A new multimodal planning algorithm based on PRM

. The robot motion planning problem has a unique multimodal structure, where the space of feasible configuration consists of intersecting submanifolds. The planning problem of reconfigurable robot leg motions was considered as a multimodal structure composed of intersecting submanifolds in different dimensions. After that, a new multimodal planning algorithm based on PRM was proposed. Simulation results showed that the new algorithm had a shorter running time than PRM in different modes.


Introduction
The PRM (Probabilistic Roadmap) planning algorithm is applied for high-dimensional configuration space motion planning under geometrically complex feasibility constraints [1] . With a suitable sampling configuration of probability measure, the connectivity of the robot's feasible space is approximated and connected by simple paths. However, the PRM planning algorithm is not suitable for multimodal structures consisting of intersecting submanifolds in different dimensions [2] . This multimodal structure appears in the planning problem of manipulating leg motions of reconfigurable robots [3][4] . In this structure, each submanifold corresponds to a mode. A set of fixed contact points is maintained between the robots and the environment (or between the robots). The planning algorithm consists of discrete modal sequences and continuous single-mode paths (namely joint spatial motions for contact changes). In a multimodal structure, the PRM planning algorithm cannot explain whether a path exists or not. When a single-modal query fails, the feasibility of the query cannot be judged. It needs more time to query whether the path exists. Therefore, the work put forward MM-PRM for finite multimodal problems. The new algorithm builds a cross-modal PRM by sampling configurations in transitions between modalities (corresponding to intersections of submanifolds). If all modes are extensible when being restricted to insert submanifolds, the new algorithm, as the classical PRM planner, will find a feasible path with an improved convergence rate. It is determined that the expected running time is finite, with finite variance.

PRM planning algorithm and non-extended space
The PRM planning algorithm approximates the connectivity of a feasible subset of robot configuration space ] using a configuration roadmap (called milestone) connected by simple paths (usually straight lines). Meanwhile, the concept of expansion degree is introduced to characterize the convergence rate of the roadmap.
The expansion degree of ] is measured by three parameters , , . These values depend only on the relative volume of some subset of ] , rather than the dimensionality. If all three parameters have positive values, then ] is extended; otherwise, ] is non-extended [5][6]. In all extension spaces, the query probability of the basic PRM planning algorithm decreases exponentially as the roadmap grows. Larger , , H D E leads to a larger convergence rate.
In a multimodal structure, ] is non-extended if a cusp is included ( 0 H ). However, PRM still works well in a position far from the cusp because the structure is expanded to maintain its connectivity after removing the tiny areas around the cusp. If it contains the regions in different dimensions ( 0 D or 0 E ), then ] is also non-extended. For example, a PRM planner cannot find a path connecting the 2D regions even in an arbitrary time in a space consisting of two 2D regions connected by a 1D curve.
Variable dimensions are inherent in the multimodal problem structure. If the number of modalities is finite, then the submanifolds with joint forms can be enumerated from the problem definition. However, the number can be enormous.

Definition of robot multimodal problem
A hybrid system must be planned in major planning problems, where the state space consists of a discrete finite (or countably finite) set called the modal space. In other words, the system moves between sets and modalities. Depending on different current modalities, the robot is constrained to traverse some lower-dimensional subspace of the actual space.
A robotic system with contact moves between modalities, where each modality defines a contact state. For example, this concept was applied for leg movements in Ref. [10], where the modality refers to a set of footholds constraining the foot on the ground. The mutual transition of the modalities ensures walking. Manipulation planning also exhibits a multimodal structure. When a robot pushes an object at a fixed point of contact, it traverses the same modality. However, the robot has to go through the specified modal without contact before reaching the modal of new points. Different from traditional robotic motion planning algorithms, multimodal planner reaches the goal through discrete mode conversion sequence, thus achieving continuous path within each modality.
It is considered that a robot can move between finite sets ¦ of modalities, where each ] is the transition region between \ and \ ; q the transition configuration. A modality has two feasibility constraints.
(1) Dimensionality reduction constraint is often expressed as Eq.
( ) 0 q \ G , which defines submanifold \ \ as a set of configurations satisfying these constraints.
(2) Volume reduction constraint is often expressed as . This may lead to null of \ ] . Otherwise, it has the same dimension but a smaller volume. In

Expected runtime
If there is a solution, then the MM-PRM algorithm will find a path with a probability of 1, which is probabilistically complete (see Ref. [2]). However, this condition is not sufficient The expected runtime is limited. The definition of variance is applied to obtain Eq. (2), thus restricting the variance in runtime.

Simulation
The robot moves in the three-dimensional space (see Fig. 1). However, it only moves in the two-dimensional plane of the cube due to obstacles. The cube provides four surfaces to the configuration space (the rest do not belong to the space), where there are two rectangular barriers to form a narrow channel.

Conclusion
The work introduced the traditional PRM planning algorithm. After that, the planning problem of reconfigurable robot leg motions was considered as a multimodal structure composed of intersecting submanifolds in different dimensions to propose an MM-PRM planning algorithm. Simulation results showed that MM-PRM had a shorter runtime than PRM in different modes.