Appunti di Spacecraft attitude dynamics

CONTROL SYSTEM DESIGN

REVIEW OF CLASSICAL DESIGN TECHNIQUES

INTRODUCTION

The S/C attitude dynamics is non-linear but as first step we'll perform a linear analysis in a simplified configuration where we consider a single axis rotation.

If the S/C is required to have a large pointing accuracy (e.g. because it's got enormous solar panels) this attitude must be strictly defined and maintained against disturbance torques and passive stabilization is not sufficient, so we equip it with an active attitude control system.

S/C STRUCTURE

• Body of the S/C - contains all the payload instruments, must be very rigid as withstands residual forces.
• Flexible appendages - antennas, solar panels, booms built from light materials and easily bent.

We model the body as a rigid central body with moment of inertia J and the appendages as massless beams of length L bearing empty B.j carrying a tip mass m:

Tc -> control torque

Td -> disturbance torque

If we suppose to have n d.o.f. pendulums we obtain a one degree of freedom system with d₁(t)

T = 1/2 J₀ ḋ₁² + 1/2 m l² ḋ₁² + 1/2 m(l ė₁)² = 1/2 (J₀ + 2m e l) ė₁²

V = 0 (m like rigid case)

δW_nc = δT_c + δT_d + δQ_d = δQ_d - virtual work due external and disturbance torque

Recalling that the Lagrange function L = T(θ, θ̇) - V(θ) we write the Lagrange equation:

d/dt ( ∂L/∂ḋ₁ ) - ∂L/∂θ = Q_θ => J₀ ë = T_c + T_d equation of motion in the time domain

LAPLACE DOMAIN AND TRANSFORM

Instead of working in the time domain we can work in the Laplace (frequency) domain.

DEF (LAPLACE TRANSFORM) → L{f(t)} = F(s) = ∫₀^∞ f(t) e^-st dt

where s = σ + jω ∈ ℂ (complex variable) ω = frequency [rad/s]

So in our case we have:

Θ(s) = L{d(t)} and D(s) = L^-1{Θ(s)}

PROPERTIES OF L

• L{a f(t) + b g(t)} = a F(s) + b G(s) → linearity - superposition
• L{ d/dt^j f(t) } = s f(s) - f(0) → derivative

Properties of the Delta Function

δ(t) = 0 for t ≠ 0

∫_-∞^∞ δ(t) dt = 1 (unit area)

∫_-∞^∞ f(t) δ(t-t₀) dt = f(t₀) (sampling property)

δ(t-t₀) = δ(t₀) δ(t-t₀)

It's possible to compute the system's response to any arbitrary input u(t) by using the impulse response h(t).

- We can decompose the given signal into a sum of elementary components and, by superposition, we conclude that the overall signal is the sum of the response to the elementary signals.

y(t) = ∫₀^∞ h(t-τ) u(τ) dτ → Convolution Integral

Let's suppose that the input signal for an LTI system is a short pulse p(t) and the corresponding output signal is h(t).

If the input is scaled to k₀ p(t), then, by the scaling property of superposition, the output response will be k₀ h(t).

If we delay the short pulse in time by τ, the input has

1)

G(s) = μ

│G│dB

0

│μ│κ→1

ω

∠G

0

μ→0

μ 1.

But we notice that to have good noise rejection we should have |L(jω)| > 1 for low ω