Attitude Control Of Rigid Body Research Articles

Generalized homogeneous rigid-body attitude control

The attitude tracking problem for a full-actuated rigid body in 3D is studied using a system model based on Lie algebra so(3). A nonlinear homogeneous controller is designed to track a smooth attitude trajectory in a finite/fixed time. A global settling time estimate is obtained, which is easily adjustable by tuning the homogeneity degree. The input-to-state stability of the control system with respect to measurement noises and additive perturbations is studied. Simulations illustrating the performance of the proposed algorithm are presented.

Rigid-Body Attitude Control, Synchronization, and Bipartite Consensus Using Feedback Reshaping

Improved attitude control laws based on a feedback reshaping strategy using rotation matrices are proposed for attitude control of a single rigid body on and for multibody attitude synchronization on . For attitude control of a single body the proposed control law improves the closed-loop response for initial principal rotation angles near 180° while producing a globally continuous control torque. For the problem of multibody attitude synchronization, an alternative feedback reshaping strategy is proposed, which destabilizes nonconsensus equilibria and produces attitude synchronization with almost global stability and globally continuous control. Finally, a previously proposed bipartite consensus algorithm based on the concept of a structurally balanced graph with antagonistic interactions represented by negative edge weights is applied for the first time to rigid-body attitude consensus control with feedback reshaping in which the consensus attitudes and angular velocities are divided into two distinct groups. Simulations serve as illustrative examples of each of these problems and confirm the advantages of the proposed strategies. For multibody synchronization the focus is on ring graph topologies for which locally stable nonconsensus equilibria arise when using an existing attitude consensus control protocol in the literature.

Constrained control for systems on matrix Lie groups with uncertainties

AbstractIn this paper, the constrained control of systems evolving on matrix Lie groups with uncertainties is considered. The proposed methodology is composed of a nominal Model Predictive Control (MPC), and a feedback controller. The previous work on the control of systems on manifolds is applied to design the nominal MPC, which generates the nominal trajectory. In the nominal MPC, the state and input constraints on the Lie group are transformed into the constraints on the Euclidean space. While to deal with uncertainties, the feedback control used to track the nominal trajectory is designed directly on the Lie group. The tracking error in the feedback control is proved to be bounded in invariant sets, which are further used to revise the constraints in nominal MPC. We prove that the input‐to‐state stability of the entire system under the proposed control methodology with respect to the disturbances can be achieved. The proposed methodology is applied to the constrained attitude control of rigid bodies with uncertainties. In the application example, the detailed mathematical proof and the comparative numerical simulation are presented to demonstrate the feasibility of the proposed methodology.

Robust adaptive learning for attitude control of rigid bodies with initial alignment errors

In the attitude tracking control tasks of rigid-body systems, initial alignment errors are recognized as one of the main factors that hinder the effective controller design and thus the control accuracy improvement. To overcome this difficulty, we develop a novel adaptive iterative learning control (ILC) scheme to achieve the attitude tracking control tasks with high precision for rigid bodies. Owing to the newly employed deadzone mechanism in the parametric updating law, the proposed adaptive ILC scheme is able to deal with initial alignment errors properly without imposing extra requirements to the system dynamics. Moreover, by utilizing the quaternion-based description in the system dynamics, various uncertainties, including external iteration-varying disturbances and unknown system parameters, can be addressed together by a single scalar estimation scheme. The convergence of the proposed adaptive ILC scheme is analyzed rigorously with a Lyapunov-like theory by virtue of newly introducing a modified composite energy function. In addition, to demonstrate its effectiveness, the proposed adaptive ILC method is applied to the attitude tracking of a spacecraft, which shows excellent control performances despite various uncertainties.

Tube-based attitude control of rigid-bodies with magnitude-bounded disturbances

This paper is concerned with the guaranteed cost control (GCC) problem of nonlinear rigid-body attitude systems subject to magnitude-bounded disturbances, for which a novel tube-based control framework is developed based on two independent aspects: nominal performance and disturbance attenuation. In the nominal case of no disturbances, a GCC method is proposed such that the nominal closed-loop system is asymptotically stable with the optimal performance. To achieve approximate performance in the case of disturbances, an auxiliary error control scheme is constructed for steering the original attitude system with disturbances to track the ideal trajectory that is generated in the nominal case. Simulation results are provided that highlight the guaranteed cost and disturbance attenuation properties of the proposed method.

Adaptive Deep Learning for High-Dimensional Hamilton--Jacobi--Bellman Equations

Computing optimal feedback controls for nonlinear systems generally requires solving Hamilton-Jacobi-Bellman (HJB) equations, which are notoriously difficult when the state dimension is large. Existing strategies for high-dimensional problems often rely on specific, restrictive problem structures, or are valid only locally around some nominal trajectory. In this paper, we propose a data-driven method to approximate semi-global solutions to HJB equations for general high-dimensional nonlinear systems and compute candidate optimal feedback controls in real-time. To accomplish this, we model solutions to HJB equations with neural networks (NNs) trained on data generated without discretizing the state space. Training is made more effective and data-efficient by leveraging the known physics of the problem and using the partially-trained NN to aid in adaptive data generation. We demonstrate the effectiveness of our method by learning solutions to HJB equations corresponding to the attitude control of a six-dimensional nonlinear rigid body, and nonlinear systems of dimension up to 30 arising from the stabilization of a Burgers'-type partial differential equation. The trained NNs are then used for real-time feedback control of these systems.

Rigid-body attitude control guaranteeing finite-time convergence

This study proposes an effective solution to the problem of attitude control for a rigid body satisfying angular velocity constraint as well as providing fault-tolerant capability. More specifically, a finite-time sliding surface containing attitude quaternion and angular velocity is first defined. Then, a novel tan-type prescribed performance control (PPC) with simple structure is presented to confine the sliding surface within a predefined performance boundary. Not only the attitude quaternion and angular velocity are indirectly constrained, but also it is thoroughly proved that the rotation velocity constraint is met even when severe actuators faults occur. The closed-loop attitude system is confirmed to be finite-time stable in the sense of Lyapunov stability. Numerical simulations clearly illustrate the effectiveness and usefulness of the suggested finite-time PPC despite actuator faults and environmental disturbances.

Rigid-Body Pose Hybrid Control Using Dual Quaternions: Global Asymptotic Stabilization and Robustness

A hybrid feedback control scheme is proposed for stabilization of rigid-body dynamics (position, orientation, and velocities) using unit dual quaternions, in which the dual quaternions and velocities are used for feedback. Specifically, both set-point stabilization and tracking control are addressed in this work. It is well known that rigid-body attitude control is subject to topological constraints, which often result in discontinuous control to avoid the unwinding phenomenon. In contrast, the hybrid scheme allows the controlled system to be robust in the presence of uncertainties, which would otherwise cause chattering about the point of discontinuous control while also ensuring acceptable closed-loop response characteristics. The stability of the closed-loop system is guaranteed through a Lyapunov analysis and the use of an invariance principle for hybrid systems. Simulation results for a rigid-body model are presented to illustrate the performance of the proposed hybrid dual-quaternion feedback control schemes.

Model Predictive Tracking Control for Invariant Systems on Matrix Lie Groups via Stable Embedding Into Euclidean Spaces

For controller design for systems on manifolds embedded in Euclidean space, it is convenient to utilize a theory that requires a single global coordinate system on the ambient Euclidean space rather than multiple local charts on the manifold or coordinate-free tools from differential geometry. In this article, we apply such a theory to design model predictive tracking controllers for systems whose dynamics evolve on manifolds and illustrate its efficacy with the fully actuated rigid body attitude control system.

Uniform Exponential Stability Result for the Rigid-Body Attitude Tracking Control Problem

Uniform exponential stability (UES) is proved for the rigid-body attitude control problem using a proportional-derivative (PD) controller with feedforward terms associated with the desired attitude state. Whereas this so-called “PD+” controller is classical and has been extensively analyzed in the literature for more than two decades, thus far it has only been shown to deliver uniform aysmptotic stability for the resulting closed-loop system. This paper parameterizes the kinematics through the three-dimensional modified Rodrigues parameter, assumes perfect measurement of the full-state, and guarantees a stronger UES result. It should be emphasized that no additional restrictions on the reference trajectory or high-gain feedback assumptions are placed in achieving this stronger stability result for the closed-loop system. The design of a new Lyapunov function is shown to permit UES, which further lends itself to robustness analysis in the possible presence of bounded unknown external disturbance torques. Saliently, this new Lyapunov function naturally extends to the classical Gibbs–Rodrigues parameterization of the attitude kinematics. The controller is also tested in simulation to compare the result with practical performance.

Redundancy implies robustness for bang‐bang strategies

SummaryWe develop in this paper a method ensuring robustness properties of bang‐bang strategies, for general nonlinear control systems. Our main idea is to add bang arcs in the form of needle‐like variations of the control. With such bang‐bang controls having additional degrees of freedom, steering the control system to some given target amounts to solving an overdetermined nonlinear shooting problem, what we do by developing a least‐square approach. In turn, we design a criterion to measure the quality of robustness of the bang‐bang strategy, based on the singular values of the end‐point mapping, and which we optimize. Our approach thus shows that redundancy implies robustness, and we show how to achieve some compromises in practice, by applying it to the attitude control of a 3d rigid body.

Dynamically Scaled Immersion and Invariance Adaptive Control for Euler–Lagrange Mechanical Systems

This paper addresses adaptive control of specific Euler–Lagrange systems: rigid-body attitude control and the -link robot manipulator. For each problem, the model parameters are unknown but the lower bound of the smallest eigenvalue of the inertia matrix is assumed to be known. The dynamic scaling immersion and invariance adaptive controller is proposed to stabilize the system without employing a filter for the regressor matrix. A scalar scaling factor is instead implemented to overcome the integrability obstacle that arises in immersion and invariance adaptive control design. First, a filter-free controller is proposed for the attitude problem such that the rate feedback gain is proportional to the square of the scaling factor in the tracking error dynamics. The gain is then shown to be bounded through state feedback while achieving stabilization of the tracking error. A similar approach for the dynamic gain design is applied to a filter-dependent controller, where a filter for the angular rate is used to build a parameter estimator. The proposed design is also applied to robot manipulator systems. Spacecraft attitude and two-link planar robot tracking problems are considered to demonstrate the performance of the controllers through simulations.

Mitigating the curse of dimensionality: sparse grid characteristics method for optimal feedback control and HJB equations

We address finding the semi-global solutions to optimal feedback control and the Hamilton–Jacobi–Bellman (HJB) equation. Using the solution of an HJB equation, a feedback optimal control law can be implemented in real-time with minimum computational load. However, except for systems with two or three state variables, using traditional techniques for numerically finding a semi-global solution to an HJB equation for general nonlinear systems is infeasible due to the curse of dimensionality. Here we present a new computational method for finding feedback optimal control and solving HJB equations which is able to mitigate the curse of dimensionality. We do not discretize the HJB equation directly, instead we introduce a sparse grid in the state space and use the Pontryagin’s maximum principle to derive a set of necessary conditions in the form of a boundary value problem, also known as the characteristic equations, for each grid point. Using this approach, the method is spatially causality free, which enjoys the advantage of perfect parallelism on a sparse grid. Compared with dense grids, a sparse grid has a significantly reduced size which is feasible for systems with relatively high dimensions, such as the 6-D system shown in the examples. Once the solution obtained at each grid point, high-order accurate polynomial interpolation is used to approximate the feedback control at arbitrary points. We prove an upper bound for the approximation error and approximate it numerically. This sparse grid characteristics method is demonstrated with three examples of rigid body attitude control using momentum wheels.

Guidance and Attitude Control of Unstable Rigid Bodies With Single-Use Thrusters

In this paper, we introduce a novel computationally efficient and practical algorithm for robust guidance and attitude control of unstable rigid bodies using the discrete fixed-magnitude, fixed-duration actuation representative of single-use thrusters. New guidance and attitude continuous control laws are developed based on the nonlinear rigid-body motion model, where the objective is to intercept a target and maintain the near zero angle of attack. A methodology is then introduced to adapt the continuous control system to discrete impulsive forces with constant magnitude and duration. An actuation management scheme is presented based on the equivalence of the total impulse of the continuous control law and thruster force. Realistic time delays and noise distributions are included in a sensor fusion algorithm based on the extended Kalman filter. Simultaneous control for course corrections to ensure the target interception and stabilization of the vehicle attitude is presented through Monte Carlo simulations with various uniformly distributed direction and velocity errors.

Attitude control of rigid body with inertia uncertainty and saturation input

In this paper, the attitude control problem of rigid body is addressed with considering inertia uncertainty, bounded time-varying disturbances, angular velocity-free measurement, and unknown non-symmetric saturation input. Using a mathematical transformation, the effects of bounded time-varying disturbances, uncertain inertia, and saturation input are combined as total disturbances. A novel finite-time observer is designed to estimate the unknown angular velocity and the total disturbances. For attitude control, an observer-based sliding-mode control protocol is proposed to force the system state convergence to the desired sliding-mode surface; the finite-time stability is guaranteed via Lyapunov theory analysis. Finally, a numerical simulation is presented to illustrate the effective performance of the proposed sliding-mode control protocol.

Rigid Body Attitude Control Based on a Manifold Representation of Direction Cosine Matrices

Autonomous systems typically actively observe certain aspects of their surroundings, which makes them dependent on a suitable controller. However, building an attitude controller for three degrees of freedom is a challenging task, mainly due to singularities in the different parametrizations of the three dimensional rotation group SO(3). Thus, we propose an attitude controller based on a manifold representation of direction cosine matrices: In state space, the attitude is globally and uniquely represented as a direction cosine matrix R ∈ SO(3). However, differences in the state space, i.e., the attitude errors, are exposed to the controller in the vector space ℝ3. This is achieved by an operator, which integrates the matrix logarithm mapping from SO(3) to so(3) and the map from so(3) to ℝ3. Based on this representation, we derive a proportional and derivative feedback controller, whose output has an upper bound to prevent actuator saturation. Additionally, the feedback is preprocessed by a particle filter to account for measurement and state transition noise. We evaluate our approach in a simulator in three different spacecraft maneuver scenarios: (i) stabilizing, (ii) rest-to-rest, and (iii) nadir-pointing. The controller exhibits stable behavior from initial attitudes near and far from the setpoint. Furthermore, it is able to stabilize a spacecraft and can be used for nadir-pointing maneuvers.

Sliding Mode Active Vibration Control and Three Axis Attitude Control using Four Reaction Wheels

The most important problems in satellites are their energy supply and decreasing weight. One of the solutions is using flexible appendages. Flexibility can supply both energy and weight loss in satellites. Also other appendages like antennas and mechanical arms can be flexible. But flexibility can produce some vibrations. These vibrations cause complexity attitude control systems due to flexible appendages. In this paper, the attitude control of flexible satellite by using four reaction wheels in presence of gravity gradient and orbit frequency has been considered. The dynamic of flexible appendages have been derived from energy equations and Lagrange method. Also, momentum management and minimization of the momentum have been employed to avoid of reaction wheels saturation. Then, the control command law using the quaternion error vector for attitude control of rigid body and sliding mode control (SMC) for active suppuration of flexible appendages have been used. Finally the performance of control system between presence and absence of reaction wheels has been compared.

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.

Attitude control of multiple rigid bodies with uncertainties and disturbances

Decentralized attitude synchronization and tracking control for multiple rigid bodies are investigated in this paper. In the presence of inertia uncertainties and environmental disturbances, we propose a class of decentralized adaptive sliding mode control laws. An adaptive control strategy is adopted to reject the uncertainties and disturbances. Using the Lyapunov approach and graph theory, it is shown that the control laws can guarantee a group of rigid bodies to track the desired time-varying attitude and angular velocity while maintaining attitude synchronization with other rigid bodies in the formation. Simulation examples are provided to illustrate the feasibility and advantage of the control algorithm.

Port-Hamiltonian Formulation of Rigid-Body Attitude Control

The aim of this paper is to present a port-Hamiltonian (pH) formulation of the rigidbody attitude control problem, therefore enhancing the set of available tools for its modeling and control. First, a pH formulation of both dynamics and kinematics equations is presented. Second, a standard energy-balancing passivity-based controller (EB-PBC) is used for set-point tracking. Third, the controlled system is endowed with a dynamical extension to achieve set-point tracking without measuring the angular velocities. As a conclusive remark, it is showed under specific assumptions that these three results can be achieved regardless the coordinate representation in use. Additional examples follow to motivate the adoption of the pH formulation.