Group Of Rigid Body Rotations Research Articles

Geometric Integral Attitude Control on SO(3)

This article proposes a novel integral geometric control attitude tracking scheme, utilizing a coordinate-free representation of attitude on the Lie group of rigid body rotations, SO(3). This scheme exhibits almost global asymptotic stability in tracking a reference attitude profile. The stability and robustness properties of this integral tracking control scheme are shown using Lyapunov stability analysis. A numerical simulation study, utilizing a Lie Group Variational Integrator (LGVI), verifies the stability of this tracking control scheme, as well as its robustness to a disturbance torque. In addition, a numerical comparison study shows the effectiveness of the proposed geometric integral term, when compared to other state-of-the-art attitude controllers. In addition, software-in-the-loop (SITL) simulations show the advantages of utilizing the proposed attitude controller in PX4 autopilot compared to using PX4’s original attitude controller.

Almost global finite-time stabilization of rigid body attitude dynamics using rotation matrices

Summary This work considers continuous finite-time stabilization of rigid body attitude dynamics using a coordinate-free representation of attitude on the Lie group of rigid body rotations in three dimensions, SO(3). Using a Holder continuous Morse–Lyapunov function, a finite-time feedback stabilization scheme for rigid body attitude motion to a desired attitude with continuous state feedback is obtained. Attitude feedback control with finite-time convergence has been considered in the past using the unit quaternion representation. However, it is known that the unit quaternion representation of attitude is ambiguous, with two antipodal unit quaternions representing a single rigid body attitude. Continuous feedback control using unit quaternions may therefore lead to the unstable unwinding phenomenon if this ambiguity is not resolved in the control design, and this has adverse effects on actuators, settling time, and control effort expended. The feedback control law designed here leads to almost global finite-time stabilization of the attitude motion of a rigid body with Holder continuous feedback to the desired attitude. As a result, this control scheme avoids chattering in the presence of measurement noise, does not excite unmodeled high-frequency structural dynamics, and can be implemented with actuators that can only provide continuous control inputs. Numerical simulation results for a spacecraft in low Earth orbit, obtained using a Lie group variational integrator, confirm the theoretically obtained stability and robustness properties of this attitude feedback stabilization scheme. Copyright © 2015 John Wiley & Sons, Ltd.

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.

DYNAMIC MODELLING, ESTIMATION, AND CONTROL FOR PRECISION POINTING OF AN ATMOSPHERIC BALLOON PLATFORM

In this paper we consider the dynamic modelling, estimation and control of an atmospheric balloon platform. The platform is modelled as a rigid body constrained to move with a three dimensional pendulum. We investigate the dynamics of the system and derive the equations of motion from first principles. A nonlinear estimator that evolves on the special orthogonal group of rigid-body rotations, denoted SO(3), is implemented and used in conjunction with a proportional derivative (PD) compensator to control the yaw angle of the platform. A simulation is conducted, and the results demonstrate successful estimation and yaw control.

Cartesian stiffness for wrist joints: analysis on the Lie group of 3D rotations and geometric approximation for experimental evaluation

This paper is concerned with the analysis and the numerical evaluation from experimental measurements of the static, Cartesian stiffness of wrist joints, in particular the human wrist. The primary aim is to extend from Euclidean spaces to so(3), the group of rigid body rotations, previous methods for assessing the end-point stiffness of the human arm, typically performed via a robotic manipulandum. As a first step, the geometric definition of Cartesian stiffness from current literature is specialised to the group so(3). Emphasis is placed on the choice of the unique, natural, affine connection on so(3) which guarantees symmetry of the stiffness matrix in presence of conservative fields for any configuration, also out of equilibrium. As the main contribution of this study, a coordinate-independent approximation based on the geometric notion of geodesics is proposed which provides a working equation for evaluating stiffness directly from experimental measurements. Finally, a graphical representation of the stiffness is discussed which extends the ellipse method often used for end-point stiffness visualisation and which is suitable to compare stiffness matrices evaluated at different configurations.

Adaptive Identification on the Group of Rigid-Body Rotations and its Application to Underwater Vehicle Navigation

This paper reports a novel stable adaptive identifier on the group of rigid-body rotations, and its application to a sensor-calibration problem arising in underwater vehicle navigation. The problem addressed is the identification of an unknown rigid-body rotation map from input-output data. General least-squares (LS) and adaptive identification techniques are commonly employed to identify general linear maps from input-output data, but do not guarantee that the resulting identified map is a rigid-body rotation. At present, an LS singular value decomposition approach is the standard method for identification constrained to the group of rigid-body rotations. This paper reports the first exact adaptive identifier on the group of rigid-body rotations, together with a proof of asymptotic stability. Techniques for navigating underwater vehicles are reviewed, and the Doppler-alignment calibration problem is posed. The reported adaptive identifier is employed to solve this problem, and performance of this adaptive identifier is evaluated on both laboratory and field experimental data. The results reported herein compare favorably with results obtained via previously reported LS techniques. The methodology reported herein may be of broader interest because of its applicability to more general problems in the identification, dynamics, and control on the group of rigid-body motions