Angular Velocity Of Rigid Body Research Articles

An Algorithm for Rigid-Body Angular Velocity and Attitude Estimation Based on Isotropic Accelerometer Strapdowns

A novel algorithm for the estimation of rigid-body angular velocity and attitude—the most challenging part of pose-and-twist estimation—based on isotropic accelerometer strapdowns, is proposed in this paper. Quaternions, which employ four parameters for attitude representation, provide a compact description without the drawbacks brought about by other representations, for example, the gimbal lock of Euler angles. Within the framework of quaternions for rigid-body angular velocity and attitude estimation, the proposed methodology automatically preserves the unit norm of the quaternion, thus improving the accuracy and efficiency of the estimation. By virtue of the inherent nature of isotropic accelerometer strapdowns, the centripetal acceleration is filtered out, leaving only its tangential counterpart, to be estimated and updated. Meanwhile, using the proposed integration algorithm, the angular velocity and the quaternion, which are dependent only on the tangential acceleration, are calculated and updated at appropriate sampled instants for high accuracy. This strategy, which brings about robustness, allows for relatively large time-step sizes, low memory demands, and low computational complexity. The proposed algorithm is tested by simulation examples of the angular velocity and attitude estimation of a free-rotating brick and the end-effector of an industrial robot. The simulation results showcase the algorithm with low errors, as estimated based on energy conservation, and high-order rate of convergence, as compared with other algorithms in the literature.

Global attitude estimation using single delayed vector measurement and biased gyro

Time delay in attitude determination systems is unavoidable and it is usually caused by low-quality data sampling, poor sensor synchronization, momentary sensor outage or operational restrictions of sensors. Despite its performance-degrading effect, time delay has been widely neglected in the existing attitude estimation methods. This paper presents a general framework for design of attitude estimators with and without SO(3) manifold restriction by assuming that measurement of a single vector measurement is available with time delay. Also, measurement of rigid-body angular velocity is considered available with an unknown bias. The proposed estimator guarantees global and asymptotic convergence of attitude estimate to its true value. The observer gain is calculated by solving a delay-dependent matrix differential equation. Solvability of the differential equation depends on an observability condition related to the reference vectors. To illustrate performance of the proposed observer, we present simulation of a satellite system where the single delayed vector measurement is provided by a magnetometer.

Rigid Body Attitude Control With Delayed Attitude Measurement

Time delay in the evaluation of the attitude of a rigid body can significantly hinder the performance of the attitude control system. In this brief, we propose a dynamically smooth control law that guarantees asymptotic convergence of the rigid body attitude and angular velocity to their desired values in the presence of delayed attitude measurement. We assume that the moment-of-inertia matrix of the body is unknown. The control law includes a matrix gain that is computed by solving a delay-dependent and time-varying matrix differential equation. It is shown that the solvability of the matrix differential equation depends on a uniform controllability condition, which can be verified a priori and based on available information on reference vectors. Finally, the simulation example illustrates the performance of the proposed controller in the case of a satellite system and in comparison with another existing controller.

Rigid body attitude estimation based on the Lagrange–d’Alembert principle

Estimation of rigid body attitude and angular velocity without any knowledge of the attitude dynamics model is treated using the Lagrange–d’Alembert principle from variational mechanics. It is shown that Wahba’s cost function for attitude determination from two or more non-collinear vector measurements can be generalized and represented as a Morse function of the attitude estimation error on the Lie group of rigid body rotations. With body-fixed sensor measurements of direction vectors and angular velocity, a Lagrangian is obtained as the difference between a kinetic energy-like term that is quadratic in the angular velocity estimation error and an artificial potential obtained from Wahba’s function. An additional dissipation term that depends on the angular velocity estimation error is introduced, and the Lagrange–d’Alembert principle is applied to the Lagrangian with this dissipation. A Lyapunov analysis shows that the state estimation scheme so obtained provides stable asymptotic convergence of state estimates to actual states in the absence of measurement noise, with an almost global domain of attraction. These estimation schemes are discretized for computer implementation using discrete variational mechanics. A first order Lie group variational integrator is obtained as a discrete-time implementation. In the presence of bounded measurement noise, numerical simulations show that the estimated states converge to a bounded neighborhood of the actual states.

New exact solution of Euler’s equations (rigid body dynamics) in the case of rotation over the fixed point

A new exact solution of Euler’s equations (rigid body dynamics) is presented here. All the components of angular velocity of rigid body for such a solution differ from both the cases of symmetric rigid rotor (which has two equal moments of inertia: Lagrange’s or Kovalevskaya’s case), and from the Euler’s case when all the applied torques are zero, or from other well-known particular cases. The key features are the next: the center of mass of rigid body is assumed to be located at meridional plane along the main principal axis of inertia of rigid body, besides, the principal moments of inertia are assumed to satisfy to a simple algebraic equality. Also, there is a restriction at choosing of initial conditions. Such a solution is also proved to satisfy to Euler–Poinsot equations, including invariants of motion and additional Euler’s invariant (square of the vector of angular momentum is a constant). So, such a solution is a generalization of Euler’s case.

A Nonlinear Program for Angular-Velocity Estimation From Centripetal-Acceleration Measurements

Measuring the trajectory of a rigid body in space is commonly done using an inertial measurement unit composed of one triaxial accelerometer and one triaxial gyroscope. When the rigid body undergoes high accelerations, it is often preferable to resort to an array of accelerometers rather than the traditional accelerometer-gyroscope combination, an approach that is now common in crashworthiness and other biomechanics applications. In this paper, we present an algorithm for the estimation of the rigid-body angular velocity from the centripetal components of the accelerations measured by the array of accelerometers. The proposed algorithm and the others available in the literature were benchmarked using the accelerometer array octahedral constellation of twelve accelerometers. In the reported testing conditions, the proposed method is slightly more robust than any other based on centripetal-acceleration measurements.

Estimating the angular velocity of a rigid body moving in the plane from tangential and centripetal acceleration measurements

Two methods are available for the estimation of the angular velocity of a rigid body from point-acceleration measurements: (i) the time-integration of the angular acceleration and (ii) the square-rooting of the centripetal acceleration. The inaccuracy of the first method is due mainly to the accumulation of the error on the angular acceleration throughout the time-integration process, which does not prevent that it be used successfully in crash tests with dummies, since these experiments never last more than one second. On the other hand, the error resulting from the second method is stable through time, but becomes inaccurate whenever the rigid body angular velocity approaches zero, which occurs in many applications. In order to take advantage of the complementarity of these two methods, a fusion of their estimates is proposed. To this end, the accelerometer measurements are modeled as exact signals contaminated with bias errors and Gaussian white noise. The relations between the variables at stake are written in the form of a nonlinear state-space system in which the angular velocity and the angular acceleration are state variables. Consequently, a minimum-variance-error estimate of the state vector is obtained by means of extended Kalman filtering. The performance of the proposed estimation method is assessed by means of simulation. Apparently, the resulting estimation method is more robust than the existing accelerometer-only methods and competitive with gyroscope measurements. Moreover, it allows the identification and the compensation of any bias error in the accelerometer measurements, which is a significant advantage over gyroscopes.

Optimal Control of Rigid Body Rotation Around Center of Mass

The problem of axisymmetric rigid body angular velocity vector maneuvering in the body-fixed frame is considered. Control actuators are provided by two pairs of reactive jets with constant torque directions, and with possibly unequal jet characteristics. The transfer time from the initial to endpoint in the body-fixed frame of angular velocities is not specified. Both the initial and endpoint are arbitrary and given in advance. The problem is solved in a closed analytical form, using sufficient conditions of optimality.

Energy-Based Stabilization of Angular Velocity of Rigid Body in Failure Configuration

Energy-Based Stabilization of Angular Velocity of Rigid Body in Failure Configuration

EFFECTS OF STRUCTURAL DEFORMATIONS OF THE CRANK-SLIDER MECHANISM ON THE ESTIMATION OF THE INSTANTANEOUS ENGINE FRICTION TORQUE

The focus on the current study is to assess the effects of structural deformations of the crankshaft/connecting-rod/piston mechanism on the computation of the instantaneous engine friction torque. This study is performed in a fully controlled environment in order to isolate the effects of structural deformations from those of measurement errors or noise interference. Therefore, a detailed model, accounting for the rigid and flexible motions of the crank-slider mechanism and including engine component friction formulations, is considered in this study. The model is used as a test bed to generate the engine friction torque,Tfa, and to predict the rigid and flexible motions of the system in response to the cylinder gas pressure. The torsional vibrations and the rigid body angular velocity of the crankshaft, as predicted by the detailed model of the crank-slider mechanism, are used along with the engine load torque and the cylinder gas pressure in the (P–ω) method to estimate the engine friction torque,Tfe. This method is well suited for the purpose of this study because its formulation is based on the rigid body model of the crank-slider mechanism. The digital simulation results demonstrate that the exclusion of the structural deformations of the crank-slider mechanism from the formulation of the (P–ω) method leads to an overestimation of the engine friction torque near the top-dead-center (TDC) position of the piston under firing conditions. Moreover, for the remainder of the engine cycle, the estimated friction torque exhibits large oscillations and takes on positive numerical values as if it is inducing energy into the system. Thus, the adverse effects of structural deformations of the crank-slider mechanism on the estimation of the engine friction torque greatly differ in their nature from one phase of the engine cycle to another.

Rigid body rotation evolution due to a disturbing torque which is known in a body frame

This paper deals with the problem of rigid body rotation evolution under the action of a torque which depends on the projections of rigid body angular velocity onto the body frame only. This is the so calledrigid body with self-excitation. If the magnitude of the torque is arbitrary, the solution to the problem is known in the simplest cases only. The problem with small self-excitation is considered in the paper. This problem is of great importance from the standpoint of a number of applications in spacecraft and rigid body dynamics. Perhaps the most important example is spinstabilized spacecraft rotation evolution due to an internal disturbing torque. In the case of a small torque, it is possible to use perturbation methods to obtain complete analytical and geometrical descriptions of the rotation. In this paper, the common approach to investigating the problem of rigid body rotation under the action of a small torque known in the body frame is described. Using this approach, two problems are solved: the problem of maintaining the angular velocity of a gyroscope using a control torque quadratic in the angular velocity (the problem of Grammel for the case of a small torque), and the problem of asymmetric spacecraft rotation evolution with an electric propulsion thruster.

Optimal Control of Rigid Body Angular Velocity with Quadratic Cost

In this paper we consider the problem of obtaining optimal controllers which minimize a quadratic cost function for the rotational motion of a rigid body. We are not concerned with the attitude of the body and we consider only the evolution of the angular velocity as described by Euler's equations. We obtain conditions which guarantee the existence of linear stabilizing optimal and suboptimal controllers. These controllers have a very simple structure.

Optimal Control of the Rotation of an Axisymmetric Rigid Body: The General Case

The problem of an axisymmetric rigid body angular velocity vector maneuvering in the body-fixed frame is considered. Control actuators are provided by two pairs of reactive jets with constant directions of their torques. Characteristics of these jets could be unequal. The time of the process of a transfer from the initial to the end point in the body-fixed frame of angular velocities is not prescribed. Both initial and end points could be arbitrary but given in advance. The problem is solved in the closed analytical form. Sufficient conditions of optimality are used.

Decentralized sliding mode control for flexible link robots

A technique using augmented sliding mode control for robust, real-time control of flexible multiple link robots is presented. For the purpose of controller design, the n-link, n-joint robot is subdivided into n single joint, single link subsystems. A sliding surface for each subsystem is specified so as to be globally, asymptotically stable. Each sliding surface contains rigid-body angular velocity, angular displacement and flexible body generalized velocities. The flexible body generalized accelerations are treated as disturbances during the controller design. This has the advantage of not requiring explicit equations for the flexible body motion. The result is n single input, single output controllers acting at the n joints of the robot, controlling rigid body angular displacement and providing damping for flexible body modes. Furthermore, the n controllers can be operated in parallel so that compute speed is independent of the number of links, affording real-time, robust, control.

Nutational stability and passive control of spinning rockets with internal mass motion

Necessary and sufficient conditions for asymptotic stability are developed for a symmetric spinning rocket with axial forcing and dissipative internal mass motion. Results show that spin about the axis of least inertia is asymptotically stable provided that three conditions are satisfied: 1) the axial thrust magnitude is sufficiently large, 2) the restoring forces for the internal mass motion are sufficiently large, and 3) the internal mass motion occurs aft of the system mass center. Under these same conditions, spin about the axis of largest inertia may be unstable. Numerical studies show that a passive control device with these characteristics can be used to attenuate the coning of prolate spinners such as those transfer orbit boost vehicles that have been plagued by unexpectedly pronounced coning instabilities. Nomenclature c = damping constant for particle motion F = magnitude of thrust (assumed constant) /i, /i, 73 = moments of inertia of (symmetric) rigid body for axes #1, #2, and 53, respectively k = spring constant for elastic restraint on particle M = mass of rigid body m = mass of particle li = m/(m -f M) Q = nominal spin rate of rigid body about #3 0)1,0)2 = components of rigid-body angular velocity for axes B and B2, respectively