Performance and evolution of biological and engineered motors and devices used for locomotion
Drosophila thorax Cummins turbo diesel
Jim Marden
Dept. of Biology
Penn State University jhm10@psu.edu

“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 F max 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

Initial question: How and why does flight performance vary among animal species?
Log force (N) = 1.75 + 0.99 log flight motor mass (Kg) r 2 = 0.99
M 1.0
Striking features:
- Mass 1.0
scaling
- one line fits all
- little effect of variation in phylogeny, wing morphology, or physiology
- why?
Marden 1987; J. Exp. Biol. 130, 235-258

What about other types of motors? - How do they compare?
Swimmers
Jets
Rotary electric
Linear electric Pistons
Runners
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 motor independent of all non-motor payload some less precise motor mass examples: mammalian limb mass; total fish myotome musculature
(not perfect, but close enough)
Marden & Allen 2002; PNAS 99, 4161-4166

What about other types of motors? - How do they compare?
Swimmers
Runners
Rotary electric
Log force (N) = 1.74 + 0.99 log flight motor mass (Kg)
Linear electric
Force = 2πMG
Pistons
Jets
- Mass 1.0
scaling
- one line fits all mean = 57 N/Kg; SD = 14 mean abs dev.=0.07 log units
- little effect of variation in materials or mechanisms
Maximum specific force (N kg -1 )
Marden & Allen 2002; PNAS 99, 4161-4166

Common reactions to these data:
1.This cannot be right
2.Surely one could design a more forceful motor at a given mass
-1.0
-1.5
-2.0
Insects (Marden 1987)
Euglossine bees (Dillon & Dudley, 2004)
MIT microjet (Epstein et al. 2000)
-2.5
-3.0
-3.5
-5.0
-4.5
Log
10
-4.0
-3.5
Motor mass (Kg)
-3.0
-2.5
- Other investigators find same result
- A completely novel modern design (MIT microjet) that aimed for much higher specific force conforms exactly

A second scaling regime: anchored translational motors and rockets
Single molecules Muscles
Linear actuators
Log force (N) = 2.95 + 0.667 log motor mass (Kg)
Winches
M 0.67
M 1.0
Rockets
- 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

Why these two scaling regimes?
Hypotheses:
Mass 2/3 for translational motors: steady uniaxial force loads
Actuator F max
Rocket F max
F max
α Area
α Critical Stress (N/m
α Nozzle area
2 )
Mass 1 for locomotion motors:
- Multiaxial stress, fatigue, probabilistic failure
F max
α Stress gradient (N/m 3 )
F max
α Volume
(Marden, 2005)
- Scaling of optimal locomotion performance
(Bejan & Marden, 2006)

Fatigue theory: load-life relationships
Uniaxial loading:
N = a (σ ult
/ σ) b
N = lifespan number of cycles
σ ult
= ultimate uniaxial stress
σ = applied stress
Multiaxial loading:
N = a (C/ P) b
N = lifespan number of cycles
C = load that causes failure in 1 cycle
P = applied load
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)
Norton (2000) Machine Design, An Integrated Approach

Load-life in an animal example
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
Conclusion: Animal motors conform to general form of load-life theory
Marden (2005) J. Experimental Biology; 208, 1653

Evidence for low cycle fatigue in locomotion motors operating above about 57 N/Kg
Marden (2005) J. Experimental Biology; 208, 1653

Location of transportation motors on the load-life curve
Jet turbine lifespan Distribution of motor F max
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 = (W
1
+ W
2
) / L where W
1 is vertical energy loss per cycle (vertical deflections of the body or medium)
W
2 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 ≈ L b
Find d(W/L)/dV = 0 and associated frequency and force output
Theory predictions for running, swimming and flying
V opt
≈ g 1/2 ρ b
-1/6 M b
1/6
Freq opt
≈ g 1/2 ρ b
1/6 M b
-1/6
Force opt
≈ gM b
Bejan & Marden (2006) J. Exper. Biol. 209, 238

V opt
≈ g 1/2 ρ b
-1/6 M b
1/6
Freq opt
Force opt
≈ g 1/2 ρ b
≈ gM b
1/6 M b
-1/6
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 = M 0.83
Force outptut of the optimized locomotor system should scale as
M 1.0
, as observed for diverse motors
(actuators plus attached levers)
How is the remaining M 1/6 gap in force scaling between oscillatory actuator force output and integrated system force output solved?

The lever system of the dragonfly flight motor
Fulcrum
Wing
F out
Simple model for torque conservation : F dyn d
1
= F out d
2
Empirical measurement across 8 species: determine the mass scaling for each of these terms
Schilder & Marden 2004; J. Exp. Biol. 207, 767-76

Result:
F out
= F dyn d
1
/ d
2
M 1.04
α M 0.83 M 0.54
M -0.31
Conclusions from our dragonfly case study:
- Static actuator force output scales as expected: M 2/3
- Dynamic force output of the actuator scales as predicted (M 2/3 + 1/6 = M 0.83
)
- Force output of the integrated system scaled as M 1 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
(M 0.54
) is the way that the gap in force scaling was solved
M 0.67
M 1.0
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
5
0
-5
-10
M 0.67
M 1.0
The two lines cross at 4400 Kg
Prediction: M 1 scaling cannot continue at masses above 4400Kg because these integrated systems would generate forces equal to the static limit of their actuators
-20 -15 -10 -5 log
10
Motor mass (Kg)
0 5
Marden & Allen, 2002

Testing this prediction with piston engines
Burmeister & Wain K98MC-C
1.9 million Kg
Magnum XL15A
165 g

As predicted, force output and geometry of piston engines changes dramatically at a mass of approximately 4400Kg
4400 kg
4400 kg
M 0.67
M 1.0
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