CC BY
107Programming for Modular Reconfigurable Robots
Anna Gorbenko Vladimir Popov
Department of Mathematics and Department of Mathematics and
Mechanics Mechanics
Ural State University Ural State University
Ekaterinburg, Russia, 620083 Ekaterinburg, Russia, 620083
Email: gorbenko.aa@gmail.com Email: Vladimir.Popov@usu.ru
Abstract—Composed of multiple modular robotic units, self-reconfigurable modular robots are metamorphic systems that can autonomously rearrange the modules and form different configurations for dynamic environments and tasks. Selfreconfiguration is to solve how to change connectivity among modules to transform the robot from the current configuration into the goal configuration within the restrictions of physical implementation. The existing reconfiguration algorithms used different methods, such as divide-and-conquer, graph matching etc, to reduce the reconfiguration cost. However, the optimal solution with least reconfiguration steps has never been reached. The optimal reconfiguration planning problem of finding the least number of reconfiguration steps to transform between two configurations is NP-complete. In this paper we describe an approach to solve this problem. This approach is based on constructing a logical models for considered problem.
I. Introduction
Modular robotics has been the subject of much interest in the research community [1]. Using large numbers of simple modules to replace one complicated, special-purpose device provides benefits in terms of flexibility, robustness, and manufacturing cost. The challenge in these systems lies in controlling large numbers of low-powered, unreliable modules. Motion planning and shape formation for these systems is the main problem of such a difficult challenge.
Metamorphic robotic systems [2] can be viewed as a large swarm of connected robots which collectively act as a single entity. Potential applications of metamorphic systems composed of a large number of modules include:
• obstacle avoidance in highly constrained and unstructured environments;
• growing structures composed of modules to form bridges, buttresses, and other civil structures in times of emergency;
• envelopment of objects, such as recovering satellites from space;
• performing inspections in constrained environments such as nuclear reactors.
Self-reconfiguring robots were first proposed in [3]. In this planar system modules were heterogeneous and semi-autonomous. other research focused on homogeneous systems with non-autonomous modules in two dimensions [4] - [7] and three dimensions [8] - [10]. In this type of system the modules are not capable of acting independently, and thus
must remain connected. Five types of modular reconfigurable robotic systems have been proposed in the literature:
• robots in which modules are reconfigured using external intervention, e.g. [11] - [14];
• cellular robotic systems in which a heterogeneous collection of independent specialized modules are coordinated, e.g. [15] - [18];
• swarm intelligence in which there are generally no physical connections between modules, e.g. [19] - [22];
• modular robots composed of a few basic elements which can be composed into complex systems and used for various modes of locomotion, e.g. [23] - [25];
• fractal systems composed of modules with zero kinematic mobility, but which can walk over each other in discrete quanta due to changes in the polarity of magnetic fields, e.g. [5], [26].
In the present work, a metamorphic robotic system is a collection of independently controlled mechatronic modules, each of which has the ability to connect, disconnect, and climb over adjacent modules, e.g. [6]. A metamorphic system can dynamically reconfigure by the locomotion of modules over their neighbors. Thus they can be viewed as a collection of connected modular robots which act together to perform the given task. composed of multiple modular robotic units, self-reconfigurable modular robots are metamorphic systems that can autonomously rearrange the modules and form different configurations for dynamic environments and tasks.
Modular reconfigurable robot programming can be substantially more challenging than normal robot programming due to:
• scale / number of modules;
• concurrency and asynchronicity, both in physical interactions and potentially at the software level;
• the local scope of information naturally available at each module.
For modular reconfigurable robots it is developed several specialized programming languages (e.g. [27], [28]). However, existing programming methods show relatively poor performance for reconfiguration planning problems. Note that reconfiguration planning problems play a central role for modular robots (e.g. [29] - [40]). Solutions for such problems lies at the heart of any control system of modular robots. Performance
of such solutions is the base factor for the performance of the whole control system. So, the main challenges for modular robotic systems is an efficient planner. It has long been recognized that traditional methods are unsuitable due to the large search space and the blocking constraints imposed by realizable module design. To ease the planning problem, many groups have proposed different kinds of metamodules, groups of modules that act as a unit for planning or motion execution purposes, each specific to a particular module design [7], [38], [41], [42].
Poor performance for reconfiguration planning problems is not surprising, since such problems are computationally hard. In particular, in proved that the optimal reconfiguration planning problem of finding the least number of reconfiguration steps to transform between two configurations (oRP) is NP-complete. Therefore, we need some intelligent solution for this problem. However, the applying of distributed algorithms or any iterative procedure requires a great exchange of information between modules. This leads to the loss of solution accuracy and reduce performance. Therefore, it is desirable to solve oRP in a separate intelligent module which would generate a final solution represented by simple instructions. Note that the centralization of oRP solution allows to use some remote computing resources and makes the performance independent from computing resources of modules. When using such approach, programming of individual modules consists in
• processing of sensory information;
• transmission of sensory information;
• motor control;
• receiving instructions for actuators.
In this paper we describe an approach to solve oRP problem. This approach is based on constructing a logical models for considered problem.
II. Optimal Reconfiguration Planning Problem
Self-reconfiguration is to solve how to change connectivity among modules to transform the robot from the current configuration into the goal configuration within the restrictions of physical implementation. Depending on the hardware design, reconfiguration algorithms fall into two groups:
• reconfiguration for lattice-type modular robot and reconfiguration for chain-type modular robot. In lattice-type robot, modules lie in 2D or 3D grids;
• the reconfiguration is achieved through discrete movements of modules detaching from the current lattice location, moving along and surface of the robot and docking at the adjacent cells.
Example reconfiguration work includes [32] - [40] etc. In chain-type robots, modules can form moving chains and loops of any graph topology, and the reconfiguration is achieved through “connect” and “disconnect” operations between modules along with the joint motion of chains composed of several modules. Due to its difficulty, the chain-type reconfiguration has received less attention. Existing algorithms include [43] -
[47] etc. The different geometric arrangement of modules between lattice-type and chain-type modular robots makes their reconfiguration planning mechanisms fundamentally different. The work in this paper is more focused on chain-type reconfiguration. For simplicity, we will use the term “modular robots” or simply “robots” to denote “the chain-type modular robots”, and use “reconfiguration” to denote “chain-type reconfiguration” in the following.
The existing reconfiguration algorithms used different methods, such as divide-and-conquer [43], graph matching [44] etc, to reduce the reconfiguration cost. However, the optimal solution with least reconfiguration steps has never been reached. In [48] proved that the optimal reconfiguration planning problem of finding the least number of reconfiguration steps to transform between two configurations is NP-complete.
A. Configuration Representation
Before defining the optimal reconfiguration planning problem, we would describe representation of robot’s configuration first. Two robots with the same graph topology can function differently if the modules are connected via different connectors (see e.g. [48]). To fully represent a robot’s configuration, a special graph called C-Graph (Connector-Graph) is proposed in [48]. C-Graph is the extension of normal graph with differentiated connecting points. Each node has a finite number of ports that are internally labeled corresponding to the connectors of a module. A connection between module u’s connector i and module v’s connector j corresponds to an edge (i, j) between u and v.
In principle we could represent a robot's configuration as a C-Graph
G = (V,E),
V = {v[1], v[2],..., v[n]},
E = {e[1], e[2],..., e[m]},
where:
• each node v [i] e V represents the set
v[i] = {v[i, 1], v[i, 2],. .., v[i,pi]}
of connecting points of ith module, where pi is the
number of connecting points of ith module;
• each edge
e[j] = (v[ii,/i],v[i2,/2]) e E
represents a connection between module i1’s connector 11 and module i2’s connector 12, where
1 < il < n, 1 < i2 < n,
1 < ll < Pi! , 1 < 12 < Pi2 .
B. Reconfiguration Actions
The two elementary reconfiguration actions are:
• making new connections;
• disconnecting current connections between modules for connectivity rearrangement.
The robot can bend its body through module joints, so any two modules with free connectors can potentially be aligned and dock with each other.
C. Optimal Reconfiguration Planning Problem
The reconfiguration planning problem is defined as how modules in one configuration rearrange into another using several sets of reconfiguration actions. Basically, what connections to make and what connections to disconnect so as to reconfigure from arbitrary one shape to another? Without loss of generality, we will always assume that the number of modules in the initial configuration is the same as that in the goal configuration.
During the reconfiguration process, the reconfiguration actions are most time- and energy-consuming, so it is a common practice to aim at minimizing the number of reconfiguration steps, i.e. the number of connect actions plus the number of disconnect actions. Therefore, the optimal reconfiguration planning problem is to find the least number of reconfiguration steps to transform from the initial configuration into the goal configuration.
Since the number of physical connections is predefined in the initial and goal configurations, the number of connect actions is fixed once the number of disconnect action is known, and vice versa. So we get that the optimal reconfiguration planning problem is to find the either one of the following metrics:
• least number of connect actions;
• least number of disconnect actions;
• least number of reconfiguration steps (i.e., the number of connect actions plus the number of disconnect actions).
For given two connected C-Graphs
I =(V,Ei)
and
G = (V,E2)
we say that there exists a reconfiguration plan with at most k reconfiguration steps if and only if there exists a sequence of r < k connect and disconnect actions such that starting from I we obtain G and applying each of this connect and disconnect actions we obtain a connected C-Graph. The decision version of optimal reconfiguration planning problem is formulated as the following problem.
Optimal reconfiguration planning problem (ORP):
Instance: C-Graphs I = (V, E1) and G = (V, E2), a given integer k.
Question: Whether there exists a reconfiguration plan for C-Graphs I and G with at most k reconfiguration steps?
III. Logical model of ORP
The propositional satisfiability problem (PSAT) is a core problem in mathematical logic and computing theory. Propo-sitional satisfiability is the problem of determining if the variables of a given boolean function can be assigned in such a way as to make the formula evaluate to true. PSAT was the first known NP-complete problem, as proved by Stephen cook in 1971 [49]. Until that time, the concept of an NP-complete problem did not even exist. considered also different variants of the satisfiability problem. For instance, Satisfiability (SAT) is the problem of determining if the variables of a given boolean function in conjunctive normal form can be assigned in such a way as to make the formula evaluate to true. In practice, the satisfiability problem is fundamental in solving many problems in automated reasoning, computer-aided design, computer-aided manufacturing, machine vision, database, robotics, integrated circuit design, computer architecture design, and computer network design. Traditional methods treat the satisfiability problem as a discrete, constrained decision problem.
A. Reduction to PSAT consider a set of c-Graphs
{G[q] = (V, E[q] | 0 < q < k},
where
E[q] = {efe ^efo 2L..., efe mq]}
each edge
e[q, j] = (v[ii,1i],v[i2,/2]) e E[q]
represents a connection between module ii’s connector /i and module i2's connector /2, where
1 < ii < n, 1 < i2 < n, 1 < /i < Pi1 , 1 < /2 < Pi2 .
Let G[0] = I, G[k] = G. Now consider a set of boolean variables
{x[q, ii, i2, *3, i4] | 0 < q < k, 1 < ii < n,
1 < i2 < Pi1, 1 < i3 < n, 1 < i4 < Pi3 }.
Suppose that relation
x[q, ii, i2, i3, i4] = 1
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(v[ii,i2],v[i3,i4]) e E[q].
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Clearly, ^1 is not in conjunctive normal form. Using the distributive law, we can obtain from ^1 a boolean function in conjunctive normal form but this function will be have exponential size. In some sense it is a good news. The propositional satisfiability problem seems to become easier if boolean functions are restricted to those in disjunctive normal form. This is because such a formula is satisfiable if and only if some clause is satisfiable, and a conjunctive clause is satisfiable if and only if it does not contain both x and —x for some variable x. This can be checked in polynomial time. Correspondently, the propositional satisfiability problem seems to become harder if boolean functions are restricted to those in conjunctive normal form. From this point of view the impossibility of polynomial reduction from ^1 to a boolean function in conjunctive normal form is a good news.
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It is easy to see that ^[q] ^ ^"[q], where
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IV. Conclusion and Experimental Results
In recent years, many optimization methods, parallel algorithms, and practical techniques have been developed for solving the satisfiability problem (see [50]). In particular, proposed several genetic algorithms [51] - [54]. Considered hybrid algorithms in which the approach of genetic algorithms combined with local search [55].
Modern propositional satisfiability solvers are usually designed to solve SAT formula encoded in conjunctive normal form (CNF). Stochastic local search techniques have been successful in solving propositional satisfiability problems encoded in CNF. Recently complete solvers have shown that there are advantages to tackling propositional satisfiability problems in a more expressive natural representation, since the conversion to CNF can lose problem structure and introduce significantly more variables to encode the problem. CNF solvers can be disadvantageous for problems which are more naturally encoded as arbitrary propositional formula. The conversion to CNF form may increase the size of the formula exponentially, or significantly reduce the strength of the formulation. The translation may introduce many new variables which increases the size of the raw valuation space through which the solver must search. Recently, interest has arisen in designing non-clausal satisfiability algorithms (see e.g. [56] - [63]).
Relatively high efficiency demonstrated by algorithms based solely on local search. Of course, these algorithms require exponential time at worst. But they can relatively quick receive solutions for many boolean functions. Therefore, it is natural to use a reduction to different variants of the satisfiability problem to solve computational hard problems.
Encoding problems as Boolean satisfiability and solving them with very efficient satisfiability algorithms has recently caused considerable interest. In particular, local search algorithms have given impressive results on many problems. For example, there are several ways of SAT-encoding constraint satisfaction [64] - [73], clique [74], planning [75] - [95], and colouring problems [74], [96] - [98]. The maximum cut, vertex cover and maximum independent set problems can be reduced to MAX-2-SAT [99] - [101]. There are a number of implicit reductions from the Hamiltonian cycle problem to the satisfiability (SAT) problem (see [74], [102], [103]).
In previous section we obtain an implicit reduction from the optimal reconfiguration planning problem of finding the least number of reconfiguration steps to transform between two configurations to some variants of satisfiability: PSAT, SAT. We create a generator of special hard and natural instances for the optimal reconfiguration planning problem of finding the least number of reconfiguration steps to transform between two configurations. We use algorithms from [104]. Also we design our own genetic algorithm for SAT which based on algorithms from [104]. We use heterogeneous cluster based on three clusters (Cluster uSu, Linux, 8 calculation nodes, Intel Pentium IV 2.40GHz processors; umt, Linux, 256 calculation nodes, Xeon 3.00GHz processors; um64, Linux, 124 calculation nodes, AMD Opteron 2.6GHz bi-processors)
[105]. For computational experiment we create special hard test sets and natural test sets. Special hard test sets based on ideas from [106]. Natural test sets based on ideas from [48]. In tests we consider systems consisted from approximately 400 of modular robots.
Each test was run on a cluster of at least 100 nodes. For special hard test sets: the maximum solution time was 16 hours; the average time to find a solution was 33.2 minutes; the best time was 116 seconds. For natural test sets: the maximum solution time was 9 hours; the average time to find a solution was 9.8 minutes; the best time was 9 seconds. Based on our experiments we can say that considered model can be used as an efficient planner.
Acknowledgment
The work was partially supported by Grant of President of the Russian Federation MD-1687.2008.9 and Analytical Departmental Program "Developing the scientific potential of high school” 2.1.1/1775.
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