a physically intuitive haptic drumstick

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A Physically Intuitive Haptic Drumstick

Edgar Berdahl, Bill Verplank, and Julius O. Smith IIICenter for Computer Research in Music and Acoustics

—Gunter Niemeyer

Telerobotics Laboratory—

Stanford University—

International Computer Music ConferenceCopenhagen, Denmark

August, 2007

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Outline

Explain how drum rolls are played

Develop a physical model

Implement the model dynamics on a haptic display

Alter the dynamics to make it easier to play drum rolls

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Drum Rolls

◮ Drummers can control drum rolls at rates up to 30Hz.

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Drum Rolls

◮ Drummers can control drum rolls at rates up to 30Hz.◮ The human neuromuscular system has a reaction time of

over 100ms.

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Drum Rolls

◮ Drummers can control drum rolls at rates up to 30Hz.◮ The human neuromuscular system has a reaction time of

over 100ms.◮ How is this possible?!

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Drum Rolls

◮ Drummers can control drum rolls at rates up to 30Hz.◮ The human neuromuscular system has a reaction time of

over 100ms.◮ How is this possible?!

1. The bouncing effect: the drum membrane “throws” the stickback at the drummer.

0 0.5 1 1.5−0.005

0

0.005

0.01

0.015

0.02

0.025

Time [sec]

Dis

plac

emen

t [m

]

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Drum Rolls

◮ Drummers can control drum rolls at rates up to 30Hz.◮ The human neuromuscular system has a reaction time of

over 100ms.◮ How is this possible?!

1. The bouncing effect: the drum membrane “throws” the stickback at the drummer.

2. Impedance modulation: the drummer can alter theimpedance of his or her hand in real time.

K Routhand

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Outline

Explain how drum rolls are played

Develop a physical model

Implement the model dynamics on a haptic display

Alter the dynamics to make it easier to play drum rolls

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Above The Drum Membrane

zmg*

handK RoutDrum membrane

Figure: Drumstick dynamics for z > 0

◮ For the purposes of investigating drum rolls, we model thedrumstick as a bouncing ball with mass m.

◮ The spring and dashpot are commuted to the end.◮ Drummer can adjust rest position zh0 of the spring Khand .◮ Letting zss = zh0 − mg∗/Khand , we have

mz + Rout z + Khand (z − zss) = 0.

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“Inside” The Drum Membrane

◮ At DC, a drumstick being pressed into a drum membranebehaves like a linear spring Kcoll .

◮ There is still damping Rin due to losses in the hand andcollision.

−z

*mg

handK Rin

collKDrum Membrane

Figure: Drumstick dynamics for z < 0

mz + Rinz + Khand (z − zss) + Kcollz = 0

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Coefficient Of Restitution

◮ Let β be the coefficient of restitution (COR) ofdrumstick/membrane collisions.

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Coefficient Of Restitution

◮ Let β be the coefficient of restitution (COR) ofdrumstick/membrane collisions.

◮ For a collision beginning at any time tin and ending at anytime tout , if z(tin) = v0, then z(tout) = −βv0.

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Coefficient Of Restitution

◮ Let β be the coefficient of restitution (COR) ofdrumstick/membrane collisions.

◮ For a collision beginning at any time tin and ending at anytime tout , if z(tin) = v0, then z(tout) = −βv0.

◮ Perfectly elastic collisions: β = 1

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Coefficient Of Restitution

◮ Let β be the coefficient of restitution (COR) ofdrumstick/membrane collisions.

◮ For a collision beginning at any time tin and ending at anytime tout , if z(tin) = v0, then z(tout) = −βv0.

◮ Perfectly elastic collisions: β = 1◮ Inelastic collisions: 0 < β < 1

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Coefficient Of Restitution

◮ Let β be the coefficient of restitution (COR) ofdrumstick/membrane collisions.

◮ For a collision beginning at any time tin and ending at anytime tout , if z(tin) = v0, then z(tout) = −βv0.

◮ Perfectly elastic collisions: β = 1◮ Inelastic collisions: 0 < β < 1◮ β ≈ exp −Rinπ√

4mKcoll−R2in

Summary Of Model Dynamics

◮ Above membrane (z > 0, COR α):mz + Rout z + Khand (z − zss) = 0

Summary Of Model Dynamics

◮ Above membrane (z > 0, COR α):mz + Rout z + Khand (z − zss) = 0

◮ “Inside” membrane (z < 0, COR β):mz + Rinz + Khand (z − zss) + Kcollz = 0

Summary Of Model Dynamics

◮ Above membrane (z > 0, COR α):mz + Rout z + Khand (z − zss) = 0

◮ “Inside” membrane (z < 0, COR β):mz + Rinz + Khand (z − zss) + Kcollz = 0

z

z

.

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Outline

Explain how drum rolls are played

Develop a physical model

Implement the model dynamics on a haptic display

Alter the dynamics to make it easier to play drum rolls

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Haptic Drumstick

◮ We use the three DOF Model T PHANTOM robotic arm1.

1From SensAble Technologies, see http://www.sensable.com.

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Sound Synthesis

Figure: Rectilinear 2D Mesh

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Sound Synthesis

Figure: Rectilinear 2D Mesh

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Outline

Explain how drum rolls are played

Develop a physical model

Implement the model dynamics on a haptic display

Alter the dynamics to make it easier to play drum rolls

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Altering The Drumstick Dynamics

◮ Noise and deterministic chaos are not physically intuitive.

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Altering The Drumstick Dynamics

◮ Noise and deterministic chaos are not physically intuitive.◮ Stable limit cycles, or self-sustaining attractive oscillations,

describe biological oscillations such as the heart beating.

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Altering The Drumstick Dynamics

◮ Noise and deterministic chaos are not physically intuitive.◮ Stable limit cycles, or self-sustaining attractive oscillations,

describe biological oscillations such as the heart beating.◮ Limit cycles also manifest themselves in bowed strings,

vibrating reeds, and drum rolls.

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Altering The Drumstick Dynamics

◮ Noise and deterministic chaos are not physically intuitive.◮ Stable limit cycles, or self-sustaining attractive oscillations,

describe biological oscillations such as the heart beating.◮ Limit cycles also manifest themselves in bowed strings,

vibrating reeds, and drum rolls.◮ Goal: Assist the drummer in playing drum rolls.

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Altering The Drumstick Dynamics

◮ Noise and deterministic chaos are not physically intuitive.◮ Stable limit cycles, or self-sustaining attractive oscillations,

describe biological oscillations such as the heart beating.◮ Limit cycles also manifest themselves in bowed strings,

vibrating reeds, and drum rolls.◮ Goal: Assist the drummer in playing drum rolls.◮ We can help bring about self-oscillations with

1. system delay

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Altering The Drumstick Dynamics

◮ Noise and deterministic chaos are not physically intuitive.◮ Stable limit cycles, or self-sustaining attractive oscillations,

describe biological oscillations such as the heart beating.◮ Limit cycles also manifest themselves in bowed strings,

vibrating reeds, and drum rolls.◮ Goal: Assist the drummer in playing drum rolls.◮ We can help bring about self-oscillations with

1. system delay2. a hysteretic spring Kcoll

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Altering The Drumstick Dynamics

◮ Noise and deterministic chaos are not physically intuitive.◮ Stable limit cycles, or self-sustaining attractive oscillations,

describe biological oscillations such as the heart beating.◮ Limit cycles also manifest themselves in bowed strings,

vibrating reeds, and drum rolls.◮ Goal: Assist the drummer in playing drum rolls.◮ We can help bring about self-oscillations with

1. system delay2. a hysteretic spring Kcoll

3. negative damping Rout

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Altering The Drumstick Dynamics

◮ Noise and deterministic chaos are not physically intuitive.◮ Stable limit cycles, or self-sustaining attractive oscillations,

describe biological oscillations such as the heart beating.◮ Limit cycles also manifest themselves in bowed strings,

vibrating reeds, and drum rolls.◮ Goal: Assist the drummer in playing drum rolls.◮ We can help bring about self-oscillations with

1. system delay2. a hysteretic spring Kcoll

3. negative damping Rout

4. forcing the drumstick in the z-dimension every time that thestick enters the simulated membrane

h(t) =m∆vpulse

τe−t/τ

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Forced Pulses Can Induce Limit Cycles

d

*

z

. zpulsestarts

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Forced Pulses Can Induce Limit Cycles

d

*

z

. zpulsestarts

◮ Here we choose each pulse to be of constant magnitude.

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Forced Pulses Can Induce Limit Cycles

d

*

z

. zpulsestarts

◮ Here we choose each pulse to be of constant magnitude.◮ But is d stable?

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Related Discrete-Time System

e

**

pulsestarts

dp

U

P(q)Eq

z

. z

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Related Discrete-Time System

e

**

pulsestarts

dp

U

P(q)Eq

z

. z

◮ P(·) is a Poincare map.

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Related Discrete-Time System

e

**

pulsestarts

dp

U

P(q)Eq

z

. z

◮ P(·) is a Poincare map.◮ Let vi is the velocity of the drumstick tip at the end of

the i th collision with the drum membrane.

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Related Discrete-Time System

e

**

pulsestarts

dp

U

P(q)Eq

z

. z

◮ P(·) is a Poincare map.◮ Let vi is the velocity of the drumstick tip at the end of

the i th collision with the drum membrane.◮ We analyze the stability of the closed orbit d by analyzing

the stability of:vi+1 = P(vi) = αβvi + β∆vpulse

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Applying Pulses With Constant Magnitude

◮ Choose ∆vpulse = k/β, where k is constant

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Applying Pulses With Constant Magnitude

◮ Choose ∆vpulse = k/β, where k is constant

◮ vi+1 = P(vi) = αβvi + k

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Applying Pulses With Constant Magnitude

◮ Choose ∆vpulse = k/β, where k is constant

◮ vi+1 = P(vi) = αβvi + k = α(β + kαvi

)vi

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Applying Pulses With Constant Magnitude

◮ Choose ∆vpulse = k/β, where k is constant

◮ vi+1 = P(vi) = αβvi + k = α(β + kαvi

)vi

◮ At small amplitudes, the effective COR β = (β + kαvi

) > 1.

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Applying Pulses With Constant Magnitude

◮ Choose ∆vpulse = k/β, where k is constant

◮ vi+1 = P(vi) = αβvi + k = α(β + kαvi

)vi

◮ At small amplitudes, the effective COR β = (β + kαvi

) > 1.

◮ At large amplitudes, the effective COR β = (β + kαvi

) ≈ β.

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Applying Pulses With Constant Magnitude

◮ Choose ∆vpulse = k/β, where k is constant

◮ vi+1 = P(vi) = αβvi + k = α(β + kαvi

)vi

◮ At small amplitudes, the effective COR β = (β + kαvi

) > 1.

◮ At large amplitudes, the effective COR β = (β + kαvi

) ≈ β.◮ α < 1 and β < 1 ⇒ αβ < 1

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Applying Pulses With Constant Magnitude

◮ Choose ∆vpulse = k/β, where k is constant

◮ vi+1 = P(vi) = αβvi + k = α(β + kαvi

)vi

◮ At small amplitudes, the effective COR β = (β + kαvi

) > 1.

◮ At large amplitudes, the effective COR β = (β + kαvi

) ≈ β.◮ α < 1 and β < 1 ⇒ αβ < 1

◮ ⇒ limi→∞vi = k

1−αβ

∆

= vlc is positive and finite.

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Applying Pulses With Constant Magnitude

◮ Choose ∆vpulse = k/β, where k is constant

◮ vi+1 = P(vi) = αβvi + k = α(β + kαvi

)vi

◮ At small amplitudes, the effective COR β = (β + kαvi

) > 1.

◮ At large amplitudes, the effective COR β = (β + kαvi

) ≈ β.◮ α < 1 and β < 1 ⇒ αβ < 1

◮ ⇒ limi→∞vi = k

1−αβ

∆

= vlc is positive and finite.◮ The discrete-time system is stable.

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Applying Pulses With Constant Magnitude

◮ Choose ∆vpulse = k/β, where k is constant

◮ vi+1 = P(vi) = αβvi + k = α(β + kαvi

)vi

◮ At small amplitudes, the effective COR β = (β + kαvi

) > 1.

◮ At large amplitudes, the effective COR β = (β + kαvi

) ≈ β.◮ α < 1 and β < 1 ⇒ αβ < 1

◮ ⇒ limi→∞vi = k

1−αβ

∆

= vlc is positive and finite.◮ The discrete-time system is stable.◮ P(·) is a Poincare map ⇒ d is a stable limit cycle

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Conclusions

◮ Informal testing suggests that the altered dynamics make iteasier to play drum rolls.

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Conclusions

◮ Informal testing suggests that the altered dynamics make iteasier to play drum rolls.

◮ Drum roll limit cycles are guaranteed to be stable.

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Conclusions

◮ Informal testing suggests that the altered dynamics make iteasier to play drum rolls.

◮ Drum roll limit cycles are guaranteed to be stable.◮ Drummers can increase the drum roll rate by increasing

Khand or decreasing zh0 as in traditional drum roll playing.

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Conclusions

◮ Informal testing suggests that the altered dynamics make iteasier to play drum rolls.

◮ Drum roll limit cycles are guaranteed to be stable.◮ Drummers can increase the drum roll rate by increasing

Khand or decreasing zh0 as in traditional drum roll playing.◮ The previous point suggests that the new musical instrument

is physically intuitive–i.e., the new instrument supportsphysical interactions that are familiar to traditionalperformers of traditional drums.

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Thank You!

Questions?