Control Barrier Certificates Overview

• Control Barrier Certificates (CBCs) are defined as safety constraints on control inputs that ensure occlusion avoidance by maintaining a safe distance from obstacles.
• Probabilistic CBCs (PrCBCs) transform chance constraints into deterministic quadratic conditions, robustly handling measurement noise and uncertainty.
• Integrating CBCs with Model Predictive Control (MPC) enables real-time control adjustments by filtering unconstrained inputs through a quadratic program ensuring safety margins.

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

1. 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

$$\mathcal{H}⁽ᶜ⁾ = \{ (\boldsymbol{s}_i, \boldsymbol{s}_o) \in \mathbb{R}² : h⁽ᶜ⁾_{i,o}(\boldsymbol{s},\boldsymbol{s}_o) \geq 0, \forall i \}$$

1. 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.

1. 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

$$P((\boldsymbol{s}, \boldsymbol{s}_o) \in \mathcal{H}⁽ᶜ⁾) \geq \sigma$$

1. 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.

1. 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.

1. 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.