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We propose a hybrid formulation of the linear inverted pendulum model for bipedal locomotion, where the foot switches are triggered based on the center of mass position, removing the need for pre-defined footstep timings. Using a concept similar to reference spreading, we define nontrivial tracking error coordinates induced by our hybrid model. These coordinates enjoy desirable linear flow dynamics and rather elegant jump dynamics perturbed by a suitable extended class ${\mathcal K}_\infty$ function of the position error. We stabilize this hybrid error dynamics using a saturated feedback controller, selecting its gains by solving a convex optimization problem. We prove local asymptotic stability of the tracking error and provide a certified estimate of the basin of attraction, comparing it with a numerical estimate obtained from the integration of the closed-loop dynamics. Simulations on a full-body model of a real robot show the practical applicability of the proposed framework and its advantages with respect to a standard model predictive control formulation.

Mountain slopes are perfect examples of harsh environments in which humans are required to perform difficult and dangerous operations such as removing unstable boulders, dangerous vegetation or deploying safety nets. A good replacement for human intervention can be offered by climbing robots. The different solutions existing in the literature are not up to the task for the difficulty of the requirements (navigation, heavy payloads, flexibility in the execution of the tasks). In this paper, we propose a robotic platform that can fill this gap. Our solution is based on a robot that hangs on ropes, and uses a retractable leg to jump away from the mountain walls. Our package of mechanical solutions, along with the algorithms developed for motion planning and control, delivers swift navigation on irregular and steep slopes, the possibility to overcome or travel around significant natural barriers, and the ability to carry heavy payloads and execute complex tasks. In the paper, we give a full account of our main design and algorithmic choices and show the feasibility of the solution through a large number of physically simulated scenarios.

In the homogeneous manifold $\mathbb{E}(-1,\tau),$ for $0

Trajectory Optimization (TO) and Reinforcement Learning (RL) are powerful and complementary tools to solve optimal control problems. On the one hand, TO can efficiently compute locally-optimal solutions, but it tends to get stuck in local minima if the problem is not convex. On the other hand, RL is typically less sensitive to non-convexity, but it requires a much higher computational effort. Recently, we have proposed CACTO (Continuous Actor-Critic with Trajectory Optimization), an algorithm that uses TO to guide the exploration of an actor-critic RL algorithm. In turns, the policy encoded by the actor is used to warm-start TO, closing the loop between TO and RL. In this work, we present an extension of CACTO exploiting the idea of Sobolev learning. To make the training of the critic network faster and more data efficient, we enrich it with the gradient of the Value function, computed via a backward pass of the differential dynamic programming algorithm. Our results show that the new algorithm is more efficient than the original CACTO, reducing the number of TO episodes by a factor ranging from 3 to 10, and consequently the computation time. Moreover, we show that CACTO-SL helps TO to find better minima and to produce more consistent results.

We present a clear and practical way to characterize the parabolicity of a complete immersed surface that is invariant with respect to a Killing vector field of the ambient space., Comment: 12 pages, 2 figures

In recent years, advanced model-based and data-driven control methods are unlocking the potential of complex robotics systems, and we can expect this trend to continue at an exponential rate in the near future. However, ensuring safety with these advanced control methods remains a challenge. A well-known tool to make controllers (either Model Predictive Controllers or Reinforcement Learning policies) safe, is the so-called control-invariant set (a.k.a. safe set). Unfortunately, for nonlinear systems, such a set cannot be exactly computed in general. Numerical algorithms exist for computing approximate control-invariant sets, but classic theoretic control methods break down if the set is not exact. This paper presents our recent efforts to address this issue. We present a novel Model Predictive Control scheme that can guarantee recursive feasibility and/or safety under weaker assumptions than classic methods. In particular, recursive feasibility is guaranteed by making the safe-set constraint move backward over the horizon, and assuming that such set satisfies a condition that is weaker than control invariance. Safety is instead guaranteed under an even weaker assumption on the safe set, triggering a safe task-abortion strategy whenever a risk of constraint violation is detected. We evaluated our approach on a simulated robot manipulator, empirically demonstrating that it leads to less constraint violations than state-of-the-art approaches, while retaining reasonable performance in terms of tracking cost, number of completed tasks, and computation time., Comment: 7 pages, 3 figures, 3 tables, 2 pseudo-algo, conference

In this work, we consider the complex control problem of making a monopod reach a target with a jump. The monopod can jump in any direction and the terrain underneath its foot can be uneven. This is a template of a much larger class of problems, which are extremely challenging and computationally expensive to solve using standard optimisation-based techniques. Reinforcement Learning (RL) could be an interesting alternative, but the application of an end-to-end approach in which the controller must learn everything from scratch, is impractical. The solution advocated in this paper is to guide the learning process within an RL framework by injecting physical knowledge. This expedient brings to widespread benefits, such as a drastic reduction of the learning time, and the ability to learn and compensate for possible errors in the low-level controller executing the motion. We demonstrate the advantage of our approach with respect to both optimization-based and end-to-end RL approaches.

Trajectory optimization under uncertainties is a challenging problem for robots in contact with the environment. Such uncertainties are inevitable due to estimation errors, control imperfections, and model mismatches between planning models used for control and the real robot dynamics. This induces control policies that could violate the contact location constraints by making contact at unintended locations, and as a consequence leading to unsafe motion plans. This work addresses the problem of robust kino-dynamic whole-body trajectory optimization using stochastic nonlinear model predictive control (SNMPC) by considering additive uncertainties on the model dynamics subject to contact location chance-constraints as a function of robot's full kinematics. We demonstrate the benefit of using SNMPC over classic nonlinear MPC (NMPC) for whole-body trajectory optimization in terms of contact location constraint satisfaction (safety). We run extensive Monte-Carlo simulations for a quadruped robot performing agile trotting and bounding motions over small stepping stones, where contact location satisfaction becomes critical. Our results show that SNMPC is able to perform all motions safely with 100% success rate, while NMPC failed 48.3% of all motions.

Let $\mathbb{E}$ be a connected and orientable Riemannian 3-manifold with a non-singular Killing vector field $\xi\in\mathfrak{X}(\mathbb{E})$ whose associated one-parameter group of the isometries of $\mathbb{E}$ acts freely and properly on $\mathbb{E}$. Then, there is a Killing submersion from $\mathbb{E}$ onto a connected and orientable surface $M$ whose fibers are the integral curves of $\xi$. In this setting we solve the Dirichlet problem for minimal Killing graphs over certain unbounded domains of $M$, taking piecewise continuous boundary values. We obtain general Collin-Krust type estimates. In the particular case of the Heisenberg group, we prove a uniqueness result for minimal Killing graphs with bounded boundary values over a strip. We also prove that isolated singularities of Killing graphs with prescribed mean curvature are removable., Comment: 18 pages, 8 figures

We consider a Riemannian submersion from a 3-manifold $\mathbb{E}$ to a surface $M$, both connected and orientable, whose fibers are the integral curves of a Killing vector field without zeros, not necessarily unitary. We solve the Jenkins-Serrin problem for the minimal surface equation in $\mathbb{E}$ over a relatively compact open domain $\Omega\subset M$ with prescribed finite or infinite values on some arcs of the boundary under the only assumption that the same value $+\infty$ or $-\infty$ cannot be prescribed on two adjacent components of $\partial\Omega$ forming a convex angle. The domain $\Omega$ can have reentrant corners as well as closed curves in its boundary. We show that the solution exists if and only if some generalized Jenkins-Serrin conditions (in terms of a conformal metric in $M$) are fulfilled. We develop further the theory of divergence lines to study the convergence of a sequence of minimal graphs. We also provide maximum principles that guarantee the uniqueness of the solution. Finally, we obtain new examples of minimal surfaces in $\mathbb{R}^3$ and in other homogeneous $3$-manifolds., Comment: 42 pages, 13 figures

We develop a conformal duality for spacelike graphs in Riemannian and Lorentzian three-manifolds that admit a Riemannian submersion over a Riemannian surface whose fibers are the integral curves of a Killing vector field, which is timelike in the Lorentzian case. The duality swaps mean curvature and bundle curvature and sends the length of the Killing vector field to its reciprocal while keeping invariant the base surface. We obtain two consequences of this result. On the one hand, we find entire graphs in Lorentz-Minkowski space $\mathbb{L}^3$ with prescribed mean curvature a bounded function $H\in C^\infty(\mathbb{R}^2)$ with bounded gradient. On the other hand, we obtain conditions for existence and non existence of entire graphs which are related to a notion of critical mean curvature., Comment: 20 pages, 2 figures. Some references updated in version 2

Nowadays, there is growing interest in applying Artificial Intelligence (AI) on board Earth Observation (EO) satellites for time-critical applications, such as natural disaster response. However, the unavailability of raw satellite data currently hinders research on lightweight pre-processing techniques and limits the exploration of end-to-end pipelines, which could offer more efficient and accurate extraction of insights directly from the source data. To fill this gap, this work presents a novel methodology to automate the creation of datasets for the detection of target events (e.g., warm thermal hotspots) or objects (e.g., vessels) from Sentinel-2 raw data and other multispectral EO pushbroom raw imagery. The presented approach first processes the raw data by applying a pipeline consisting of spatial band registration and georeferencing of the raw data pixels. Then, it detects the target events by leveraging event-specific state-of-the-art algorithms on the Level-1C products, which are mosaicked and cropped on the georeferenced correspondent raw granule area. The detected events are finally re-projected back onto the corresponding raw images. We apply the proposed methodology to realize THRawS (Thermal Hotspots in Raw Sentinel-2 data), the first dataset of Sentinel-2 raw data containing warm thermal hotspots. THRawS includes 1090 samples containing wildfires, volcanic eruptions, and 33,335 event-free acquisitions to enable thermal hotspot detection and general classification applications. This dataset and associated toolkits provide the community with both an immediately useful resource as well as a framework and methodology acting as a template for future additions. With this work, we hope to pave the way for research on energy-efficient pre-processing algorithms and AI-based end-to-end processing systems on board EO satellites., Comment: 17 Pages

Quadruped robots are machines intended for challenging and harsh environments. Despite the progress in locomotion strategy, safely recovering from unexpected falls or planned drops is still an open problem. It is further made more difficult when high horizontal velocities are involved. In this work, we propose an optimization-based reactive Landing Controller that uses only proprioceptive measures for torque-controlled quadruped robots that free-fall on a flat horizontal ground, knowing neither the distance to the landing surface nor the flight time. Based on an estimate of the Center of Mass horizontal velocity, the method uses the Variable Height Springy Inverted Pendulum model for continuously recomputing the feet position while the robot is falling. In this way, the quadruped is ready to attain a successful landing in all directions, even in the presence of significant horizontal velocities. The method is demonstrated to dramatically enlarge the region of horizontal velocities that can be dealt with by a naive approach that keeps the feet still during the airborne stage. To the best of our knowledge, this is the first time that a quadruped robot can successfully recover from falls with horizontal velocities up to 3 m/s in simulation. Experiments prove that the used platform, Go1, can successfully attain a stable standing configuration from falls with various horizontal velocity and different angular perturbations., Comment: 8 pages, 5 figures, 2 tables, submitted to ral, accompanying video at https://youtu.be/KnmNbhkOKWI

Safety is often the most important requirement in robotics applications. Nonetheless, control techniques that can provide safety guarantees are still extremely rare for nonlinear systems, such as robot manipulators. A well-known tool to ensure safety is the Viability kernel, which is the largest set of states from which safety can be ensured. Unfortunately, computing such a set for a nonlinear system is extremely challenging in general. Several numerical algorithms for approximating it have been proposed in the literature, but they suffer from the curse of dimensionality. This paper presents a new approach for numerically approximating the viability kernel of robot manipulators. Our approach solves optimal control problems to compute states that are guaranteed to be on the boundary of the set. This allows us to learn directly the set boundary, therefore learning in a smaller dimensional space. Compared to the state of the art on systems up to dimension 6, our algorithm resulted to be more than 2 times as accurate for the same computation time, or 6 times as fast to reach the same accuracy., Comment: This work has been submitted to the IEEE for possible publication. Copyright may be transferred without notice, after which this version may no longer be accessible

In a world designed for legs, quadrupeds, bipeds, and humanoids have the opportunity to impact emerging robotics applications from logistics, to agriculture, to home assistance. The goal of this survey is to cover the recent progress toward these applications that has been driven by model-based optimization for the real-time generation and control of movement. The majority of the research community has converged on the idea of generating locomotion control laws by solving an optimal control problem (OCP) in either a model-based or data-driven manner. However, solving the most general of these problems online remains intractable due to complexities from intermittent unidirectional contacts with the environment, and from the many degrees of freedom of legged robots. This survey covers methods that have been pursued to make these OCPs computationally tractable, with specific focus on how environmental contacts are treated, how the model can be simplified, and how these choices affect the numerical solution methods employed. The survey focuses on model-based optimization, covering its recent use in a stand alone fashion, and suggesting avenues for combination with learning-based formulations to further accelerate progress in this growing field., Comment: submitted for initial review; comments welcome

This paper presents a novel algorithm for the continuous control of dynamical systems that combines Trajectory Optimization (TO) and Reinforcement Learning (RL) in a single framework. The motivations behind this algorithm are the two main limitations of TO and RL when applied to continuous nonlinear systems to minimize a non-convex cost function. Specifically, TO can get stuck in poor local minima when the search is not initialized close to a "good" minimum. On the other hand, when dealing with continuous state and control spaces, the RL training process may be excessively long and strongly dependent on the exploration strategy. Thus, our algorithm learns a "good" control policy via TO-guided RL policy search that, when used as initial guess provider for TO, makes the trajectory optimization process less prone to converge to poor local optima. Our method is validated on several reaching problems featuring non-convex obstacle avoidance with different dynamical systems, including a car model with 6D state, and a 3-joint planar manipulator. Our results show the great capabilities of CACTO in escaping local minima, while being more computationally efficient than the Deep Deterministic Policy Gradient (DDPG) and Proximal Policy Optimization (PPO) RL algorithms., Comment: 8 pages, 8 figures. Submitted to IEEE RA-L