introduction to humanoid robotics - seattle robotics · introduction to humanoid robotics by dr....

Introduction to Humanoid Robotics

by

Dr. Rawichote Chalodhorn (Choppy) Humanoid Robotics Lab, Neural System Group,

Dept. of Computer Science & Engineering, University of Washington.

RoboCup soccer

The brain controlled humanoid robot

Outline

• History of humanoid robots• Humanoid robots today• Androids• Analytical approaches of bipedal

locomotion• Learning to walk through imitation• Future of humanoid robotics

Atomaton: Leonardo's robot

1495

Atomaton: The Japanese tea serving doll

18th century to 19th century

Honda E series

Honda P series

P1 P2 P3

1993 1996 1997

AIST / Kawada Industries : HRP series

HRP-4C

Androids

Prof. Hiroshi Ishiguro

• A robot that closely resembles

a human• Human robot interaction

Zero-Moment Point (ZMP)

Zero-moment point1010//8888Hiro Hirukawa The Onassis Foundation Lecture Series 2006

What is ZMP (Zero Moment Point)?What is ZMP (Zero Moment Point)?

Vukobratovic and Stepanenco, 1972

Center of Pressure

•ZMP NEVER leaves the support polygon!•ZMP can be measured by force sensors in feet.

!

Chapter 2

The Zero-Moment Point

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2.1 Definition

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We can maintain balance as long as the ZMP is staying within the support polygon.

[Vukobratovic et al. 1972]

Derivation of the ZMP

Vectors of a walking mechanism

[Takanishi et al., 1989]

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Derivation of the ZMP

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[Huang et al., 2001]

ZMP equations with external forces and moments

A simplified version of ZMP equations

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[Kajita et al. 2003]

The Cart-Table Model

In this paper we introduce a novel walking patterngeneration that allows arbitrary foot placements as amixture of the ZMP based and the inverted pendulumbased approaches. It is also shown that by using thepreview controller, we can take into account of theprecise multibody dynamics although our method isbased on a simple inverted pendulum model.

2 Dynamic Models of Biped Robot

2.1 3D Linear Inverted Pendulum Modeand Zero-moment point

When we apply a constraint control to an invertedpendulum such that the mass should move along anarbitrary defined plane, we obtain a simple linear dy-namics called the Three-Dimensional Linear InvertedPendulum Mode (3D-LIPM)[9, 11]. We take Cartesiancoordinates as shown in Figure 2 and specify the x-axisas the ordinal walking direction. The constraint planeis represented with given normal vector (kx, ky,!1)and z intersection zc as

z = kxx + kyy + zc. (1)

If the constraint plane is horizontal (kx = ky = 0),the dynamics under the constraint control is given by

y =g

zcy ! 1

mzc!x, (2)

x =g

zcx +

1mzc

!y, (3)

where m is the mass of the pendulum, g is gravityacceleration and !x, !y are the torques around x-axisand y-axis respectively.

Even in the case of the sloped constraint wherekx, ky "= 0, we can obtain the same dynamics by ap-plying additional constraint

!xx + !yy = 0, (4)

for the input torques.Eqs. (2) and (3) are linear equations. The only pa-

rameter which governs those dynamics is zc, i.e., the zintersection of the constraint plane and the inclinationof the plane never a!ects the horizontal motion.

For the 3D-LIPM with the horizontal constraint(kx = ky = 0), we can easily calculate the zero-moment point (ZMP), which is widely used in bipedrobot research [13],

px = ! !y

mg,

py =!x

mg, (5)

Figure 2: A pendulum under constraint

where (px, py) is the location of the ZMP on the floor.By substituting Eqs. (5) to the 3D-LIPM ((2) and(3)) we obtain

y =g

zc(y ! py), (6)

x =g

zc(x ! px). (7)

2.2 ZMP equations and cart-table model

To control the ZMP, it should be the outputs of thesystem while it appears as the inputs of the 3D-LIPMin the last section. Therefore, we rewrite Eqs. (6) and(7) to have the ZMP as their outputs as

py = y ! zc

gy, (8)

px = x ! zc

gx. (9)

In the following part of this paper, we will refer theabove equations as the ZMP equations.

Figure 3 shows a suggestive model directly corre-sponds to these equations. It depicts a running cartof mass m on a pedestal table whose mass is negligible(we need two sets of a cart on a table for the motionof x and y).

As shown in the figure, the foot of the table is toosmall to let the cart stay on the edge. However, if thecart accelerates with a proper rate, the table can keepupright for a while. At this moment, the ZMP existsinside of the table foot. Since the moment around theZMP must be zero, we have

!zmp = mg(x ! px) ! mxzc = 0. (10)

We can verify that this yields the same equation toEq. (9).

1621

1414//8888Hiro Hirukawa The Onassis Foundation Lecture Series 2006

CartCart--Table ModelTable Model

• A running cart on a mass-less table

• The table has a small support area

M

cz

x0

x

O0ZMP =!

x!!

x

cz

xp

m

O

Figure 3: A cart-table model

3 Walking pattern generation for givenZMP

3.1 Pattern generation as an inverseproblem

When we represent a robot as the cart-table modeland give the the cart motion as the trajectory of thecenter of mass (CoM) of the robot, we can easily cal-culate the resulted ZMP by using the ZMP equationsEqs. (8) and (9).

On the other hand, a walking pattern generation isthe inverse problem of this. That is, the cart motionshould be calculated from the given ZMP trajectory,which is determined by the desired footholds and stepperiod.

Takanishi et al. proposed to solve this problem byusing Fourier Transformation [14]. By applying theFast Fourier Transformation (FFT) to the ZMP refer-ence, the ZMP equations can be solved in frequencydomain. Then the inverse FFT returns the resultedCoM trajectory into time domain.

Kagami, Nishiwaki et al. proposed a method tosolve this problem in the discrete time domain [15].They showed the ZMP equation can be discretized asa trinomial expression, and it can be e!ciently solvedby an algorithm of O(N) for the given reference dataof size N .

Both methods are proposed as batch processes thatuse a ZMP reference of certain period and generatethe corresponding CoM trajectory. To generate con-tinuous walking pattern for a long period, they mustcalculate entire trajectory by o"-line or must connectthe piece of trajectories calculated from the ZMP ref-erence divided into short segments.

3.2 ZMP control as a servo problem

Let us define a new variable ux as the time deriva-tive of the horizontal acceleration of CoM.

d

dtx = ux (11)

Regarding ux as the input of Eq. (9), we can translatethe ZMP equation into a strictly proper dynamicalsystem as

ddt

!

"xxx

#

$ =

!

"0 1 00 0 10 0 0

#

$

!

"xxx

#

$ +

!

"001

#

$ ux

px =%

1 0 !zc/g&!

"xxx

#

$ .

(12)

For Eq. (8), we define uy and obtain the system of thesame form.

By using the dynamics of Eq. (12) we can con-struct a walking pattern generator as a ZMP trackingcontrol system (Figure 4). The system generates theCoM trajectory such that the resulted ZMP followsthe given reference. However, we must consider an

xuZMP

reference

Servo

Controller

Dynamic ZMP

equation (12)

p

refp

x

+ ! p

ZMP

CoM

Figure 4: Pattern generation as ZMP tracking control

0 0.5 1 1.5 2 2.5!0.05

0

0.05

0.1

0.15

0.2

0.25

0.3

x [m

]

time [s]

zmp ref

x

Output

Input

x

ZMP

Figure 5: ZMP and CoM trajectory

interesting feature of this problem as follows. Figure5 illustrates the ideal trajectories of the ZMP and the

1622

Why CoM?





z = kxx + kyy + zc. (1)


y =g

zcy ! 1

mzc!x, (2)

x =g

zcx +

1mzc

!y, (3)



!xx + !yy = 0, (4)




px = ! !y

mg,

py =!x

mg, (5)



y =g

zc(y ! py), (6)

x =g

zc(x ! px). (7)



py = y ! zc

gy, (8)

px = x ! zc

gx. (9)




!zmp = mg(x ! px) ! mxzc = 0. (10)


1621





z = kxx + kyy + zc. (1)


y =g

zcy ! 1

mzc!x, (2)

x =g

zcx +

1mzc

!y, (3)



!xx + !yy = 0, (4)




px = ! !y

mg,

py =!x

mg, (5)



y =g

zc(y ! py), (6)

x =g

zc(x ! px). (7)



py = y ! zc

gy, (8)

px = x ! zc

gx. (9)




!zmp = mg(x ! px) ! mxzc = 0. (10)


1621

ZMP





z = kxx + kyy + zc. (1)


y =g

zcy ! 1

mzc!x, (2)

x =g

zcx +

1mzc

!y, (3)



!xx + !yy = 0, (4)




px = ! !y

mg,

py =!x

mg, (5)



y =g

zc(y ! py), (6)

x =g

zc(x ! px). (7)



py = y ! zc

gy, (8)

px = x ! zc

gx. (9)




!zmp = mg(x ! px) ! mxzc = 0. (10)


1621

The ZMP of the Cart-Table Model

Dynamics equations





z = kxx + kyy + zc. (1)


y =g

zcy ! 1

mzc!x, (2)

x =g

zcx +

1mzc

!y, (3)



!xx + !yy = 0, (4)




px = ! !y

mg,

py =!x

mg, (5)



y =g

zc(y ! py), (6)

x =g

zc(x ! px). (7)



py = y ! zc

gy, (8)

px = x ! zc

gx. (9)




!zmp = mg(x ! px) ! mxzc = 0. (10)


1621





z = kxx + kyy + zc. (1)


y =g

zcy ! 1

mzc!x, (2)

x =g

zcx +

1mzc

!y, (3)



!xx + !yy = 0, (4)




px = ! !y

mg,

py =!x

mg, (5)



y =g

zc(y ! py), (6)

x =g

zc(x ! px). (7)



py = y ! zc

gy, (8)

px = x ! zc

gx. (9)




!zmp = mg(x ! px) ! mxzc = 0. (10)


1621

ZMP Model-based Feedback controller configuration[Kajita et al. 2003]

2424//8888Hiro Hirukawa The Onassis Foundation Lecture Series 2006

Configuration of the Feedback ControllerConfiguration of the Feedback Controller

Gyro

Accelerometer

Force/torque

sensor

Joint servo

Kalmanfilter

Bodyinclination

Stabilizer

References of Joint angles,

body inclination and ZMP

Balancer Module

encoder

ZMPZMP

calculator

Learning Approach[Chalodhorn et al. 2006]

Learning to walk through imitation

Dimension reduction[Grochow et al. 2004]Gaussian process latent variable model

Learning through low-dimensional spaces

[Chalodhorn et al. 2005]

!

Figure 6. Optimization result for a walking motion pattern in a low-dimensional subspace based on an action subspace embed-ding.

We summarize below the entire optimization and action selection process:

1) Use PCA to represent in a reduced 3D space the ini-tial walking gait data from human motion capture.

2) Employ the non-linear embedding algorithm for pa-rameterization of the gait.

3) Start the learning process by projecting actions back to the original joint space and executing the corre-sponding sequence of servo motor commands in the Webots HOAP-2 robot simulator [Webots, 2004].

4) Use the sensory and motor inputs from the previous step to update the sensory-motor predictor as de-scribed in Section 4 where the state vector is given by the gyroscope signal of each axis and the action

variables are !,r and h in the low-dimensional sub-

space. 5) Use the learned model to estimate actions according

to the model predictive controller framework de-scribed above (Figure 5).

6) Execute computed actions and record sensory (gyro-scope) feedback.

7) Repeat steps 4 through 6 until a satisfactory gait is obtained.

6 Experimental Results

Figure 7. Motion pattern scaling. The target motion pattern is

scaled down until it can produce a stable motion to start the motion

optimization process.

This section explains how the optimization methodology in the previous section is used in conjunction with the mocap data. From our study of the motion pattern in the reduced subspace, we found that we can scale up and down the mo-tion pattern and get similar humanoid motion patterns ex-cept for changes in the magnitude of motion. When we scale down the pattern in the reduced subspace, it produces a smaller movement of the humanoid robot, resulting in s-maller changes in dynamics during motion. Our strategy is to scale down the pattern until we find a dynamically stable motion and start learning at that point. We apply the motion optimization method in Section 5 to the scaled-down pattern until its dynamic performance reaches an optimal point; then we scale up the trajectory of the optimization result toward the target motion pattern. In our experiments, we found that a scaling down of 0.3 of the original motion pat-tern is typically stable enough to start the learning process. Our final optimization result obtained using this procedure is shown as a trajectory of red circles in Figure 7. It corre-sponds to about 80% of the full scale motion.

-5

0

5

-3-2-101234

-4

-2

0

2

4

Target pattern

Final result Scale 0.7

Scale 0.3

Scale 0.5

-5

0

5-4 -2 0 2 4

-4

-2

0

2

4 Original data

Embed function

Optimized dataSearch space

-5

0

5-4 -2 0 2 4

-4

-2

0

2

4 Original data

Embed function

Optimized dataSearch space

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0 50 100 150 200 250 300-2

-1

0

1

2Gyroscope signal of X-axis

0 50 100 150 200 250 300-2

-1

0

1

2Gyroscope signal of Y-axis

0 50 100 150 200 250 300-2

-1

0

1

2Gyroscope signal of Z-axis

Original : RMS = 0.3795Optimized : RMS = 0.0533



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Learning by imitation results

Intrinsically low-dimensional humanoid motion

Conclusion

• Humanoid robots are used as a

research tool in several scientific areas.

• Although the initial aim of humanoid

research was to build better orthosis

and prosthesis for human beings.

http://en.wikipedia.org/wiki/Orthosis

http://en.wikipedia.org/wiki/Orthosis

http://en.wikipedia.org/wiki/Prosthesis

http://en.wikipedia.org/wiki/Prosthesis

Future of Humanoid robots

THANK YOU

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