Efstathios Bakolas

Ahmed Khalil, Mohamed Safwat, Efstathios Bakolas

This work proposes a novel Alternating Direction Method of Multipliers (ADMM)-based Ensemble Kalman Inversion (EKI) algorithm for solving constrained nonlinear model predictive control (NMPC) problems. First, stage-wise nonlinear inequality constraints in the NMPC problem are embedded via an augmented Lagrangian with nonnegative slack variables. We then show that the resulting unconstrained augmented-Lagrangian primal subproblem admits a Bayesian interpretation: under independent Gaussian virtual observations, its minimizers coincide with MAP estimators, enabling solution via EKI. However, since the nonnegativity constraint on the slacks is a hard constraint not naturally encoded by a Gaussian model, our proposed algorithm yields a two-block ADMM scheme that alternates between (i) an inexact primal step that minimizes the augmented-Lagrangian objective (implemented via EKI rollouts), (ii) a nonnegativity projection for the slacks, and (iii) a dual ascent step. To balance exploration and convergence, an annealing schedule tempers sampling covariances while a penalty schedule increases constraint enforcement over outer iterations, encouraging global search early and precise constraint satisfaction later. We evaluate the proposed controller on a 6-DOF UR5e manipulation benchmark in MuJoCo, comparing it against DIAL-MPC (an iterative MPPI variant) as the arm traverses a cluttered tabletop environment.

Augustinos D. Saravanos, Isin M. Balci, Arshiya Taj Abdul et al.

This paper studies the problem of steering large-scale multi-agent stochastic linear systems between Gaussian distributions under probabilistic collision avoidance constraints. We introduce a family of \textit{distributed covariance steering (DCS)} methods based on the Alternating Direction Method of Multipliers (ADMM), each offering different trade-offs between conservatism and computational efficiency. The first method, Full-Covariance-Consensus (FCC)-DCS, enforces consensus over both the means and covariances of neighboring agents, yielding the least conservative safe solutions. The second approach, Partial-Covariance-Consensus (PCC)-DCS, leverages the insight that safety can be maintained by exchanging only partial covariance information, reducing computational demands. The third method, Mean-Consensus (MC)-DCS, provides the most scalable alternative by requiring consensus only on mean states. Furthermore, we establish novel convergence guarantees for distributed ADMM with iteratively linearized non-convex constraints, covering a broad class of consensus optimization problems. This analysis proves convergence to stationary points for PCC-DCS and MC-DCS, while the convergence of FCC-DCS follows from standard ADMM theory. Simulations in 2D and 3D multi-agent environments verify safety, illustrate the trade-offs between methods, and demonstrate scalability to thousands of agents.

Srinath Tankasala, Can Pehlivanturk, Efstathios Bakolas et al.

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.

Vrushabh Zinage, Efstathios Bakolas

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.

Steven Patrick, Efstathios Bakolas

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.

Andrei Marchidan, Efstathios Bakolas

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.

Isin M. Balci, Abhishek Halder, Efstathios Bakolas

In this work, we analyze the properties of the solution to the covariance steering problem for discrete time Gaussian linear systems with a squared Wasserstein distance terminal cost. In our previous work, we have shown that by utilizing the state feedback control policy parametrization, this stochastic optimal control problem can be associated with a difference of convex functions program. Here, we revisit the same covariance control problem but this time we focus on the analysis of the problem. Specifically, we establish the existence of solutions to the optimization problem and derive the first and second order conditions for optimality. We provide analytic expressions for the gradient and the Hessian of the performance index by utilizing specialized tools from matrix calculus. Subsequently, we prove that the optimization problem always admits a global minimizer, and finally, we provide a sufficient condition for the performance index to be a strictly convex function (under the latter condition, the problem admits a unique global minimizer). In particular, we show that when the terminal state covariance is upper bounded, with respect to the Löwner partial order, by the covariance matrix of the desired terminal normal distribution, then our problem admits a unique global minimizing state feedback gain. The results of this paper set the stage for the development of specialized control design tools that exploit the structure of the solution to the covariance steering problem with a squared Wasserstein distance terminal cost.

Jaemin Lee, Seung Hyeon Bang, Efstathios Bakolas et al.

This paper proposes an MPC-based controller to efficiently execute multiple hierarchical tasks for underactuated and constrained robotic systems. Existing task-space controllers or whole-body controllers solve instantaneous optimization problems given task trajectories and the robot plant dynamics. However, the task-space control method we propose here relies on the prediction of future state trajectories and the corresponding costs-to-go terms over a finite time-horizon for computing control commands. We employ acceleration energy error as the performance index for the optimization problem and extend it over the finite-time horizon of our MPC. Our approach employs quadratically constrained quadratic programming, which includes quadratic constraints to handle multiple hierarchical tasks, and is computationally more efficient than nonlinear MPC-based approaches that rely on nonlinear programming. We validate our approach using numerical simulations of a new type of robot manipulator system, which contains underactuated and constrained mechanical structures.

Jaemin Lee, Efstathios Bakolas, Luis Sentis

In this work, we propose a trajectory generation method for robotic systems with contact force constraint based on optimal control and reachability analysis. Normally, the dynamics and constraints of the contact-constrained robot are nonlinear and coupled to each other. Instead of linearizing the model and constraints, we directly solve the optimal control problem to obtain the feasible state trajectory and the control input of the system. A tractable optimal control problem is formulated which is addressed by dual approaches, which are sampling-based dynamic programming and rigorous reachability analysis. The sampling-based method and Partially Observable Markov Decision Process (POMDP) are used to break down the end-to-end trajectory generation problem via sample-wise optimization in terms of given conditions. The result generates sequential pairs of subregions to be passed to reach the final goal. The reachability analysis ensures that we will find at least one trajectory starting from a given initial state and going through a sequence of subregions. The distinctive contributions of our method are to enable handling the intricate contact constraint coupled with system's dynamics due to the reduction of computational complexity of the algorithm. We validate our method using extensive numerical simulations with a legged robot.

Jaemin Lee, Efstathios Bakolas, Luis Sentis

This paper presents a trajectory generation method for contact-constrained robotic systems such as manipulators and legged robots. Contact-constrained systems are affected by the interaction forces between the robot and the environment. In turn, these forces determine and constrain state reachability of the robot parts or end effectors. Our study subdivides the trajectory generation problem and the supporting reachability analysis into tractable subproblems consisting of a sampling problem, a convex optimization problem, and a nonlinear programming problem. Our method leads to significant reduction of computational cost. The proposed approach is validated using a realistic simulated contact-constrained robotic system.