Author

# Sneha Gajbhiye

Other affiliations: University of Macau

Bio: Sneha Gajbhiye is an academic researcher from Indian Institute of Technology Bombay. The author has contributed to research in topic(s): Nonholonomic system & Equations of motion. The author has an hindex of 5, co-authored 10 publication(s) receiving 71 citation(s). Previous affiliations of Sneha Gajbhiye include University of Macau.

##### Papers

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TL;DR: In this article, the dynamic model of a spherical robot with rolling constraints is derived using Lagrangian reduction theory defined on symmetry groups, which is achieved by applying Hamilton's variation principle on a reduced Lagrangians and then imposing the nonholonomic constraints.

Abstract: Mechanical systems with rolling constraints form a class of nonholonomic systems. In this paper we derive the dynamic model of a spherical robot, which has been designed and realized in our laboratory, using Lagrangian reduction theory defined on symmetry groups. The reduction is achieved by applying Hamilton's variation principle on a reduced Lagrangian and then imposing the nonholonomic constraints. The equations of motion are in the Euler-Poincare form and are equivalent to those obtained using Lagrange-d'Alembert's principle.

20 citations

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TL;DR: In this article, the authors present the modeling and local equilibrium controllability analysis of a spherical robot that is internally actuated by a pendulum mechanism, and derive the dynamic model of the system using Lagrangian reduction and the variational principle.

Abstract: Summary
In this paper, we present the modeling and local equilibrium controllability analysis of a spherical robot. The robot consists of a spherical shell that is internally actuated by a pendulum mechanism. The rolling motion of the sphere manifests itself as a nonholonomic constraint in the modeling. We derive the dynamic model of the system using Lagrangian reduction and the variational principle. We first compute the Lagrangian and identify the symmetry with respect to a group action. The system Lagrangian and the rolling constraint are invariant with respect to the group isotropy and hence permit a reduced dynamic formulation termed as the nonholonomic ‘Euler-Poincare’ equation with advected dynamics. Using Lie brackets and symmetric products of the potential and control vector fields, local configuration accessibility and local (fiber) equilibrium controllability are presented. Copyright © 2015 John Wiley & Sons, Ltd.

14 citations

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TL;DR: In this article, tracking control laws for two different objectives of a nonholonomic system -a spherical robot - using a geometric approach are presented for orientation and contact position tracking using a right transport map for the angular velocity error.

Abstract: 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.

10 citations

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TL;DR: In this paper, the Lagrange-d'Alembert principle was used to derive the equations of motion for the wheeled inverted pendulum subject to nonholonomic constraints.

Abstract: This paper develops the equations of motion in the reduced space for the wheeled inverted pendulum, which is an underactuated mechanical system subject to nonholonomic constraints. The equations are derived from the Lagrange-d'Alembert principle using variations consistent with the constraints. The equations are first derived in the shape space, and then, a coordinate transformation is performed to get the equations of motion in more suitable coordinates for the purpose of control.

9 citations

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TL;DR: This paper presents tracking control laws for two different objectives of a nonholonomic system - a spherical robot - using a geometric approach that addresses orientation tracking using a modified trace potential function and contact positiontracking using a transport map for the angular velocity error.

Abstract: 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.

5 citations

##### Cited by

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TL;DR: In this paper, free and controlled motions of a spherical robot of combined type moving by displacing the center of mass and by changing the internal gyrostatic momentum are studied. But the authors focus on the control of the robot in the absence of control actions.

Abstract: This paper is concerned with free and controlled motions of a spherical robot of combined type moving by displacing the center of mass and by changing the internal gyrostatic momentum. Equations of motion for the nonholonomic model are obtained and their first integrals are found. Fixed points of the reduced system are found in the absence of control actions. It is shown that they correspond to the motion of the spherical robot in a straight line and in a circle. A control algorithm for the motion of the spherical robot along an arbitrary trajectory is presented. A set of elementary maneuvers (gaits) is obtained which allow one to transfer the spherical robot from any initial point to any end point.

44 citations

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TL;DR: A novel contribution of this note is a smooth, asymptotically stabilizing geometric control law for position and reduced attitude, which corresponds to an equilibrium submanifold of dimension one, which is the best possible dimension of a smoothly stabilized equilibrium sub manifold.

Abstract: Geometric control of a spherical robot rolling on a horizontal plane with three independent inertia disc actuators is considered in this note. The dynamic model of the spherical robot in the geometric framework is used to establish the strong accessibility and small-time local controllability properties. Smooth stabilizability to an equilibrium fails for the nonholonomic spherical robot. A novel contribution of this note is a smooth, asymptotically stabilizing geometric control law for position and reduced attitude, which corresponds to an equilibrium submanifold of dimension one. From Brockett's condition, this is the best possible dimension of a smoothly stabilized equilibrium submanifold. We also present a novel smooth global tracking controller for tracking position trajectories.

31 citations

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TL;DR: In this paper, a model of a controlled spherical robot with an axisymmetric pendulum-type actuator with a feedback system suppressing the pendulum's oscillations at the final stage of motion was developed.

Abstract: In this paper, we develop a model of a controlled spherical robot with an axisymmetric pendulum-type actuator with a feedback system suppressing the pendulum’s oscillations at the final stage of motion. According to the proposed approach, the feedback depends on phase variables (the current position and velocities) and does not depend on the type of trajectory. We present integrals of motion and partial solutions, analyze their stability, and give examples of computer simulation of motion using feedback to illustrate compensation of the pendulum’s oscillations.

24 citations

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TL;DR: This paper uses the method of Controlled Lagrangians for the stabilization of an equilibrium characterized by the length of the driven path, the orientation, and the pitch angle, and designs a speed control law based on the results.

Abstract: The paper deals with the energy-based stabilization and speed control of a wheeled inverted pendulum, which is an underactuated, unstable mechanical system subject to nonholonomic constraints. We use the method of Controlled Lagrangians for the stabilization of an equilibrium characterized by the length of the driven path, the orientation, and the pitch angle. The approach is systematic and very intuitive, for it is physically motivated. Based on the stabilization results, we design a speed control law. After the presentation of the model under nonholonomic constraints in Lagrangian representation, we provide an elegant solution to the matching equations for kinetic and potential energy shaping for the considered system. Simulations show the applicability of the method, and the comparison with a linear controller emphasizes its performance.

21 citations

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TL;DR: The simulation results show the significant performance of the proposed NMPC in stabilization of the spherical shell from every initial configuration to every desired position and orientation even in the uncontrollable region.

Abstract: Control of nonholonomic spherical mobile robot is a generalization of the classical ball-plate problem which is still challenging in robotic researches. In this paper, point stabilization of a nonholonomic spherical mobile robot actuated by two internal rotors is investigated. Since every kinematic trajectory is not always dynamically realizable for the spherical robot driven by two actuators, the mathematical model of the robot is derived based on the angular momentum conservation principle. The controllability of the robot is evaluated based on the obtained model and the uncontrollable configurations as well as their geometrical meaning are specified. To simultaneous control of position and orientation of the robot, a nonlinear model predictive control (NMPC) is developed for the first time and the stability analysis is performed through using Lyapunov stability theorem. The performance of the designed control system is assessed through computer simulations in different test conditions. The simulation results show the significant performance of the proposed NMPC in stabilization of the spherical shell from every initial configuration to every desired position and orientation even in the uncontrollable region. Considering additive bounded noises, the robust stabilization of the nonholonomic spherical robot by the NMPC is also assessed in simulations. A spherical mobile robot driven by two perpendicular rotors is investigated.The equations of motion are derived using angular momentum conservation principle.The controllability of the model is analyzed and its uncontrollable configurations are specified.A NMPC is designed to control the position and orientation of the spherical shell simultaneously.The stability of designed controller is proven and its performance is evaluated.

20 citations