Autonomous Satellite Rendezvous via Hybrid Feedback Optimization

TL;DR

Simulations show that this approach provides up to a 98.4\% reduction in the magnitude of disturbances across a range of simulations, which illustrates the viability of hybrid feedback optimization for autonomous satellite rendezvous.

Abstract

As satellites have proliferated, interest has increased in autonomous rendezvous, proximity operations, and docking (ARPOD). A fundamental challenge in these tasks is the uncertainties when operating in space, e.g., in measurements of satellites' states, which can make future states difficult to predict. Another challenge is that satellites' onboard processors are typically much slower than their terrestrial counterparts. Therefore, to address these challenges we propose to solve an ARPOD problem with feedback optimization, which computes inputs to a system by measuring its outputs, feeding them into an optimization algorithm in the loop, and computing some number of iterations towards an optimal input. We focus on satellite rendezvous, and satellites' dynamics are modeled using the continuous-time Clohessy-Wiltshire equations, which are marginally stable. We develop an asymptotically stabilizing controller for them, and we use discrete-time gradient descent in the loop to compute inputs to them. Then, we analyze the hybrid feedback optimization system formed by the stabilized Clohessy-Wiltshire equations with gradient descent in the loop. We show that this model is well-posed and that maximal solutions are both complete and non-Zeno. Then, we show that solutions converge exponentially fast to a ball around a rendezvous point, and we bound the radius of that ball in terms of system parameters. Simulations show that this approach provides up to a 98.4\% reduction in the magnitude of disturbances across a range of simulations, which illustrates the viability of hybrid feedback optimization for autonomous satellite rendezvous.

Figures (8)

• Figure 1: The target and chaser satellites are orbiting around the earth and the green line defines a rendezvous trajectory. The axes of each satellite are defined as follows: the radial vectors $x_t$ and $x_c$ point away from the earth, the tangential vectors $y_t$ and $y_c$ point in the direction each satellite is moving along its orbit, and the out-plane vectors $z_t$ and $z_c$ are perpendicular to the orbital planes formed by $x_t/y_t$ and $x_c/y_c$, respectively.
• Figure 2: Diagram of the hybrid system model of feedback optimization $\mathcal{H}_{\text{FO}}$ from \ref{['eq:hybridFO']} with the stabilized CW equations and in-the-loop optimization. The optimization algorithm receives the sampled output $\mathbf{y}_s$ and computes the next input $\mathbf{u}$ with timing determined by the timers $\tau_{c}$ and $\tau_{g}$ reaching zero.
• Figure 3: A visual representation of the evolution of the input $\mathbf{u}$. There are $\alpha(p)$ iterations of gradient descent when computing the $(p+1)^{\textnormal{th}}$ input. The first two changes in the input occur at the hybrid times $(t_{\alpha(0) + 1}, \alpha(0) + 1)$ and $(t_{\alpha(0) + \alpha(1) + 2}, \alpha(0) + \alpha(1) + 2)$. The timer $\tau_{c}$ reaches zero for the second time at the same time that $\tau_{g}$ reaches zero. This occurrence can trigger either a case (i) jump followed by a case (ii) jump or a case (ii) jump followed by a case (i) jump.
• Figure 4: Convergence of the state $\mathbf{x}$ under the dynamics $\mathcal{H}_{\text{FO}}$ from the initial condition in \ref{['eq:initconds1']}. The value of $\mathbf{x}(t, j)$ is asymptotically close to the chosen rendezvous point $\tilde{\mathbf{x}}(t)$, and the asymptotic value of $\|\mathbf{x}(t, j) - \tilde{\mathbf{x}}(t)\|$ is approximately $8.2\cdot 10^{-2}$ meters.
• Figure 5: Convergence of the rendezvous error $\|\mathbf{x}(t, j) - \tilde{\mathbf{x}}(t)\|$ for the perturbed system $\mathcal{H}_{\text{FO}}^{\rho}$ from the initial condition in \ref{['eq:initconds1']} with $\theta = 1.0$ and varying values of $\kappa$. The perturbed system attains an asymptotic error of at most $\sim$$4.39$ meters even though there are perturbations to the dynamics of $\tau_{c}$ and $\tau_{g}$.

Theorems & Definitions (18)

• Definition 1: Hybrid Basic Conditions Hybridbook
• Theorem 1: Stabilizing Controller Gains
• proof
• Lemma 1
• proof
• Proposition 1: Completeness of Maximal Solutions
• proof
• Lemma 2: Input Convergence Rate
• proof
• Proposition 2: Convergence of $\mathcal{H}_{\text{FO}}$