$\mathcal{O}_{i,\alpha}^{M}$. In other words, there exist locations
$\mathbf{b}$ and $\mathbf{e}$, where
$\mathbf{b},\mathbf{e}\in\partial\mathcal{O}_{i,\alpha}^{M}$ such that the
line segment
$\mathcal{L}_{s}(\mathbf{b},\mathbf{e})\subset\mathcal{L}_{s}(\mathbf{h}_{\beta},\mathbf{0})\cap\mathcal{O}_{i,\alpha}^{M}$,
as shown in Fig. 21. We further consider two more sub-cases based on the
distance between the target $\mathbf{0}$ and the modified obstacle
$\mathcal{O}_{i,\alpha}^{M}$. Note that, as per Assumption 1,
$d(\mathbf{0},\mathcal{O}_{i,\alpha}^{M})>r_{a}.$

Case A1: When $d(\mathbf{0},\mathcal{O}_{i,\alpha}^{M})\geq\beta,$ as shown in
Fig. 21a, one has
$\lVert\mathbf{h}_{\beta}\rVert-\lVert\mathbf{p}_{\beta}\rVert\geq\lVert\mathbf{h}_{\beta}\rVert-\lVert\mathbf{e}\rVert+\lVert\mathbf{b}\rVert-\lVert\mathbf{p}_{\beta}\rVert.$

Since
$\mathbf{h}_{\beta}\in\partial\mathcal{D}_{\beta}(\mathcal{O}_{i,\alpha}^{M}),\\{\mathbf{b},\mathbf{e}\\}\subset\partial\mathcal{O}_{i,\alpha}^{M}$
and $d(\mathbf{0},\mathcal{O}_{i,\alpha}^{M})\geq\beta$, one has
$\lVert\mathbf{h}_{\beta}\rVert-\lVert\mathbf{e}\rVert\geq\beta$ and
$\lVert\mathbf{b}\rVert\geq\beta+\lVert\mathbf{p}_{\beta}\rVert.$ Hence,

$\displaystyle\lVert\mathbf{h}_{\beta}\rVert-\lVert\mathbf{p}_{\beta}\rVert\geq
2\beta=2(\beta-r_{a})+2r_{a},$ $\displaystyle
d(\mathbf{h}_{\beta},\mathcal{B}_{2(\beta-r_{a})}(\mathbf{p}_{\beta}))\geq
2r_{a}.$ (63)

Case A2: When $r_{a}<d(\mathbf{0},\mathcal{O}_{i,\alpha}^{M})<\beta,$ as shown
in Fig. 21b, one has
$\lVert\mathbf{h}_{\beta}\rVert-\lVert\mathbf{p}_{\beta}\rVert\geq\lVert\mathbf{h}_{\beta}\rVert-\lVert\mathbf{e}\rVert+\lVert\mathbf{b}\rVert-\lVert\mathbf{p}_{\beta}\rVert.$

Since
$\mathbf{h}_{\beta}\in\partial\mathcal{D}_{\beta}(\mathcal{O}_{i,\alpha}^{M}),\\{\mathbf{b},\mathbf{e}\\}\subset\partial\mathcal{O}_{i,\alpha}^{M}$,
$r_{a}<d(\mathbf{0},\mathcal{O}_{i,\alpha}^{M})<\beta$ and by construction, as
shown in Fig. 21b, one has
$\lVert\mathbf{h}_{\beta}\rVert-\lVert\mathbf{e}\rVert\geq\beta$ and
$\lVert\mathbf{b}\rVert\geq\beta-\lVert\mathbf{p}_{\beta}\rVert=r_{a}+\lVert\mathbf{p}_{r_{a}}\rVert.$
Hence, as
$\lVert\mathbf{p}_{r_{a}}\rVert=d(\mathbf{0},\mathcal{D}_{r_{a}}(\mathcal{O}_{i,\alpha}^{M}))$,
one has

$\displaystyle\lVert\mathbf{h}_{\beta}\rVert-\lVert\mathbf{p}_{\beta}\rVert\geq
2\beta-2\lVert\mathbf{p}_{\beta}\rVert=2\lVert\mathbf{p}_{r_{a}}\rVert+2r_{a},$
$\displaystyle
d(\mathbf{h}_{\beta},\mathcal{B}_{2d(\mathbf{0},\mathcal{D}_{r_{a}}(\mathcal{O}_{i,\alpha}^{M}))}(\mathbf{p}_{\beta}))>2r_{a}.$
(64)

Case B: When $\mathbf{h}_{\beta}\in\mathcal{G}$, where the set
$\mathcal{G}:=\left(\partial\mathcal{D}_{\beta}(\mathcal{O}_{i,\alpha}^{M})\cap\mathcal{CH}(\mathbf{0},\mathcal{D}_{r_{a}}(\mathcal{O}_{i,\alpha}^{M}))\right)\setminus\mathcal{CH}(\mathbf{0},\mathcal{O}_{i,\alpha}^{M})$.
In other words, when the line segment joining the locations
$\mathbf{h}_{\beta}$ and the target at the origin does not intersect with the
interior of the modified obstacle $\mathcal{O}_{i,\alpha}^{M}$ i.e.,
$\mathcal{L}_{s}(\mathbf{h}_{\beta},\mathbf{0})\cap(\mathcal{O}_{i,\alpha}^{M})^{\circ}=\emptyset,$
as shown in Fig. 21. To proceed with the proof, we use the following fact
which states that the projection of the target location onto the set
$\mathcal{G}$ always belongs to the intersection between the boundary of the
conic hull
$\mathcal{CH}(\mathbf{0},\mathcal{D}_{r_{a}}(\mathcal{D}_{r_{a}}(\mathcal{O}_{i,\alpha}^{M})))$
and the set $G$.

Fact 2: The set
$\mathcal{PJ}(\mathbf{0},\mathcal{G})\subset\mathcal{G}\cap\partial\mathcal{CH}(\mathbf{0},\mathcal{D}_{r_{a}}(\mathcal{O}_{i,\alpha}^{M})).$

Sketch of the proof: The proof is by contradiction. We assume that there
exists a location $\mathbf{x}$ in the set
$\mathcal{PJ}(\mathbf{0},\mathcal{G})$ that does not belong to the
intersection between the set $\mathcal{G}$ and the boundary of the conic hull
to the set $\mathcal{D}_{r_{a}}(\mathcal{O}_{i,\alpha}^{M})$ i.e.,
$\mathbf{x}\in\mathcal{PJ}(\mathbf{0},\mathcal{G})$ and
$\mathbf{x}\notin\mathcal{G}\cap\partial\mathcal{CH}(\mathbf{0},\mathcal{D}_{r_{a}}(\mathcal{O}_{i,\alpha}^{M}))$.
We know that the curve $\mathcal{G}$ belongs to the boundary of the
$\beta-$dilated modified obstacle i.e.,
$\mathcal{G}\subset\partial\mathcal{D}_{\beta}(\mathcal{O}_{i,\alpha}^{M})$.
As a result, there exists a partial section of the boundary of the modified
obstacle $\mathcal{O}_{i,\alpha}^{M}$, let say
$\mathcal{M}\subset\partial\mathcal{O}_{i,\alpha}^{M}$ with
$\mathbf{x}\in\mathcal{D}_{\beta}(\mathcal{M})$ such that this curve
$\mathcal{M}$ belongs to the relative complement of the conic hull
$\mathcal{CH}(\mathbf{0},\mathcal{O}_{i,\alpha}^{M})$ with respect to the
conic hull
$\mathcal{CH}(\mathbf{0},\mathcal{D}_{r_{a}}(\mathcal{O}_{i,\alpha}^{M}))$
i.e.,
$\mathcal{M}\in\mathcal{CH}(\mathbf{0},\mathcal{D}_{r_{a}}(\mathcal{O}_{i,\alpha}^{M}))\setminus\mathcal{CH}(\mathbf{0},\mathcal{O}_{i,\alpha}^{M})$.
However, by construction of the set $\mathcal{G}$, the intersection between
the modified obstacle $\mathcal{O}_{i,\alpha}^{M}$ and the region
$\mathcal{CH}(\mathbf{0},\mathcal{D}_{r_{a}}(\mathcal{O}_{i,\alpha}^{M}))\setminus\mathcal{CH}(\mathbf{0},\mathcal{O}_{i,\alpha}^{M})$
must be an empty set. Therefore, we arrive at a contradiction.

According to Fact 2, the location
$\mathbf{h}_{\beta}\in\mathcal{G}\cap\partial\mathcal{CH}(\mathbf{0},\mathcal{D}_{r_{a}}(\mathcal{D}_{r_{a}}(\mathcal{O}_{i,\alpha}^{M})))$,
as shown in Fig. 22. Similar to case A, we consider two more sub-cases based
on the distance between the target $\mathbf{0}$ and the modified obstacle
$\mathcal{O}_{i,\alpha}^{M}$.

Case B1: When $d(\mathbf{0},\mathcal{O}_{i,\alpha}^{M})\geq\beta,$ as shown in
Fig 22a, one has

$\lVert\mathbf{h}_{\beta}\rVert-\lVert\mathbf{p}_{\beta}\rVert=\lVert\mathbf{h}_{\beta}\rVert-\lVert\mathbf{h}_{r_{a}}\rVert+\lVert\mathbf{h}_{r_{a}}\rVert-\lVert\mathbf{p}_{\beta}\rVert.$
(65)

Since the location
$\mathbf{h}_{\beta}\in\partial\mathcal{D}_{\beta}(\mathcal{O}_{i,\alpha}^{M})$
and the location
$\mathbf{h}_{r_{a}}\in\partial\mathcal{D}_{r_{a}}(\mathcal{O}_{i,\alpha}^{M})$,
$\lVert\mathbf{h}_{\beta}\rVert-\lVert\mathbf{h}_{r_{a}}\rVert\geq\beta-
r_{a}$. Let $\mathbf{a}\in\partial\mathcal{O}_{i,\alpha}^{M}$ be the location
such that the lines $\mathcal{L}(\mathbf{a},\mathbf{h}_{r_{a}})$ and
$\mathcal{L}(\mathbf{h}_{r_{a}},\mathbf{0})$ are perpendicular, as shown in
Fig. 21a. Note that, as the line $\mathcal{L}(\mathbf{h}_{r_{a}},\mathbf{0})$
is tangent to the set $\mathcal{D}_{r_{a}}(\mathcal{O}_{i,\alpha}^{M})$ at
$\mathbf{h}_{r_{a}}$, the distance $d(\mathbf{h}_{r_{a}},\mathbf{a})=r_{a}.$
Hence, one has

$\displaystyle\lVert\mathbf{h}_{r_{a}}\rVert$
$\displaystyle=\sqrt{\lVert\mathbf{a}\rVert^{2}-d(\mathbf{h}_{r_{a}},\mathbf{a})^{2}}\geq\sqrt{d(\mathbf{0},\mathcal{O}_{i,\alpha}^{M})^{2}-r_{a}^{2}}.$
(66)

After substituting (66) in (65), one gets

$\lVert\mathbf{h}_{\beta}\rVert-\lVert\mathbf{p}_{\beta}\rVert\geq\beta-
r_{a}+\sqrt{d(\mathbf{0},\mathcal{O}_{i,\alpha}^{M})^{2}-r_{a}^{2}}-\lVert\mathbf{p}_{\beta}\rVert.$

Since $d(\mathbf{0},\mathcal{O}_{i,\alpha}^{M})\geq\beta$,
$\lVert\mathbf{p}_{r_{a}}\rVert-\lVert\mathbf{p}_{\beta}\rVert=\beta-r_{a}.$
Hence, one has

$\displaystyle\lVert\mathbf{h}_{\beta}\rVert-\lVert\mathbf{p}_{\beta}\rVert\geq
2(\beta-
r_{a})+\sqrt{d(\mathbf{0},\mathcal{O}_{i,\alpha}^{M})^{2}-r_{a}^{2}}-\lVert\mathbf{p}_{r_{a}}\rVert,$
$\displaystyle d(\mathbf{h}_{\beta},\mathcal{B}_{2(\beta-
r_{a})}(\mathbf{p}_{\beta}))\geq\sqrt{d(\mathbf{0},\mathcal{O}_{i,\alpha}^{M})^{2}-r_{a}^{2}}-\lVert\mathbf{p}_{r_{a}}\rVert.$
(67)

Case B2: When $r_{a}<d(\mathbf{0},\mathcal{O}_{i,\alpha}^{M})<\beta,$ as shown
in Fig. 22b. According to (65), (66) and the fact that
$\lVert\mathbf{h}_{\beta}\rVert-\lVert\mathbf{h}_{r_{a}}\rVert\geq\beta-
r_{a}$, one has

$\lVert\mathbf{h}_{\beta}\rVert-\lVert\mathbf{p}_{\beta}\rVert\geq\beta-
r_{a}+\sqrt{d(\mathbf{0},\mathcal{O}_{i,\alpha}^{M})^{2}-r_{a}^{2}}-\lVert\mathbf{p}_{\beta}\rVert,$

Since $r_{a}<d(\mathbf{0},\mathcal{O}_{i,\alpha}^{M})<\beta$,
$\lVert\mathbf{p}_{\beta}\rVert+\lVert\mathbf{p}_{r_{a}}\rVert=\beta-r_{a}.$
Hence, one has

$\displaystyle\lVert\mathbf{h}_{\beta}\rVert-\lVert\mathbf{p}_{\beta}\rVert\geq
2\lVert\mathbf{p}_{r_{a}}\rVert+\sqrt{d(\mathbf{0},\mathcal{O}_{i,\alpha}^{M})^{2}-r_{a}^{2}}-\lVert\mathbf{p}_{r_{a}}\rVert,$
$\displaystyle
d(\mathbf{h}_{\beta},\mathcal{B}_{2\lVert\mathbf{p}_{r_{a}}\rVert}(\mathbf{p}_{\beta}))\geq\sqrt{d(\mathbf{0},\mathcal{O}_{i,\alpha}^{M})^{2}-r_{a}^{2}}-\lVert\mathbf{p}_{r_{a}}\rVert.$
(68)

Now, considering all the obstacles i.e., for all $i\in\mathbb{I}$, according
to (63), (64), (67) and (68), one has

$d(\mathbf{h}_{\beta},\mathcal{B}_{2\delta}(\mathbf{p}_{\beta}))\geq\epsilon_{h},$
(69)

where $\delta=\min\left\\{\beta-
r_{a},\underset{i\in\mathbb{I}}{\min}\\{d(\mathbf{0},\mathcal{O}_{i,\alpha}^{M})-r_{a}\\}\right\\}=\min\\{\beta-
r_{a},d(\mathbf{0},\mathcal{O}_{\mathcal{W}}^{M})-r_{a}\\}$ and $\epsilon_{h}$
is evaluated as

$\displaystyle\epsilon_{h}$
$\displaystyle=\min\left\\{2r_{a},\underset{i\in\mathbb{I}}{\min}\left\\{\sqrt{d(\mathbf{0},\mathcal{O}_{i,\alpha}^{M})^{2}-r_{a}^{2}}-d(\mathbf{0},\mathcal{D}_{r_{a}}(\mathcal{O}_{i,\alpha}^{M}))\right\\}\right\\},$
$\displaystyle=\min\\{2r_{a},\sqrt{d(\mathbf{0},\mathcal{O}_{\mathcal{W}}^{M})^{2}-r_{a}^{2}}-d(\mathbf{0},\mathcal{D}_{r_{a}}(\mathcal{O}_{\mathcal{W}}^{M}))\\}.$
(70)

It is clear that for $n>0$, the function
$f(k)=\sqrt{k^{2}-n^{2}}-(k-n),d\in[n,\infty),$ is monotonically increasing
and $\underset{k\rightarrow\infty}{\lim}f(k)=n$. Hence, according to
Assumption 1, as $d(\mathbf{0},\mathcal{O}_{\mathcal{W}}^{M})>r_{a}>0$, one
has

$\sqrt{\left(d(\mathbf{0},\mathcal{O}_{\mathcal{W}}^{M})^{2}-r_{a}^{2}\right)}-d(\mathbf{0},\mathcal{D}_{r_{a}}(\mathcal{O}_{\mathcal{W}}^{M}))<r_{a},$

irrespective of the locations of obstacles relative to the target location. As
a result,

$\displaystyle\epsilon_{h}$
$\displaystyle=\min\\{2r_{a},\sqrt{d(\mathbf{0},\mathcal{O}_{\mathcal{W}}^{M})^{2}-r_{a}^{2}}-d(\mathbf{0},\mathcal{D}_{r_{a}}(\mathcal{O}_{\mathcal{W}}^{M}))\\}$
$\displaystyle=\sqrt{\left(d(\mathbf{0},\mathcal{O}_{\mathcal{W}}^{M})^{2}-r_{a}^{2}\right)}-d(\mathbf{0},\mathcal{D}_{r_{a}}(\mathcal{O}_{\mathcal{W}}^{M})).$

####  -H2 Proof of claim 2

In this proof, our goal is to show that the set
$\mathcal{H}_{\mathbf{p}}=\mathcal{B}_{\delta}(\mathbf{p})\cap\mathcal{N}_{\gamma}(\mathcal{D}_{r_{a}}(\mathcal{O}_{i,\alpha}^{M}))$
belongs to the always exit region $\mathcal{R}_{a}$, for any location
$\mathbf{p}\in\mathcal{PJ}(\mathbf{0},\partial\mathcal{D}_{\beta}(\mathcal{O}_{i,\alpha}^{M}))$
and the parameter
$\delta=\min\\{d(\mathbf{0},\mathcal{O}_{\mathcal{W}}^{M})-r_{a},\beta-
r_{a}\\}$, where $\beta\in[r_{a},r_{a}+\gamma_{s}]$. Based on the distance of
the target location from the modified obstacle $\mathcal{O}_{i,\alpha}^{M}$,
we consider two cases as follows:

Case 1: When $d(\mathbf{0},\mathcal{O}_{i,\alpha}^{M})\geq\beta$, as shown in
Fig. 21a and Fig. 22a. Since
$d(\mathbf{0},\mathcal{O}_{i,\alpha}^{M})\geq\beta$, for any location
$\mathbf{p}\in\mathcal{PJ}(\mathbf{0},\mathcal{D}_{\beta}(\mathcal{O}_{i,\alpha}^{M}))$,
the Euclidean ball
$\mathcal{B}_{\lVert\mathbf{p}\rVert+\delta}(\mathbf{0})\subset\mathcal{B}_{d(\mathbf{0},\mathcal{D}_{r_{a}}(\mathcal{O}_{i,\alpha}^{M}))}(\mathbf{0})$,
where $\delta=\min\\{\beta-
r_{a},d(\mathbf{0},\mathcal{O}_{i,\alpha}^{M})-r_{a}\\}=\beta-r_{a}.$ Hence,
for any location
$\mathbf{p}\in\mathcal{PJ}(\mathbf{0},\mathcal{D}_{\beta}(\mathcal{O}_{i,\alpha}^{M}))$,
the Euclidean ball
$\mathcal{B}_{\delta}(\mathbf{p})\subset\mathcal{B}_{d(\mathbf{0},\mathcal{D}_{r_{a}}(\mathcal{O}_{i,\alpha}^{M}))}(\mathbf{0})$.
Moreover,
$\mathcal{B}_{d(\mathbf{0},\mathcal{D}_{r_{a}}(\mathcal{O}_{i,\alpha}^{M}))}(\mathbf{0})\cap\left(\mathcal{D}_{r_{a}}(\mathcal{O}_{i,\alpha}^{M})\right)^{\circ}=\emptyset$.
Hence, according to (28), the set
$\mathcal{H}_{\mathbf{p}}\subset\mathcal{R}_{a}$.

Case 2: When $r_{a}<d(\mathbf{0},\mathcal{O}_{i,\alpha}^{M})<\beta$, as shown
in Fig. 21b and Fig. 22b. According to Lemmas 2 and 3,
$\textbf{card}(\mathcal{PJ}(\mathbf{0},\mathcal{D}_{r_{a}}(\mathcal{O}_{i,\alpha}^{M})))=1$,
i.e., $\Pi(\mathbf{0},\mathcal{D}_{r_{a}}(\mathcal{O}_{i,\alpha}^{M}))$ is
unique. Since, as per Lemma 2,
$\textbf{reach}(\mathcal{D}_{r_{a}}(\mathcal{O}_{i,\alpha}^{M}))=\alpha-
r_{a}$, according to [26, Lemma 4.5], $(\mathcal{B}_{\alpha-
r_{a}}(\mathbf{q}))^{\circ}\cap\mathcal{D}_{r_{a}}(\mathcal{O}_{i,\alpha}^{M})=\emptyset,$
where the location
$\mathbf{q}=\Pi(\mathbf{0},\mathcal{D}_{r_{a}}(\mathcal{O}_{i,\alpha}^{M}))+(\alpha-
r_{a})\frac{\mathbf{q}-\Pi(\mathbf{0},\mathcal{D}_{r_{a}}(\mathcal{O}_{i,\alpha}^{M}))}{\lVert\mathbf{q}-\Pi(\mathbf{0},\mathcal{D}_{r_{a}}(\mathcal{O}_{i,\alpha}^{M}))\rVert}.$
Now, notice that, for the location
$\mathbf{p}\in\partial\mathcal{D}_{\beta}(\mathcal{O}_{i,\alpha}^{M})\cap\mathcal{L}_{s}(\mathbf{q},\Pi(\mathbf{0},\mathcal{D}_{r_{a}}(\mathcal{O}_{i,\alpha}^{M})))$,
$\mathcal{B}_{\delta}(\mathbf{p})\subset\mathcal{B}_{\alpha-
r_{a}}(\mathbf{q})$, where $\delta=\min\\{\beta-
r_{a},d(\mathbf{0},\mathcal{O}_{i,\alpha}^{M})-r_{a}\\}=d(\mathbf{0},\mathcal{O}_{i,\alpha}^{M})-r_{a}<\alpha-
r_{a}$. Hence, if one shows that this location
$\mathbf{p}\in\mathcal{PJ}(\mathbf{0},\partial\mathcal{D}_{\beta}(\mathcal{O}_{i,\alpha}^{M}))$
and
$\textbf{card}(\mathcal{PJ}(\mathbf{0},\partial\mathcal{D}_{\beta}(\mathcal{O}_{i,\alpha}^{M})))=1$,
then, according to (28), it is straightforward to notice that the set
$\mathcal{H}_{\mathbf{p}}\subset\mathcal{R}_{a}$, where the set
$\mathcal{H}_{\mathbf{p}}$ is defined in claims 1 and 2.

To this end, let us assume that
$d(\mathbf{0},\mathcal{D}_{r_{a}}(\mathcal{O}_{i,\alpha}^{M}))=\eta$, where
$r_{a}<\eta<\beta<\alpha.$ Notice that,
$\mathbf{0}\in\mathcal{L}(\Pi(\mathbf{0},\mathcal{D}_{r_{a}}(\mathcal{O}_{i,\alpha}^{M})),\mathbf{q}).$
Since,
$\mathbf{p}\in\mathcal{L}_{s}(\mathbf{q},\Pi(\mathbf{0},\mathcal{D}_{r_{a}}(\mathcal{O}_{i,\alpha}^{M})))\cap\partial\mathcal{D}_{\beta}(\mathcal{O}_{i,\alpha}^{M}),$
the distance
$d(\mathbf{p},\Pi(\mathbf{0},\mathcal{D}_{r_{a}}(\mathcal{O}_{i,\alpha}^{M})))=\beta-
r_{a}.$ Therefore, the distance $d(\mathbf{0},\mathbf{p})=\beta-\eta.$
Moreover, according to Lemmas 1 and 3, the set
$\mathcal{M}:=(\mathcal{O}_{i,\alpha}^{M}\oplus\mathcal{B}_{\alpha}(\mathbf{0}))^{c}$
has the reach equal to $\alpha$, i.e., $\textbf{reach}(\mathcal{M})=\alpha.$
Now, notice that, for
$\beta\in[r_{a},r_{a}+\gamma_{s}],\partial\mathcal{D}_{\beta}(\mathcal{O}_{i,\alpha}^{M})=\partial\mathcal{D}_{\alpha-\beta}(\mathcal{M})$.
Hence, $\mathbf{0}\in\partial\mathcal{D}_{\alpha-\eta}(\mathcal{M})$ and
$\mathbf{p}\in\partial\mathcal{D}_{\alpha-\beta}(\mathcal{M})$. Then,
according to Lemma 2,
$\textbf{card}(\mathcal{PJ}(\mathbf{0},\partial\mathcal{D}_{\alpha-\beta}(\mathcal{M})))=\textbf{card}(\mathcal{PJ}(\mathbf{0},\partial\mathcal{D}_{\beta}(\mathcal{O}_{i,\alpha}^{M})))=1$.
Moreover, as $d(\mathbf{0},\mathbf{p})=\beta-\eta,$ the location
$\mathbf{p}\in\mathcal{PJ}(\mathbf{0},\partial\mathcal{D}_{\beta}(\mathcal{O}_{i,\alpha}^{M}))$.

Figure 21: The diagrammatic representation for the Case A in the proof of
Lemma 36. (a) $d(\mathbf{0},\mathcal{O}_{i,\alpha}^{M})\geq\beta$, (b)
$r_{a}<d(\mathbf{0},\mathcal{O}_{i,\alpha}^{M})<\beta.$ Figure 22: The
diagrammatic representation for the Case B in the proof of Lemma 36. (a)
$d(\mathbf{0},\mathcal{O}_{i,\alpha}^{M})\geq\beta$, (b)
$r_{a}<d(\mathbf{0},\mathcal{O}_{i,\alpha}^{M})<\beta.$

###  -I Proof of Lemma 8

First we prove that the union of the flow and jump sets covers exactly the
obstacle-free state space $\mathcal{K}$. For $m=0$, according to (33) and
(32), by construction we have
$\mathcal{F}_{0}^{\mathcal{W}}\cup\mathcal{J}_{0}^{\mathcal{W}}=\mathcal{V}_{r_{a}}.$
Similarly, for $m\in\\{-1,1\\}$, according to (37) and (34), by construction
we have
$\mathcal{F}_{m}^{\mathcal{W}}\cup\mathcal{J}_{m}^{\mathcal{W}}=\mathcal{V}_{r_{a}}$.
Inspired by [32, Appendix 11], the satisfaction of the following equation:

$\mathcal{F}_{m}^{\mathcal{W}}\cup\mathcal{J}_{m}^{\mathcal{W}}=\mathcal{V}_{r_{a}},m\in\mathbb{M},$
(71)

along with (38) and (39) implies $\mathcal{F}\cup\mathcal{J}=\mathcal{K}.$

Now, inspired by [32, Appendix 1], for the hybrid closed-loop system (42),
with data $\mathcal{H}=(\mathcal{F},\mathbf{F},\mathcal{J},\mathbf{J}),$
define $\mathbf{S}_{\mathcal{H}}(\mathcal{K})$ as the set of all maximal
solutions $\xi$ to $\mathcal{H}$ with $\xi(0,0)\in\mathcal{K}$. Each
$\xi\in\mathbf{S}_{\mathcal{H}}(\mathcal{K})$ has range $\text{rge
}\xi\subset\mathcal{K}=\mathcal{F}\cup\mathcal{J}.$ Furthermore, if for each
$\xi(0,0)\in\mathcal{K}$ there exists one solution and each
$\xi\in\mathbf{S}_{\mathcal{H}}(\mathcal{K})$ is complete, then the set
$\mathcal{K}$ will be in fact forward invariant [33, Definition 3.3]. To that
end using [28, Proposition 6.10], we show the satisfaction of the following
viability condition:

$\mathbf{F}(\mathbf{x},\mathbf{h},m)\cap\mathbf{T}_{\mathcal{F}}(\mathbf{x},\mathbf{h},m)\neq\emptyset,\forall(\mathbf{x},\mathbf{h},m)\in\mathcal{F}\setminus\mathcal{J},$
(72)

which will allow us to establish the completeness of the solution $\xi$ to the
hybrid closed-loop system (42). In (72), the notation
$\mathbf{T}_{\mathcal{F}}(\mathbf{x},\mathbf{h},m)$ denotes the tangent cone
to the set $\mathcal{F}$ at $(\mathbf{x},\mathbf{h},m)$, as defined in Section
12.

Let $(\mathbf{x},\mathbf{h},m)\in\mathcal{F}\setminus\mathcal{J}$, which
implies by (38) and (39) that
$(\mathbf{x},\mathbf{h})\in(\mathcal{F}_{m}^{\mathcal{W}}\setminus\mathcal{J}_{m}^{\mathcal{W}})\times\mathcal{V}_{r_{a}}$
for some $m\in\mathbb{M}$. For
$\mathbf{x}\in(\mathcal{F}_{m}^{\mathcal{W}})^{\circ}\setminus\mathcal{J}_{m}^{\mathcal{W}},m\in\mathbb{M},$
$\mathbf{T}_{\mathcal{F}}(\xi)=\mathbb{R}^{2}\times\mathcal{HP}\times\\{0\\},$
where the set $\mathcal{HP}$ is given by

$\mathcal{HP}=\begin{cases}\begin{matrix}[l]\mathbb{R}^{2},&\text{if
}\mathbf{h}\in(\mathcal{V}_{r_{a}})^{\circ},\\\
\mathcal{P}_{\geq}(\mathbf{0},(\mathbf{h}-\Pi(\mathbf{h},\mathcal{O}_{\mathcal{W}}^{M}))),&\text{if
}\mathbf{h}\in\partial\mathcal{V}_{r_{a}},\end{matrix}\end{cases}$ (73)

where, according to Lemma 3, for $\mathbf{h}\in\partial\mathcal{V}_{r_{a}}$,
the projection $\Pi(\mathbf{h},\mathcal{O}_{\mathcal{W}}^{M})$ is unique.
Since, according to (42), $\dot{\mathbf{h}}=\mathbf{0}$,
$\dot{\mathbf{h}}\in\mathcal{HP}$ and (72) holds,

For $m=0$, according to (32) and (33), one has

$\partial\mathcal{F}_{0}^{\mathcal{W}}\setminus\mathcal{J}_{0}^{\mathcal{W}}\subset\partial\mathcal{D}_{r_{a}}(\mathcal{O}_{\mathcal{W}}^{M})\cap\mathcal{R}_{e},$
(74)

and according to Lemma 3, for
$\mathbf{x}\in\partial\mathcal{F}_{0}^{\mathcal{W}}\setminus\mathcal{J}_{0}^{\mathcal{W}}$,
the set $\mathcal{PJ}(\mathbf{x},\mathcal{O}_{\mathcal{W}}^{M})$ is a
singleton i.e., the projection $\Pi(\mathbf{x},\mathcal{O}_{\mathcal{W}}^{M})$
is unique. Hence,
$\forall\mathbf{x}\in\partial\mathcal{F}_{0}^{\mathcal{W}}\setminus\mathcal{J}_{0}^{\mathcal{W}}$,

$\mathbf{T}_{\mathcal{F}}(\mathbf{x},\mathbf{h},0)=\mathcal{P}_{\geq}(\mathbf{0},(\mathbf{x}-\Pi(\mathbf{x},\mathcal{O}_{\mathcal{W}}^{M})))\times\mathcal{HP}\times\\{0\\},$
(75)

where $\mathcal{HP}$ is defined in (73). Also, according to (23a), for $m=0$,
$\mathbf{u}(\mathbf{x},\mathbf{h},0)=-\kappa_{s}\mathbf{x},\kappa_{s}>0$.
According to (27),
$\mathbf{u}(\mathbf{x},\mathbf{h},0)\in\mathcal{P}_{\geq}(\mathbf{0},(\mathbf{x}-\Pi(\mathbf{x},\mathcal{O}_{\mathcal{W}}^{M})))$
and, according to (42), $\dot{\mathbf{h}}=\mathbf{0}\in\mathcal{HP}$. As a
result, viability condition (72) holds for $m=0$.

For $m\in\\{-1,1\\}$, according to (37) and (34) one has

$\partial\mathcal{F}_{m}^{\mathcal{W}}\setminus\mathcal{J}_{m}^{\mathcal{W}}\subset\partial\mathcal{D}_{r_{a}}(\mathcal{O}_{\mathcal{W}}^{M}),$
(76)

and according to Lemma 3, for
$\mathbf{x}\in\partial\mathcal{F}_{m}^{\mathcal{W}}\setminus\mathcal{J}_{m}^{\mathcal{W}},m\in\\{-1,1\\}$,
the set $\mathcal{PJ}(\mathbf{x},\mathcal{O}_{\mathcal{W}}^{M})$ is a
singleton i.e., the projection $\Pi(\mathbf{x},\mathcal{O}_{\mathcal{W}}^{M})$
is unique and the circle
$\mathcal{B}_{\lVert\mathbf{x}-\Pi(\mathbf{x},\mathcal{O}_{\mathcal{W}}^{M})\rVert}(\mathbf{x})$
intersect with $\partial\mathcal{O}_{\mathcal{W}}^{M}$ only at the location
$\Pi(\mathbf{x},\mathcal{O}_{\mathcal{W}}^{M}).$ Hence,
$\forall\mathbf{x}\in\partial\mathcal{F}_{m}^{\mathcal{W}}\setminus\mathcal{J}_{m}^{\mathcal{W}},m\in\\{-1,1\\},$

$\mathbf{T}_{\mathcal{F}}(\mathbf{x},\mathbf{h},0)=\mathcal{P}_{\geq}(\mathbf{0},(\mathbf{x}-\Pi(\mathbf{x},\mathcal{O}_{\mathcal{W}}^{M})))\times\mathcal{HP}\times\\{0\\},$
(77)

where $\mathcal{HP}$ is defined in (73). Also for $m\in\\{-1,1\\}$, according
to (23a),
$\mathbf{u}(\mathbf{x},\mathbf{h},m)=\kappa_{r}\mathbf{v}(\mathbf{x},m),\kappa_{r}>0.$
Since, according to (24),
$\mathbf{u}(\mathbf{x},\mathbf{h},m)^{\intercal}(\mathbf{x}-\Pi(\mathbf{x},\mathcal{O}_{\mathcal{W}}^{M}))=0$
and, according to (42), $\dot{\mathbf{h}}=\mathbf{0}\in\mathcal{HP},$ the
viability condition in (72) holds true for $m\in\\{-1,1\\}.$

Hence, according to [28, Proposition 6.10], since (72) holds for all
$\xi\in\mathcal{F}\setminus\mathcal{J}$, there exists a nontrivial solution to
$\mathcal{H}$ for each initial condition in $\mathcal{K}$. Finite escape time
can only occur through flow. They can neither occur for $\mathbf{x}$ in the
set $\mathcal{F}_{-1}^{\mathcal{W}}\cup\mathcal{F}_{1}^{\mathcal{W}}$, as this
set is bounded as per definition (37), nor for $\mathbf{x}$ in the set
$\mathcal{F}_{0}^{\mathcal{W}}$ as this would make
$\mathbf{x}^{\intercal}\mathbf{x}$ grow unbounded, and would contradict the
fact that $\frac{d}{dt}(\mathbf{x}^{\intercal}\mathbf{x})\leq 0$ in view of
the definition of $\mathbf{u}(\mathbf{x},\mathbf{h},0)$. Therefore, all
maximal solutions do not have finite escape times. Furthermore, according to
(42), $\mathbf{x}^{+}=\mathbf{x},$ and from the definition of the update law
in (40) and (41), it follows immediately that
$\mathbf{J}(\mathcal{J})\subset\mathcal{K}$. Hence, the solutions to the
hybrid closed-loop system (42) cannot leave $\mathcal{K}$ through jump and, as
per [28, Proposition 6.10], all maximal solutions are complete.

###  -J Proof of Theorem 1

Forward invariance and stability: The forward invariance of the obstacle-free
set $\mathcal{K}$, for the hybrid closed-loop system (42), is immediate from
Lemma 8. We next prove stability of $\mathcal{A}$ using [28, Definition 7.1].
Since the modified set $\mathcal{V}_{r_{a}}$ is closed by construction (16),
and according to Assumption 1 and (32),
$\mathbf{0}\in(\mathcal{V}_{r_{a}})^{\circ}$, $\exists\bar{\delta}>0,$ such
that
$\mathcal{B}_{\bar{\delta}}(\mathbf{0})\cap\mathcal{J}_{0}^{\mathcal{W}}\cap(\mathcal{D}_{r_{a}}(\mathcal{O}_{\mathcal{W}}^{M}))^{\circ}=\emptyset.$
Hence, for each $\delta\in[0,\bar{\delta}],$ the set
$\mathcal{S}:=\mathcal{B}_{\delta}(\mathbf{0})\times\mathcal{V}_{r_{a}}\times\mathbb{M}$
is forward invariant because the component
$\mathbf{x}\in\mathcal{B}_{\delta}(\mathbf{0})$ of solutions evolves, after at
most one jump, in the move-to-target mode
$\mathbf{\dot{x}}=-\kappa_{s}\mathbf{x},\kappa_{s}>0.$ Consequently, similar
to [32, Appendix 2], the stability of $\mathcal{A}$ for the hybrid closed-loop
system (42) is immediate from [28, Definition 7.1]. Next, we proceed to
establish convergence properties of the origin.

Convergence: For the closed-loop system (42), consider a solution $\xi$
operating in the move-to-target mode i.e., for some $(t_{0},j_{0})\in\text{dom
}\xi$, $\xi(t_{0},j_{0})\in\mathcal{F}_{0}$. If
$\xi(t,j)\notin\mathcal{J}_{0},\forall(t,j)\succeq(t_{0},j_{0}),$ then under
the influence of the stabilizing control of the form
$\mathbf{u}(\mathbf{x},\mathbf{h},0)=-\kappa_{s}\mathbf{x},\kappa_{s}>0,$ the
$\mathbf{x}$ component of the solution will converge to the origin. On the
other hand, assume that there exists $(t_{1},j_{1})\succeq(t_{0},j_{0})$ such
that $\xi(t_{1},j_{1})\in\mathcal{J}_{0}$, then the solution will enter the
obstacle-avoidance mode. According to (40),
$\mathbf{h}(t_{1},j_{1}+1)=\mathbf{x}(t_{1},j_{1})$ and
$m(t_{1},j_{1}+1)\in\\{-1,1\\}$ i.e., the $\mathbf{x}$ component of the
solution $\xi$ will move either in the clockwise direction $(m=1)$ or in the
counter-clockwise direction $(m=-1)$. To that end, we use the following lemma
to show that if the solution $\xi$ to the hybrid closed-loop system (42),
which has been initialized in the move-to-target mode, evolves in the
obstacle-avoidance mode, then it will always enter in the jump set of that
mode only through the set
$\mathcal{ER}_{m}^{\mathbf{h}}\times\mathcal{V}_{r_{a}}\times\\{-1,1\\}$,
where, according to (35), the $\mathbf{x}$ component of the solution $\xi$ is
closer to the target location than the current hit point.

######  Lemma 10.

Let Assumptions 1 and 2 hold, and the parameter $\bar{\epsilon}$, used in
(35), is chosen as per Lemma 36, then for all solutions
$\xi=(\mathbf{x},\mathbf{h},m)$ to the hybrid closed-loop system (42) with
$\xi(t,j-1)\in\mathcal{J}_{0}$, $\xi(t,j)\in\mathcal{F}_{l}$, for some
$l\in\\{-1,1\\}$ and $(t,j)\in\text{dom }\xi,$ there exists $(p,j)\in\text{dom
}\xi$, $(p,j)\succ(t,j)$ such that following conditions hold true:

  1. 1.

$\xi(p,j)\in\mathcal{ER}_{l}^{\mathbf{h}(t,j)}\times\mathcal{V}_{r_{a}}\times\\{l\\}\subset\mathcal{J}_{l},$

  2. 2.

$\xi(v,j)\in\mathcal{F}_{l},\forall(v,j)\in([t,p)\times j)$.

###### Proof.

See Appendix -K. ∎

According to Lemma 10, $\exists(t_{2},j_{1}+1)\succeq(t_{1},j_{1}+1)$ such
that
$\xi(t_{2},j_{1}+1)\in\mathcal{ER}_{m(t_{1},j_{1}+1)}^{\mathbf{h}(t_{1},j_{1}+1)}\times\mathcal{V}_{r_{a}}\times\\{-1,1\\}\subset\mathcal{J}_{m(t_{1},j_{1}+1)}.$
Notice that, according to (34) and (35),
$\lVert\mathbf{x}(t_{2},j_{1}+1)\rVert<\lVert\mathbf{h}(t_{1},j_{1}+1)\rVert$,
where, according to (40), $\mathbf{h}(t_{1},j_{1}+1)=\mathbf{x}(t_{1},j_{1})$
i.e., when any solution $\xi$, while operating in the obstacle-avoidance mode,
enters in jump set of the respective mode, the $\mathbf{x}$ component of the
solution $\xi$ is always closer to the origin than when it entered in the
obstacle-avoidance mode. Hence, with consecutive application of Lemma 10 along
with the fact that the $\mathbf{x}$ component of the solution $\xi$ moves
straight towards the origin while operating in the move-to-target mode, one
can conclude convergence of the $\mathbf{x}$ component of the solution $\xi$
towards the origin.

###  -K Proof of Lemma 10

We consider a connected modified obstacle
$\mathcal{O}_{i,\alpha}^{M}\subset\mathcal{O}_{\mathcal{W}}^{M}$, for some
$i\in\mathbb{I}$, as stated in Lemma 4, where $\alpha$ is selected according
to Lemma 5. Let
$\xi(t,j)\in\mathcal{F}_{l}^{\mathcal{W}}\cap\mathcal{N}_{\gamma}(\mathcal{D}_{r_{a}}(\mathcal{O}_{i,\alpha}^{M}))\times\mathcal{V}_{r_{a}}\times\\{l\\}$,
for some $l\in\\{-1,1\\}$. Since $m(t,j-1)=0$ and $m(t,j)\in\\{-1,1\\}$,
according to (32),
$\mathbf{x}(t,j)=\mathbf{h}(t,j)\in\partial\mathcal{D}_{\beta}(\mathcal{O}_{i,\alpha}^{M})\cap\mathcal{J}_{0}^{\mathcal{W}},$
where $\beta\in[r_{a},r_{a}+\gamma_{s}]$. Next we state a fact which shows
that the $\mathbf{x}$ component of the solution $\xi$, evolving in the
$\gamma-$neighbourhood of the $r_{a}-$dilated modified obstacle
$\mathcal{D}_{r_{a}}(\mathcal{O}_{i,\alpha}^{M})$, under the influence of only
the avoidance control vector (24), will never leave the
$\gamma-$neighbourhood.

Fact 3: Under Assumptions 1 and 2, consider the following flow-only system:

$\mathbf{\dot{x}}=\kappa_{r}\mathbf{v}(\mathbf{x},m),\mathbf{x}\in\mathcal{N}_{\gamma}(\mathcal{D}_{r_{a}}(\mathcal{O}_{i,\alpha}^{M})),$
(78)

for some $m\in\\{-1,1\\},$ where $\kappa_{r}>0$ and $\mathbf{v}(\mathbf{x},m)$
is given by (24). If
$\mathbf{x}(t_{0})\in\partial\mathcal{D}_{\beta_{1}}(\mathcal{O}_{i,\alpha}^{M}),$
$\beta_{1}\in[r_{a},r_{a}+\gamma]$, for some $t_{0}\geq 0$, then
$\mathbf{x}(t)\in\partial\mathcal{D}_{\beta_{1}}(\mathcal{O}_{i,\alpha}^{M})$
for all $t\geq t_{0}.$

###### Proof.

According to Lemma 3, for
$\mathbf{x}\in\partial\mathcal{D}_{\beta_{1}}(\mathcal{O}_{i,\alpha}^{M}),\beta_{1}\in[r_{a},r_{a}+\gamma]$,
$\textbf{card}(\mathcal{PJ}(\mathbf{x},\mathcal{O}_{i,\alpha}^{M}))=1$. Hence,
according to [26, Lemma 4.5], the vector
$\mathbf{x}-\Pi(\mathbf{x},\mathcal{O}_{i,\alpha}^{M})$ is normal to the set
$\partial\mathcal{D}_{\beta}(\mathcal{O}_{i,\alpha}^{M})$ at $\mathbf{x}.$ As
a result, the tangent cone to the set
$\partial\mathcal{D}_{\beta_{1}}(\mathcal{O}_{i,\alpha}^{M})$ at
$\mathbf{x}\in\partial\mathcal{D}_{\beta_{1}}(\mathcal{O}_{i,\alpha}^{M})$ is
given by

$\mathbf{T}_{\partial\mathcal{D}_{\beta_{1}}(\mathcal{O}_{i,\alpha}^{M})}(\mathbf{x})=\mathcal{P}(\mathbf{0},\mathbf{x}-\Pi(\mathbf{x},\mathcal{O}_{i,\alpha}^{M})).$
(79)

Since
$\mathbf{v}(\mathbf{x},m)^{\intercal}(\mathbf{x}-\Pi(\mathbf{x},\mathcal{O}_{i,\alpha}^{M}))=0$,
$\mathbf{v}(\mathbf{x}(t),m)\in\mathbf{T}_{\partial\mathcal{D}_{\beta_{1}}(\mathcal{O}_{i,\alpha}^{M})}(\mathbf{x}(t)),$
for all $t\geq t_{0},$ which implies that the solution $\mathbf{x}(t)$ to the
flow-only system (78) belongs to the set
$\partial\mathcal{D}_{\beta_{1}}(\mathcal{O}_{i,\alpha}^{M})$ for all $t\geq
t_{0}$. ∎

According to (23a), for $l\in\\{-1,1\\},$
$\mathbf{u}(\mathbf{x},\mathbf{h},l)=\kappa_{r}\mathbf{v}(\mathbf{x},l),\kappa_{r}>0$.
Since $\xi(t,j)\in\mathcal{F}_{l},$ for some $l\in\\{-1,1\\}$, with
$\mathbf{x}(t,j)\in\partial\mathcal{D}_{\beta}(\mathcal{O}_{i,\alpha}^{M}),\beta\in[r_{a},r_{a}+\gamma_{s}],$
according to Fact 3, the $\mathbf{x}$ component of the solution $\xi$ will
evolve along the curve
$\partial\mathcal{D}_{\beta}(\mathcal{O}_{i,\alpha}^{M})$. Since the obstacles
$\mathcal{O}_{i},i\in\mathbb{I}\setminus\\{0\\},$ are compact and the closed
boundary obstacle $\mathcal{O}_{0}$ has finite boundary,
$\exists(p,j)\in\text{dom }\xi$, $(p,j)\succ(t,j)$ such that at $(p,j)$, the
$\mathbf{x}$ component of the solution $\xi$ will enter the set
$\mathcal{H}_{\mathbf{p}}=\mathcal{B}_{\delta}(\mathbf{p})\cap\mathcal{N}_{\gamma}(\mathcal{D}_{r_{a}}(\mathcal{O}_{i,\alpha}^{M}))$,
where $\delta=\min\\{\beta-
r_{a},d(\mathbf{0},\mathcal{O}_{\mathcal{W}}^{M})-r_{a}\\}$. Since the
parameter $\bar{\epsilon}\in(0,\epsilon_{h}]$, where $\epsilon_{h}$ is defined
in (36), according to Lemma 36,
$\mathbf{x}(p,j)\in\mathcal{ER}_{l}^{\mathbf{h}(t,j)}$. Hence, according to
(34) and (38),
$\xi(p,t)\in\mathcal{ER}_{l}^{\mathbf{h}}\times\mathcal{V}_{r_{a}}\times\\{l\\}\subset\mathcal{J}_{l},$
and condition 1 in Lemma 10 holds true.

Also, since $\xi(t,j)\in\mathcal{F}_{l}$, for some $l\in\\{-1,1\\},$ with
$\mathbf{x}(t,j)\in\partial\mathcal{D}_{\beta}(\mathcal{O}_{i,\alpha}^{M}),\beta\in[r_{a},r_{a}+\gamma_{s}],$
according to Fact 3, the $\mathbf{x}$ component of the solution $\xi$ evolves
in the region
$\mathcal{N}_{\gamma}(\mathcal{D}_{r_{a}}(\mathcal{O}_{i,\alpha}^{M}))$, along
the curve $\partial\mathcal{D}_{r_{a}+\beta}(\mathcal{O}_{i,\alpha}^{M}),$
until it enters the set $\mathcal{ER}_{l}^{\mathbf{h}}$, which guarantees that
condition 2 in Lemma 10 holds true.

###  -L Proof of Proposition 44

Since the obstacles $\mathcal{O}_{i},i\in\mathbb{I}\setminus\\{0\\}$, are
convex and compact, and the pairwise distance between any two obstacles
$d(\mathcal{O}_{i},\mathcal{O}_{j})>2r_{a},\forall i,j\in\mathbb{I},i\neq j$,
the $r_{a}-$dilated versions of the obstacles do not intersect with each other
i.e.,
$d(\mathcal{D}_{r_{a}}(\mathcal{O}_{i}),\mathcal{D}_{r_{a}}(\mathcal{O}_{j}))>0,\forall
i,j\in\mathbb{I},i\neq j$. This ensures the pathwise connectedness of the
interior of the obstacle-free workspace w.r.t. the center of the robot
$\mathcal{W}_{r_{a}}^{\circ}$. In addition, since
$\mathbf{0}\in\mathcal{W}_{r_{a}}^{\circ},$ Assumption 1 is satisfied.

Define $\bar{\alpha}=\underset{i,j\in\mathbb{I},i\neq
j}{\min}d(\mathcal{O}_{i},\mathcal{O}_{j})/2$. Since the pairwise distance
between any two obstacles is greater than $2r_{a}$, it follows that
$\bar{\alpha}>r_{a}$, and for any $\alpha\in(r_{a},\bar{\alpha}]$, the
$\alpha-$dilated versions of the obstacles do not intersect with each other
i.e.,
$d(\mathcal{D}_{\alpha}(\mathcal{O}_{i}),\mathcal{D}_{\alpha}(\mathcal{O}_{j}))>0$.
Then, it is straightforward to see that for any location
$\mathbf{x}\in\mathcal{D}_{\alpha}(\mathcal{O}_{\mathcal{W}})\setminus\mathcal{O}_{\mathcal{W}}^{\circ},$
there is a unique closest point on the boundary of the set
$\mathcal{D}_{\alpha}(\mathcal{O}_{\mathcal{W}}).$ Since the distance
$d(\mathcal{D}_{\alpha}(\mathcal{O}_{i}),\mathcal{D}_{\alpha}(\mathcal{O}_{j}))>0$,
the set
$\mathcal{W}_{\alpha}=\mathbb{R}^{2}\setminus\mathcal{D}_{\alpha}^{\circ}(\mathcal{O}_{\mathcal{W}})$.
Hence, one has
$\textbf{reach}(\mathcal{W}_{\alpha})\geq\alpha,\forall\alpha\in(r_{a},\bar{\alpha}]$.
Finally, by virtue of Fact 1, Assumption 2 is satisfied.

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