Proceedings - 5th IMA Conference on Mathematics in Defence 23 November 2017, The Royal Military Academy Sandhurst, Camberley, UK

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1 3 November 7, The Royal Military Academy Sadhurst, Camberley, UK Eteral Fake Costraits Iterpolatio: the ed of Ruge pheomeo with high degree polyomials relyig o equispaced odes Applicatio to aerial robotics motio plaig By Nicolas Belager Airbus, Iovatio Departmet, Marigae, Frace Abstract The preset paper itroduces a ew approimatio method that suppresses the Ruge pheomeo with high degree polyomials relyig o equispaced odes. The priciple cosists i usig eteral costraits of the type P ()= eterally to the iterpolatio iterval. As those costraits do ot have ay validity regardig the Ruge fuctio behavior we call them Eteral Fake Costraits (EFC). We demostrate that the EFC method ca be of greater accuracy tha Splie iterpolatio. We the apply the EFC priciple to a simple aerial robotics motio plaig problem ad demostrate the iterest for liear system optimizatio uder kiematics costraits ad preset the associated results. We the ope the door to directios to improve ad formalize EFC method ad propose other applicatios i aerial robotics motio plaig.. Itroductio For more tha a cetury, the Ruge pheomeo [] has fasciated geeratios of approimatio theory mathematicias. Major advaces have bee made through the last two decades by researchers like J.P. Boyd [3] [4] [5] or Rodrigo Platte [6] [7] to deeply uderstad ad defeatig this couter ituitive pheomeo. Nevertheless, the geeral spread idea is that high order polyomial iterpolatio with equispaced odes is ot suitable to approimate the Ruge fuctio as demostrated by Trefethe []. To mitigate the Ruge problem, it is commo to use the Chebyshev odes for Lagrage iterpolatio which are kow as the best solutio to iterpolate the fuctio with polyomials while atteuatig largely the wild oscillatios occurrig ear the limits of the iterpolatio iterval. This solutio ca be satisfyig for mathematicias, but a lot of operatioal problems require the eeds of equidistat odes which is ot the case with Chebyshev odes. Thus, i order to overcome the Ruge pheomeo while keepig the iterest of polyomials i the solutio, Dechao & al [8] eplored piecewise fuctios such as splies. The results preseted i this study are very accurate ad ca serve as a very good referece for the method that we aim to itroduce. The authors measured the efficiecy of differet splies methods such as : atural cubic splies, parabolically termied cubic splies, etrapolated cubic splies. Usig the least squares error idicator : E K k K ( y ˆ k k y k ) The results obtaied i [8] are recpitualed i the Table with + equidistat odes:

2 3 November 7, The Royal Military Academy Sadhurst, Camberley, UK Natural Cubic Splie Etrapolated cubic splie Parabolically termiated cubic splie PseudoIverse Cubic splie (PCS) TABLE. Results obtaied with splies iterpolatio i De Chao ad al. [8] Aother recet work [9] based o a Immuity Geetic Algorithm (IGA) applied to a parametric curve has demostrated that it was possible to solve the Ruge pheome by learig methods. The results preseted with the IGA method state of E error of for equidistat samples. The recet results obtaied with the PseudoIverse Cubic Splie (PCS) iterpolatio [8] ad with the IGA [9] will serve as a referece for our approach. I the preset paper we itroduce a ew method which approimate the Ruge fuctio f()=/(+5 ) withi [-;] by the meas of a sigle high order degree polyomial with equispaced odes. The iterest of the method goes beyod the pure mathematical eercise as we demostrate the added value that it brigs for optimizatio problems. We illustrate it through a case of path plaig optimizatio for a aerial mobile robot.. Eteral Fake Costraits Iterpolatio Cosiderig the Ruge pheomeo ad the wild oscillatios that occur ear the edpoits of the iterpolatio iterval, we propose a ew approach. Our idea is to artificially icrease the degree of the iitial Lagrage iterpolatio polyomial relyig o equispaced odes. To do so, we eted the liear system represeted by k equatios used to build the Lagrage iterpolatio polyomial with equispaced odes. These equatios which ca be see as Lagrage iterpolatio cotributors i our system are of the type: a a. a f ( ) (where f is the Ruge fuctio). Li Li Li Where the are the equidistat k odes (k is a odd umber) i the iterval [-:] L i To those k equatios we add m equatios (m is a eve umber ad k + m = + ) that we use to stabilize the Ruge pheomeo oscillatios. Those m equatios impose to the polyom P EFC () that we wat to fid, the costraits that its secod derivative is equal to i m/ poits located withi a small iterval ]--; -] ad m/ equatios of the same type located eterally ad symmetrically o the iterval [; +[ where >. We call those costraits Eteral Fake Costraits (EFC) because regardig the behaviour of the secod derivative of the Ruge fuctio which equatio is 5 5 f Ruge "( ) is ever equal to 3 (5 ) (5 ) withi the small itervals ]--; -] ad [; +[as it ca be see o Figure : FIGURE. Distributio of secod derivative of the Ruge fuctio

3 3 November 7, The Royal Military Academy Sadhurst, Camberley, UK The EFC equatios brought ito the liear system guaratee that there will be o ew oscillatios geerated by the icrease of the iterpolatio polyomial degree. Ideed, itroducig additioal costraits o the secod order derivative of the polyomial guaratee that the ew oscillatios due to the degree icrease will be geerated eterally to the iterval [-;]. So those kid of costraits are iterestig because they ca perform stabilizatio of the iteral wild oscillatios due to Ruge pheomeo while ecludig of the iterpolatio iterval potetial additioal oscillatios itroduced by the polyomial degree icrease. Thus the global liear system that we build with the k+m equatios has the followig form: 3 L L Lk.( )..( )..( ). EFC EFC EFC m L L Lk ( ).( ). ( ).( ). ( ).( ). 3 EFC 3 EFC 3 EFC m L L EFC EFC EFC m L L Lk a f ( L ) a f ( L ) f ( L ) k. a a The challege is the to determiig m the efficiet umber of Eteral Fake Costraits ad S the set of abscissa positios of the fake costraits : S={ } ;; EFC EFCm The work that we performed has ot yet led to a formal determiatio of those variables. However the study has a empirical feature at this stage, the results we have bee able to provide are of iterest for future optimizatio problems.the liear system that we have formed is ill coditioed which does ot facilitate the research of the best iterpolatio polyomial. We have ot eplored a large rage of EFC sizes ad we have ot eplored all the possibilities i terms of EFC i distributio over the adjacet itervals as well. That is why we thik that the EFC priciple still holds a big potetial of improvemet. We preset i the followig sectio, the results that we obtaied for 5 equidistat iterpolatio odes ad 46 EFC poits: FIGURE. Compariso of the Ruge fuctio ad EFC Iterpolatio for 5 equidistat odes The results obtaied o Fig. with oly 5 equidistat odes which would correspod to a degree 4 for the iterpolatio polyomial are quite impressive. Ideed the method allows us to reach a accuracy of which is ot that far from the results obtaied by Dechao & al. [8] with the splie iterpolatio with 9 equidistat poits. It has to be oticed that the EFC method ca geerate implicit crossigs betwee the iterpolatio curve ad the Ruge fuctio. This is the case at the abscissas -.9 ad.9 which are ot iterpolatio odes. Aother remarkable poit is the etreme sesitivity of the approimatio to the space betwee

4 3 November 7, The Royal Military Academy Sadhurst, Camberley, UK the EFC abscissas. I the preseted case, the local miimum foud has a icrease of te times accuracy for a etremely small variatio ( - ) betwee the abscissa where we apply the EFC: the distace betwee the EFC abscissa is called the EFC space. 4 The results obtaied for 9 equidistat odes o Fig.3 still improve i the way of beig better tha the splie results as we foud a local miimum at.6-4 compare to for the splie study. We did very few attempts of varyig the umber of EFC so there is certaily potetial to improve the accuracy of the results i the future. It has to be metioed that the implicit EFC crossigs still remai as for the 5 equidistat odes iterpolatio, but the implicit crossigs have lightly moved to the abscissas.3 ad -.3. FIGURE 3. Compariso of the Ruge fuctio ad EFC Iterpolatio for 9 equidistat odes The results we obtaied for equidistat odes ad 6 EFC o Fig.4 : are lightly less accurate tha those obtaied by [8] i the splie study (5.6-5 ). The eplaatio certaily comes from the limited tests we had with differet umber of EFC sets. A larger eploratio of EFC sizes ad their associated spaces betwee fid better local miima. EFC i should have bee doe to FIGURE 4. Compariso of the Ruge fuctio ad EFC Iterpolatio for equidistat odes At least, for equidistat odes ad 5 EFC we obtai oce agai a better result tha the oe obtaied with the splie approimatio. Our approimatio reaches (see Figure 5.) where the splie method obtais for the same equidistat poits. The origiality of the result we obtaied comes from the offset of the first two EFC i which are ot positioed o the boud of the iterpolatio iterval but slightly moved away to abscissas : ad This result highlights aother ais of improvemet which deals with the potetial of o costat EFC space (whe the distace betwee EFC abscissa is ot costat).

5 3 November 7, The Royal Military Academy Sadhurst, Camberley, UK 5 FIGURE 5. Compariso of the Ruge fuctio ad EFC Iterpolatio for equidistat odes Last but ot least we had a attempt for the same problem with equidistat odes, to d P( ) itegrate few Fake Costraits of the same type but iterally to the iterpolatio d iterval [-;]. With itroducig those 6 iteral fake costraits we obtaied a accuracy of times superior to splie method : The list of abscissas used for the fake costraits is provided i Table. List of abscissas used for Eteral Fake Costraits ad 6 Iteral Fake Costraits (i bold),7,4,,88,75,6,49,36,3,,97,84,7,58,45,3,9,6,993,98,967 -,7 -,4 -, -,88 -,75 -,6 -,49 -,36 -,3 -, -,97 -,84 -,7 -,58 -,45 -,3 -,9 -,6 -,993 -,98 -,967 - TABLE. equidistat odes 44 EFC 6 IFC This prelimiary work demostrates the iterest of maagig the variatios of a polyomial o a domai by the use of eteral costraits. The iterest of this priciple has bee validated o a famous eample : the Ruge pheomeo which has bee fied. We illustrate the iterest of the method for motio plaig applicatios i aerial robotics. 3. Applicatio of Eteral Fake Costraits to aerial mobile robotics for motio plaig optimizatio We itroduce here a simple aerial robotics motio plaig problem. The idea is to geerate a lateral chage i a horizotal plae with a small UAV. The trajectory geeratio is performed by the use of a polyomial P(). The costraits imposed at the poit M s start of the trajectory ad at M f ed of the trajectory are simple, they are epressed i a referetial liked to the iitial positio ad directio. The liear system to be solved is the followig: P( s )= P( f )=y f P ( s )= P ( f )= () P ( s )= P ( f )= (curvature=)

6 3 November 7, The Royal Military Academy Sadhurst, Camberley, UK Regardig the kiematics model that we established for the UAV, at a costat speed of km/h we limit the curvature of the trajectory at K ma <7-3 m -, calculated from : y"( ) K( ), the maimum derivative of the curvature at 3 ( y' ( )) dk <6.5-3 m - s - dk ad () <5-3 at the begiig of the trajectory. dt ma dt With 6 costraits to fulfil, it is required a 5 degree polyomial: 6 P()=a a a a + a +a Ad with coordiates for the edig poit M f (;4), solvig the liear system () brigs up the followig solutio : a 5 =7.5 - ; a 4 = ; a 3 =5-5 ; a =; a =; a = We show o the Figure 6 the correspodig trajectory ad its curvature distributio o Figure 7. FIGURE 6. 5 degree polyomial trajectory FIGURE 7. Curvature distributio for 5 degree polyomial This is a simple solutio to the problem. But with a 5 degree polyomial ad 6 costraits we have o latitude to optimize the trajectory. Icreasig the polyomial degree to itroduce ew costraits o the trajectory to optimize it i a better way ivolves some additioal oscillatios risk. That is why people geerally prefer piecewise methods as Splie to get rid of the oscillatios problem of a sigle polyomial (see with [8] to solve the Ruge pheomeo). But the iterest of the sigle polyomial is that the trajectory is deeply cotiuous. Let s ow itroduce two Eteral Fake Costraits of the type : P ( EFC )= symetrically outside of the trajectory iterval. The first iterest of those costraits is that they guaratee the additioal oscillatios ivolved by the icrease of the polyomial degree will be remaied eterally to the domai of iterest, i our case : [;]. Eplaatio : As oscillatios are due to a chage of cocavity which occurs whe f ()=, imposig oly costraits of this type guaratee that o additioal oscillatios will be geerated withi the bouds of the iterpolatio iterval The secod iterest of itroducig two EFC i the liear system is that we brig some additioal latitude to optimize the trajectory i the way that positioig the EFC at differet abscissa will modify the behaviour of the polyomial withi the bouds that defie the trajectory problem. To be coviced about this assertio we propose to solve the liear system

7 3 November 7, The Royal Military Academy Sadhurst, Camberley, UK correspodig to 6 trajectory cotraits formed i the system () plus two EFC. The trajectory polyomial becomes of degree 7 ad the correspodig system to be solved is the: P( s )= P( f )=y f P ( s )= P ( f )= P ( s )= P ( f )= P ( EFC )= P ( EFC )= () We the solve the above system with 3 differet combiatios of EFC to show their direct ifluece o the trajectory. EFC =-. EFC =. Polyomial coefficiets: a 7= ; a 6 = ; a 5 = a 4 = ; a 3 = EFC =- EFC = Polyomial coefficiets: Polyomial coefficiets: a 7= ; a 6 = ; a 5 = ; a 4 = ; a 3 = EFC =- EFC =4 Polyomial coefficiets: a 7 = ; a 6 = ; a 5 = ; a 4 = ; a 3 = Those three 7 degree polyomials are plotted o the Figure 8. It is show the ifluece of the EFC positioig i the trajectory geeratio. We ca observe o the graph that positioig the EFC et to the bouds of the trajectory iterval (cya curve) slows dow the begiig ad the ed of the curve to capture imposed coditios i M f. O the other side, positioig the EFC far from the bouds of the trajectory iterval brigs the 7 degree polyomial (red curve) et to the 5 degree polyomial. 7 FIGURE 8. 7 degree polyomials with differet EFC FIGURE 9. Curvature distributio for 7 degree polyomials We ca aalyse the matchig of the trajectory costraits o the Figure 9 where the curvature distributio is plotted for the three curves. This graph shows that whe the EFC are close to

8 3 November 7, The Royal Military Academy Sadhurst, Camberley, UK dk the bouds (cya curve) the derivative of the curvature () is close to ad the d maimum of the curvature is the highest of the three curves; i the preset problem this costrait is slightly eceeded (>7. -3 m - ). At the same time the derivative of the curvature eceeds the maimum authorized (> m - s - ). It comes that we caot select this curve as a possible curve to fulfil the costraits o the trajectory. I the Table 3. We metio the other mai variables for the differet EFC polyomials ad the 5 degree polyomial. The derivative of the curvature is calculated aalytically followig the below equatio: dk dk d dk dk v cos v cos( Ata( y'( )) dt d dt d d EFC -; 4 EFC -; EFC -.;. Poly. Deg 5 K ma dk dt ma dk () dt ds 5.6 m 5.96 m 6.3 m 5.57 m TABLE 3. Values of differet mai variables for EFC polyomials As we ca see i the Table 3, 5 degree polyomial ad 7 degree polyomial with EFC (- ;4) largely eceed the costraits o the derivative of the curvature especially at the begiig of the trajectory. I our case the oly possible solutio is the 7 degree polyomial with EFC (-;) despite oe of the costraits are saturated. For sure the above solutios are close oe to the other (see close curviliear abscissas values ds) but our goal was to poit out the fact that motio plaig ca be maaged differetly by the use of EFC which is somethig ew. Icreasig the umber of EFC is possible ad offers ew possibilities to fulfill the costraits imposed to the trajectory. We ca imagie etedig the priciple to parametric curves by imposig costraits to egative time t< or time over the limit of trajectory t>t limit to maage the motio plaig i a better way providig more fleibility to the optimizatio Coclusio We itroduced i the preset work a ew approimatio method based o Eteral Fake Costraits ad high degree polyomial iterpolatio to solve the famous Ruge pheomeo raised i 9 by the mathematicia. The EFC used i our work imposed secod derivative of the polyomials to be. the We compared our EFC method with Splie piecewise method. It has demostrated that we were able to reach very good approimatio with few equispaced odes (5 poits) which was ot feasible with Splie iterpolatio. Util equispaced odes we had globally better results tha the Splie method. Settig up few iteral fake costraits immediately icrease the accuracy of the method by times o the computatios doe for equispaced odes ( -8 accuracy) ad this result ca be discussed. Ayway there is a lot of work ahead to establish a real formal EFC iterpolatio method as several questios have to be ivestigated: ifluece of the umber of EFC, space betwee EFC odes to be aalyzed ad mastered: costat or variable EFC space, other types of EFC like direct eteral costraits of C or C types which is possible but their domai of use has to be formalized aalytically to avoid udesirable oscillatios, miig up EFC ad IFC of C types. Last but ot least, the system is strogly ill-coditioed ad this umerical istability is certaily a major difficulty if ot a limitatio to approimatig comple fuctios. But this priciple ca have a rich rage of applicatios like: umerical itegratio, polyomial regressio, Splie approimatio methods improvemet ad motio plaig optimizatio. O this last

9 3 November 7, The Royal Military Academy Sadhurst, Camberley, UK applicatio, we demostrated o a simple case how the EFC ca help to optimize the trajectory i terms of costraits fulfilmet. A particular domai of iterest for EFC method is certaily motio plaig problems usig approaches with parametric curves havig polyomial curvature like of the form: k( s) as a s a a 9 REFERENCES [] C. Ruge, Uber empirische Fuktioe ad die Iterpolatio zwische aquidistate Ordiate. Z. Math. Phys. 46:4 43, 9 [] L. N. Trefethe, Joural of approimatio theory 65, 47-6, 99 [3] J. P. Boyd, J. R. Og, Epoetially-coverget strategies for defeatig the Ruge pheomeo for the approimatio of o-periodic fuctios, part I: sigle-iterval schemes, Commu. Comput. Phys., vol. 5, o. -4, pp , 9. [4] J. P. Boyd, Defeatig the Ruge pheomeo for equispaced polyomial iterpolatio via Tikhoov regularizatio, Appl. Math. Lett., vol.5, o. 6, pp , 99. [5] J. P. Boyd, Luis F. Alfaro, Hermite fuctio iterpolatio o a fiite uiform grid: Defeatig the Ruge pheomeo ad replacig radial basis fuctios Appl. Mat. Lett., vol. 6, o., pp , 3. [6] Be Adcock ad Rodrigo B. Platte, A Mapped Polyomial Method for High-Accuracy Approimatios o Arbitrary Grids, SIAM Joural o Numerical Aalysis > Volume 54, Issue 4, (6 pages), 6 [7] Platte R., Driscoll T., Polyomials ad potetial theory for Gaussia radial basis fuctios iterpolatio. SIAM J. Numer. Aal. 43, pages , 5 [8] Dechao Che, Tiajia Qiao, Hogzhou Ta, Migmig Li ad Yuog Zhag, IEEE 7 th Iteratioal Coferece o Computatioal Sciece ad Egieerig, 4 [9] Hogwei Li, Lijie Su, Searchig globally optimal parameter sequece for defeatig Ruge pheomeo by immuity geetic algorithm, Applied Mathematics ad Computatio 64 (5) 85 98, Elsevier, 5