous jumping motions for quadrupedal robots. The core of our framework is a stance controller, which combines a manually designed acceleration controller with a learned residual policy. As the acceleration controller warm starts policy for efficient training, the trained policy overcomes the limitation of the acceleration controller and improves the jumping stability. In addition, a low-level whole-body controller converts the body pose command from the stance controller to motor commands. After training in simulation, our framework can be deployed directly to the real robot, and perform versatile, continuous jumping motions, including omni-directional jumps at up to 50cm high, 60cm forward, and jump-turning at up to 90 degrees. Please visit our website for more results: https://sites.google.com/view/learning-to-jump.
Sessile droplets coalescing on superhydrophobic substrates spontaneously jump from the surface. In this process, the excess surface energy available at the initiation of coalescence overcomes the minimal surface adhesion and manifests as sufficient kinetic energy to propel the droplets away from the substrate. Here, we show that the coalescence induced droplet jumping velocity is significantly curtailed if the superhydrophobic substrate is flexible in nature. Through detailed experimental measurements and numerical simulations, we demonstrate that the droplet jumping velocity and jumping height can be reduced by as much as 40 % and 64%, respectively, by synergistically tuning the substrate stiffness and substrate frequency. We show that this hitherto unexplored aspect of droplet coalescence jumping can be gainfully exploited in water harvesting from dew and fog harvesting. Additionally, through an exemplar butterfly wing substrate, we demonstrate that this effect is likely to manifest on many natural superhydrophobic substrates due to their inherent flexibility.
Performing highly agile dynamic motions, such as jumping or running on uneven stepping stones has remained a challenging problem in legged robot locomotion. This paper presents a framework that combines trajectory optimization and model predictive control to perform robust and consecutive jumping on stepping stones. In our approach, we first utilize trajectory optimization based on full-nonlinear dynamics of the robot to generate periodic jumping trajectories for various jumping distances. A jumping controller based on a model predictive control is then designed for realizing smooth jumping transitions, enabling the robot to achieve continuous jumps on stepping stones. Thanks to the incorporation of MPC as a real-time feedback controller, the proposed framework is also validated to be robust to uneven platforms with unknown height perturbations and model uncertainty on the robot dynamics.
A discontinuous model of computation called one-way jumping finite automata was defined by H. Chigahara et. al. This model was a restricted version of the model jumping finite automata. One-way jumping finite automata change their states after deleting a letter of an input and jump only in one direction. Allowing a state to delete a subword instead of a letter, we define a new model generalized circular one-way jumping finite automata. These automata work on an input in a circular manner. Similar to one-way jumping finite automata, generalized circular one-way jumping finite automata also jump only in one direction. We show that this newly defined model is powerful than one-way jumping finite automata. We define new variants(right and left) of the model generalized circular one-way jumping finite automata and compare them. We also compare the newly defined model with Chomsky hierarchy. Finally, we explore closure properties of the model.
This work aims to push the limits of agility for bipedal robots by enabling a torque-controlled bipedal robot to perform robust and versatile dynamic jumps in the real world. We present a reinforcement learning framework for training a robot to accomplish a large variety of jumping tasks, such as jumping to different locations and directions. To improve performance on these challenging tasks, we develop a new policy structure that encodes the robot's long-term input/output (I/O) history while also providing direct access to a short-term I/O history. In order to train a versatile jumping policy, we utilize a multi-stage training scheme that includes different training stages for different objectives. After multi-stage training, the policy can be directly transferred to a real bipedal Cassie robot. Training on different tasks and exploring more diverse scenarios lead to highly robust policies that can exploit the diverse set of learned maneuvers to recover from perturbations or poor landings during real-world deployment. Such robustness in the proposed policy enables Cassie to succeed in completing a variety of challenging jump tasks in the real world, such as standing long jumps, ju
Generating dynamic jumping motions on legged robots remains a challenging control problem as the full flight phase and large landing impact are expected. Compared to quadrupedal robots or other multi-legged robots, bipedal robots place higher requirements for the control strategy given a much smaller footprint. To solve this problem, a novel heuristic landing planner is proposed in this paper. With the momentum feedback during the flight phase, landing locations can be updated to minimize the influence of uncertainties from tracking errors or external disturbances when landing. To the best of our knowledge, this is the first approach to take advantage of the flight phase to reduce the impact of the jump landing which is implemented in the actual robot. By integrating it with a modified kino-dynamics motion planner with centroidal momentum and a low-level controller which explores the whole-body dynamics to hierarchically handle multiple tasks, a complete and versatile jumping control framework is designed in this paper. Extensive results of simulation and hardware jumping experiments on a miniature bipedal robot with proprioceptive actuation are provided to demonstrate that the pro
We present a design, model, and control for a novel jumping-flying robot that is called PogoDrone. The robot is composed of a quadrotor with a passive mechanism for jumping. The robot can continuously jump in place or fly like a normal quadrotor. Jumping in place allows the robot to quickly move and operate very close to the ground. For instance, in agricultural applications, the jumping mechanism allows the robot to take samples of soil. We propose a hybrid controller that switches from attitude to position control to allow the robot to fall horizontally and recover to the original position. We compare the jumping mode with the hovering mode to analyze the energy consumption. In simulations, we evaluate the effect of different factors on energy consumption. In real experiments, we show that our robot can repeatedly impact the ground, jump, and fly in a physical environment.
For any real $x$ and any integer $k\ge1$, we say that a set $\mathcal{D}_{k}$ of $k$ distinct integers is a $k$-tuple jumping champion if it is the most common differences that occurs among $k+1$ consecutive primes less than or equal to $x$. For $k=1$, it's known as the jumping champion introduced by J. H. Conway. In 1999 A. Odlyzko, M. Rubinstein, and M. Wolf announced the Jumping Champion Conjecture that the jumping champions greater than 1 are 4 and the primorials 2, 6, 30, 210, 2310,.... They also made a weaker and possibly more accessible conjecture that any fixed prime $p$ divides all sufficiently large jumping champions. These two conjectures were proved by Goldston and Ledoan under the assumption of appropriate forms of the Hardy-Littlewood conjecture recently. In the present paper we consider the situation for any $k\ge2$ and prove that any fixed prime $p$ divides every element of all sufficiently large $k$-tuple jumping champions under the assumption that the Hardy-Littlewood prime $k+1$-tuple conjecture holds uniformly for $\mathcal{D}_k\subset[2,\log^{k+1}x]$. With a stronger form of the Hardy-Littlewood conjecture, we also proved that, for any sufficiently large $k$-tu