The Linear Biped Model and Application to Humanoid Estimation and Control Benjamin Stephens Carnegie Mellon University Monday June 29, 2009 Introduction 2 Motivation Robotics Find simple models for complex systems Develop algorithms that use simple models to make humanoid control simpler Better way to understand and explain dynamic balance and locomotion Human Physiology Evaluating biomechanical models Understand and prevent falls, which can lead to

hip/wrist fractures. 3 Take-Home Message The Linear Biped Model is a simple model of balance that can describe a wide range of behaviors and be directly applied to humanoid robot estimation and control 4 Outline Modeling Balance Overview Linear Biped Model Orbital Energy Control Lateral Foot Placement Control Humanoid Robot Center of Mass Estimation Feed-forward Control

Future Work Conclusion 5 Modeling Balance 6 Intro to Modeling Balance Sum of forces Center of pressure Fy Base of support Fy FL Fg Feq Feq

FR 7 Fg FR Feq Feq FL center of pressure center of pressure Linear Inverted Pendulum Model Features: y All mass concentrated at CoM Massless legs

Does not move vertically Linear I mgL sin (Linearize) mLy mgy g g y y y ycop L mg L 8 L F

ycop y Kajita, S.; Tani, K., "Study of dynamic biped locomotion on rugged terrain-derivation and application of the linear inverted pendulum mode," IEEE International Conference on Robotics and Automation, vol.2, pp.1405-1411, 1991. 0 Stability of Linear Inverted Pendulum Whats the best we can do? Apply maximum allowable force to the ground Move center of pressure to edge of base of support g g y max y y max L

mL L mL y d y d k 9 Benjamin Stephens, "Humanoid Push Recovery," The IEEE-RAS 2007 International Conference on Humanoid Robots, Pittsburgh, PA, November 2007 The Linear Biped Model Weighted sum of the dynamics due to two linear inverted pendulum models (rooted at the feet) y my FLY FRY mg FLZ FRZ F FLY L LZ y y L

L L u F FRY R RZ y y R L L u mg y yL wR u mg y yR my wL L L L L mg wL mg wR mg wL wR 1 10

R wR u L wLu L FR R FRZ wR mg FLZ wL mg Benjamin Stephens, " Energy and Stepping Control of Linear Biped Model in the Coronal Plane," Submitted to The IEEE-RAS 2009 International Conference on Humanoid Robots. FL yR y L yL

The Double Support Region We define the Double Support Region as a fixed fraction of the stance width. y D W 1 yD wL 2D 0 wR 1 wL 11 , y D ,

y D , y D L FR R FL y 0 2D 2W L 2d

Dynamics of Double Support The dynamics during double support simplify to a simple harmonic oscillator u mg y yL wR u mg y yR my wL L L L L mg D y y W D y y W my R L L 2 DL g y R L

D W y mL DL g y y L mL LIPM Dynamics 12 y g u y L mL 1

Stability of the Linear Biped Model Whats the best we can do? Apply maximum allowable force to the ground Move center of pressure to edge of base of support u u gg mgd g mgd yy max y y max LL mL mL L mL d

FZ R wR mgd L wL mgd R L u mgd 13 Phase Space of LiBM y Double Support Region y FFR R yR 14 Location of feet

yL FLFL Controlling Balance 15 Static Balance Control Goal: Return to a state of static balance (zero velocity) Strategies: 16 Periodic Balance Goal: Balance while moving in a cyclic motion, returning to the cycle if perturbed. y y

17 Slow Fast Swaying Swaying Marching in Place or Walking Orbital Energy Control Orbital Energy: 1 2 g 2 E y y 2 2L Solution is a simple harmonic oscillator: g 2 LEd

1 2 g 2 y y Ed 0 y sin t 2 2L g L We control the energy: e Ed E 18 g e Ke 0 y y y Ke 0 L g

y y Ky Ed E L 19 Energy Control Trajectories 0.4 0.3 0.2 y-vel 0.1 0 -0.1 -0.2

-0.3 -0.4 20 -0.2 -0.15 -0.1 -0.05 0 y-pos 0.05 0.1 0.15

0.2 Stepping Control Because we define double support region, when to step is pre-determined, we only have to decide how far to step x2 u1 21 yR y x0 u0

x1 DSP region moves y L y N-Step Controller Because DSP region is fixed, we know when to take a step, only need to decide where N-Step lookahead over a set foot step distances 2 cost K1 stance _ width K 2 y Benefits: Very fast Works for any desired energy Recovers from Pushes Stabilizes position 22 2 23

24 Application to Humanoid Balance 25 Humanoid Applications Linear Biped Model predicts gross body motion and determines a set of forces that can produce that motion State Estimation Combine sensors to predict important features, like center of mass motion. Feed-Forward Control Perform force control to generate the desired ground contact forces. 26

Robot Sensing Overview High Level Controller PROCESS NOISE Joint Level Controller State Estimate Robot MEASUREMENT NOISE Estimate Fusion & Filter Position Measurement Kinematics

Model Potentiometers MEASUREMENT NOISE Flatness Calculation Force/Torque Sensors Force Measurement Robot Model Acceleration Measurement Acceleration Estimate Joint Torques

IMU MEASUREMENT NOISE Center of Mass Filtering A (linear) Kalman Filter can combine multiple measurements to give improved position and velocity center of mass estimates. Joint Kinematics Hip Accelerometer Kalman Filter Periodic Periodic Humanoid CoM

State Humanoid Balance Balance Feet Force Sensors NOTE: Because we measure force, we should also be able to estimate push/disturbance magnitudes 28 29 Feed-Forward Force Control LiBM can be used for feedforward control of a complex biped system. Torques can be generated by force control of the CoM with respect to each foot

L J LT FL R J RT FR J R (q) J L (q) Additional controls are applied to bias towards a home pose and to keep the torso vertical. FR FL 30 0.1 velocity 0.05 0

-0.05 -0.1 -0.02 31 -0.01 0 0.01 position 0.02 0.03 Movie Summary

32 Conclusion The Linear Biped Model is a simple model of balance that can describe a wide range of behaviors and be directly applied to humanoid robot estimation and control Joint Kinematics y Slow Swaying Fast Swaying y Periodic Periodic CoM State Humanoid

Humanoid Balance Balance Kalman Filter Hip Accel Force Sensors L y FR R FL

J R (q) L J L (q) Marching in Place or Walking FR FL 33 Future Work x y 3D Linear Biped Model FLy FLx

Refine Robot Behaviors z Foot Placement Push Recovery x y FLz FRy FRz Walking Robust Control/Estimation Sliding Mode Control of LiBM FRx Rx

Ry Lx Push Force Estimation Online LiBM Parameter Estimation/Adaptation 34 Ly The End Thanks to Research Committee Members: Chris Atkeson Jessica Hodgins Martial Herbert Stuart Anderson Questions? 35 37

Dynamic Constraints d FZ RSP DSP LSP mgd R wR mgd L L wL mgd u mgd

u R L y R mgd 2D 38 Friction Constraints on LiBM FLY FLZ L mg y yL L L L L mg

L mg L y y L L L mg L y y L L u mg y yL L L L L mg mg L y y L u mg L y y L mg L y y R u mg L y y R L mgd L L mgd Double Support Right

Support R L Left Support mgd L R 2W mgd y L mg L y y L L L mg L y y L

Hybrid Orbital Energy In DSP region, we use the same energy equation as before, x is relative to half way between feet 1 g 2 E DSP x 2 x 2 2L In SSP region, we use the orbital energy, x is relative to stance foot 1 g 2 ESSP x 2 x

2 2L Energy at middle of SSP determines curve DSP region!