In this paper we present a hybrid feedback approach to solve the navigation problem of a point mass in the n-dimensional space containing an arbitrary number of ellipsoidal shape obstacles. The proposed hybrid control algorithm guarantees both global asymptotic stabilization to a reference and avoidance of the obstacles. The intuitive idea of the proposed hybrid feedback is to switch between two modes of control: stabilization and avoidance. The geometric construction of the flow and jump sets of the proposed hybrid controller, exploiting hysteresis regions, guarantees Zeno-free switching between the stabilization and the avoidance modes. Simulation results illustrate the performance of the proposed hybrid control approach for 2-dimensional and 3-dimensional scenarios.
Obstacle avoidance between polytopes is a challenging topic for optimal control and optimization-based trajectory planning problems. Existing work either solves this problem through mixed-integer optimization, relying on simplification of system dynamics, or through model predictive control with dual variables using distance constraints, requiring long horizons for obstacle avoidance. In either case, the solution can only be applied as an offline planning algorithm. In this paper, we exploit the property that a smaller horizon is sufficient for obstacle avoidance by using discrete-time control…
In this letter we study variational obstacle avoidance problems on complete Riemannian manifolds. The problem consists of minimizing an energy functional depending on the velocity, covariant acceleration and a repulsive potential function used to avoid a static obstacle on the manifold, among a set of admissible curves. We derive the dynamical equations for extrema of the variational problem, in particular on compact connected Lie groups and Riemannian symmetric spaces. Numerical examples are presented to illustrate the proposed method.
In this paper we study a path planning problem from a variational approach to collision and obstacle avoidance for multi-agent systems evolving on a Riemannian manifold. The problem consists of finding non-intersecting trajectories between the agent and prescribed obstacles on the workspace, among a set of admissible curves, to reach a specified configuration, based on minimizing an energy functional that depends on the velocity, covariant acceleration and an artificial potential function used to prevent collision with the obstacles and among the agents. We apply the results to examples of a p…
Hamburg University of Technology（汉堡工业大学）；Institute of Mechanics and Ocean Engineering（力学与海洋工程研究所）
The goal of this thesis is to propose the combination of Control-Barrier-Functions (CBF) with Model-Predictive-Control (MPC) resulting in the novel Model-Predictive-Control-Barrier-Function (MPCBF). It can be shown, that the performance of the MPCBF surpasses the performance of the CBF due to the increased time horizon of the MPC. Moreover, the MPCBF was applied to a quadrotor, a system strongly in need of fast and predictive control. Using the MPCBF, the quadrotor was able to avoid obstacles, which the CBF failed to avoid due to the relative speed of the obstacle. The results of this work are…
Kiel University（基尔大学）；AVL Software and Functions GmbH（AVL软件与功能有限公司）
A path-following collision-avoidance model predictive control (MPC) method is proposed which approximates obstacle shapes as convex polygons. Collision-avoidance is ensured by means of the signed distance function which is calculated efficiently as part of the MPC problem by making use of a dual formulation. The overall MPC problem can be solved by standard nonlinear programming (NLP) solvers. The dual signed distance formulation yields, besides the (dual) collision-avoidance constraints, norm, and consistency constraints. A novel approach is presented that combines the arising norm equality w…
This paper studies the reduction by symmetry of a variational obstacle avoidance problem. We derive the reduced necessary conditions in the case of Lie groups endowed with a left-invariant metric, and for its corresponding Riemannian homogeneous spaces by considering an alternative variational problem written in terms of a connection on the horizontal bundle of the Lie group. A number of special cases where the obstacle avoidance potential can be computed explicitly are studied in detail, and these ideas are applied to the obstacle avoidance task for a rigid body evolving on SO$(3)$ and for th…
We study the reduction by symmetry for optimality conditions in optimal control problems of left-invariant affine multi-agent control systems, with partial symmetry breaking cost functions for continuous-time and discrete-time systems. We recast the optimal control problem as a constrained variational problem with a partial symmetry breaking Lagrangian and obtain the reduced optimality conditions from a reduced variational principle via symmetry reduction techniques in both settings, continuous-time, and discrete-time. We apply the results to a collision and obstacle avoidance problem for mult…
This paper introduces a class of time-varying controllers for Euler-Lagrange systems such that the convergence occurs at an arbitrary finite time, independently of initial conditions, and free of chattering. The proposed controller is based on a mapping technique and is designed in two steps: First, a conventional (obstacle avoidance) asymptotically stable controller is specified for the nominal system; then, by a simple substitution, a prescribed-time (obstacle avoidance) controller is achievable for the perturbed system. It is proved that the proposed scheme is uniformly prescribed-time stab…
We study variational obstacle avoidance problems on complete Riemannian manifolds and apply the results to the construction of piecewise smooth curves interpolating a set of knot points in systems with impulse effects. We derive the dynamical equations for extrema in the variational problem, and show the existence of minimizers by using lower-continuity arguments for weak convergence on an infinite-dimensional Hilbert manifold. We then provide conditions under which it is possible to ensure that the extrema will safely avoid a given obstacle within some desired tolerance.
UC Berkeley（加州大学伯克利分校）；Ford Research and Advanced Engineering（福特研究与高级工程部门）
We propose a Model Predictive Control (MPC) for collision avoidance between an autonomous agent and dynamic obstacles with uncertain predictions. The collision avoidance constraints are imposed by enforcing positive distance between convex sets representing the agent and the obstacles, and tractably reformulating them using Lagrange duality. This approach allows for smooth collision avoidance constraints even for polytopes, which otherwise require mixed-integer or non-smooth constraints. We consider three widely used descriptions of the uncertain obstacle position: 1) Arbitrary distribution wi…
We employ the proximal averaged Newton-type method for optimal control (PANOC) to solve obstacle avoidance problems in real time. We introduce a novel modeling framework for obstacle avoidance which allows us to easily account for generic, possibly nonconvex, obstacles involving polytopes, ellipsoids, semialgebraic sets and generic sets described by a set of nonlinear inequalities. PANOC is particularly well-suited for embedded applications as it involves simple steps, its implementation comes with a low memory footprint and its fast convergence meets the tight runtime requirements of fast dyn…
We study the problem of target stabilization with robust obstacle avoidance in robots and vehicles that have access only to vision-based sensors for the purpose of realtime localization. This problem is particularly challenging due to the topological obstructions induced by the obstacle, which preclude the existence of smooth feedback controllers able to achieve simultaneous stabilization and robust obstacle avoidance. To overcome this issue, we develop a vision-based hybrid controller that switches between two different feedback laws depending on the current position of the vehicle using a hy…
This paper studies a variational obstacle avoidance problem on complete Riemannian manifolds. That is, we minimize an action functional, among a set of admissible curves, which depends on an artificial potential function used to avoid obstacles. In particular, we generalize the theory of bi-Jacobi fields and biconjugate points and present necessary and sufficient conditions for optimality. Local minimizers of the action functional are divided into two categories and subsequently classified, with local uniqueness results obtained in both cases.
We consider the problem of computing safety regions, modeled as nonconvex backward reachable sets, for a nonlinear car collision avoidance model with time-dependent obstacles. The Hamilton-Jacobi-Bellman framework is used. A new formulation of level set functions for obstacle avoidance is given and sufficient conditions for granting the obstacle avoidance on the whole time interval are obtained, even though the conditions are checked only at discrete times. Different scenarios including various road configurations, different geometry of vehicle and obstacles, as well as fixed or moving obstacl…
This text presents the proofs of the technical facts underlying theoretical justification of the convergence and performance of the novel algorithm for reactive navigation of differential drive wheeled robots in dynamic uncertain environments. The algorithm restricts neither the natures nor the motions of the obstacles, they need not be rigid but conversely may deform. It does not consume data about the velocities, shapes, sizes, or orientations of the obstacles, and does not need a map of the environment or recognition of individual obstacles. The only information about the scene is the curre…
Optimal Control of Incompressible Ideal Flows with Obstacle Avoidance
IE School of Science and Technology, IE University（IE科学与技术学院，IE大学）；Department of Mathematics, University of Michigan（密歇根大学数学系）；Centre for Automation and Robotics (CSIC-UPM)（自动化与机器人中心（西班牙国家研究委员会-马德里理工大学））
The standard Dynamic Programming (DP) formulation can be used to solve Multi-Stage Optimization Problems (MSOP's) with additively separable objective functions. In this paper we consider a larger class of MSOP's with monotonically backward separable objective functions; additively separable functions being a special case of monotonically backward separable functions. We propose a necessary and sufficient condition, utilizing a generalization of Bellman's equation, for a solution of a MSOP, with a monotonically backward separable cost function, to be optimal. Moreover, we show that this propose…
In this paper, we propose a new control design scheme for solving the obstacle avoidance problem for nonlinear driftless control-affine systems. The class of systems under consideration satisfies controllability conditions with iterated Lie brackets up to the second order. The time-varying control strategy is defined explicitly in terms of the gradient of a potential function. It is shown that the limit behavior of the closed-loop system is characterized by the set of critical points of the potential function. The proposed control design method can be used under rather general assumptions on p…
