USING PHYSICAL MODELS IS NECESSARY TO GUARANTEE STABILE ANALOG HAPTIC FEEDBACK FOR ANY USER AND HAPTIC DEVICE

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1 USING PHYSICAL MODELS IS NECESSARY TO GUARANTEE STABILE ANALOG HAPTIC FEEDBACK FOR ANY USER AND HAPTIC DEVICE Edgar Berdahl Association pour la Création et la Recherche sur les Outils d Expression (ACROE) and the Center for Coputer Research in Music and Acoustics (CCRMA) Edgar.Berdahl@iag.fr Jean-Loup Florens Association pour la Création et la Recherche sur les Outils d Expression (ACROE) and ICA Laboratory Grenoble Institute of Techn., France Jean-Loup.Florens@iag.fr Claude Cadoz Association pour la Création et la Recherche sur les Outils d Expression (ACROE) and ICA Laboratory Grenoble Institute of Tech., France Claude.Cadoz@iag.fr ABSTRACT It ight be easy to iagine that physical odels only represent a sall portion of the universe of appropriate force feedback controllers for haptic new edia; however, we argue the contrary in this work, in which we apply creative physical odel design to re-exaine the science of feedback stability. For exaple, in an idealized analog haptic feedback control syste, if the feedback corresponds to a passive physical odel, then the haptic control syste is guaranteed to be stable, as we show. Furtherore, we argue that it is in fact necessary that the feedback corresponds to a passive physical odel. Otherwise, there exists a passive userhaptic device transfer function that can drive the feedback control syste unstable. To siplify the atheatics, we ake several assuptions, which we discuss throughout the paper and reexaine in an appendix. The work iplies that besides all of the known advantages of physical odels, we can argue that we should eploy only the for designing haptic force feedback. For exaple, even though granular synthesis has traditionally been ipleented using signal odeling ethods, we argue that physical odeling should still be eployed when controlling granular synthesis with a haptic force-feedback device. 1.1 Physical Modeling 1. INTRODUCTION In the field of sound and usic coputing, there is already a strong history of physical odeling. The ost basic physical odeling approach is to study the physics of a usical instruent, and then to siulate the physical equations in a coputer to synthesize sound [1, 2, 3]. However, besides erely iitating pre-existing usical instruents, new virtual instruents can be designed with a coputer by siulating the acoustics of hypothetical situations [4], creating a etaphorisation of real instruents. Of partic- Copyright: c 2011 Edgar Berdahl et al. This is an open-access article distributed under the ters of the Creative Coons Attribution 3.0 Unported License, which perits unrestricted use, distribution, and reproduction in any ediu, provided the original author and source are credited. ular iportance is also that sounds generated using physical odels tend to be physically plausible, enhancing the listener s percept due to failiarity [5, 6]. Physical odels can also be eployed for real-tie interaction. Here the perceptual advantages can be augented by the apparent physical reality of the siulation. For exaple, when interacting with a virtual acoustical object, if the user changes the interaction point and the sound changes appropriately, the iersiveness of the user s experience is enhanced, as well as the quality of the generated sound. This property is not iediately offered by techniques such as sapling, unless the usical instruent s sound is sapled at all the possible interaction points and typically also at any different excitation velocities, which can require recording and storing large aounts of data. By eploying appropriate environents for generating large-scale physical odels, coposers can even create entire pieces using the physical odeling paradig. For instance, initial conditions for ass trajectories can control the evolution of a piece, or coplex inner siulated dynaics can also control tibres, notes, phrases, and even whole oveents [7]. 1.2 Haptic Force-Feedback Interaction According to the Ergotic Function On a philosophical level, Claude Cadoz already defined three functions according to which a user can interact with an environent (physical or virtual). The first function is the episteic function, which pertains priarily to knowing, for which a user can use the eyes, ears, or kinaesthetic and tactile touch receptors. The second function is the seiotic function, which users eploy for transitting sybolic inforation by way of the voice and body language. In contrast, when a user exchanges significant echanical energy with the environent by way of gesticulating, he or she uses the third, ergotic function for interaction [8]. For instance, eploying a tool to defor an object or ove it is ergotic. Bowing a string or playing a dru is also ergotic. In ergotic interaction, the user not only infors and transfors the world, but the world also infors and transfors the user. This is in soe sense a consequence of Newton s third law: for every force, there is an equal and opposite reaction force.

2 Velocity V(s) User K(s) Haptic Feedback G(s) K0 Controller + U(s) Force F(s) Figure 2. User G(s) connected with haptic controller K 0 K(s) in feedback. Figure 1. Handholdinghapticdevice. The ergotic function can be substituted by neither the episteic nor seiotic function. In total when the ergotic, seiotic, and episteic functions are siulated, it is possible for a user to ultiately engage in instruental interaction using a haptic device controlled by a physical odel [9]. In the reainder of the paper, we will argue that we should eploy physical odels for prograing haptic force-feedback controllers, effectively iplying also that the ergotic function and instruental interaction should be siulated using physical odels. We begin with a ore foral discussion of feedback control. 2. FEEDBACK CONTROL Feedback systes with active eleents have the potential to becoe unstable. For exaple, acoustic feedback fro aloudspeakerintotheicrophoneofapublicaddress(pa) syste can cause the PA syste to becoe unstable. Typically howling sets in, where one or a few sinusoids steadily increase in volue until the PA syste cannot becoe any louder or is reconfigured. Howling instability is unpleasant for listeners and should be avoided. Force feedback haptic devices, such as the one shown in Figure 1, can siilarly becoe unstable. A circuit calculates a feedback force as a function of the orientation of the device (and its history), and the feedback force is exerted on the device. If the haptic feedback control syste becoes unstable, then the haptic device can begin to ove about erratically [10]. The haptic device can be daaged, other objects in the vicinity of the device can be daaged, and if the device is particularly strong, the user could even be injured. While instability can be interesting for art, it is clearly iportant to be able to design haptic feedback systes that are guaranteed to be stable. For siplicity of the atheatics and analysis, we initially assue zero feedback control delay, infinite control bandwidth, tie-invariance, linearity, and zero initial conditions to arrive at a practical result, even under real conditions such as digital haptic feedback control (see Appendices A.1-A.3 and B for ore discussion of the assuptions). Consequently, the analog feedback control syste can be represented in the Laplace s-doain. For a tie-doain function g(t), theright-sidedlaplacetransforis G(s) =L {g(t)} = 0 g(t)e st dt. (1) Ablockdiagraforthefeedbackcontrolsysteisshown in Figure 2. V (s) represents the velocity of the user s hand coupled to the haptic device, 1 and F (s) represents the force exerted on the haptic device. In the absence of additional external forces, F (s) = U(s), wheretheinus sign is used to ephasize that the syste is designed to operate using negative feedback. K(s) represents the haptic feedback control filter, where the scalar loop gain K 0 has been separated out (see Figure 2). Because we eploy a negative feedback configuration, we take the gain K 0 0. (2) 3. USER-DEVICE TRANSFER FUNCTION G(s) Since the user s hand and the haptic device are physical devices, they can always be represented using a physical odel G(s) = V (s)/f (s). Furtherore, since we are considering the linear, tie-invariant case, the user s hand is stationary and holding the end of the haptic device with aconstantgrip. Thereissoefrictionatallfrequencies, eaning that we assue the hand coupled to the haptic device can only dissipate energy, never create it. 2 Consequently, in the absence of feedback control, the force F (s) and velocity V (s) of the haptic device will never be far enough out of phase with one another to create energy. Matheatically, this iplies that G(s) < 90 (3) s=j2πf for all frequencies f in Hz [12]. (Matheatically speaking, G(s) is strictly positive real. For ore inforation, consult Appendix D.) 1 For convenience, we eploy velocities rather than positions. With the zero initial conditions assuption, one can easily convert between velocity and position by integrating and in the other direction by differentiating. 2 Technically speaking, it could be possible for a user to intentionally destabilize soe passive physical odels by continually addingenergyat low frequencies; however, users see usually to be sensible enough not to do this [11].

3 4. SUFFICIENCY OF PHYSICAL MODELS FOR HAPTIC FEEDBACK CONTROL For the oent, assue that the controller K(s) is deterined using a passive physical odel. That is, the odel consists of passive eleents and no energy sources. For exaple, the odel ight consist only of asses, springs, and viscous dapers (or equivalently capacitors, inductors, and resistors) all with non-negative coefficients. Then by an analogous arguent to the one in the prior section, we have that K(s) 90 (4) s=j2πf g k g Figure 3. Siplifiedphysicalodelofuser-device. R g for all f since K(s) is positive real (see Appendix D) [12]. Note that we allow frequencies at which there is zero daping this could for exaple happen if there were no dapers/resistors and would result in an angle of 90 or 90. Next, we show that the net control syste is guaranteed to be stable; however, to do so, we need to first introduce a criterion for deterining the stability of the control syste. 4.1 Revised Bode Stability Criterion Since neither K(s) nor G(s) has any unstable poles, we can eploy the Revised Bode Stability Criterion to state that the feedback syste is stable if no candidate unstable frequency f u exists for which: K 0 K(s)G(s) 1 (5) s=j2πfu and ( K 0 K(s)G(s) ) s=j2πfu = 180 n(360 ), (6) for any integer n [13]. In other words, the syste is stable if there is no frequency at which a sinusoid traveling all the way around the loop could interfere perfectly constructively with itself (due to (6)) with the agnitude of that sinusoid increasing with every loop (due to (5)). 4.2 Proof Of Stability Since (2), (3) and(4) hold,there is no frequency f for which (6) can hold. Thus,we have showed that the haptic feedback control syste ust be stable if controlled by aphysicalodel,foranypassiveuser-devicetransferfunction G(s). Roughlyspeaking,thestabilityisindependent of the choice of haptic device and what the user is doing. 4.3 Unconditional Stability Note that the stability is also independent of the agnitude of K 0.Inotherwords,thecontrolgaincanbearbitrarily large! This rearkable property is known as unconditional stability [14, 15]. It indicates that under our ideal assuptions, the values of the physical odel do not atter the haptic control syste is guaranteed always stable. 5. EXAMPLE We now illustrate the proof of stability using a concrete exaple of a user touching a virtual resonator. Figure 4. Magnitudeandphaseofuser-deviceG(s) s=jω. 5.1 User-Device G(s) Although the user can exert forces at will on the resonator, we consider these forces as inputs to the control syste and not as part of the feedback loop, so we do not need to odel the when exaining the stability of the feedback loop. Hence, we provide a siple physical odel of a user coupled to a haptic device in Figure 3. Theass g represents the ass of the user s hand coupled to the haptic device. The user s hand in conjunction with the haptic device presents a stiffness of k g and viscous daping R g.much ore coplex odels could be eployed at this point, but it is not necessary for the illustrative purposes of this paper [16]. We can use the odel to find G(s): G(s) = V (s) F (s) = s g s 2. (7) + R g s + k g The odel values could vary significantly, so for exaple we eploy approxiate paraeters obtained by averaging results fro a subject test, in which subjects gripped ahapticdevicewithagripforceofabout9n[11]. In other words, we chose g = 143 g, R g = 5 N/(/s), and k g =0.538 N/. The phase response of the user coupled to the haptic device is shown in Figure 4. Asrequired by (3), the phase response lies within the range ( ) as illustrated in Figure 4, botto. 5.2 Controller The physical odel for the controller is shown in Figure 5. Theusicalresonatorhasass v =4ganddaping coefficient R v =0.01 N/(/s), setting the exponential decay tie constant to 0.8 sec. To ake the resonance frequency approxiately 300Hz, we choose stiffness k v =14.2N/. To liit the force that the haptic

4 k v v Figure 5. Physical odel eployed to derive controller K 0 K(s). k c R v set of passive, linear physical odels with collocated input and output is the sae as the set of passive, linear transfer functions K(s) (see [12]andAppendixD). Indeed both of these sets share the sae phase relationship described by (4). Otherwise if K(s) does not correspond to a passive linear physical odel, there exists a passive user-device G(s) for which the feedback syste can be driven unstable in other words, a passive user and haptic device could be found for which the haptic control syste would be unstable. The proof of necessity is too long to be included in this conference paper without eliinating the exaples [17]. Nevertheless in suary, it is a proof by construction that is analogous to choosing G(s) such that (6) holdsatsoe frequency f u,andthenincreasingthescalargainofg(s) until (5)alsoholds MAIN RESULT Thus we have arrived at what we consider to be a rather rearkable result: Figure 6. Magnitude and phase of controller K 0 K(s) s=jω. device ust display, an additional spring is incorporated into the odel k c =3N/. Solving the equations of otion and converting to the Laplace doain, we arrive at the following, fro which it can be seen that k c plays a role siilar to the loop gain: K 0 K(s) = U(s) V (s) = k c s v s 2 + R v s + k v v s 2 + R v s +(k v + k c ). (8) The agnitude response of the controller is shown in Figure 6 (top), which shows that the resonance frequency is increased slightly to 330Hz due to the presence of k c.there is also an anti-resonance frequency still at approxiately 300Hz. However, because the controller represents a physical odel, its phase response still lies with in the range [ ]asspecifiedby(4), even though it coes close to its allowable boundaries in Figure 6 (botto). Hence, as proved in Section 4.2, no candidate unstable frequency f u exists satisfying (5) and(6), so the control syste is guaranteed stable. This will hold for any passive physical odel eployed to specify the controller K 0 K(s), whichisconvenientforusicalpractice. 6. NECESSITY OF PHYSICAL MODELS FOR HAPTIC FEEDBACK CONTROL Aong the sound and usic coputing counity, it appears not to be known that in the following sense, it is in fact necessary forthehapticcontroller K(s) tocorrespond to a passive physical odel. The reason for this is that the if stable feedback control of a haptic device is desired for applications in new edia, then we argue that designers should not start by siply eploying any arbitrary feedback, rather they should design the feedback using physical odels. This has further iplications particularly in sound and usic coputing. When eploying a haptic device to control traditionally non-physical odeling sound synthesis engines, a physical odeling approach should nonetheless be eployed. This is one reason why ACROE designed the CORDIS-ANIMA physical odeling language that incorporates passive physical odeling eleents such as the ass, spring, friction, conditional link, etc. for siulating the ergotic function and enabling instruental interaction [18]. For instance, if one were to ipleent haptic feedback control of granular synthesis [3], a good approach would be to odel the grains as sall asses flowing along a river. An independent external force would cause each ass to vibrate according to its own audio grain signal. Then the user could dip into the river using the haptic device via a conditional link, and the output audio signal would be generated by easuring the force exerted upon the haptic device. We would recoend this approach not only because of our positive experiences with physical odels, but also because of the arguents in this paper. 3 The proof involves counting the nuber of possible clockwise loops around the -1 point of the Nyquist plot of K 0 K(s)G(s). Onekeyrealization is that any counterclockwise loop would iply that either K(s) or G(s) were open-loop unstable, which would violate the assuptions, so there cannot be any counterclockwise loops. It is also necessary to observe that 1) a transfer function can never have a negative nuber of poles or zeros and 2) offsetting a transfer function by a coplex constant does not change its poles.

5 8. CONCLUSION Physical odeling for new edia can indeed be a creative activity, and now having re-discovered this approach through a scientific stability analysis, we hope to have provided new insight for design of new edia. We have attepted to present soe results fro the echanical engineering literature [17] in a way that is accessible to the sound and usic coputing counity. In re-presenting this work, we have gathered new perspective on the stability of feedback control systes and re-affired our enthusias for physical odels. Acknowledgents We are grateful to the reviewers for their insightful coents, which we have attepted to integrate to ake the paper as accessible as possible. Reviewers and readers interested in designing digital feedback controllers for plants with a specified delay ay wish to consult work by Florens et al [19]. Finally, Edgar Berdahl would like to thank J. Edward Colgate for personal correspondence and Günter Nieeyer for inforing hi about the prior work [17]. A. NOTES ON ASSUMPTIONS We now argue that the general results are still practically valid in light of the assuptions we ade in Section 2. A.1 User Is Tie-Varying In practice, G(s) changes with tie according to what the user is doing. For exaple, while copleting certain tasks, the user changes the obility of his or her hand [20, 21, 11]. However, this is in fact the point of the present paper inorder toensure stableperforance for any arbitrary userdevice obility G(s),it is necessary that K(s) correspond to a passive physical odel. Hence, we argue that K(s) ay as well be designed using a passive physical odel to guarantee stable perforance [18]. A.2 Non-Ideal Feedback Characteristics In practice, all feedback control systes have liited bandwidth. In addition, digital control systes exhibit additional delay in the control loop. Consequently, real controllers cannot perfor significant feedback control at especially high frequencies. However, in practice, one observes that the siple theory in this paper predicts relevant aspects of practical perforance, as long as one inserts a viscoelastic (optionally nonlinear) eleent in between the haptic device control point and the physical odel, such as k c (see Figure 5). This reduces the agnitude of the feedback control at high frequencies where practical digital control delay can be particularly probleatic (see Appendix B) [10, 16]. It could also be argued that a digitized version of an ideally passive physical odel ay no longer be forally passive. However, we argue that the physical odel should siply be discretized appropriately so that it causes an input-output delay of precisely one digital tie unit. In practice then, this tie unit is aligned with the converter sapling and results in no additional, unnecessary delay that could further affect the passivity [22, 23]. We relate the discussion here also to teleoperation, in which a aster haptic device and a slave haptic device are linked together using force-feedback. Signal transission delay between two separate locations can cause even ore significant delay than digital sapling. In this case, the controller cannot for a good odel of a siple daper and spring to link the devices together. However, this delay can be cleverly absorbed into the controller odel by incorporating a vibrating string (or equivalently an electrical transission line) into the controller odel [24]. Again, we discover a solution based on physical odels! A.3 User Is Nonlinear In practice, a real user is nonlinear. However, the user is also dissipative. If the haptic feedback controller is passive and corresponds to a physical odel, then even if the physical odel is nonlinear, by the conservation of energy, the energy in the feedback control syste ust dissipate over tie in the absence of external excitation. Hence, although the necessity of the nonlinear case is apparently unproven, 4 conservation of energy proves sufficiency, andit is also very practical to design nonlinear haptic feedback controllers using nonlinear physical odels [18]. A.4 Unstable Perforance Could Be Desirable In Soe Situations In soe situations, artists ay desire to create control systes that are unstable. In fact, the E-Bow and Sustainiac are successful coercial products that drive vibrating guitar strings unstable in a controllable anner [25, 26]. Siilarly, bowed strings, any wind instruents, and soe dru roll techniques incorporate self-oscillations that have becoe accepted as sounding usical. Even the unstable Haptic Dru enables a perforer to play arbitrarily coplex dru rolls or dru rolls at superhuanly fast speeds [27]. Hence, there are soe nonlinear situations where physical odels will not be necessary for ipleentation, but they nevertheless see to be sufficient given the very wide range of physical phenoena that could be odeled. Unstable behavior can be created using external energy source eleents in physical odeling or negative dapers, or siilar effects can be obtained by setting initial energetic conditions for objects. Indeed creativity causes us to rethink the science of physical odeling, so possible future work could soeday involve studying necessity and sufficiency proofs for eploying (nonpassive) physical odels to ipleent unstable haptic usical siulations. A.5 Transparency of Haptic Rendering In this paper, we have considered only the stability of the control syste according to classical passivity theory [17]. However, we have not considered how transparently the 4 Personal correspondence with Ed Colgate on Jan. 17, 2011

6 physical odel is presented to the user through the haptic feedback control syste. In the classical representation, iproving transparency (i.e. accuracy) requires increasing control gains, which can haper the stability of digital control systes (see Section B). For this reason, Florens et al. have introduced a new ethod for deriving haptic feedback control systes, which consider the stability and transparency concurrently [19]. The result is a whole new paradig for deriving haptic feedback controllers. The ethod involves odeling the coupling of the user-device to the physical odel (called a teporary hybrid syste or THS) andadjustingodelparaeterstoachieveoptiu dynaics [19]. We are actively carrying out further research in this doain. B. DIGITAL DELAY In practice, it is usually ore practical to eploy digital feedback instead of analog feedback, especially because coputers are now so widely available and inexpensive. However, digital feedback control always causes delay in the control loop, which is due to Figure 7. Magnitude and phase of digital controller K 0 K(s) s=jω. analog-to-digital conversion (ADC), coputation tie, possible additional delay due to operating syste, interrupt, and bus echaniss on the coputer, and digital-to-analog conversion (DAC). F ext Although the delay is always longer than half of one sapling interval T,thisisneverthelessaconvenientapproxiation, assuing a conventional ipleentation of the control loop eleents [28, 23]. 5 Hence, (8) becoesthe following: K 0 K(s) = k C s v s 2 + R v s + k v e st/2 v s 2 + R v s +(k v + k c ). (9) The sapling interval T has no effect on the agnitude of (9)(seeFigure7,top).However,thephaseresponsefor T =1s is shown in the thinner line of Figure 7 (botto). There is a linear trend to further and further negative phases, which causes the phase response to dip beneath the allowable liit -90.ThephaseresponseforT =0.1 s is shown in the thicker line of Figure 7 (botto). It reains uch closer to the liit, but for sufficiently high frequencies falls outside of the allowable range, preventing the digital controller fro perfectly calculating feedback equivalent to a delayless, analog physical odel. Nevertheless, it turns out that both for T = 1 s and T = 0.1 s, the control syste is still stable for these paraeters. However, given the digital control delay, it is possible to drive the syste unstable by increasing k c, which is analogous to eliinating the stabilizing copliant spring k c and attepting to bind the haptic device 5 Converters ipleented using siga-delta odulation are usually not fast enough, so for low-latency feedback control successive approxiation ADCs and resistor-ladder DACs are often used. Figure 8. Physical analog odel for granular synthesis using ten trapped asses. directly to the resonator itself. However, this would be contrary to our approach of prooting stability by inserting the copliant eleent k c at the interaction point. Indeed, for a ore precise arguent but specific to a less interesting scenario for sound and usic coputing, the reader should read the work by Diolati et al. [16]. Typically the shorter T can be ade, the further k c can be increased. C. REAL-TIME USER INTERACTION WITH MORE COMPLEX EXAMPLE To deonstrate the viability of the technology, even for digital feedback control, we briefly present an exaple with real-tie user interaction. A physical odel for a kind of granular synthesis is shown in Figure 8, inwhichten grain asses fly back and forth vertically between a echanical ground (below) lined with linear contact springs and squared-nonlinearity contact springs along a rigid bar (top) coupled to the user s hand by the haptic device. Despite the specificness of this odel, any other physical odel scenarios could be eployed to ipleent granular-

7 Soe further properties of positive real and strictly positive real functions are [30]: 1. 1/ K(s) is positive real. 2. 1/ G(s) is strictly positive real. Figure 9. Snapshotfrovisualoutputofdeonstration. type synthesis. In the present exaple, an external force acts on each ass, causing it to vibrate according to input sounds. The force exerted on the haptic device is easured, highpass filtered (not shown), and passed to the audio output, as represented by the loudspeaker scheatic sybol in Figure 8, topright. For such coplicated odels, visual feedback is also helpful. Hence, we incorporated visual feedback in a deonstration video. 6 AsnapshotofthevideoisshowninFigure 9. Ratherthanadjustingtheaplitude,frequency,density, grain length, and siilar signal paraeters, as is typical in granular synthesis [3], these sonic characteristics are adjustable by physical eans such as the position of the haptic device, force applied to the haptic device, the stiffness and daping of the hand, etc. D. POSITIVE REAL FUNCTIONS For the atheatically inded, we present soe inforation on positive real functions for describing atheatically passive systes. It underscores their equivalence to passive physical odels. Positive real functions were introduced in 1931 for synthesizing transfer functions corresponding to electrical analog circuits [12]. Since then, a rational function K(s) has usually been defined to be positive real if and only if K(s) is real when s is real, and Re{ K(s)} 0 for all s such that Re{s} 0. and siilarly, a rational function G(s) has usually been defined to be strictly positive real if G(s + ɛ) is positive real for all real ɛ>0 [29]. However, for our purposes it is uch ore convenient to use the following equivalent definitions in ters of the angle along the frequency axis. We define the rational function K(s) to be positive real if and only if K(j2πf) 90 for all frequencies f, and siilarly the rational function G(s) is strictly positive real if and only if G(j2πf) < 90 for all frequencies f [29]. 6 eberdahl/copmusic/gcg.4v 3. If K(s) represents either the driving point ipedance or driving point obility of a syste, eaning that the sensor and actuator ust be collocated, then the syste is passive as seen fro the driving point. In other words, if K(s) is positive real, then it corresponds to a passive, linear physical odel, and vice versa. 4. If G(s) represents either the driving point ipedance or driving point obility of a syste, eaning that the sensor and actuator ust be collocated, then the syste is dissipative as seen fro the driving point. In other words, if G(s) is strictly positive real, then it corresponds to a dissipative, linear physical odel, and vice versa. 5. K(s) and G(s) are stable. 6. K(s) and G(s) are iniu phase. 7. The relative degrees of K(s) and G(s) ust be less than No atter what causal tie-doain function f(t) is used to excite the driving point, the velocity response v(t) will be such that 0 f(t)v(t)dt 0. E. REFERENCES [1] C. Cadoz, A. Luciani, and J.-L. Florens, Synthèse usicale par siulation des écanises instruentaux, Revue d acouqistique, vol.59,pp ,1981. [2] J. Sith III, Synthesis of bowed strings, in Proceedings of the International Coputer Music Conference, Venice, Italy, [3] C. Roads, The Coputer Music Tutorial. Cabridge, MA: MIT Press, February [4] N. Castagne and C. Cadoz, Creating usic by eans of physical thinking : The usician oriented Genesis environent, in Proc. 5th Internat l Conference on Digital Audio Effects, Haburg,Gerany,Sept.2002, pp [5] N. Castagné andc.cadoz, Agoals-basedreview of physical odeling, in Proceedings of the International Coputer Music Conference, Barcelona,Spain, Sept [6] I. Peretz, D. Gaudreau, and A.-M. Bonnel, Exposure effects on usic preference and recognition, Meory and Cognition, vol.26,no.5,pp ,1998.

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