Ravi Banavar

Matthew Hampsey, Pieter van Goor, Ravi Banavar et al.

Underwater vehicles are naturally modelled as rigid bodies on SE(3) subjected to added mass effects. The passivity of the Hamiltonian structure of the system can be exploited to design energy-based stabilising controllers, however, the extension of these control designs to tracking control is not trivial since the error system for the classical error formulations is not itself Hamiltonian. In this paper, we show that a novel choice of error function leads to error dynamics that are Hamiltonian. We go on to derive an energy-based tracking control for a fully coupled model of a submersible vehicle. Asymptotic convergence of the control scheme is proved and the control is demonstrated in a simulation study of the Blue Robotics BlueROV2 Heavy submersible.

Mishal Assif, Ravi Banavar, Anthony Bloch et al.

In this article we introduce a variational approach to collision avoidance of multiple agents evolving on a Riemannian manifold and derive necessary conditions for extremals. The problem consists of finding non-intersecting trajectories of a given number of agents, 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 among the agents. The results are validated through numerical experiments on the manifolds $\mathbb{R}^{2}$ and $S^2$.

Karmvir Singh Phogat, Debasish Chatterjee, Ravi Banavar

This article solves an optimal control problem arising in attitude control of a spacecraft under state and control constraints. We first derive the discrete-time attitude dynamics by employing discrete mechanics. The orientation transfer, with initial and final values of the orientation and momentum and the time duration being specified, is posed as an energy optimal control problem in discrete-time subject to momentum and control constraints. Using variational analysis directly on the Lie group SO(3), we derive first order necessary conditions for optimality that leads to a constrained two point boundary value problem. This two point boundary value problem is solved via a novel multiple shooting technique that employs a root finding Newton algorithm. Robustness of the multiple shooting technique is demonstrated through a few representative numerical experiments.

Shruti Kotpalliwar, Pradyumna Paruchuri, Karmvir Singh Phogat et al.

In this article we present a geometric discrete-time Pontryagin maximum principle (PMP) on matrix Lie groups that incorporates frequency constraints on the controls in addition to pointwise constraints on the states and control actions directly at the stage of the problem formulation. This PMP gives first order necessary conditions for optimality, and leads to two-point boundary value problems that may be solved by shooting techniques to arrive at optimal trajectories. We validate our theoretical results with a numerical experiment on the attitude control of a spacecraft on the Lie group SO(3).

Sudin Kadam, Ravi Banavar

We analyse weak and strong controllability notions for the locomotion of the 3-link Purcell's swimmer, the simplest possible swimmer at low Reynolds number from a geometric framework. After revisiting a purely kinematic form of the equations, we apply an extension of Chow's theorem to analyze controllability in the strong and weak sense. Further, the connection form for the symmetric version of the Purcell's swimmer is derived, based on which, the controllability analysis utilizing the Abelian nature of the structure group is presented. The novelty in our approach is the usage of geometry and the principal fiber bundle structure of the configuration manifold of the system to arrive at strong and weak controllability notions.

Sudin Kadam, Sneha Gajbhiye, Ravi Banavar

This paper presents a geometric framework for analysis of dynamics of flapping wing micro aerial vehicles (FWMAV) which achieve locomotion in the special Euclidean group SE(3) using internal shape changes. We review the special structure of the configuration manifold of such systems. This work addresses to extend the work in geometric locomotion to the aerial locomotion problem. Furthermore, there seems to be limited work in modelling of flapping wing bodies in a geometric framework. We derive the dynamic model of the FWMAV using Lagrangian reduction theory defined on symmetry groups. The reduction is achieved by applying Hamilton's variation principle on a reduced Lagrangian. The resultant dynamics is governed by the Euler-Poincare and Euler-Lagrange equations.

Karmvir Singh Phogat, Ravi Banavar, Debasish Chatterjee

The Wheeled Inverted Pendulum (WIP) is a nonholonomic, underactuated mechanical system, and has been popularized commercially as the {\it Segway}. Designing optimal control laws for point-to-point state-transfer for this autonomous mechanical system, while respecting momentum and torque constraints as well as the underlying manifold, continues to pose challenging problems. In this article we present a successful effort in this direction: We employ geometric mechanics to obtain a discrete-time model of the system, followed by the synthesis of an energy-optimal control based on a discrete-time maximum principle applicable to mechanical systems whose configuration manifold is a Lie group. Moreover, we incorporate state and momentum constraints into the discrete-time control directly at the synthesis stage. The control is implemented on a WIP with parameters obtained from an existing prototype; the results are highly encouraging, as demonstrated by numerical experiments.

Karmvir Singh Phogat, Debasish Chatterjee, Ravi Banavar

In this article we derive a Pontryagin maximum principle (PMP) for discrete-time optimal control problems on matrix Lie groups. The PMP provides first order necessary conditions for optimality; these necessary conditions typically yield two point boundary value problems, and these boundary value problems can then solved to extract optimal control trajectories. Constrained optimal control problems for mechanical systems, in general, can only be solved numerically, and this motivates the need to derive discrete-time models that are accurate and preserve the non-flat manifold structures of the underlying continuous-time controlled systems. The PMPs for discrete-time systems evolving on Euclidean spaces are not readily applicable to discrete-time models evolving on non-flat manifolds. In this article we bridge this lacuna and establish a discrete-time PMP on matrix Lie groups. Our discrete-time models are derived via discrete mechanics, (a structure preserving discretization scheme,) leading to the preservation of the underlying manifold over time, thereby resulting in greater numerical accuracy of our technique. This PMP caters to a class of constrained optimal control problems that includes point-wise state and control action constraints, and encompasses a large class of control problems that arise in various field of engineering and the applied sciences.

Mishal Assif P K, Debasish Chatterjee, Ravi Banavar

We establish a geometric Pontryagin maximum principle for discrete time optimal control problems on finite dimensional smooth manifolds under the following three types of constraints: a) constraints on the states pointwise in time, b) constraints on the control actions pointwise in time, c) constraints on the frequency spectrum of the optimal control trajectories. Our proof follows, in spirit, the path to establish geometric versions of the Pontryagin maximum principle on smooth manifolds indicated in [Cha11] in the context of continuous-time optimal control.

Tejaswi K. C., Srikant Sukumar, Ravi Banavar

The notion of feedback integrators permits Euclidean integration schemes for dynamical systems evolving on manifolds. Here, a constructive Lyapunov function for the attitude dynamics embedded in an ambient Euclidean space has been proposed. We then combine the notion of feedback integrators with the proposed Lyapunov function to obtain a feedback law for the attitude control system. The combination of the two techniques yields a domain of attraction for the closed loop dynamics, where earlier contributions were based on linearization ideas. Further, the analysis and synthesis of the feedback scheme is carried out entirely in Euclidean space. The proposed scheme is also shown to be robust to numerical errors.

Tejas Kotwal, Roshail Gerard, Ravi Banavar

In a series of papers, D. E. Chang, et al., proved and experimentally demonstrated a phenomenon they termed "damping-induced self-recovery". However, these papers left a few questions concerning the observed phenomenon unanswered - in particular, the effect of the intervening lubricant-fluid and its viscosity on the recovery, the abrupt change in behaviour with the introduction of damping, a description of the energy dynamics, and the curious occurrence of overshoots and oscillations and its dependence on the control law. In this paper we attempt to answer these questions through theory. In particular, we derive an expression for the infinite-dimensional fluid-stool-wheel system, that approximates its dynamics to that of the better understood finite-dimensional case.

Arkadeep Saha, Pieter van Goor, Ravi Banavar

In Landmark-Inertial Simultaneous Localisation and Mapping (LI-SLAM), the positions of landmarks in the environment and the robot's pose relative to these landmarks are estimated using landmark position measurements, and measurements from the Inertial Measurement Unit (IMU). However, the robot and landmark positions in the inertial frame, and the yaw of the robot, are not observable in LI-SLAM. This paper proposes a nonlinear observer for LI-SLAM that overcomes the observability constraints with the addition of intermittent GNSS position and magnetometer measurements. The full-state error dynamics of the proposed observer is shown to be both almost-globally asymptotically stable and locally exponentially stable, and this is validated using simulations.

Pranav Gupta, Ravi Banavar, Anastasia Bizyaeva

Local bifurcation analysis plays a central role in understanding qualitative transitions in networked nonlinear dynamical systems, including dynamic neural network and opinion dynamics models. In this article we establish explicit bounds of validity for the classification of bifurcation diagrams in two classes of continuous-time networked dynamical systems, analogous in structure to the Hopfield and the Firing Rate dynamic neural network models. Our approach leverages recent advances in computing the bounds for the validity of Lyapunov-Schmidt reduction, a reduction method widely employed in nonlinear systems analysis. Using these bounds we rigorously characterize neighbourhoods around bifurcation points where predictions from reduced-order bifurcation equations remain reliable. We further demonstrate how these bounds can be applied to an illustrative family of nonlinear opinion dynamics on k-regular graphs, which emerges as a special case of the general framework. These results provide new analytical tools for quantifying the robustness of bifurcation phenomena in dynamics over networked systems and highlight the interplay between network structure and nonlinear dynamical behaviour.

Shreyas Bharadwaj, Bamdev Mishra, Cyrus Mostajeran et al.

This paper discusses robustness guarantees for online tracking of time-varying subspaces from noisy data. Building on recent work in optimization over a Grassmannian manifold, we introduce a new approach for robust subspace tracking by modeling data uncertainty in a Grassmannian ball. The robust subspace tracking problem is cast into a min-max optimization framework, for which we derive a closed-form solution for the worst-case subspace, enabling a geometric robustness adjustment that is both analytically tractable and computationally efficient, unlike iterative convex relaxations. The resulting algorithm, GeRoST (Geometrically Robust Subspace Tracking), is validated on two case studies: tracking a linear time-varying system and online foreground-background separation in video.

Mishal Assif P K, Debasish Chatterjee, Ravi Banavar

We treat the so-called scenario approach, a popular probabilistic approximation method for robust minmax optimization problems via independent and indentically distributed (i.i.d) sampling from the uncertainty set, from various perspectives. The scenario approach is well-studied in the important case of convex robust optimization problems, and here we examine how the phenomenon of concentration of measures affects the i.i.d sampling aspect of the scenario approach in high dimensions and its relation with the optimal values. Moreover, we perform a detailed study of both the asymptotic behaviour (consistency) and finite time behaviour of the scenario approach in the more general setting of nonconvex minmax optimization problems. In the direction of the asymptotic behaviour of the scenario approach, we present an obstruction to consistency that arises when the decision set is noncompact. In the direction of finite sample guarantees, we establish a general methodology for extracting `probably approximately correct' type estimates for the finite sample behaviour of the scenario approach for a large class of nonconvex problems.

Yashaswini Murthy, Ravi Banavar

The need to develop models to predict the motion of microrobots, or robots of a much smaller scale, moving in fluids in a low Reynolds number regime, and in particular, in non Newtonian fluids, cannot be understated. The article develops a Lagrangian based model for one such mechanism - a two-link mechanism termed a microscallop, moving in a low Reynolds number environment in a non Newtonian fluid. The modelling proceeds through the conventional Lagrangian construction for a two-link mechanism and then goes on to model the external fluid forces using empirically based models for viscosity to complete the dynamic model. The derived model is then simulated for different initial conditions and key parameters of the non Newtonian fluid, and the results are corroborated with a few existing experimental results on a similar mechanism under identical conditions. Lastly, with a view to implementing control algorithms we explore accessibility of the system at certain configurations.

Sudin Kadam, Kedar Joshi, Naman Gupta et al.

Locomotion at low Reynolds numbers is a topic of growing interest, spurred by its various engineering and medical applications. This paper presents a novel prototype and a locomotion algorithm for the 3-link planar Purcell's swimmer based on Lie algebraic notions. The kinematic model based on Cox theory of the prototype swimmer is a driftless control-affine system. Using the existing strong controllability and related results, the existence of motion primitives is initially shown. The Lie algebra of the control vector fields is then used to synthesize control profiles to generate motions along the basis of the Lie algebra associated with the structure group of the system. An open loop control system with vision-based positioning is successfully implemented which allows tracking any given continuous trajectory of the position and orientation of the swimmer's base link. Alongside, the paper also provides a theoretical interpretation of the symmetry arguments presented in the existing literature to generate the control profiles of the swimmer.

Aradhana Nayak, Ravi Banavar

In this article, we propose a control law for almost-global asymptotic tracking (AGAT) of a smooth reference trajectory for a fully actuated simple mechanical system (SMS) evolving on a Riemannian manifold which can be embedded in a Euclidean space. The existing results on tracking for an SMS are either local, or almost-global, only in the case the manifold is a Lie group. In the latter case, the notion of a configuration error is naturally defined by the group operation and facilitates a global analysis. However, such a notion is not intrinsic to a Riemannian manifold. In this paper, we define a configuration error followed by error dynamics on a Riemannian manifold, and then prove AGAT. The results are demonstrated for a spherical pendulum which is an SMS on $S^2$ and for a particle moving on a Lissajous curve in $\mathbb{R}^3$.

Sudin Kadam, Ravi Banavar

Micro-robotics at low Reynolds number has been a growing area of research over the past decade. We propose and study a generalized 3-link robotic swimmer inspired by the planar Purcell's swimmer. By incorporating out-of-plane motion of the outer limbs, this mechanism generalizes the planar Purcell's swimmer, which has been widely studied in the literature. Such an evolution of the limbs' motion results in the swimmer's base link evolving in a 3-dimensional space. The swimmer's configuration space admits a trivial principal fiber bundle structure, which along with the slender body theory at the low Reynolds number regime, facilitates in obtaining a principal kinematic form of the equations. We derive a coordinate-free expression for the local form of the kinematic connection. A novel approach for local controllability analysis of this 3-dimensional swimmer in the low Reynolds number regime is presented by employing the controllability results of the planar Purcell's swimmer. This is followed by control synthesis using the motion primitives approach. We prove the existence of motion primitives based control sequence for maneuvering the swimmer's base link whose motion evolves on a Lie group. Using the principal fiber bundle structure, an algorithm for point to point reconfiguration of the swimmer is presented. A set of control sequences for translational and rotational maneuvers is then provided along with numerical simulations.

Sudin Kadam, Ravi Banavar

We present a generalized, 3 dimensional version of the Purcell's swimmer which is a planar mechanism locomoting at low Reynlods number regime. We use Cox theory and resistive force theory to come up with the forces acting on the system. We finally come up with a purely kinematic form of the system's equations.