1# Quotient Space Planning Framework {#quotientSpacePlanning}
2
3[TOC]
4
5We use the Quotient Space Planning Framework to formalize planning on different abstractions levels.
6Abstraction levels are defined as [QuotientSpaces](https://en.wikipedia.org/wiki/Quotient_space_(topology)) which are lower-dimensional abstractions of the configuration space.
7A simple example is a rigid body in the plane with the configuration space \f$SE(2)\f$.
8We can abstract this configuration space by projecting onto \f$\mathbb{R}^2\f$, the QuotientSpace of positions of the rigid body.
9
10## Admissible QuotientSpace Projections
11
12QuotientSpace projections are many-to-one mappings, which map subsets of the
13configuration space to a point of a QuotientSpace. The QuotientSpace, as the
14configuration space, contains feasible and infeasible configurations. We like to
15disallow projections which map feasible configurations onto infeasible
16Quotientspace configurations --- because we could thereby abstract away valid
17feasible paths.
18
19A QuotientSpace projection thus projects an infeasible subset onto an infeasible
20point, but never a feasible configuration onto an infeasible configuration. Such
21projection are called admissible.
22
23For the rigid body example, we get an admissible projection by nesting a smaller
24robot in the larger one. Below we show a simple planning problem, where a rigid
25body needs to move from an initial start configuration (green) to a designated
26goal configuration (red). This problem can be abstracted by nesting a disk in
27the rigid body (right Figure).
28
29\htmlonly
30<div class="row">
31  <img src="images/quotient/rigidbody2d_1.png" class="col-xs-6 col-xs-offset-3">
32  <img src="images/quotient/rigidbody2d_2.png" class="col-xs-6 col-xs-offset-3">
33</div>
34</div>
35\endhtmlonly
36
37The 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.
38
39\htmlonly
40<div class="row">
41  <img src="images/quotient/planar_manipulator_3dofs.png" class="col-xs-6 col-xs-offset-3">
42  <img src="images/quotient/planar_manipulator_2dofs.png" class="col-xs-6 col-xs-offset-3">
43</div>
44</div>
45\endhtmlonly
46
47In practice, an admissible projection requires that any constraint on the
48QuotientSpace is also a constraint on the ConfigurationSpace or any
49high-dimensional QuotientSpace. In that sense, QuotientSpace projections are
50similar to constraint relaxations.
51
52## Spurious Paths
53
54Our main problem when planning with QuotientSpaces are spurious paths. A
55spurious path is a path on a quotient space which cannot be lifted to the
56configuration space (A QuotientSpace path can be lifted iff there exists a
57feasible path on the ConfigurationSpace which when projected onto the
58QuotientSpace will yield the QuotientSpace path). An example is shown below. On
59the left we have a solution path for the nested disk going below the obstacle.
60This solution path, however, cannot be lifted to the configuration space --- the
61L-shaped robot cannot pass below the obstacle. A feasible path on the
62ConfigurationSpace is shown on the right, which goes above the obstacle. This
63path is a projection of the feasible path on the 3-dimensional
64ConfigurationSpace down onto the 2-dimensional QuotientSpace.
65
66\htmlonly
67<div class="row">
68  <img src="images/quotient/rigidbody2d_3.png" class="col-xs-6 col-xs-offset-3">
69  <img src="images/quotient/rigidbody2d_4.png" class="col-xs-6 col-xs-offset-3">
70</div>
71</div>
72\endhtmlonly
73
74## Probabilistic Completeness
75
76To plan with a sequence of QuotientSpaces, we have developed a new algorithm
77called the QuotientSpace Rapidly-exploring random tree (QRRT) algorithm.
78ompl::geometric::QRRT is probabilistically complete when used with admissible
79projections, i.e. it can solve any planning problem which has a solution. In
80particular, QRRT can deal with spurious paths. It does so by sampling random
81vertices from a lower-dimensional QuotientSpace, and projecting them into the
82configuration space.
83
84## Why Use Quotient Space Planning
85
86- Because it is more general than lower-dimensional projections like
87  ompl::base::ProjectionEvaluator:
88  - (1) We can handle a finite number of sequential projections instead of just one.
89  - (2) We can guarantee probabilistic completeness when used with admissible projections
90  - (3) We can project onto many manifold spaces instead of just euclidean
91    space \f$\mathbb{R}^N\f$.
92- Because it can be much faster compared to other planning algorithms.
93
94### Hypercube Benchmark
95
96To see how much faster QuotientSpace planning can be, we provide a hypercube
97demo, which you can run yourself using the [Quotient Space HyperCube File.](QuotientSpacePlanningHyperCube_8cpp_source.html)
98
99For demonstration, we change the number of dimensions of the hypercube from
1006 to 8 to 10 and finally to 12 (using a narrow passage
101of 0.1).
102
103#### 6-dimensional HyperCube
104
105Our results show that ompl::geometric::PRM
106performs best with 0.103 seconds and ompl::geometric::QRRT on second place
107with 0.111 seconds.
108
109~~~{.txt}
110Finished Benchmark (Runtime:10, RunCount:5)
111Placement <Rank> <Time (in Seconds)> <Success (in Percentage)>
112--------------------------------------------------------------------------------
113Place <1> Time: <0.103107> %Success: <100> (PRM) <-- Winner
114Place <2> Time: <0.111702> %Success: <100> (QuotientSpaceRRT[3lvl])
115Place <3> Time: <5.33306> %Success: <60> (STRIDE)
116Place <4> Time: <9.86457> %Success: <20> (RRT)
117Place <5> Time: <10.0467> %Success: <0> (KPIECE1)
118Place <6> Time: <10.069> %Success: <0> (EST)
119--------------------------------------------------------------------------------
120~~~
121
122#### 8-dimensional HyperCube
123
124Comparing 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.
125
126~~~{.txt}
127Finished Benchmark (Runtime:10, RunCount:5)
128Placement <Rank> <Time (in Seconds)> <Success (in Percentage)>
129--------------------------------------------------------------------------------
130Place <1> Time: <0.65329> %Success: <100> (QuotientSpaceRRT[4lvl]) <-- Winner
131Place <2> Time: <10.0419> %Success: <0> (PRM)
132--------------------------------------------------------------------------------
133~~~
134
135#### 10-dimensional HyperCube
136
137ompl::geometric::QRRT even performs well when we increase the dimensions further to 10.
138
139~~~{.txt}
140Finished Benchmark (Runtime:10, RunCount:5)
141Placement <Rank> <Time (in Seconds)> <Success (in Percentage)>
142--------------------------------------------------------------------------------
143Place <1> Time: <1.84356> %Success: <100> (QuotientSpaceRRT[5lvl]) <-- Winner
144--------------------------------------------------------------------------------
145~~~
146
147#### 12-dimensional HyperCube
148
149The algorithm comes to a limit when we increase the dimensionality further to 12.
150
151~~~{.txt}
152Finished Benchmark (Runtime:10, RunCount:5)
153Placement <Rank> <Time (in Seconds)> <Success (in Percentage)>
154--------------------------------------------------------------------------------
155Place <1> Time: <9.11489> %Success: <20> (QuotientSpaceRRT[6lvl]) <-- Winner
156--------------------------------------------------------------------------------
157~~~
158
159## Want to learn more?
160
161### Tutorials
162
163Check out the [tutorial](quotientSpacePlanningTutorial.html).
164
165### Demos
166
167Check out the [demos](group__demos.html).
168
169### Papers
170
171- A Orthey and M Toussaint, _Rapidly-Exploring Quotient-Space Trees: Motion Planning using Sequential Simplifications_, 2019. Also available at <https://arxiv.org/abs/1906.01350>.
172
173- A Orthey, A Escande and E Yoshida, _Quotient Space Motion Planning_, ICRA, 2018. Also available at <https://arxiv.org/abs/1807.09468>.
174