1. Fast and Robust Legged Locomotion Sean Bailey Mechanical Engineering Design Division Advisor: Dr. Mark Cutkosky May 12, 2000

2. Intro Biomimesis Design AnalysisConclusions Overview Introduction Functional Biomimesis Robot Design Model Analysis Conclusions

3. Intro Biomimesis Design AnalysisConclusions Fast, Robust Rough Terrain Traversal Why? –Mine clearing –Urban Reconnaissance Why legs? Basic Design Goals –1.5 body lengths per second –Hip-height obstacles –Simple

4. Intro Biomimesis Design AnalysisConclusions Traditional Approaches to Legged Systems Statically stable –Tripod of support – –Slow –Rough terrain Dynamically stable –No support requirements – –Fast –Smooth terrain

5. Intro Biomimesis Design AnalysisConclusions Biological Example Death-head cockroach Blaberus discoidalis Fast –Speeds of up to 10 body/s Rough terrain –Can easily traverse fractal terrain of obstacles 3X hip height Stability –Static and dynamic

6. Functional Biomimesis “Biomimetic” configuration Extract fast rough terrain locomotion capabilities Too complex! Intro Biomimesis Design AnalysisConclusions Biomimesis Options

7. Intro Biomimesis Design AnalysisConclusions Biological Inspiration Control heirarchy –Passive component –Active component

8. Intro Biomimesis Design AnalysisConclusions Is Passive Enough? Passive Dynamic Stabilization –No active stabilization –Geometry –Mechanical system properties

9. Intro Biomimesis Design AnalysisConclusions Cockroach Geometry Passive Compliant Hip Joint Effective Thrusting Force Functional Biomimesis Damped, Compliant Hip Flexure Embedded Air Piston Robot Implementation Geometry Rotary Joint Prismatic Joint

10. Intro Biomimesis Design AnalysisConclusions Sprawlita Mass -.27 kg Dimensions - 16x10x9 cm Leg length - 4.5 cm Max. Speed - 39cm/s 2.5 body/sec Hip height obstacle traversal

11. Intro Biomimesis Design AnalysisConclusions Movie Compliant hip Alternating tripod Stable running Obstacle traversal

12. Intro Biomimesis Design AnalysisConclusions Mechanical System Properties Prototype: Empirically tuned properties Design for behavior ? Mechanical System Properties Modeling

13. Intro Biomimesis Design AnalysisConclusions “Simple” Model Body has 3 planar degrees of freedom –x, z, theta –mass, inertia 3 massless legs (per tripod) –rotating hip joint - damped torsional spring –prismatic leg joint - damped linear spring –6 parameters per leg 18 parameters to tune - TOO MANY! Full 3D model Planar model Symmetry assumption K, B,  nom k, b, nom

14. Intro Biomimesis Design AnalysisConclusions Simplest Locomotion Model Body has 2 planar degrees of freedom –x, z –mass 4 massless legs –freely rotating hip joint –prismatic leg joint - damped linear spring –3 parameters per leg 6 parameters to tune, assuming symmetry g g k, b, nom Biped Quadruped Biped

15. Intro Biomimesis Design AnalysisConclusions Time-Based Mode Transitions –Clock-driven motor pattern –“Groucho running” 1 One “reset” mode –Two sets of legs - Two modes –Symmetric - treat as one mode Mode initial conditions –Nominal leg angles –Instant passive component compression Modeling assumptions 1 McMahon, et al 1987 Leg Set 2 1 2 1 State Time x 0 = state trajectory Stride Period TT T T g

16. Intro Biomimesis Design AnalysisConclusions Time-Based Mode Transitions –Clock-driven motor pattern –“Groucho running” 1 One “reset” mode –Two sets of legs - Two modes –Symmetric - treat as one mode Mode initial conditions –Nominal leg angles –Instant passive component compression Modeling assumptions 1 McMahon, et al 1987 Leg Set 2 1 2 1 State Time x 0 = state trajectory Stride Period g t = 2T - TT T T

17. Intro Biomimesis Design AnalysisConclusions Time-Based Mode Transitions –Clock-driven motor pattern –“Groucho running” 1 One “reset” mode –Two sets of legs - Two modes –Symmetric - treat as one mode Mode initial conditions –Nominal leg angles –Instant passive component compression Modeling assumptions 1 McMahon, et al 1987 Leg Set 2 1 2 1 State Time x 0 = state trajectory Stride Period t = 2T + TT T T g

18. Intro Biomimesis Design AnalysisConclusions Time-Based Mode Transitions –Clock-driven motor pattern –“Groucho running” 1 One “reset” mode –Two sets of legs - Two modes –Symmetric - treat as one mode Mode initial conditions –Nominal leg angles –Instant passive component compression Modeling assumptions 1 McMahon, et al 1987 Leg Set 2 1 2 1 State Time x 0 = state trajectory Stride Period t = 2T + 1 / 3 T TT T T g

19. Intro Biomimesis Design AnalysisConclusions Time-Based Mode Transitions –Clock-driven motor pattern –“Groucho running” 1 One “reset” mode –Two sets of legs - Two modes –Symmetric - treat as one mode Mode initial conditions –Nominal leg angles –Instant passive component compression Modeling assumptions 1 McMahon, et al 1987 Leg Set 2 1 2 1 State Time x 0 = state trajectory Stride Period t = 2T + 2 / 3 T TT T T g

20. Intro Biomimesis Design AnalysisConclusions Time-Based Mode Transitions –Clock-driven motor pattern –“Groucho running” 1 One “reset” mode –Two sets of legs - Two modes –Symmetric - treat as one mode Mode initial conditions –Nominal leg angles –Instant passive component compression Modeling assumptions 1 McMahon, et al 1987 Leg Set 2 1 2 1 State Time x 0 = state trajectory Stride Period t = 3T - TT T T g

21. Intro Biomimesis Design AnalysisConclusions Time-Based Mode Transitions –Clock-driven motor pattern –“Groucho running” 1 One “reset” mode –Two sets of legs - Two modes –Symmetric - treat as one mode Mode initial conditions –Nominal leg angles –Instant passive component compression Modeling assumptions 1 McMahon, et al 1987 Leg Set 2 1 2 1 State Time x 0 = state trajectory Stride Period t = 3T + TT T T g

22. Intro Biomimesis Design AnalysisConclusions Time-Based Mode Transitions –Clock-driven motor pattern –“Groucho running” 1 One “reset” mode –Two sets of legs - Two modes –Symmetric - treat as one mode Mode initial conditions –Nominal leg angles –Instant passive component compression Modeling assumptions 1 McMahon, et al 1987 Leg Set 2 1 2 1 State Time x 0 = state trajectory Stride Period t = 3T + 1 / 3 T TT T T g

23. Intro Biomimesis Design AnalysisConclusions Non-linear analysis tools Discrete non-linear system Fixed points –numerically integrate to find –exclude horizontal position information = state trajectory = fixed points x k+1 = x k = x* Leg Set 2 1 2 1 State Time x 0 = state trajectory Stride Period TT T T

24. Intro Biomimesis Design AnalysisConclusions Non-linear analysis tools Floquet technique –Analyze perturbation response –Digital eigenvalues via linearization - examine stability –Use selective perturbations to construct M matrix = nominal trajectory Numerically Integrate

25. Intro Biomimesis Design AnalysisConclusions Non-linear analysis tools Floquet technique

26. Perturbation Response Intro Biomimesis Design AnalysisConclusions

27. Intro Biomimesis Design AnalysisConclusions Relationships –damping vs. speed and “robustness” –stiffness, leg angles, leg lengths, stride period, etc Use for design –select mechanical properties –select other parameters Insight into the mechanism of locomotion 6.577.588.599.5 10 1.2 1.4 1.6 1.8 2 2.2 2.4 2.6 2.8 0.04 0.045 0.05 0.055 0.06 0.065 0.07 0.075 Damping (N-s/m) Recovery RateHorizontal Velocity X_dot (m/s) 1/max[eig(M)] Analysis trends

28. Intro Biomimesis Design AnalysisConclusions Design Example Robustness Speed Stiffness Damping Stiffness Damping Stiffness Damping Speed = 0Speed = 13 cm/sSpeed = 23.5 cm/s

29. Intro Biomimesis Design AnalysisConclusions Locomotion Insight Statically Unstable Region Initial condition Mode Equilibrium Trajectory Leg Extension Limit Leg Pre- Compressions Body tends towards equilibrium point Parameters and mechanical properties determine how

30. Intro Biomimesis Design AnalysisConclusions Current leg systems are either fast or can handle rough terrain Biology suggests emphasis on good mechanical design –enhances capability –simplifies control Purely clock-driven systems can be fast and robust Floquet technique can be used to indicate locomotion robustness Trends can be established to improve design and provide insight Summary and Conclusions

31. Intro Biomimesis Design AnalysisConclusions Future Work Extend findings and insights to more complex models Develop easily modeled 4th generation robot Utilize sensor feedback in high level control Examine other behaviors

32. Thanks! Center for Design Research Dexterous Manipulation Lab Rapid Prototyping Lab Mark Cutkosky Jorge Cham, Jonathan Clark Intro Biomimesis Design AnalysisConclusions