This thesis investigates the nonlinear attitude control of an underactuated rigid-body spacecraft system in the body-orbital and inertial frames. This thesis studies the attitude control of an underactuated rigid body spacecraft system using nonlinear control methods and rigorous stability analysis.
Rationale and Motivations
In Section 1.1, the rationale behind the research on the undervalidated control system is explained, as well as the motivations behind the methodologies reviewed. A literature review of relevant topics on weak control systems is covered in Section 1.2.
Literature Review
Underactuated systems
This problem is mainly concerned with finding the control signals that stabilize a spacecraft, where the desired error in the attitude and angular velocity of the spacecraft system is zero. In comparison to this thesis, the pseudospectral optimal control theory approach is considered to optimize the spacecraft state trajectory and implemented using model predictive control, a methodological approach that has not been considered in the literature.
Problem Statement and Research Objectives
- Body-orbital relative attitude stabilization for an underactu-
- Inertial attitude stabilization for an underactuated rigid-body
- Output feedback control of an Underactuated rigid-body
- Lyapunov stability of observer-based nonlinear control system 7
The stabilization of the inertial attitude of an underactuated rigid-body spacecraft system under time-varying disturbances and uncertainties using nonlinear control is investigated. The problem of achieving attitude convergence of an output-feedback system for an underactuated rigid-body spacecraft system is investigated.
Research Overview
Euclidean space
The set of all n-dimensional vectors x = [x1, · · · ,xn]T defines the n-dimensional Euclidean space denoted by Rn, where x1, · · · ,xn are real-valued numbers. We denote R as a representation of the one-dimensional Euclidean space consisting of all real numbers.
Vector and matrix norms
Let A ∈ Rm×n be an m×n matrix of real elements defining a linear mapping y= AxfromRnintoRm, the induced p-norm of Ais defined by.
Topology
Manifolds
Charts and maps
The mapτM :TM→ Mdefined byτM([c]m) = missing the tangent bundle projection ofM. Let U be an open set subset of Rn and let f :U →Rm. A pair (Uα, φα) is called a coordinate scheme of M atq ∈ M, where Uα is an open set of M containing the pointq, and the homeomorphismφα is a bijection of Uα.
Vector fields
Geometrically, this means that all trajectories that have passed fort =0 in a certain neighborhood Vofq can be collected into a single differential map as seen in Figure 2.1. For proof of Theorem 2.1 and Theorem 2.2, please see sections “Lipschitz conditions” and “Existence and properties of solutions of the system” in Hurewicz.
Homogeneity of Functions
Furthermore, it can be shown that since f1 is continuous, homogeneous and positive-definite, f1 is radially unbounded and therefore E is a bounded set.
Lyapunov Stability of Systems
Finite-time stability
The origin of the system is (locally) finite-time stable if it is Lyapunov stable and finite-time convergent in a neighborhood U0 ⊂ U of the origin. When U =U0 =Rn, the origin of the system is said to be globally finite-time stable.
Optimal Control
The optimal control problem
A nonlinear observer is used to estimate the angular velocity of the spacecraft to complete an output feedback control law. While most of the existing research in the literature on spacecraft attitude control uses full state feedback (attitude and angular velocity measurements are available) for formulating the control law as such in the works of [G. Godard and K. Kumar, 2010; Godard, K. Kumar and A. Zou,2013;.
Problem Formulation
Coordinate frames
Spacecraft attitude kinematics and dynamics
It is assumed that the spacecraft is actuated only about the pitch and yaw axes, so that. If the under-actualized axis is the pitch axis, then due to the nature of the spacecraft attitude dynamics, three-axis stability is not directly achievable.
Control objective
Design of Underactuated Attitude Control Law
Control design for unactuated states
Control design for actuated states
The conclusion can be drawn by applying similar arguments as in the proof of Lemma 3.1. The relative angular velocity ωbn is replaced in the control law (3.35) by an estimate ˆωbn from a finite timekeeper defined in the next section.
Observer design
Now a strict Lyapunov function will be constructed for the nominal system corresponding to (3.53) (with f ≡ d˜ ≡ 0). While for a given limit ¯ω, it is proved that (3.68) holds in the open group U(ω¯), it will be useful to find an open subgroup of Vα contained in U(ω¯), since this will then become an invariant set, and in the absence of disturbanced, will be within the field of attraction of the observer.
Main Result
Anticipating similar properties of finite-time observers, this is why the parameter α has been included in the proposed observer. In the case α=1, a result similar to Lemma 4 can be obtained by using ¯V(x˜)directly (without the need for Vα), and in this case the observer becomes asymptotically convergent in the absence of perturbations.
Numerical Examples
- Nominal disturbance-free case
- Nominal case with disturbances
- All uncertainties and disturbances
- Effect of α parameter
The purpose of the example in this subsection is twofold: (1) to demonstrate the performance of the observer-based control law under ideal conditions (zero orbital eccentricity, perfect knowledge of the spacecraft inertia matrix, no external perturbations except gravity -gradient); (2) to compare the performance of ideal continuous-time (unsampled data) observer-based control in the absence of measurement noise with observer-based control of sampled data with noisy measurements. While the transient responses are similar over the considered range of α, at steady state (as seen in Figure 3.8), the attitude error initially decreases as α decreases from 1, but then increases as α becomes smaller.
Introduction
The stability of the combined controller-observer system is rigorously proven to show the local ultimate limit of all signals. Numerical examples are produced in Section 4.4 to demonstrate the effectiveness of the control-observer system for inertial attitude stabilization.
Problem Formulation
Attitude kinematics and dynamics
Also presented in section 4.2 is the definition of the control objective, the control design for non-actuated and activated states, the observer design and the main results. To accommodate an angular velocity observer, the control input is distributed asu = un+∆u, where un is an ideal state feedback control law, while ∆ure represents the deviation from un due to the angular velocity estimation error.
Control objective
It is assumed that a nominal inertia matrix Jn ∈ R3×3 of the spacecraft is known, such that J = Jn+∆J, where ∆J ∈ R3×3 is the uncertainty of the inertia matrix. It is assumed that the spacecraft is driven only about the roll and pitch axes, such that.
Control design for unactuated states
By defining the dilation δλr(x) = (λq1,λq2,λ2q3,λ2ωz,λωx,λωy), it is straightforward to verify that the nominal vector field f(x,un) is homogeneous of degree zero with respect to δλr, provided that the input sunx(x,t) and uny(x,t) of the state feedback control are homogeneous of degree 1. We now need to find constraints on e > 0 such that W(η,t) is a strict Lyapunov function to note , that ˜ Water(∂V/∂η˜ )fηis both continuous, periodic in the period T=2π and homogeneous of degree 4 with respect toδλr¯, from corollary 1 we get.
Control design for actuated states
However, to ensure that f(x,un) is homogeneous of degree zero with respect to δrλ, we must do this in such a way thatunx(x,t)anduny(x,t) are homogeneous of degree 1. Since vx ,vy are homogeneous of degree 1 with respect to δλ¯r (which is compatible with δrλ), it is clear that unx(x,t) anduny(x,t) are homogeneous of degree 1 with respect to δrλ.
Observer design
It can be verified that gα is homogeneous of degree k= α−1 with respect to the dilationδrλe(α)(x˜) = [λq˜vT,λαω˜T]T. While for a given bound ¯χ it has been established that (4.72) holds on the open set U(χ¯), it will be useful to find an open sublevel set of Vα contained in U(χ¯), since this will then become an invariant set, and in the absence of uncertainty and disturbance, be within the domain of attraction of the observer.
Main Result
From Lemmas 4.4 and 4.6 it can be deduced that x(t) cannot reach the limit of Bχ(0) before ˜x(t) reaches the limit of Bη(0), and vice versa.
Numerical Examples
Nominal disturbance-free case
The purpose of the example in this subsection is twofold: (1) to demonstrate the performance of the observer-based control law under ideal conditions (perfect knowledge of the spacecraft inertial matrix, no external perturbations other than gravity gradient); (2) to compare the performance of the ideal continuous-time (unsampled data) observer-based control in the absence of measurement noise with the sampled-data observer-based control with noisy measurements. Having established this, the examples in the remaining subsections are for the sample data controller only.
Nominal case with disturbances
It can be seen that the attitude and angular velocity responses are similar, with the sampled-data responses exhibiting some noise, which is due to the presence of noise on the measurements. the finite-time observer-based controller.
All uncertainties and disturbances
The initial true pose is obtained as qv(0) = asin(φ/2), q4(0) = cos(φ/2), where ∈ S2 is generated using MATLAB's “randn” function, followed by normalization, and φ ∈ Ris generated using MATLAB's “randn” function, with standard deviation of π/3, corresponding to an initial pose error of 180 degrees (3-sigma). The initial estimated inertial angular velocity ˆω(0) is generated using MATLAB's “randn” function, with a standard deviation of 1.4×π/(3×180), which corresponds to an initial estimated angular velocity of 1.4 degrees /second (3-sigma ).
Chapter Summary
Attitude stabilization of a spacecraft using two torques supplied by reaction wheels aligned about the major axes of the spacecraft is investigated in this chapter. The full dynamics of the spacecraft with two reaction wheels is not controllable [Krishnan, McClamroch, and Reyhanoglu, 1995].
Problem Formulation
- Attitude kinematics and dynamics
- Optimal control
- Pseudospectral discretization
- Model predictive control
- Stability analysis
The total angular momentum contribution of the reaction wheel array,h, with inertia Jwcan be expressed as. The total angular momentum of the spacecraft RW system with respect to FI, written inFbis written as.
Numerical Simulations
The control signals are then interconnected and used as control inputs to the underpowered spacecraft plant. As seen in Figure 5.5, by t = 0.9 orbit, the optimal control solver has generated ten interpolated control signals for the MPC.
Chapter Summary
Research overview of the spacecraft attitude control
Overview of Chapter 3
Overview of Chapter 4
Overview of Chapter 5
A subset C⊂Si is called closed if its complement S\{C} is open. ii) For a pointnp⊂S, the neighborhood is an open setU ⊂For whichp⊂U. iii). This is called the subspace topology. xii) IfB⊂ A ⊂ S, then intA(B) denotes the interior of B in the subspace topology in A.
Differential maps of a trajectory
Suppose there exists a continuously differentiable positive definite function V(x) on the neighborhood U1⊂U of the origin, such that. Kumar , 2011 ] used a higher order sliding mode that required prior knowledge of the perturbations to be considered, which is not the case for the full state feedback control law developed here.
Coordinate Topologies
It should be noted that Jtis is only used in the numerical propagation of the true attitude dynamics. Global Asymptotic Stabilization of Attitude and Angular Norms of an Underactualized Nonsymmetric Rigid Body”.
Attitude and angular velocity responses in nominal disturbance-free
Attitude and angular velocity responses in nominal case with distur-
Control torque in nominal case with disturbances
Finally, numerical examples are presented of a sample data implementation of the proposed observer-based control law that demonstrates robustness to uncertainties in the spacecraft inertial matrix, non-zero orbital eccentricity and measurement noise, as well as convergence to large initial attitude errors. It should be noted that Jt is only used in the numerical reproduction of the true attitude dynamics.
Attitude and angular velocity responses with inertia matrix uncer-
Effect of α on the transient response
Effect of α on attitude error
Effect of α on the steady-state response
For the selection of the gains, we can first consider the mean reduced closed-loop system corresponding to (4.15) given by. Numerical examples are presented of a sample data implementation of the proposed observer-based control law that demonstrates robustness to uncertainties in the spacecraft inertial matrix, non-zero orbital eccentricity and measurement noise, as well as convergence to large initial attitude errors.
Attitude and angular velocity responses in nominal disturbance-free
Attitude and angular velocity responses in nominal case with distur-
Control torque in nominal case with disturbances
The feasibility of the generated optimal control can be validated by numerically integrating the under-performance federated dynamics of the spacecraft system using a 4th-order Runge-Kutta routine. The spacecraft dynamics are integrated using a 4th order Runge-Kutt with a standard step size of 0.1 seconds.
Attitude and angular velocity responses with inertia matrix uncer-
Optimal MPC Control Interpolation
MPC continuous dynamics control signals
2 A generalized new time-varying attitude control law that achieves convergence in inertial attitude for an underactuated rigid body spacecraft system. 3 Implementation of predictive control of an optimal model for an underactuated spacecraft using pseudospectral optimal control theory with a federated dynamics model to achieve stabilization for an underactuated rigid-body, two-reaction-wheel spacecraft system.
MRP response
Angular momentum response
1 A new nonlinear observer-based attitude control law that achieves attitude convergence without any angular velocity measurements for a nadir-pointing, un-deactivated rigid-body spacecraft system in the presence of time-varying perturbations, non-zero orbital eccentricity, inertia matrix uncertainties, measurement noise in a sample data implementation. Spacecraft attitude control and stabilization: applications of geometric control theory to rigid body models.
