Page  00000363 A PHYSICALLY INTUITIVE HAPTIC DRUMSTICK Edgar Berdahl Bill Verplank Julius O. Smith III Giinter Niemeyer Stanford University Center for Computer Research in Music and Acoustics ABSTRACT since neurosci objects is gov We motivate and discuss the design of a physically intu- action models itive haptic drumstick. A new musical instrument is phys- in learning mu ically intuitive if the physics of haptic interaction are simi- verification. I lar to those supported by a traditional musical instrument. new musical i We suggest that physically intuitive new musical instru- fer by virtue o ments may help performers transfer motor skills from familiar, traditional musical instruments. Both actively controlled acoustic instruments and many haptic instruments are physically intuitive. We design a physically intuitive 1.1. Actively haptic drumstick. Simple models of drumstick dynamics and drumstick/membrane collisions are explained and An actively co implemented on a high-resolution haptic device. Next, particular trad we create a new musical instrument by altering the hap- structural acoi tic drumstick dynamics in a physically intuitive manner. that the instru We focus on drum rolls. In particular, we alter the haptic the physical c drumstick dynamics to assist performers in playing single- instrument, th handed drum rolls. Finally, we analyze the stability of the tion with the in altered system dynamics using a Poincar6 map. the instrument 1. INTRODUCTION 1.2. Physical Consider the differences between a professional new musical instrument performance and a professional violin per- Haptic or forc formance. The violinist has had years if not a lifetime to in use since th master the motor skills used to play the violin. In con- can be progra trast, the new musical instrument performer has perhaps pending on th learned foreign motor skills for a new interface under time limits of the h constraints. While the learning process undoubtedly pro- tem can be rea vides the new instrument performer with fresh ideas, he bility of hapti or she must largely ignore the vast portions of motor skills ment designei obtained while mastering other less relevant instruments. argues that "p Instrument designers can avoid handicapping new in- it can distract strument performers by leveraging their prior motor skills. humans" may One possible solution involves designing interfaces sup- ing, paralyzin porting physical interactions that are already familiar to new musical i performers. We call such interfaces physically intuitive. curse of progr This design criterion is similar to the common goal in itive haptic m haptics of using a haptic device to accurately simulate the interactions si physics of various real-world interactions. The haptics lit- strument supp erature suggests that motor skills may be transferred from the design of haptic interfaces to real-world situations when the physics begin by desi~ of interaction are accurately modeled [7]. We are more haves similarl interested in the reverse-the transfer of skills from a tra- we alter the ha ditional musical instrument to a new, haptic musical in- tic drumsticki strument. We argue that the reverse should be plausible ing drum rolls Stanford University Telerobotics Laboratory entists believe that human interaction with erned by internal conceptual physical inter[10]. Nevertheless, concrete experiments isical instruments need to be carried out for n the following, we discuss some kinds of nstruments that aim to promote skill transf their physical intuitiveness. Controlled Acoustic Musical Instruments introlled musical instrument is based upon a itional acoustic musical instrument, whose ustics are altered with feedback control so ment sounds very different [3] [2]. Due to onstraints revolving around the traditional e physics of the performer's basic interacistrument remain similar. As a consequence, is physically intuitive. ly Intuitive Haptic Musical Instruments e-feedback musical instruments have been e 1970's if not earlier [4]. Such instruments immed to exert forces on the performer dee performer's position in space. Within the iardware and software, any dynamical syslized. This seemingly limitless programmac instruments provides the musical instruwith considerable freedom. Perry Cook rogrammability is a curse" in this sense as musical instrument designers, and "normal find especially exotic instruments "frustratg, or offensive" [5]. We suggest that the instrument designer may help mitigate the ammability by considering physically intuusical instruments, which support physical milar to those that a traditional musical inorts. In the following sections, we explain a physically intuitive haptic drumstick. We gning a haptic musical instrument that bey to a standard drumstick and drum. Then ptic instrument slightly, creating a new hapnstrument that assists the performer in playwhile still remaining physically intuitive. 363

Page  00000364 SR Drum membrane hand out Figure 1. Drumstick dynamics for z > 0 2. MODEL OF DRUMSTICK DYNAMICS Before designing a physically intuitive haptic drumstick, we need to develop a physical model of a drummer playing a drum roll. Consider the double stroke roll in which the drummer throws a stick at the membrane, allows it to bounce twice, retracts it, and then repeats the action with the other stick. The manner in which the drum membrane throws the stick back toward the drummer helps facilitate both the bouncing and retracting actions. Because we are concerning ourselves primarily with relatively fast drum rolls, we assume that 0 does not change much (see Figure 1). This simplification allows us to linearize the rotation of the drumstick tip, and so we model the vertical motion of the drumstick tip as a bouncing ball with mass m. Given the rotational inertia of the stick and the position where it is held, it is possible to derive the equivalent mass m of the ball and gravity g* such that forces on the ball can be mapped directly to forces on the tip of the drumstick. 2.1. Above The Drum Membrane The hand grasping the stick acts as both a rotational spring and a rotational damper at the butt of the stick (not shown). For simplicity, we linearize these elements and commute them to the drumstick tip, representing them by Khand and Rout (see Figure 1). By changing the grasp of the stick, the drummer can change Khand and Rout. This is known as passive impedance modulation and allows drummers to play drum rolls at rates up to 30Hz, even though the human neuromuscular system has a reaction time of over 100ms [6]. By considering that the drummer may also adjust the rest position zho of the spring, in effect exerting a force on the stick, we may let Zss zho - mg* /Khand and write the equation of motion in the absence of collisions with the drum membrane (i.e. for z > 0): m2 + Routz + Khand(z - zss) = 0 (1) 2.2. Simple Collision Model The physics-based approach to modeling stick/membrane collisions involves the coefficient of restitution (COR) /3. For a collision beginning at any time tin and ending at any time tout, if x(tin) = v0o, then z(tout) = -/3o. On the other hand, in haptics, the standard method for modeling a collision involves a spring-like penalty force implemented I 1Drum Membrane -z Ko ol - -- - - - -j mg* Khand l Rin Figure 2. Drumstick dynamics for z < 0 by a spring with constant Kcoii > Khand. There must also be some damping factor Rin due to losses absorbed by the hand and the collision, so we arrive at the dynamics for z < 0 (see Figure 2): m2 + RinZ + Khand(z - zss) + KcollZ = 0 (2) The term Zss causes the COR /3 to become weakly dependent on the velocity i(tin) at the beginning of a collision. However, since the collision is quick, the terms involving Rin and Kcoii dominate. Neglecting the other terms, we may solve the differential equation analytically to arrive at -Rin - /3 exp /4mKcoi - Rin (3) 2.3. Upward Soft Collisions Noting the similarity between (1) and (2), we suspect that we may also determine a COR a for the "soft collision" against the hand in the upward direction. However, since the spring constant Khand is relatively small, the collision takes longer, and so we can no longer neglect Zss. Consequently, a(z(tout)) is strongly dependent on the velocity Z(tout) at the beginning of each "soft collision," so the expression for a is more complicated than the expression for /3. Nevertheless, in simulations of (1) for reasonable system parameters, we have verified that a remains roughly constant for bounces of similar amplitude. 3. HAPTIC DRUMSTICK To construct the haptic drumstick we desire an appropriate haptic display for implementing the equations of motion. Haptic drumsticks have previously been implemented using single DOF haptic displays [1].' We use the three DOF Model T PHANTOM haptic device 2 because it has relatively strong motors-for moderately strong strokes, the motors are able to render stiffnesses comparable to a typical drumhead. The performer holds the drumstick-like 1 http://web.media.mit.edu/~grindlay/FielDrum.html 2 From SensAble Technologies, see http://www.sensable.com. 364

Page  00000365 Figure 3. Graphical display (upper left) and PHANTOM robotic arm (lower right) stylus in one hand as shown in Figure 3. Using standard drumming techniques, he or she can for instance play a double stroke drum roll with the help of a traditional drum and drumstick controlled by the other hand (not shown). The sound synthesis engine consists of a digital waveguide mesh modeling waves propagating in a drum membrane [9]. The force of the drumstick on the spring Kcoll is fed into the nearest node in the mesh. This way, striking the modeled drum at different positions results in different sounds. Note that the haptic forces are implemented by (1) and (2) do not depend on the sound synthesis engine. A video demonstration is available online. 3 4. ALTERING THE DRUMSTICK DYNAMICS There are many ways in which the dynamics could be altered. We consider alterations involving noise and deterministic chaos to be less desirable because they are not physically intuitive. In contrast, stable limit cycles, which are self-sustaining attractive oscillations, describe the behavior of biological oscillators such as the heart. Limit cycles also manifest themselves in bowed strings, vibrating reeds, and drum rolls. Most drummers use two hands to play drum rolls. This limits the types of patterns that they can play. However, by inducing limit cycle behavior, we make it easy for a drummer to single-handedly play a drum roll. In informal tests, we determined that implementing system delay, a hysteretic spring Kcoii, or negative damping Rout make the haptic drumstick more likely to self-oscillate, but not in a particularly physically intuitive way. Indeed, such mechanical elements do not occur readily in nature. A more physically intuitive solution involves forcing the drumstick in the positive z-direction by the pulse h(t) every time the stick enters the simulated membrane, where pulse sstarts Y Figure 4. Example system trajectories and Poincare map enough pulse, the pulse h(t) causes the velocity of the drumstick tip to change by Avpis. Playing a drum roll with these altered dynamics feels somewhat like holding a stick against the spokes of a rotating bicycle wheel. 4.1. Directly Altering The COR Since we can effectively change the velocity of the drumstick after the collision, by choosing Avps = -7(tin)// for some 7 > 0, we can obtain a new COR3 = 3 + 7. With 3 > 1, we can counter the damping due to the hand. 4.2. Applying Pulses With Constant Magnitude Choosing Avpis to be constant presents a superior alternative because the total energy in the system becomes limited. This safety mechanism prevents drummers from inadvertently damaging the robotic arm. Consider that for vibrations at small amplitudes, the effective /3 is large, and that for vibrations at large amplitudes, the effective /3 is approximately equal to 3. In the next section, we show that choosing Avpl, constant leads to stable limit cycle behavior. 5. STABILITY ANALYSIS 5.1. Analysis We analyze the altered dynamics with a Poincare map, which allows the stability of a closed orbit in a continuoustime system to be determined from the stability of a related discrete-time system [8]. Let z = (z, ) e R2 describe the system state, and let the system flow O (zo) describe the current state given an initial state zo at t seconds in the past. Call d the periodic orbit of the system in the phase plane of z, and consider the semi-infinite line E, which is crossed once per cycle by system trajectories (see Figure 4). h(t) = mAvpl, etl/ T (4) and r 2 ms. These force pulses are superimposed with the forces described by (1) and (2). Assuming a quick 3 http://ccrma.stanford.edu/eberdahl/Projects/HapticDrumstick E = {(z,) = e, > 0} (5) For simplicity, we take e > 0 to be arbitrarily small. We assume that zss < 0 so that after leaving the mem 365

Page  00000366 brane, the drumstick will eventually strike it again. To simplify the analysis, we restrict U to be a small enough neighborhood of p on E so that a is approximately constant (see section 2.3). We further assume that the pulse h(t) is short enough that it always ends before the trajectory intersects E. If we then define the map P: U -> E so that for q eGU P(q)= 0,()(q) (6) where T(q) is the time for the orbit O (q) based at q to first return to E, P(.) is a Poincar6 map. Consequently, we may analyze the stability of the closed orbit d by analyzing the stability of the discrete-time system P(vi), where vi is the velocity of the drumstick tip at the end of the ith collision with the drum membrane. vi+1 = P(vi) = a3vi +,Avpls (7) 5.2. Directly Altering The COR For the alteration where Avpis = z(t,)/ = yavi//3, vi+1 = P(vi) = a(/3 + y)vi. (8) If the alteration of the dynamics is configured with -y such that a(/3 + y) = 1, then (7) describes a marginally-stable system, so the drumstick will oscillate at constant amplitude. However, this behavior is not stable-any parameter deviation will lead to a decaying oscillation or a growing oscillation. Since P(.) is a Poincar6 map, the closed orbit d is an unstable limit cycle. Nevertheless, a drummer may stabilize this system by adjusting a in real time using passive impedance modulation. 5.3. Applying Pulses With Constant Magnitude When Avpi, is held constant, (7) describes a discrete-time linear system driven by a constant input. With the exception of the pulse h(t), collisions with the membrane and upward soft collisions against the hand are dissipative. This means that a < 1 and 13 < 1. Then a/3 < 1, so (7) describes a stable discrete-time system. Since P(.) is a Poincar6 map, the closed orbit d is a stable limit cycle. We can also use (7) to easily calculate other properties of the system. Since Avpis is constant, vi will approach the steady-state vi'. vic can be employed to calculate the height to which the tip of the drumstick jumps during one cycle. 6. CONCLUSION In informal tests, we found that the altered dynamics made playing single-handed drum rolls easy for both 1) directly altering the COR and 2) applying pulses with constant magnitude. The main advantage in applying pulses with constant magnitude is that drum roll limit cycles are guaranteed to be stable. However, both types of altered dynamics allow drummers to increase the drum roll rate by increasing Khamd or decreasing zho as in traditional drum roll playing [6]. This means that the new musical instrument is physically intuitive, which we believe should facilitate skill transfer for performers of traditional drums. 7. REFERENCES [1] Bennett, P. D. "Hap-Kit: A Haptic Interface for a Virtual Drum-Kit," The Symposium for Cybernetics Annual Research Projects, University of Reading, June, 2004. [2] Berdahl, E. and J. 0. Smith III, "Methods for Inducing Unusual Dynamics in Acoustic Musical Instruments," IEEE Conference on Control Applications, Singapore, Oct. 1-3, 2007. [3] Besnainou, C. "Transforming the Voice of Musical Instruments by Active Control of the Sound Radiation," International Symposium on Active Noise and Vibration Control, Fort Lauderdale, FL, Dec. 2-4, 1999. [4] Cadoz, C. et al, "Artistic Creation and Computer-Interactive Multisensory Simulation Force Feedback Gesture Transducers," Proceedings of the 2003 Conference on New Interfaces for Musical Expression, pp. 235 -246, Montreal, Canada, 2003. [5] Cook, P. "Principles for Designing Computer Music Controllers," ACM CHI Workshop in New Interfaces for Musical Expression, Seattle, Washington, pp. 1-4, 2001. [6] Hajian, A., R. Howe, et al, "Drum roll: Increasing bandwidth through passive impedance modulation," Proceedings of the IEEE International Conference on Robotics and Automation, Albuquerque, New Mexico, pp. 2294-9, April 20 - 25, 1997. [7] Huang, F., B. Gillespie, et al, "Human Adaptation to Interaction Forces in Visuo-Motor Coordination," IEEE Transactions on Neural Systems and Rehabilitation Engineering, Vol. 14, No. 3, pp. 390-397, September, 2006. [8] Sastry, S. Nonlinear Systems. Springer-Verlag, New York, 1999. [9] Smith III, J. 0. Physical Audio Signal Processing: For Virtual Musical Instruments and Audio Effects, Sept. 2007, [Online] http://ccrma.stanford.edu/~jos/pasp/ [10] Wolpert D. and M. Kawato, "Multiple paired forward and inverse models for motor control,~ Neural Networks, vol.11, no.7-8, pp.131'7-29, Oct.-Nov. 1998. 366