Partitioning de Bruijn graphs into fixed-length cycles for robot identification and tracking

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• 作者：Tony Grubman^a ; ^b ; ^{tony.grubman@monash.edu" class="auth_mail" title="E-mail the corresponding author} ; Y. Ahmet Şekercioğlu^a ; ^c ; ^{asekerci@utc.fr" class="auth_mail" title="E-mail the corresponding author} ; David R. Wood^b ; ^{david.wood@monash.edu" class="auth_mail" title="E-mail the corresponding author}
• 关键词：Graph theory ; Robot network ; de Bruijn graph ; Graph decomposition ; Pose estimation ; Linear feedback shift register
• 刊名：Discrete Applied Mathematics
• 出版年：2016
• 出版时间：20 November 2016
• 年：2016
• 卷：213
• 期：Complete
• 页码：101-113
• 全文大小：1076 K

文摘

We propose a new camera-based method of robot identification, tracking and orientation estimation. The system utilises coloured lights mounted in a circle around each robot to create unique colour sequences that are observed by a camera. The number of robots that can be uniquely identified is limited by the number of colours available, class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0166218X1630230X&_mathId=si7.gif&_user=111111111&_pii=S0166218X1630230X&_rdoc=1&_issn=0166218X&md5=66cac516cdfb5c9a325d28be1239ba36" title="Click to view the MathML source">qclass="mathContainer hidden">class="mathCode">$q$, the number of lights on each robot, class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0166218X1630230X&_mathId=si8.gif&_user=111111111&_pii=S0166218X1630230X&_rdoc=1&_issn=0166218X&md5=b0b83dc83401f0bee968bbdad2437d73" title="Click to view the MathML source">kclass="mathContainer hidden">class="mathCode">$k$, and the number of consecutive lights the camera can see, class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0166218X1630230X&_mathId=si9.gif&_user=111111111&_pii=S0166218X1630230X&_rdoc=1&_issn=0166218X&md5=786e63d4f1842b513b1209e514940e58" title="Click to view the MathML source">ℓclass="mathContainer hidden">class="mathCode">$\ell$. For a given set of parameters, we would like to maximise the number of robots that we can use. We model this as a combinatorial problem and show that it is equivalent to finding the maximum number of disjoint class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0166218X1630230X&_mathId=si8.gif&_user=111111111&_pii=S0166218X1630230X&_rdoc=1&_issn=0166218X&md5=b0b83dc83401f0bee968bbdad2437d73" title="Click to view the MathML source">kclass="mathContainer hidden">class="mathCode">$k$-cycles in the de Bruijn graph class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0166218X1630230X&_mathId=si11.gif&_user=111111111&_pii=S0166218X1630230X&_rdoc=1&_issn=0166218X&md5=0d973d82bfd2a56efdce6f952e31ff6b" title="Click to view the MathML source">dB(q,ℓ)class="mathContainer hidden">class="mathCode">$dB(q,\ell )$.

We provide several existence results that give the maximum number of cycles in class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0166218X1630230X&_mathId=si11.gif&_user=111111111&_pii=S0166218X1630230X&_rdoc=1&_issn=0166218X&md5=0d973d82bfd2a56efdce6f952e31ff6b" title="Click to view the MathML source">dB(q,ℓ)class="mathContainer hidden">class="mathCode">$dB(q,\ell )$ in various cases. For example, we give an optimal solution when class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0166218X1630230X&_mathId=si13.gif&_user=111111111&_pii=S0166218X1630230X&_rdoc=1&_issn=0166218X&md5=1a27acc0ae6b4a69081e9f005d2a3ec9" title="Click to view the MathML source">k=q^ℓ−1class="mathContainer hidden">class="mathCode">$k=q^{\ell -1}$. Another construction yields many cycles in larger de Bruijn graphs using cycles from smaller de Bruijn graphs: if class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0166218X1630230X&_mathId=si11.gif&_user=111111111&_pii=S0166218X1630230X&_rdoc=1&_issn=0166218X&md5=0d973d82bfd2a56efdce6f952e31ff6b" title="Click to view the MathML source">dB(q,ℓ)class="mathContainer hidden">class="mathCode">$dB(q,\ell )$ can be partitioned into class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0166218X1630230X&_mathId=si8.gif&_user=111111111&_pii=S0166218X1630230X&_rdoc=1&_issn=0166218X&md5=b0b83dc83401f0bee968bbdad2437d73" title="Click to view the MathML source">kclass="mathContainer hidden">class="mathCode">$k$-cycles, then class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0166218X1630230X&_mathId=si16.gif&_user=111111111&_pii=S0166218X1630230X&_rdoc=1&_issn=0166218X&md5=ed32f4bd6ce5174542de3e75eda04a1c" title="Click to view the MathML source">dB(q,tℓ)class="mathContainer hidden">class="mathCode">$dB(q,t\ell )$ can be partitioned into class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0166218X1630230X&_mathId=si17.gif&_user=111111111&_pii=S0166218X1630230X&_rdoc=1&_issn=0166218X&md5=06af7a4f0c775d489eda6d46c6e1112f" title="Click to view the MathML source">tkclass="mathContainer hidden">class="mathCode">$tk$-cycles for any divisor class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0166218X1630230X&_mathId=si18.gif&_user=111111111&_pii=S0166218X1630230X&_rdoc=1&_issn=0166218X&md5=97f99d944f13e5191d9cc3ca910e8ae7" title="Click to view the MathML source">tclass="mathContainer hidden">class="mathCode">$t$ of class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0166218X1630230X&_mathId=si8.gif&_user=111111111&_pii=S0166218X1630230X&_rdoc=1&_issn=0166218X&md5=b0b83dc83401f0bee968bbdad2437d73" title="Click to view the MathML source">kclass="mathContainer hidden">class="mathCode">$k$. The methods used are based on finite field algebra and the combinatorics of words.