A Terradynamics for Legged Locomotion on Granular Media

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Information about A Terradynamics for Legged Locomotion on Granular Media
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Published on February 5, 2014

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Tingnan Zhang, Chen Li, and Daniel Goldman, School of Physics, Georgia Institute of Technology, University of California at Berkeley. Paper 80960_0

12/20/2013 A Terradynamics for Legged Locomotion on Granular Media Tingnan Zhang*, Chen Li*†, and Daniel I. Goldman* *School of Physics, Georgia Institute of Technology †University of California at Berkeley Li, Zhang, Goldman, Science (2013) 1

12/20/2013 Many natural, particulate media can flow under stress mud Martian soil JPL sand debris 2

12/20/2013 Flowing substrates are challenging to move on JPL Car on sand Tank on soil Rover on Martian soil Difficult to gain purchase without slipping for wheeled and tracked vehicles alike Kumagai (2004), IEEE Spectrum RHex on dirt/mud Kod*lab Lizard vs. snake by BBC Ghost crab Slowed 50 3

12/20/2013 Challenges: Limb-ground interaction is complex Zebra-tailed lizard X-ray video, slowed 50× 5 cm Li, Hsieh, and Goldman, J. Exp. Biol. (2012) SandBot (RHex-class) 10 cm Slowed 10× Li, Umbanhowar, Komsuoglu, Koditschek, and Goldman, PNAS (2009) Complicated morphology + kinematics 4

12/20/2013 Challenges: No comprehensive force models In fluids, Navier-Stokes equations + moving boundary conditions Vogel (1996), Life in moving fluids Flying Swimming Dickinson et al. (2000), Science Comprehensive force models are lacking for general particulate media 5

12/20/2013 Is terramechanics applicable? Classical terramechanics can accurately and quickly predict forces and performance for (large) wheeled and tracked vehicles Based on penetration resistance, pressuresinkage, and shear resistance tests, not developed for legged locomotion penetrometer Terramechanics for legged locomotion ? bevameter M. G. Bekker (1960), Off-the-road locomotion, research and development in terramechanics J. Y. Wong (2010), Terramechanics and off-road vehicle engineering 6

12/20/2013 Discrete Element Method dynaRoACH (10 cm, 20 g) on 3 mm glass particles elasticity dissipation multi-body dynamic simulation coupled to DEM friction Maladen, Ding, Umbanhowar, Kamor, and Goldman, J. Roy. Soc. Interface (2011) Zhang, Qian, Li, Masarati, Hoover, Birkmeyer, Pullin, Fearing, and Goldman, Intl. J. Robotic. Res. (2013) Pros: Accurate (One simulation could take a few days) Cons: Slow, impractical for large scales 7

12/20/2013 Continuum model approach? Hypothesis: Linear superposition of independent element forces predicts net forces Vertical plane • Inspired by resistive force theory for low Re number swimmers • Valid in non-inertial regime (negligible particle inertia) • Works for sand-swimming in horizontal plane Lauga & Powers, Rev. Prog. Phys. (2009) Maladen, Ding, Li, Goldman, Science (2009) 8

12/20/2013 Measuring stresses using a plate element – Video taken at boundary for illustration – Force measured in the bulk – v = 0.01 m/s – Video played 10 faster Total force ~ 1 mm poppy seeds (below surface) Extraction Fluidization Fully immersed and far from bottom (above surface) Stresses are hydrostatic-like z (cm) 9

12/20/2013 Stresses per unit depth vs. orientation, movement direction Vertical Horizontal Black curves: z,x =0 Complex dependence 10

12/20/2013 Net forces on c-leg: Experiment vs. model leg is divided into 30 segments Net force Segmental force (on a larger scale) ~ 1 mm poppy seeds – Video taken at boundary for illustration – Force measured in the bulk – v = 0.01 m/s – Video played 10 faster Fz experiment Fz model Fx experiment Fx model (rad) (rad) 11

12/20/2013 Net forces on c-leg: Experiment vs. model leg is divided into 30 segments Net force Segmental force (on a larger scale) ~ 1 mm poppy seeds – Video taken at boundary for illustration – Force measured in the bulk – v = 0.01 m/s – Video played 10 faster Fz model Fz experiment F (N) Fx experiment (rad) F (N) Fx model (rad) 12

12/20/2013 Applicability to granular media of various particle size, density, friction, and compaction Poppy seeds loosely packed Generic stress profiles closely packed 0.3 mm glass particles loosely packed closely packed 3 mm glass particles Single measurement with an off-the-shelf penetrometer closely packed (Photo credit: Sarah Sharpe) Stress profiles and model accuracy are generic 13

12/20/2013 Application on natural sands Yuma sand under microscope 0.06-3mm Experimental measurement Prediction using generic profile Yuma sand Palm sand z (cm) 14

12/20/2013 Using resistive force model to predict legged locomotion Xplorer (150g) – Legs of similar friction to plate element – Leg speeds < 0.6 m/s (non-inertial regime) – Motion mostly confined in the vertical plane 10 cm – Each body plate and leg is divided into 30 elements – Total force F and torque are calculated using resistive force model – Body movement is calculated by: Multibody dynamic simulator (MBDyn) Ghiringhelli et al., Nonlinear Dynamics (1999) 15

12/20/2013 Robot moving on granular media using c-legs f = 2.0 Hz, slowed 5 c-leg Experiment Simulation 16

12/20/2013 Terradynamics is accurate and efficient Predicts speed Predicts ground reaction forces Much faster than DEM e.g. 10 seconds vs. 30 days for 1 second of locomotion on a bed of 5,000,000 poppy seeds (~106 times speed-up) 17

12/20/2013 RFT wheel test (in collaboration with MIT) Horizontal bearing • • In collaboration with Dr. Karl Iagnemma’s group at MIT. Experiments performed by Carmine Senatore from MIT and Mark Kingsbury from Crab lab. Force spring Vertical bearing Photo by carmine 18

12/20/2013 MER wheel on fluidized bed 19

12/20/2013 Wheels and testing conditions McMaster Small McMaster Large 3D Printed MIT Smooth Diameter [mm] 152.4 203.2 145 (to lug tips) 260 Width [mm] 44.5 50.8 76.2 160 Fz Tested [N] 7 20 10, 18 60, 120 Loose Loose and Compact Loose and compact (only for 18 N) Loose and Compact Terrain State Tested (Poppy seeds) McMaster Small McMaster Large 3D Printed MIT Smooth (approx. to scale) 20

12/20/2013 Drawbar vs. slip ratio in experiment and model Experiment WR RFT 10 Drawbar [N] 5 0 -5 -0.5 0 0.5 Slip 21

12/20/2013 Experiment WR RFT 10 3D Printed Fz = 18 N Compact Drawbar [N] 5 0 -5 -0.5 0 0.5 Slip 0.8 25 Sinkage [mm] Torque [Nm] 0.6 0.4 0.2 15 10 0 -0.2 20 -0.6 -0.4 -0.2 0 Slip 0.2 0.4 0.6 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 Slip 22

12/20/2013 Experiment WR RFT 6 4 McMaster Large Fz = 20 N Compact Drawbar [N] 2 0 -2 -4 -6 -8 -10 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 Slip 35 1 30 Sinkage [mm] Torque [Nm] 0.8 0.6 0.4 0.2 25 20 15 0 10 -0.2 -0.6 -0.4 -0.2 0 Slip 0.2 0.4 0.6 5 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 Slip 23

12/20/2013 Summary 1. Developed a resistive force model in the vertical plane for legged locomotion on granular media (for slow intrusions) 2. Resistive force model predicts forces (without any fitting parameters) on intruders of complex morphology and kinematics 3. Resistive force model + multi-body simulation predicts legged robot performance 4. RFT is able to predict wheel performance under a wide range of conditions. 24

12/20/2013 Acknowledgements: Yang Ding, Nick Gravish, Paul Umbanhowar, Gareth Meirion-Griffith, and Hal Komsuoglu for discussion. Jeff Shen for robot modification. Pierangelo Masarati for MBDyn support. Sarah Sharpe for taking the photos of granular materials. Paul Umbanhowar and Hamid Marvi for assistance with natural sand collection. Funded by: Burrough’s Wellcome Fund, ARL MAST CTA, ARO, NSF PoLS and Miller Research Fellowship (C.L.). 25

12/20/2013 Starting point: level, uniform, dry granular media 1 cm Granular media (e.g., sand and gravel): collections of discrete particles that interact through dissipative, repulsive contact forces Nedderman (1992), Statics and Kinematics of Granular Materials A convenient model flowing substrate for locomotion studies: representative, relevant, relatively simple, controllable ~ 1 mm poppy seeds A fluidized bed prepares repeatable packing states Air flow Jackson (2000), The Dynamics of Fluidized Particles Air flow 26

12/20/2013 3 McMaster Small Fz = 7 N Loose Experiment WR RFT 2 Drawbar [N] 1 0 -1 -2 -3 -4 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 Slip 0.3 30 0.25 25 Sinkage [mm] Torque [Nm] 0.2 0.15 0.1 0.05 0 20 15 10 -0.05 -0.5 0 Slip 0.5 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 Slip 27

12/20/2013 Drawbar [N] 5 Experiment WR RFT McMaster Large Fz = 20 N Loose 0 -5 -10 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 Slip 1.2 50 1 Sinkage [mm] Torque [Nm] 0.8 0.6 0.4 0.2 40 30 20 0 10 -0.2 -0.6 -0.4 -0.2 0 Slip 0.2 0.4 0.6 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 Slip 28

12/20/2013 Experiment WR RFT 6 4 McMaster Large Fz = 20 N Compact Drawbar [N] 2 0 -2 -4 -6 -8 -10 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 Slip 35 1 30 Sinkage [mm] Torque [Nm] 0.8 0.6 0.4 0.2 25 20 15 0 10 -0.2 -0.6 -0.4 -0.2 0 Slip 0.2 0.4 0.6 5 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 Slip 29

12/20/2013 3 Experiment WR RFT 2 Drawbar [N] 1 3D Printed Fz = 10 N Loose 0 -1 -2 -3 -4 -5 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 Slip 25 0.4 Sinkage [mm] 20 Torque [Nm] 0.3 0.2 0.1 15 10 0 -0.1 -0.6 -0.4 -0.2 0 Slip 0.2 0.4 0.6 -0.5 0 0.5 Slip 30

12/20/2013 Experiment WR RFT Drawbar [N] 5 3D Printed Fz = 18 N Loose 0 -5 -10 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 Slip 35 30 Sinkage [mm] Torque [Nm] 0.6 0.4 0.2 25 20 15 0 10 -0.2 -0.6 -0.4 -0.2 0 Slip 0.2 0.4 0.6 -0.5 0 0.5 Slip 31

12/20/2013 Experiment WR RFT 10 3D Printed Fz = 18 N Compact Drawbar [N] 5 0 -5 -0.5 0 0.5 Slip 0.8 25 Sinkage [mm] Torque [Nm] 0.6 0.4 0.2 15 10 0 -0.2 20 -0.6 -0.4 -0.2 0 Slip 0.2 0.4 0.6 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 Slip 32

12/20/2013 30 Experiment L Experiment C WR L WR C RFT L RFT C Drawbar [N] 20 10 0 MIT Wheel Fz = 60 N Compact/Loose -10 -20 -30 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 50 Slip 4 40 Sinkage [mm] Torque [Nm] 3 2 1 0 30 20 10 -1 -0.5 0 Slip 0.5 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 Slip 33

12/20/2013 Experiment L Experiment C WR L WR C RFT L RFT C 40 Drawbar [N] 20 0 MIT Wheel Fz = 120 N Compact/Loose -20 -40 -60 -0.6 -0.4 -0.2 0 0.2 0.4 8 60 50 Sinkage [mm] 6 Torque [Nm] 0.6 Slip 4 2 40 30 20 0 10 -2 -0.6 -0.4 -0.2 -0.5 0 0.5 0 0.2 0.4 0.6 Slip Slip 34

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