Dynamic Control Barrier Function (D-CBF)

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The CBF value §

• The value $h(x,t)=\Vert x-c_{obs}(t){\Vert }_2-r_{obs}(t)$
• x represents the state of the system (position, angle, velocity…)
• $c_{obs}(t)$ is the center of the obstacles
• $r_{obs}(t)$ is the radius of the obstacles
• $\Vert .{\Vert }_2$ is the euclidean distance
• CBF ensures the distance between the system and obstacles always bigger than the radius of the obstacles to avoid collision

Gradient of CBF with Respect to $x$ §

• Using the chain rule: $\frac{\partial h}{\partial x}=\frac{x-c_{obs}(t)}{\Vert x-c_{obs}(t)\Vert }$

Explicit Time Derivative of CBF §

• Using the chain rule: $\frac{\partial h}{\partial t}=\frac{\partial \Vert x-c_{obs}(t){\Vert }_2}{\partial t}-\frac{\partial r_{obs}(t)}{\partial t}$
• The first term is the time derivative of the distance: $\frac{\partial \Vert x-c_{obs}(t){\Vert }_2}{\partial t}=\frac{\Vert x-c_{obs}(t+\partial t){\Vert }_2-\Vert x-c_{obs}(t){\Vert }_2}{\partial t}$
• The second term is the time derivative of the obstacle radius: $\frac{\partial r_{obs}(t)}{\partial t}$

Approximating Time Derivatives Using Finite Differences §

• Using the forward difference method: $\frac{\partial f}{\partial t}(t)\approx \frac{f(t+\delta t)-f(t)}{\delta t}$
• For the obstacle center: $\frac{\partial c_{obs}(t)}{\partial t}\approx \frac{c_{obs}(t+\delta t)-c_{obs}(t)}{\delta t}$
• For the obstacle radius: $\frac{\partial r_{obs}(t)}{\partial t}\approx \frac{r_{obs}(t+\delta t)-r_{obs}(t)}{\delta t}$