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