An efficient and accurate computational approach is proposed for optimal attitude control of a rigid body. The problem is formulated directly as a discrete time optimization problem using a Lie group variational integrator. Discrete necessary conditions for optimality are derived, and an efficient computational approach is proposed to solve the resulting two point boundary value problem. The use of geometrically exact computations on SO(3) guarantees that this optimal control approach has excellent convergence properties even for highly nonlinear large angle attitude maneuvers. Numerical results are presented for attitude maneuvers of a 3D pendulum and a spacecraft in a circular orbit.
This paper provides new results for a tracking control of the attitude dynamics of a rigid body. Both of the attitude dynamics and the proposed control system are globally expressed on the special orthogonal group, to avoid complexities and ambiguities associated with other attitude representations such as Euler angles or quaternions. By selecting an attitude error function carefully, we show that the proposed control system guarantees a desirable tracking performance uniformly for nontrivial rotational maneuvers involving a large initial attitude error. In a special case where the desired att…
Stability proof of the controller proposed in the conference paper "Attitude and Angular Velocity Tracking for a Rigid Body using Geometric Methods on the Two-Sphere", DOI: 10.1109/ECC.2015.7331033. This proof must be studied together with Sec. III-C in the aforementioned paper.
This paper studies a rigid body attitude tracking control problem with attitude measurements only, when angular velocity measurements are not available. An angular velocity observer is constructed such that the estimated angular velocity is guaranteed to converge to the true angular velocity asymptotically from almost all initial estimates. As it is developed directly on the special orthogonal group, it completely avoids singularities, complexities, or discontinuities caused by minimal attitude representations or quaternions. Then, the presented observer is integrated with a proportional-deriv…
Two nonlinear observers for velocity-aided attitude estimation, relying on gyrometers, accelerometers, magnetometers, and velocity measured in the body-fixed frame, are proposed. As opposed to state-of-the-art body-fixed velocity-aided attitude observers endowed with local properties, both observers are (almost) globally asymptotically stable, with very simple and flexible tuning. Moreover, the roll and pitch estimates are globally decoupled from magnetometer measurements.
Optimal control problems are formulated and efficient computational procedures are proposed for attitude dynamics of a rigid body with symmetry. The rigid body is assumed to act under a gravitational potential and under a structured control moment that respects the symmetry. The symmetry in the attitude dynamics system yields a conserved quantity, and it causes a fundamental singularity in the optimal control problem. The key feature of this paper is its use of computational procedures that are guaranteed to avoid the numerical ill-conditioning that originates from this symmetry. It also prese…
A global model is presented that can be used to study attitude maneuvers of a rigid spacecraft in a circular orbit about a large central body. The model includes gravity gradient effects that arise from the non-uniform gravity field and characterizes the spacecraft attitude with respect to the uniformly rotating local vertical local horizontal coordinate frame. An accurate computational approach for solving a nonlinear boundary value problem is proposed, assuming that control torque impulses can be applied at initiation and at termination of the maneuver. If the terminal attitude condition is …
A continuous solution is proposed to the problem of uniform global exponential estimation of the angular velocity for rigid bodies by means of direct attitude measurements. The proposed observer is designed on $\mathbb{R}^3\times\mathbb{R}^3\times\mathbb{R}^3$, and not on the inherent geometry $\mathtt{SO}(3)\times\mathbb{R}^3$, which allows for a simple structure of the observer, being a copy of the system plus some correction terms, and a straightforward classical Lyapunov stability proof. The performances of the observer are illustrated by means of a simple simulation scenario.
A time optimal attitude control problem is studied for the dynamics of a rigid body. The objective is to minimize the time to rotate the rigid body to a desired attitude and angular velocity while subject to constraints on the control input. Necessary conditions for optimality are developed directly on the special orthogonal group using rotation matrices. They completely avoid singularities associated with local parameterizations such as Euler angles, and they are expressed as compact vector equations. In addition, a discrete control method based on a geometric numerical integrator, referred t…
We address the problem of attitude stabilization of a rigid body, in which neither the angular velocity nor the instantaneous measurements of the attitude are used in the feedback, only body vector measurements are needed. The design of the controller is based on an angular velocity observer-like system, where a first order linear auxiliary system based directly on vector measurements is introduced. The introduction of gain matrices provides more tuning flexibility and better results compared with existing works. The proposed controller ensures almost global asymptotic stability. The performan…
Motivated by attitude control and attitude estimation problems for a rigid body, computational methods are proposed to propagate uncertainties in the angular velocity and the attitude. The nonlinear attitude flow is determined by Euler-Poincaré equations that describe the rotational dynamics of the rigid body acting under the influence of an attitude dependent potential and by a reconstruction equation that describes the kinematics expressed in terms of an orthogonal matrix representing the rigid body attitude. Uncertainties in the angular velocity and attitude are described in terms of ellips…
In this paper, an event-triggered Reinforcement Learning (RL) method is proposed for the optimal attitude consensus of multiple rigid body networks with unknown dynamics. Firstly, the consensus error is constructed through the attitude dynamics. According to the Bellman optimality principle, the implicit form of the optimal controller and the corresponding Hamilton-Jacobi-Bellman (HJB) equations are obtained. Because of the augmented system, the optimal controller can be obtained directly without relying on the system dynamics. Secondly, the self-triggered mechanism is applied to reduce the co…
There are several attitude estimation algorithms in existence, all of which use local coordinate representations for the group of rigid body orientations. All local coordinate representations of the group of orientations have associated problems. While minimal coordinate representations exhibit kinematic singularities for large rotations, the quaternion representation requires satisfaction of an extra constraint. This paper treats the attitude estimation and filtering problem as an optimization problem, without using any local coordinates for the group of rotations. An attitude determination a…
A deterministic attitude estimation problem for a rigid body in a potential field, with bounded attitude and angular velocity measurement errors is considered. An attitude estimation algorithm that globally minimizes the attitude estimation error is obtained. Assuming that the initial attitude, the initial angular velocity and measurement noise lie within given ellipsoidal bounds, an uncertainty ellipsoid that bounds the attitude and the angular velocity of the rigid body is obtained. The center of the uncertainty ellipsoid provides point estimates, and the size of the uncertainty ellipsoid me…
A deterministic attitude estimation problem for a rigid body in an attitude dependent potential field with bounded measurement errors is studied. An attitude estimation scheme that does not use generalized coordinate representations of the attitude is presented here. Assuming that the initial attitude, angular velocity and measurement noise lie within given ellipsoidal bounds, an uncertainty ellipsoid that bounds the attitude and the angular velocity of the rigid body is obtained. The center of the uncertainty ellipsoid provides point estimates, and its size gives the accuracy of the estimates…
Estimation of rigid body attitude motion is a long-standing problem of interest in several applications. This problem is challenging primarily because rigid body motion is described by nonlinear dynamics and the state space is nonlinear. The extended Kalman filter and its several variants have remained the standard and most commonly used schemes for attitude estimation over the last several decades. These schemes are obtained as approximate solutions to the nonlinear optimal filtering problem. However, these approximate or near optimal solutions may not give stable estimation schemes in genera…
Optimal control problems are formulated and efficient computational procedures are proposed for combined orbital and rotational maneuvers of a rigid body in three dimensions. The rigid body is assumed to act under the influence of forces and moments that arise from a potential and from control forces and moments. The key features of this paper are its use of computational procedures that are guaranteed to preserve the geometry of the optimal solutions. The theoretical basis for the computational procedures is summarized, and examples of optimal spacecraft maneuvers are presented.
We consider the classical problem of estimating the attitude and gyro biases of a rigid body from vector measurements and a triaxial rate gyro. We propose a simple "geometry-free" nonlinear observer with guaranteed uniform global asymptotic convergence and local exponential convergence; the stability analysis, which relies on a strict Lyapunov function, is rather simple. The excellent behavior of the observer is illustrated through a detailed numerical simulation.
A deterministic attitude estimator for a rigid body under an attitude dependent potential is studied. This estimator requires only a single direction measurement to a known reference point at each measurement instant. The measurement cannot completely determine the attitude, but an attitude estimation scheme based on this measurement is developed; a feasible set compatible with the measurement is described and it is combined with an attitude dynamics model to obtain an attitude estimate. The attitude is globally represented by a rotation matrix, and the uncertainties are described by ellipsoid…
This paper presents a new geometric adaptive control system with state inequality constraints for the attitude dynamics of a rigid body. The control system is designed such that the desired attitude is asymptotically stabilized, while the controlled attitude trajectory avoids undesired regions defined by an inequality constraint. In addition, we develop an adaptive update law that enables attitude stabilization in the presence of unknown disturbances. The attitude dynamics and the proposed control systems are developed on the special orthogonal group such that singularities and ambiguities of …
This paper presents four types of tracking control systems for the attitude dynamics of a rigid body. First, a smooth control system is constructed to track a given desired attitude trajectory, while guaranteeing almost semi-global exponential stability. It is extended to achieve global exponential stability by using a hybrid control scheme based on multiple configuration error functions. They are further extended to obtain robustness with respect to a fixed disturbance using an integral term. The resulting robust, global exponential stability for attitude tracking is the unique contribution o…
This paper presents global tracking strategies for the attitude dynamics of a rigid body. It is well known that global attractivity is prohibited for continuous attitude control systems on the special orthogonal group. Such topological restriction has been dealt with either by constructing smooth attitude control systems that exclude a set of measure zero in the region of attraction, or by introducing hybrid control systems to obtain global asymptotic stability. This paper proposes alternative attitude control systems that are continuous in time to achieve exponential stability, where the regi…
We propose a simple nonlinear observer for estimating the attitude and velocity of a rigid body from the measurements of specific acceleration, angular velocity, magnetic field (in body axes), and linear velocity (in body axes). It is uniformly globally exponentially convergent, and also enjoys other nice properties: global decoupling of pitch and roll estimation from magnetic measurements, good local behavior, and easy tuning. The observer is "geometry-free", in the sense that it respects only asymptotically the rotational geometry. The good behavior of the observer, even when the measurement…
The finite-time attitude synchronization problem is considered in this paper, where the rotation of each rigid body is expressed using the axis-angle representation. Two discontinuous and distributed controllers using the vectorized signum function are proposed, which guarantee almost global and local convergence, respectively. Filippov solutions and non-smooth analysis techniques are adopted to handle the discontinuities. Sufficient conditions are provided to guarantee finite-time convergence and boundedness of the solutions. Simulation examples are provided to verify the performances of the …
This paper investigates tracking controls for an arbitrary number of cooperating quadrotor unmanned aerial vehicles with a suspended rigid body payload. Assuming that a rigid body is connected to multiple quadrotors by rigid massless links, control systems for quadrotors are constructed such that the rigid body asymptotically follows a given desired position and attitude trajectory. These are developed in a coordinate-free fashion to avoid singularities and complexities associated with local parameterizations.
The existing attitude controllers (without angular velocity measurements) involve explicitly the orientation (\textit{e.g.,} the unit-quaternion) in the feedback. Unfortunately, there does not exist any sensor that directly measures the orientation of a rigid body, and hence, the attitude must be reconstructed using a set of inertial vector measurements as well as the angular velocity (which is assumed to be unavailable in velocity-free control schemes). To overcome this \textit{circular reasoning}-like problem, we propose a velocity-free attitude stabilization control scheme relying solely on…
We present a uniformly globally exponentially stable hybrid angular velocity observer for rigid body systems designed directly on $SO(3)\times\mathbb{R}^3$. The global exponential stability result makes this observer a good candidate for a controller-observer combination with a guaranteed separation property. Simulation results are provided to demonstrate the effectiveness of the proposed hybrid observer as a part of an attitude stabilization scheme.
This paper solves the rendezvous problem for a network of underactuated rigid bodies such as quadrotor helicopters. A control strategy is presented that makes the centres of mass of the vehicles converge to an arbitrarily small neighborhood of one another. The convergence is global, and each vehicle can compute its own control input using only an on-board camera and a three-axis rate gyroscope. No global positioning system is required, nor any information about the vehicles' attitudes.
This paper addresses the problem of estimating the attitude of a rigid body, which is subject to high accelerations and equipped with inertial measurement unit (IMU) and sensors providing the body velocity (expressed in the reference frame attached to the body). That issue can be treated differently depending on the level of confidence in the measurements of the magnetometer of the IMU, particularly with regard to the observation of the inclination component with respect to the vertical direction, rendering possible to describe the interaction with gravity. Two cases are then studied: either (…
This paper proposes two novel nonlinear attitude filters evolved directly on the Special Orthogonal Group SO(3) able to ensure prescribed measures of transient and steady-state performance. The tracking performance of the normalized Euclidean distance of attitude error is trapped to initially start within a large set and converge systematically and asymptotically to the origin from almost any initial condition. The convergence rate is guaranteed to be less than the prescribed value and the steady-state error does not exceed a predefined small value. The first filter uses a set of vectorial mea…
This paper is focused on tracking control for a rigid body payload, that is connected to an arbitrary number of quadrotor unmanned aerial vehicles via rigid links. A geometric adaptive controller is constructed such that the payload asymptotically follows a given desired trajectory for its position and attitude in the presence of uncertainties. The coupled dynamics between the rigid body payload, links, and quadrotors are explicitly incorporated into control system design and stability analysis. These are developed directly on the nonlinear configuration manifold in a coordinate-free fashion t…
In this work, we utilize discrete geometric mechanics to derive a 2nd-order variational integrator so as to simulate rigid body dynamics. The developed integrator is to simulate the motion of a free rigid body and a quad-rotor. We demonstrate the effectiveness of the simulator and its accuracy in long term integration of mechanical systems without energy damping. Furthermore, this work deals with the geometric nonlinear control problem for rigid bodies where backstepping controller is designed for full tracking of position and orientation. The attitude dynamics and control are defined on \text…
This paper proposes an integrated design optimization framework for the gain schedules and bending filter for the longitudinal control of a rocket during its ascent flight. Dynamic models representing the pitch/yaw motion of a rocket considering the elements such as the rigid body dynamics, aerodynamics, sloshing, bending, sensor/actuator, and flight computer are introduced. The linear proportional and differential (PD) control law with scheduled (time-varying) gains and bending filter parameters are identified as key decision variables for stabilizing the pitch/yaw motion of the rocket. The i…
The general objective of this Ph.D. thesis is to study the dynamics and control of rigid and flexible spacecraft supported by a high-fidelity numerical simulation environment. The demand for greater attitude pointing precision, attitude maneuvering or recovery with the increased use of lightweight and flexible materials necessitates the consideration of flexible dynamics in the control strategy. These highly nonlinear dynamics which increase the order of the system are extremely difficult to model with high degree of accuracy. A general model for attitude and flexible dynamics for a class of s…
In this work, asymptotically stable state estimation schemes are proposed for rigid body motion, using the framework of geometric mechanics. Rigorous stability analyses of the estimation schemes presented here guarantee the nonlinear stability of these schemes. The stability of these schemes does not depend on the characteristics of the sensor measurement noise or external disturbances. In addition, they are robust to initial errors in the state estimates and do not need to be re-tuned when sensor noise properties change. In the first part of this dissertation, estimation of rigid body states …
A minimum-time reorientation of an axisymmetric rigid spacecraft controlled by three torques is studied. The orientation of the body is modeled such that the attitude kinematics are representative of a spin-stabilized spacecraft. The optimal control problem considered is shown to have a switching control structure. Moreover, under certain assumptions, the solutions contain segments that lie on a singular arc. A numerical optimization study is performed using a recently developed method that is designed to accurately solve bang-bang and singular optimal control problems. The optimality conditio…
The problem of rigid body pose estimation is treated in discrete-time via discrete Lagrange-d'Alembert principle and discrete Lyapunov methods. The position and attitude of the rigid body are to be estimated simultaneously with the help of vision and inertial sensors. For the discrete-time estimation of pose, the continuous-time rigid body kinematics equations are discretized appropriately. We approach the pose estimation problem as minimising the energies stored in the errors of estimated quantities. With the help of measurements obtained through optical sensors, artificial rotational and tra…
We propose a tracking control law for the fully actuated rigid body system in the presence of any unknown constant disturbance by employing quaternions with the stable embedding technique and Lyapunov stability theory. The stable embedding technique extends the attitude dynamics from the set of unit quaternions to the set of quaternions, which is a Euclidean space, such that the set of unit quaternions is an invariant set of the extended dynamics. Such a stable extension of the system dynamics to a Euclidean space allows us to employ well studied Lyapunov techniques in Euclidean spaces such as…
In this work, we are interested in tree-shaped networks of freely vibrating beams which are geometrically exact (GEB) -- in the sense that large motions (deflections, rotations) are accounted for in addition to shearing -- and linked by rigid joints. For the intrinsic GEB formulation, namely that in terms of velocities and internal forces/moments, we derive transmission conditions and show that the network is locally in time well-posed in the classical sense. Applying velocity feedback controls at the external nodes of a star-shaped network, we show by means of a quadratic Lyapunov functional …
This paper presents the design of a geometric trajectory tracking controller for the cooperative task of two quadrotor UAVs (unmanned aerial vehicles) carrying and transporting a rigid bar, which is attached to the quadrotors via inflexible elastic cables. The elasticity of the cables together with techniques of singular perturbation allows a reduction in the model to that of a similar model with inelastic cables. In this reduced model, we design a controller such that the rod exponentially tracks a given desired trajectory for its position and attitude, under some assumptions on initial error…
This letter studies a distributed collision avoidance control problem for a group of rigid bodies on a sphere. A rigid body network, consisting of multiple rigid bodies constrained to a spherical surface and an interconnection topology, is first formulated. In this formulation, it is shown that motion coordination on a sphere is equivalent to attitude coordination on the 3-dimensional Special Orthogonal group. Then, an angle-based control barrier function that can handle a geodesic distance constraint on a spherical surface is presented. The proposed control barrier function is then extended t…
