Terms Of Exponential Coordinates Research Articles

Integrated 6-DOF Orbit-Attitude Dynamical Modeling and Control Using Geometric Mechanics

The integrated 6-DOF orbit-attitude dynamical modeling and control have shown great importance in various missions, for example, formation flying and proximity operations. The integrated approach yields better performances than the separate one in terms of accuracy, efficiency, and agility. One challenge in the integrated approach is to find a unified representation for the 6-DOF motion with configuration space SE(3). Recently, exponential coordinates of SE(3) have been used in dynamics and control of the 6-DOF motion, however, only on the kinematical level. In this paper, we will improve the current method by adopting exponential coordinates on the dynamical level, by giving the relation between the second-order derivative of exponential coordinates and spacecraft’s accelerations. In this way, the 6-DOF motion in terms of exponential coordinates can be written as a second-order system with a quite compact form, to which a broader range of control theories, such as higher-order sliding modes, can be applied. For a demonstration purpose, a simple asymptotic tracking control law with almost global convergence is designed. Finally, the integrated modeling and control are applied to the body-fixed hovering over an asteroid and verified by a simulation, in which absolute motions of the spacecraft and asteroid are simulated separately.

Adaptive sliding mode control for spacecraft body-fixed hovering in the proximity of an asteroid

An adaptive sliding mode control scheme for autonomous body-fixed hovering maneuvers of a rigid spacecraft in the proximity of an asteroid involving parameter uncertainties and time-varying external disturbances is proposed in the framework of geometric mechanics. The kinematics and dynamics models of the six-degree-of-freedom relative motion of the spacecraft about the asteroid are described using the Lie group SE(3), which is the set of positions and orientations of the rigid spacecraft in three-dimensional Euclidean space. The relative configuration (position and orientation) between the spacecraft and asteroid is described in terms of exponential coordinates on the Lie group of rigid body motions. An adaptive sliding mode control scheme on the Lie group SE(3), where the bounds of parametric uncertainties and disturbances are not required in advance, is proposed to guarantee the asymptotic stability of states for position and attitude stabilization in the presence of parametric uncertainties and external disturbances. Detailed design principles and a rigorous stability analyses are provided. Numerical simulation results demonstrate the effectiveness of the proposed control scheme for autonomous body-fixed hovering over an asteroid with relatively low control inputs.

Finite-time control for spacecraft body-fixed hovering over an asteroid

A finite-time control scheme for autonomous body-fixed hovering of a rigid spacecraft over a tumbling asteroid is presented. The relative configuration between the spacecraft and asteroid is described in terms of exponential coordinates on the Lie group SE(3), which is the configuration space for the spacecraft. With a Lyapunov stability analysis, the finite-time convergence of the proposed control scheme for the closed-loop system is proved. Numerical simulations validate the performance of the proposed control scheme.

Spacecraft Coupled Tracking Maneuver Using Sliding Mode Control with Input Saturation

AbstractThis paper presents a sliding mode control scheme on for autonomous spacecraft formation flying via a virtual leader state trajectory. The configuration space for a spacecraft is the Lie group SE(3), which is the set of positions and orientations of the rigid spacecraft in three-dimensional (3D) Euclidean space. A virtual leader trajectory, in the form of natural attitude and translational (orbital) motion of a satellite, is generated offline. Each spacecraft tracks a desired relative configuration with respect to the virtual leader in a decentralized and autonomous manner to achieve the desired formation. The relative configuration between each spacecraft and the virtual leader is described in terms of exponential coordinates on SE(3). A new feedback control scheme is proposed for coupled translational and rotational maneuver using a new sliding surface, which is defined as the exponential coordinates and velocity tracking errors such that the sliding surface converges to zero without explicit re...

Almost global asymptotic tracking control for spacecraft body-fixed hovering over an asteroid

Almost global asymptotic tracking control for autonomous body-fixed hovering of a rigid spacecraft over an asteroid is proposed in the framework of geometric mechanics. The configuration space for the spacecraft is the Lie group SE(3), which is the set of positions and orientations of the rigid spacecraft in three-dimensional Euclidean space. The relative motion with respect to the spacecraft is assumed to be available through the spacecraft on-board navigation. The spacecraft tracks a desired relative configuration with respect to an asteroid in an autonomous manner. The relative configuration between the spacecraft and the asteroid is described in terms of exponential coordinates on the Lie group of rigid body motions. A continuous-time feedback tracking control using these exponential coordinates and the relative velocities is presented to perform coupled translational and rotational maneuver over an asteroid in the presence of control force saturation. A Lyapunov analysis guarantees that the spacecraft asymptotically converges to the desired trajectory. Numerical simulation results demonstrate the asymptotic tracking control achieve autonomous body-fixed hovering over a selected asteroid.

Asymptotic Tracking Control for Spacecraft Formation Flying with Decentralized Collision Avoidance

This paper presents a tracking control scheme for spacecraft formation flying with a decentralized collision-avoidance scheme, using a virtual leader state trajectory. The configuration space for a spacecraft is the Lie group , which is the set of positions and orientations in three-dimensional Euclidean space. A virtual leader trajectory, in the form of attitude and orbital motion of a virtual satellite, is generated offline. Each spacecraft tracks a desired relative configuration with respect to the virtual leader in an autonomous manner, to achieve the desired formation. The relative configuration between a spacecraft and the virtual leader is described in terms of exponential coordinates on . A continuous-time feedback tracking control scheme is designed using these exponential coordinates and the relative velocities. A Lyapunov analysis guarantees that the spacecraft asymptotically converge to their desired state trajectories. This tracking control scheme is combined with a decentralized collision-avoidance control scheme generated from artificial potentials for each spacecraft, which includes information of relative positions of other spacecraft within communications range. Asymptotic convergence to the desired trajectory with this combined control law is demonstrated using a Lyapunov analysis. Numerical simulation results verify the successful application of this tracking control scheme to a formation maneuver with decentralized collision avoidance.

Kinematic and dynamic modeling of spherical joints using exponential coordinates

Traditional Euler angle-based methods for the kinematic and dynamic modeling of spherical joints involve highly complicated formulas that are numerically sensitive, with complex bookkeeping near local coordinate singularities. In this regard, exponential coordinates are known to possess several advantages over Euler angle representations. This paper presents several new exponential coordinate-based formulas and computational procedures that are particularly useful in the modeling of mechanisms containing spherical joints. Computationally robust procedures are derived for evaluating the forward and inverse formulas for the angular velocity and angular acceleration in terms of exponential coordinates. We show that these formulas simplify the parametrization of joint range limits for spherical joints, and lead to more compact equations in the forward and inverse dynamic analysis of mechanisms containing spherical joints.