arXiv subjects

Efstathios Bakolas

Publications and source records attributed to Efstathios Bakolas.

At least 37 records · Page 2Linked to original sources

arXiv2023

Disturbance Observer-based Robust Integral Control Barrier Functions for Nonlinear Systems with High Relative Degree

In this paper, we consider the problem of safe control synthesis of general controlled nonlinear systems in the presence of bounded additive disturbances. Towards this aim, we first construct a governing augmented state space model consisting of the equations of motion of the original system, the integral control law and the nonlinear disturbance observer. Next, we propose the concept of Disturbance Observer based Integral Control Barrier Functions (DO-ICBFs) which we utilize to synthesize safe control inputs. The characterization of the safe controller is obtained after modifying the governing integral control law with an additive auxiliary control input which is computed via the solution of a quadratic problem. In contrast to prior methods in the relevant literature which can be unnecessarily cautious due to their reliance on the worst case disturbance estimates, our DO-ICBF based controller uses the available control effort frugally by leveraging the disturbance estimates computed by the disturbance observer. By construction, the proposed DO-ICBF based controller can ensure state and input constraint satisfaction at all times. Further, we propose Higher Order DO-ICBFs that extend our proposed method to nonlinear systems with higher relative degree with respect to the auxiliary control input. Finally, numerical simulations are provided to validate our proposed approach.

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arXiv2023

Two-Player Reconnaissance Game with Half-Planar Target and Retreat Regions

This paper is concerned with the reconnaissance game that involves two mobile agents: the Intruder and the Defender. The Intruder is tasked to reconnoiter a territory of interest (target region) and then return to a safe zone (retreat region), where the two regions are disjoint half-planes, while being chased by the faster Defender. This paper focuses on the scenario where the Defender is not guaranteed to capture the Intruder before the latter agent reaches the retreat region. The goal of the Intruder is to minimize its distance to the target region, whereas the Defender's goal is to maximize the same distance. The game is decomposed into two phases based on the Intruder's myopic goal. The complete solution of the game corresponding to each phase, namely the Value function and state-feedback equilibrium strategies, is developed in closed-form using differential game methods. Numerical simulation results are presented to showcase the efficacy of our solutions.

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arXiv2022

Covariance Steering of Discrete-Time Linear Systems with Mixed Multiplicative and Additive Noise

In this paper, we study the covariance steering (CS) problem for discrete-time linear systems subject to multiplicative and additive noise. Specifically, we consider two variants of the so-called CS problem. The goal of the first problem, which is called the exact CS problem, is to steer the mean and the covariance of the state process to their desired values in finite time. In the second one, which is called the ``relaxed'' CS problem, the covariance assignment constraint is relaxed into a positive semi-definite constraint. We show that the relaxed CS problem can be cast as an equivalent convex semi-definite program (SDP) after applying suitable variable transformations and constraint relaxations. Furthermore, we propose a two-step solution procedure for the exact CS problem based on the relaxed problem formulation which returns a feasible solution, if there exists one. Finally, results from numerical experiments are provided to show the efficacy of the proposed solution methods.

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arXiv2022

Koopman Operator Based Modeling and Control of Rigid Body Motion Represented by Dual Quaternions

In this paper, we systematically derive a finite set of Koopman based observables to construct a lifted linear state space model that describes the rigid body dynamics based on the dual quaternion representation. In general, the Koopman operator is a linear infinite dimensional operator, which means that the derived linear state space model of the rigid body dynamics will be infinite-dimensional, which is not suitable for modeling and control design purposes. Recently, finite approximations of the operator computed by means of methods like the Extended Dynamic Mode Decomposition (EDMD) have shown promising results for different classes of problems. However, without using an appropriate set of observables in the EDMD approach, there can be no guarantees that the computed approximation of the nonlinear dynamics is sufficiently accurate. The major challenge in using the Koopman operator for constructing a linear state space model is the choice of observables. State-of-the-art methods in the field compute the approximations of the observables by using neural networks, standard radial basis functions (RBFs), polynomials or heuristic approximations of these functions. However, these observables might not providea sufficiently accurate approximation or representation of the dynamics. In contrast, we first show the pointwise convergence of the derived observable functions to zero, thereby allowing us to choose a finite set of these observables. Next, we use the derived observables in EDMD to compute the lifted linear state and input matrices for the rigid body dynamics. Finally, we show that an LQR type (linear) controller, which is designed based on the truncated linear state space model, can steer the rigid body to a desired state while its performance is commensurate with that of a nonlinear controller. The efficacy of our approach is demonstrated through numerical simulations.

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arXiv2022

Neural Koopman Control Barrier Functions for Safety-Critical Control of Unknown Nonlinear Systems

We consider the problem of synthesis of safe controllers for nonlinear systems with unknown dynamics using Control Barrier Functions (CBF). We utilize Koopman operator theory (KOT) to associate the (unknown) nonlinear system with a higher dimensional bilinear system and propose a data-driven learning framework that uses a learner and a falsifier to simultaneously learn the Koopman operator based bilinear system and a corresponding CBF. We prove that the learned CBF for the latter bilinear system is also a valid CBF for the unknown nonlinear system by characterizing the $\ell^2$-norm error bound between these two systems. We show that this error can be partially tuned by using the Lipschitz constant of the Koopman based observables. The CBF is then used to formulate a quadratic program to compute inputs that guarantee safety of the unknown nonlinear system. Numerical simulations are presented to validate our approach.

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arXiv2022

Vector Field-based Collision Avoidance for Moving Obstacles with Time-Varying Elliptical Shape

This paper presents an algorithm for local motion planning in environments populated by moving elliptical obstacles whose velocity, shape and size are fully known but may change with time. We base the algorithm on a collision avoidance vector field (CAVF) that aims to steer an agent to a desired final state whose motion is described by a double integrator kinematic model. In addition to handling multiple obstacles, the method is applicable in bounded environments for more realistic applications (e.g., motion planning inside a building). We also incorporate a method to deal with agents whose control input is limited so that they safely navigate around the obstacles. To showcase our approach, extensive simulations results are presented in 2D and 3D scenarios.

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arXiv2022

Neural Koopman Lyapunov Control

Learning and synthesizing stabilizing controllers for unknown nonlinear control systems is a challenging problem for real-world and industrial applications. Koopman operator theory allows one to analyze nonlinear systems through the lens of linear systems and nonlinear control systems through the lens of bilinear control systems. The key idea of these methods lies in the transformation of the coordinates of the nonlinear system into the Koopman observables, which are coordinates that allow the representation of the original system (control system) as a higher dimensional linear (bilinear control) system. However, for nonlinear control systems, the bilinear control model obtained by applying Koopman operator based learning methods is not necessarily stabilizable. Simultaneous identification of stabilizable lifted bilinear control systems as well as the associated Koopman observables is still an open problem. In this paper, we propose a framework to construct these stabilizable bilinear models and identify its associated observables from data by simultaneously learning a bilinear Koopman embedding for the underlying unknown control affine nonlinear system as well as a Control Lyapunov Function (CLF) for the Koopman based bilinear model using a learner and falsifier. Our proposed approach thereby provides provable guarantees of asymptotic stability for the Koopman based representation of the unknown control affine nonlinear control system as a bilinear system. Numerical simulations are provided to validate the efficacy of our proposed class of stabilizing feedback controllers for unknown control-affine nonlinear systems.

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arXiv2022

Exact SDP Formulation for Discrete-Time Covariance Steering with Wasserstein Terminal Cost

In this paper, we present new results on the covariance steering problem with Wasserstein distance terminal cost. We show that the state history feedback control policy parametrization, which has been used before to solve this class of problems, requires an unnecessarily large number of variables and can be replaced by a randomized state feedback policy which leads to more tractable problem formulations without any performance loss. In particular, we show that under the latter policy, the problem can be equivalently formulated as a semi-definite program (SDP) which is in sharp contrast with our previous results that could only guarantee that the stochastic optimal control problem can be reduced to a difference of convex functions program. Then, we show that the optimal policy that is found by solving the associated SDP corresponds to a deterministic state feedback policy. Finally, we present non-trivial numerical simulations which show the benefits of our proposed randomized state feedback policy derived from the SDP formulation of the problem over existing approaches in the field in terms of computational efficacy and controller performance.

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arXiv2022

Constrained Covariance Steering Based Tube-MPPI

In this paper, we present a new trajectory optimization algorithm for stochastic linear systems which combines Model Predictive Path Integral (MPPI) control with Constrained Covariance Steering (CSS) to achieve high performance with safety guarantees (robustness). Although MPPI can be used to solve complex nonlinear trajectory optimization problems, it may not always handle constraints effectively and its performance may degrade in the presence of unmodeled disturbances. By contrast, CCS can handle probabilistic state and / or input constraints (e.g., chance constraints) and also steer the state covariance of the system to a desired positive definite matrix (control of uncertainty) which both imply that CCS can provide robustness against stochastic disturbances. CCS, however, suffers from scalability issues and cannot handle complex cost functions in general. We argue that the combination of the two methods yields a class of trajectory optimization algorithms that can achieve high performance (a feature of MPPI) while ensuring safety with high probability (a feature of CCS). The efficacy of our algorithm is demonstrated in an obstacle avoidance problem and a circular track path generation problem.

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arXiv2022

Smooth time optimal trajectory generation for drones

In this paper, we address a minimum-time steering problem for a drone modeled as point mass with bounded acceleration, across a set of desired waypoints in the presence of gravity. We first provide a method to solve for the minimum-time control input that will steer the point mass between two waypoints based on a continuous-time problem formulation which we address by using Pontryagin's Minimum Principle. Subsequently, we solve for the time-optimal trajectory across the given set of waypoints by discretizing in the time domain and formulating the minimum-time problem as a nonlinear program (NLP). The velocities at each waypoint obtained from solving the NLP in the discretized domain are then used as boundary conditions to extend our two-point solution across those multiple waypoints. We apply this planning methodology to execute a surveying task that minimizes the time taken to completely explore a target area or volume. Numerical simulations and theoretical analyses of this new planning methodology are presented. The results from our approach are also compared to traditional polynomial trajectories like minimum snap planning.

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arXiv2022

Feedback Strategies for Hypersonic Pursuit of a Ground Evader

In this paper, we present a game-theoretic feedback terminal guidance law for an autonomous, unpowered hypersonic pursuit vehicle that seeks to intercept an evading ground target whose motion is constrained in a one-dimensional space. We formulate this problem as a pursuit-evasion game whose saddle point solution is in general difficult to compute onboard the hypersonic vehicle due to its highly nonlinear dynamics. To overcome this computational complexity, we linearize the nonlinear hypersonic dynamics around a reference trajectory and subsequently utilize feedback control design techniques from Linear Quadratic Differential Games (LQDGs). In our proposed guidance algorithm, the hypersonic vehicle computes its open-loop optimal state and input trajectories off-line and prior to the commencement of the game. These trajectories are then used to linearize the nonlinear equations of hypersonic motion. Subsequently, using this linearized system model, we formulate an auxiliary two-player zero-sum LQDG which is effective in the neighborhood of the given reference trajectory and derive its feedback saddle point strategy that allows the hypersonic vehicle to modify its trajectory online in response to the target's evasive maneuvers. We provide numerical simulations to showcase the performance of our proposed guidance law.

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arXiv2021

Minimum-fuel Spacecraft Rendezvous based on Sparsity Promoting Optimization

In this paper, we consider the classical spacecraft rendezvous problem in which the so-called active spacecraft has to approach the target spacecraft which is moving in an elliptical orbit around a planet by using the minimum possible amount of fuel. Instead of using standard convex optimization tools which can be computationally expensive, we use modified versions of the Iteratively Reweighted Least Squares (IRLS) algorithm from compressive sensing to compute sparse optimal control sequences which minimize the fuel consumption for both thrust vectoring and orthogonal vectoring (active) spacecraft. Numerical simulations are performed to verify the efficacy of our approach.

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arXiv2021

Optimal Strategies for Guarding a Compact and Convex Target Set: A Differential Game Approach

We revisit the two-player planar target-defense game initially posed by Isaacs where a pursuer (or defender) attempts to guard a target set from an attack by an evader (or attacker). This paper builds on existing analytical solutions to games of defending a simple shape of target area to develop a generalized and extended solution to the same game with a compact convex target set with smooth boundary. Isaacs' method is applied to address the game of kind and games of degree. A geometric solution approach is used to find the barrier surface that demarcates the winning sets of the players. A value function coupled with a set of optimal state feedback strategies in each winning set is derived and proven to correspond to the saddle point solution of the game. The proposed solutions are illustrated by means of numerical simulations.

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arXiv2021

Guarding a Target Set from a Single Attacker in the Euclidean Space

This paper addresses a two-player target defense game in the $n$-dimensional Euclidean space where an attacker attempts to enter a closed convex target set while a defender strives to capture the attacker beforehand. We provide a complete and universal differential game-based solution which not only encompasses recent work associated with similar problems whose target sets have simple, low-dimensional geometric shapes, but can also address problems that involve nontrivial geometric shapes of high-dimensional target sets. The value functions of the game are derived in a semi-analytical form that includes a convex optimization problem. When the latter problem has a closed-form solution, one of the value functions is used to analytically construct the barrier surface that divides the state space of the game into the winning sets of players. For the case where the barrier surface has no analytical expression but the target set has a smooth boundary, the bijective map between the target boundary and the projection of the barrier surface is obtained. By using Hamilton-Jacobi-Isaacs equation, we verify that the proposed optimal state feedback strategies always constitute the game's unique saddle point whether or not the optimization problem has a closed-form solution. We illustrate our solutions via numerical simulations.

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arXiv2021

Collision Avoidance Using Spherical Harmonics

In this paper, we propose a novel optimization-based trajectory planner that utilizes spherical harmonics to estimate the collision-free solution space around an agent. The space is estimated using a constrained over-determined least-squares estimator to determine the parameters that define a spherical harmonic approximation at a given time step. Since spherical harmonics produce star-convex shapes, the planner can consider all paths that are in line-of-sight for the agent within a given radius. This contrasts with other state-of-the-art planners that generate trajectories by estimating obstacle boundaries with rough approximations and using heuristic rules to prune a solution space into one that can be easily explored. Those methods cause the trajectory planner to be overly conservative in environments where an agent must get close to obstacles to accomplish a goal. Our method is shown to perform on-par with other path planners and surpass these planners in certain environments. It generates feasible trajectories while still running in real-time and guaranteeing safety when a valid solution exists.

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arXiv2021

Distributed Coverage Control of Multi-Agent Networks with Guaranteed Collision Avoidance in Cluttered Environments

We propose a distributed control algorithm for a multi-agent network whose agents deploy over a cluttered region in accordance with a time-varying coverage density function while avoiding collisions with all obstacles they encounter. Our algorithm is built on a two-level characterization of the network. The first level treats the multi-agent network as a whole based on the distribution of the locations of its agents over the spatial domain. In the second level, the network is described in terms of the individual positions of its agents. The aim of the multi-agent network is to attain a spatial distribution that resembles that of a reference coverage density function (high-level problem) by means of local (microscopic) interactions of its agents (low-level problem). In addition, as the agents deploy, they must avoid collisions with all the obstacles in the region at all times. Our approach utilizes a modified version of Voronoi tessellations which are comprised of what we refer to as Obstacle-Aware Voronoi Cells (OAVC) in order to enable coverage control while ensuring obstacle avoidance. We consider two control problems. The first problem which we refer to as the high-level coverage control problem corresponds to an interpolation problem in the class of Gaussian mixtures (no collision avoidance requirement), which we solve analytically. The second problem which we refer to as the low-level coverage control problem corresponds to a distributed control problem (collision avoidance requirement is now enforced at all times) which is solved by utilizing Lloyd's algorithm together with the modified Voronoi tessellation (OAVC) and a time-varying coverage density function which corresponds to the solution of the high-level coverage control problem. Finally, simulation results for coverage in a cluttered environment are provided to demonstrate the efficacy of the proposed approach.

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arXiv2021

Greedy Decentralized Auction-based Task Allocation for Multi-Agent Systems

We propose a decentralized auction-based algorithm for the solution of dynamic task allocation problems for spatially distributed multi-agent systems. In our approach, each member of the multi-agent team is assigned to at most one task from a set of spatially distributed tasks, while several agents can be allocated to the same task. The task assignment is dynamic since it is updated at discrete time stages (iterations) to account for the current states of the agents as the latter move towards the tasks assigned to them at the previous stage. Our proposed methods can find applications in problems of resource allocation by intelligent machines such as the delivery of packages by a fleet of unmanned or semi-autonomous aerial vehicles. In our approach, the task allocation accounts for both the cost incurred by the agents for the completion of their assigned tasks (e.g., energy or fuel consumption) and the rewards earned for their completion (which may reflect, for instance, the agents' satisfaction). We propose a Greedy Coalition Auction Algorithm (GCAA) in which the agents possess bid vectors representing their best evaluations of the task utilities. The agents propose bids, deduce an allocation based on their bid vectors and update them after each iteration. The solution estimate of the proposed task allocation algorithm converges after a finite number of iterations which cannot exceed the number of agents. Finally, we use numerical simulations to illustrate the effectiveness of the proposed task allocation algorithm (in terms of performance and computation time) in several scenarios involving multiple agents and tasks distributed over a spatial 2D domain.

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arXiv2021

Collision Avoidance for Unmanned Aerial Vehicles in the Presence of Static and Moving Obstacles

This paper presents a new collision avoidance procedure for unmanned aerial vehicles in the presence of static and moving obstacles. The proposed procedure is based on a new form of local parametrized guidance vector fields, called collision avoidance vector fields, that produce smooth and intuitive maneuvers around obstacles. The maneuvers follow nominal collision-free paths which we refer to as streamlines of the collision avoidance vector fields. In the case of multiple obstacles, the proposed procedure determines a mixed vector field that blends the collision avoidance vector field of each obstacle and assumes its form whenever a pre-defined distance threshold is reached. Then, in accordance to the computed guidance vector fields, different collision avoidance controllers that generate collision-free maneuvers are developed. Furthermore, it is shown that any tracking controller with convergence guarantees can be used with the avoidance controllers to track the streamlines of the collision avoidance vector fields. Finally, numerical simulations demonstrate the efficacy of the proposed approach and its ability to avoid collisions with static and moving pop-up threats in three different practical scenarios.

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