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

Robust Dynamic Locomotion Background and Motivation Biological Inspiration 0.07 0.08 0.09 0.1 0.11 0.12 0.13 0.14 0.15 Trajectory Modeling Approach Hexapedal Prototypes Conclusions And Future Work Biology Modeling Prototypes

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

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

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

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

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

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

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 0.07 0.08 0.09 0.1 0.11 0.12 0.13 0.14 0.15 0.01 0.02 0.03 Simulation Response over 0.01 step Trajectory X (meters) Z (meters) Intro

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

Modeling – Non-linear 2 DOF Simulations show that for a wide range of system parameters… …trajectories converge to stable periodic motion… …despite large disturbances 0.83 0.84 0.85 0.86 0.87 0.88 0.005 0.01 0.015 0.02 0.025 Perturbation Response over 3 Mode Transitions X (meters) Z (meters) Nominal Orbit Perturbed Trajectory 0.07 0.08 0.09 0.1 0.11 0.12 0.13 0.14 0.15 0.03 Simulation Response over 0.01 step Trajectory Modeling

Modeling – Floquet Analysis Behavior is confirmed by Floquet analysis Small perturbation analysis Floquet multipliers indicate attractiveness of periodic motion 6.5 7 7.5 8 8.5 9 9.5 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) Robustness Horizontal Velocity X_dot (m/s) 1/max[eig(M)] = nominal trajectory Modeling

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

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

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

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

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

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

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

Prototype - Movie >O> Prototypes

Conclusions Findings from biomechanics suggests that robust dynamic locomotion… …can be achieved without sensory feedback Prototypes and simulations confirm fast, stable performance 0.07 0.08 0.09 0.1 0.11 0.12 0.13 0.14 0.15 0.01 0.02 0.03 Simulation Response over 0.01 step Trajectory X (meters) Z (meters) Future

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

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