Piyush Grover

Designing Trajectories in a Planet-Moon Environment Using the Controlled Keplerian Map

The design of fuel-efficient trajectories that visit different moons of a planetary system is best handled by breaking the problem up into multiple three-body problems. This approach, called the patched three-body approach, has received considerable attention in recent years and has proved to lead to substantial fuel savings compared with the traditional patched-conic approach. We consider the problem of designing fuel-efficient multimoon orbiter spacecraft trajectories in the Jupiter―Europa―Ganymede spacecraft system with realistic transfer times. First, fuel- optimal (i.e., near-zero-fuel) trajectories without the use of any control are determined, but turn out to be infeasible due to the very long transfer times involved. We then describe a methodology that exploits the underlying structure of the dynamics of the two three-body problems, that is, the Jupiter―Europa spacecraft and Jupiter―Ganymede spacecraft, using the Hamiltonian structure-preserving Keplerian map approximations derived earlier and small control inputs in the form of instantaneous ΔV to get trajectories with times of flight on the order of months rather than several years. A typical trajectory constructed using the control algorithm can complete the mission in about 10% of the time of flight of an uncontrolled trajectory.

Model-free control framework for multi-limb soft robots

The deformable and continuum nature of soft robots promises versatility and adaptability. However, control of modular, multi-limbed soft robots for terrestrial locomotion is challenging due to the complex robot structure, actuator mechanics and robot-environment interaction. Traditionally, soft robot control is performed by modeling kinematics using exact geometric equations and finite element analysis.

On optimal performance of nonlinear energy sinks in multiple-degree-of-freedom systems

Abstract We study the problem of optimizing the performance of a nonlinear spring–mass–damper attached to a class of multiple-degree-of-freedom systems. We aim to maximize the rate of one-way energy transfer from primary system to the attachment, and focus on impulsive excitation of a two-degree-of-freedom primary system with an essentially nonlinear attachment. The nonlinear attachment is shown to be able to perform as a ‘nonlinear energy sink’ (NES) by taking away energy from the primary system irreversibly for some types of impulsive excitations. Using perturbation analysis and exploiting separation of time scales, we perform dimensionality reduction of this strongly nonlinear system. Our analysis shows that efficient energy transfer to nonlinear attachment in this system occurs for initial conditions close to homoclinic orbit of the slow time-scale undamped system, a phenomenon that has been previously observed for the case of single-degree-of-freedom primary systems. Analytical formulae for optimal parameters for given impulsive excitation input are derived. Generalization of this framework to systems with arbitrary number of degrees-of-freedom of the primary system is also discussed. The performance of both linear and nonlinear optimally tuned attachments is compared. While NES performance is sensitive to magnitude of the initial impulse, our results show that NES performance is more robust than linear tuned mass damper to several parametric perturbations. Hence, our work provides evidence that homoclinic orbits of the underlying Hamiltonian system play a crucial role in efficient nonlinear energy transfers, even in high dimensional systems, and gives new insight into robustness of systems with essential nonlinearity.

Optimal transport over nonlinear systems via infinitesimal generators on graphs

We present a set-oriented graph-based computational framework for continuous-time optimal transport over nonlinear dynamical systems. We recover provably optimal control laws for steering a given initial distribution in phase space to a final distribution in prescribed finite time for the case of non-autonomous nonlinear control-affine systems, while minimizing a quadratic control cost. The resulting control law can be used to obtain approximate feedback laws for individual agents in a swarm control application. Using infinitesimal generators, the optimal control problem is reduced to a modified Monge-Kantorovich optimal transport problem, resulting in a convex Benamou-Brenier type fluid dynamics formulation on a graph. The well-posedness of this problem is shown to be a consequence of the graph being strongly-connected, which in turn is shown to result from controllability of the underlying dynamical system. Using our computational framework, we study optimal transport of distributions where the underlying dynamical systems are chaotic, and non-holonomic. The solutions to the optimal transport problem elucidate the role played by invariant manifolds, lobe-dynamics and almost-invariant sets in efficient transport of distributions in finite time. Our work connects set-oriented operator-theoretic methods in dynamical systems with optimal mass transportation theory, and opens up new directions in design of efficient feedback control strategies for nonlinear multi-agent and swarm systems operating in nonlinear ambient flow fields.

Optimal perturbations for nonlinear systems using graph-based optimal transport

Abstract We formulate and solve a class of finite-time transport and mixing problems in the set-oriented framework. The aim is to obtain optimal discrete-time perturbations in nonlinear dynamical systems to transport a specified initial measure on the phase space to a final measure in finite time. The measure is propagated under system dynamics in between the perturbations via the associated transfer operator. Each perturbation is described by a deterministic map in the measure space that implements a version of Monge–Kantorovich optimal transport with quadratic cost. Hence, the optimal solution minimizes a sum of quadratic costs on phase space transport due to the perturbations applied at specified times. The action of the transport map is approximated by a continuous pseudo-time flow on a graph, resulting in a tractable convex optimization problem. This problem is solved via state-of-the-art solvers to global optimality. We apply this algorithm to a problem of transport between measures supported on two disjoint almost-invariant sets in a chaotic fluid system, and to a finite-time optimal mixing problem by choosing the final measure to be uniform. In both cases, the optimal perturbations are found to exploit the phase space structures, such as lobe dynamics, leading to efficient global transport. As the time-horizon of the problem is increased, the optimal perturbations become increasingly localized. Hence, by combining the transfer operator approach with ideas from the theory of optimal mass transportation, we obtain a discrete-time graph-based algorithm for optimal transport and mixing in nonlinear systems.

Learning to control partial differential equations: Regularized Fitted Q-Iteration approach

This paper formulates a class of partial differential equation (PDE) control problems as a reinforcement learning (RL) problem. We design an RL-based algorithm that directly works with the state of PDE, an infinite dimensional vector, thus allowing us to avoid the model order reduction, commonly used in the conventional PDE controller design approaches. We apply the method to the problem of flow control for time-varying 2D convection-diffusion PDE, as a simplified model for heating, ventilating, air conditioning (HVAC) control design in a room.

Data-driven gain computation in the feedback particle filter

The recently introduced feedback particle filter (FPF) is a control-oriented particle filter (PF), aimed at estimation of nonlinear/non-Gaussian systems. The FPF controls each particle using feedback from the measurements and is resampling free, which is in contrast to conventional PFs based on importance sampling. The control gains are computed by solving boundary value problems. In general, numerical approximations are required and it is an open question how to properly compute the approximate solution. This paper outlines a novel method inspired by high-dimensional data-analysis techniques. Based on the time evolution of the particle cloud, we compute values of the gain function for each particle. We exemplify applicability and highlight the benefits of the approach on a two-body problem.

Efficient estimation and uncertainty quantification in space mission dynamics

The problem of efficient and accurate orbit estimation of space trajectories is discussed. For highly sensitive low-fuel trajectories designed to exploit the complex nonlinear dynamics of the three-body problem , it is vital to have accurate state estimation during maneuvers and ability to deal with irregular observation update times. For instance, in Halo-orbit insertion and station keeping maneuvers, state estimation errors can propagate quickly. In this paper, we combine an efficient probability propagation method with a homotopy-based posterior computation method. The resulting particle filter is highly accurate even in highly nonlinear regime with intermittent observations, and yet an order of magnitude or more efficient than a generic particle filter implementation.

A mean-field game model for homogeneous flocking

Empirically derived continuum models of collective behavior among large populations of dynamic agents are a subject of intense study in several fields, including biology, engineering, and finance. We formulate and study a mean-field game model whose behavior mimics an empirically derived nonlocal homogeneous flocking model for agents with gradient self-propulsion dynamics. The mean-field game framework provides a non-cooperative optimal control description of the behavior of a population of agents in a distributed setting. In this description, each agents state is driven by optimally controlled dynamics that result in a Nash equilibrium between itself and the population. The optimal control is computed by minimizing a cost that depends only on its own state and a mean-field term. The agent distribution in phase space evolves under the optimal feedback control policy. We exploit the low-rank perturbative nature of the nonlocal term in the forward-backward system of equations governing the state and control distributions and provide a closed-loop linear stability analysis demonstrating that our model exhibits bifurcations similar to those found in the empirical model. The present work is a step towards developing a set of tools for systematic analysis, and eventually design, of collective behavior of non-cooperative dynamic agents via an inverse modeling approach.