Control of Magnetic Microrobot Teams for Temporal Micromanipulation Tasks

Control of Magnetic Microrobot Teams for Temporal Micromanipulation Tasks

Yiannis Kantaros   Benjamin Johnson   Sagar Chowdhury   David J. Cappelleri   and Michael M. Zavlanos   Y, Kantaros and M. Zavlanos are with the Department of Mechanical Engineering & Material Science, Duke University, Durham, NC, 27708 USA, e-mail: {yiannis.kantaros,michael.zavlanos}@duke.edu. B. Johnson, S. Chowdhury, and D. Cappelleri are with the School of Mechanical Engineering, Purdue University, West Lafayette, IN, 47907 USA, e-mail: {john1360,sagar353,dcappell}@purdue.edu. This work was supported by NSF grants IIS-1358446 and IIS-1302283. Opinions expressed are those of the authors and do not necessarily reflect opinions of the sponsors.
Abstract

In this paper, we present a control framework that allows magnetic microrobot teams to accomplish complex micromanipulation tasks captured by global Linear Temporal Logic (LTL) formulas. To address this problem, we propose an optimal control synthesis method that constructs discrete plans for the robots that satisfy both the assigned tasks as well as proximity constraints between the robots due to the physics of the problem. Our proposed algorithm relies on an existing optimal control synthesis approach combined with a novel technique to reduce the state-space of the product automaton that is associated with the LTL specifications. The synthesized discrete plans are executed by the microrobots independently using local magnetic fields. Simulation studies show that the proposed algorithm can address large-scale planning problems that cannot be solved using existing optimal control synthesis approaches. Moreover, we present experimental results that also illustrate the potential of our method in practice. To the best of our knowledge, this is the first control framework that allows independent control of teams of magnetic microrobots for temporal micromanipulation tasks.

Keywords: Path Planning for Multiple Mobile Robot Systems, Micro/Nano Robots, Temporal Logic Planning, Optimal Control Synthesis

I Introduction

Manipulation of microscale objects can be characterized by a number of unique features: (1) The manipulator has to be small relative to the workspace such that it does not cover the entire workspace, (2) The mass of the microscale object is very small which makes the surface related forces (e.g. surface tension, drag, viscous forces etc.) dominate the inertial forces (e.g. weight of the object), (3) Highly parallelized operation requires the robots/manipulators to be capable of coordinating with each other in manipulating a single object or multiple objects in a sequential manner. The ability to manipulate such objects autonomously in a sequential manner can potentially revolutionize the micro-assembly operations where a microscale component will be moved with a micro-robot to the workspace to be assembled with its counterpart carried by another robot to create a heterogenous product. These unique features give rise to a new set of constraints on the design of the robots as well as provide unique opportunities for the development of novel actuation and planning techniques.

In this paper, we propose a novel control framework for teams of magnetic microrobots responsible for accomplishing temporal micromanipulation tasks. We have utilized an 8 8 planar magnetic coil array from our previous work [1, 2, 3] and developed a new 11 11 planar magnetic coil array for use as an actuation system for independently controlling multiple microrobots. Each coil can be switched on or off independently creating a local magnetic field to only actuate the microrobot in its vicinity. Therefore, the coil array is suitable for our control framework to achieve temporal actuation of microrobots. This magnetic substrate partitions the workspace into square cells giving rise to a Finite Transition System (FTS) that models the robot motion. The micromanipulation tasks that the robots need to accomplish are captured using global Linear Temporal Logic (LTL) formulas that also incorporate proximity constraints that the robots need to satisfy during their motion, that are due to the physics of the problem. To generate high-level plans for the robots, we propose a novel optimal control synthesis algorithm that combines an existing temporal planning method with a new technique that reduces the state-space of the product automaton associated with the LTL statement.Given the transition systems that abstract robot mobility, the key idea is to create smaller transition systems that may not be as expressive as the original ones but they can still generate motion plans that satisfy the assigned LTL tasks. In this way, our method can solve large planning problems arising from large magnetic coil arrays, large numbers of robots, and complex temporal tasks, which is not possible using existing optimal control synthesis methods. Finally, given discrete high-level plans for the robots, we control the magnetic fields within each cell in the workspace so that the robots can navigate across cells as specified by the discrete high-level plans. We present local magnetic field actuation experiments that illustrate our approach. To the best of our knowledge, this is the first control framework for magnetic microrobots accomplishing temporal micromanipulation tasks that has also been successfully demonstrated in practice. A preliminary description of the proposed planning method is summarized in the extended abstract [4]. Compared to [4], we analyze the complexity of the proposed algorithm while extensive simulation and experimental studies are provided.

I-a Related Work

Contact based manipulation where two to four micromanipulators with point contacts are coordinated to grasp an object and transport it to a goal location have been demonstrated in [5, 6, 7, 8, 9, 10]. Rather than using multiple manipulators, a flexible microgripper attached to a multi DOF manipulator can be used to realize a pick-and-place operations as demonstrated in [11, 12]. Manipulator and micro-gripper based manipulation for micro-assembly operations based on micro snap fasteners has also been demonstrated [13, 14, 15, 16]. In many of these applications, several parts need to be soldered for making stable electrical connections. Venkatesan and Cappelleri [17] have replaced one of the micromanipulators in their assembly cell with a soldering iron along with an automated solder feed mechanism to realize a flexible soldering station for micro-assembly operations. These pick-and-place approaches are suitable for fast serialized operations where the gripper or manipulators are designed to handle only one type of object.

Another powerful approach for micromanipulation tasks is to replace the fixed manipulator with mobile microrobots. Each microrobot can be equipped with a customized gripper suitable for handling a particular object. Due to their small size, global actuation fields (e.g. magnetic field, optical field, physiological energy) are typically utilized as actuation mechanisms for microscale robots [18]. However, actuating multiple robots with a global field is challenging as it is difficult to apply individual and/or independent control on the robots. To address this challenge, optical manipulation methods have been recently developed where a single laser beam is split into multiple paths with the help of a spatial light modulator or high speed mirror technology [19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29]. However, optical manipulation is only suitable for objects that are less than 10 in size due to the very small actuation force (on the order of pico-Newtons) that can be generated by the laser beam.

Magnetic fields can generate forces as high as on the order of micro-Newtons and can also be modulated into various forms with minimum effort. Hence, magnetic fields have become a very useful tool to address a variety of applications from biology [30, 31, 32, 33, 34] to micro assembly [35]. To obtain independent control, there have been various approaches that introduce heterogeneity into the robots so that they can be affected by the same field in a significantly different fashion [36, 37, 38]. In this way, controllers can be developed that can control multiple robots somewhat independently of each other with the same global input [39]. Another approach is to utilize the influence zone of a particular magnetic coil and partition the workspace accordingly so that identical robots can be controlled by dedicated coils without affecting the others [40]. Hybrid approaches, where a magnetic field actuates a robot while an electrostatic surface is selectively activated to make the other robot static have also been proposed for independent control with magnetic global fields [41]. While the approaches presented above allow for different behaviors of the microrobots in the workspace, these motions are still coupled to some degree. Therefore, they are not useful when the robots need to truly independently operate in the workspace. Moreover, these approaches are not scalable since the complexity of coupled dynamics increases as the number of robots increases in the workspace. In addition, the controller needs to account for the motion constraints of the robots that might make some manipulation tasks in the workspace unattainable by the robots. In a truly independent system, the robots will not have any motion constraints due to the nature of actuation and hence will operate under fewer constraints that will help in achieving complex manipulation tasks.

To overcome the challenges in controlling microrobots independently using global magnetic fields, Pelrine et al. [42] developed a four layer printed circuit board that generates local magnetic fields for independent control of multiple homogeneous mm-scale robots. Similarly, arrays of multiple mm-scale planar coils have been proposed in [1] that generate local magnetic fields large enough to actuate multiple small scale robots independently. This design was successfully experimentally demonstrated for autonomous navigation in [2, 3]. Chowdhury et al. [43, 44] proposed a similar small scale array of current carrying wires arranged on a planar substrate. The array can be provided with current individually to actuate micron sized robots independent of each other.

The planning algorithm proposed in [3, 44] computes the waypoints for each robot separately, without considering the other robots, and does not consider proximity constraints between the robots due to the physics of the problem, e.g., the magnetic forces applied between robots. Hence, the waypoints computed in [3, 44] need to be recomputed frequently to account for such constraints. Moreover, the planning problems addressed in [3, 44] are restricted to point-to-point navigation and can not incorporate high-level tasks, such as sequential or temporal navigation and operation, that are typical in manipulation. Control synthesis for mobile robots under complex tasks, such as those of interest here, can be captured by Linear Temporal Logic (LTL) formulas. These build upon either bottom-up approaches when independent LTL expressions are assigned to robots [45, 46, 47] or top-down approaches when a global LTL formula describing a collaborative task is assigned to a team of robots [48, 49]. Common in the above works is that they rely on model checking theory [50, 51] to find paths that satisfy LTL-specified tasks, without optimizing task performance. Optimal control synthesis under local and global LTL specifications has been addressed in [52, 53, 54, 55, 56] and [57, 58, 59], respectively. In top-down approaches [57, 58, 59], optimal discrete plans are derived for every robot using the individual transition systems that capture robot mobility and a Non-deterministic Bchi Automaton (NBA) that represents the global LTL specification. Specifically, by taking the synchronous product among the transition systems and the NBA, a synchronous product automaton can be constructed. Then, representing the latter automaton as a graph and using graph-search techniques, optimal motion plans can be derived that satisfy the global LTL specification and optimize a cost function. As the number of robots or the size of the NBA increases, the state-space of the product automaton grows exponentially and, as a result, graph-search techniques become intractable. Consequently, these motion planning algorithms scale poorly with the number of robots and the complexity of the assigned task. To the contrary, the method we propose here scales to much larger planning problems that are typical for micromanipulation tasks.

I-B Contributions

The contributions of this work are the following. First, to the best of our knowledge, we present the first planning method for the independent control of teams of magnetic microrobots for temporal micromanipulation tasks, that has also been demonstrated successfully in practice. Second, our proposed planning method is able to solve much larger planning problems, that are typical for micromanipulation tasks, compared to existing optimal control synthesis techniques. Third, with respect to local magnetic field actuation of mobile microrobots, we have identified new diagonal static equilibrium points in the workspace that allow for additional waypoints for path planning and diagonal robot trajectories. We have also systematically characterized and utilized the surrounding coils of the robot to obtain predictable motions between workspace waypoints. Finally, the experimental results of the proposed algorithm exhibit temporal synchronization of multiple magnetic microrobots being independently controlled in the workspace for the first time. Experiments with two, three, and four microrobots are presented. Additionally, a two robot microassembly experiment has been demonstrated to show the potential for using the system for micromanipulation and assembly tasks.

The rest of this paper is organized as follows. In Sec. II, we present the theory behind using local magnetic field actuation for magnetic microrobot control. The problem we are solving is defined in Sec. III. Next, the proposed method for temporal planning for microrobot teams is described in Sec. IV followed by simulation results using the method in Sec. V. Our microrobot experimental testbed is described in Sec. VI along with validation and multi-robot LTL experiments. Finally, Sec. VII concludes the paper.

Ii Local Magnetic Field Actuation

Consider the 8 8 planar array of magnetic coils developed in [1, 2, 3] and shown schematically in Fig.1, and permanent magnetic disc-shaped microrobots with diameters smaller than the footprint of a single coil. Each magnetic coil has a radius of influence, , that corresponds to a distance from the coil center. If a robot is within a distance of from the center of the coil, it will be influenced when the coil is activated. If the magnetic robot is greater than from the coil center, the activated coil will not affect the robot.

Fig. 1: Schematic of 8 8 planar magnetic coil array. Waypoints for the developed path planning algorithm consist of each coils center and corner (workspace diagonal) positions, all marked with an “x”.

The magnetic field generated by the a single planar coil can be evaluated by considering it as a series of finite wire segments in a spiral fashion. The force on a body with magnetization due to an external magnetic flux density generated by single coil is . A disc shaped robot has the peak magnetization in the Z-direction (), and hence the value can be approximated as and the force components in the X, Y, and Z-directions computed accordingly [3].

For a constant height at which the field gradient is measured, the magnetic flux densities and gradients experienced by the magnetic robot along the Y-axis in the YZ plane are shown in Fig. 2. Therefore, for a current flowing in the positive X-axis direction, the magnetic robot experiences a force in the negative Y-axis due to negative magnetic gradients. Thus, for a series of wires laid parallel to the X-axis, with currents all flowing in the same direction, the magnetic robot will translate in the Y-direction.

Fig. 2: Magnetic field density and magnetic flux gradient on a magnetic microrobot in the Y and Z-directions as a function of distance along the Y-axis from an infinitely long wire. Current is flowing in the positive X-direction.

The currents in a spiral coil then will either repulse, force the robot outward, or attract, pull the robot towards its center, depending on the direction of current flowing through the coil. The net forces on a robot for a fixed direction of current are shown in Fig. 3(a) and (b). Each coil acts like an electromagnet. When the robot has attractive potential, it is like the coil having a magnetic moment in the same direction as that of the robot. This will attract the robot if it’s inside the bounds of the coil. Meanwhile, if the robot is out of the bounds of the coil, the coil and robot will repel each other like two magnets aligned parallel in the workspace. Therefore, a repulsive potential generated by the coil will push the magnet to the outer border of the coil.

Fig. 3: Local magnetic field actuation. (a) Repulsive force due to clockwise wire current. (b) Attractive force due to anti-clockwise current in the planar coil. (c) Coil actuation states needed to actuate a microrobot to a diagonal waypoint from a center waypoint; (d) States needed to actuate a microrobot to a center waypoint from a diagonal waypoint.

To define waypoints (goal locations) for the robot, it is important that they are at equilibrium points, i.e points at lowest potential. In our previous work, it was believed that these equilibrium points only existed at the center of each coil, limiting the waypoints for path planning and trajectories to orthogonal moves in the workspace. Here we have identified an additional equilibrium point at the corner positions of each coil (diagonal positions in the workspace). The robot remains in the center of the coil if the coil “attracts” the robot, and it remains in the diagonal (corner) position if all the surrounding coils “repel” the robot. However, to reliably move from a center waypoint to a diagonal waypoint, all nine neighboring coils of the robot are needed, as shown in Fig.3(c). The repulsing coils push the robot to the diagonal goal while the outer surrounding attracting coils help direct the robot towards the diagonal. Since the robot is outside the borders of these attracting coils, they actually repulse the robot. This makes sure that the robot moves in a predictable manner to the diagonal waypoint. In order to move from a diagonal waypoint to a center waypoint, the robot simply has to reside in the influence radius, , of the corresponding coil. It will then get pulled to the the static equilibrium at the center of the attracting coil, as shown in  (Fig. 3(d)). All of these possible center and diagonal equilibrium points in the workspace yield the potential waypoints for path planning.

Iii Problem Definition

Consider a team of microrobots residing in a magnetic workspace with waypoints/locations described in Sec. II. The positions of the microrobots and the waypoints are denoted by and , respectively, where and .

Iii-a Discrete Abstraction of Robot Mobility

Robot mobility in the workspace can be represented by a weighted Transition System (wTS) defined as follows [50]; see also Figure 4.

Definition III.1 (weighted Transition System)

A weighted Transition System associated with robot is a tuple where:

  • is the set of states. A state is associated with the presence of robot in location .

  • is the initial state of robot ;

  • is a set of actions; The available actions at state are move diagonally: up and left, up and right, down and left, down and right.

  • is the transition relation. Transition from state to exists if there is an action at state that can drive the robot to state .

  • is a cost function that assigns weights/cost to each possible transition in the wTS. For example, these costs can be associated with the distance between two states and ;

  • is the set of atomic propositions, where the atomic proposition is true if , for a sufficiently small and false, otherwise, and

  • is an observation/output function defined as , for all .

Fig. 4: Graphical depiction of a wTS as per Definition III.1. Blue dots denote the states of the wTS and red dashed lines depict the available transitions between the states. The red and black square denote two robots that have to accomplish a temporal logic task while maintaining a distance between them that is always greater than .

Next, given for all robots, the Product Transition System (PTS) is constructed, which captures all the possible combinations of robots’ states in their respective and is defined as follows [50].

Definition III.2 (Product Transition System)

Given transition systems , the weighted Product Transition System is a tuple where:

  • is the set of states,

  • is the initial state,

  • is a set of actions,

  • is the transition relation defined by the rule111The notation here is adopted from [50]. It means that if the proposition above the solid line is true, then so is the proposition below the solid line. The state stands for , where with slight abuse of notation . Also, is defined accordingly. ,

  • ,

  • is the set of atomic propositions; and

  • is an observation/output relation giving the set of atomic propositions that are satisfied at a state.

In what follows, we give definitions related to the PTS, that we will use throughout the rest of the paper. An infinite path of a PTS is an infinite sequence of states, such that , , and , , where is an index that points to the -th entry of denoted by . A finite path of a PTS can be defined accordingly. The only difference with the infinite path is that a finite path is defined as a finite sequence of states of a PTS. Given the definition of the weights in Definition III.2, the cost of a finite path , denoted by , can be defined as

(1)

In (1), stands for the number of states in . In words, the cost (1) captures the total cost incurred by all robots during the execution of the finite path . The trace of an infinite path of a PTS, denoted by , where denotes infinite repetition, is an infinite word that is determined by the sequence of atomic propositions that are true in the states along , i.e., . The trace of a transition system PTS is defined as , where is the set of all infinite paths of PTS.

Iii-B Task Specification

In what follows, we assume that the microrobot team has to accomplish a complex collaborative task encapsulated by a global LTL statement defined over the set of atomic propositions . The basic ingredients of an LTL formula are a set of atomic propositions , the boolean operators, i.e., conjunction , and negation , and two temporal operators, next and until . For the sake of brevity we abstain from presenting the derivations of other Boolean and temporal operators, e.g., always , eventually , implication , which can be found in [50]. For instance, consider the following temporal task:

(2)

where is an atomic proposition that is true only if robot is located at of the magnetic workspace. In words, the task described in (III-B) requires robot 1 to move part 1 of an object to location . Once this happens, robot 1 moves back to location , Then robot 2, moves to location to take over the task. Then, robot 2 moves part 1 of this object to location and then eventually robot 3 that carries part 2 of the considered object moves to location to finalize the assembly task.

Also, we require that as the microrobots move to accomplish the assigned temporal task, the distance between them is always greater than , where is the radius of influence of a robot with respect to another in the same workspace; see also Figure 4. Note that . Robots closer together than will attract or repel each other. This way, we ensure that the magnetic fields used for mobility of robot do not induce movement of any robot . This requirement can encapsulated by the following LTL statement.

(3)

Consequently, the robots need to navigate the magnetic workspace so that the global LTL statement

(4)

is satisfied.

Iii-C Cost of Infinite Paths

Given an LTL formula , we define the language , where is the satisfaction relation, as the set of infinite words that satisfy the LTL formula . Given such a global LTL formula and an infinite path of a PTS satisfies if and only if , which is equivalently denoted by .

In what follows, we describe how an infinite path that satisfies an LTL formula can be written in a finite representation and then, we define its cost. First, we translate the LTL formula defined over a set of atomic propositions into a Nondeterministic Bchi Automaton (NBA) that is defined as follows [60, 50].

Definition III.3

A Nondeterministic Bchi Automaton (NBA) over is defined as a tuple , where

  • is the set of states,

  • is a set of initial states,

  • is an alphabet,

  • is the transition relation, and

  • is a set of accepting/final states.

Given the PTS and the NBA that corresponds to the LTL , we can now define the Product Bchi Automaton (PBA) , as follows [50].

Definition III.4 (Product Bchi Automaton)

Given the product transition system and the NBA , the Product Bchi Automaton is a tuple where:

  • is the set of states,

  • is a set of initial states,

  • is the transition relation defined by the rule ,

  • , where and , and

  • is a set of accepting/final states.

Given the PBA, we can represent an infinite path in a finite form. Specifically, any infinite path that satisfies an LTL formula can be written in a finite representation, called prefix-suffix structure, i.e., , where the prefix part is executed only once followed by the indefinite execution of the suffix part [54, 55, 56]. The prefix part is the projection of a finite path of the PBA, i.e., a finite sequence of states of the PBA, denoted by , onto , which has the following structure

(5)

with . The suffix part is the projection of a finite path of the PBA, denoted by , onto , which has the following structure

(6)

where . Then our goal is to compute a plan

(7)

where stands for the projection on the state-space , so that the LTL formula in (4) is satisfied and the following objective function is minimized

(8)

which captures the total cost incurred by all robots during the execution of the prefix and a single execution of the suffix part. In (8), and stands for the cost of the prefix and suffix part, where the cost function is defined in (1), i.e., .

The problem that we address in this paper is summarized as follows:

Problem: Determine a motion plan for the team of microrobots such that in (4) is accomplished, i.e., the assigned temporal task captured by is accomplished and independent mobility of all robots is ensured as per , while minimizing the total cost (8).

Iv Temporal Planning for Microrobot Teams

To solve the problem as defined, known optimal control synthesis techniques can be employed that rely on graph search techniques applied to the product automaton defined in Definition III.4. In IV-A, we briefly summarize such methods. Then, in Section IV-B, we develop a modification to this algorithm that allows us to construct weighted Transitions Systems (wTS) with smaller state-spaces that generate words that satisfy the LTL formula .

Iv-a Existing Optimal Control Synthesis Methods

The problem at hand is typically solved by applying graph-search methods to the PBA, see e.g., [54, 55, 56]. Specifically, to generate a motion plan that satisfies and minimizes the cost function (8), the PBA is viewed as a weighted directed graph , where the set of nodes is indexed by the set of states , the set of edges is determined by the transition relation , and the weights assigned to each edge are determined by the function . Then, we find the shortest paths from the initial states to all reachable final states and projecting these paths onto the PTS results in the prefix parts , where . The respective suffix parts are constructed similarly by computing the shortest cycle around the -th final state. All the resulting motion plans satisfy the LTL specification . Among all these plans, we can easily compute the optimal plan that minimizes the cost function defined in (8) by computing the cost for all plans and selecting the one with the smallest cost; see e.g. [55, 56, 58, 59].

(a)
(b)
Fig. 5: (a): Blue dots and red (solid and dashed) edges represent the states and all possible transitions among these states in , respectively. Red solid edges represent the transitions among the states of when the global LTL formula is described by (IV-B). (b): Yellow (green) [orange] dots and edges represent the states and transitions that are included in at the first (second) [third] iteration of the while-loop of Algorithm 1, respectively. Given the LTL expression (IV-B), for robot 1 and, therefore, .

Iv-B Proposed Optimal Control Synthesis Approach

The optimal control synthesis algorithm discussed in Section IV-A relies on the construction of a synchronous product transition system among all robots in the network. As a result, it suffers from the state explosion problem and, therefore, it is resource demanding and scales poorly with the number of robots. Specifically, the worst case complexity of that algorithm is , since it is based on executing the Dijkstra algorithm times.

To decrease the complexity of this model checking algorithm we develop a novel method that aims to reduce the state-space of the product transition system and, consequently, the cardinality of the sets and by taking into account the atomic propositions that appear in the LTL expression . Specifically, the algorithm checks at which regions these atomic propositions are satisfied and then constructs paths towards those regions. This way, different LTL expressions will result in different wTSs. To achieve this, we construct wTSs, denoted by so that , defined in Definition III.1 and satisfy the following trace-inclusion property (see Chapter 3.2.4 in [50])

(9)

In other words, our goal is to construct that may not be as “expressive” as , since , but have smaller state-spaces and are able to generate motion plans that satisfy the LTL formula . Note that for and it holds that while both wTSs are defined over the same set of actions , the same transition rule , and the same function . The wTSs are constructed iteratively as per Algorithm 1 until they become expressive enough to satisfy the LTL formula . Specifically, the wTS constructed at iteration of Algorithm 1 for robot is denoted by .

1:  if  then
2:     ;
3:  else
4:     ;
5:  end if
6:  , , , ;
7:  while  do
8:     Pick a state from ;
9:     Construct the PTS state ;
10:      (Alg. 2);
11:     ;
12:     if  then
13:        , ;
14:     end if
15:     ;
16:  end while
Algorithm 1 Update
1:  ;
2:  for  do
3:     if  then
4:        for  do
5:           if  then
6:              Add state to ;
7:              Transition exists with cost ;
8:           end if
9:           
10:        end for
11:     end if
12:     if  then
13:        for  do
14:           if  then
15:               to , if it does not already exist;
16:              Transition exists with cost ;
17:           end if
18:        end for
19:     end if
20:  end for
Algorithm 2

Initialization

First, we present the construction of initial wTS denoted by given a global LTL expression and original wTS . Given an LTL formula we define the following sets of atomic propositions. Let be an ordered set that collects all atomic propositions associated with robot that appear in without the negation operator in front of them, including the atomic proposition that is true at . Also, let be a set that collects all atomic propositions associated with robot that appear in with the negation operator in front of them. If an atomic proposition appears in more than once, both with and without the negation operator, then it is included in both sets. For example, consider the following :

(10)

Then, and become and where denotes the atomic proposition that is satisfied at the initial state of robot .

To construct the state-space of , we view the corresponding original transition system as a weighted directed graph , where the set of vertices and the set of indices are determined by and , respectively. Weights on edges are assigned by the function . Then, we compute the shortest paths from the location where is true to the location where is true avoiding all locations associated with atomic propositions in , for all ; see Figure (a)a. All states of that appear in these shortest paths comprise the set .222The order in the sets can be selected either randomly or by checking which order results in the smaller state-space . Given , we construct the corresponding PBA defined as . Given the sets of states , we define the sets that collect states in that if visited by robot , then the LTL formula might be violated. These sets will be used for the re-construction of the PBA at iteration of Algorithm 1. The sets are defined as , . Otherwise, [line 4].

Iterative Update

The PBA constructed at iteration of Algorithm 1 may or may not satisfy the LTL specification . In the latter case all robots design new wTSs, denoted by , with states-spaces that are constructed by adding a new state to their respective , as per Algorithm 1. The candidate state to be added to is selected from the neighborhood of the states that belong to an ordered set since, intuitively, visitation of these states may lead to violation [line 2]. Specifically, to construct a new state is added to that is generated by a discrete probability density function that is bounded away from zero on the . The set that contains the -hops connected neighbors of the -th state in , denoted by , in the graph excluding the states that already belong to the state space of ; see Figure (b)b.

Once the states are generated and the state-spaces are constructed, the corresponding PBA is constructed as well. Instead of reconstructing the PBA from scratch, which is a computationally expensive step, we construct it by updating as per Algorithm 2. Specifically, is initialized as . Then, we identify all possible transitions from states to that satisfy , where , for all [lines 2-11, Alg. 2]. Similarly, next, we identify all possible transitions from to , for all [lines 12-20, Alg. 2]. If for some , all the states from have been included then the index is increased by one and is reset to one [line 14]. We then have the following result:

Proposition IV.1 (Completeness)

Assume that , i.e., that the initial transition systems can generate motions plans that satisfy . Then, Algorithm 1 will eventually construct wTSs that can generate a motion plan that satisfies .

Proof: The proof is based on the fact that at the worst case scenario, Algorithm 1 will incorporate all states from into , i.e., , for some and, therefore, , where . In such a case, we have completing the proof.

Remark IV.2 (Optimality)

Let denote the optimal cost when the motion plan is computed over . Given that , for all robots and for all , it holds that , is the state-space of . Therefore, in general, for the optimal cost computed over we have that .

Iv-C Complexity Analysis

In this section, we discuss the computational complexity of Algorithm 1. At every iteration of Algorithm 1, we check if prefix-suffix plans that satisfy the assigned LTL specification can be generated based on the respective PBA . To check that, we compute the shortest paths from the initial states to the final states of the PBA (prefix parts) and the shortest cycles around these final states (suffix parts). Viewing the constructed PBA as a graph with set of nodes and edges denoted by and , respectively, we get that the computational complexity of this process is , since it is based on executing the Dijkstra algorithm times. If a feasible plan cannot be generated by , then a new state is sampled and added to the wTSs and the corresponding PBA is constructed by Algorithm 2. The complexity of Algorithm 2 is . Therefore, the computational complexity per iteration of Algorithm 2 is . Observe that the computational complexity of the Algorithm 1 increases with respect to iterations since the size of and increases.

Finally, viewing the PBA as a graph the memory required to store it using its adjacency list is . Note that this is much smaller than the memory required to store the original PBA, since and , for all by construction. This allows us to solve larger planning problems that the existing approaches cannot manipulate due to memory requirements.

Remark IV.3 (Construction of wTS)

Note that there can be cases where the resulting transition systems, constructed by Algorithm 1, are the same as the original ones. For example, this can happen if , for all robots , i.e., if all states in the original transition systems are associated with atomic propositions that appear in the assigned task , since then . Also, if an infeasible LTL task is assigned to the robots then eventually our algorithm will eventually construct wTSs with , for all robots . In these cases, our algorithm cannot alleviate the computational cost of synthesizing motion plans.

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Fig. 6: Case Study I: Robot trajectories due to the execution of the prefix part . (a), (b), and (c) illustrate the states of the prefix part, for , , and . The projection of the states , , and onto the trace-included wTS of the red and blue robot are depicted by red and blue disks, respectively. The red and blue circles illustrate the part of the workspace that falls within the influence radius of the red and blue robot in these states. Filled in red regions are obstacles in the workspace.

V Simulation Results

In this section we present three case studies, implemented using MATLAB R2015b on a computer with Intel Core i7-2670QM at 2.2 GHz and 4 GB RAM, that illustrate our proposed algorithm. The considered case studies pertain to motion planning problems with PBA that have , , and states, respectively. Recall that the state-space of the PBA defined in Definition III.4 has states. Note that the last two case studies pertain to planning problems that cannot be solved by standard optimal control synthesis algorithms, discussed in Section  I, that rely on the explicit construction of the PBA defined in Section IV-A due to memory requirements. In fact, our Matlab implementation of the algorithm described in Section IV-A cannot synthesize plans for PBA with more than few millions of states and transitions. In all case studies, the costs , defined in Definition III.1 are associated with the distance between states. Also, we select , , and distance units for the first, second and third case study. Finally, notice that the off-the-shelf model checkers SPIN [61] and NuSMV [62] cannot be used to synthesize plans that satisfy the LTL formula in (4). The reason is that weights associated with distance need to be assigned to transitions of the wTS’s, which is not possible in either SPIN or NuSMV. Videos of the simulations for all case studies can be viewed in Supplemental Video 1.

V-a Case Study I

(a) Case Study I
(b) Case Study II
(c) Case Study III
Fig. 7: Graphical depiction of the wTS constructed for each robot for the first (a), the second (b), and the third (c) case study. The transition system for the red, blue, green, magenta, and black robot are depicted with corresponding color. In (a) the yellow diamonds and brown squares correspond to states that were added to the wTS at the first and the second iteration of Algorithm 1. Red filled regions stand for obstacles in the workspace.
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Fig. 8: Case Study I: Robot trajectories due to a single execution of the suffix part . (a)-(f) illustrate the states of the suffix part, for , , , , , and . The projection of the states , , , , , and onto the wTS of the red and blue robot are depicted by red and blue disks, respectively, in each figure. The red and blue circles illustrate the part of the workspace that falls within the influence radius of the red and blue robot, respectively.
(a) Prefix Part.
(b) Suffix Part.
Fig. 9: Case Study I: Graphical depiction of the minimum distance among robots, when the robots are in states , for all (a) and , for all (b), where and stand for the number of states in the prefix and suffix part .

In the first simulation study we consider a network of microrobots residing in a workspace, as shown in Figure 6, with , for both robots. The assigned temporal task is described by the following LTL formula:

(11)

Specifically, the task in (V-A) requires: (a) the red robot to visit location infinitely often, (b) the red robot to avoid location until it visits once location , (c