Collision Avoidance for Elliptical Agents with Control Barrier Function Utilizing Supporting Lines

In this paper, we present a collision avoidance method among elliptical agents, labeled through the index set $\mathcal{N}=\{1\cdots n\}$, in 2-D Euclidean space ${\mathbb{R}}^{2}$, as illustrated in Fig. 1. We denote the world coordinate frame as $\Sigma_{w}$. We also define the coordinate frame of agent $i$ as $\Sigma_{i}$, arranged at the center of agent $i$ so that its $x_{i}$-axis corresponds with the major axis of the ellipse. The relative pose of $\Sigma_{i}$ with respect to $\Sigma_{w}$ is described as $(\bm{p}_{i},R_{i}(\theta)):{\mathbb{R}}^{2}\times SO(2)$ with the position $\bm{p}_{i}=[p_{ix},p_{iy}]^{T}\in{\mathbb{R}}^{2}$ and the orientation

The state of agent $i$ is defined as $\bm{x}_{i}=[p_{ix},p_{iy},\theta_{i}]^{T}$. We suppose that the motion of agent $i$ can be represented according to a single integrator dynamics,

with the velocity input $[u_{ix},u_{iy}]^{T}$ and the angular velocity input $u_{i\theta}$. We denote the control input for agent $i$ as $\bm{u}_{i}=[u_{ix},u_{iy},u_{i\theta}]^{T}$. Agent $i$ occupies the elliptical region $\mathcal{E}_{i}$ described as

where the constant matrix $Q_{i}={\rm diag}(q_{ix},q_{iy})$ is defined with $q_{ix}$ and $q_{iy}$ which specify the length of the major axis and the minor axis.

We propose a control method that prevents a collision between elliptical agents described with (3). If the minimum distance between $\mathcal{E}_{i}$ and $\mathcal{E}_{j}$ is described as $w_{ij}^{*}(\bm{x}_{i},\bm{x}_{j})$, the safe set restricting collisions between agents $i$ and $j$ can be captured by the set

where this set has to be rendered forward invariant. For this goal, we leverage control barrier functions (CBFs) being introduced in the next subsection.

CBFs have been utilized for ensuring the forward invariance property to the set $\mathcal{S}$, in which the state $\bm{x}$ should be confined during the task execution of agents. We assume that a set $\mathcal{S}$ can be expressed as the superlevel set of a continuously differentiable function $h:{\mathbb{R}}^{n}\to{\mathbb{R}}$, namely, ${\mathcal{S}}=\{\bm{x}\in\mathbb{R}^{n}\mid h(\bm{x})\geq 0\}$. Then, CBF is defined as follows.

The forward invariance of the set $\mathcal{S}$ is ensured through the following corollary.

The condition (7) guaranteeing forward invariance of the set $\mathcal{S}$ can be synthesized to the control law through the optimization-based controller leveraging Quadratic Programming (QP). Let us denote the nominal input as $\bm{u}_{\mathrm{nom}}$ and wish to modify it minimally invasive way so as to satisfy the condition (7). This goal can be achieved by employing the input $\bm{u}^{*}$ derived from the following QP

We first introduce a supporting line $l_{ij}$ of agent $i$, which contacts $\mathcal{E}_{i}$ at the point $\bm{m}_{ij}$, as depicted in Fig. 2. The point $\bm{m}_{ij}$ can be defined as

with a positive definite matrix $\bar{Q}_{i}=R_{i}Q_{i}R_{i}^{T}$ and a parameter $\phi_{ij}\in(-\pi,\pi]$. Here, the parameter $\phi_{ij}$ is introduced to specify a point on the boundary of the ellipse $\mathcal{E}_{i}$, where the graphical interpretation is shown in Fig. 3. Then, the supporting line $l_{ij}$ is described as

which is determined by $\bm{x}_{i}$ and $\phi_{ij}$.

Let us derive the separation condition evaluated with the signed distance from the supporting line $l_{ij}$, which provides a positive value to a point in the different half-plane with $\mathcal{E}_{i}$, and a negative value otherwise as shown in Fig. 4. The point $\bm{n}_{ij}\in\mathcal{E}_{j}$ that minimizes the signed distance from the supporting line $l_{ij}$ is determined by

Then, the minimum signed distance from $l_{ij}$ is calculated by

which satisfies $h_{ij}(\bm{x}_{i},\bm{x}_{j},\phi_{ij})>0$ if and only if $\mathcal{E}_{i}$ and $\bm{n}_{ij}$ are in the different half-plane separated by the line $l_{ij}$ as shown in Fig. 4. In other words, if there exists $\phi_{ij}$ that fulfills $h_{ij}(\bm{x}_{i},\bm{x}_{j},\phi_{ij})>0$, then two ellipses are separated by the line $l_{ij}$. Note that $\bm{n}_{ij}\in\mathcal{E}_{j}$ is not the closest point to the supporting line $l_{ij}$ as depicted in Fig. 4(a) and (b). In the remaining of this subsection, we analyze the property of $h_{ij}(\bm{x}_{i},\bm{x}_{j},\phi_{ij})$ assuming two agents $i$ and $j$ are fixed, hence denote $h_{ij}(\bm{x}_{i},\bm{x}_{j},\phi_{ij})$ as $h_{ij}(\phi_{ij})$.

The fact that $h_{ij}(\phi_{ij})>0$ encodes the separation condition between two agents motivates us to employ $h_{ij}(\phi_{ij})$ as a CBF for ensuring collision avoidance between elliptical agents. However, the naive selection of the parameter $\phi_{ij}$, namely the line $l_{ij}$, could yield a shorter distance than the actual distance $w_{ij}^{*}$ as depicted in Fig. 2(a). Because of this difference from $w_{ij}^{*}$, $h_{ij}(\phi_{ij})$ could overestimate the risk of collisions and hence cause too conservative evasive motion. To mitigate this gap, we propose the following optimization problem that intends to rotate the supporting line $l_{ij}$ on the boundary of $\mathcal{E}_{i}$ so that $h_{ij}(\phi_{ij})$ is maximized as

where its graphical interpretation is depicted in Fig. 2(b).

While, in the next subsection, we introduce the input to $\phi_{ij}$ for maximizing $h_{ij}(\phi_{ij})$, we first establish the connection between the optimization problem (14) and the actual distance between two agents, namely $w_{ij}^{*}$. For this goal, we introduce the following optimization problem, which optimal solution $\|\bm{w}^{*}\|$ is equal to $w_{ij}^{*}$.

where $f_{k}(\bm{\xi}):=(\bm{\xi}-\bm{p}_{k})^{T}\bar{Q}_{k}^{-2}(\bm{\xi}-\bm{p}_{k})-1\leq 0$ signifies the condition $\bm{\xi}\in\mathcal{E}_{k}$. Then, the following theorem formalizes the relationship between $w_{ij}^{*}$ and $h_{ij}(\phi_{ij})$.

Theorem 1 implies that the proposed update law of the supporting line, described as the optimization problem (14) and Fig. 2, renders $h_{ij}(\phi_{ij})$ the actual distance $w_{ij}^{*}$ between two ellipses. Furthermore, because of (16), the condition $h_{ij}(\phi_{ij})>0$ provides a sufficient condition for preventing collisions even if $\phi_{ij}$ does not converge to the maximizer of (14). In the following subsection, we develop $h_{ij}$ as a CBF together with presenting the update procedure of $\phi_{ij}$.

In this subsection, we first design the collision avoidance methods between two elliptical agents $i$ and $j$, then extend the results to the case of $n$ agents. Hereafter, we regard $h_{ij}$ as a function of $\bm{x}_{i},\bm{x}_{j}$, and $\phi_{ij}$ as first defined in (13), though we have dropped the dependency of $h_{ij}$ for notational convenience.

As mentioned in the previous subsection, we require a supporting line between agents $i$ and $j$ to evaluate the separation condition $h_{ij}$. Without loss of generality, we introduce a supporting line for the agent with a lower ID number. Then, as the supporting line is to be updated for maximizing $h_{ij}$, the model of the agent (2) now has to describe the dynamics of $\phi_{ij}$ as well. Therefore, we introduce the augmented state $\bm{x}_{ij}=\left[\bm{x}_{i}^{T},~{}\bm{x}_{j}^{T},~{}\phi_{ij}\right]^{T}$ to achieve collision avoidance between two agents $i$ and $j$ ($i<j$), where the dynamics of the augmented state is

where $\bm{u}_{ij}=\left[\bm{u}_{i}^{T},\bm{u}_{j}^{T},u_{\phi_{ij}}\right]^{T}$.

As the nominal input for $\phi_{ij}$ intending to maximize $h_{ij}$, we employ the following gradient-based input

The input (31) drives $h_{ij}$ to a local maximum point, and hence preventing too conservative evasive motions caused by the difference between $w_{ij}^{*}$ and $h_{ij}$. Note that the maximizer $\phi_{ij}^{*}$ of $h_{ij}$ changes as the agent $i$ and $j$ traverse. However, as $\phi_{ij}$ is a virtual variable not dependent on the physical dynamics of the agents, we can make $\phi_{ij}$ converge fast enough with large $\gamma$ to follow the agent movement, as we will demonstrate in the simulations in Section IV-A.

Let us denote the nominal control input for the agent $i$ as $\bm{u}_{{\rm nom},i}$, which is designed to achieve its own objective, e.g., reaching the goal position. Combining this nominal control input $\bm{u}_{{\rm nom},i}$ with (31), the nominal input for the augmented state $\bm{x}_{ij}$ is expressed as $\bm{u}_{\mathrm{nom},ij}=\left[\bm{u}_{\mathrm{nom},i}^{T},~{}\bm{u}_{\mathrm{nom},j}^{T},~{}u_{\mathrm{nom},\phi_{ij}}\right]^{T}$.

The goal of the collision avoidance strategy is to allow agents to execute their tasks encoded by $\bm{u}_{{\rm nom},i}$ while ensuring collision never occurs between agents. To achieve this objective, we propose the collision avoidance methods that render the following set $\hat{\mathcal{S}}_{ij}$ forward invariant.

Since $w_{ij}^{*}\geq h_{ij}$ holds from Theorem 1, the set $\mathcal{S}_{ij}$ in (4) is also ensured to be forward invariant if we guarantee the forward invariance of $\hat{\mathcal{S}}_{ij}$. Similar to (8), this strategy can be achieved by $\bm{u}_{ij}^{*}$ derived from the following QP integrating $h_{ij}$ as a CBF:

Note that the constraint (33b) is derived by calculating the Lie derivative of $h_{ij}$ along $f_{ij}=\bm{0}_{7\times 1}$ and $g_{ij}=I_{7}$ in (30).

As the proposed CBF $h_{ij}$ prevents collision between elliptical agents $i$ and $j$, we need to confirm whether there always exists the control input that satisfies (6) in the Definition 1 and it provides us the forward invariance property.

Theorem 2 signifies there always exists the control input $\bm{u}_{i},\bm{u}_{j}\in{\mathbb{R}}^{3}$ that renders the set $\hat{\mathcal{S}}_{ij}$ forward invariant. Therefore, the collision is prevented if the state $\bm{x}_{ij}$ is in the safe set $\hat{\mathcal{S}}_{ij}$ at the initial time. Note that if two ellipses do not collide with each other at the initial time, we can easily specify the initial $\phi_{ij}$ that satisfies $h_{ij}>0$ by any heuristic approach.

We then extend the collision avoidance method (33) to the scenario with $n$ agents. Similar to the case of the two agents, we need to introduce a supporting line for each pair of agents, resulting in ${}_{n}C_{2}$ numbers of CBFs for the whole system. Since the agent with a smaller ID in a pair owns a supporting line, a vector augmenting $\phi_{ij}$ of agent $i$ is described as $\bm{\phi}_{i}=[\phi_{i\hskip 1.13809pti+1},\phi_{i\hskip 1.13809pti+2},...,\phi_{i\hskip 1.13809ptn}]^{T}$ with $\bm{\phi}_{n}=\emptyset$. In addition, we introduce a vector combining $h_{ij}$ as $\bm{h}_{i}=[h_{i\hskip 1.13809pti+1},h_{i\hskip 1.13809pti+2},...,h_{i\hskip 1.13809ptn}]^{T}$. Then, the ensemble state and CBFs of the system are expressed as $\bm{x}=\left[\bm{x}_{1}^{T},\bm{x}_{2}^{T},...,\bm{x}_{n}^{T},\bm{\phi}_{1}^{T},\bm{\phi}_{2}^{T},...,\bm{\phi}_{n-1}^{T}\right]^{T}$ and $\bm{h}=\left[\bm{h}_{1}^{T},\bm{h}_{2}^{T},..,\bm{h}_{n-1}^{T}\right]^{T}$, respectively. With the introduced ensemble vectors, the collision avoidance method for $n$ agents can be achieved by $\bm{u}^{*}$ derived from

where $f=\bm{0}_{(3n+_{n}C_{2})\times 1}$ and $g_{ij}=I_{(3n+_{n}C_{2})}$ since we assume a single integrator model.

We first demonstrate our proposed algorithm with the two elliptical agents, the sizes of which are characterized by $Q_{1}=\mathrm{diag}(0.4,0.2)$ for a red agent and $Q_{2}=\mathrm{diag}(0.6,0.2)$ for a blue agent, respectively. The initial condition of the simulation is depicted in Fig. 5(a), with the initial pose $x_{1}(0)=\left[0,1,{-\pi}/{4}\right]^{T}$, $x_{2}(0)=\left[2,0.1,0\right]^{T}$. Note that we randomly chose the initial $\phi_{12}(0)$ from the range that yields a supporting line $l_{12}$ separating two ellipses. Two agents traverse the environment so that they intersect around the center of the field. We utilize $\alpha(h_{12})=10h_{12}$ for an extended class $\mathcal{K}$ function in CBF conditions and $u_{\mathrm{nom},\phi_{12}}=20\left({\partial h_{12}}/{\partial\phi_{12}}\right)$ for a nominal input to $\phi_{12}$.

The snapshots of the simulations are presented in Fig. 5, which illustrate a supporting line incorporated in agent 1’s CBF as a black line. In addition, the distance between the supporting line and the blue ellipse is depicted as a green line. The snapshots demonstrate that the supporting line on agent $i$ is updated so that it separates two agents without causing any conservative transition in their evasive trajectories. The value of $h_{12}$ is illustrated in Fig. 6 together with the optimal solution of (15), namely the actual distance between two agents. Although $w_{12}^{*}$ and $h_{12}$ differ at the initial time since we set $\phi_{12}$ randomly, the proposed gradient-based input (31) successfully drives $h_{12}$ to the actual distance $w_{12}^{*}$. Furthermore, Fig. 6 verifies that the input (31) rotates the supporting line so that $h_{12}$ follows the transition of the actual distance between two ellipses. We can also confirm that the value of $h_{12}$ keeps in the positive value, hence, collision avoidance is achieved.

In the next simulation, we demonstrate our proposed algorithm with the four elliptical agents, the sizes of which are specified by $Q_{1}=\mathrm{diag}(0.3,0.15)$ for a red agent, $Q_{2}=\mathrm{diag}(0.4,0.2)$ for a blue agent, $Q_{3}=\mathrm{diag}(0.4,0.2)$ for a green agent and $Q_{4}=\mathrm{diag}(0.6,0.3)$ for an orange agent, respectively. The initial condition of the simulation is depicted in Fig. 7(a), with the initial pose $x_{1}(0)=\left[-0.1,1.1,-{\pi}/{4}\right]^{T}$, $x_{2}(0)=\left[1.9,-1.1,-{\pi}/{4}\right]^{T}$, $x_{3}(0)=\left[-0.1,-1.1,{5\pi}/{4}\right]^{T}$ and $x_{4}(0)=\left[1.9,1.1,{5\pi}/{4}\right]^{T}$. All four agents traverse the environment so that they intersect around the center of the field. We utilize $\alpha(\bm{h})=10\bm{h}$ for an extended class $\mathcal{K}$ function in CBF conditions and $u_{\mathrm{nom},\phi_{ij}}=20\left({\partial h_{ij}}/{\partial\phi_{ij}}\right)$ for a nominal input to each $\phi_{ij}$.

The snapshots of the simulation are presented in Fig. 7, where each agent is depicted with its trajectory. As illustrated in Fig. 7(b), all agents move straightly toward their goal positions until their distances become closer at the center. Then, the proposed methods modify the nominal input minimally invasive way so that the collision is avoided, as illustrated in Fig. 7(c) and (d). Note that the proposed CBF achieves the collision-free trajectories while changing the attitude of the agents in Fig. 7(c). Fig. 8 depicts the transitions of $h_{ij}$, where the proposed methods keep them in the positive value.