DETC2016/MESA INFORMATION FUSION IN POLAR COORDINATES

Transcription

1 Proeedings of the ASME 6 Interntionl Design Engineering Tehnil Conferenes & Computers nd Informtion in Engineering Conferene IDETC/CIE 6 August -, 6, Chrlotte, USA DETC6/MESA INFORMATION FUSION IN POLAR COORDINATES Hui Dong Shool of Mehnil Engineering nd Automtion Fuzhou University Fuzhou, Fujin, 356, Chin Emil: dhuihit@gmilom Gregory Chirikjin Deprtment of Mehnil Engineering Johns Hopkins University Bltimore, MD 8, USA Emil:greg@jhuedu ABSTRACT Mny sensing modlities used in robotis ollet informtion in polr oordintes For mobile robots nd utonomous vehiles these modlities inlude rdr, sonr, nd lser rnge finders In the ontext of medil robotis, ultrsound imging nd CT both ollet informtion in polr oordintes Moreover, every sensing modlity hs ssoited noise Therefore when the position of point in spe is estimted in the referene frme of the sensor, tht position is repled by probbility density expressed in polr oordintes If the sensor moves from one lotion to nother nd the sme point is sensed, then the two ssoited probbilities n be fused together to obtin better estimte thn either one individully Here we derive the e- qutions for this fusion proess in polr oordintes The result involves the omputtion of integrls of three Bessel funtion We derive new reurrene reltions for the effiient omputtion of these Bessel-funton integrls to id in the informtion-fusion proess INTRODUCTION The definition of informtion fusion is proposed s: Informtion fusion is the study of effiient methods for utomtilly or semi-utomtilly trnsforming informtion from different soures nd different points in time into representtion tht provides effetive support for humn or utomted deision mking [] The informtion provided by sensors is lwys im- Address ll orrespondene to this uthor preiseness tht is ffeted by ssoited noise in the mesurements Informtion fusion lgorithms should be ble to exploit the redundny nd impreise informtion to redue effets of noise Let,, denote three points in the plne nd let x i j denote the reltive position vetor from point i to point j The length of this vetor is r i j x i j x i j x i j Reltive to the x-xis of world oordinte system, the vetor x i j mkes ounterlokwise mesured ngle ϕ i j s shown in Figure Suppose tht point is observed by sensor t point nd t point The sensors t lotions nd my be different sensors tht re fixed t these lotions, or they my be the sme sensor tht hs moved between these lotions Assoited with these sensors re mesurement probbility densities f (x ) nd f (x ) of the point t position Rell tht in generl probbility densities f : X R stisfy the onditions f (x) nd X f (x)dx nd in our se X R nd dx is the Lebesgue mesure in Crtesin oordintes Our gol is to fuse the probbility densities f (x ) nd f (x ) in order to obtin the best estimte of the position of point reltive to point The true position of this point is denoted s x, wheres x n be ny hypothetil vlue of it Copyright 6 by ASME

2 Sine x x x, () the fused density will be of the form f 3 (x ) α f (x ) f (x ) () where for fixed x x f (x ) f (x ) nd f (x ) f (x x ) (3) nd α is sling prmeter tht ensures tht f 3 (x ) is probbility density If the mesurement devie is urte, we would expet either the men or the mode (or both) of f i (x i ) to be t or ner the true vlue x i Eq () n be viewed s form of Byesin fusion wherein one of the mesurement probbility densities is onsidered s the prior, nd the other is likelihood, nd their produt results in posterior probbility density Tht is, in generl Byes rule sttes tht FIGURE Desription of Vribles nturl representtion of mesurement probbilities is the Fourier-Bessel desription f (x) f (r,ϕ) n e inϕ ˆf n (p)j n (pr)pd p () This onsists of Fourier series in the ϕ diretion nd Hnkel trnsform in the r diretion, nd the funtions { ˆf n (p) n Z, p R } define f (x), nd n be reovered from f (x) using the formul of mesurement probbilities is the Fourier-Bessel desription Therefore, f (θ,x) f (θ x) f (x) f (x θ) f (θ) f (θ x) f (x θ) f (θ) f (x) ˆf n (p ) π π In polr oordintes () nd (3) give f (r,ϕ)e in ϕ J n (p r)rdrdϕ (5) f 3 (r,ϕ ) α f (r,ϕ ) f (r,ϕ ) (6) In our se θ x, f (θ x) f 3 (x ), f (θ) f (x ), nd f (θ x)/ f (x) α f (x ) Our problem hs slightly more struture thn the generl Byesin fusion senrio, sine α n be omputed expliitly s α ( f f )( x ) where denotes the onvolution ( f f )(x) f (y) f (x y)dy R nd f (x) f ( x) For sensing modlities tht ollet position dt in polr oordintes, x [r osϕ,r sinϕ] T, By mnipulting expressions in Wtson s lssi work on Bessel funtions [5], it n be shown tht e in ϕ m e in ϕ J n (p r ) J n +m(p r )J m (p r )e imϕ (7) where in the ontext of our problem, r r is the true fixed distne between the sensor positions, nd ϕ sine both sensors lie on the x xis Here J n (p r ) denotes Bessel funtions of the first kind lter The reson for using n nd p here rther thn n nd p will beome ler Copyright 6 by ASME

3 Therefore, if we know the sensor mesurements of point tken from lotion, ie, f (r,ϕ ), then we n onvert this to representtion t point using (7) to fuse the two mesurements To do this, we use () nd nd f (r,ϕ ) f (r,ϕ ) e inϕ n n e in ϕ ˆf n () (p)j n (pr )pd p ˆf () n (p )J n (p r )p d p Substituting (7) into the seond of these nd using (6) expresses f 3 (r,ϕ ) s f 3 (r,ϕ ) e inϕ n ˆf () α n (p ) m ˆf () n (p)j n (pr )pd p e in ϕ n J n +m(p r )J m (p r )e imϕ p d p We ompute ˆf (3) n (p ) with (5), where f (r,ϕ) f 3 (r,ϕ ) The result is ˆf (3) n (p ) π f 3 (r,ϕ )e in ϕ J π n (p r )r dr dϕ αδ n+n,n +m J m (p r ) n n α n n m ˆf () n (p) ˆf () n (p ) J n (pr )J n (p r )J n +m(p r )r dr pp d pd p ˆf () n (p) ˆf () n (p )J n+n n (p r ) J n (pr )J n (p r )J n+n n (p r )r dr pp d pd p (9) This n be simplified s (8) where we denote the three-term Bessel integrl s m n l J m (r)j n (br)j l (r)r dr () Remrkbly, suh integrls pper not to hve been studied in muh detil in the lssil literture, nd only reently hve been investigted [3] In order to filitte the omputtion of ˆf (3) n (p ), nd hene the resulting posterior density f 3 (r,ϕ ), we investigte effiient wys to ompute these integrls There re some obvious symmetries nd redundnies in the integrl in () For exmple, olumns n be permutted simply beuse hnging the order of the slr multiplition of the Bessel funtions in the integrl will not hnge the resulting vlue Moreover, sine,b, > we n hnge the vribles of integrtion s follows Let r r, we n get dr dr, r r nd dr dr Substituting into () then gives m n l when nd m n J m ( r ) J n ( b r ) J l ( r ) r dr m n l () b m n l J b m (r)j n (br)j l ()rdr δ( b) δ l,, (3) where δ l, denotes Kroneker delt funtion nd δ( b) is Dir delt funtion In the following setions we will mke use of the lssil reltions [5] J m(x) [J m (x) J m+ (x)] () J m (x) x m [J m+(x) + J m (x)] (5) ˆf (3) n (p ) α n n ( n n n + n n p p p ˆf () n (p) ˆf () n (p )J n+n n (p r ) ) pp d pd p () J m (x) ( ) m J m (x) (6) (x m J m (x)) x m J m (x) (7) 3 Copyright 6 by ASME

4 nd the relted ft d[(x) m J m (x)] d m(x) m xj m (x) + (x) m J (x)x m(x) m x x m [J m+(x) + J m (x)] +(x) m [J m (x) J m+ (x)]x m x m+ J m (x) These will llow us to develop reurrene reltions for the three- Bessel-funtion integrls bsed on the lssil reurrene reltions for the Bessel funtions themselves To strt the reurrene reltions we seek losed-form expressions for some speil ses, whih is the subjet of the following setion STARTING VALUES FOR RECURSIONS In speil ses the three-bessel-funtion integrl in () hve losed-form solutions whih n be used to initite reursions for omputing the other vlues In this setion we fous on these speil ses, nd then in Setion we develop reurrene reltions We begin by observing the known losed-form integrl [] f (,b,,α) If we set α in (8) then f (,b,,) J α (r)j α (br)j α (r)r α dr (8) [ ( b) ] α [( + b) ] α 3α πγ(α + )(b)α J (r)j (r)j (r)rdr [ ( b) ] [( + b) ] πγ( ), (9) whih is losed-form expression for prtiulr three-besselfuntion integrl Moreover, if we set α, then f (,b,,) J (r)j (br)j (r)dr r [J (r) + J (r)]j (br)j (r)dr ( ) + () If we tke the prtil derivtive of this with respet to, nd use the properties of Bessel funtions listed erlier, then f (,b,,) [J (r) J (r)]j (br)j (r)rdr () By ombining the bove equtions s () + (), we get the losed-form expression f + f 3 + b b πγ( 3)(b) b + b + b () And similrly, by ombining the bove equtions s () (), we get the losed-form expression f f 6b + b b πγ( 3 )( b) b + b + b Now we set α, nd evlute f (,b,,) (3) J (r)j (br)j (r)r dr, () whih is lso speil se of the losed-form expression in (8) Multiplied by on both sides of the eqution () gives f (,b,,) (r) J (r)j (br)j (r)r 3 dr (5) We then tke the prtil derivtive with respet to for both sides to give f + f r 3 J (r)j (br)j (r)r 3 dr (6) where denotes / Then dividing by on both sides of the eqution (6) gives f + f rj (r)j (br)j (r)r dr (7) Copyright 6 by ASME

5 Then tking the prtil derivtion with respet to on both sides of the Eqution(7) gin, we n get f + f + f r J (r)j (br)j (r)r dr (8) 3 f + f More generlly, if we let f be defined s f (,b,,α) J α(r)j α (br)j α (r)r α dr, nd we multiply by α then we get α f (,b,,α) (r) α J α (r)j α (br)j α (r)r α dr (9) Tking the prtil of (9) with respet to, nd letting A α f (,b,,α) the result is A α r α+ J α (r)j α (br)j α (r)r α dr (3) Dividing by on both sides of (3) nd letting A A, we get A (r) α J α (r)j α (br)j α (r)r α+ dr (3) Moreover, tking the prtil derivtive with respet to nd dividing (9) α times, we n get α α A α A α (3) Doing the sme lultions on both nd b in (), b f (r)j (r)(br)j (br)j (r)r 3 dr (33) then tking the derivtive of for both sides of (33), b f + b f r J (r)(br)j (br)j (r)r 3 dr (3) Dividing by on both sides of the eqution (3), Then tking the derivtive of b for both sides of the Eqution(35), f + b f b + f + b f b J (r)br J (br)j (r)r dr (36) Dividing by b on both sides of the eqution (36), f b + f b + f b + f b J (r)j (br)j (r)rdr (37) We wnt J α(r)j α (br)j α (r)r α dr to hnge to J m(r)j n (br)j l (r)rdr by tking derivtives (b) α f (r) α J α (r)(br) α J α (br)(r) α J α (r)r α dr (38) let B [(b)α f ], m α α, n α α, l α α 3 B B B α B α r α+α J m (r)(br) α J α (br)(r) α J α (r)r α dr B α + B α b b B α +α B α +α b b r α+α J m (r)r α+α J n (br)(r) α J α (r)r α dr B α +α + B α +α B α +α +α 3 B α +α +α 3 r α+α J m (r)r α+α J n (br)r α+α 3 J l (r)r α dr if α + α + α + α + α + α 3 α, m + n + l α, we n get b f + b f J (r)(br)j (br)j (r)r dr (35) m n l B α +α +α 3 (39) 5 Copyright 6 by ASME

6 The losed-form expressions for speil three-besselfuntion integrls omputed in this setion n be used s the initil vlues in the reursive omputtion of other integrls of this form The reurrene reltions used in these reursive omputtions re derived in the following setion RECURRENCE RELATIONS Then We define g(,b,,m,n,l) J m (r)j n (r)j l (r)dr () We use integrtion by prts to ompute J m (r)j n (br)j l (r)dr J m (r)j n (br)j l (r)r r [J m(r)j n (br)j l (r)]rdr [J m(r)j n (br)j l (r) + bj m (r)j n(br)j l (r) + J m (r)j n (br)j l (r)]rdr [ m n l m + n l m n l m n + l + b b ] m n l m n l + + the right side of () equl the right side of (), then () g(,b,,m,n,l) r m [J m+(r) + J m (r)]j n (br)j l (r)dr [ ] m + n l m n l +, m () ( m m + n l ) + ( m + m n l ) b m n + l b m n l + m n l + m n l (5) nd if we do the sme with b insted of, we n get We n get the similr result by omputing the b nd g(,b,,m,n,l) br n J m(r)[j n+ (br) + J n (br)]j l (r)dr b [ ] m n + l m n l +, n () ( b n b m n + l ) + ( b n + b m n l ) m + n l m n l + m n l + m n l (6) nd with, g(,b,,m,n,l) r l J m(r)j n (br)[j l+ (r) + J l ]dr [ ] m n l + m n l + l (3) ( l m n l + ) + ( l + m n l ) m + n l m n l + b m n + l b m n l (7) 6 Copyright 6 by ASME

7 Rewriting (5), (6) nd (7) together in mtrix form b m b b n b m + b b n + b b l l + we ompute the determinnt of mtrix, let b m M b b n l det(m) m + n l m n + l ( ) m n l + m n l m n l ( ) m n l bl + bm + bn b 8lmn (8) (9) we wnt det(m) bl +bm+bn b nd mln From the funtions we know, if l + m + n nd m,n,l,must hs M mke M M I m + b let N b n + b l + Multiplying M on both sides of (8), m + n l m n l ( ) m n + l ( M N m n l ) (5) m n l + m n l l D 3 (l + m + n ) l D 33 l + m + n D 3 bl (l + m + n ) MODELS FOR POLAR MEASUREMENT PROBABILI- TIES For idel sensors, the mesurement probbility density models in Crtesin oordintes would be f k (x k ) δ(x k x k ) for k, where δ( ) is the Dir delt funtion for R We n onvert these to polr oordintes s f k (r k,ϕ k ) r k δ(r k r k )δ(ϕ k ϕ k ) This n be expressed in Fourier-Bessel expnsion by substituting into (5) to get (k ) ˆf n (p ) ϕ π e in k J n (p r k ) (5) Of ourse, if the sensors mde ext mesurements, there would be no need to perform fusion As n opposite extreme, onsider the se where rnge sensor hs bsolutely no bering informtion This would orrespond to the Figure where the lotion of the observed objet ould eqully be loted t the two intersetion points This sort of rnge-only informtion orresponds to the pdf in rel spe with Fourier-Bessel oeffiients (5) with only n nd ll other terms bsent As Let D M N, then m D l + m + n D bm (l + m + n ) m n D 3 D (l + m + n ) b(l + m + n ) n D l + m + n D n 3 b(l + m + n ) FIGURE Fusion of Two Rnge-Only Mesurements 7 Copyright 6 by ASME

8 more relisti model for the detetion probbility for the point x in the plne desribed in polr oordintes, (r,ϕ ), we use bndlimited pproximtion of the idel Dir delt In other words, we set ll of the oeffiients in (5) to zero exept for (k ) { ˆf n (p ) n,,} This hs the effet of smering out the delt funtion in the ϕ diretion (k ) We set { ˆf n (p ) n,,} for the sensor informtion The probbility distribution funtion (PDF) of informtion when sensor t point is shown in Figure 3, here the monitoring position is t point ( r, ϕ) (,) The PDF of sensor informtion t point is shown in Figure, where the distne between point nd point r 3 we n see PDF of diret multiplition of f (x) nd f (x) in Figure 5 The PDF is omputed by our informtion fusion lgorithm tht is shown in Figure 6 Compring with the results of informtion fusion in Figure 5 nd 6, the lotion of monitoring point is more urte nd ler in Figure 6 It proofs the useful nd effetive of Our informtion fusion lgorithm y(m) x(m) FIGURE PDF of f (x) when the sensor t point, the model for detetion probbility for the point ( r, ϕ) (, ), the distne between point nd point r y(m) 5 y(m) x(m) FIGURE 3 PDF of f (x) when the sensor t point, the model for detetion probbility for the point ( r, ϕ) (,) FIGURE x(m) Fused PDF by diret multiplition of f (x) nd f (x) ACKNOWLEDGMENTS This work ws supported by NSF Grnt RI-Medium: I- IS695 nd the Ntionl Institute of Generl Medil Sienes of the NIH under wrd number RGM3 CONCLUSIONS AND FUTURE WORK We present method to fuse probbilities orresponding to noisy mesurement models in polr oordintes At the ore of this pproh is the observtion tht integrls of produts of three Bessel funtions rise We derive reurrene reltions for these integrls in order to effiiently evlute the fused distributions REFERENCES [] Khleghi B, Khmis A, Krry F O, et l Multisensor dt fusion: A review of the stte-of-the-rt[j] Informtion Fusion, 3, (): 8- [] Andrews, GE, Askey, R, Roy, R, Speil Funtions, Enylopedi of Mthemtis nd Its Applitions, Vol 7, Cmbridge University Press, Copyright 6 by ASME

9 6 5 y(m) x(m) FIGURE 6 Resulting Fused PDF when f (x) is shifted in the e diretion by (In this se the two irles kiss t point) [3] Auluk, SKH, On the Integrl of the Produt of Three Bessel Funtions over n Infinite Domin Fourier-Spe Representtion of Nonliner Dynmis of Continuous Medi in Cylindril Geometry, The Mthemti Journl,Vol, pp-39,, [] Chirikjin, GS, Kytkin, AB, Engineering Applitions of Nonommuttive Hrmoni Anlysis, CRC Press, Bo Rton, FL [5] Wtson, G N, A tretise on the theory of Bessel funtions Cmbridge University Press, Copyright 6 by ASME