Spatiotemporal Tube

Dynamical Systems (DS) are reactive motion policies representing vector fields trained with theoretical guarantees of stability and convergence. To ensure safety during deployment in unknown environments they must be locally reshaped, either through modulation or geometric control barrier function strategies. However, depending on the geometry of the obstacles and the complexity of the DS, these local strategies can lead the system to unavoidable collisions or spurious attractors. In this work, we certify safety with a value function drawn from the notion of backward reachability tube, which measures the worst-case safety along a rollout trajectory of the nominal DS. Usually, such a value function is intractable for a controlled system due to curse of dimensionality. We show that in the DS-based learning-from-demonstration setting, the absence of a control input collapses the reachability problem to a deterministic rollout, and the presence of certain stability conditions truncates the infinite horizon to a finite one, resulting in a well-defined value function. We further show that the value function we devised is the maximal forward-invariant subset of the obstaclefree region for the nominal DS flow. The application of this certificate function is validated across five DS constructions - analytical, Neural ODE, diffeomorphic latent space, LPV-DS, SE(3)and validate it on a Franka manipulator. Modulation and geometric CBFs also suffer from saddle point in cases of headon approach towards an unsafe zone. We show that CBF-on-V avoids this pitfall entirely.

Least-restrictive safety filters based on Hamilton-Jacobi reachability provide strong safety guarantees by overriding a nominal controller only when the system reaches the boundary of the set of unsafe states defined as a Backward Reachable Tube (BRT). These guarantees, however, rely on the continuous-time nature of the underlying formulation. In practice, robotic systems apply control at discrete sampling intervals, which creates a mismatch where the system may jump into the unsafe BRT between updates, allowing failures that are theoretically avoidable. This work introduces a principled solution based on a one-step expanded BRT that predicts all states capable of reaching the true BRT within a single timestep. By using this expanded boundary as the activation condition for the safety filter, safety interventions occur early enough to ensure correctness under discrete-time execution. We formulate this expanded set as a modified reachability problem and compute it using standard continuous-time solvers.

Hybrid dynamical systems provide a powerful modeling framework for robotic systems, particularly in contact-rich environments. However, ensuring safety and performance in such systems remains challenging due to the intricate coupling between continuous dynamics and discrete mode transitions. In this work, we extend classical Hamilton-Jacobi (HJ) reachability analysis, a formal verification method for continuous-time nonlinear systems, to hybrid dynamical systems. Our framework characterizes safe sets for hybrid systems through a generalized value function defined over both discrete and continuous states while accounting for control constraints and model uncertainty. We additionally provide a numerical algorithm to compute this value function. Building on these safe sets, we propose two different mechanisms to integrate performance objectives. First, we introduce a hybrid least-restrictive safety filter that intervenes on both the discrete and continuous components of a nominal controller only when necessary to avoid unsafe states, thereby preserving nominal behavior whenever possible. Second, we formulate and compute hybrid backward reach-avoid tubes, enabling the simultaneous enforcement of safety and goal-reaching behavior, an extension not previously addressed within hybrid HJ reachability. This enables the synthesis of continuous and discrete control policies that guarantee both safety and task completion. We validate our framework through simulation studies and real-world experiments on a quadrupedal robot, demonstrating its effectiveness in hybrid mode planning and safety-critical applications.

Soft eversion robots, also known as vine robots, have attracted growing interest for navigation and inspection tasks, including minimally invasive medical applications [1]. A vine robot consists of a thin, flexible, inextensible tube folded inward that everts and grows forward when pressurized. This tip-growth enables navigation with minimal friction, making vine robots well suited for complex environments such as the human colon [2]. While their inherent softness allows passive conforma- tion to curved pathways in confined spaces, navigation performance strongly depends on environmental inter- actions, including contact angle and the length of un- constrained deployed material [3], [4]. Sharp directional changes, such as those in the sigmoid colon, often limit passive growth and necessitate active steering. Existing solutions include distributed artificial muscles [5] or dedicated tip-based steering mechanisms [6]. In addition, many applications require payload delivery, such as sensors and tools [7], [8]. Within the ERC Synergy project EndoTheranostics, this motivates the development of vine robots capable of delivering micro- surgical tools during growth. Prior work has integrated working channels within the vine body [8], [9], but these approaches constrain tool size, introduce friction, and limit access to the environment to the robot tip. In this work, we propose a multi-vine architecture in which two vine robots are coupled to an externally integrated working channel via soft mounting tips [10]. Independent vine actuation enables active tip steering while advancing the working channel without embed- ding it within the vine bodies Figure 1. Experiments demonstrate sharp steering of nearly 90 degrees during growth, highlighting the potential of this architecture for versatile medical and non-medical applications.

This work proposes an actuator-aware spatiotemporal tube (STT) synthesis framework to accomplish temporal reach-avoid-stay (T-RAS) tasks for an unknown nonlinear multi-input and multi-output (MIMO) system under actuator constraints. Existing STT synthesis methods address actuator saturation after the tube generation either through repeated online re-optimization or controller redesign. Instead, the proposed framework incorporates actuator constraints directly into the tube synthesis process. The STT centerline and width are parameterized using Bernstein polynomial basis functions, whose convex-hull property enables sample-free enforcement of geometric and derivative constraints. By analyzing the worst-case closed-loop error dynamics of an approximation-free prescribed performance controller (PPC) used for STT tracking, we derive a linear actuator feasibility constraint. The constraints are embedded directly in terms of the tubes' Bernstein control points into the STT synthesis optimization for actuator-feasible tube generation, eliminating the need for online re-optimization or controller redesign. A simulation study on an omnidirectional mobile robot performing a T-RAS task shows that the proposed framework adheres to the prescribed actuator limits throughout the task and reduces required control effort by approximately $50\%$ compared with an existing STT synthesis method.

Robot sensors and electronic equipment are prone to failure in harsh environments. With water hydraulic drive, thin and long tubes enable remote operation without actuator-side sensors. Furthermore, the elasticity of the tubes reduces the impedance of the joints (actuators), benefiting robot tasks involving unexpected contact with the environment or vibrations. However, owing to the low impedance and limited camera visibility, accurately positioning the joint (or end effector) to the target location under varying load conditions is challenging. This study proposes a novel method that employs water-filled flexible tubes to enable the transmission of driving power and actuator-side information to and from the actuator, respectively, without actuator-side sensors. By modeling volumetric loss during transmission based on pressure fluctuations and incorporating minor air entrapment, simultaneous power transmission and position estimation is achieved through a tube up to 50 m. Thus, it becomes possible to use a feedback control framework that was previously difficult to implement in sensorless systems. Experimental validation confirms stable position control of a sensorless water hydraulic cylinder under varying loads. Furthermore, a field parameter-identification method accounts for tube and air entrainment variability without requiring actuator-side sensors. These contributions promote reliable remote control of robots in harsh environments.

Safe navigation in dynamic environments is challenging when system dynamics are unknown and actuator inputs are limited. Existing methods either rely on accurate models, require online optimization, or do not explicitly account for input constraints. This paper presents a real-time control framework for unknown Euler-Lagrange systems that guarantees finite-time reach-avoid-stay (FT-RAS) specifications while respecting actuator limits. We extend the spatiotemporal tube (STT) framework by incorporating input constraints into the controller design and derive offline-verifiable feasibility conditions that relate the available control authority to the tube design and uncertainty bounds. The resulting framework is approximation-free and computationally efficient, making it suitable for real-time implementation. The proposed approach is validated through simulations on a mobile robot, a quadrotor, and a spacecraft, together with hardware experiments on a mobile robot, demonstrating safe navigation while satisfying actuator constraints.

This paper presents a Spatiotemporal Tube (STT)-based control framework for general unknown nonlinear Euler-Lagrange (EL) systems subject to input constraints, with the objective of satisfying Signal Temporal Logic (STL) specifications, where confinement of the system trajectory within the STT guarantees the satisfaction of the corresponding STL task. For both single and multi-agent scenarios, the STT corresponding to each agent is modeled as a time-varying ball, whose center and radius are jointly parameterized using a physics-informed neural network (PINN). The robustness metric associated with the STL specification corresponding to the agents is incorporated into the training process as a loss function, enabling the learned tube to encode task-level temporal requirements. For a multi-agent scenario, we introduce an additional robustness metric corresponding to the global task, which, when satisfied, ensures the tubes do not collide with each other. To ensure that the system trajectory remains within the learned STT and thereby satisfies the local and global STL specifications, we propose a control strategy that explicitly accounts for input constraints. In particular, a closed-form control law is developed to keep the trajectory inside the tube while regulating the motion of the tube by enforcing bounds on its evolution depending on the input constraints of the system. The proposed approach has been validated over several case studies.

The high-angle-of-attack flip maneuver of reusable launch vehicles presents significant challenges for robust trajectory optimization due to the combined effects of highly nonlinear dynamics, aerodynamic uncertainties, and actuator saturation. This paper presents a differentiable physics framework for saturation-aware robust trajectory optimization. At its core, a Differentiable Particle Tube Control (DPTC) scheme is developed to optimize uncertainty evolution through an ensemble-based distribution shaping strategy. State uncertainty is represented by a Lagrangian particle ensemble, while hard actuator projection operators are embedded directly into the computational graph, enabling the joint optimization of the nominal feedforward trajectory and a time-varying feedback policy via end-to-end backpropagation. The proposed framework is evaluated against an automatic differentiation-based Successive Convexification (AD-SCvx) baseline combined with a conventional covariance steering feedback strategy. Six-degree-of-freedom Monte Carlo simulations demonstrate that, although the baseline achieves nominal fuel-optimal solutions, its unconstrained feedback formulation becomes susceptible to actuator saturation under aerodynamic disturbances, leading to degraded closed-loop robustness. In contrast, the proposed DPTC framework proactively performs a constraint-aware performance trade-off by relaxing spatial tracking to preserve critical control authority. These results demonstrate that integrating differentiable physics with ensemble-based optimization provides an effective and practical framework for robust guidance in highly constrained aerospace flight systems.

We present STT-LfD, a unified Learning from Demonstration (LfD) framework that integrates motion learning with control for unknown Euler-Lagrange systems. Unlike traditional decoupled approaches that track a fixed reference, the proposed method treats demonstrations as a data-driven safety specification. Using heteroscedastic Gaussian Processes, STT-LfD learns Spatiotemporal Tubes (STTs) as an intent envelope that capture time-varying precision requirements of a task. A closed-form feedback controller then enforces these learned constraints while respecting actuator limits, without requiring explicit system identification. The approach preserves the temporal structure of demonstrations, remains computationally efficient, and avoids explicit system identification. Hardware experiments on a mobile robot and a 7-DOF manipulator show that it outperforms baselines in robustness to disturbances and computational speed.

Due to the limited endurance of embedded energy sources such as lithium-polymer (LiPo) batteries, the flight duration and operational range of unmanned aerial vehicles (UAVs) are severely constrained. Although energy-efficient trajectory planning and control have been widely studied, most existing approaches rely on accurate system models and computationally expensive optimization procedures. This paper proposes a model-free online iterative learning (IL) framework to minimize energy consumption. Without requiring explicit models of UAV dynamics or energy consumption, the proposed method improves energy efficiency while maintaining a low computational cost. The per-iteration computational complexity is O(n), where n denotes the number of path points. In the tested cases, the proposed method is approximately 50--60 times faster than the model-based IPOPT benchmark. Simulation results and real-world flight experiments across multiple UAV platforms validate the effectiveness, computational efficiency, and practical applicability of the proposed approach.

Artificial dynamic tactile sensing requires sensitivity, robustness, and compliance, yet existing technologies face trade-offs when scaling to large-area arrays, compounded by wiring complexity and cost. Here, we report a passive distributed paradigm using deep sub-wavelength acoustic waveguides that decouples performance from structural flexibility. Elastic-membrane-capped Helmholtz resonators interconnected by spring-reinforced microtubes form an enclosed network with invariant acoustic transmission under macroscopic bending. By sparsely embedding microphones, the system achieves real-time localization (4 mm highest spatial resolution; >99% accuracy in a 4 microphones 64-node sensing array) and waveform reconstruction of low-frequency signals (<100 Hz). Fast Continuous Wavelet Transform and a lightweight neural network enable inference within 5.5 ms. We demonstrate conformable prototypes-fingertip arrays, a tactile glove, and large-area skins-detecting stimuli from single-hair contact to 5-mg particle impacts, arterial pulse waves, feather touches, and finger contact. This establishes a scalable, flexible, low-cost paradigm for next-generation human-machine interfaces.

Spatio-Temporal Video Grounding aims to localize object tubes based on textual queries. While recent methods have achieved remarkable success, they mainly focus on high-quality(HQ) inputs, neglecting the widespread presence of low-quality(LQ) videos in real-world scenarios. Although tuning methods like LoRA can adapt to degraded inputs, they inevitably disrupt pre-trained knowledge. To address this, we propose Null-Space Tuning (NST). This framework exploits the geometric property that adding vectors within the null-space of frozen weights to the layer input does not affect the output. Leveraging this, NST injects learnable residuals into input features that can be selectively invisible to the pre-trained backbone. Specifically, NST combines the Quality-Adaptive Unit and Dual-Space Reparameterization to synthesize these residuals by confining components for HQ inputs to the null-space, while directing restoration components for LQ inputs to the non-null space. As the frozen weights eliminate null-space components, we effectively rectify degraded inputs while preserving pre-trained knowledge for HQ inputs. Extensive experiments show that NST outperforms state-of-the-art methods on our Mixed-Quality benchmark.

Autonomous spacecraft docking requires control policies that simultaneously ensure collision avoidance and target reachability under coupled, high-dimensional translational-rotational dynamics. Hamilton-Jacobi (HJ) reachability provides formal reach-avoid guarantees, but classical solvers are limited to low-dimensional systems. Learning-based approaches have begun to scale HJ analysis, yet they struggle in reach-avoid settings, especially where goal and failure sets are tightly coupled, as in docking. We propose a learning-based Backward Reach-Avoid Tube (BRAT) framework that addresses this challenge by tightly integrating HJ structure with MPC-based supervision. In the offline phase, we train a neural approximation of the HJ value function using PDE-based losses augmented with curriculum-driven MPC supervision, which provides informative value targets and stabilizes training in regions where purely PDE-based methods fail. In the online phase, the learned value function is deployed through two real-time controllers: (i) a value gradient-driven controller, and (ii) a value-function-augmented terminal MPC that explicitly enforces reachability at the horizon. We evaluate the proposed method on a 6D planar docking problem against grid-based ground truth and then scale to the full 13D system. Across both settings, our approach outperforms existing methods in success rate and computational efficiency.

We present a soft corrugated tube sensor designed to estimate strain in each half segment. When air flows through the tube, the internal corrugated cavities induce pressure oscillations that excite the tube's standing wave resonance mode, generating an acoustic tone. Stretching the tube affects both the resonance mode frequency, due to changes in overall length, and the frequency-flow speed relationship, due to variations in cavity width, which is particularly useful for local strain estimation. By sweeping flow rates in a controlled manner, we collected resonance frequency data across flow speeds under various local stretch conditions, enabling a machine learning algorithm (gradient boosting regressor) to estimate segmental strain with high accuracy. The dual-period tube design (3.1 mm and 4.18 mm corrugation periods) achieved a mean absolute error (MAE) of 0.8 mm, while the single-period tube (3.1 mm) provided a satisfactory MAE of 1 mm. Testing on a mannequin finger demonstrated the sensor's capability to differentiate multi-joint configurations, showing its potential for estimating non-uniform deformations in soft bodies.

This paper presents a two-stage trajectory planning framework for a multi-UAV rigid-payload cascaded transportation system, aiming to address planning challenges in densely cluttered environments. In Stage I, an Enhanced Tube-RRT* algorithm is developed by integrating active hybrid sampling and an adaptive expansion strategy, enabling rapid generation of a safe and feasible virtual tube in environments with dense obstacles. Moreover, a trajectory smoothness cost is explicitly incorporated into the edge cost to reduce excessive turns and thereby mitigate cable-induced oscillations. Simulation results demonstrate that the proposed Enhanced Tube-RRT* achieves a higher success rate and effective sampling rate than mixed-sampling Tube-RRT* (STube-RRT*) and adaptive-extension Tube-RRT* (AETube-RRT*), while producing a shorter optimal path with a smaller cumulative turning angle. In Stage II, a convex quadratic program is formulated by considering payload translational and rotational dynamics, cable tension constraints, and collision-safety constraints, yielding a smooth, collision-free desired payload trajectory. Finally, a centralized geometric control scheme is applied to the cascaded system to validate the effectiveness and feasibility of the proposed planning framework, offering a practical solution for payload attitude maneuvering in densely cluttered environments.

Tubular tree structures, such as blood vessels and airways, are essential in human anatomy and accurately tracking them while preserving their topology is crucial for various downstream tasks. Trexplorer is a recurrent model designed for centerline tracking in 3D medical images but it struggles with predicting duplicate branches and terminating tracking prematurely. To address these issues, we present Trexplorer Super, an enhanced version that notably improves performance through novel advancements. However, evaluating centerline tracking models is challenging due to the lack of public datasets. To enable thorough evaluation, we develop three centerline datasets, one synthetic and two real, each with increasing difficulty. Using these datasets, we conduct a comprehensive evaluation of existing state-of-the-art (SOTA) models and compare them with our approach. Trexplorer Super outperforms previous SOTA models on every dataset. Our results also highlight that strong performance on synthetic data does not necessarily translate to real datasets. The code and datasets are available at https://github.com/RomStriker/Trexplorer-Super.