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This article presents an attitude tracking control scheme with Hölder continuity and finite-time stability. The first part of this article discusses and compares the features of first-order multivariable Hölder-continuous systems with coupled-scalar sliding-mode systems. The advantages of Hölder-continuous systems over sliding-mode systems are presented from the perspectives of control continuity and noise robustness. Thereafter, a Hölder-continuous second-order differentiator is presented with its stability and robustness properties. This is followed by its use in an attitude tracking control scheme, which is covered in the second part of the article. The proposed tracking control scheme is designed directly on the state-space of rigid-body rotational motion, which is the tangent bundle of the Lie group of 3D rotations. The control scheme design, its stability, and its robustness properties are obtained through Lyapunov stability analyses. The proposed Hölder-continuous design is compared with three comparable sliding-mode designs. Numerical simulations on a simulated CubeSat demonstrate the performance of the proposed control scheme and compare it with the sliding-mode control schemes. The numerical simulations also compare the proposed control scheme with other state-of-the-art sliding-mode control approaches in existing research publications. The comparison results demonstrate that the proposed Hölder-continuous attitude control scheme exhibits lower control efforts and tracking control errors over these sliding-mode control schemes in simulations that incorporate actuator dynamics and measurement uncertainties. 

This article presents an estimation scheme for a rotating rigid body in the presence of unknown disturbance torque and unknown bias in angular velocity measurements. The attitude, angular velocity and disturbance torque are estimated from on-board control inputs, landmark vector measurements, and angular velocity measurements. The estimated attitude evolves directly on the special orthogonal group SO(3) of rigid body rotations. A Lyapunov analysis is given to prove that the proposed estimation scheme is almost globally Lyapunov stable in the absence of measurement noise and dynamic disturbance. The estimation scheme is discretized as a geometric integrator for practical implementation. The geometry-preserving properties of this numerical integrator preserve the Lie group structure of the configuration space, and give long time numerical stability. Numerical simulations demonstrate the stability and robustness properties of the proposed scheme. 

We consider estimation of motion on spheres as a variational problem. The concept of variational estimation for mechanical systems is based on application of variational principles from mechanics, to state estimation of mechanical systems evolving on configuration manifolds. If the configuration manifold is a symmetric space, then the overlying connected Lie group of which it is a quotient space, can be used to design nonlinearly stable observers for estimation of configuration and velocity states from measurements. If the configuration manifold is a sphere, then it can be globally represented by an unit vector. We illustrate the design of variational observers for mechanical systems evolving on spheres, through its application to estimation of pointing directions (reduced attitude) on the regular sphere S^2. 

Variational estimation of a mechanical system is based on the application of variational principles from mechanics to state estimation of the system evolving on its configuration manifold. If the configuration manifold is a Lie group, then the underlying group structure can be used to design nonlinearly stable observers for estimation of configuration and velocity states from measurements. Measured quantities are on a vector space on which the Lie group acts smoothly. We formulate the design of variational observers on a general finite-dimensional Lie group, followed by the design and experimental evaluation of a variational observer for rigid body motions on the Lie group SE(3). 

This article presents an extended state observer for a vehicle modeled as a rigid body in three-dimensional translational and rotational motions. The extended state observer is applicable to a multi-rotor aerial vehicle with a fixed plane of rotors, modeled as an under-actuated system on the state-space TSE(3), the tangent bundle of the six-dimensional Lie group SE(3). This state-space representation globally represents rigid body motions without singularities. The extended state observer is designed to estimate the resultant external disturbance force and disturbance torque acting on the vehicle. It guarantees stable convergence of disturbance estimation errors in finite time when the disturbances are constant, and finite time convergence to a bounded neighborhood of zero errors for time-varying disturbances. This extended state observer design is based on a Hölder-continuous fast finite time stable differentiator that is similar to the super-twisting algorithm, to obtain fast convergence. Numerical simulations are conducted to validate the proposed extended state observer. The proposed extended state observer is compared with other existing research to show its advantages. A set of experimental results implementing disturbance rejection control using feedback of disturbance estimates from this extended state observer is also presented. 

This work analyzes and develops some fundamental results for attitude consensus control of a network of rigid-body vehicles, considered a multi-agent rigid body system (MARBS). The system is analyzed using a full rigid body dynamics model on TSO(3) for each vehicle (agent) in the network. Therefore, the state space of the system is TSO(3)^N, where N is the number of vehicles. Attitude synchronization control laws for each vehicle to reach a consensus attitude with zero angular velocity for a particular type of network are obtained, using a Morse-Lyapunov function. Some fundamental results on equilibria of the network under these attitude consensus control laws are obtained. We show that unlike cooperative control of multi-agent systems with highly simplified dynamics models for agents, like point particles or unicycles where the state space of the dynamics is modeled as a vector space, there are multiple equilibrium solutions possible for attitude consensus control laws for a MARBS with dynamics on TSO(3)^N. Further, the number of equilibria depends on the network graph topology. This is followed by numerical simulation results for two different network graphs, which show this network control framework to be effective in obtaining attitude consensus. 

This article presents a framework for model-free control design for mechanical systems without velocity measurements and with an unknown dynamics, considered as a bounded disturbance input. The system states consist of zeroth-order (e.g position) and first-order (e.g velocity) vectors, but only the zeroth-order states are the measured outputs. This model-free control framework is based on a first-order signal differentiator and a finite-time stable extended state observer that simultaneously estimates the states and the bounded disturbance input in real time with guaranteed bounds on accuracy of the estimates. The estimates provided by this observer are used to track a desired output trajectory and compensate the disturbance in real time. Overall nonlinear stability and robustness of the observer is shown theoretically and verified through numerical simulations. The proposed method can be applied to second-order systems and their teams, like mobile robots, unmanned aerial vehicles, unmanned (under)water vehicles and space vehicles. 

The rigid body attitude estimation problem is treated using the discrete-time Lagrange-d'Alembert principle. Three different possibilities are considered for the multi-rate relation between angular velocity measurements and direction vector measurements for attitude: 1) integer relation between sampling rates, 2) time-varying sampling rates, 3) non-integer relation between sampling rates. In all cases, it is assumed that angular velocity measurements are sampled at a higher rate compared to the inertial vectors. The attitude determination problem from two or more vector measurements in the body-fixed frame is formulated as Wahba's problem. At instants when direction vector measurements are absent, a discrete-time model for attitude kinematics is used to propagate past measurements. A discrete-time Lagrangian is constructed as the difference between a kinetic energy-like term that is quadratic in the angular velocity estimation error and an artificial potential energy-like term obtained from Wahba's cost function. An additional dissipation term is introduced and the discrete-time Lagrange-d'Alembert principle is applied to the Lagrangian with this dissipation to obtain an optimal filtering scheme. A discrete-time Lyapunov analysis is carried out to show that the optimal filtering scheme is asymptotically stable in the absence of measurement noise and the domain of convergence is almost global. For a realistic evaluation of the scheme, numerical experiments are conducted with inputs corrupted by bounded measurement noise. These numerical simulations exhibit convergence of the estimated states to a bounded neighborhood of the actual states. 

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.