We document two environment-generating distributions used for sampling random 2D maps. The first generates random hallway environments based on a Markov chain and the second generates random forest environments based on the Poisson distribution

Richter, Charles

Roy, Nicholas

Vega-Brown, William

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Computer Science and Artificial Intelligence LaboratoryTechnical Reportm a s s a c h u s e t t s  i n s t i t u t e  o f  t e c h n o l o g y,  c a m b r i d g e ,  m a  0 213 9  u s a  —  w w w. c s a i l . m i t . e d uMIT-CSAIL-TR-2015-014 April 27, 2015Markov Chain Hallway and Poisson Forest Environment Generating DistributionsCharles Richter, William Vega-Brown, and Nicholas RoyMarkov Chain Hallway and Poisson ForestEnvironment Generating DistributionsCharles Richter, William Vega-Brown, and Nicholas Roy1 Markov Chain Hallway DistributionRandom 2D hallway-type maps can be sampled from a generating distributionbased on a Markov chain, which allows the frequency of turns and other charac-teristics to be embedded in the state-transition probabilities. Sampling a historyof states in the Markov chain corresponds to sampling a sequence of straighthallway segments and turns, which are then used to construct a map.To generate hallways that are unable to cross over themselves, we require thatthe Markov chain alternate between pairs of left turns and pairs of right turns(i.e. LEFT-LEFT-RIGHT-RIGHT-LEFT-LEFT, etc). Therefore, the state representa-tion must store which type of turn preceded each straight segment, and whichtype of straight segment preceded each turn, regardless of how many segmentsago the previous turn or straight segment occurred. Let s ∈ {1, 2, 3, 4, 5, 6, 7, 8}be the state of the Markov chain. The eight states are:1. STRAIGHT segment oriented left, following a #5 LEFT TURN2. STRAIGHT segment oriented up, following a #7 RIGHT TURN3. STRAIGHT segment oriented up, following a #6 LEFT TURN4. STRAIGHT segment oriented right, following a #8 RIGHT TURN5. LEFT TURN segment, following a #3 STRAIGHT segment6. LEFT TURN segment, following a #4 STRAIGHT segment7. RIGHT TURN segment, following a #1 STRAIGHT segment8. RIGHT TURN segment, following a #2 STRAIGHT segmentThe transition probability matrix is:T =pss 0 0 0 0 0 pst 00 pss 0 0 0 0 0 pst0 0 pss 0 pst 0 0 00 0 0 pss 0 pst 0 0pts 0 0 0 0 0 ptt 00 0 pts 0 ptt 0 0 00 pts 0 0 0 0 0 ptt0 0 0 pts 0 ptt 0 0(1)where pss, pts, ptt, and pst specify the probability of a straight segment followinga straight segment, a straight segment following a turn, a turn following a1Figure 1: Randomly generated Markov hallway map with pst = 0.4 and ptt = 0.Figure 2: Randomly generated Poisson forest map with λ = 0.05 trees · m−2and r = 1 m.turn, and a turn following a straight segment, respectively. To ensure thatthe rows of T sum to one, we must have pss + pst = 1 and pts + ptt = 1. Ifdesired, we can prevent U-turns (two turns following one another without astraight segment in between) by setting ptt = 0, implying that pts = 1. Weinitialize the Markov chain to begin with one of the eight possible states, andfor k = 1, 2, . . . , N−1, we sample sk+1 from the discrete probability distributiondefined by the (sk)th row of T , where N is the number of segments (i.e., thelength of the hallway). Figure 1 shows an example of a randomly sampledMarkov hallway with pst = 0.4 and ptt = 0.2 Poisson Forest DistributionRandom 2D forest-type environments can be generated using the Poisson dis-tribution. Let A be the area of a rectangular forest. The number of trees in theforest, X, is distributed X ∼ Poisson(λA), where λ is the mean number of treesper unit area. The sampled number of trees are then placed uniformly at ran-dom within the forest, with radius r. The Poisson process can be modified usingrejection sampling during the tree-placement phase to enforce a minimum dis-tance between tree centers if desired. Figure 2 shows an example of a randomlygenerated Poisson forest with λ = 0.05 trees ·m−2 and r = 1 m.2

Markov Chain Hallway and Poisson Forest Environment Generating Distributions

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Markov Chain Hallway and Poisson Forest Environment Generating Distributions

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