This study proposes an adaptive graph algorithm for collision-free motion planning of articulated robots in dynamic environments. For this purpose, deformations of the configuration space were analyzed according to the changes of the workspace using various simulations. Subsequently, we adopted the principles of gas motion dynamics in our adaptation algorithm to address the issue of the deformation of the configuration space. The proposed algorithm has an adaptation mechanism based on expansive repulsion and sensory repulsion, and it can be performed to provide the entire adaptation using distributed processing. The simulation results confirmed that the proposed method allows the adaptation of the roadmap graph to changes of the configuration space.
Robot motion planning in dynamic environments has been regarded by researchers as a challenging problem. The motion planning problem, particularly for articulated robots, is known to be difficult because of the geometrical and algebraic complexities resulting from the increase in the degrees of freedom (DOFs) of the robot. The representative statement of this motion planning problem is known as the configuration space formulation. The key function of a configuration space is to represent the robot as a point in an appropriate space and map the obstacles in this space. This mapping transforms the problem of planning the motion of a dimensioned object into a planning problem for the motion of a point. It also makes the motion constraints of the robot more explicit [
For collision-free manipulation, especially in dynamic environments, planning algorithms must overcome additional challenging problems [
Although extensive work has been performed on robot motion planning, most of the widely used algorithms utilize sampling-based methodologies. These sampling-based methods construct geometric graphs in collision-free configuration spaces to obtain solutions to motion planning problems; moreover, they employ a variety of strategies for generating samples and connecting the samples with paths. Sampling-based algorithms, such as
In this paper, we propose an algorithm for geometric graphs that can adapt to a changeable configuration space topology. We contrived this algorithm via inspiration from recent investigations, such as multiagent systems [
The
Mapping relations between the workspace and the configuration space.
Dynamic environments are workspaces in which various changes can occur. Changes in the workspace lead to deformation of the configuration space; this relationship between the two spaces is determined by the robot kinematics equation
Obstacles in the configuration space change in size according to the change of the workspace. The size of an obstacle is determined by the distance between the coordinate of the robot base and that of the obstacle in the workspace. Figure
Free configuration space ratio with respect to the distance of the robot base from an obstacle in the workspace.
Figure
Scaling deformation of the configuration space caused by the movement of a workspace object. (a) No object in the configuration space; (b) appearance of the configuration space object; (c) expansion of the configuration space object by movement toward the robot in the workspace.
The position of an obstacle in the configuration space is determined by the direction of the obstacle in the workspace with respect to the robot base coordinate. Thus, if the obstacle position changes while preserving the distance from the robot base in the workspace, a translation motion of the obstacle occurs in the configuration space. Figure
Translation of a configuration space object generated by a moving workspace object. (a) Object at 0°; (b) positive translation of the configuration space object (object at 90°); (c) negative translation of the configuration space object (object at −90°).
Obstacles in the configuration space consist of the union set of all obstacles converted from the workspace. In general, there is no merging phenomenon in the workspace because changes in the workspace are rigid body motions without a scale change. However, in the configuration space, there can be a merging phenomenon with respect to the change of scale and position of the obstacles. Figure
Merging of configuration space objects caused by moving workspace objects. (a) No configuration space object; (b) appearance of configuration space objects; (c) merging of configuration space objects by movement in the workspace.
In the mobile manipulation problem, which is a challenging task in robot motion planning, the robot position and direction change by the movement of the mobile robot. If the robot moves by turning, the obstacles near the robot move by turning in the opposite direction in the workspace; thus, it can be predicted that all these obstacles move in the same direction. In addition, if the robot base coordinate moves by the motion of the mobile base, some obstacles will be closer, and some will be farther from the robot base. In this case, some obstacles in the configuration space will be expanded, and some will be contracted, according to their distances. Therefore, in mobile manipulation problems involving the movement of the mobile base, the environments become dynamic even when the workspace is a fixed environment. Figure
Expansion and contraction of configuration space objects by the movement of the robot base. (a) Robot placed at (0, 0); (b) robot moves to (24, 23); (c) robot moves to (−21, −23).
In dynamic environments, the deformations of the configuration space are combinations of the three nonlinear deformations of
Another important problem that complicates motion planning in dynamic environments is the processing time required for the reconstruction of a modified configuration space [
Most robot motion planning algorithms have been derived from sampling-based algorithms, such as PRM and RRT; these methods describe the complex configuration space with geometric graphs through the construction of nodes and edges. In the PRM method, after a learning phase in which a graph for the entire configuration space is constructed, the optimal path between a given starting point and a goal point can be obtained from the learned graph. In contrast, the RRT method identifies a reachable path by expanding the graph using a tree-type approach. In the PRM method, the obtained path is the optimal path based on the previously constructed graph; the PRM method cannot be applied in dynamic environments because the graph must be reconstructed if the configuration space changes. Because the RRT method constructs a new graph for every path-finding query, it can be applied in changeable environments. However, the path obtained by the RRT method is not optimal and requires considerable time for constructing the new graph. In addition, sampling-based algorithms perform tests for the construction of nodes and edges; particularly, tests for edge construction require significantly more time because collisions for line segments are determined by tests performed for numerous points on the edge. With the purpose of resolving these problems, a faster edge-test algorithm that uses the upper bound of robot motion has been proposed [
In this study, we attempt to solve the problem of motion planning in dynamic environments. For this purpose, we propose an adaptation algorithm of geometric graphs, called the
The proposed adaptive roadmap algorithm performs the adaptation process by changing the structure of the graph according to the deformation of the configuration space. To adjust the roadmap to changeable environments, each node in the graph moves automatically to the free configuration space. Then, the node updates its edge information with respect to the changed node information. These processes are distributed and performed in parallel for all nodes. Hence, this algorithm can adapt the roadmap to the change of the configuration space by changing the graph structure. In the proposed method, some new concepts, such as the neighbor node, neighbor radius, and sensing node, are introduced.
Definitions used are as follows:
Figure
Neighbor node and sensing node. (a) Neighbor node for neighbor radius
The roadmap method is an abstract representation of the complex topology of the configuration space, with geometric graphs that consist of nodes and edges. In the previous section, we confirmed that a radical deformation of the configuration space can appear in dynamic environments. When changes occur in the workspace, they create various deformations of the configuration space; then, the constructed roadmap becomes useless. Thus, in this research, we propose an adaptation algorithm for roadmaps by adjusting the graph topology to the changed environments.
In the proposed adaptation algorithm, we adopted the motion dynamics of gases [
The expansive repulsion factor expresses an interaction between nodes that can repel other nodes within the effective close area. Nodes in the effective close area are defined as neighbor nodes, and the threshold value that determines neighbor nodes is the neighbor radius
Expansive repulsion factor between two nodes in a neighbor area.
If the distance
The expansive repulsion factor results from all such interactions between neighbor nodes. Thus, using the repulsive potential function
Sensory repulsion represents the repulsive interactions from sensing node
Sensory repulsion factors generated by sensing nodes.
The sensing nodes generate repulsion factors in the opposition direction for node
The adaptation mechanism of the proposed adaptive roadmap algorithm is based on two repulsion factors: the expansive repulsion factor
Interaction factors in a node. (a) Expansive repulsion from neighbor nodes; (b) sensory repulsion from sensing nodes that detect collisions; (c) total internal interaction factors; (d) movement of a node.
Figure
An example of the adaptive roadmap.
Flowchart for processing the adaptive roadmap algorithm.
To verify the feasibility of the proposed adaptive roadmap method, simulations were performed in a simplified test environment. If the robot has three or more DOFs, the free configuration space becomes a complex high-dimensional space, which is difficult to visualize and identify. Thus, we considered a test environment with a two-link robot with a two-dimensional configuration space. In the test simulations, we set the number of nodes
Each joint of the robot was set to have an operating limit range of −170°–170°; thus, the configuration space had the same topology as the Euclidean space. If the two-link robot has no operating limit, the configuration space is a torus-type space; in the case of higher dimensions, the configuration space becomes a more complex space that is difficult to handle. Thus, we assumed that the robot had operating limits to simplify the configuration space, because most actual robots have joint limits.
Figure
Test workspace and the corresponding configuration space. (a) Two-link robot and obstacles in workspace; (b) configuration space for the workspace in (a).
Figure
Running procedures of the adaptive roadmap algorithm. (a) Initial state; (b) state of the roadmap after the 5th iteration; (c) state of the roadmap after the 10th iteration; (d) state of the roadmap after the 100th iteration.
Figure
Motion planning results and trajectories of robot motion for two test cases (start motion (green); goal motion (red)). (a) Roadmap graph and the planned motion (blue) in the configuration space for the collision-free path from [155°, 78°] to [−85°, −39°]; (b) motion trajectory in the workspace for (a); (c) roadmap graph and the planned motion (blue) in the configuration space for the collision-free path from [−9°, 10°] to [−160°, −33°]; (d) motion trajectory in the workspace for (c).
Next, we tested the behavior of the adaptive roadmap algorithm for changes of the workspace. Figure
Changed workspace and the corresponding configuration space. (a) Two-link robot and obstacles in the workspace; (b) configuration space of the workspace shown in (a).
Figure
Adaptation procedures for the changed configuration space. (a) Initial state; (b) state of the roadmap after the 5th iteration; (c) state of the roadmap after the 10th iteration; (d) state of the roadmap after the 100th iteration.
Figure
Motion planning results and trajectories of robot motion for two test cases (start motion (green); goal motion (red)). (a) Roadmap graph and the planned motion (blue) in the configuration space for the collision-free path from [−155°, 35°] to [37°, 157°]; (b) motion trajectory in the workspace for (a); (c) roadmap graph and the planned motion (blue) in the configuration space for the collision-free path from [−4°, 59°] to [−165°, 62°]; (d) motion trajectory in the workspace for (c).
Subsequently, the behavior of the adaptive roadmap was observed in dynamic environments. To generate continuous changes of the workspace, the positions of all obstacles were modified slightly and randomly for every simulation step, and considerable obstacle position changes were applied at every 50th step.
Figure
Simulation results of the adaptive roadmap for various conditions of the configuration space volume (red). (a) Total expansive repulsion (green); (b) coverage of the roadmap (gray).
Figure
Topologies of the adaptive roadmap at equilibrium for various conditions of the free configuration space ratio. (a) 89.6%; (b) 69.3%; (c) 79.5%; (d) 59.9%; (e) 59.3%; (f) 54.1%; (g) 57.8%; (h) 48.68%; (i) 38.7%.
In this paper, we propose an algorithm that can adapt to a radically changeable configuration space. For this purpose, we first analyzed the deformation of the configuration space with respect to the change of the workspace. To address the issue of the deformation of the configuration space, we presented an adaptation algorithm for the roadmap graph based on distributed processing. In this algorithm, each node in the graph moves in the free configuration space by detecting the deformation of the configuration space; then, the node updates its edge information with respect to the changed node information. The adaptation mechanism of the proposed algorithm is based on two basic repulsion factors: the expansive repulsion factor and the sensory repulsion factor. Moreover, the adaptive roadmap can perform the entire adaptation by distributed processing of the complex procedure in each node. To verify the effectiveness of the proposed method, simulations of a continuously changing workspace were performed in a test environment of a two-link manipulator case. The simulation results confirmed that the graph could perform the entire adaptation with respect to the change of the configuration space using the proposed algorithm.
As future work, high-dimensional configuration space and complex problems, such as redundancy and mobile manipulation for actual robots, will be considered. Thus, to overcome these problems, we should deliberately utilize massively parallel processing technology using GPUs. Moreover, the application of the current results to the study of complex systems will require further investigation. The adaptive roadmap algorithm is strongly related to the complex system that interacts with many independent agents internally. Therefore, the further development of the adaptive algorithm by combining it with complex systems science should be considered as important future work.
The authors declare that there is no conflict of interests regarding the publication of this paper.
This research was financially supported by the Ministry of Trade, Industry and Energy (MOTIE) and Korea Institute for Advancement of Technology (KIAT) through the Promoting Regional Specialized Industry.