introduction to biped walking lecture 1 background, simple dynamics, and control
TRANSCRIPT
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Introduction to Biped Walking
Lecture 1
Background, simple dynamics, and control
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Some Sample Videos
• Human Walk.avi
• Hubo straight leg.avi
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Human Leg Anatomy
Torso
Hip, 3DOF
Knee, 1DOF
Ankle, 2DOF
Toes, ~2 DOF
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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.
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Walking Cycle (2D)
Kim, Jung-Yup (2006)
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Stages
Kim, Jung-Yup (2006)
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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)
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Block Diagram of KHR-2
Kim, Jung-Yup (2006)
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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
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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
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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
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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”
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Joint Motor Controller Basics
• DC brush motors
• Harmonic drive gear reduction
• Simple governing equations
• Inefficient at low speeds
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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
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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
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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
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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
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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
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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)
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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)
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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
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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)(
)(
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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
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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
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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
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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)
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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
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Timing of walking cycle
• Short double support phase (<10% of half cycle)
• Knee compression and extension
• Short landing phase
Kim, Jung-Yup (2006)
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Trajectory Parameters
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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