---
title: Control Barrier Certificates Overview
url: https://www.emergentmind.com/topics/control-barrier-certificates-cbcs
type: topic
---

# Control Barrier Certificates Overview

(PrCBCs) under chance constraints. These certificates encode safety conditions as constraints on the control input.

2. Formulating the Occlusion Avoidance Condition  
   • Given a feature point state \( \boldsymbol{s}_i \) and the obstacle’s center \( \boldsymbol{s}_o \) (both in the normalized image plane), the occlusion-free condition is defined by the function  
   \[
   h⁽ᶜ⁾_{i,o}(\boldsymbol{s},\boldsymbol{s}_o) = \|\boldsymbol{s}_i - \boldsymbol{s}_o\|² - R_n²
   \]
   where \( R_n \) is the obstacle’s radius in the normalized image plane. The safe (or admissible) set is then  
   \[
   \mathcal{H}⁽ᶜ⁾ = \{ (\boldsymbol{s}_i, \boldsymbol{s}_o) \in \mathbb{R}² : h⁽ᶜ⁾_{i,o}(\boldsymbol{s},\boldsymbol{s}_o) \geq 0, \forall i \}
   \]

3. Classical Control Barrier Certificates (CBCs)  
   • Without measurement uncertainty, one enforces occlusion avoidance by ensuring that the time derivative of \( h⁽ᶜ⁾ \) is nonnegative when entering the “safety margin.” The admissible control space is defined as:  
   \[
   \mathcal{B}(\boldsymbol{s},\boldsymbol{s}_o) = \{ \boldsymbol{V}_c \in \mathcal{U} \mid \dot{h}⁽ᶜ⁾_{i,o}(\boldsymbol{s},\boldsymbol{s}_o, \boldsymbol{V}_c) + \gamma h⁽ᶜ⁾_{i,o}(\boldsymbol{s},\boldsymbol{s}_o) \geq 0, \forall i \}
   \]
   • The derivative of the barrier function is given by  
   \[
   \dot{h}⁽ᶜ⁾_{i,o}(\boldsymbol{s},\boldsymbol{s}_o, \boldsymbol{V}_c) = 2(\boldsymbol{s}_i - \boldsymbol{s}_o)^\top (L_{s_i} - L_o) \boldsymbol{V}_c - 2R_n L_{or} \boldsymbol{V}_c
   \]
   where \( L_{s_i}, L_o, \) and \( L_{or} \) are interaction matrices derived from the camera model. 

4. Accounting for Measurement Uncertainty: Chance Constraints  
   • Typically, pixel coordinates are noisy, given measurements 
   \[
   \hat{\boldsymbol{s}}_i = \boldsymbol{s}_i + \boldsymbol{w}_i \quad \text{and} \quad \hat{\boldsymbol{s}}_o = \boldsymbol{s}_o + \boldsymbol{w}_o
   \]
   with \( \boldsymbol{w}_i, \boldsymbol{w}_o \sim N(0,\Sigma) \).  
   • To ensure that occlusion avoidance holds with high probability (confidence level \( \sigma \in (0,1) \)), it is required that 
   \[
   P((\boldsymbol{s}, \boldsymbol{s}_o) \in \mathcal{H}⁽ᶜ⁾) \geq \sigma
   \]

5. Formulation of Probabilistic Control Barrier Certificates (PrCBCs)  
   • The PrCBC transforms the chance constraint into a deterministic quadratic control constraint. From the chance constraint 
   \[
   P(\boldsymbol{V}_c \in \mathcal{B}(\boldsymbol{s},\boldsymbol{s}_o)) \geq \sigma
   \]
   it derives a deterministic condition as 
   \[
   \|\Delta\boldsymbol{s}\|² + \frac{2}{\gamma} \Delta\boldsymbol{s}^\top (\Delta L - 4R_n L_r) \boldsymbol{V}_c \geq 2R_n² + \frac{1}{\gamma²} \boldsymbol{V}_c^\top (\Delta L^\top \Delta L) \boldsymbol{V}_c + 4e²
   \]
   where \( \Delta\boldsymbol{s} = \hat{\boldsymbol{s}}_i - \hat{\boldsymbol{s}}_o \), \( \Delta L = L_{s_i} - L_o \), and \( e = \Phi^{-1}(\sigma) \), the quantile of the standard normal distribution. This leads to the constraint 
   \[
   \boldsymbol{V}_c^\top A_{i,o}^\sigma \boldsymbol{V}_c + \boldsymbol{b}_{i,o}^\sigma \boldsymbol{V}_c + c_{i,o} \leq 0, \, \forall i
   \]
   where \( A_{i,o}^\sigma, \boldsymbol{b}_{i,o}^\sigma, \) and \( c_{i,o} \) are derived terms from the transformation.

6. Integration with Model Predictive Control (MPC)  
   • The approach integrates MPC to generate an unconstrained control sequence \( \boldsymbol{V}_c^{(\text{mpc})} \). The control input is filtered through a Quadratic Program (QP):
   \[
   \boldsymbol{V}_c^* = \arg\min_{\boldsymbol{V}} \|\boldsymbol{V} - \boldsymbol{V}_c^{(\text{mpc})}\|²
   \]
   subject to 
   \[
   \boldsymbol{V} \in \mathcal{S}^\sigma(\hat{\boldsymbol{s}}, \hat{\boldsymbol{s}}_o), \quad \|\boldsymbol{V}\| \leq V_{\text{max}}
   \]
   ensuring the final control \( \boldsymbol{V}_c^* \) respects the safety constraints while executing the MPC's planned strategy.

7. Simulation Results and Practical Implications  
   • Simulations reveal that CBCs under perfect conditions achieve occlusion avoidance; however, under noise, PrCBCs ensure robust avoidance. The PrCBC successfully retains minimum distances, maintaining predefined safety thresholds across trials.

The PrCBC formulation here transforms a chance-constrained safety requirement into a deterministic quadratic control condition, offering robust IBVS under uncertainty. This integration allows MPC strategies to be adapted in real-time, effectively handling dynamic environments and measurement imperfection.

Source: https://www.emergentmind.com/topics/control-barrier-certificates-cbcs