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Constraint-free discretized manifold-based path planner

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Abstract

Autonomous robotic path planning in partially known environments, such as warehouse robotics, deals with static and dynamic constraints. Static constraints include stationary obstacles, robotic and environmental limitations. Dynamic constraints include humans, robots and dis/appearance of anticipated dangers, such as spills. Path planning consists of two steps: First, a path between the source and target is generated. Second, path segments are evaluated for constraint violation. Sampling algorithms trade memory for maximal map representation. Optimization algorithms stagnate at non-optimal solutions. Alternatively, detailed grid-maps view terrain/structure as expensive memory costs. The open problem is thus to represent only constraint-free, navigable regions and generating anticipatory/reactive paths to combat new constraints. To solve this problem, a Constraint-Free Discretized Manifolds-based Path Planner (CFDMPP) is proposed in this paper. The algorithm’s first step focuses on maximizing map knowledge using manifolds. The second uses homology and homotopy classes to compute paths. The former constructs a representation of the navigable space as a manifold, which is free of apriori known constraints. Paths on this manifold are constraint-free and do not have to be explicitly evaluated for constraint violation. The latter handles new constraint knowledge that invalidate the original path. Using homology and homotopy, path classes can be recognized and avoided by tuning a design parameter, resulting in an alternative constraint-free path. Path classes on the discretized constraint-free manifold characterize numerical uniqueness of paths around constraints. This designation is what allows path class characterization, avoidance, and querying of a new path class (multiple classes with tuning), even when constraints are simply anticipatory.

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Data Availability

Not applicable

Code Availability

Not applicable

Notes

  1. https://investor.costco.com/corporate-profile-2.

  2. https://github.com/petercorke/robotics-toolbox-matlab.

Abbreviations

CFDM:

Constraint-free discretized manifold

CFDMPP:

Constraint-free discretized manifolds-based path planner

CFM:

Constraint-free manifold

CS:

Constraint set

DC:

Dynamic constraint

DCS:

Disjoint constraint set

EA:

Evolutionary algorithm

EF:

Evaluative factor

HBM:

Homotopy based method

ICE:

Inner constraint edge

IFTM:

Inverse function theorem for manifolds

MBM:

Model based methods

MPC:

Model predictive control

NF:

Navigation function

OA:

Optimization algorithm

OCE:

Outer constraint edge

PPM:

Path planning manifold

PPP:

Path planning problem

PPS:

Path planning space

PPPS:

Primary path planning space

PRM:

Probabilistic road map

PSM:

Product smooth manifold

RRT:

Rapidly exploring random tree

SA:

Sampling algorithm

SC:

Static constraint

SM:

Smooth manifold

SPPS:

Secondary path planning space

TM:

Topological manifold

UGV:

Unmanned ground vehicle

\(\mathcal {C}\) :

Configuration space

\(\mathcal{C}\mathcal{O}\) :

Convex obstacle region in configuration space

\({CS}_{r}\) :

Constraint set r

\({DCS}_{r}\) :

Disjoint constraint set r

DCSs :

Set of all disjoint constraint sets of all path planning spaces

\({DCSs}^{(\cdot )}\) :

Set of all disjoint constraint sets of space \((\cdot )\)

\(n_{ML}\) :

Number of SCs classified as mechanical limits (ML)

\(n_{NG}\) :

Number of SCs classified as no-go zones (NG)

\(n_{{OB}}\) :

Number of SCs classified as obstacles (OB)

\(n_{SG}\) :

Number of SCs classified as singularities (SG)

\(n_{{SC}}\) :

Total number of SCs

\({{\textbf {node}}}^{start}\) :

Starting point in CFDM

\({{\textbf {curr}}}\) :

Current node in CFDM

\({{\textbf {nbr}}}\) :

Neighbour node in CFDM

\({\mathcal {P}}\) :

Path planning space

\({\mathcal {P}}^{{mechanical\, limits}}\) :

\(\subseteq \mathcal {P}\) Space corresponding to the mechanical limits of the robot (e.g., joint limits)

\({\mathcal {P}}^{{no-go\, zones}}\) :

\(\subseteq \mathcal {P}\) Space corresponding to the no-go zones

\({\mathcal {P}}^{{obstacles}}\) :

\(\subseteq \mathcal {P}\) Space corresponding to the obstacles

\({\mathcal {P}}^{{singularities}}\) :

\(\subseteq \mathcal {P}\) Space corresponding to the singularities of the robot

\({\mathcal {P}}^{{constraint}}\) :

\(\equiv \mathcal {P}^{{mechanical\, limits}} \cup \mathcal {P}^{{no-go-zones}} \cup \mathcal {P}^{{obstacles}} \cup \mathcal {P}^{{singularities}}\) Space corresponding to all constraints

\(\mathcal {P}^{{free}}\) :

\(\equiv \mathcal {P} \setminus \mathcal {P}^{\text {constraint}}\) Constraint-free Space

\({{\textbf {p}}}\) :

\(\in \mathcal {P}\) A point in \(\mathcal {P}\)

\({{\textbf {p}}}^{{start}}\) :

\(\in \mathcal {P}\) Starting point in \(\mathcal {P}\)

\({{\textbf {p}}}^{{goal}}\) :

\(\in \mathcal {P}\) Destination point in \(\mathcal {P}\)

\({{\textbf {p}}}_{{path}}\) :

\(\in \mathcal {P}\) Path connecting \({\textbf {p}}^{\text {start}}\) to \({\textbf {p}}^{\text {goal}}\) in \(\mathcal {P}\)

\({\mathcal {S}}_{{surface}}\) :

A Surface

\({SC}_{k}\) :

Static constraint k

\({{\textbf {T}}}_p\mathcal {S}\) :

Tangent Space at a point p on a surface

\({\mathcal{T}\mathcal{S}}\) :

Topological space of set \(\mathcal {S}\) and topology \(\mathcal {T}\)

\(\mathcal {W}\) :

Workspace

\(\Delta _{{CS}}\) :

Global proximity threshold between constraints

\(\Delta ^{ij}_{{CS}}\) :

Measured proximity between constraints i and j

\(\delta\) :

Boundary. For instance, \(\delta x\) denotes the boundary of x

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This work was partially supported by the Natural Sciences and Engineering Research Council of Canada (NSERC). Grant RGPIN-2014-06512.

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    Sindhu Radhakrishnan & Wail Gueaieb

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All authors contributed to the study conception and design. Material preparation, data collection and analysis were performed by SR. The first draft of the manuscript was written by SR and all authors commented on previous versions of the manuscript. All authors read and approved the final manuscript.

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Radhakrishnan, S., Gueaieb, W. Constraint-free discretized manifold-based path planner. Int J Intell Robot Appl 7, 810–855 (2023). https://doi.org/10.1007/s41315-023-00300-3

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