Quotient Space Planning Framework

We use the Quotient Space Planning Framework to formalize planning on different abstractions levels. Abstraction levels are defined as QuotientSpaces which are lower-dimensional abstractions of the configuration space. A simple example is a rigid body in the plane with the configuration space $$SE(2)$$. We can abstract this configuration space by projecting onto $$\mathbb{R}^2$$, the QuotientSpace of positions of the rigid body.

QuotientSpace projections are many-to-one mappings, which map subsets of the configuration space to a point of a QuotientSpace. The QuotientSpace, as the configuration space, contains feasible and infeasible configurations. We like to disallow projections which map feasible configurations onto infeasible Quotientspace configurations — because we could thereby abstract away valid feasible paths.

A QuotientSpace projection thus projects an infeasible subset onto an infeasible point, but never a feasible configuration onto an infeasible configuration. Such projection are called admissible.

For the rigid body example, we get an admissible projection by nesting a smaller robot in the larger one. Below we show a simple planning problem, where a rigid body needs to move from an initial start configuration (green) to a designated goal configuration (red). This problem can be abstracted by nesting a disk in the rigid body (right Figure).

The same can be done to any robot. To abstract a 3-dof manipulator (Left Figure), one could remove the last link to obtain a 2-dof manipulator (Right Figure) which corresponds to a QuotientSpace projection from a 3-dimensional space to a 2-dimensional space.

In practice, an admissible projection requires that any constraint on the QuotientSpace is also a constraint on the ConfigurationSpace or any high-dimensional QuotientSpace. In that sense, QuotientSpace projections are similar to constraint relaxations.

# Spurious Paths

Our main problem when planning with QuotientSpaces are spurious paths. A spurious path is a path on a quotient space which cannot be lifted to the configuration space (A QuotientSpace path can be lifted iff there exists a feasible path on the ConfigurationSpace which when projected onto the QuotientSpace will yield the QuotientSpace path). An example is shown below. On the left we have a solution path for the nested disk going below the obstacle. This solution path, however, cannot be lifted to the configuration space — the L-shaped robot cannot pass below the obstacle. A feasible path on the ConfigurationSpace is shown on the right, which goes above the obstacle. This path is a projection of the feasible path on the 3-dimensional ConfigurationSpace down onto the 2-dimensional QuotientSpace.

# Probabilistic Completeness

To plan with a sequence of QuotientSpaces, we have developed a new algorithm called the QuotientSpace Rapidly-exploring random tree (QRRT) algorithm. ompl::geometric::QRRT is probabilistically complete when used with admissible projections, i.e. it can solve any planning problem which has a solution. In particular, QRRT can deal with spurious paths. It does so by sampling random vertices from a lower-dimensional QuotientSpace, and projecting them into the configuration space.

# Why Use Quotient Space Planning

• Because it is more general than lower-dimensional projections like ompl::base::ProjectionEvaluator:
• (1) We can handle a finite number of sequential projections instead of just one.
• (2) We can guarantee probabilistic completeness when used with admissible projections
• (3) We can project onto many manifold spaces instead of just euclidean space $$\mathbb{R}^N$$.
• Because it can be much faster compared to other planning algorithms.

## Hypercube Benchmark

To see how much faster QuotientSpace planning can be, we provide a hypercube demo, which you can run yourself using the Quotient Space HyperCube File.

For demonstration, we change the number of dimensions of the hypercube from 6 to 8 to 10 and finally to 12 (using a narrow passage of 0.1).

### 6-dimensional HyperCube

Our results show that ompl::geometric::PRM performs best with 0.103 seconds and ompl::geometric::QRRT on second place with 0.111 seconds.

Finished Benchmark (Runtime:10, RunCount:5)
Placement <Rank> <Time (in Seconds)> <Success (in Percentage)>
--------------------------------------------------------------------------------
Place <1> Time: <0.103107> %Success: <100> (PRM) <-- Winner
Place <2> Time: <0.111702> %Success: <100> (QuotientSpaceRRT[3lvl])
Place <3> Time: <5.33306> %Success: <60> (STRIDE)
Place <4> Time: <9.86457> %Success: <20> (RRT)
Place <5> Time: <10.0467> %Success: <0> (KPIECE1)
Place <6> Time: <10.069> %Success: <0> (EST)
--------------------------------------------------------------------------------

### 8-dimensional HyperCube

Comparing only the two winning algorithms, we see that ompl::geometric::PRM does not scale well to 8 dimensions with zero solved runs, while ompl::geometric::QRRT performs well with 0.653 seconds for solving every single run.

Finished Benchmark (Runtime:10, RunCount:5)
Placement <Rank> <Time (in Seconds)> <Success (in Percentage)>
--------------------------------------------------------------------------------
Place <1> Time: <0.65329> %Success: <100> (QuotientSpaceRRT[4lvl]) <-- Winner
Place <2> Time: <10.0419> %Success: <0> (PRM)
--------------------------------------------------------------------------------

### 10-dimensional HyperCube

ompl::geometric::QRRT even performs well when we increase the dimensions further to 10.

Finished Benchmark (Runtime:10, RunCount:5)
Placement <Rank> <Time (in Seconds)> <Success (in Percentage)>
--------------------------------------------------------------------------------
Place <1> Time: <1.84356> %Success: <100> (QuotientSpaceRRT[5lvl]) <-- Winner
--------------------------------------------------------------------------------

### 12-dimensional HyperCube

The algorithm comes to a limit when we increase the dimensionality further to 12.

Finished Benchmark (Runtime:10, RunCount:5)
Placement <Rank> <Time (in Seconds)> <Success (in Percentage)>
--------------------------------------------------------------------------------
Place <1> Time: <9.11489> %Success: <20> (QuotientSpaceRRT[6lvl]) <-- Winner
--------------------------------------------------------------------------------