Adding Inertia and Mass to Test Stability Predictions in Rapid Running Insects

1. Adding Inertia and Mass to Test Stability Predictions in Rapid Running Insects Talia Yuki Moore*, Sam Burden, Shai Revzen, Robert Full PolyPEDAL Lab University of California Berkeley taliayuki@berkeley.edu

2. Natural Changes in Moment of Inertia (Gerald and Buff Corsi, Visuals Unlimited) (Tim Flach Stone/Getty Images) Animals compensate for large changes in mass and moment of inertia. (Flagstaffotos) (Pauline Smith)

3. Differences in Body Mass & Form (Aivar Mikko) (Sophia Moore) (http://dcydiary.blogspot.com) (http://academic.ru) (John S. Reid) Animals have evolved diverse and successful body forms that differ in mass and moment of inertia.

4. Different View of Stability Sagittal Plane Horizontal Plane

5. Instability in the Horizontal Plane

6. Instability in the Horizontal Plane

7. Lateral Leg Spring (LLS) Template Animal Bouncing Side to Side 3 Legs Acting as One Schmitt & Holmes, (2000)

8. Model Parameters k L d m I β - leg stiffness - leg length - center of pressure position - body mass - inertia - leg angle β k d m Schmitt & Holmes (2000)

9. Input Parameters k = 2.25 Nm β = 1 rad I = 2.04 10-7 kgm2 L = 0.1 m m = 0.0025 kg β k Schmitt, Holmes, Garcia, Razo & Full (2001)

10. Model State Variables q v Velocity d qHeading w d Body orientation wRotational velocity Schmitt, Holmes, Garcia, Razo & Full (2002)

11. Self-Stabilization Passive, mechanical self-stabilizing with minimal neural feedback Heading Velocity Orientation Rotational Velocity Schmitt, Holmes,Garcia, Razo & Full (2002)

12. Vary Body Mass Animal Stability of Body Orientation & Rotational Velocity to Lateral Perturbation 1.0 Less Stable Perturbation remaining per stride [Eigenvalue, ] 0.8 0.6 0.4 More Stable 0.2 2 3 5 1 4 Nondimensional Body Mass Schmitt, Holmes, Garcia, Razo & Full (2000)

13. Tuning for Self-Stabilization Animal  Vary Leg Angle - Stride Length 1.0 Perturbation remaining per stride [Eigenvalue, ] Animal Less Stable 0.8 0.6 0.4 More Stable 0.2 0.8 0.9 1.1 1.3 1 1.2 Nondimensional Leg angle Vary Leg Length- Sprawl 1.0 Perturbation remaining per stride [Eigenvalue, ] Less Stable 0.8 0.6 0.4 More Stable 0.2 0.005 0.01 0.015 Vary Leg Stiffness Nondimensional Leg length 1.0 Perturbation remaining per stride [Eigenvalue, ] Less Stable Animal 0.8 0.6 0.4 More Stable 0.2 0 1 2 3 4 Schmitt, Holmes, Garcia, Razo & Full (2000) Nondimensional Spring stiffness

14. Animal & Inertia Animal Moment of Inertia Hypothesis: A cockroach with added mass and increased moment of inertia will recover from perturbations slower and be unstable. 1.0 Perturbation remaining per stride [Eigenvalue, ] Less Stable 0.8 0.6 0.4 More Stable 0.2 0 1 2 0.5 1.5 Nondimensional Moment of Inertia Schmitt, Holmes, Garcia, Razo & Full (2000) 14

15. Changing Moment of Inertia & Mass Control Inertia Added Mass 40% 90% 90% Added Inertia 20% 30% 960% Mass Treatment Each cockroach was its own control Inertia 1.0 Perturbation remaining per stride [Eigenvalue, ] Less Stable 0.8 0.6 Mass 0.4 Control More Stable 0.2 1 0 0.5 1.5 2 Non-Dimensional Moment of Inertia

16. Evidence for Mechanical Feedback Rapid Impulse Perturbation Device Recovery begins <10ms after perturbation Challenges fastest neural reflexes Slowed 30X Jindrich and Full (2002)

17. Lateral Perturbation Platform accelerates laterally at 0.6±0.1 g in a 0.1 sec interval providing a 50±3 cm/sec specific impulse, then maintains velocity. camera diffuser mirror magnetic lock animal motion cart Cockroach runs at: 31±6 cm/sec Stride Frequency: 12.5±1.7 Hz trackway cart motion pulley rail mass cable elastic ground

18. Lateral Perturbation Measured: 1. Distal tarsal (foot) position 2. Pitch, roll, yaw 3. Forward, lateral, rotational velocity 4. Heading, body orientation Cart impulse Criteria for trial rejection: 1. >15° deviation in heading pre-perturbation 2. Contact with the cart sides 3. >50% Change in forward velocity pre-perturbation Equal and opposite impulse on animal

19. Lateral Perturbation Experiment Real time

20. Leg and Body Tracking Slowed 40x Cart Velocity

21. Compare Response to Pre-Perturbation Behavior Onset of Perturbation Raw Data χ Model χ Residual χ Phase

22. Residual Orientation Peak Perturbation Inertia Changes Body Orientation Less Animals Recover Orientation

23. Residual Forward Velocity Fore Peak Perturbation Aft All Treatments Decrease Speed

24. Horizontal Plane Instability Increase Moment of Inertia Limits Maneuverability 35% Decrease in Speed Reject Lateral Leg Spring Prediction Increased Moment of Inertia Treatment Recovers & Does Not Lead to Instability Limit Maneuverability Decrease Speed Carrier et al. 2001 J. Experimental Biology

25. Residual Roll Peak Perturbation Roll From Impulse Lean Into Impulse Animals Overcompensate in Recovery Mass Rolls Most 25

26. Residual Pitch Nose down Peak Perturbation Nose up Mass Pitches More than Inertia Animals Remain Pitched Down in Recovery

27. Residual Lateral Velocity Peak Perturbation Inertia Lateral Velocity Changes Less Animals Overcompensate & Move Into Perturbation

28. Residual Lateral Tarsal Position Peak Perturbation Animals Overcompensate & Place Feet as if to Resist Next Perturbation Inertia Recovery Slower

29. Overcompensation in Humans Welch and Ting (2009)

30. Feedback Response Feedback - Mechanical, Neural or Both? Mechanical Feedback No Frequency Change Perturbation Perturbation Tarsal Fore-Aft Position Residual Phase Time Time Frequency Change Neural Feedback Tarsal Fore-Aft Position Perturbation Perturbation Residual Phase Frequency Change Time Time Revzen, Bishop-Moser, Spence, Full (2007)

31. Residual Phase Response No Frequency Change Supports Mechanical Feedback Peak Perturbation Frequency Change Supports Neural Feedback Mechanical Feedback Followed by Neural Feedback to the Central Pattern Generator

32. Conclusions 1. Changes in body mass and form affect response to perturbations. Mechanical feedback important early in response. 2. Increased moment of inertia reduces and delays response to perturbation, but limits maneuverability. 3. Passive horizontal plane model (Lateral Leg Spring) is insufficient to explain response to lateral perturbations. Higher degree of freedom models needed.

33. Spring Loaded Inverted Pendulum (SLIP) Three Dimensional Models Spring-Loaded Inverted Pendulum (SLIP) Lateral Leg Spring (LLS) Lateral Leg Spring (LLS) Seipel 2005

34. Conclusions 1. Changes in body mass and form affect response to perturbations. Mechanical feedback important early in response. 2. Increased moment of inertia reduces and delays response to perturbation, but limits maneuverability. 3. Passive horizontal plane model (Lateral Leg Spring) is insufficient to explain response to lateral perturbations. Higher degree of freedom models needed. 4. Hexapods overcompensate in recovery perhaps providing greater stability to another perturbation from the same direction. Neural feedback to CPG may assist. 5. Placement of payload in legged robots can learn from nature.

35. Acknowledgements Guidance, Input, and Advice: Berkeley Biomechanics Group Prof. Robert Full PolyPEDAL Lab Think Tanks, Matlab Wizards: Sam Burden Shai Revzen Tarsus Trackers: Debbie Li Brian McRae Cockroach Wrangler: Jessie Ding