Introduction to Biped Walking

Lecture 1

Background, simple dynamics, and control

Some Sample Videos

• Human Walk.avi

• Hubo straight leg.avi

Human Leg Anatomy

Torso

Hip, 3DOF

Knee, 1DOF

Ankle, 2DOF

Toes, ~2 DOF

Building Blocks of Biped Walking

• Dynamic modeling• Trajectory generation• Inverse kinematic model• Trajectory error

controllers• Additional failure mode

controllers• Mechatronics • Programming

• Provides virtual experimentation platform

• The ideal path that the hips and feet follow.

• Specifies the joint movements to make feet and hips follow the trajectory

• Specify how the joints should move to compensate for trajectory error.

• Adjusts the trajectory to compensate for nonidealities.

• The structure and implementation and the limitations thereof

• Reading sensors, processing and filtering their data, sending joint position commands.

Walking Cycle (2D)

Kim, Jung-Yup (2006)

Stages

Kim, Jung-Yup (2006)

Controllers

• Damping Controller reduces reactive oscillations to swinging legs

• ZMP controller minimizes ankle torque and optimizes hip trajectory

• Landing controller limits impact forces at foot, controls foot angle

• Torso/pelvis controllers follow prescribed trajectory

• Tilt-over controller adjusts foot placement if ZMP becomes unstable

• Landing position controller adjusts foot landing to compensate for excess angular velocity

Kim, Jung-Yup (2006)

Block Diagram of KHR-2

Kim, Jung-Yup (2006)

Balance Control

• Controls Center of mass location– Prevents tiltover– Controls foot placement during landings

• Consists of:– Torso sway damping controller– ZMP controller– Foot placement controller– Foot Landing Controller

Single Support Vibration Modeling

• Compliance between ankle and torso

• Model robot body as lumped mass

• Model flexible parts and joints as spring

• Use Torque along X axis of ankle to counteract motion

• Linearize with small angle

0)()sin(2 Tukmgml

Vibration Damping Control

• Apply Laplace Transform

• Factor out Θ(s) and U(s) to form transfer function

• Substitute to find TF of Torque wrt input angle

)()(1

)()())()()(

)(

)(

)()()(

))()()()()(

)

22

2

22

2

22

2

22

2

22

22

2

sU

lg

mlk

s

lg

ssU

lg

mlk

s

mlk

sUsU

lg

mlk

s

mlk

sUsk(sT

lg

mlk

s

mlk

sU

s

sksmlmglskU

sUsk( ssmlsmglsT

uk(θ θmlT = mglθ

Damping Controller

• Substitute β= K/ml2−g/l

α=K/ml2

• Apply derivative feedback of error

• Simulation shows effect of damping on vibrations

• (See )“vibdamp.mdl”

Joint Motor Controller Basics

• DC brush motors

• Harmonic drive gear reduction

• Simple governing equations

• Inefficient at low speeds

Joint Motor Controller

bKv

iT

dt

diLRiKvV

out

Motor Voltage/Speed constant (V-s/rad)

Output Torque (N-m)

Rotor Inductance (Henry)

Rotor Resistance (Ω)

Input Voltage (V)

Motor equivalent viscous friction (N-m-s)

Current (Amp)

Block Diagram of System

Effects of Motor on Control

• Torque limit due to R– torque inversely

proportional to speed– High current (and

heat) at zero speed

rout JbKv

iT

dt

diLRiKvV

rout JKv

iT

RiV

,0

Ankle model with motor

• Assume simple inverted pendulum

• Combine electrical and mechancal ODE’s

sinmglmgT

bKv

iT

NKRiV

out

out

v

22

2

2

)1(sin

)sin(

)sin(

Rml

NRb

l

g

RKml

V

NKmglbNmlRKV

imglbNmlK

v

vv

v

Zero Moment Point

• Point about which sum of inertia and gravitational forces = 0

• Requires no applied moment to attain instantaneous equilibrium

• Control objective: minimize horizontal distance between COM and ZMP

x

g

0, rr MF

Single Support Model

• Divide ZMP control into 2 planes

• Track hip center to ZMP• Requires dynamic model or

experiment to determine model parameters

• Pole placement compensator

• (See “ZMP.mdl”)

Kim, Jung-Yup (2006)

Double inverted pendulum

Foot Landing Placement

• IMU measures X and Y angular velocity

• Hip sway monitored by trajectory controllers

• Excess angular velocity reduced by widening landing stance

• Reduced angular velocity maintains hip trajectory

Kim, Jung-Yup (2006)

Landing Problem

• Foot landing causes impact and shock to system

• Dynamics of shock are difficult to model

• Large reaction forces• Angular momentum

controlled with 1 ankle

Before After

v2

v’1=0

v’2

v1

Fz(t)

M(t)

Simplified Collision Dynamics

• Governing Formulas

• Impact Energy Losses

• Power Input

ImpactBefore After

v2

v1

221

Lmvdmv

tFvm

21

22

21

22

)1cos(cos2

)(2

vm

T

vvm

T

stridefTimp

T

s

impP

Deriving the ideal model

• Ideal mass-spring-damper

• mT≈53kg (hubo’s mass)

• c, k = model constants• Form transfer function• Solve numerically

)(tfkyycym

mT

y

c k

mk

smc

s

mk

sU

sY

2)(

)(

Dynamic Model of knee

• Lump mass of torso at hip• Lagrange method to derive

dynamics • Add artificial damping to

reduce simulation noise• Use PID control to stabilize

),( yx

22 ,T

11,T

22 , lm

11, lm

mT

2lc

1lc

Knee Inverse Kinematics

• Need to solve θi(x,t) (i=1,2)

• Desired path along y axis (x=0)

• Setup constraint equations & solve

• Apply as input to model

),( yx

2,

2

)cos()cos(

)sin()sin(

21

21

21

yll

ll

22 ,T

11,T

22 , lm

11, lm

)(t

Trajectory Generation

“Goal” Control

• Needs no knowledge of model

• Low computation overhead

• Non-optimal path

Trajectory Feedforward

• Requires mathematical model

• Input conditioned for system

• Requires online computation

• Allows path optimization

Hubo’s Hip Trajectory

• Y=A*sin(ωt)– A=sway amplitude

– Ω= stride frequency (rad/s)

• Simplifies frequency domain design

• X=c*A1cos (ωt)+(1-c)A2*t

– A2=A1*π/(2 ω)

• c controls start/end velocity

• Amplitude A1 controls step length

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10.2

0.25

0.3

0.35

0.4

0.45

0.5

0.55

0.6

0.65Trajectory of Hip: X direction

dist

ance

(mm

)

time (s)

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1Trajectory of Hip: Y direction

dist

ance

(mm

)

time (s)

Basic foot trajectory

• Continuous function of t

• 0 velocity at each full cycle

• Velocity adjustable by linear component

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

20

40

60

80

100

120

140

160

180

200Trajectory of Foot: X direction

dist

ance

(mm

)

time (s)

200/(2)*(2t/N-sin(2t/N))

Cycloid function

Timing of walking cycle

• Short double support phase (<10% of half cycle)

• Knee compression and extension

• Short landing phase

Kim, Jung-Yup (2006)

Trajectory Parameters

What’s Next

Biped Design Procedure

• Concepts• Dynamic modeling• Simulations• Trajectory generation

Next Lecture:

• Fundamentals of dynamics

• Fundamentals of controls• 2d dynamic modeling• Implementing posture

control systems• Basic X and Z axis

trajectories