A Modular Self-Reconfigurable Bipartite Robotic System: Implementation and Motion Planning

Transcription

1 A Modular Self-Reconfigurable Bipartite Robotic System: Implementation and Motion Planning CEM ÜNSAL, HAN KILIÇÇÖTE, AND PRADEEP K. KHOSLA Institute for Complex Engineered Systems, Carnegie Mellon University, Pittsburgh, PA, , USA {unsal kiliccote Abstract. In this manuscript, we discuss I-Cubes, a class of modular robotic system that is capable of reconfiguring itself in 3-D to adapt to its environment. This is a bipartite system, i.e., a collection of (i) active elements for actuation, and (ii) passive elements acting as connectors. Active elements (links) are 3-DOF manipulators that are capable of attaching/detaching from/to the passive elements (cubes), which can be positioned and oriented using links. Self-reconfiguration capability enables the system to perform locomotion tasks over difficult terrain; the shape and size can be changed according to the task. This paper describes the design of the system, and 3-D reconfiguration properties. Specifics of the hardware implementation, results of the experiments with the current prototypes, our approach to motion planning and problems related to 3-D motion planning are given. Keywords: Modular robots, self-reconfiguration, metamorphic robots, collective robotics, mechatronics 1. Introduction Statically stable gaits that are currently available for mobile robots include wheels, threads, and similar mechanisms. Although capable of moving relatively faster in level ground, such robots cannot climb stairs, or move over obstacles. A robot that can move with relative ease on flat terrain and an ability to climb over large obstacles is yet to be designed. On the other hand, recent progress in technologies such as electronics component design and micro electromechanical systems enabled research thrusts on small mobile robots, and multi-sensor/ multi-actuator systems. Applications for groups of small mobile robots accomplishing tasks in unstructured environments and narrow spaces are slowly emerging. Drawing from the earlier research efforts on modular robots (e.g., [3,6,15]) and current research on small mobile robots (e.g, Millibots [7]), we propose a modular self-reconfigurable group of robots that consists of two modules with different characteristics. A sufficient number of modules combined as a single entity would be capable of self-reconfiguring themselves into defined shapes. Self-reconfiguration capability in three dimensions provides a new type of locomotion gait that may be combined with other capabilities. A large group of modules that can change its shape according to the locomotion, manipulation or sensing task at hand will then be capable of transforming into a snake-like robot to travel inside a air duct or tunnel, a legged robot to move on unstructured terrain, a climbing robot that can climb walls or move over large obstacles, a flexible manipulator for space applications, or an extending structure to form a bridge. Creating a modular system with identical elements has several advantages over large and complex robotic systems. The units can be mass-produced, and their homogeneity can provide faster production at a lower cost. A large system consisting of many elements is less prone to mechanical and electrical failures, since it would capable of replacing nonfunctioning elements by removing them from the group and reconfiguring its elements. Homogeneous groups of modules that are capable of self-reconfiguring into different shapes also provide a manufacturing solution at the design phase where identical elements are considered, while providing a modular system that can be re-arranged for different tasks. To have the advantages given above, a modular system must have several essential properties, such as geometric, physical and mechanical compatibility among individual modules. Furthermore, several design issues need to be considered for such a system

2 to be autonomous. Essential properties of our particular system as well as design issues relating to the implementation are given in Section 2. Previous research on modular robotic systems includes serial link manipulators that can be designed according to the task specification [15], design of kinematic structures that can be modularly synthesized [12], and cellular systems as self-organizing manipulators [6]. These and similar ideas on modularity has been applied to modular structures that are capable of selfreconfiguring into desired shapes. Previous twodimensional systems include Inchworm [9], selforganizing robots [8] moving in vertical planes, selfrepairing modular machine [21], and metamorphing robots [13] moving in horizontal plane. Recent threedimensional systems are Polypod/Polybot [19, 20] and CONRO [2] that can combine different gaits and the self-reconfiguring molecule [10] and the selfreconfigurable structure [11] that are both capable of moving in any direction using neighboring elements as pivot points. The system described here is a self-reconfiguring bipartite system that enables the separation of the components that provide computation, sensing and power from the components that provide actuation, in order to combine different gaits and task-oriented modules with self-reconfiguration capabilities. In the next section, we introduce our approach to modular self-reconfigurable robotic systems, the I-Cubes (or ICES-Cubes), defining its characteristics and advantages. Section 3 illustrates simple examples of reconfiguration in three-dimensional space. Section 4 describes the hardware implementation and discusses related design issues. Sections 5 and 6 define our approach to the motion planning for this system, while Section 7 gives the initial algorithms and results. Section 8 discusses the hardware implementation and motion planner, defining possible improvements, with Section 9 concluding the paper with current contributions and future extension. 2. A 3-D Modular Self-Reconfigurable System I-Cubes are a class of modular self-reconfigurable system with two distinct elements. This bipartite system is a collection of independently controlled mechatronic modules (called links) and passive connection elements (called cubes). The links are capable of connecting to and disconnecting from the cube faces. A link can therefore move from one cube face to another, from one cube to another neighboring cube, or move a cube while attached to another (Figures 1 and 2). We envision that all active (link) and passive (cube) modules are capable of permitting power and information flow to attached modules. A group of links and cubes does in fact form a dynamic three-dimensional (pseudo-)graph links are the edges and the cubes are the vertices. When the links move, the structure and the shape of the 3-D system change. In this self-reconfiguring system: Active elements can be independently controlled; only cubes attached to the moving end of a link are affected by link motions. All elements are physically, mechanically, and computationally compatible, i.e., any link can connect to any cube, and the cubes have attachment points to receive the link connectors. Cube positions fit a cubic lattice to guarantee interlocking of neighboring element, i.e., distance from one cube to another is constant while in position to accept the links. All elements form a single connected (pseudo-) graph where all links are connected to at least one cube, and all cubes are connected to at least one link. Active elements have sufficient degrees of freedom to complete motions in three-dimensional space. Since all the actuation for self-reconfiguration, with the exception of the attachment mechanism, is provided by the links, cubes can be used to provide computation, sensing and power resources. If the modules are designed to exchange power and information, the cubes can be equipped with batteries, microprocessors, and sensing modules to create a collective intelligence while the links becomes the muscles of the system. Furthermore, it is possible to remove some of the attachment points on the cubes to provide these modules with different and faster gaits, such as wheeled or treaded locomotion. Specifically, we envision small mobile robots that can reposition themselves to form a group that is capable of changing its gait in order to move over obstacles that a single element cannot overtake (Section 3). Similar scenarios that require reconfiguration include climbing stairs and traversing pipes. The required attachment points between the links and the cubes need to be mechanically feasible to hold multiple elements together and must be designed to transfer power and information between elements. In the following sections, we will introduce the specifications of these elements and the attachment mechanism Geometric Design and Link Actuation Figure 1 shows two links connecting three cubes. If the length of a cube edge is taken as d, the links should have four essential sections of length d/2, d, d, and d/2. 2

3 Three rotation degrees of freedom for the links are provided by the joint (J2) between two middle sections of length d, and the joints located at both ends sections of length d/2 (J1, J3). Joints J1 and J3 are both assumed capable of providing continuous 360-degree rotations, while J2 can only rotate 270 degrees. In order to keep the cubic lattice formed by the cubes intact, the distance between the cubes must be exactly d, while in position to accept the links. Therefore, the links must be designed to provide this exact distance when the two middle sections are closed to touch (See the link on the right in Figure 1). The design properties and the attachment capabilities given above the links to: move from one face to another face of a cube. move one cube while attached to another cube. move from one cube to another. J1 d J2 d Figure 1. Geometric requirements for the system. 1 rotate 1 rotate attach 2 J3 3 detach d/2 4 rotate 1 rotate detach 3 Figure 2. Link motions. (c) 4 rotate 2 attach All these motions require links to be capable of attaching to the cube faces, and performingmiddle and end joint rotations. Note that it is also possible for a link to move a cube by rotating its middle joint, although not shown here. The separation of the system into two different modules increases the minimal distance required for identical elements to move in three dimensions. We call this distance the group resolution. In a system where the cubes could attach to and move over neighboring cubes (e.g., 3-D selfreconfigurable structure in [11]), the group resolution A d B would be d, the size of a unit. In this system, however, the action displaces the elements by 2d. The 3-D space is separated into two disjoint regions that are occupied by the links and the cubes (The region for the cubes is discontinuous). The 3-D space can be visualized as the combination of infinitely many grids of cells of length d. A cube can occupy only one of these cells, while there are six cells that a link can occupy, assuming discretized states separated by 90-degree joint angles as shown in Figure Design of the Links and the Cubes The links have three worm-wheel gear mechanisms driven by small servos to provide continuous rotation at the end joints, and 270-degree rotation at the middle joint (Figure 3). All servos are coupled to worm gears driving the wheels. At the end joints, the wheels are placed on a shaft that the connection pieces are attached. At the middle joint, the servo and the worm are located on one side of the link body, while the wheel and its shaft are connected to the other side. Thus, rotating the servos will rotate the connectors at the end sections, or rotate one side of the link with respect to the other at the middle. Figure 3 shows the design of the link equipped with the servos, and gear structures. The middle wheel shaft is coupled to the piece holding the single servo, while the middle worm is coupled with one of the servos in the other side. The distance between joint shafts is again equal to d. There are two main advantages of using worm-wheel structure coupled with servos. First, the rotational speed of the servo is reduced by the ratio of the gear mechanism. (However, higher values will cause the link to move slowly). Similarly,the torque provided by the servo is increased by the same ratio. Second, the worm-wheel system is an energy efficient solution for actuation. Since the wheel cannot drive the worm, the servos do not have to be powered all the time to hold the links in a specific position. The cubes are passive elements that consist of at most six attachment points for the link connectors. They are not capable of moving by themselves; a cube attached to a link can either be rotated, translated in a plane, or act as a pivot point for a moving link. The role of the cube depends on the position and motion of the link as well as connections of the modules. We have designed individual faceplates with male and female matching edges. These can be used on a platform to test the links or combined to form cubes with different numbers and types of attachment mechanisms. Alternative mechanisms are described in [16]; a newer version can be found in [18]. Figure 3 shows CAD image of a cube retrofitted with five 3

4 attachment points for cross-shaped link connectors that we are currently using. The system is based on twistand-lock mechanical behavior, and the details are given in the next section. When six faceplates are combined to form a cube, the maximum available volume inside the cube for computational elements and power source is approximately 151cm 3, or 40% of the total volume of the cube excluding the walls. Note that one edge of a cube is 8cm. that the connector clears the cube face while approaching the attachment point. Note that the connector does not approach the cube face from the normal, but following a curve that is normal to the faceplate only at the surface of the supporting layer below the cube surface. The edges of the connector are also rounded to facilitate sliding into locked position Figure 3. CAD images: The link is shown with two different connectors (servo holders expanded); the cube is shown with five faceplates Twist-and-Lock Attachment Mechanism A self-reconfiguring modular robotic system must consist of elements that are capable of attaching/detaching themselves to/from neighboring elements. This attachment mechanism must be energy efficient. Actuation for the latching mechanism should be limited, preferably for a few seconds during the locking phase. Any actuation method enabling the modules to lock must be designed to provide a statically and dynamically stable system that holds the elements together. An ideal candidate for such a system is an attachment mechanism that is in locked position while idle, and can easily switch to unlocked position for a short period during attachment/detachment. The mechanism should not open without actuation. To obtain such a mechanism, the latching motion on the cube side must be decoupled from the locking motion of the link connector. The aim is to create a lock that will only change state when actuated, and that needs to be actuated only when a state change is needed. We have designed a cross-shaped link connector for the links that enters and twists inside an opening on the cube face (Figure 4). After entering the similarly shaped opening, the link connector will twist to lock in place. This motion will limit the motion of the link away from the cube surface, while the upward motion of the two locking pegs will stop the free rotation of the link end with respect to the cube. This mechanism requires a single actuator to move the pegs up and down for latching (See 3 in Figure 4). The edges of the opening on the faceplate are chamfered to guarantee Figure 4. Cross-shaped connector with twist-and-lock mechanism. The two pegs hold the cross-shaped connector in locked position when pushed upward, and are actuated by a servo pulling and pushing them via a rod connecting the two pegs. Therefore, the rotation of the link shaft in clockwise direction results in two different actions depending on the lock state: (un)locking the connector to the cube, or rotating the cube in place. The cross-shaped connector must be correctly oriented with respect to the opening on the faceplate before attempting to latch Design Issues The 3-D system described here is designed to overcome common problems in a modular selfreconfigurable system. Capabilities such as efficient power consumption, on-board power resources, information and power transfer between elements, and autonomous motion, are required in order to have a feasible practical implementation. Our design of actuators for the links and the connection mechanisms significantly reduce the power consumption. The worm-wheel structure and the latching elements are to be actuated only when a link or locking motion is required. Available space inside cubes can be used for batteries to power all actuators, sensing and control modules. Simple electromechanical connectors can provide multiple lines for power and serial communications while keeping continuous link rotations. The surface of the cross-shaped attachment piece will be the connection point for power and communication lines. We have equipped one of the links with encoders and custom-made switches to provide position feedback on the link joints to reach our goal: semi-autonomous reconfiguration. 4

5 While designing the system, we assumed that one link can only move a cube at a time. There might also be situations where a link is holding a cube in place while another using the cube as a pivot point. This situation is encountered in small groups (See Section 3). The main constraint in designing such a system is the weight of the individual elements. The choice of using two different types of modules complicates the design and implementation of the system, also doubling the group resolution. However, it presents an extension to a task-oriented, self-powered robotic system. Furthermore, this design approach enables us to reduce the minimum number of elements for stable motion. Self-reconfiguring robots with substrate reconfiguration [1] usually require a large number of modules to create stable gaits. A group of four links and four cubes is capable of moving in three dimensions for most situations (e.g., moving over obstacles, translating without tripping over). In situations that a statically stable configuration cannot be found, it may be possible to use a free link (with only one end attached) to support the structure. If one link is capable of moving itself and an attached cube, simultaneous motions in 3-D are possible for the cubes. Combining several of these motions in sequence, it is possible for a group of links and cubes to change shape and/or move in certain direction D Reconfiguration Combining several suitable link actions such as detaching from and attaching to cubes, and joint rotations into a sequence provides solutions for a group of cubes and links to move and self-reconfigure from one position/shape to another. Figure 5 shows a group of three links and three cubes (we call this group 3L3C) on the ground traveling from left to right. The sequence of states (i.e., configurations) from left to right show the changes in the cube/link positions. Gray color indicates active elements that are moved to reach the next state. Numbers between states give the total number of 90-degree rotations to be completed by the active link. Although the given sequence of actions may not be feasible for an actual implementation due to static and dynamical equilibrium constraints, we present the case for its simplicity. Note that there are many alternative solutions combining individual link rotations in a different sequence, combining simultaneous link motions, which probably result in faster group movement, and solutions with link rotations other than 90-degree increments. However, these solutions are difficult to illustrate with still pictures, and our simulation program currently considers only 90-degree motions. As seen in Figure 5, the group returns to its initial configuration after thirty 90-degree rotations of links. This sequence of actions moves the group to the right by 4d. This sequence is in fact the solution generated by the simulation described in Section 7 when the final condition of the cubes are defined as 4 units to the right of the initial configuration. (1) (2) (11) (1) (2) (4) (4) (3) Figure 5. A group of three links and three cubes in linear motion (image sequence left-to-right). To illustrate more complex motion sequences, we present snapshots of a possible scenario for a group of four links and four cubes (4L4C). As shown in Figure 6, this 4L4C group is capable of moving to a higher surface (e.g., stair climbing) by reconfiguring itself. Connections between elements are kept such that the system forms a single connected graph at any time. This enables modules to exchange information and power during rearrangement. There are several time intervals, where multiple links move simultaneously. Furthermore, the links do not have to complete 90- degree motions to, for example, detach from a cube or move from one cube face to another (See Section 4.2 for an evaluation). There are few positions where a link moves on a pivot cube held in the air by another cube. Tests on current prototypes show that a middle joint of a link is strong enough to hold a cube and a nonmoving link. Also, note that the cube faces initially on the ground and several other faces (i.e., attachment points) are not used during reconfiguration sequence. It may be possible to find other solution sequences that minimize the number of faces used in the reconfiguration process. As seen in the final image, the cubes are still oriented with the same faces looking up at the end of the action sequence. To guarantee this result, some of the cubes need to be re-oriented in the earlier phases of the solution sequence, as seen in third and fifth images. Note that this sequence for the problem is generated manually. We believe that the combination of four cubes and four links is the minimal group that is feasible for 3-D motion and self-reconfiguration. Increasing the number of modules in the group will result in more stable and capable reconfiguration. 5

6 (c) (d) (e) (f) Figure 6. A 4L4C group reconfiguring to move over an obstacle. Another example that combines self-reconfiguration with faster locomotion capability is given in Figure 7. Since the cubes are passive elements that do not contribute to the reconfiguration motions with the exception of the locking mechanism, they can be equipped with capabilities that provide different gaits (such as wheel or treads) and task-oriented modules such as sensors. In Figure 7, the leftmost robot in the first image includes a camera directed at the wall. This robot is obviously not capable of seeing what is behind this obstacle. Assuming these wheeled robots are capable of carrying one or more links, they can move into a position to form a single entity. If initial actions forming the group are possible, then the newly formed group can self-reconfigure into a tower. For an initial configuration and a dependent sequence of actions, it is possible to move the robot with the camera on top of others. The required number of faces with attachment points for each robot is three or four for this specific scenario. These must include the face initially on top. (c) (d) Figure 7. A group of four links and four cubes capable of locomotion forming a tower. This scenario illustrates an important characteristic of this robotic system. A heterogeneous group of small robots combines individual robot capabilities with selfreconfiguration to complete a task that would not be possible with individual robots of relatively small size. It is also possible for some robots to carry multiple links in order to enable other teammates to roam about with out additional payload (a link) while carrying out tasks that require only individual capabilities. Furthermore, the only geometric constraint on the shape of the robots is that the distance and orientation of the attachment points be such that they can receive the link connectors. 4. Hardware Implementation The designs given in Section 2 were implemented using plastic link and cube bodies assembled with off the shelf mechanical and electrical components. The body of the link, faceplates for cubes and additional pieces required for attaching mechanical components are created using P1500 prototyping plastic with Genisys fused deposition modeling machine. Files generated on CAD programs can be sent to this 3-D printer for creating complete pieces that require minimal assembly. The printer resolution is 0.3mm. The design of the links and the cubes is completed using ProEngineer engineering design program. Off the shelf elements are measured and created in ProEngineer (Figure 3) in order to design and test assemblies of the robotic elements. All components are combined into multiple assemblies to test compatibility and unrestricted motion of all modules. Actuation of the worm-wheel gear mechanism is provided by small-size high-torque servo. These servos are placed on the link body and coupled to the worm shaft with custom pieces. The feedback circuit and mechanical stops in the servo gearbox are removed to obtain continuous 360-degree turns on the worm shaft. Mechanical components such as gears, worms, shafts, bearings and collars are available from several companies selling high precision components. The available sizes of these off the shelf components impose another constraint on the size of the prototypes. A link equipped with three sets of servo, worm-wheel gear pairs with torque/speed ratio of 1:40, complete with couplers, collars and bearings measures approximately 326gr (Figure 8a). The length of an edge of the cube is taken as 8cm; the length from the axis of one end joint to another is 16cm, when the middle joint angle is at 0 degree. One side of the link body is shifted slightly down to enable the link to complete a 270- degree turn around its middle joint. The servos on the link are held in place with plastic holders located between the link and the servos. Servo heads are coupled to the worm shaft with a custom-designed 6

7 piece. Figure 8b shows a cube with cross-shaped connection mechanisms. Servos are used to move the locking pegs into place. One faceplate for the cube including the actuator, metal bars, and joints measures approximately 48gr. actuating the attachment mechanisms on the platform as well as the controllers for the actuators. As seen in Figure 9, the link attached to the horizontal faceplate rotates and moves to the faceplate on the vertical wall. After locking its free end to this vertical attachment point, the other end detaches and a series of actions move the link to the second vertical faceplate. The link is then capable of transferring itself from one vertical connection point to another. This demonstration shows that the links are capable of transferring from one cube to another while the locking mechanisms enables the links to attach to cubes. Figure 8. Link prototype with two different connectors, and cube with two faces removed. The use of servos and other off the shelf mechanical components introduces a lower limit on the size of the links. New prototypes with different attachment mechanisms that will enable us to scale the link and the cube sizes down are tested to solve this problem. The size and consequently the weight of the elements directly affect the torque requirements on the actuators. Other details of the hardware implementation are given in [16]. A never version is discussed in [18]. One of our link prototypes has 24-CPR mechanical encoders attached to worm shafts at both ends and a custom-designed switching mechanism on the middle shaft providing position feedback on the rotation of the joints. Feedback signal resolution is 360/(24 40) = degree for the end joints, and 90 degree for middle joint. These modules will be connected to an on-board controller Experiments The prototypes described above are used to test the feasibility of the system on a platform with horizontal and vertical attachment points. These tests show that the links are capable of moving from one cube to another or from one cube face to another. We also present the result of an experiment where two links interchange a cube with four attachment points. Figure 9 shows a link that moves between three attachment points on the platform. The command signals and power are external. A 6V battery is used to power all three servos on the link, and the servos (c) (d) Figure 9. A link moving from horizontal attachment point to vertical attachment points. Results of an experiment with one link and one cube are shown in Figure 10. As seen in the pictures, the link is capable of translating itself from one cube face to another. Initially attached to the top faceplate, the link moves to one of the side faceplates, and then translates to another side faceplate. It finally returns to the top face of the cube. Note that at the end of this sequence of actions, the link end attached to top faceplate is different. As seen from the pictures, the link does not have to complete 90-degree motions to move from one cube face to another. In Figure 11, two links and a cube with four faceplates are shown. Control signals to the cube actuators are external; all actuators are powered with a single 6V battery. As seen in the pictures, the link on the right moves the cube to the correct position by a series of joint rotations. The cube is rotated to reorient the faceplate for the second link connector to attach. The second link then moves into position and attaches its free end to the cube face. The mechanism on the cube face is activated to lock the second link connector in place. First link detaches from the cube with a similar sequence of actions in reverse order, and the second link is able to move the cube to the next position. 7

8 states are the states where the moving end of the link is attached to a cube. The path is evaluated for zero initial and final velocity and accelerations. The lift-off and set-down points chosen as (-81, 90) and (-81, 180) guarantee that the cross-shaped connector clears the attachment point on the cube before rotating joint B. Initial [-90, 90] Top view [-72,135] 1 4 B Final position [ ] 2 (c) (d) 4 1 Figure 10. A link moving over a cube. 2 Figure 12. Initial, intermediate and final positions of a link moving over a cube; The α βrepresentation of the configuration, and the link path that clears the cube. (c) Figure 12b shows the αβ-space representation of the situation. The axes of the plot correspond to the joint angles α and β. The regions indicated with numbers correspond to the cubes. These regions define the αβsubspaces occupied by the cubes: if the link is found to be touching a cube for a given pair (α, β), this αβ-point is marked as occupied, hence the regions. Note that for an αβ-point to be marked as such, at least one of the points on the link that are checked against the cube(s) should coincide with the space occupied by the cube(s). For each position (i.e., (α, β) pair), several points on the links are checked for collision. The lines in Figure 12a represent the outer surfaces and edges on the link that need to clear the cubes for collision-free paths. As seen in Figure 12b, a motion path passing through the defined lift-off and set-down points clears cube 3 with a center joint angle that is slightly greater than 72 degrees. The αβ-point (-72, 135) shown in Figure 12 roughly corresponds to third picture in Figure 10. Note that the resolution of αβ-space is 0.5 degrees. (d) Figure 11. Two links transferring a 4-faced cube Link Manipulation Link motions defined previously included only 90degree actions. The motion planning algorithms described in the following sections also consider 90degree actions. However, our hardware experiments support our expectations that it is possible for a link to move between cubes without completing 90-degree motions, as shown in Figure 10. Hardware tests and the simulation of a link with different cube configurations indicate that a link can clear the neighboring cubes and the cube it is attached to, at small angles of the middle joint. Figure 12a shows initial and final positions of a link attached to a cube from the top. Since end B of the link is fixed, the link trajectory can be represented with a pair (α, β) where α is the rotation of the center joint and βis the rotation of the joint B. The ranges are [-90, 180] and [0, 360] respectively. The initial and final 5. Abstraction: Discrete State Representation In this section, we introduce a grid representation of the group consisted of m cubes and n links. For a given combination, there are multiple states that the system can take depending on the relative positions of the 8

9 cubes, links and link shapes. In fact, the number of possible states will possibly grow exponentially with the increasing number of cubes and/or links. A group of m cubes and n links can be represented with a threedimensional grid of size d d d where d = (2m-1) + 4. The number 2m - 1 indicates the size of the longest possible formation where all the cubes aligned on a straight line. The cell grid of size (2m-1) (2m-1) (2m- 1) defines the possible cells the cubes can occupy. This grid should be extended by two cells in each direction to completely define the cases where the links extend beyond the cells occupied by the cubes (Figure 13). Each cell in this d d d grid may be occupied only by a single cube or link. Each cube can occupy only one cell, while the links may occupy two or three cells depending on their shapes, as seen in Figure 13. We discretize the link positions at 90-degree increments for simplicity. Global origin g z y = 0 3 x 3 grid (m = 2) x Link pos. A = [0 2 3] C = [1 2 3] B = [0 2 3] 7 x 7 grid Cube pos. [2 0 0] Link pos. A = [2 0 1] C = [1 0 1] B = [1 0 2] 2x2x2 grid on eight corners are never occupied Cube pos. = [0 0 2] Figure 13. Grid definition, and two of four possible link shapes for a given position: link looking up and down. The cells in the 3-D grid can be represented by their local coordinates where the origin is defined as the cell closer to a global origin. If the position of this grid in global coordinates is known, global positions of the links and the cubes can be evaluated using the local coordinate system defined on the grid. Definition 1: The states defined above can be S = C, L, f, d where: represented with a quadruple { } C is the set of vectors c i representing the positions of the cubes in state coordinates, L is the set of couples l i = { p i, s i } with a triple of vectors p i = { p ia, p ic, p ib } defining the position of the three cells a link occupies, and a triple s i = { s ia, s ic, s ib } indicating the orientation of the link segment at given cells, f is a binary relation between C and L, and is a subset of the Cartesian product C L. It is defined by an m n connection matrix where each row represent a cube and each column a link, with elements f ij indicating a cube-link connection with a value of 1. Each row in f should have at least one and at most six 1 s; each column should have one or two 1 s. d = 2m + 3 is the size of the grid containing all possible configurations of the cubes and the links. Definition 2: For a given state definition represented by S, it is possible to define a pseudo-graph G = { V, E, f } where V is the set of vertices associated with the cubes in set C, and E defines the set of edges associated with the links in set L, and f is the m n connection matrix. Each element in C (or L) corresponds to an element of V (or E). In G, some of the edges (links) do not connect to a vertex (cube), and some are attached to the same vertex. Definition 3: We define a couple N { S, g} = of the state S and its global position g and call it a node. The global position of a state is given by the location of the cell closest to the global origin. Using g, and c i (l i ) it is possible to define the global position of a cube (link). A given problem with initial and final configurations N, and can then be represented as a pair { } initial N final may have multiple solutions. For a given state S, there are 24 actions for each link. These actions are given in Table 1. Letters A and B do not indicate a specific end of the link, but rather describes whether the end is the one closer to the origin or not. Letter A defines the link end that is closer to the origin. Since using a Euclidean definition for distance to the global origin create a problem in distinguishing the ends, we use a function that compares the positions of the ends in x, y, and z axes separately, in the order given. The actions a ij ( i = 1,...,24, j = 1,..., n) are relations from one state to another, and are associated with translation t ij indicating possible change in global coordinates of the node N associated with the state S. For example, the action in Figure 2 is detach others at A, fix B, B + where all the links except the active link connected to the moving cube are detached, the end B is assumed to be fixed, and the joint at end B is rotated clockwise. If the cube attached to end A is the cube with smallest x- coordinate before the action, then this action will be associated with a translation [2 0 0] T indicating that the newly formed state is 2 units away from the previous state in x direction. 9

10 Actions a ij are actually a group of actuation commands for one or more attachment mechanism on the cubes, and for servos on the links. The end that is fixed is also defined. In actual implementation, the definition of the fixed link directly follows from the configuration. For a link to move, the desired location and the path of the link (and the cube that may be attached) must be clear. In other words, cells associated with the new location, and path of the link (and cube), and some of the cells next to the initial and final cube positions should not be occupied by modules or obstacles. There is a different set of cells to be checked for different action types for a given link or link-cube pair. Table 1. Link actions. Actions 1-12 Actions Det. A, fix B, Ctr + Det. A, fix B, A + Det. others at A, fix B, Ctr + Det. A, fix B, A - Det. B, fix A, Ctr + Det. others at A, fix B, A + Det. others at B, fix A, Ctr + Det. others at A, fix B, A Det. A, fix B, Ctr + Det. B, fix A, A + Det. others at A, fix B, Ctr - Det. B, fix A, A - Det. B, fix A, Ctr - Det. others at B, fix A, A + Det. others at B, fix A, Ctr Det. others at B, fix A, A Det. A, fix B, B + Det. B, fix A, B + Det. A, fix B, B - Det. B, fix A, B - Det. others at A, fix B, B + Det. others at B, fix A, B + Det. others at A, fix B, B Det. others at B, fix A, B An action a ij acting on state S will affect p j, s i and may affect c k and kth row of f if f jk = 1, thus creating a new state S new. The translation t ij associated with the action a ij may have non-zero elements if there is a cube relocation that changes the definition of the cube closer to the origin. For a given initial state S o of m cubes and n links located at g (i.e., node N o ), it is possible to generate all possible positions/shapes for the mlnc group given. Definition 4: We call the graphical representation of all the states with the action and translation pairs (a ij, t ij ) between states a state transition diagram. Figure 14 shows part of the transition diagram for 1L2C case, where each edge may indicate multiple actiontranslation pairs. Table 2 gives the total number of possible states in a state transition diagram for different combinations of cubes and links under different attachment constraints. As seen from the table, the number of possible states increases exponentially with the number of cubes and links. Note that for a given position of cubes, there may be several states represented by the same connection matrix f, but have different link positions and shapes. Each connection between states on the transition diagram represents one or more actions and their associated state-to-state translation. For example, as seen in Figure 14, there are four possible actions from state #1 to state #4, each associated with a different translation value. y=2 z S1 g = [20 2] a1,11 a1,19 a1,12 a1,20 x z S4 x [0 0 2] [0 0 0] g = [20 2] + [2 0 2] [2 0 0] Figure 14. Transitions from state #1 and action/translation definitions from state #1 to state #4. Table 2. Number of state for different configurations. Conf. Description States 1L1C One link, one cube 84 1L2C One link, two cubes, connected group 36 1L2Cunat t One link, two cubes, detachment possible L2C Two links, two cubes L3Cgnd2 Three links, three cubes, connected group, two cubes always on the ground * * State generation interrupted at states; approximately 8000 states were not investigated. While solving a path planning problem with given initial and final states, one has to generate a graph of all possible nodes N i evaluating all possible actions on all possible links for each possible state. This new graph of nodes is significantly larger than the transition diagram where the states are considered. As stated in [4], finding a solution with minimal number of actions corresponds to a shortest path problem on this graph of nodes. The evaluation of all nodes and a solution to shortest path problem on this graph are computationally very complex: exponential growth in the number of nodes is much more significant in our case due to the 3-D nature of I-Cubes. However, finding upper bounds for the number of link rotations or number of cube moves for a given problem may be possible, if the approach in [5] can be applied to 3-D. Investigation of the values in Table 2 indicate that finding a solution to a path problem involving large number of modules would be computationally expensive and time consuming. If the initial and final configurations given in a problem are distant, the 10

11 number of moves in the solution will be large. Consequently, the graph of nodes that needs to be generated during solution will extend beyond memory and processing capabilities of an average computer as well as realistic time limits for an on-line application. For the simple problem with initial and final states shown in Figure 15, the total number of nodes generated ranges from 645 to A total of 346 (or 297) of these nodes are evaluated before reaching the desired position and shapes. Note that the solver employs an heuristic algorithm (A*) for generating the nodes using number of actions for cost function, and the sum of distances to the desired cube locations for estimate of additional cost. This configuration is not statically feasible, but given here to illustrate the issues related to motion planning. The minimum number of actions is 11. A breadth-first search generates 2063 and evaluates 297 nodes that are in close proximity to the initial and desired positions before reaching the solution node. Considering the number of possible states (36), the displacement between initial and final states ([4 2 0]), and the total number of nodes generated by the heuristic algorithm, the complexity of the problem for larger groups becomes obvious. The approach defined in the next section is therefore used in order to represent the planning problem in a simplified, tractable form. Figure 15. Initial and final states for a 1L2C problem. 6. Motion Planning: Divide-and-Conquer Method Finding a sequence of actions for a given motion planning problem involving m cubes and n links where m, n > 2 is very difficult using the state representation given in the previous section for large groups. However, it may be possible to divide the problem into sub-problems that can be quickly solved. If we are to generate a sequence of actions a ~ from a given node N o to a desired node N f, the elements on a ~ can be regrouped into subsequences so that, the difference between the states at the beginning and end of each subsequence is a single cube motion. If there is a solution from N o to N f, and this solution requires k cube moves, then the solution a ~ can be divided into k subsequences that are solutions to k less complex problems. Since only one cube will be moving in each of these sub-problems, it is possible to take a state of actual nlmc problem and create one or more sub-problems that consider only pairs of cubes. These pairs considered for inclusion into the sub-problem would be either: the cube that needs to be moved, and the cube that acts as a pivot point for the active link(s) or, the cube that a link is detached from or attached to, and the cube that has other end of the link. The state A in Figure 16 shows a 4L4C group that is moving to the right. The cube M needs to be moved to the indicated position. However, the required rotation of the link between cubes M and P is not possible, due to additional links on cube M. Furthermore, additional links on M cannot be detached because of the connectivity requirement. However, the sequence of actions given in Table 3 will enable cube M to move its desired position. The sub-problem definitions used to solve this motion-planning problem are also given in this table. Once a decision on link(s) and the cube to be moved is made, the locations of the cubes in the next state S i+1 (or node N i+1 ) are known. The generation of the subproblem from the definitions of N i and N i+1 is straightforward. If a solution to the sub-problem is found, the new positions and shapes are incorporated into current state S i to obtain the new state S i+1. Table 3. Definition of the sub-problems and associated parameters for solution given in Figure 16. # Transition Sub-problem Associated cubes # of moves Result 1 1L2Cunatt Attach link to cube O M, O, and I 1 B 2 1L2Cunatt Detach link from cube O M, O, and K 1 3 1L2Cunatt Attach link to cube P M, P, and K 2 C 4a 1L2Cunatt Detach link from cube P M, P, and K 1 5a 1L2Cunatt Attach link to cube Q P, Q, and K 2 D 4b 1L2Cunatt Detach link from cube P M, P, and L 1 5b 1L2Cunatt Attach link to cube Q P, Q, and L 10 Not shown 6 1L2Cunatt Detach link from cube M M, O, and I 1 E 7 1L2C Move cube M to desired position M, P, and K 2 F 11

12 I L M P M O K J K Q A B C D E F Figure 16. A 4L4C group moving to the right. We have found that three state transition diagrams 1L2C, 1L2Cunatt, and 2L2C can be used to evaluate all possible cube motions using suitable combinations of cubes and links 1. For sub-problems using 1L2C transition diagram, a pair of cubes where one is the moving cube and the other is the pivot cube (one of many possibilities) is chosen. For 2L2C, the pair of cubes consists of a moving cube and a pivot cube with at least one link initially attached to both cubes. Pivot cubes are not moved during solution. For 1L2Cunatt solutions, both cubes are fixed in their positions. Note that only the sub-problems associated with 1L2Cunatt transition diagram may leave a cube without any link connections, and therefore may create a non-connected pseudo-graph G of the links (edges) and cubes (vertices). The definition of a connected pseudo-graph is given below. Definition 5: A pseudo-graph G { V, E, f } = of state S is connected if there is at least one path pkl = {( e1, e2,..., ez ), ei E, i = 1,.., z} where each pair of edges (e j, e j+1 ) is connected to the same vertex v t, i.e., f t,j = f t,j+1 = 1 for some 1 = t = m, for any pair of vertices ( v, v ) V V with f k,1 = f l,z = 1. k l The actions a ij associated with the sub-problem include three low-level definitions for detachment, fixed end, and (c) rotation (Table 1). The first definition includes low-level actions such as detaching other links on the cube connected to end A (or B). This can be interpreted as keeping the active link attached in 1L2C cases, and detaching the second link on the cube (if any) in 2L2C case. Other links in an actual path planning problem that are not part of the simplified sub-problem, are detached before entering the sub-problem in earlier phase of the solution. Link and cubes that are not considered for a sub-problem are 1 Transition diagram 1L2C is actually a subset of the diagram 1L2Cunatt. We can also divide any solution sequence into smaller sequences using multiple sub-problems involving 1L2Cunat, instead of 2L2C. However, 1L2C and 2L2C diagrams enable fast solutions. taken into account as obstacles. We also need to guarantee that the sub-solutions generated by the transition diagrams does not affect the connectivity of the pseudo-graph G, i.e., no link nor a cube is separated from the rest of the group. Theorem 1: For a given problem with initial and final nodes {N o, N F } with a connected pseudo-graph G o for the initial state S o, a solution generated using state transition diagrams 1L2C, 2L2L, and 1L2Cunatt with the condition that the pseudo-graph representation of two cubes to be detached correspond to two vertices with at least two distinct paths, will result in a final state S F with a connected pseudo-graph G F. The details of the proof of this and related lemmas/theorems on path planning can be found in [17]. If the conditions above are met, any sub-solution generated by transition diagrams will keep the group of modules connected if it initially is. We can then state that a group of modules will be connected at all times, if a solution can be generated by an algorithm that can choose a suitable group of cubes and links for each sub-solution. If it is possible to generate solutions to the sub-problems of suitably chosen cube-link combinations, the overall solution can be obtained by concatenating all these solutions. Finding an algorithm that generates a sequence of motions from a given state to a desired state is obviously very difficult, given the number of possibilities that needs to be evaluated along the solution path. However, if such a planning algorithm can be written, it will operate on individual cube actions at different steps toward the desired state and position, instead of the state representations that combines the position and shape information on all cubes and links. The search space with cube motion definitions is much simpler, although no straightforward solution exists with the exception of few cases similar to ones we present in the next section. On the other hand, considering all possible configurations for a group of m cubes and n links is very difficult, if not impossible, due to the fast growth of the number of possibilities because of the number of elements, different shapes that the elements can take, and the 3-D nature of the problem. For every cube motion chosen to be evaluated by the decision making algorithm, a sub-problem with initial and final states from one of the state transition diagrams 1L2C, 2L2L, and 1L2Cunatt can be quickly generated and solved in limited time steps. Our initial tests using the above defined transition diagrams gave promising results.