Performance and evolution of biological and engineeredmotors and devices used for locomotionJim MardenDept. of BiologyPenn State University

jhm10@psu.eduDrosophila thoraxCummins 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) r2 = 0.99Marden 1987; J. Exp. Biol. 130, 235-258M1.0Initial question: How and why does flight performance vary among animal species?Log10 Maximum force output (N)

Marden & Allen 2002; PNAS 99, 4161-4166Data 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)SwimmersRunnersRotary electricLinearelectricPistonsJetsWhat about other types of motors? - How do they compare?

Marden & Allen 2002; PNAS 99, 4161-4166SwimmersRunnersRotary electricLinearelectricPistonsJetsWhat about other types of motors? - How do they compare?

Common reactions to these data:

This cannot be rightSurely one could design a more forceful motor at a given mass Other investigators find same result A completely novel modern design (MIT microjet) that aimed for much higher specific force conforms exactly

Single moleculesMusclesLinear actuatorsWinchesRocketsLog 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 variabilityMarden & Allen 2002; PNAS 99, 4161-4166A second scaling regime: anchored translational motors and rockets

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

Mass1 for locomotion motors:

- Multiaxial stress, fatigue, probabilistic failure Fmax Stress gradient (N/m3) Fmax Volume(Marden, 2005)

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

Why these two scaling regimes?Hypotheses:

Fatigue theory: load-life relationships

N = a (ult / )bN = lifespan number of cyclesult = ultimate uniaxial stress = applied stress

Uniaxial loading:Multiaxial loading:N = a (C/ P)bN = 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 exampleMarden (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/KgMarden (2005) J. Experimental Biology; 208, 1653

Location of transportation motors on the load-life curveMarden (2005) J. Experimental Biology; 208, 1653

An entirely different approach: Physics theory for force production that minimizes work (energy loss) per distanceW / 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 b -1/6 Mb 1/6

Freqopt g1/2 b 1/6 Mb -1/6

Forceopt gMb

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

Vopt g1/2 b -1/6 Mb 1/6

Freqopt g1/2 b 1/6 Mb -1/6

Forceopt gMbCycle 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 = M0.83

Force 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 d2Empirical measurement across 8 species: determine the mass scaling for each of these termsThe lever system of the dragonfly flight motorWingFulcrumFoutSchilder & Marden 2004; J. Exp. Biol. 207, 767-76

M1.04 M0.83 M0.54 M-0.31Fout = Fdyn d1 / d2Conclusions 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:Schilder & Marden 2004; J. Exp. Biol. 207, 767-76Conclusion: 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 intersectThe two linescross 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

M0.67M1.0Marden & Allen, 2002

Testing this prediction with piston enginesMagnum XL15A 165 gBurmeister & Wain K98MC-C 1.9 million Kg

As predicted, force output and geometry of piston engines changes dramatically at a mass of approximately 4400KgM0.67M1.0log10 Maximum force output (N)log10 ratio of piston diameter to stroke length4400 kg4400 kgMarden & 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