Ravi N. Banavar

Total Energy Shaping Control of Mechanical Systems: Simplifying the Matching Equations Via Coordinate Changes

Total energy shaping is a controller design methodology that achieves (asymptotic) stabilization of mechanical systems endowing the closed-loop system with a Lagrangian or Hamiltonian structure with a desired energy function - that qualifies as Lyapunov function for the desired equilibrium. The success of the method relies on the possibility of solving two PDEs which identify the kinetic and potential energy functions that can be assigned to the closed loop. Particularly troublesome is the partial differential equation (PDE) associated to the kinetic energy which is nonlinear and inhomogeneous and the solution, that defines the desired inertia matrix, must be positive-definite. In this note, we prove that we can eliminate or simplify the forcing term in this PDE by modifying the target dynamics and introducing a change of coordinates in the original system. Furthermore, it is shown that, in the particular case of transformation to the Lagrangian coordinates, the possibility of simplifying the PDEs is determined by the interaction between the Coriolis and centrifugal forces and the actuation structure. The examples of pendulum on a cart and Furutas pendulum are used to illustrate the results.

Motion analysis of a spherical mobile robot

A path planning algorithm for a spherical mobile robot rolling on a plane is presented in this paper. The robot is actuated by two internal rotors that are fixed to the shafts of two motors. These are in turn mounted on the spherical shell in mutually orthogonal directions. The system is nonholonomic due to the nonintegrable nature of the rolling constraints. Further, the system cannot be converted into a chained form, and neither is it nilpotent nor differentially flat. So existing techniques of nonholonomic path planning cannot be applied directly to the system. The approach presented here uses simple geometrical notions and provides numerically efficient and intuitive solutions. We also present the dynamic model and derive motor torques for execution of the algorithm. Along the proposed paths, we achieve dynamic decoupling of the variables making the algorithm more suitable for practical applications.

An LQG/H/sub /spl infin// controller for a flexible manipulator

An LQG/H/sub /spl infin// controller synthesis is presented for a flexible one-link manipulator with noncolocated sensing. The challenge of the problem lies in the nonminimum phase characteristics and the ill-conditioned nature of the plant. The inner-loop LQG controller provides adequate damping to the flexible modes and numerically conditions the system while the outer-loop H/sub /spl infin// controller provides stability in the face of unstructured perturbation. With only tip position sensing and using a fourth-order model, a controller is synthesized. It is tested with an eighth-order model of the plant and is found to satisfy both performance and stability requirements.

Controllability and point-to-point control of 3-DOF planar horizontal underactuated manipulators

This paper presents the controllability properties of an underactuated 3R manipulator and the application of a discontinuous point-to-point control law for a three-degree-of-freedom (DOF) PPR manipulator. In the first half we present a controllability analysis at various actuator configurations of the manipulator. A novel contribution is that we use a single co-ordinate system to analyse the controllability at all the three possible configurations. In the second half we present a discontinuous feedback control law for point-to-point control of a 3 DOF PPR underactuated manipulator moving in a horizontal plane. Simulation results are presented that demonstrate the effectiveness of the controller.

Risk-sensitive filters for recursive estimation of motion from images

In this paper, an extended risk-sensitive filter (ERSF) is used to estimate the motion parameters of an object recursively from a sequence of monocular images. The effect of varying the risk factor /spl theta/ on the estimation error is examined. The performance of the filter is compared with the extended Kalman filter (EKF) and the theoretical Cramer-Rao lower bound. When the risk factor /spl theta/ and the uncertainty in the measurement noise are large, the initial estimation error of the ERSF is less than that of the corresponding EKF The ERSF is also found to converge to the steady state value of the error faster than the EKF. In situations when the uncertainty in the initial estimate is large and the EKF diverges, the ERSF converges with small errors. In confirmation with the theory, as /spl theta/ tends to zero, the behavior of the ERSF is the same as that of the EKF.

PROPORTIONAL NAVIGATION THROUGH PREDICTIVE CONTROL

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Almost global attitude stabilization of a rigid body for both internal and external actuation schemes

Recent developments in the attitude control of a rigid body include the development of almost globally stabilizing coordinate-free control laws. However, all of these results focus on external actuation. In this paper, we observe that the frequently used way of stabilization by error functions can also be given a Hamiltonian interpretation. We show that there exists a class of control laws which are applicable for both external and internal actuation. We also construct such control laws, motivated by the potential energy of a spinning top.

Extremum seeking loops with quadratic functions: estimation and control

Extremum-seeking (also peak-seeking ) controllers are designed to operate at an a priori unknown set-point that extremizes the value of a performance function. Traditional approaches to the problem assume a time-scale separation between the gradient computation and function minimization and the system dynamics. The work here, in contrast, assumes that the performance function can be approximated by a quadratic function with a finite number of parameters. These parameters are estimated on-line and the extremum seeking controller operates based on these estimated values. A significant advantage of a quadratic function is that it allows the peak-seeking control loop to be reduced to a linear system. For such a loop, the wealth of linear system analysis and synthesis tools can be employed. First, the control loop is analysed assuming that the parameters in the function are known (full information case) and then when the parameters are estimated on line (the partial information case).

Nonlinear control synthesis for asymptotic stabilization of the swing equation using a controllable series capacitor via Immersion and Invariance

A nonlinear control law is proposed to asymptotically stabilize a single machine infinite bus (SMIB) system based on immersion and invariance (I&I) control strategy. The actuator used is a controllable series capacitor (CSC). The SMIB system is described using the nonlinear second order swing equation model, and the CSC is modeled using a first order system. The control objective here is to approximate the complete third order system with a second order dynamics, for which we have an asymptotically stabilizing control law.

A mixed norm performance measure for the design of multirate filterbanks

A mixed norm performance measure is presented to design the synthesis filters of a multirate filterbank. The mixed norm performance measure is based on the energy as well as the peak value of the error signal. Mathematically, the performance measure minimizes the l/sub 2/-norm of the error signal subject to the l/sub /spl infin//-norm of the error being bounded by some positive value v (this imposes a bound on the peak value of the error signal). The design problem is shown to be that of a mixed /spl Hscr//sub 2///spl Hscr//sub /spl infin// optimization problem. The theory of linear matrix inequalities (LMIs) offers a tractable solution to such multiobjective synthesis problems. The synthesis filters designed with the new performance measure are compared with those obtained by similar induced norm minimization techniques in terms of degree of reconstruction, order of the synthesis filters, SNR, and aliasing distortion.