Nonholonomic Motion Planning for a Free-Falling Cat Using Quasi-Newton Method

The motion planning problem of a free-falling cat is investigated. Nonholonomicity arises in a free-falling cat subject to nonintegrable velocity constraints or nonintegrable conservation laws. When the total angular momentum is zero, the rotational motion of the cat subjects to nonholonomic constraints. The equation of dynamics of a free-falling cat is obtained by using the model of two symmetric rigid bodies. The control of system can be converted to the motion planning problem for a driftless system. Based on the input parameterization, the continuous optimal control problem is transformed into the discrete one. The quasi-Newton method of motion planning for nonholonomic multibody system is proposed. The effectiveness of the numerical algorithm is demonstrated by numerical simulation

A Hamilton-Jacobi Formulation for Time-Optimal Paths of Rectangular Nonholonomic Vehicles

We address the problem of optimal path planning for a simple nonholonomic vehicle in the presence of obstacles. Most current approaches are either split hierarchically into global path planning and local collision avoidance, or neglect some of the ambient geometry by assuming the car is a point mass. We present a Hamilton-Jacobi formulation of the problem that resolves time-optimal paths and considers the geometry of the vehicle

Virtual Structure Based Formation Tracking of Multiple Wheeled Mobile Robots: An Optimization Perspective

Today, with the increasing development of science and technology, many systems need to be optimized to find the optimal solution of the system. this kind of problem is also called optimization problem. Especially in the formation problem of multi-wheeled mobile robots, the optimization algorithm can help us to find the optimal solution of the formation problem. In this paper, the formation problem of multi-wheeled mobile robots is studied from the point of view of optimization. In order to reduce the complexity of the formation problem, we first put the robots with the same requirements into a group. Then, by using the virtual structure method, the formation problem is reduced to a virtual WMR trajectory tracking problem with placeholders, which describes the expected position of each WMR formation. By using placeholders, you can get the desired track for each WMR. In addition, in order to avoid the collision between multiple WMR in the group, we add an attraction to the trajectory tracking method. Because MWMR in the same team have different attractions, collisions can be easily avoided. Through simulation analysis, it is proved that the optimization model is reasonable and correct. In the last part, the limitations of this model and corresponding suggestions are given

Feedback Synthesis for Controllable Underactuated Systems using Sequential Second Order Actions

This paper derives nonlinear feedback control synthesis for general control affine systems using second-order actions---the needle variations of optimal control---as the basis for choosing each control response to the current state. A second result of the paper is that the method provably exploits the nonlinear controllability of a system by virtue of an explicit dependence of the second-order needle variation on the Lie bracket between vector fields. As a result, each control decision necessarily decreases the objective when the system is nonlinearly controllable using first-order Lie brackets. Simulation results using a differential drive cart, an underactuated kinematic vehicle in three dimensions, and an underactuated dynamic model of an underwater vehicle demonstrate that the method finds control solutions when the first-order analysis is singular. Moreover, the simulated examples demonstrate superior convergence when compared to synthesis based on first-order needle variations. Lastly, the underactuated dynamic underwater vehicle model demonstrates the convergence even in the presence of a velocity field.Comment: 9 page

Contact aware robust semi-autonomous teleoperation of mobile manipulators

In the context of human-robot collaboration, cooperation and teaming, the use of mobile manipulators is widespread on applications involving unpredictable or hazardous environments for humans operators, like space operations, waste management and search and rescue on disaster scenarios. Applications where the manipulator's motion is controlled remotely by specialized operators. Teleoperation of manipulators is not a straightforward task, and in many practical cases represent a common source of failures. Common issues during the remote control of manipulators are: increasing control complexity with respect the mechanical degrees of freedom; inadequate or incomplete feedback to the user (i.e. limited visualization or knowledge of the environment); predefined motion directives may be incompatible with constraints or obstacles imposed by the environment. In the latter case, part of the manipulator may get trapped or blocked by some obstacle in the environment, failure that cannot be easily detected, isolated nor counteracted remotely. While control complexity can be reduced by the introduction of motion directives or by abstraction of the robot motion, the real-time constraint of the teleoperation task requires the transfer of the least possible amount of data over the system's network, thus limiting the number of physical sensors that can be used to model the environment. Therefore, it is of fundamental to define alternative perceptive strategies to accurately characterize different interaction with the environment without relying on specific sensory technologies. In this work, we present a novel approach for safe teleoperation, that takes advantage of model based proprioceptive measurement of the robot dynamics to robustly identify unexpected collisions or contact events with the environment. Each identified collision is translated on-the-fly into a set of local motion constraints, allowing the exploitation of the system redundancies for the computation of intelligent control laws for automatic reaction, without requiring human intervention and minimizing the disturbance of the task execution (or, equivalently, the operator efforts). More precisely, the described system consist in two different building blocks. The first, for detecting unexpected interactions with the environment (perceptive block). The second, for intelligent and autonomous reaction after the stimulus (control block). The perceptive block is responsible of the contact event identification. In short, the approach is based on the claim that a sensorless collision detection method for robot manipulators can be extended to the field of mobile manipulators, by embedding it within a statistical learning framework. The control deals with the intelligent and autonomous reaction after the contact or impact with the environment occurs, and consist on an motion abstraction controller with a prioritized set of constrains, where the highest priority correspond to the robot reconfiguration after a collision is detected; when all related dynamical effects have been compensated, the controller switch again to the basic control mode

A Partially Randomized Approach to Trajectory Planning and Optimization for Mobile Robots with Flat Dynamics

Motion planning problems are characterized by huge search spaces and complex obstacle structures with no concise mathematical expression. The fixed-wing airplane application considered in this thesis adds differential constraints and point-wise bounds, i. e. an infinite number of equality and inequality constraints. An optimal trajectory planning approach is presented, based on the randomized Rapidly-exploring Random Trees framework (RRT*). The local planner relies on differential flatness of the equations of motion to obtain tree branch candidates that automatically satisfy the differential constraints. Flat output trajectories, in this case equivalent to the airplane's flight path, are designed using Bézier curves. Segment feasibility in terms of point-wise inequality constraints is tested by an indicator integral, which is evaluated alongside the segment cost functional. Although the RRT* guarantees optimality in the limit of infinite planning time, it is argued by intuition and experimentation that convergence is not approached at a practically useful rate. Therefore, the randomized planner is augmented by a deterministic variational optimization technique. To this end, the optimal planning task is formulated as a semi-infinite optimization problem, using the intermediate result of the RRT(*) as an initial guess. The proposed optimization algorithm follows the feasible flavor of the primal-dual interior point paradigm. Discretization of functional (infinite) constraints is deferred to the linear subproblems, where it is realized implicitly by numeric quadrature. An inherent numerical ill-conditioning of the method is circumvented by a reduction-like approach, which tracks active constraint locations by introducing new problem variables. Obstacle avoidance is achieved by extending the line search procedure and dynamically adding obstacle-awareness constraints to the problem formulation. Experimental evaluation confirms that the hybrid approach is practically feasible and does indeed outperform RRT*'s built-in optimization mechanism, but the computational burden is still significant.Bewegungsplanungsaufgaben sind typischerweise gekennzeichnet durch umfangreiche Suchräume, deren vollständige Exploration nicht praktikabel ist, sowie durch unstrukturierte Hindernisse, für die nur selten eine geschlossene mathematische Beschreibung existiert. Bei der in dieser Arbeit betrachteten Anwendung auf Flächenflugzeuge kommen differentielle Randbedingungen und beschränkte Systemgrößen erschwerend hinzu. Der vorgestellte Ansatz zur optimalen Trajektorienplanung basiert auf dem Rapidly-exploring Random Trees-Algorithmus (RRT*), welcher die Suchraumkomplexität durch Randomisierung beherrschbar macht. Der spezifische Beitrag ist eine Realisierung des lokalen Planers zur Generierung der Äste des Suchbaums. Dieser erfordert ein flaches Bewegungsmodell, sodass differentielle Randbedingungen automatisch erfüllt sind. Die Trajektorien des flachen Ausgangs, welche im betrachteten Beispiel der Flugbahn entsprechen, werden mittels Bézier-Kurven entworfen. Die Einhaltung der Ungleichungsnebenbedingungen wird durch ein Indikator-Integral überprüft, welches sich mit wenig Zusatzaufwand parallel zum Kostenfunktional berechnen lässt. Zwar konvergiert der RRT*-Algorithmus (im probabilistischen Sinne) zu einer optimalen Lösung, jedoch ist die Konvergenzrate aus praktischer Sicht unbrauchbar langsam. Es ist daher naheliegend, den Planer durch ein gradientenbasiertes lokales Optimierungsverfahren mit besseren Konvergenzeigenschaften zu unterstützen. Hierzu wird die aktuelle Zwischenlösung des Planers als Initialschätzung für ein kompatibles semi-infinites Optimierungsproblem verwendet. Der vorgeschlagene Optimierungsalgorithmus erweitert das verbreitete innere-Punkte-Konzept (primal dual interior point method) auf semi-infinite Probleme. Eine explizite Diskretisierung der funktionalen Ungleichungsnebenbedingungen ist nicht erforderlich, denn diese erfolgt implizit durch eine numerische Integralauswertung im Rahmen der linearen Teilprobleme. Da die Methode an Stellen aktiver Nebenbedingungen nicht wohldefiniert ist, kommt zusätzlich eine Variante des Reduktions-Ansatzes zum Einsatz, bei welcher der Vektor der Optimierungsvariablen um die (endliche) Menge der aktiven Indizes erweitert wird. Weiterhin wurde eine Kollisionsvermeidung integriert, die in den Teilschritt der Liniensuche eingreift und die Problemformulierung dynamisch um Randbedingungen zur lokalen Berücksichtigung von Hindernissen erweitert. Experimentelle Untersuchungen bestätigen, dass die Ergebnisse des hybriden Ansatzes aus RRT(*) und numerischem Optimierungsverfahren der klassischen RRT*-basierten Trajektorienoptimierung überlegen sind. Der erforderliche Rechenaufwand ist zwar beträchtlich, aber unter realistischen Bedingungen praktisch beherrschbar

On Centroidal Dynamics and Integrability of Average Angular Velocity

In the literature on robotics and multibody dynamics, the concept of average angular velocity has received considerable attention in recent years. We address the question of whether the average angular velocity defines an orientation framethat depends only on the current robot configuration and provide a simple algebraic condition to check whether this holds. In the language of geometric mechanics, this condition corresponds to requiring the flatness of the mechanical connection associated to the robotic system. Here, however, we provide both a reinterpretation and a proof of this result accessible to readers with a background in rigid body kinematics and multibody dynamics but not necessarily acquainted with differential geometry, still providing precise links to the geometric mechanics literature. This should help spreading the algebraic condition beyond the scope of geometric mechanics,contributing to a proper utilization and understanding of the concept of average angular velocity.Comment: 8 pages, accepted for IEEE Robotics and Automation Letters (RA-L

Hamilton-Jacobi Theory for Degenerate Lagrangian Systems with Holonomic and Nonholonomic Constraints

We extend Hamilton-Jacobi theory to Lagrange-Dirac (or implicit Lagrangian) systems, a generalized formulation of Lagrangian mechanics that can incorporate degenerate Lagrangians as well as holonomic and nonholonomic constraints. We refer to the generalized Hamilton-Jacobi equation as the Dirac-Hamilton-Jacobi equation. For non-degenerate Lagrangian systems with nonholonomic constraints, the theory specializes to the recently developed nonholonomic Hamilton-Jacobi theory. We are particularly interested in applications to a certain class of degenerate nonholonomic Lagrangian systems with symmetries, which we refer to as weakly degenerate Chaplygin systems, that arise as simplified models of nonholonomic mechanical systems; these systems are shown to reduce to non-degenerate almost Hamiltonian systems, i.e., generalized Hamiltonian systems defined with non-closed two-forms. Accordingly, the Dirac-Hamilton-Jacobi equation reduces to a variant of the nonholonomic Hamilton-Jacobi equation associated with the reduced system. We illustrate through a few examples how the Dirac-Hamilton-Jacobi equation can be used to exactly integrate the equations of motion.Comment: 44 pages, 3 figure

An Efficient Spatial-Temporal Trajectory Planner for Autonomous Vehicles in Unstructured Environments

As a core part of autonomous driving systems, motion planning has received extensive attention from academia and industry. However, real-time trajectory planning capable of spatial-temporal joint optimization is challenged by nonholonomic dynamics, particularly in the presence of unstructured environments and dynamic obstacles. To bridge the gap, we propose a real-time trajectory optimization method that can generate a high-quality whole-body trajectory under arbitrary environmental constraints. By leveraging the differential flatness property of car-like robots, we simplify the trajectory representation and analytically formulate the planning problem while maintaining the feasibility of the nonholonomic dynamics. Moreover, we achieve efficient obstacle avoidance with a safe driving corridor for unmodelled obstacles and signed distance approximations for dynamic moving objects. We present comprehensive benchmarks with State-of-the-Art methods, demonstrating the significance of the proposed method in terms of efficiency and trajectory quality. Real-world experiments verify the practicality of our algorithm. We will release our codes for the research communit