1. Robust Dynamic Locomotion Through Feedforward-Preflex Interaction
Jorge G. Cham, Sean A. Bailey, Mark R. Cutkosky
Center for Design Research
Stanford University
2000 ASME International Mechanical
Engineering Congress and Expo
November 9, 2000
This work is supported by ONR and NSF

2. Robust Dynamic Locomotion Background and Motivation Biological
Inspiration
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Trajectory
Modeling
Approach
Hexapedal
Prototypes
Conclusions
And Future
Work
Biology
Modeling
Prototypes

3. Motivation Hazardous tasks for humans
Access to areas inaccessible to wheeled vehicles
Legged animals are faster and more agile in rough terrain
Intro

4. Motivation Most current robots have neither simplicity of wheels…
…nor versatility and speed of legged animals
Dante Robot
Raibert Monopod
Intro

5. Motivation Statically-stable robots
Robust by maintaining at least three legs on the ground
Limited speed
Dante Robot
Raibert Monopod
Intro

6. Motivation Dynamically-stable robots
Fast locomotion that is stable over time
Limited robustness and versatility
Dante Robot
Raibert Monopod
Intro

7. Motivation Robust and Dynamic
Robustness: Rapid convergence to desirable behavior steady-state despite large disturbances
Dynamic: significant transfers of kinetic and potential energies
Deathhead Cockroach
Intro

8. Recent Work Passively-stable walking (McGeer, 1990)
Self-stabilizing running (Ringrose, 1997)
Rhex (Saranli, 2000)
Intro

9. Hypothesis
Robust and Dynamic locomotion can be achieved with no sensory feedback…
Disturbance-rejection is a property of the mechanical system…
…tuned to a feedforward (open loop) activation
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Simulation Response over 0.01 step
Trajectory
X (meters)
Z (meters)
Intro

10. Biological Inspiration
Up to 50 body-lengths per second
Traverse terrain with obstacle three times height of center of mass
Prof. Robert Full, Berkeley Polypedal Lab
Biology

11. Biological Inspiration
When transitioning from flat to rough terrain…
…impulses sent to the muscles did not noticeably change
Similar activation despite large changes in events
Flat
terrain
Rough
terrain
Biology

12. Biological Inspiration
Implies exclusion of sensory feedback
No precise foot-placement or “follow-the-leader gait”
But still able to traverse rough terrain..!
Biology

13. Preflexes Passive properties of the mechanical system…
…that stabilize and reject disturbances
Immediate response
No delays associated with sense-compute-command loops
Mechanical
System
(muscles, limbs)
Environment
Feedback
(Preflexes)
Feedforward
Motor Pattern
Passive Dynamic
Self-Stabilization
Locomotion
Biology

14. Preflexes – Self-Stabilizing Posture
Sprawled posture
Individual leg function
Front legs decelerate, hind legs accelerate
Self-correcting forces with respect to the geometry
Biology

15. Preflexes – Visco-elastic Properties
Exoskeleton and muscle properties
Compliance
Damping
Hysteresis loop
@10Hz
Biology

16. Control Hierarchy
Neural System
(CPG)
Preflexes provide immediate stabilization for repetitive task
Reflexes and neural feedback adapt to changing conditions…
…through the feedforward pattern
Feedforward
Motor Pattern
Sensory
Feedback
(Reflexes)
Mechanical
System
(muscles, limbs)
Mechanical
Feedback
(Preflexes)
Environment
Passive Dynamic
Self-Stabilization
Locomotion
Biology

17. Modeling Initial attempts at characterizing stability and performance…
…of a feedforward activation pattern…
…applied to a properly designed passive mechanical system
Modeling

18. Modeling - Mode Transitions
Tripod 2
Tripod 1
Locomotion is a series of transitions between modes
Here, modes are determined by the feedforward pattern…
…especially if we don’t account for a flight phase
Modeling

19. Modeling – Linear systems
Show that feedforward mode transitions…
…result in stable, converging periodic motion
Modeling

20. Modeling – Non-linear 2 DOF
Servo
Simple model
Opposing legs with passive properties
At fixed times, legs are given an impulse extension
Sagittal
plane
k, b, nom
Planar
“Quadruped”
simplified
model
Modeling

21. Modeling – Non-linear 2 DOF
Leg Set 2 1
State
Time x
= state trajectory
Stride Period
t = 2T+ T g
At beginning of mode…
Modeling

22. Modeling – Non-linear 2 DOF
Leg Set 2 1
State
Time x
= state trajectory
Stride Period
t = 2T + 1/3T T g
At beginning of mode…
…the mass moves…
Modeling

23. Modeling – Non-linear 2 DOF
Leg Set 2 1
State
Time x
= state trajectory
Stride Period
t = 2T + 2/3T T g
At beginning of mode…
…the mass moves…
…according to the mode’s dynamics
Modeling

24. Modeling – Non-linear 2 DOF
Leg Set 2 1
State
Time x
= state trajectory
Stride Period
t = 3T- T g
At a fixed time…
Modeling

25. Modeling – Non-linear 2 DOF
Leg Set 2 1
State
Time x
= state trajectory
Stride Period
t = 3T+ T g
At a fixed time…
…the system transitions to the new mode…
…carrying the state conditions into the next mode
Modeling

26. Modeling – Non-linear 2 DOF
Simulations show that for a wide range of system parameters…
…trajectories converge to stable periodic motion…
…despite large disturbances
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Perturbation Response over 3 Mode Transitions
X (meters)
Z (meters)
Nominal Orbit
Perturbed Trajectory
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Simulation Response over 0.01 step
Trajectory
Modeling

27. Modeling – Floquet Analysis
Behavior is confirmed by Floquet analysis
Small perturbation analysis
Floquet multipliers indicate attractiveness of periodic motion
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Damping (N-s/m)
Robustness
Horizontal
Velocity
X_dot (m/s)
1/max[eig(M)]
= nominal trajectory
Modeling

28. Modeling – System Behavior
“Chasing an equilibrium”
Equilibrium changes at fixed times according to activation pattern
System parameters influence trajectory within mode
Statically
Unstable
Region
Initial
condition
Mode
Equilibrium
Trajectory
Leg
Extension
Limit
Leg Pre-
Compressions
Modeling

29. Prototypes Built prototypes based on biological principles described
No active sensing
Fixed cycle of tripod activation
time
Feedforward Activation
Pattern
Prototypes

30. Prototype - Design Sprawled Posture Leg function Compliant joints
Center of
Mass
Forward
Direction
5.5 cm
Top view
Side view
Prototypes

31. Prototype - Design Passive compliant hip joint in sagittal plane
Rotary Joint
Active
Thrusting
Force
Passive compliant hip joint in sagittal plane
Piston thrusts along direction of hip
Prototypes

32. Prototype - Fabrication
Fabrication for robustness
Active components embedded inside structure
Integrated soft-hard materials in joints
Shape Support Material
Embed servos & wiring
Deposit and Shape
Body Material
Cycle of Integrated
Fabrication of Robot
Prototypes

33. Prototype - Performance
Dynamic running
Speeds of up to 3 body-lengths per second (40 cm/sec)
Prototypes

34. Prototype - Performance
Obstacles of hip-height
Slopes of up to 18 deg.
Prototypes

35. Prototype - Movie
>O>
Prototypes

36. Conclusions
Findings from biomechanics suggests that robust dynamic locomotion…
…can be achieved without sensory feedback
Prototypes and simulations confirm fast, stable performance
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Simulation Response over 0.01 step
Trajectory
X (meters)
Z (meters)
Future

37. Future Work Characterize role of system properties…
…to design for appropriate performance
Using higher level feedback (reflexes) for adaptation
Future

38. Questions? Acknowledgements
ONR, NSF
Jonathan Clark, Pratik Nahata, Ed Froehlich
Stanford DML and RPL
Intro
Biology
Modeling
Prototypes
Future