Performance and evolution of biological and engineered motors and devices used for locomotion

1. 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

2. “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

3. 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

4. Initial question: How and why does flight performance vary among animal species? Log force (N) = 1.75 + 0.99 log flight motor mass (Kg) Striking features: - Mass1.0 scaling - one line fits all - little effect of variation in phylogeny, wing morphology, or physiology - why? r2 = 0.99 M1.0 Log10 Maximum force output (N) Marden 1987; J. Exp. Biol. 130, 235-258

5. What about other types of motors? - How do they compare? Jets Linear electric Pistons Swimmers Rotary electric 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

6. What about other types of motors? - How do they compare? Jets Linear electric Pistons Swimmers Rotary electric Runners Log force (N) = 1.74 + 0.99 log flight motor mass (Kg) - Mass1.0 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 Force = 2πMG Number of motors Maximum specific force (N kg-1) Marden & Allen 2002; PNAS 99, 4161-4166

7. Common reactions to these data: • This cannot be right • Surely one could design a more forceful motor at a given mass Log10 Maximum force (N) Log10 Motor mass (Kg) • Other investigators find same result • A completely novel modern design (MIT microjet) that aimed for • much higher specific force conforms exactly

8. A second scaling regime: anchored translational motors and rockets M0.67 M1.0 Muscles Single molecules Winches Linear actuators 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

9. Why these two scaling regimes? Hypotheses: Mass2/3 for translational motors: steady uniaxial force loads Actuator Fmaxα Critical Stress (N/m2) Rocket Fmaxα Nozzle area Fmaxα Area Mass1 for locomotion motors: - Multiaxial stress, fatigue, probabilistic failure Fmaxα Stress gradient (N/m3) Fmaxα Volume (Marden, 2005) - Scaling of optimal locomotion performance (Bejan & Marden, 2006)

10. Fatigue theory: load-life relationships Uniaxial loading: N = a (σult / σ)b N = lifespan number of cycles σult = 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

11. Load-life in an animal example Generalized 1 kg motor from scaling equation max load = 890 N, a =1and 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

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

13. Location of transportation motors on the load-life curve Jet turbine lifespan Distribution of motor Fmax Marden (2005) J. Experimental Biology; 208, 1653

14. 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 Mb1/6 Freqopt ≈ g1/2 ρb1/6 Mb-1/6 Forceopt ≈ gMb Bejan & Marden (2006) J. Exper. Biol. 209, 238

15. Vopt ≈ g1/2 ρb-1/6 Mb1/6 Freqopt ≈ g1/2 ρb1/6 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 = M0.83 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 actuator force output and integrated system force output solved?

16. The lever system of the dragonfly flight motor Wing Fulcrum Fout Simple model for torque conservation : Fdyn d1 = Fout d2 Empirical measurement across 8 species: determine the mass scaling for each of these terms Schilder & Marden 2004; J. Exp. Biol. 207, 767-76

17. Result: Fout = Fdyn d1 / d2 M1.04α M0.83 M0.54 M-0.31 • Conclusions from • our 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 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

18. Prediction regarding the very largest motors: Function and design must change where the two scaling lines intersect 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

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

20. As predicted, force output and geometry of piston engines changes dramatically at a mass of approximately 4400Kg 4400 kg 4400 kg log10 Maximum force output (N) log10 ratio of piston diameter to stroke length M0.67 M1.0 Marden & Allen, 2002

21. 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