Real-Time Sun-Avoidance Satellite
Attitude Control with Safety Guarantees

A Unified CLF and HOCBF Formulation

Xinkun Li , Yuhan Liu* , Guangfu Ma , Chuanjiang Li , Ming Li

1 Dept. of Control Science & Engineering, Harbin Institute of Technology, Harbin, China

2 Division of Decision and Control Systems, KTH Royal Institute of Technology, Stockholm, Sweden

* Corresponding author: yhliu@hit.edu.cn

OUTLINE

In-Orbit Safety-Critical Control Motivation

03/15

1

In-Orbit Irreversibility & Photosensitive Payload Protection

In-Orbit Irreversibility: Spacecraft operate in harsh, unserviceable orbital environments where physical repairs post-launch are 100% impossible. Safety-critical control is vital for mission survival.

Photosensitive Sensor Blindness: Star trackers, optical telescopes, and infrared detectors suffer permanent CCD thermal damage or image degradation if intense direct sunlight enters the optical boresight.

Coupled Physical Constraints: Spacecraft attitude reorientation must simultaneously satisfy high-order pointing exclusion cones, maximum angular velocity limits, and motor actuator torque saturation bounds.

Summary

Paper Abstract & Core Objectives

Unified Safety Strategy: Develops a safety-critical attitude controller incorporating sun avoidance ($m=2$), angular velocity ($m=1$), and torque bounds into a structured CLF-HOCBF formulation.

Mathematical Proofs: The controller guarantees provable asymptotic stability (Theorem 1) and forward invariance of the pointing safe set (Lemma 2).

Microsecond Flight Execution: Enables real-time Quadratic Programming (QP) optimization with an average solution time of $41.06\,\mu\text{s}$, suitable for onboard flight computers.

SECTION 1: RELATED WORK

Existing Safety Control Paradigms & Limitations

04/15

APF

Artificial Potential Field (APF) Method & Limitations

Advantages: Generates real-time control torque output using attractive forces to target and repulsive forces from obstacles.

Fatal Limitations: Heuristic method lacking rigorous safety proofs. Induces severe control input torque oscillations near obstacle boundaries, risking fluid-damped structural damage. Prone to getting trapped in local equilibrium points (local minima) when target pointing is near forbidden zones.

OPT

Trajectory Planning & Optimization Methods

Advantages: Pre-plans or optimizes global attitude trajectories to avoid hazardous exclusion zones.

Fatal Limitations: Computationally complex iterative non-convex optimization imposing heavy, unacceptable CPU overhead on onboard flight computers. Decoupled planning and tracking stages render it incapable of handling real-time dynamic safety enforcement.

SECTION 2.1: KINEMATICS & GEOMETRY

Attitude Kinematics & Sun-Avoidance Geometry

05/15

1

Quaternion Error Kinematics (Eq. 1-8)

Unit Quaternion (Eq. 1): $\mathbf{q} = [q_0, \mathbf{q}_v^\top]^\top \in \mathbb{S}^3$ avoids gimbal lock singularities in attitude representation.

Error Quaternion $\mathbf{q}_e$ & Rotation Matrix (Eq. 2):

\begin{aligned} \mathbf{q}_e &= \mathbf{q}_d^{-1} \otimes \mathbf{q} = [q_{e0}, \mathbf{q}_{ev}^\top]^\top \\ \mathbf{R}(\mathbf{q}_e) &= (q_{e0}^2 - \|\mathbf{q}_{ev}\|^2)\mathbf{I}_3 + 2\mathbf{q}_{ev}\mathbf{q}_{ev}^\top - 2q_{e0}\mathbf{q}_{ev}^\times \end{aligned}

Error Kinematics (Eq. 8-9): For target $\mathbf{q}_d$, vector kinetics $\dot{\mathbf{q}}_{ev}$ and error rate $\dot{\boldsymbol{\omega}}_e$:

\dot{\mathbf{q}}_{ev} = \frac{1}{2}(q_{e0}\mathbf{I}_3 + \mathbf{q}_{ev}^\times)\boldsymbol{\omega}_e, \quad \dot{\boldsymbol{\omega}}_e = \dot{\boldsymbol{\omega}} - \boldsymbol{\omega}_e^\times \mathbf{R}(\mathbf{q}_e)\boldsymbol{\omega}_d

Fig 1

Geometric Representation

Fig. 1. Geometric representation of sun-avoidance pointing constraint

SECTION 2.2: DYNAMICS & CONSTRAINTS

Control-Affine State Space & Triple Physical Constraints

06/15

1

Control-Affine State Space (Eq. 10-12)

State Vector & Domain: Define attitude error state vector $\mathbf{x} = [\mathbf{q}_{ev}^\top, \boldsymbol{\omega}_e^\top]^\top \in \mathcal{X} \subset \mathbb{R}^6$ on manifold $\mathcal{X} = \mathbb{S}^3 \times \mathbb{R}^3$.

System Equation of Motion (Eq. 11):

\dot{\mathbf{x}} = f(\mathbf{x}) + g(\mathbf{x})\mathbf{u}

Drift Vector Field $f(\mathbf{x})$ (Eq. 12):

f(\mathbf{x}) = \begin{bmatrix} f_q(\mathbf{x}) \\ f_\omega(\mathbf{x}) \end{bmatrix} = \begin{bmatrix} \frac{1}{2}(q_{e0}\mathbf{I}_3 + \mathbf{q}_{ev}^\times)\boldsymbol{\omega}_e \\ -\mathbf{J}^{-1}(\boldsymbol{\omega}^\times \mathbf{J}\boldsymbol{\omega}) \end{bmatrix}

Control Input Matrix $g(\mathbf{x})$ (Eq. 12):

g(\mathbf{x}) = \begin{bmatrix} g_q(\mathbf{x}) \\ g_\omega(\mathbf{x}) \end{bmatrix} = \begin{bmatrix} \mathbf{0}_{3\times 3} \\ \mathbf{J}^{-1} \end{bmatrix}

2

Triple Physical Constraints (Eq. 13-16)

(a) Sun Exclusion Cone $h_q(\mathbf{x})$ (Eq. 13-14):

h_q(\mathbf{x}) = \cos\theta_{safe} - \cos\theta \ge 0, \quad \cos\theta = \frac{\mathbf{s}^\top \mathbf{R}(\mathbf{q})\mathbf{n}}{\|\mathbf{s}\|\|\mathbf{n}\|}

Safe set $\mathcal{C}_q = \{\mathbf{x} : h_q(\mathbf{x}) \ge 0\}$ with minimum safe angle $\theta_{safe} = 30.0^\circ$.

(b) Angular Velocity Limits (Eq. 15):

h_{\omega i}(\mathbf{x}) = \omega_{max}^2 - \omega_i^2 \ge 0, \quad \|\boldsymbol{\omega}\|_\infty \le \omega_{max} = 10.0^\circ/\text{s}

(c) Motor Torque Saturation (Eq. 16):

\mathbf{u}_{min} \preceq \mathbf{u} \preceq \mathbf{u}_{max} = [6, 10, 6]^\top\,\text{N}\cdot\text{m}

SECTION 3.1: PRELIMINARIES

Theoretical Foundations: CLF & HOCBF Concepts

07/15

1

Control Lyapunov Function (CLF)

Lemma 1 (Sontag, 1989): A continuously differentiable, positive definite function $V(\mathbf{x})$ is a CLF if $\mathcal{L}_g V(\mathbf{x}) = \mathbf{0} \implies \mathcal{L}_f V(\mathbf{x}) < 0$.

To enforce exponential convergence, the CLF condition is strengthened to (Eq. 18):

\dot{V}(\mathbf{x}) = \mathcal{L}_f V(\mathbf{x}) + \mathcal{L}_g V(\mathbf{x})\mathbf{u} < -\gamma V(\mathbf{x})

where $\gamma$ is a class-$\mathcal{K}$ function guaranteeing exponential tracking convergence.

2

Higher-Order CBF (HOCBF)

Relative Degree $m$: Smallest positive integer such that $\mathcal{L}_g \mathcal{L}_f^{m-1} h(\mathbf{x}) \not\equiv \mathbf{0}$.

Recursive Virtual Variables (Eq. 20): intermediate variables defined as:

\psi_k(\mathbf{x}) = \dot{\psi}_{k-1}(\mathbf{x}) + \beta_k(\psi_{k-1}(\mathbf{x})), \quad \psi_0(\mathbf{x}) = h(\mathbf{x})

Lemma 2 (Xiao & Belta, 2022): If $\psi_k(\mathbf{x}(0)) > 0$ for $k=1,\dots,m$, then $\psi_m(\mathbf{x}(t)) \ge 0 \implies \mathbf{x}(t) \in \mathcal{C}$ for all $t \ge 0$.

SECTION 3.2: CLF STABILITY

Theorem 1: Candidate CLF & Schur Complement Proof

08/15

Thm 1

Candidate CLF & Schur Proof (Eq. 22)

Candidate Lyapunov function $V(\mathbf{x}_e)$ for error state $\mathbf{x}_e = [\mathbf{q}_{ev}^\top, \boldsymbol{\omega}_e^\top]^\top \in \mathbb{R}^6$:

V(\mathbf{x}_e) = \frac{\eta}{2}\mathbf{q}_{ev}^\top \mathbf{q}_{ev} + \frac{\lambda}{2}\boldsymbol{\omega}_e^\top \boldsymbol{\omega}_e + \frac{\epsilon}{2}\mathbf{q}_{ev}^\top \boldsymbol{\omega}_e = \frac{1}{2}\mathbf{x}_e^\top \mathbf{M} \mathbf{x}_e

Theorem 1 (Positive Definiteness Condition):

$V(\mathbf{x}_e)$ is a valid candidate CLF when $\epsilon < 2\sqrt{\lambda \eta}$.
Proof (Schur Complement): $\mathbf{M} = \begin{bmatrix} \eta \mathbf{I}_3 & \frac{\epsilon}{2}\mathbf{I}_3 \\ \frac{\epsilon}{2}\mathbf{I}_3 & \lambda \mathbf{I}_3 \end{bmatrix} \succ 0 \iff \lambda \eta \mathbf{I}_3 \succ \frac{\epsilon^2}{4}\mathbf{I}_3 \iff \epsilon < 2\sqrt{\lambda\eta}$.

2

Lie Derivatives & Exponential CLF (Eq. 24-27)

Input coefficient: $\mathcal{L}_g V = -\left(\lambda \boldsymbol{\omega}_e^\top + \frac{\epsilon}{2}\mathbf{q}_{ev}^\top\right)\mathbf{J}^{-1}$ (Eq. 24).

When $\mathcal{L}_g V = \mathbf{0} \implies \boldsymbol{\omega}_e^\top = -\frac{\epsilon}{2\lambda}\mathbf{q}_{ev}^\top \implies \mathcal{L}_f V = -\frac{\epsilon}{4\lambda}q_{e0}\|\mathbf{q}_{ev}\|^2 (2\eta - \frac{\epsilon^2}{2\lambda}) < 0$ (Eq. 26).

\underbrace{\mathcal{L}_f V(\mathbf{x}_e) + \gamma V(\mathbf{x}_e)}_{a(\mathbf{x})} + \underbrace{\mathcal{L}_g V(\mathbf{x}_e)}_{b(\mathbf{x})}\mathbf{u} \le \zeta

SECTION 3.3: HOCBF SAFETY

Sun Avoidance Relative Degree Proof ($m=2$) & Lemma 2

09/15

1

Relative Degree Proof $m=2$ (Eq. 29-31)

Pointing barrier function $h_q(\mathbf{x}) = \cos\theta_{safe} - \cos\theta$. 1st Lie derivative vanishes (Eq. 29):

\mathcal{L}_g h_q(\mathbf{x}) = \mathbf{0}_{1\times 3} \implies \text{Relative degree } m > 1

2nd Lie derivative term is non-zero (Eq. 31):

\mathcal{L}_g \mathcal{L}_f h_q(\mathbf{x}) = -\frac{\mathbf{s}^\top \mathbf{R}(\mathbf{q})\mathbf{n}^\times}{\|\mathbf{s}\|\|\mathbf{n}\|} \mathbf{J}^{-1} \not\equiv \mathbf{0}_{1\times 3} \implies m = 2

Lem 2

Lemma 2 Forward Invariance (Eq. 32)

Construct 2nd-order HOCBF condition $\psi_2(\mathbf{x}) \ge 0$ (Eq. 32):

\psi_2(\mathbf{x}) = \underbrace{\mathcal{L}_f^2 h_q + (\beta_1+\beta_2)\mathcal{L}_f h_q + \beta_1\beta_2 h_q}_{c_q(\mathbf{x})} + \underbrace{\mathcal{L}_g \mathcal{L}_f h_q}_{d_q(\mathbf{x})}\mathbf{u} \ge 0

Lemma 2 (Forward Invariance of Safe Set $\mathcal{C}_q$):

If $\psi_1(\mathbf{x}(0)) \ge 0$ and control torque $\mathbf{u}$ satisfies $c_q(\mathbf{x}) + d_q(\mathbf{x})\mathbf{u} \ge 0$ for all $t \ge 0$, then pointing safe set $\mathcal{C}_q$ is strictly forward-invariant forever!

SECTION 3.4 - 3.5: CONTROLLER

Velocity CBF ($m=1$) & Unified Convex QP Controller

10/15

1

Angular Velocity CBF ($m=1$) (Eq. 33-35)

Velocity barrier $h_{\omega i}(\mathbf{x}) = \omega_{max}^2 - \omega_i^2 \ge 0$ ($m=1$). Compact matrix inequality (Eq. 35):

\mathbf{c}_\omega(\mathbf{x}) + \mathbf{d}_\omega(\mathbf{x})\mathbf{u} \succeq \mathbf{0}_{3\times 1}

Clamps 3-axis angular velocities within $\pm 10.0^\circ/\text{s}$ to protect flexible solar arrays.

QP

Unified Convex QP Optimization (Eq. 36)

Unifying CLF, HOCBF, velocity CBF, and torque saturation into convex QP (Eq. 36):

\min_{\mathbf{u}\in\mathbb{R}^3, \, \zeta\in\mathbb{R}} \;\; \frac{1}{2}\mathbf{u}^\top \mathbf{u} + \frac{1}{2} r \zeta^2

\text{s.t.} \quad \begin{bmatrix} a(\mathbf{x}) + b(\mathbf{x})\mathbf{u} \le \zeta \\ c_q(\mathbf{x}) + d_q(\mathbf{x})\mathbf{u} \ge 0 \\ \mathbf{c}_\omega(\mathbf{x}) + \mathbf{d}_\omega(\mathbf{x})\mathbf{u} \succeq \mathbf{0}_{3\times 1} \\ \mathbf{u}_{min} \preceq \mathbf{u} \preceq \mathbf{u}_{max}, \;\; \zeta \ge 0 \end{bmatrix}

Slack Mechanism ($r=10^6$): Relaxes tracking under extreme maneuvers while keeping safety 100% strictly enforced!

SECTION 4.2: SIMULATION RESULTS

Sensor Pointing Trajectory Benchmark (Fig. 2)

11/15

Fig 2

Sensor Pointing Trajectories

Fig. 2. (a) Inertial frame pointing & (b) Sensor pointing polar plot

2

Comparative Analysis of Baselines

Proposed CLF-CBF (Blue Solid): Maneuvers smoothly along the $30.0^\circ$ boundary with zero safety boundary violation.

Standard CLF (Orange Dashed): Penetrates the forbidden cone, destroying optical detectors.

APF Method (Green Dotted): Trapped in local minima near obstacle where repulsive and attractive forces cancel out.

SECTION 4.2: SIMULATION RESULTS

Quaternion Convergence & Angular Velocity (Fig. 3-4)

12/15

Fig 3

Quaternion Error History

Fig. 3. (a) Vector $\mathbf{q}_v$ & (b) Error $1 - |q_{e0}| \to 10^{-3}$

Fig 4

Angular Velocity Limit

Fig. 4. Angular velocity profiles: $\pm 10.0^\circ/\text{s}$ limit compliance

SECTION 4.2 - 4.3: SIMULATION & CPU

Control Torques & Flight CPU Speed (Fig. 5 & Table)

13/15

Fig 5

Control Input Torque

Fig. 5. Control torque signals: CLF-CBF smooth vs APF severe chatter

CPU

Microsecond Flight CPU Benchmark

Hardware: Intel i7-13700HX, 16GB DDR5, $100\,\text{Hz}$ Loop.

Execution Metric	Time (μs)	Flight CPU Assessment
Mean QP Time	41.06 μs	Ideal for $100\,\text{Hz} - 1000\,\text{Hz}$ loops
Std. Dev.	17.53 μs	Extremely low jitter
WCET Worst	341.30 μs	Consumes only $3.4\%$ of $10\,\text{ms}$ period

Conclusion & Future Research

Core Contributions

1

Unified CLF-HOCBF strategy encoding sun avoidance ($m=2$), angular velocity bounds ($m=1$), and torque saturation in a single QP

2

Provable asymptotic stability (Theorem 1) and forward invariance of safe set (Lemma 2)

3

Microsecond computation: mean 41.06 us, worst-case 341.30 us at 100 Hz

4

Eliminates APF local minima and control oscillations

Future Research

Flexible spacecraft dynamics with vibration constraints in CBFs
Multi-sensor exclusion zones with moving celestial bodies
Complex multi-constraint mission scenarios

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Real-Time Sun-Avoidance Satellite
Attitude Control with Safety Guarantees

Table of Contents

In-Orbit Safety-Critical Control Motivation

Existing Safety Control Paradigms & Limitations

Attitude Kinematics & Sun-Avoidance Geometry

Control-Affine State Space & Triple Physical Constraints

Theoretical Foundations: CLF & HOCBF Concepts

Theorem 1: Candidate CLF & Schur Complement Proof

Sun Avoidance Relative Degree Proof ($m=2$) & Lemma 2

Velocity CBF ($m=1$) & Unified Convex QP Controller

Sensor Pointing Trajectory Benchmark (Fig. 2)

Quaternion Convergence & Angular Velocity (Fig. 3-4)

Control Torques & Flight CPU Speed (Fig. 5 & Table)

Conclusion & Future Research

Thank You for Your Attention!

SciBabel-SI v1.0: Scientific Simultaneous Interpretation Model

Real-Time Sun-Avoidance SatelliteAttitude Control with Safety Guarantees

Table of Contents

In-Orbit Safety-Critical Control Motivation

Existing Safety Control Paradigms & Limitations

Attitude Kinematics & Sun-Avoidance Geometry

Control-Affine State Space & Triple Physical Constraints

Theoretical Foundations: CLF & HOCBF Concepts

Theorem 1: Candidate CLF & Schur Complement Proof

Sun Avoidance Relative Degree Proof ($m=2$) & Lemma 2

Velocity CBF ($m=1$) & Unified Convex QP Controller

Sensor Pointing Trajectory Benchmark (Fig. 2)

Quaternion Convergence & Angular Velocity (Fig. 3-4)

Control Torques & Flight CPU Speed (Fig. 5 & Table)

Conclusion & Future Research

Thank You for Your Attention!

SciBabel-SI v1.0: Scientific Simultaneous Interpretation Model

Real-Time Sun-Avoidance Satellite
Attitude Control with Safety Guarantees