Machine Learning and Robot Perception - Bruno Apolloni et al (Eds) Part 6 pdf

Subsampling the signal by factor 2, then reconstructing the signal at this level; b Estimating: Using the RLSM to process the linear velocity signal and the angular velocity signal obt

To subband a signal, Discrete Wavelet Transform is used As shown in Fig 3.4, h (n) and g (n) are a lowpass filter and a highpass filter, respec-tively The two filters can halve the bandwidth of the signal at this level Fig 3.4 also shows the DWT coefficients of the higher frequency compo-nents at each level

As a result, the raw signal is preprocessed to have the desired low quency components The multiresolution approach from discrete wavelet analysis will be used to decompose the raw signal into several signals with different bandwidths This algorithm makes the signal, in this case, the raw angular velocity signal passes through several lowpass filters At each level it passes the filter, and the bandwidth of the signal would be halved Then the lower frequency component can be obtained level by level

fre-The algorithm can be described as the following procedures:

(a) Filtering: Passing the signal through a lowpass Daubechies filter

with bandwidth which is the lower half bandwidth of the signal at the last level Subsampling the signal by factor 2, then reconstructing the signal at this level;

(b) Estimating: Using the RLSM to process the linear velocity signal and

the angular velocity signal obtained from the step (a) to estimate the matic length of the cart

kine-(c) Calculating: Calculating the expectation of the length estimates and

the residual

(d) Returning: Returning to (a), until it can be ensured that e~ is ing

increas-(e) Comparing: Comparing the residual in each level, take the estimate of

length at a level, which has the minimum residual over all the levels, as the most accurate estimate

The block diagram of DWMI algorithm is shown in Fig 3.5

Fig 3.5 Block Diagram of Model Identification Algorithm

3.4 Convergence of Estimation

In this section, the parameter estimation problem in time domain is lyzed in frequency domain The estimation convergence means that the es-timate of the parameter can approximately approach the real value, if the measurement signal and the real signal have an identical frequency spec-trum First, we convert the time based problem into frequency domain through Fourier Transform

ana-The least square of estimation residual can be described by

dt t v t v

0

2

)) ( ) ( (

~ (3.4.1)

and the relationships can be designed as follows:

) ( )

v ˜ ZT , (3.4.2)

)(ˆ)

v ˜ZM , (3.4.3)

L L

Lˆ '  , (3.4.4)

), ( ) ( )

Z  ' (3.4.5)

) ( ) ( )

FM T  ' , (3.4.6)

L is the true value of the length, and Lˆis the estimate of the length in

least square sense v (t) is the true value of the linear velocity, v ˆ t( ) is the estimate of the linear velocity, ZT(t ) is the true value of Tc, ZM(t ) is the measurements of Tc and 'Z(t)are measurement additive noise sig-nal of Tc, respectively Z (Z)

e~ in least square sense The estimation residual is in terms of the

fre-quency domain form of 'F(Z)the error signal 'Z(t) Hence, the lem is turned into describing the relation between the e~ and 'F(Z).The following lemma provides a conclusion that functions with a cer-tain form are increasing functions of a variable Based on the lemma, a theorem can be developed to prove that e~ is a function of 2

prob-) (

L L

' which has the same form as in the lemma Thus, the estimation error decreases, as the residual is reduced

Lemma: Let ::(f,f) and

2

) (

) ( ) ( (

Z Z Z

Z

Z

d F

d F F

X

M M

(3.4.7)

()

((2

X d F

d F

L e

func-Proof: First, we try to transfer the problem to real space through

simplify-ing X Since 'F(Z)is orthogonal to Z (Z)

M

0 ) ( )

³:FZ Z F Z d Z

M (3.4.9) Simplifying the integrals

( ) (

³

³

³:FZ Z d Z : FZ Z d Z  :' F Z d Z

T M

2 2

2

) ( )

( )

(

These two questions can move out some terms in X, it is clear that X is

a real function as

2 2 2

2

) ) ) ( )

( (

) ( (

Z

Z Z

F

d F X

F

T

2 2

) ( 1

)

e~ can be expressed in teams of X

Z Z S

Z Z Z

Z Z

Z S

Z

Z

d F

X

X X

L

d F X

d F

X d F

2 2

2 2

) ( 1

) 1

(

2

) ) ( )

( )

( (

It can be written as

X d F

L

S

ZZ

Z

2

)(

~

2 2

³:

Let f(X) e~, then

0 4

) ( )

(

2 2

!

X

d F

L X

S

Z Z

|

³

:

, f ( X) is an increasing function of X

Finally, e~ is an increasing function of X If ~ e 0, X 0

The lemma provides a foundation to prove 2

)(

L

L

' will reach a

mini-mum value when the estimation residual e~ takes a minimum value

Theorem: Given 'F:ZoC, C is a complex space, when e~ takes a minimum value, 2

) (

L L

' also takes a minimum value

Proof: Consider the continuous case:

dt t v t v t v t v

0

2 2

] ) ( ) ( ) ( ) ( [

~

Given ::(f,f), according to Parseval’s Equation,

Z Z Z

Z Z

From (3.4.3) and Linear properties of Fourier Transform, it can be ily seen that

eas-Z Z S

Z Z Z

Z

d F

d F

F L F

L e

v

T

2

ˆ 2

ˆ 2

)(2

1

)()(ˆ2)(ˆ(2

ˆ2ˆ

2

w

w

L L

e

v

M M

The above equation implies that the solution of Lˆ can be expressed as

0 ) ( ) ( )

( ˆ

³: L FZ Z FZ Z Fv Z d Z

M M

Using (3.4.2), the above equation implies that the solution of Lˆ can be expressed as

˜

Z Z

Z Z Z

Z Z

Z Z Z

Z

Z Z

Z Z

d F

d F F

L

d F

d F F

L

M

T M

M

2 2

) (

) ( ) (

) (

) ( ) ( ˆ

Z Z Z Z

Z

Z Z

d F

d F F

F L

L L

M

M M

2 2

|)(

|

)()(

|)(

|

(3.4.14)

There exists the relation between the estimation error ( )

L L

' in the time domain and the measurement error ('F(Z)) in frequency domain,

Z Z

Z Z Z

Z

Z

d F

d F F

L L

M

M

2

|)(

|

)()(

(3.4.15)

Note that ifX is defined in the beginning of the section, then

((

)(

)()((

2

~

2 2

2 2

Z Z Z Z

Z

Z Z

Z Z Z S

Z Z

Z Z Z

d F F

F F

d F

d F F

L

e

M M

M m

(3.4.16)

We define:

dt t t

t t

dt t

0

2 2

2

Applying Parserval’s Equation to the error signal 'Z yields

Z Z Z

Z

Z Z Z

Z

Z Z

Z Z

Z Z

Z

Z Z

Z Z

Z Z

Z Z

d F F

F

d F

d F

dw F

F F

d F

d F

M M

M T

T M

M T

: :

) ( ) ( )(

( 2

) ( )

(

) ( ) ( 2 ) ( )

( )

(

2 2

2 2

2

Therefore,

Z Z Z Z

Z Z Z

Z

Z Z

d F F

F

d F

))()

((

2

2 2

'

'

( ( 2

Z Z Z

It can be easily seen that e~has the same form as in the lemma, then e~

is an increasing function of X , for different 'F , when e~takes a

mini-mum value, 2

)(

L

L

' also takes a minimum value Since the minimum value

of e~ is equal to 0, the ( ) 2

L L

' will approach 0 as well The residual of the estimation is convergence and the estimation error goes to 0, as the two frequency spectra are identical

3.5 Experimental Implementation and Results

The proposed method has been tested using a Mobile Manipulation System consisting of a Nomadic XR4000 mobile robot, and a Puma560 robot arm attached on the mobile robot A nonholonomic cart is gripped by the end-effector of the robot arm as shown in Fig 3.1 There are two PCs in the mobile platform, one uses Linux as the operating system for the mobile ro-bot control and the other uses a real time operating system QNX for the control of the Puma560 The end-effector is equipped with a Jr3force/torque sensor

In order to identify the model of the cart, two types of interaction tween mobile manipulator and the cart are planned First, the robot pushes the cart back and forward without turning the cart The sensory measure-ment of the acceleration and the force applied to the cart can be recorded Second, the cart was turned left and right alternatively to obtain the sen-sory measurements of the position of the point A and the orientation of the cart The mass and length estimation are carried out on different carts of varying length and mass

be-3.5.1 Mass Estimation

To estimate the mass of the cart, the regular recursive Least Square Method (LSM) is used The measured acceleration signal and the meas-ured signal of the pushing force contain independent white noise Hence, the estimation should be unbiased The estimate of the mass of the cart can

be obtained directly by LSM

Fig 3.6, 3.7, 3.8 indicate the mass estimation process At the beginning, the estimation is oscillating, however, a few seconds later, the estimation became stable The mass estimation results are listed in Table 3.2, which indicates that the mass estimation errors, normally, less than 15%

0 5 10 15 20 25 0

Fig 3.8 Mass Estimation, for m = 30kg

Table 3.2 Mass Estimation Results

Mass Estimate Error(kg) Error(%)

45.0 49.1 4.1 9.1% 55.0 62.2 7.2 13.1% 30.0 26.8 3.2 10.7%

3.5.1 Length Estimation

According to the proposed method, the algorithm filters the raw signal to

have different bandwidths For different frequency ranges of the signal,

re-cursive Least Square Method is used for parameter identification The

ex-perimental results of length estimation are shown by the graphs below

Corresponding to the frequency components of the angular velocity signal at different lower ranges, ) ]

2

1(,0( S˜ level There are maximally 13 estimation stages in this estimation, therefore the index of the levels ranges

from 1 to 13

Figures 3.9, 3.10, 3.11 and 3.12 show the estimation processes at 9th-12

levels for L=1.31m and L=0.93m The tends of variance P at all the levels

show that the recursive least square method makes the estimation error creasing in the estimation process For some frequency ranges, the estima-tion errors are quite large, and at those levels (For example, 11th

level, which results in the best estimate

0.5 1 1.5 2

Time

0 5 10 15 20 25 30 0

0.2 0.4 0.6 0.8 1

0 0.2 0.4 0.6 0.8 1

0.5 1 1.5 2 2.5 3 3.5 4 4.5 5

3.5.3 Verification of Proposed Method

Figures 3.13, 3.14, 3.15 indicate e~ and the parameter estimation errors at different levels, in case of L=0.93m, 1.31m, and 1.46m, respectively The horizontal axes represent the index of the estimation level, as shown in Figs 3.13, 3.14, 3.15 The vertical axes of the figures represent the absolute value of relative estimation error, and the value of e~

' for L=0.93m

0 2 4 6 8 10 12 14 0

' for L=1.46m

The figures show the different estimation performances at different

lev-els The relationship between the estimation errors and the filtering levels

can be found

Figures 3.13, 3.14, 3.15 indicate that e~ and the estimation error, delta

L, have the same feature of changing with respect to the levels The

esti-mation reaches the minimum ' 10.5%,7.9%and2.6%

L

L

at level 11,

10 and 10, respectively At the same level, the residual e~ is also

mini-mized Thus, minimizinge~, which can be computed line by the

on-board computer, becomes the criterion for optimizing the estimation

The figures also show that after the estimation level at which the

esti-mation error takes a minimum value, the value of e~ and the estimation

er-ror are increasing, due to lack of the normal frequency components of the

true signal (serious distortion) at the further levels of low pass filtering It

also indicates that the true signal component of the measurement resides in

certain bandwidth at low frequency range

To estimate the kinematic length of a cart, the proposed method and

traditional RLSM are used The estimates by DWMI Algorithm, according

to the proposed method, and the estimates by traditional RLSM without

preprocessing the raw data are listed in Table 3.3 It can be seen that the

estimation error by RLSM method is about 80%90%, while the DWMI

method can reduce the estimation error to about 10% This is a significant

improvement of estimation accuracy

Table 3.3: Comparison of Length Estimation Results

LS DWMI Length

(m)

)(

ˆ m

L error L ˆ m( ) error0.93 0.0290 -96% 1.0278 10.5%

1.14 0.128 -89.3% 1.061 -7.0%

1.31 0.1213 -90% 1.415 7.9%

1.46 0.1577 -89% 1.50 2.6%

3.6 Conclusion

In this chapter, in order to solve the online model learning problem, a crete Wavelet based model Identification method has been proposed The method provides a new criterion to optimize the parameter estimations in noisy environment by minimizing the least square residual When the un-known noises generated by sensor measurements and numerical operations are uncorrelated, the least square residual is a monotonically increasing function of estimation error Based on this, the estimation convergence theory is created and proved mathematically This method offers signifi-cant advantages over the classical least square estimation methods in model identification for online estimation without prior statistical knowl-edge of measurement and operation noises The experimental results show the improved estimation accuracy of the proposed method for identifyingthe mass and the length of a nonholonomic cart by interactive action in cart pushing,

Dis-Robotic manipulation has a wide range of applications in complex and dynamic environments Many applications, including home care, search, rescue and so on, require the mobile manipulator to work in unstructured environments Based on the method proposed in this chapter, the task model can be found by simple interactions between the mobile manipula-tor and the environment This approach significantly improves the effec-tiveness of the operations

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