Performance and evolution of biological and engineered

motors and devices used for locomotion

Jim MardenDept. of Biology

Penn State University

jhm10@psu.edu

Drosophila thorax Cummins turbo diesel

“Specifying the actuation is a key step in the design process of a robot. This includes the choice and sizing of actuation technology.” Chevallereau et al., 2003

Objectives:

- Show major regimes of mass scaling of performance

- Examine why these scaling regimes exist

- Try to understand why there is such remarkable consistency of Fmax in locomotion motors that is independent of materials and mechanisms

- Show some theory for convergent evolution of motor performance

-Argue that these results provide design objectives and figures of merit that could be helpful for design and evaluation of robots

Striking features: - Mass1.0 scaling

- one line fits all

- little effect of variation in phylogeny, wing morphology, or physiology

- why?

Log force (N) = 1.75 + 0.99 log flight motor mass (Kg)

.

-5

-4

-3

-2

-1

0

1

-7 -6 -5 -4 -3 -2 -1 0

log10 Motor mass (Kg)

Insects Birds Bats

r2 = 0.99

Marden 1987; J. Exp. Biol. 130, 235-258

M1.0

Initial question: How and why does flight performance vary among animal species?

Log

10 M

axim

um f

orce

out

put (

N)

Marden & Allen 2002; PNAS 99, 4161-4166

Data that we compiled:

Force: mean force vector over one or more complete stroke cycles

- for torque motors we divided out shaft radius

Motor mass: as near as possible, the mass of the motorindependent of all non-motor payload

some less precise motor mass examples: mammalian limb mass; total fish myotome musculature

(not perfect, but close enough)

Swimmers Runners Rotary electric

Linearelectric Pistons

Jets

What about other types of motors? - How do they compare?

Marden & Allen 2002; PNAS 99, 4161-4166

Swimmers Runners Rotary electric

Linearelectric Pistons

Jets

What about other types of motors? - How do they compare?

.

10

20

30

30 50 70 90 110Maximum specific force (NKg-1)

- Mass1.0 scaling - one line fits all

mean = 57 N/Kg; SD = 14mean abs dev.=0.07 log units

- little effect of variation in materials or mechanisms

Log force (N) = 1.74 + 0.99 log flight motor mass (Kg)

Num

ber

of m

otor

s

Force = 2πMG

Maximum specific force (N kg-1)

Common reactions to these data:

1.This cannot be right2.Surely one could design a more forceful motor at a given mass

-3.5

-3.0

-2.5

-2.0

-1.5

-1.0

-5.0 -4.5 -4.0 -3.5 -3.0 -2.5

Insects (Marden 1987)

Euglossine bees (Dillon & Dudley, 2004)

MIT microjet (Epstein et al. 2000)

Log10 Motor mass (Kg)

Log

10 M

axim

um f

orce

(N

)

- Other investigators find same result- A completely novel modern design (MIT microjet) that aimed for much higher specific force conforms exactly

Single molecules Muscles

Linear actuators

Winches

Rockets

Log force (N) = 2.95 + 0.667 log motor mass (Kg)

- Mass 0.67 scaling - one line fits all

mean abs dev. = 0.28 log units

- little systematic effect of variation in

materials or mechanisms,but more variability

Marden & Allen 2002; PNAS 99, 4161-4166

M0.67

M1.0

A second scaling regime: anchored translational motors and rockets

Mass2/3 for translational motors: steady uniaxial force loads Actuator Fmax α Critical Stress (N/m2) Rocket Fmax α Nozzle area Fmax α Area

Mass1 for locomotion motors:

- Multiaxial stress, fatigue, probabilistic failure Fmax α Stress gradient (N/m3) Fmax α Volume

(Marden, 2005)

- Scaling of optimal locomotion performance(Bejan & Marden, 2006)

Why these two scaling regimes?

Hypotheses:

Fatigue theory: load-life relationships

N = a (σult / σ)b

N = lifespan number of cyclesσult = ultimate uniaxial stressσ = applied stress

Uniaxial loading:

Multiaxial loading:

N = a (C/ P)b

N = lifespan number of cyclesC = load that causes failure in 1 cycleP = applied load

Norton (2000) Machine Design, An Integrated Approach

Theory: accumulation of small defects limits N (i.e. high cycle fatigue)Reality: when small defects cause significant deformations, friction increases and failure is rapid (i.e. transition from high cycle to low cycle fatigue)

Generalized 1 kg motor from scaling equation max load = 890 N, a =1 and b= 3

Hummingbird empirical data (Chai & Millard, 1997) 100 N/kg, 15 wingbeats 67 N/kg, 35 wingbeats 33 N/kg, fly 10% of an entire day = thousands of cycles

Load-life in an animal example

Marden (2005) J. Experimental Biology; 208, 1653

Conclusion: Animal motors conform to general form of load-life theory

Evidence for low cycle fatigue in locomotion motorsoperating above about 57 N/Kg

Marden (2005) J. Experimental Biology; 208, 1653

Jet turbine lifespan Distribution

of motor Fmax

Location of transportation motors on the load-life curve

Marden (2005) J. Experimental Biology; 208, 1653

An entirely different approach: Physics theory for force production that minimizes work (energy loss) per distance

W / L = (W1 + W2) / L

where W1 is vertical energy loss per cycle (vertical deflections of the body or medium) W2 is horizontal loss per cycle (friction)

Approach: Ignore constants on the order of 1 Ignore elastic storage and recovery Analyze in terms of mass scaling

Apply where vertical deflections ≈ Lb

Find d(W/L)/dV = 0 and associated frequency and force output

Theory predictions for running, swimming and flying

Vopt ≈ g1/2 ρb -1/6 Mb 1/6

Freqopt ≈ g1/2 ρb 1/6 Mb -1/6

Forceopt ≈ gMb Bejan & Marden (2006) J. Exper. Biol. 209, 238

Vopt ≈ g1/2 ρb -1/6 Mb 1/6

Freqopt ≈ g1/2 ρb 1/6 Mb -1/6

Forceopt ≈ gMb

Cycle time scales as M 1/6

= more time within cycles to generate force

There are time dependences in force generation (Carnot cycles are not square), and so we expect dynamic forces of actuators working in an oscillatory fashion within optimized locomotor systems to generate force ouptut scaling as M 2/3 + 1/6 = M0.83Force outptut of the optimized

locomotor system should scale as M1.0, as observed for diverse motors (actuators plus attached levers)

How is the remaining M1/6 gap in force scaling between oscillatory actuatorforce output and integrated system force output solved?

Simple model for torque conservation : Fdyn d1 = Fout d2

Empirical measurement across 8 species: determine the mass scaling for each of these terms

The lever system of the dragonfly flight motor

Wing

FulcrumFout

Schilder & Marden 2004; J. Exp. Biol. 207, 767-76

M1.04 α M0.83 M0.54 M-0.31

Fout = Fdyn d1 / d2

Conclusions fromour dragonfly case study:

- Static actuator force output scales as expected: M2/3

- Dynamic force output of the actuator scales as predicted (M 2/3 + 1/6 = M0.83)

- Force output of the integrated system scaled as M1 and close to the 60N/Kg common upper limit (set by fatigue life?) -Departure from geometric similarity in the mass scaling of the internal lever arm length (M0.54) is the way that the gap in force scaling was solved

Result:

M0.67

M1.0

Schilder & Marden 2004; J. Exp. Biol. 207, 767-76

Conclusion: level geometry combines with time dependency of force to change the basic M2/3 force output of actuators to M1 force output of integrated systems

Prediction regarding the very largest motors:

Function and design must change where the two scaling lines

intersect

-10

-5

0

5

-20 -15 -10 -5 0 5

log10 Motor mass (Kg)

The two lines

cross at 4400 Kg

Prediction: M1 scaling cannot continue at masses above 4400Kg because these integrated systems would generate forces equal to the static limit of their actuators

M0.67

M1.0

Marden & Allen, 2002

Testing this prediction with piston engines

Magnum XL15A

165 g

Burmeister & Wain K98MC-C

1.9 million Kg

.

log10 Motor mass (Kg)

A

log10 Motor mass (Kg)

-2

-1

0

1

6420

B8

7

6

5

4

3

2

1

6420

As predicted, force output and geometry of piston engines changes dramatically at a mass of

approximately 4400Kg

M0.67

M1.0

log1

0 M

axim

um f

orce

out

put (

N)

log1

0 ra

tio

of p

isto

n di

amet

er to

str

oke

leng

th

4400 kg 4400 kg

Marden & Allen, 2002

Conclusion: These fundamental functional regimes can provide general design objectives, targets, and figures of merit for novel systems like robots.

This knowledge can be used to avoid making large mistakes, i.e. systems with short life expectancies, poor energy efficiency, insufficient or excessive force generation capacity

The End