PHYSICS-ENHANCED L-SYSTEMS

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1 PHYSICS-ENHANCED L-SYSTEMS Hansrud Noser 1, Stephan Rudolph 2, Peter Stuck 1 1 Department of Informatcs Unversty of Zurch, Wnterthurerstr. 190 CH-8057 Zurch Swtzerland noser(stuck)@f.unzh.ch, 2 Department for Statcs and Dynamcs of Aerospace Structures Unversty of Stuttgart, Pfaffenwaldrng 27 D Stuttgart Germany rudolph@sd.un-stuttgart, ABSTRACT In computer graphcs and engneerng many classes of complex objects can be desgned wth L-systems. We present a concept for enhancng tmed and parametrc L-systems wth physcs. Ths smplfes consderably the physcally correct desgn of certan classes of computer anmatons or techncal objects modelled by producton rules. The focus s on structural extensons n tmed and parametrc L-system theory necessary for constrant propagaton management for the treatment of herarchcal objects and on physcs enhanced grammar-language extensons. The proposed concept s llustrated wth a desgn model ncorporatng the statcs of arbtrary tree structures. Keywords: L-systems, rewrtng, physcs, computer graphcs, desgn, anmaton, engneerng, conceptual desgn. 1. INTRODUCTION In physcs- and engneerng-based computer graphcs smulaton, the soluton of large systems of equatons s common practce. Very often the symbolc soluton of large numbers of constrants further complcates the stuaton. A careful nformaton management becomes necessary n order to guarantee solvablty and success. In [YR99] the authors report that the conceptual desgn phase s rather mportant n the engneerng desgn process, and despte ts mportance, ths desgn stage s the least understood and only few supportng tools exst [FRS96]. Approaches for supportng the conceptual Desgn phase are descrbed n [D94]. The objectve of our work s to develop a computerbased framework that supports selected classes of desgn models n order to mprove and reduce the tme need n the conceptual engneerng desgn phase. The basc dea s to use rewrtng systems for defnng such desgn models. Rewrtng systems are synonym for L-systems, L-grammars, or so-called producton systems [PL90]. Rewrtng s a technque for buldng complex objects by successvely replacng parts of a smple ntal object usng a set of rewrtng rules or productons. In computer graphcs, an L-system descrbes a 3D object by an axom and a set of producton rules, whch can be called a grammar snce t descrbes the structure of the object. From the axom and the rules - the grammar - the computer can derve n subsequent teratons 3D objects of a gven structure (or grammar). Furthermore, by usng the concept of turtle graphcs, the symbolc objects can be vsualsed. Tradtonally, L-systems (Lndenmayer-systems) [PL90, PHM93, PJM94] are used for effcent plant and fractal modellng. In [NT99] we descrbe a behavoural anmaton system where producton rules are not only used to defne growth and topology of objects, but also behavour and

2 anmaton of objects n real-tme. In ths paper we suggest to extend such a behavoural anmaton system to a physcs-enhanced L-system applcaton. Our work s nspred from [RN00] where we descrbed a concept on engneerng desgn generaton wth XML-based knowledge-enhanced parametrc grammars. The man focus of the project s to extend a tmed and parametrc L-system applcaton [NT99] n such a way, that t produces not only the geometry but also the correspondng system of equatons or constrants that wll have to be solved automatcally for a gven target applcaton. Rewrtng systems are well suted for embeddng geometry and physcs n the same prmtves, and combnng them to objects of predefned desgn models. In partcular, they -support the constructon of a famly of smlar objects that assure solvablty -are well suted for the constructon of objects wth smlar elements such as plants, brdges, aeroplanes, etc. -can be used for defnng geometry and the correspondng system of equatons -exhbt a hgh data amplfcaton factor because of ther rule-based defnton of objects. nternal representaton of the grammars n applcatons that use the exstng XML supportng Java packages. Whle the descrbed method s best suted for the purpose of conceptual engneerng desgn wth human nteracton, however, t s not suted for realtme applcatons based on tmed L-systems. In ths work we descrbe how we can use smlar prncpals n tmed parametrc L-systems sutable for real-tme applcatons. In order to mprove speed we tend to focus on partcular desgn models and to mplement drectly the treatment of physcs n the applcaton, nstead of mantanng a unversal approach sutable for large classes of engneerng. We mantan the optonal output of the set of equatons n order to be compatble wth the work of [YR99]. Ths concept reduces the generalty of the applcaton, but t makes t sutable for real-tme applcatons and user frendler, as most physcs can be hdden to desgners. User L-system Parser/Compler 2. CONCEPT In order to enhance tmed L-systems wth physcs, we propose a general concept for ncorporatng desgn models. The man dea s to assocate not only geometry but also physcs to symbols. When the desgner uses them n rules for constructng certan classes of objects from a gven desgn model, the applcaton should be able to vsualse the object, as well as to compute the correct physcs. In [RN00] we proposed a generc concept supportng the conceptual desgn phase of engneerng applcaton. In a XML based parametrc grammar smlar to a parametrc L-system [PL90], the user can frst assocate equatons to symbols and then use them n rules for defnng objects. At teraton and nterpretaton of the rules the object was vsualsed, and the applcaton generates automatcally the set of equatons of the correspondng physcal descrpton of the object. These equatons can be solved automatcally by technques descrbed n [YR99]. Thus, an essental support of a conceptual desgn phase for engneerng problems s assured. XML was chosen to get a standardsed representaton of the L- system-lke parametrc grammar. Ths new standard smplfes representaton, edton, parsng, and Iterator L-system applcaton Axom Rules Symbolc object (turtle program) Symbol procedures Interpreter Turtle procedures Archtecture of the tmed and parametrc L- system applcaton that s the startng pont of our work Fgure 1 The startng pont of our work s the real-tme L- system applcaton descrbed n [NT99]. Fg.1 llustrates ts man components. A user desgns an object by the correspondng L-system. The parser and the compler convert t nto an nternal representaton of the axom and the rules. Then, for each frame, the terator terates the symbolc object, whch corresponds to the axom at the begnnng of a smulaton, accordng to the rules. After the teraton, the nterpreter nterprets ths symbolc turtle program by usng procedures of the turtle and the symbol

3 modules, and produces the current frame of the vrtual scene. The L-system we use s tmed, parametrc, condtonal, stochastc, and evolutonary. Tmed symbols of the alphabet of an L-system depend on tme. They have a local age, and they can only be replaced by a rule, f they have reached ther maxmal age. Ths tme dependency s necessary to model contnuous anmaton. The L-system s also parametrc and condtonal. The parameters allow the parent symbols (left sde of producton rule) to pass ther parameters to ther chldren (rght sde of rule) and to modfy them. Wth each rule we can assocate a condton that trggers the rule f t s true. In general, condtons depend on the parameters of the parent symbol and some envronment functons. Symbols can also have growth or evoluton functons (attrbutes) that determne the growth or behavour of the symbol durng ther exstence. Such functons can descrbe the geometrc growth of an object, or the movement of a camera n space, for nstance. Growth or evoluton functons can depend on the local age, the global tme, the symbol parameters, and other functons. In a real-tme L-system based applcaton, at each frame (tme step), the symbolc object s frst terated and then nterpreted. At an teraton step, each symbol of the symbolc object that has reached ts maxmal age and whose rule s trggered by a Boolean expresson (the condton), s replaced by the rght sde of the rule. Otherwse, only ts local age s ncreased by the tme step of the anmaton loop. At an nterpretaton step the symbolc object s nterpreted from left to rght. The symbolc object can be consdered as the actual turtle program, whch bulds and controls the vrtual envronment of the current frame. Each symbol of ths program corresponds to a more or less hgh level procedure wth a gven semantcs. Propertes of tmed and parametrc L-systems have been descrbed here for clarty. In the next sectons we propose new features for enhancng L-systems wth physcs. Frst, we descrbe a concept for managng parent-chld relatonshps between relevant parts of a desgn model. Such relatonshps are needed for mutual nteracton of body-parts. Then, we focus on a concept for ntroducng partcular desgn models nto L-systems. Relatonshp Mangement In most tradtonal L-system based anmaton systems the formal graphcs object s represented by a lnear symbol strng (lst), whch corresponds to the turtle program that vsualses somethng when nterpreted. In ths formal or symbolc object no parent-chld relatonshps between relevant symbols are explctly mantaned for further use, such as constrant propagaton. But n tree-lke objects, for example, consstng of rgd rods that are rgdly lnked, and that propagate forces and moments to ther parent elements, parent-chld relatonshps are needed for computng forces and moments n the tree. Therefore we need a mechansm n L-system based applcatons enablng us to control parent-chld relatonshps of certan symbols. One possblty to solve ths problem s to mplement graph rewrtng, where rules and formal objects exst as graph nstances. Here we can perform all algorthms and computatons drectly on these graph nstances where ponters lnk parents and chldren explctly. But there exst many symbol-strng-lke mplementatons of L-system-based applcatons. Therefore, we propose an extenson of the concept to manage parent-chld relatons for such types of applcatons. We need only three new elements, namely: - a parent-chld relatonshp table (reltable) - a functon getidofleftneghbour() - a functon schldof(parent). The parent-chld relatonshp table reltable s a global table mantanng ts state from frame to frame durng anmaton. It contans chld dentfer and parent dentfer felds that express our parent-chld relatonshps. We also ntroduce a functon schldof(parent) that makes an entry nto reltable. The argument of the functon sets the parent dentfer feld of reltable, and the chld dentfer feld s set wth the symbol dentfer of the symbol n whch the functon s called. Ths functon s only called n parameter expressons of symbols that are only evaluated once at a dervaton step, where also the correspondng symbol dentfer s created. We stll need a second functon, whch s called only n parameter expressons, and whch serves to get a symbol dentfer. The proposed functon getidofleftneghbour() returns the symbol dentfer of the th left neghbour of the symbol where the functon has been called. Ths functon enables us to mport symbol dentfers nto the formal parameter space of L-systems. Wth these three new elements rule desgners have now the possblty to defne n a very flexble manner parent-chld relatonshps of relevant symbols. An example s shown n secton three.

4 Integraton of Desgn Models We propose to ntroduce a desgn model by mplementng fve new elements (see Fg.2), namely - a desgn model table (desgnmodeltable) - a desgn model symbol - a functon createdesgnmodelelement() - a desgn model equaton wrter - a desgn model equaton solver. The desgn model table defnes all the attrbutes of the elements of a gven desgn model that are necessary to descrbe ther physcs. The key feld of ths table s the symbol dentfer. be called n parameter expressons of the desgn model symbol. The desgn model equaton wrter s used to output the system of equatons descrbng the physcs of the desgn model. Ths procedure needs the relatonshp table and the desgn model table as nput. It can be called at the end of an nterpretaton step at each frame. The desgn model equaton solver computes the soluton of the actual set of equatons descrbng the state of the on-gong anmaton. As nput t uses reltable and desgnmodeltable. It updates all relevant felds of desgnmodeltable that can be used by symbols for vsualsaton n a subsequent nterpretaton step. Iterator Interpreter 3. EXAMPLE OF A DESIGN MODEL schldof createdesgnmodelelement reltable desgnmodeltable nput nput result Equaton solver nput nput Ths secton llustrates the proposed concept of enhancng tmed and parametrc L-systems by realsng a partcular desgn model, namely the statcs of arbtrary rgd tree constructs wth a fxed root and free chldren (see Fg.3). For any tree structure and appled external, tme dependent forces that are desgned wth rules, the statcs wll be computed automatcally by the applcaton. Moreover, the geometry, the resultng forces and moments at the jonts are dsplayed at each frame. Equaton wrter output leaf Set of equatons Archtecture of extensons Fgure 2 The alphabet of the L-system applcaton has to be extended by one ore several desgn model symbols, whch enable desgners to set certan physcal attrbutes of the correspondng desgn model. These user-defned attrbutes correspond to growth or evoluton functons of the symbols and can depend on tme. To add a new element to desgnmodeltable, we use the functon createdesgnmodelelement(), whch can root Desgn model of arbtrary 3D tree structures wth a fxed root and free leaves Fgure 3 Ths smple example s well suted to llustrate our proposed approach. Frst, we descrbe the physcal model of the desgn model. Then, we focus on the structure of the partcular desgn model table treetab

5 and the algorthms of the equaton solver and the optonal equaton wrter. Fnally, we dscuss an L- system representng a bnary tree. Desgn model of Rgd Trees An element of our tree structure s a rgd rod wth rgd jonts propagatng forces and moments to the root (see Fg.4). G P l +1 Accordng to Fg.4, we need the followng data felds for our tree desgn model table (treetab): symbolid: The dentfer of the symbol massdensty: The mass densty of the symbol M[3]: The moment vector F[3] : The chld force vector G[3] : The external force vector l[3]: The rod unt vector len: The rod length P[3]: The rod poston vector r : The attack pont for gravty x[3] : The attack pont for G P P : P +1 : m : f : F +1 : r x f F +1 poston vector of the root sde part poston vector of the chld sde mass densty of the element Force vector caused by the mass of the element and gravty actng at r Force vector propagated by the chld and actng at l External force actng on the element at G : the poston x computed attrbutes: l = P +1 - P,, r = 0.5* l A prmtve rod element of a tree structure Fgure 4 Eq. 1 shows the force actng on a jont of a rod element n the tree. It contans the sum of the contrbuton of all chldren, the external force, and ts own contrbuton caused by gravty and ts mass. F = f + G + F (1) k k= chldofnode The moment, whch acts on the jont and whch s propagated to the parent, s gven by Eq. 2. M = r f + x l F + G + M k k k= chldofnode k= chldofnode (2) Note that Eq. 1 and Eq. 2 contan the contrbutons of ther chldren that are agan parents of ther chldren, etc. Therefore, when solvng the resultng system of equatons, all nteractons are propagated from the leaves to the root. Ths means, that ths model s physcally correct. The root, for nstance, wll see all the contrbutons of the whole tree elements. As a next step we ntroduce the desgn model symbol called rod. Its evoluton functons determne the attrbutes length, radus, mass-densty, x, external force -, that a desgner can defne. The start poston P of a rod s gven by the current poston of the turtle. Its drecton s determned by the turtle s headng vector. When nterpreted, frst the evoluton functons are evaluated n order to update the correspondng attrbutes n the desgn model table treetab. Then, the rod, the external force, and the computed moment and jont force are drawn. The equaton solver has to compute the forces and moments accordng to Eq. 1 and Eq. 2. Every parent can have multple chldren. Therefore, the forces and moments for the rod elements can be calculated n reverse sense by startng from leaves and gong back to the root. A possble soluton s to mark frst all lnes n the relatonshp table wth the topologcal dstance of the chldren from the root. Then, n a second step we can calculate all the forces and moments startng from the hghest dstances down to the root. The code of Fg.5 accomplshes ths task. It s executed after a complete teraton and nterpretaton step. resetdstancemarks(reltable, 0) maxlength = markwthdstance(reltable) for = maxlength downto 0 for all reltable.chldid wth =reltable.dstancemark calculate force and moment wth all chld contrbutons Pseudo code of the equaton solver Fgure 5 After an nterpretaton step we can also wrte the system of equatons n a text fle for further external processng. Fg.6 llustrates the pseudo code of the equaton wrter. Varables are ndexed by the correspondng symbol-dentfer. Parts of the equatons that can be evaluated are calculated

6 drectly. The algorthm of Fg. 5 outputs the equatons. These equatons can also be solved automatcally by constrant propagaton technques descrbed n [YR99], whch are used n conceptual desgn n engneerng. For our real-tme applcaton, however, t s more effcent to mplement drectly our proposed numercal soluton of Fg.5. for all rod elements el of treetab do { /* computaton of numercal values */ = el.symbolid; gravty = (0,el.massDensty*el.len*el.radus 2 *P*9.91, 0); F1 = gravty + el.g; M1 = el.r gravty + el.x el.g; =, length=, radus=1, massdensty=1) push rot (angle = a) z ( x 0 = x 0-1, =getidofleftneghbour(3), = *b) pop push rot (angle = -a) z ( x 0 = x 0-1, =getidofleftneghbour(7), = *b) pop } /* The output strngs are generated and ntalzed */ Strng F = F_ + + = + F1; Strng M = M_ + + = + M1; /* The chld contrbutons are added as varables */ for all chlds j = chldofparent(reltable, ); F = F + F_ + j; M = M el.r + F_ + j + + M_ + j; } prnt F; prnt M; Pseudo code of the L-system of a bnary tree Fgure 7 Let us now apply the terator on ths L-system and look at the produced symbolc objects. Fg.8 shows two teratons of the axom. At teraton the rght sde of the unque rule replaces the dummy symbol z of the current symbolc object. For each rght-sde symbol a unque dentfer s created that ndexes t n the symbolc object. Code of the equaton wrter that produces a text fle wth the set of equatons descrbng the statcs of a tree structure Fgure 6 L-system of a Bnary Tree The L-system of a bnary tree wthout external forces s gven by the pseudo code of Fg.7. The axom conssts of a germ-lke symbol z wth three parameters that correspond to the maxmal number of teratons, the root dentfer (-1), and the ntal length (5) of a rod element passed by the unque rule to ts rght sde symbols. Note, that the parameter s used to propagate the parent-dentfer to ts chldren. The functon getidofleftneghbour() puts the parent-dentfer nto the parameter space of the L-system. Constants a = 30 /* rotaton angle */ b = 0.8 /* rod length reducton factor */ Axom z =4, =-1, =5) Rule1: z s replaced f (t>maxage) and >0) by rod ( x 0 =schldof( ), =createtreetableelement(), Axom z 1 =5, =-1, =5) Frst teraton rod 2 =schldof(-1), =createtreetableel(), =5, length=5, radus=1, massdensty=1) push 3 rot 4 (angle=30) z 5 ( x 0 =4, =2, =4) pop 6 push 7 rot 8 (angle=-30) z 9 =4, =2, =4) pop 10 Second teraton rod 2 =schldof(-1), =createtreetableel(), =5, length=5, radus=1, massdensty=1) push 3 rot 4 (angle=30) rod 11 =schldof(-1), =createtreetableel(), =4, length=5, radus=1, massdensty=1) push 12 rot 13 (angle=30) z 14 =3, =11, =3.2) pop 15 push 16 rot 17 (angle=-30) z 18 =3, =11, =3.2) pop 19 pop 6 push 7 rot 8 (angle=-30) rod 20 =schldof(-1), =createtreetableel(), =4, length=5, radus=1, massdensty=1) push 21 rot 22 (angle=30) z 23 =3, =20, =3.2) pop 24 push 25 rot 26 (angle=-30) z 27 =3, =20, =3.2) pop 28 pop 10 The symbolc objects of two teratons of the axom Fgure 8

7 coordnate system, but also the force and moment vectors at ts root-sde jont. The symbol can access the computed resultng force and moment n the desgn model table (treetab) by ts unque symbol dentfer. Please note that ths desgn model only treats the statcs of rgd tree structures, and therefore, the effects of forces and moments cannot be vsualsed as deformatons or movements. A desgn model of the dynamcs of tree structures s much more complex. Its mplementaton s left to future work. The forces and moments caused by gravty are drawn as vectors startng from the jonts of the rod elements. Vsualzaton of the formal object, the forces, and the moments after four teratons of the axom Fgure 9 A spral composed of rod elements Fgure 11 A non-flat bnary tree wth vsualzed forces and moments Fgure 10 Accordng to the rule and the parameter x 0 of the symbol z the L-system s terated four tmes. Fg. 9 llustrates the vsualsed tree object after four teratons. The nterpreter of the L-system applcaton (Fgure 1) s responsble for the vsualsaton of the symbolc object (Fgure 8) at a gven anmaton tme. It nterprets each symbol accordng to ts semantcs and ts parameters. The rod symbol draws not only a cylnder n the turtle s The Fg.10 to 12 show other examples of the statc tree desgn model wth one fxed root and free leaves. Each fgure s defned by an approprate L- system, contanng rod symbols of the statc-treedesgn-model. A detaled descrpton of all of these L-systems s beyond the scope of ths paper. The examples serve only to llustrate the fact that, once a desgn model s mplemented, t s possble to nvestgate, vsualse, and analyse a multtude of nstances of a gven desgn model. In all fgures gravty s actng n vertcal drecton. Only n Fg.12 there are external forces that are vsualsed as thck vectors. They are attackng at each rod element of the quad tree. As a last llustraton of the descrptve power of L- systems, Fgure 13 shows the L-system of the sprallke object of Fgure 11 consstng of 29 lnked rod elements. The only rule Rule1 adds at each tme unt a rod element and adjusts the turtle by two rotatons n order to get a spral-lke shape of the fnal object.

8 llustrate our proposed concept wth a smple desgn model dealng wth the statcs of arbtrary trees. Future work wll focus on the ntegraton of more desgn models such as the dynamcs of trees, partcle systems, and most especally, desgn models contanng general parent-chld relatonshps wth cycles as they are found n most techncal constructons. REFERENCES [D94] Dym C. L., Engneerng Desgn: A Synthess of Vews, Cambrdge Unversty Press, A tree wth four branches at each node. External forces attack at each rod element Axom z =29, =-1, =5) Fgure 12 Rule1: z s replaced f (t>1) and >0) by rod ( x 0 =schldof( ), =createtreetableelement(), =, length=, radus= /9, massdensty=3) rotup (angle =37) ptchdown (angle = 28) z ( x 0 = x 0-1, =getidofleftneghbour(3), = * 0.9) L-system of the spral-lke object that s llustrated n Fgure 11. Ths object conssts of 29 lnked rod elements. The spral-lke shape s obtaned by the two turtle rotatons rotup and ptchdown after each rod element. 4. CONCLUSIONS Fgure 13 It s a vsonary fact that the tght couplng of physcs- and engneerng-based smulaton tasks wth computer-graphcs vsualsaton procedures bears a hgh potental n the feld of advanced conceptual system desgn and development. The presented work ams n ths drecton. It shows how to extend realtme L-system based applcatons n such a way, that they produce not only geometry but also the correspondng system of equatons that are solved automatcally for a gven target applcaton. We [FRS96]Fertg K. W., Reddy Y. S., Smth D. E., Constrant Management Methodology for Conceptual Desgn Tradeoff Studes, Proceedngs of the 1996 ASME Desgn Engneerng Techncal Conference and Computers n Engneerng Conference, August 18-22, Irvne, Calforna, USA, August [NT99] H. Noser, D.Thalmann, A Rule-Based Interactve Behavoral Anmaton System for Humanods, IEEE Transactons on Vsualzaton and Computer Graphcs, Vol. 5, No. 4, October-December [PHM93] P. Prusnkewcz, M.S. Hammel, E. Mjolsness, Anmaton of Plant Development, Computer Graphcs Proceedngs, SIGGRAPH 93, Annual Conference Seres, ACM Press, pp. 351, [PJM94] P. Prusnkewcz, M. James, R. Mech, Synthetc Topary, SIGGRAPH 94, Computer Graphcs Proceedngs, Annual Conference Seres, pp , [PL90] P. Prusnkewcz, A. Lndenmayer, The Algorthmc Beauty of Plants, Sprnger Verlag, [RN00] S. Rudolph, H. Noser, On Engneerng Desgn Generaton wth XML-Based Knowledge-Enhanced Grammars, Proceedngs IFIP WG5.2 Workshop on Knowledge Intensve CAD (KIC-4), Parma, Italy, May 22-24, [YR99] Yusan, H. and Rudolph, S., A Study of Constrant Management Integraton nto the Conceptual Desgn Phase, Proceedngs 25th Desgn Automaton Conference, Las Vegas, NV, September 12-15, 1999.