Planning and Control of Multi-Robot-Object Systems under Temporal Logic Tasks and Uncertain Dynamics

We focus on the task specification $\phi$ given as a Linear Temporal Logic (LTL) formula. The basic ingredients of a LTL formula are a set of atomic propositions $\Psi$ and several boolean and temporal operators. LTL formulas are formed according to the following grammar [31]: $\phi:\coloneqq\mathrm{true}\ |\ a\ |\ \phi_{1}\ \land\phi_{2}\ |\ \neg\phi\ |\ \circ\phi\ |\ \phi_{1}\mathcal{U}\phi_{2}$, where $a\in\Psi$, $\phi_{1}$ and $\phi_{2}$ are LTL formulas and $\circ$, $\mathcal{U}$ are the next and until operators, respectively. Definitions of other useful operators like $\square$ (always), $\Diamond$ (eventually) and $\Rightarrow$ (implication) are omitted and can be found at [31]. The semantics of LTL are defined over infinite words over $2^{\Psi}$. Intuitively, an atomic proposition $\psi\in\Psi$ is satisfied on a word $w=w_{1}w_{2}\dots$ if it holds at its first position $w_{1}$, i.e. $\psi\in w_{1}$. Formula $\phi$ holds true if $\phi$ is satisfied on the word suffix that begins in the next position $w_{2}$, whereas $\phi_{1}\mathcal{U}\phi_{2}$ states that $\phi_{1}$ has to be true until $\phi_{2}$ becomes true. Finally, $\Diamond\phi$ and $\square\phi$ holds on $w$ eventually and always, respectively. For a full definition of the LTL semantics, we refer the reader to [31].

Consider $N>1$ robotic agents operating in a compact $2$D workspace $\mathcal{W}\subset\mathbb{R}^{2}$ with an outer boundary with $M>0$ objects, and $\mathcal{N}\coloneqq\{1,\dots,N\}$, $\mathcal{M}\coloneqq\{1,\dots,M\}$; The objects are represented by rigid bodies whereas the robotic agents are fully actuated and holonomic, equipped with a transportation tool, such as a robotic arm. In addition, the workspace is populated with $J\in\mathbb{N}$ connected, closed sets $\{\mathfrak{O}_{j}\}_{j\in\mathcal{J}}$, with $\mathcal{J}\coloneqq\{1,\dots,J\}$, representing obstacles, and we define the free space as $\mathcal{W}_{\textup{f}}\coloneqq\mathcal{W}\backslash\bigcup_{j\in\mathcal{J}}\mathfrak{O}_{j}$.

We further assume that there exist $K>1$ points within $\mathcal{W}_{\textup{free}}$, denoted by $p_{\pi_{k}}$, corresponding to certain properties of interest (e.g., gas station, repairing area, etc.), with $\mathcal{K}\coloneqq\{1,\dots,K\}$. Since, in practise, these properties are naturally inherited to some neighborhood of the respective point of interest, we define for each $k\in\mathcal{K}$ the region of interest $\pi_{k}$, corresponding to $p_{\pi_{k}}$, as the closed ball $\pi_{k}\coloneqq\bar{\mathcal{B}}(p_{\pi_{k}},r_{\pi_{k}})\subset\mathcal{W}_{\textup{free}}$, with $r_{\pi_{k}}>0$ positive radii, $\forall k\in\mathcal{K}$. These properties of interest are expressed as boolean variables via finite sets of atomic propositions. In particular, we introduce disjoint sets of atomic propositions $\Psi_{i},\Psi^{o}_{j}$, expressed as boolean variables, that represent services provided to agent $i\in\mathcal{N}$ and object $j\in\mathcal{M}$ in $\Pi\coloneqq\{\pi_{1},\dots,\pi_{K}\}$. The services provided at each region $\pi_{k}$ are given by the labeling functions $\mathcal{L}_{i}:\Pi\rightarrow 2^{\Psi_{i}},\mathcal{L}^{o}_{j}:\Pi\rightarrow 2^{\Psi^{o}_{j}}$, which assign to each region $\pi_{k},k\in\mathcal{K}$, the subset of services $\Psi_{i}$ and $\Psi^{o}_{j}$, respectively, that can be provided in that region to agent $i\in\mathcal{N}$ and object $j\in\mathcal{M}$, respectively. In addition, we consider that the agents and the object are initially ($t=0$) in the regions of interest $\pi_{init(i)},\pi_{init_{o}(j)}$, where the functions $init:\mathcal{N}\to\mathcal{K}$, $init_{o}:\mathcal{M}\to\mathcal{K}$ specify the initial region indices. In the following, we present the modeling of the coupled dynamics of the object and the robots.

We denote by $x_{i}\in\mathbb{R}^{2}$ the position of robot $i$’s center of mass. In this work we explicitly consider the actions of robot navigation, as well as (cooperative) transportation of an object via $x_{i}$, where the joint variables of the potential mounted robotic arm are assumed to be fixed. We consider that the robotic arm joints are used only for grasping/releasing an object (when the respective mobile base is fixed), actions that we do not explicitly model. The dynamics describing the motion of robot $i$’s center of mass are given by

where $m_{i}$ is the unknown mass of robot $i$, $f_{i}:\mathbb{R}^{4}\to\mathbb{R}^{2}$ are unknown friction-like functions, $u_{i}\in\mathbb{R}^{2}$ is the control input of robot $i$, and $h_{i}\in\mathbb{R}^{2}$ is the force exerted by robot $i$ at the grasping point with object $j$ in case of contact. The aforementioned dynamics concern the cases when (i) robot $i$ is navigating to some pre-defined point, and (ii) robot $i$ is transporting, possibly collaboratively with other robots, an object $j$. In both of these cases, the joint variables of the mounted robotic arm are assumed to be constant. The procedures of grasping/releasing an object, where the robotic arm would have to be activated, are not considered here.

We consider that each robot $i$, for a given ${x}_{i}$, covers a spherical region of constant radius $r_{i}\in\mathbb{R}_{>0}$ that bounds its volume, i.e., $\bar{\mathcal{B}}({x}_{i},r_{i})$, $\forall i\in\mathcal{N}$; Moreover, we consider that the agents have specific power capabilities, which for simplicity, we associate to positive integers $\zeta_{i}>0$, $i\in\mathcal{N}$, via an analogous relation. The overall configuration is ${x}\coloneqq[{x}^{\top}_{1},\dots,{x}^{\top}_{N}]^{\top}$ $\in\mathbb{R}^{2N}$.

Regarding the objects, we denote by $x^{o}_{j}\in\mathbb{R}^{2}$ the position of the $j$th object’s center of mass, $\forall j\in\mathcal{M}$. We consider the following second-order Newton-Euler dynamics:

where $m^{o}_{j}$ is the unknown mass of object $j$, $f^{o}_{j}:\mathbb{R}^{4}\to\mathbb{R}^{2}$ are unknown friction terms, and $h^{o}_{j}\in\mathbb{R}^{2}$ are the forces exerted to the $j$th object’s center of mass, in case of contact with a robot. The overall object configuration is denoted by $x^{o}\coloneqq[x^{o}_{1},\dots,x^{o}_{M}]^{\top}$, and $\bar{x}\coloneqq[x^{\top},(x^{o})^{\top}]^{\top}$.

The functions $f_{i}$ and $f^{o}_{j}$ are assumed to satisfy the following assumption:

The aforementioned assumption is inspired by standard friction-like terms, which can be approximated by continuously differentiable velocity functions [32].

Similarly to the robots, each object’s volume is represented by the spherical set of constant radius $r^{o}_{j}\in\mathbb{R}_{>0}$, i.e., $\bar{\mathcal{B}}({x}^{o}_{j},r^{o}_{j})$, $\forall j\in\mathcal{M}$.

Next, we provide the coupled dynamics between an object $j\in\mathcal{M}$ and a subset $\mathcal{A}\subseteq\mathcal{N}$ of robots that grasp it rigidly. For these robots, it holds that $h^{o}_{j}=\sum_{i\in\mathcal{A}}h_{i}$, and since the joint variables of the robotic arms are fixed, $x^{o}_{j}=x_{i}+d^{o}_{ij}$, $\dot{x}^{o}_{j}=\dot{x}_{i}$ and $\ddot{x}^{o}_{j}=\ddot{x}_{i}$, where $d^{o}_{ij}\in\mathbb{R}^{2}$ is the constant offset between $x^{o}_{j}$ and $x_{i}$, $\forall i\in\mathcal{A}$. Therefore, one obtains that

where ${m}_{\mathcal{A},j}\coloneqq m^{o}_{j}+\sum_{i\in\mathcal{A},j}m_{i}$, ${f}_{\mathcal{A},j}\coloneqq f^{o}_{j}(x^{o}_{j},\dot{x}^{o}_{j})+\sum_{i\in\mathcal{A},j}f_{i}(x^{o}_{j}-d^{o}_{ij},\dot{x}^{o}_{j})$. Note that Assumption 1 implies that

for an unknown positive constant $\alpha_{\mathcal{A},j}$.

Regarding the volume of the coupled robots-object system, we consider that it is bounded by a sphere of radius centered at ${x}^{o}_{j}$ with constant radius $r_{\scriptscriptstyle\mathcal{A},j}\in\mathbb{R}_{>0}$, i.e., $\mathcal{B}({x}^{o}_{j},r_{\scriptscriptstyle\mathcal{A},j})$, which is large enough to cover the volume of the coupled system.

Moreover, in order to take into account the introduced robots’ power capabilities $\zeta_{i}$, $i\in\mathcal{N}$, we consider a function $\Lambda\in\{\mathsf{True},\mathsf{False}\}$ that outputs whether the robots that grasp an object are able to transport the object, based on their power capabilities. For instance, $\Lambda(m^{o}_{j},\zeta_{\mathcal{A}})=\mathsf{True}$, where $m^{o}_{j}\in\mathbb{R}_{>0}$ is the mass of object $j$ and $\zeta_{\mathcal{A}}\coloneqq[\zeta_{i}]^{\top}_{i\in\mathcal{A}}$, implies that the robots $\mathcal{A}$ have sufficient power capabilities to cooperatively transport object $j$.

Next, we define the boolean functions $\mathcal{AG}_{j}:\mathbb{R}_{\geq 0}\to 2^{\mathcal{N}_{0}}$, with $\mathcal{N}_{0}\coloneqq\mathcal{N}\cup\{0\}$, to denote which robots grasp an object $j\in\mathcal{M}$; $\mathcal{AG}_{j}=0$ means that no robots grasp object $j$. Note also that $i\in\mathcal{AG}_{j}\Leftrightarrow i\notin\mathcal{AG}_{j^{\prime}}=\mathsf{False},\forall j^{\prime}\in\mathcal{M}\backslash\{j\}$, i.e., robot $i$ can grasp at most one object at a time. We further denote $\mathcal{AG}\coloneqq[\mathcal{AG}_{1},\dots,\mathcal{AG}_{M}]^{\top}\in(2^{\mathcal{N}_{0}})^{M}$.

In the following, we use the term “entity” to refer to single robots, objects as well as systems comprised by robots that grasp an object (robots-object systems). The number of these systems depends on the variables $\mathcal{AG}$. Given a grasping configuration $\mathcal{AG}\in(2^{\mathcal{N}_{0}})^{M}$, consider $\bar{T}(\mathcal{AG})$ number of entities, indexed by the set $\mathcal{T}(\mathcal{AG})\coloneqq\{1,\dots,\bar{T}(\mathcal{AG})\}$. Each entity (robot, object, or robots-object system) is characterized by the respective configuration ($x_{i}$, $x_{j}^{o}$, $x_{j^{\prime}}^{o}$) and radius ($r_{i}$, $r^{o}_{j}$, $r_{\mathcal{A},j^{\prime}}$), respectively, which we denote for simplicity by the generic variables $x_{i}^{e}$, $r_{i}^{e}$, for all $i\in\mathcal{T}(\mathcal{AG})$.

We further define the free space for each entity $\mathcal{F}_{i}(\mathcal{AG})\coloneqq\big{(}\mathcal{W}_{\textup{f}}\backslash\bigcup_{j\in\mathcal{T}(\mathcal{AG})\backslash\{i\}}\bar{\mathcal{B}}(x_{j}^{e},r^{e}_{j})\big{)}\ominus r^{e}_{i}$, where the incorporation of $\ominus r^{e}_{i}$ enlarges the obstacles and the other entities with the radius $r_{i}^{e}$.

We give now the definitions for the transitions of the robots and the objects between the regions of interest.

Similarly, we can define the releasing action $j\xrightarrow[]{r}\ell$ for an agent $j\in\mathcal{N}$ and object $\ell\in\mathcal{M}$. Loosely speaking, the aforementioned definitions correspond to specific action primitives of the robots, namely robot navigation or object transportation, which define the navigation transition, and grasping and releasing actions. When the navigation transition from $\pi_{j_{k}}$ to $\pi_{j^{\prime}_{k}}$ corresponds to a robot navigation for robot $j\in\mathcal{N}$, we denote it by $\pi_{j_{k}}\rightarrow_{j}\pi_{j^{\prime}_{k}}$; when it corresponds to a cooperative transportation of object $j\in\mathcal{M}$ by a subset of robots $\mathcal{A}$, we denote it by $\pi_{j_{k}}\xrightarrow{T}_{\mathcal{A},j}\pi_{j^{\prime}_{k}}$.

We also assume the existence of a procedure ${\mathcal{P}}_{s}$ that outputs whether or not a set of non-intersecting spheres fits in a larger sphere as well as possible positions of the spheres in the case they fit. More specifically, given a region of interest $\pi_{k}$ and a number $\widetilde{N}\in\mathbb{N}$ of sphere radii (of robots, objects, or object-robots systems) the procedure can be seen as a function ${\mathcal{P}}_{s}\coloneqq[\mathcal{P}_{s,0},{\mathcal{P}}^{\top}_{s,1}]^{\top}$, where $\mathcal{P}_{s,0}:\mathbb{R}^{\widetilde{N}+1}_{\geq 0}\to\{\mathsf{True},\mathsf{False}\}$ outputs whether the spheres fit in the region $\pi_{k}$ whereas ${\mathcal{P}}_{s,1}$ provides possible configurations of the robots and the objects or ${0}$ in case the spheres do not fit. For instance, $P_{s,0}(r_{\pi_{2}},r_{1},r_{3},r^{o}_{1},r^{o}_{5})$ determines whether the robots $1,3$ and the objects $1,5$ fit in region $\pi_{2}$, without colliding with each other; $(x^{e}_{1},x^{e}_{2},x^{e}_{3},x^{e}_{4})=(x_{1},x_{3},{x}^{o}_{1},{x}^{o}_{5})=P_{s,1}(r_{\pi_{2}},r^{e}_{1},r^{e}_{2},r^{e}_{3},r^{e}_{4})=P_{s,1}(r_{\pi_{2}},r_{1},r_{3},r^{o}_{1},r^{o}_{5})$ provides a set of configurations such that $\bar{\mathcal{B}}_{i}(x^{e}_{i},r^{e}_{i})$ $i\in\{1,\dots,4\}$, and the pairwise intersections of these sets are empty. The problem of finding an algorithm $\mathcal{P}_{s}$ is a special case of the sphere packing problem [33]. Note, however, that we are not interested in finding the maximum number of spheres that can be packed in a larger sphere but, rather, in the simpler problem of determining whether a set of spheres can be packed in a larger sphere.

Our goal is to control the multi-robot-object system defined above such that the robots and the objects obey a given specification over their atomic propositions $\Psi_{i},\Psi^{o}_{j},\forall i\in\mathcal{N},j\in\mathcal{M}$. Given the trajectories ${x}_{i}(t),{x}^{o}_{j}(t),t\in\mathbb{R}_{\geq 0}$, of robot $i$ and object $j$, respectively, their corresponding behaviors are given by the infinite sequences

with $t_{i,\ell+1}>t_{i,\ell}\geq 0,t^{o}_{j,\ell+1}>t^{o}_{j,\ell}\geq 0,\forall\ell\in\mathbb{N}$, representing specific time stamps. The sequences $\sigma_{i},\sigma^{o}_{j}$ are the services provided to the agent and the object, respectively, over their trajectories, i.e., $\sigma_{i,\ell}\in 2^{\Psi_{i}},\sigma^{o}_{j,l}\in 2^{\Psi^{o}_{j}}$ with $\mathcal{A}_{i}({q}_{i}(t_{i,\ell}))\subset\pi_{k_{i,\ell}},\sigma_{i,\ell}\in\mathcal{L}_{i}(\pi_{k_{i,\ell}})$ and $\mathcal{O}_{j}({x}^{o}_{j}(t^{o}_{j,l}))\subset\pi_{k^{o}_{j,l}},\sigma^{o}_{j,l}\in\mathcal{L}^{o}_{j}(\pi_{k^{o}_{j,l}}),k_{i,\ell},k^{o}_{j,l}\in\mathcal{K},\forall\ell,l\in\mathbb{N},i\in\mathcal{N},j\in\mathcal{M}$, where $\mathcal{L}_{i}$ and $\mathcal{L}^{o}_{j}$ are the previously defined labeling functions. The following Lemma then follows:

The control objectives are given as LTL formulas $\phi_{i},\phi^{o}_{j}$ over $\Psi_{i},\Psi^{o}_{j}$, respectively, $\forall i\in\mathcal{N},j\in\mathcal{M}$. The LTL formulas $\phi_{i},\phi^{o}_{j}$ are satisfied if there exist behaviors $b_{i},b^{o}_{j}$ of agent $i$ and object $j$ that satisfy $\phi_{i},\phi^{o}_{j}$. We are now ready to give a formal problem statement:

Note that it is implicit in the problem statement the fact that the agents/objects starting in the same region can actually fit without colliding with each other. Technically, it holds that $\mathcal{P}_{s,0}(r_{\pi_{k}},[r_{i}]_{i\in\{i\in\mathcal{N}:\text{init}(i)=k\}},[r^{o}_{j}]_{j\in\{j\in\mathcal{M}:\text{init}_{o}(j)=k\}})=\mathsf{True}$, $\forall k\in\mathcal{K}$.

The first ingredient of our solution is the development of feedback control laws that establish the navigation transition of Def. 1, i.e., robot navigation and cooperative object transportations. We do not focus on the grasping/releasing actions (see Def. 2) and we refer to some existing methodologies that can derive the corresponding control laws (e.g., [34, 35]).

The control design is based on the integration of the adaptive control scheme and point world transformation proposed in [36] and [37], respectively. The former accommodates the uncertain robot and object dynamics and the latter deals with the complex environment obstacles. Since [37] consider single-robot navigation in complex environments, we consider here a sequential execution of the navigation and object-transportation transitions¹¹1The correctness of the multi-robot adaptive control scheme of [36] has only been proven for simple, spherical environments.. That is, only one entity is allowed to navigate from one RoI to another, viewing the rest of the entities as fixed obstacles.

Assume that the conditions of Problem 1 hold for some $t_{0}\in\mathbb{R}_{\geq 0}$, i.e., all robots and objects are located in some RoI with zero velocity. We first consider the control problem of safe navigation for a robot $i\in\mathcal{N}$ satisfying $i\notin\mathcal{AG}_{j}(t_{0})$ for all $j\in\mathcal{M}$, i.e., not grasping any objects, from region $i_{k}$ to $i^{\prime}_{k}$ (transition $\pi_{i_{k}}\rightarrow_{i}\pi_{i^{\prime}_{k}}$), with $\bar{\mathcal{B}}_{i}(x_{i}(t_{0}),r_{i})\subset\pi_{i_{k}}$ and $i_{k}$, $i^{\prime}_{k}$ $\in\mathcal{K}$. Let $x^{e}_{i}=x_{i}$. As stated before, the other entities are fixed and viewed as static obstacles by robot $i$. Moreover, to account for safety specifications, we wish the robot to avoid entering the RoI $\pi_{k}$, $k\in\mathcal{K}\backslash\{i_{k},i^{\prime}_{k}\}$. Therefore, the free space of robot $i$ becomes $\mathcal{F}_{i}=(\mathcal{W}_{\textup{f}}\backslash\mathcal{O}_{i})\ominus r_{i}$, where

Let the desired navigation goal of the robot be $x_{\textup{d,i}}\in\pi_{i^{\prime}_{k}}$, which will be provided by the procedure $\mathcal{P}_{s}$, as explained in the previous section. Next, we use the workspace transformation $\chi=\mathcal{H}(x)$ from [37] that converts the environment to the open unit disk $\mathcal{D}\coloneqq\mathcal{B}([0,0]^{\top},1)$, and the obstacles (environment obstacles, entities other than robot $i$, and RoI other than $\pi_{i_{k}}$, $\pi_{i^{\prime}_{k}}$) to $\bar{J}$ points $b_{\ell}\in\mathcal{D}$, $\ell\in\bar{\mathcal{J}}\coloneqq\{1,\dots,\bar{J}\}$, robot $i$ to the point $\chi_{i}\coloneqq\mathcal{H}(x_{i})$, and the goal of robot $i$ to $\chi_{\textup{d}}\coloneqq\mathcal{H}(x_{\textup{d,i}})$. In the subsequent analysis we omit the robot index $i$ for ease of presentation.

Let now a constant $\bar{r}>0$ satisfying

The transformed free space of the robot can be defined as $\mathcal{F}_{\mathcal{H}}\coloneqq\mathcal{D}\backslash\{b_{1},\dots,b_{\bar{J}}\}$. We define next the set $\bar{\mathcal{J}}_{0}\coloneqq\{0\}\cup\bar{\mathcal{J}}$ as well as the distances $d_{\ell}:\mathcal{F}_{\mathcal{H}}\to\mathbb{R}_{\geq 0}$, $\ell\in\bar{\mathcal{J}}_{0}$, with $d_{\ell}(\chi)\coloneqq\|\chi-b_{\ell}\|^{2}$, $\forall\ell\in\bar{\mathcal{J}}$, and $d_{0}(x)\coloneqq 1-\|\chi\|^{2}$. Note that, by keeping $d_{\ell}(\chi)>0$, $d_{0}(\chi)>0$, we guarantee that $\chi\in\mathcal{F}_{\mathcal{H}}$ and hence the safety of the robot. We also define the constant

as the minimum distance of the goal to the transformed obstacles/workspace boundary. We revisit now the notion of the $2$nd-order navigation function from [36]:

An example for $\beta$ that satisfies properties 1) and 4), is

By appropriately choosing $\tau$, only one $\beta(d_{\ell}(\chi))$, $\ell\in\bar{\mathcal{J}}_{0}$ affects the robotic agent for each $\chi\in\mathcal{F}_{\mathcal{H}}$, with ${\beta^{\prime}(d_{\ell}(\chi_{\textup{d}}))}={\beta^{\prime\prime}(d_{\ell}(\chi_{\textup{d}}))}=0$. Hence, properties 2) and 3) of Def. 3 are satisfied.

Intuitively, the obstacles and the workspace boundary have a local region of influence defined by the constant $\tau$.

To compensate for the unknown mass and friction terms of the dynamics (1), we define the estimates $\hat{m}$, $\hat{\alpha}$ of $m$ and $\alpha$ (see Assumption 1), respectively, to be used in the control design.

Given the aforementioned definition, we design a reference signal $v_{\textup{d}}:\mathcal{F}_{\mathcal{H}}\to\mathbb{R}^{2}$ for the robot velocity $\dot{x}$ as

where $J_{\mathcal{H}}(x)\coloneqq\frac{\partial\mathcal{H}(x)}{\partial x}$ is the nonsingular Jacobian matrix of $\mathcal{H}$. Next, we define the respective velocity error $e_{v}\coloneqq\dot{x}-v_{\textup{d}}(x)$, and design the control law as $u:\mathcal{F}\times\mathbb{R}^{4}\to\mathbb{R}^{2}$, with

where $k_{\varphi}$ is a positive gain, and $\hat{m}$, $\hat{\alpha}$ evolve according to

with $k_{m}$, $k_{\alpha}$ positive constants. The aforementioned control protocol guarantees safe asymptotic convergence of the robot to its goal from almost all collision-free initial conditions, i.e., except for a set of measure zero, provided that $\tau$ is sufficiently small and $k_{\varphi}>\frac{\alpha}{2}$ (see Theorem 2 in [36]). Therefore, since the convergence is asymptotic, we conclude that there exists a finite time instant $t_{i}>t_{0}$ such that $\bar{\mathcal{B}}(x_{i}(t_{i}),r_{i})\subset\pi_{i^{\prime}_{k}}$, achieving thus the transition $\pi_{i_{k}}\to_{i}\pi_{i^{\prime}_{k}}$.

We deal now with the control design for the cooperative object transportation problem. Consider an object $j\in\mathcal{M}$ grasped by a team of robots $\mathcal{A}$ at $t_{0}\in\mathbb{R}_{\geq 0}$, i.e., $\mathcal{AG}_{j}(t_{0})=\mathcal{A}$, evolving subject to the dynamics (5) and satisfying $\bar{\mathcal{B}}(x^{o}_{j}(t_{0}),r_{\mathcal{A},j})\subset\pi_{j_{k}}$, for some $j_{k}\in\mathcal{K}$. The goal is the transportation of the object to some $\pi_{j_{k}^{\prime}}$, $j_{k}^{\prime}\in\mathcal{K}$. As in the robot navigation case, we let $x^{e}_{j}=x_{j}$, $r^{e}_{j}=r^{o}_{\mathcal{A},j}$, denoting the entity consisting of object $j$ and the robots $\mathcal{A}$, viewing the rest of the entities as static obstacles. By also aiming to avoid entering the RoI $\pi_{k}$, for $k\in\mathcal{K}\backslash\{j_{k},j_{k}^{\prime}\}$, the free space of the entity $\mathcal{F}_{j}=(\mathcal{W}_{\textup{f}}\backslash\mathcal{O}_{j})\ominus r^{o}_{\mathcal{A},j}$, where

Let the desired navigation goal of the object be $x_{\textup{d},j}$, provided by the procedure $\mathcal{P}_{s}$. Similarly to the robot navigation case, we use the transformation $\chi=\mathcal{H}(x)$ to transform the environment to the open unit disk $\mathcal{D}$, the obstacles to points $b_{\ell}$, the object-robots entity to the point $\chi$, and the object goal to $\chi_{\textup{d}}=\mathcal{H}(x_{\textup{d},j})$. Next, by employing the function $\beta()$, we design a reference signal $v_{\textup{d}}:\mathcal{F}_{\mathcal{H}}\to\mathbb{R}^{2}$ for the object velocity that is identical to (9). In addition, we define the adaptation variables $\hat{m}_{\mathcal{A},j}$ and $\hat{\alpha}_{\mathcal{A},j}$ as the estimates of the unknown coupled mass and friction coefficients $m_{\mathcal{A},j}$ and ${\alpha}_{\mathcal{A},j}$ (see Assumption 1), respectively, and design the control law for the robots $\mathcal{A}$ as $u:\mathcal{F}\times\mathbb{R}^{4}\to\mathbb{R}^{2}$, with

where $\mathsf{cf}_{\ell}$ are load sharing coefficients satisfying $\mathsf{cf}_{\ell}\geq 0$, for all $\ell\in\mathcal{A}$, and $\sum_{\ell\in\mathcal{A}}\mathsf{cf}_{\ell}=1$; $k_{\varphi}$ is a positive gain, and $\hat{m}_{\mathcal{A},j}$, $\hat{\alpha}_{\mathcal{A},j}$ evolve according to (11).

By invoking the property $\sum_{\ell\in\mathcal{A}}\mathsf{cf}_{\ell}=1$ and the fact that the object-robots system is converted to a point by the transformation $\mathcal{H}$, we guarantee, similar to the robot navigation case, the safe, asymptotic convergence of the object-robots entity to $\pi_{j^{\prime}_{k}}$ from almost all collision-free initial conditions, provided that $k_{\varphi}>\frac{\alpha}{2}$. Hence, there exists a finite time instant $t_{j}>t_{0}$ such that $\bar{\mathcal{B}}(x_{j}(t_{j}),r^{o}_{\mathcal{T},j})\subset\pi_{j^{\prime}_{k}}$, achieving thus the transition $\pi_{i_{k}}\xrightarrow{T}_{\mathcal{A},j}\pi_{j^{\prime}_{k}}$.

The second part of the solution is the derivation of a high-level plan that satisfies the given LTL formulas $\phi_{i}$ and $\phi^{o}_{j}$. Thanks to (i) the proposed control laws that allow robot transitions and object transportations $\pi_{k}\rightarrow_{i}\pi_{k^{\prime}}$ and $\pi_{k}\xrightarrow{T}_{\mathcal{A},j}\pi_{k^{\prime}}$, respectively, and (ii) the off-the-self control laws that guarantee grasp and release actions $i\xrightarrow{g}j$ and $i\xrightarrow{r}j$, we can abstract the behavior of the multi-robot-object system using a finite transition system as presented in the sequel.