Appunti di Spacecraft attitude dynamics

Control system design

Review of classical design techniques

Introduction

The S/C attitude dynamics is nonlinear, but as a 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 high pointing accuracy (e.g., because it's got cameras, 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 to withstand mechanical loads.

Flexible appendages - Antennas, solar panels, booms - Built from light materials & causing flexing. We model the body as a rigid central body with the moment of inertia J_c and the appendages as massless beams of length l bearing simply a tip mass m.

Tor_c ⟶ control torque

Tor_d ⟶ disturbance torque

Control system design

The S/C attitude dynamics are nonlinear, but as a 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 high pointing accuracy (e.g., because its payload contains 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 stiff to withstand residual loads.

Flexible appendages - Antennas, solar panels, booms, built from light material + cooling system. We model the body as a rigid central box with moment of inertia J_b and the appendages as massless beams of length l, bearing simply B₃ carrying a tip mass m.

T_c -> control torque

T_d -> disturbance torque

If we suppose to have rigid appendages, we obtain a one degree of freedom system with DoF θ(t)

T = 1/2 J₀ θ̇² + 2⋅1/2 m s² 1/2 J₀ θ̇² + 2⋅1/2 m (ℓ²) θ̇² = 1/2 (J₀ + 2 m ℓ²) θ̇² (m like rigid case)

δWuc = δT₀Ic + δT d = δQd → virtual work the control and disturbance torque

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

d/dt (∂L/∂θ̇) - ∂L/∂θ = Qθ => J θ̈ = Tc + Td equation of work in the blue domain

Laplace domain and transform

Instead of verifying on the time domain, we can work in the Laplace (frequency) domain.

DEF (LAPLACE TRANSFORM) → (math expression) (∫₀^∞ f(t) e⁻ˢᵗ dt)

Where s = σ + jω ∈ ℂ (complex variable) w = frequency (rad/s)

So we can use we have :

Θ(s) = (D(1)) and D(1) = ⁻¹ (Θ(s))

PROPERTIES OF

• [a f₁(t) + b g(t)] = a f(s) + b (s) → linearity - superposition
• [ḟ (t)] = s f(s) - f(0) → of elementary
• L[₀f(t)] = s²f(0) - sf(0) - ₀f(0) → differentiation
• L[∫^t₀f(z)dz] = ¹/_sf(s) → integration

lim f(t) = lim s f(s) → final value thm

t→∞ s→0

f(0) = lim s f(s) → initial value thm

s→∞

Common Laplace Transforms

• L[(t)] = 1 → impulse function
• L[step(t)] = ¹/_s → step function
• L[ramp(t)] = ¹/_s² → ramp function
• L[e^-at step(t)] = ¹/_s+a → exponential function (a>0)
• L[sin(t) step(t)] = /_s²+² → sine function
• L[cos(t) step(t)] = ^s/_s²+² → cosine function

So cascading our case we have that:

L[_0^u]→ T_c(s) + T_d(s) = s²O(s) = T_c(s) + T_d(s)

y(s) = G(s)u(s) + (G(s)d(s)y(s) - system output

u₀(s) - system input, due to the actuator or "t" control

d(s) - disturbance in the input

G(s) - system open loop transfer function

If we consider that the output will be measured by a sensor, which will be subjected by a noise u_m(s), we get,

=> y(s) = G(s)u₀(s) + G(s)d(s) + u_m(s)

Transfer function

Transfer function - Ratio between the Laplace transform of the output of the system and the input assuming all zero initial condition.

G(s) = ^y(s)/_u0(s)

G(s) can be written as the ratio between two polynomial expressions in the complex variable s.

G(s) = N(s)/