Ravi N. Banavar

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 $S^2$.

Aradhana Nayak, Ravi N. Banavar

Almost-global orientation trajectory tracking for a rigid body with external actuation has been well studied in the literature, and in the geometric setting as well. The tracking control law relies on the fact that a rigid body is a simple mechanical system (SMS) on the $3-$dimensional group of special orthogonal matrices. However, the problem of designing feedback control laws for tracking using internal actuation mechanisms, like rotors or control moment gyros, has received lesser attention from a geometric point of view. An internally actuated rigid body is not a simple mechanical system, and the phase-space here evolves on the level set of a momentum map. In this note, we propose a novel proportional integral derivative (PID) control law for a rigid body with $3$ internal rotors, that achieves tracking of feasible trajectories from almost all initial conditions.

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.

Sneha Gajbhiye, Ravi N. Banavar, Sergio Delgado

The purpose of this article is to illustrate the role of connections and symmetries in the Wheeled Inverted Pendulum (WIP) mechanism - an underactuated system with rolling constraints - popularized commercially as the Segway, and thereby arrive at a set of simpler dynamical equations that could serve as the starting point for more complex feedback control designs. The first part of the article views the nonholonomic constraints enforced by the rolling assumption as defining an Ehresmann connection on a fiber bundle. The resulting equations are the reduced Euler-Lagrange equations, which are identical to the Lagrange d'Alembert equations of motion. In the second part we explore conserved quantities, in particular, nonholonomic momenta. To do so, we first introduce the notion of a symmetry group, whose action leaves both the Lagrangian and distribution invariant. We examine two symmetry groups - $SE (2)$ and $SE(2) \times \mathbb{S}^{1}$. The first group leads to the purely kinematic case while the second gives rise to nonholonomic momentum equations.

Nidhish Raj, Ravi N. Banavar, Abhishek et al.

We consider the problem of attitude tracking for small-scale aerobatic helicopters. A small scale helicopter has two subsystems: the fuselage, modeled as a rigid body; and the rotor, modeled as a first order system. Due to the coupling between rotor and fuselage, the complete system does not inherit the structure of a simple mechanical system. The coupled rotor fuselage dynamics is first transformed to rigid body attitude tracking problem with a first order actuator dynamics. The proposed controller is developed using geometric and backstepping control technique. The controller is globally defined on $SO(3)$ and is shown to be locally exponentially stable. The controller is validated in simulation and experiment for a 10 kg class small scale flybarless helicopter by demonstrating aggressive roll attitude tracking.

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.

Sneha Gajbhiye, Ravi N. Banavar

This paper presents tracking control laws for two different objectives of a nonholonomic system - a spherical robot - using a geometric approach. The first control law addresses orientation tracking using a modified trace potential function. The second law addresses contact position tracking using a $right$ transport map for the angular velocity error. A special case of this is position and reduced orientation stabilization. Both control laws are coordinate free. The performance of the feedback control laws are demonstrated through simulations.

Sudin Kadam, Karmvir Singh Phogat, Ravi N. Banavar et al.

In this article we present the discrete-time isoholonomic problem of the planar Purcell's swimmer and solve it using the Discrete-time Pontryagin maximum principle. The 3-link Purcell's swimmer is a locomotion system moving in a low Reynolds number environment. The kinematics of the system evolves on a principal fiber bundle. A structure preserving discrete-time kinematic model of the system is obtained in terms of the local form of a discrete connection. An adapted version of the Discrete Maximum Principle on matrix Lie groups is then employed to come up with the necessary optimality conditions for an optimal state transfer while minimizing the control effort. These necessary conditions appear as a two-point boundary value problem and are solved using a numerical technique. Results from numerical experiments are presented to illustrate the algorithm.

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.

Siddharth H. Nair, Ravi N. Banavar

This article develops variational integrators for a class of underactuated mechanical systems using the theory of discrete mechanics. Further, a discrete optimal control problem is formulated for the considered class of systems and subsequently solved using variational principles again, to obtain necessary conditions that characterize optimal trajectories. The proposed approach is demonstrated on benchmark underactuated systems and accompanied by numerical simulations.

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.

Siddharth H. Nair, Ravi N. Banavar, D. H. S. Maithripala

This article studies the dynamics and control of a novel underactuated system, wherein a plate suspended by cables and with a freely moving mass on top, whose other ends are attached to three quadrotors, is sought to be horizontally stabilized at a certain height, with the ball positioned at the center of mass of the plate. The freely moving mass introduces a 2-degree of underactuation into the system. The design proceeds through a decoupling of the quadrotors and the plate dynamics. Through a partial feedback linearization approach, the attitude of the plate and the translational height of the plate is initially controlled, while maintaining a bounded velocity along the $y$ and $x$ directions. These inputs are then synthesized through the quadrotors with a backstepping and timescale separation argument based on Tikhonov's theorem.

Aradhana Nayak, Ravi N. Banavar

Tracking the unactuated configuration variable in an underactuated system, in a global sense, has not received much attention. Here we present a scheme to do so for a pendubot - a two link robot actuated only at the first link. We propose a control law for almost-global asymptotic tracking (AGAT) of a smooth reference trajectory for the unactuated second joint of the pendubot. The control law achieves almost-global tracking for any smooth reference trajectory specified for the unactuated joint. Further, we generalize the proposed scheme to an n-link system with as many (or more) degrees of actuation than unactuation, and show that the result holds.

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.

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.

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.

Sudin Kadam, Ravi N. Banavar

The mechanism of swimming at very low Reynolds number conditions is a topic of interest to biologists and engineering community. We develop a novel kinematic model of a slender flexible swimmer which locomotes in a low Reynolds number regime. In contrast to existing techniques that model such systems as a connected set of straight, rigid links, the novelty of our technique stems from the fact that we model the swimmer with two components - one is a straight, rigid body (the head) and the other is a flexible member (the tail). Using Cox theory we model the gradient of the forces as a function of the instantaneous shape of the swimmer and its velocity. By virtue of the low inertia conditions, an expression for the translational and rotational velocity of the head is obtained for the planar motion in the form of a Lie algebra of the Special Euclidean group. We explain the principal fiber bundle structure of the configuration space of the swimmer and use that to show a weak controllability result for a type of slender flexible swimmer where the shape space is the space of all continuous curves of a given length. A set of simulation results is presented showing the variation of the swimmer head velocity for a bump function moving along the swimmer length.

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.

Klaus Albert, Karmvir Singh Phogat, Felix Anhalt et al.

The Wheeled Inverted Pendulum (WIP) is an underactuated, nonholonomic mechatronic system, and has been popularized commercially as the Segway. Designing a control law for motion planning, that incorporates the state and control constraints, while respecting the configuration manifold, is a challenging problem. In this article we derive a discrete-time model of the WIP system using discrete mechanics and generate optimal trajectories for the WIP system by solving a discrete-time constrained optimal control problem. Further, we describe a nonlinear continuous-time model with parameters for designing a closed loop LQ-controller. A dual control architecture is implemented in which the designed optimal trajectory is then provided as a reference to the robot with the optimal control trajectory as a feedforward control action, and an LQ-controller in the feedback mode is employed to mitigate noise and disturbances for ensuing stable motion of the WIP system. While performing experiments on the WIP system involving aggressive maneuvers with fairly sharp turns, we found a high degree of congruence in the designed optimal trajectories and the path traced by the robot while tracking these trajectories. This corroborates the validity of the nonlinear model and the control scheme. Finally, these experiments demonstrate the highly nonlinear nature of the WIP system and robustness of the control scheme.

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 $\mathbb{R}^3$.

Ravi N Banavar, Arjun Narayanan

The gimbal-spacecraft system, that consists of a variable speed control moment gyro (VSCMG) mounted inside a spacecraft, has been employed as an actuator for the attitude control of a spacecraft and has been much studied in the aerospace control community. Employing a Newtonian approach, the equations of motion are derived, and further study focusses on singularity issues and control law synthesis. While the geometric mechanics community has studied many mechanical systems of engineering interest, including spinning rotors (or momentum wheels) that are used as actuators, there has not been a particular effort to model and control the gimbal-spacecraft system in a geometric framework. This article serves two purposes: it presents the gimbal-spacecraft system in a geometric mechanics framework, and in particular, highlights the connection form, that could form the basis for future control design, and secondly, the exposition is of a tutorial nature whereby the willing reader, with minimal prerequisites, is introduced to the tools of differential geometry in this context.

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.