[Paper Review] Topological complexity of collision free motion planning algorithms in the presence of multiple moving obstacles

Michael Färber, Mark Grant+1 more|ArXiv.org|Sep 17, 2006

Topological and Geometric Data Analysis10 references3 citations

TL;DR

This paper computes the topological complexity of motion planning algorithms for n objects moving in R³ while avoiding multiple moving obstacles, showing that complexity grows linearly with n (2n+1 for m ≥ 2 obstacles) and is independent of the number of obstacles. Using cohomology of configuration spaces and Schwarz genus theory, it establishes tight lower and upper bounds via zero-divisor cup length and polyhedral cell structures, revealing that topological complexity remains unchanged even when obstacles move, as long as their trajectories are known in advance.

ABSTRACT

We study motion planning algorithms for collision free control of multiple objects in the presence of moving obstacles. We compute the topological complexity of algorithms solving this problem. We apply topological tools and use information about cohomology algebras of configuration spaces. The results of the paper may potentially be used in systems of automatic traffic control.

Motivation & Objective

• To determine the topological complexity of motion planning algorithms for n moving objects in R³ avoiding multiple moving obstacles with known trajectories.
• To analyze how the presence of moving obstacles affects the topological complexity of collision-free motion planning.
• To establish tight lower and upper bounds for the topological complexity using cohomological invariants and homotopy theory.
• To compare the complexity of problems with moving obstacles to those with stationary obstacles or single-point obstructions.
• To demonstrate that topological complexity depends primarily on the number of objects, not the number of obstacles, under known motion assumptions.

Proposed method

• Uses the topological complexity invariant TC(X), defined as the minimal number of local continuous rules needed to construct a motion planning algorithm on the configuration space X.
• Applies the zero-divisor cup length method via the cohomology algebra of configuration spaces F(R³−Sₘ,n), where Sₘ is the set of moving obstacle trajectories.
• Employs the Schwarz genus theory and the notion of a fiber space genus to bound TC from below using nontrivial products in H*(X;Z).
• Establishes that F(R³−Sₘ,n) is 1-connected and has the homotopy type of a CW-complex with cells corresponding to a basis of H*(F(R³−Sₘ,n);Z), enabling upper bounds via dimension-based estimates.
• Uses the fact that F(R³−Sₘ,n) is homotopy equivalent to the complement of codimension-3 affine subspaces in R³ⁿ, allowing transversality arguments for connectivity.
• Applies results from algebraic topology, including Theorem 4.4 on cohomology basis and Corollary 5.3 of [2], to derive upper bounds based on polyhedral dimension.

Experimental results

Research questions

• RQ1 What is the topological complexity of motion planning for n objects in R³ when avoiding m moving obstacles with known trajectories?
• RQ2 How does the number of moving obstacles influence the topological complexity of the motion planning problem?
• RQ3 Is the topological complexity of a system with moving obstacles equivalent to that of a system with stationary obstacles or a single point obstacle?
• RQ4 Can cohomological invariants such as zero-divisor cup length provide tight bounds on TC for configuration spaces with moving obstacles?
• RQ5 Does the topological complexity grow linearly with the number of objects, regardless of the number of obstacles?

Key findings

• The topological complexity of motion planning for n objects in R³ avoiding m ≥ 2 moving obstacles is TC(F(R³−Sₘ,n)) = 2n+1.
• For m = 1 obstacle, the complexity is TC(F(R³−Sₘ,n)) = 2n, and for m = 0 (no obstacles), it is 2n−1.
• The complexity is independent of the number of obstacles m when m ≥ 2, growing linearly only with n.
• The configuration space F(R³−Sₘ,n) is 1-connected and has the homotopy type of a CW-complex of dimension 2n when m ≥ 1.
• The lower bound of 2n+1 is established via a nontrivial product of length 2n in the cohomology algebra, specifically ∏(i=1 to n, j=n+1,n+2) (ēᵢⱼ)² ≠ 0.
• Surprisingly, the complexity of avoiding multiple moving obstacles (2n+1) is the same as that of avoiding a single point obstacle in R³ (also 2n+1), despite the intuitive increase in complexity.