radial basis function approximation
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軟體實作與計算實驗 1
Radial basis function approximation
軟體實作與計算實驗 2
K-mean Algorithm
function y=my_kmeans(X,M)mx=mean(X);Y=rand(M,2)*0.1-0.05+mx;change =1; while change < 10.^-6
D = cross_distances(X,Y)[xx ind]=min(D');
for i=1:K ind2=find(ind == i);
Y_new(i,:) = mean(X( ind2,:)); end change = mean(mean(Y-Y_new));
endreturn
軟體實作與計算實驗 3
Cross_Distance
function D=cross_distance(X,Y)M=size(Y,1);N=size(X,1);A=sum(X.^2,2)*ones(1,M);C=ones(N,1)*sum(Y.^2,2)';B=X*Y';D=sqrt(A-2*B+C);
軟體實作與計算實驗 4
Radial basis function
軟體實作與計算實驗 5
3.141
http://www.mathworks.com/matlabcentral/fileexchange/loadFile.do?objectId=157&objectType=File
軟體實作與計算實驗 6
23 4.89
軟體實作與計算實驗 7
4.23TY
軟體實作與計算實驗 8
2.23TX
軟體實作與計算實驗 9
)2.12,81.8( TT YX
oT 200
軟體實作與計算實驗 10
Forward Kinematics
323
212
11
180
180
2 frame and matlabbetween Angle1802
1 frame and matlabbetween Angle
PP
PP
P
2P
3.141 1P
3P
軟體實作與計算實驗 11
Frame positions
)sin(
)cos(
112
112
Ply
Plx
x position of frame '2'
y position of frame '2'
2P
3.141
1P
3P
1l
),( 22 yx
軟體實作與計算實驗 12
Position of frame 3
)sin(
)cos(
2223
2223
Plyy
Plxx
2P
3.141
1P
3P
1l
),( 22 yx
2l
),( 33 yx
軟體實作與計算實驗 13
Position of tool
2P
3.141
1P
3P
1l
),( 22 yx
2l
),( 33 yxx position of frame 'tool'y position of frame 'tool'
)sin(
)cos(
333
333
Plyy
Plxx
t
t
3l ),( tt yx
軟體實作與計算實驗 14
http://www.learnaboutrobots.com/forwardKinematics.htm
軟體實作與計算實驗 15
Forward Kinematics
Problem statement A schematic of a simple robot lying in the X-Y plane Robot length
Three links of the robot
Given the ang joints, where is the robot's hand?
? 2P
3.141
1P
3P
1l
),(22yx
2l ),(33yx
3l
),(ttyx
1l 2l 3l
2P1P 3P
),( tt yx
軟體實作與計算實驗 16
Forward Kinematics for planar robot
)sin()sin()sin(
)cos()cos()cos(
332211
332211
PlPlPly
PlPlPlx
t
t
2P
1P
3P
1l
3l ),( tt yx
軟體實作與計算實驗 17
223
212
11
180
180
1802
PP
PP
P
)sin()sin()sin(
)cos()cos()cos(
332211
332211
PlPlPly
PlPlPlx
t
t
t
t
yPlPlPlF
xPlPlPlF
)sin()sin()sin(),(
)cos()cos()cos(),(
332211212
332211211
軟體實作與計算實驗 18
Inverse kinematics
At the position level, the problem is stated as, "Given the desired position of the robot's hand, what must be the angles at all of the robots joints? "
軟體實作與計算實驗 19
One link
Given Xt what is the joint angle P
x
Yxt
P
軟體實作與計算實驗 20
x
Yxt
P
l
xP
l
x(P)
Plx
t
t
t
arccos
cos
solutionposition forward
)cos(
軟體實作與計算實驗 21
multiple solutions
l
xP
l
x(P)
Plx
t
t
t
arccos
cos
solutionposition forward
)cos(
o
tx
l
45
71arccos0.70P
feet 7071.0
foot one :
軟體實作與計算實驗 22
Two links
x
Y
(xt yt)
P1
P2
1l
2l
軟體實作與計算實驗 23
x
Y
(xt yt)
P1
P2
1l
2l
222tt yxB
)atan2(1t
t
y
xq
軟體實作與計算實驗 24
x
Y
(xt yt)
P1
P2
1l
2l
222tt yxB
1q
211
1
22
221
2 )2
acos(
qqP
Bl
lBlq
2q
軟體實作與計算實驗 25
x
Y
(xt yt)
P1
P2
1l
2l
B
1q
2q
)2
acos(21
222
21
ll
Bll
)(12 PP
軟體實作與計算實驗 26
Forward kinematics of two-link robot
)sin()sin(
)cos()cos(
2211
2211
PlPly
PlPlx
t
t
21,PP
tt yx ,
21,PP
tt yx ,
軟體實作與計算實驗 27
Inverse kinematics of two-link robot
22tt yxB
)atan2(1t
t
x
yq
211
1
22
221
2 )2
acos(
qqP
Bl
lBlq
)2
acos(21
222
21
ll
Bll
)(12 PP
21,PP
tt yx ,
軟體實作與計算實驗 28
Exercise
Write matlab functions to implement forward and inverse kinematics of a two-link robot
軟體實作與計算實驗 29
Example
l1=1;l2=1;p1=pi/3;p2=pi/5;[x,y]=fkin(p1,p2,l1,l2)[p1,p2]=inverse_kin(x,y,l1,l2);p1/pip2/pi
軟體實作與計算實驗 30
Nonlinear system
0
04
21
2
2
2
1
xx
xx
軟體實作與計算實驗 31
A set of nonlinear functions
0),(
04),(
21212
2
2
2
1211
xxxxf
xxxxf
軟體實作與計算實驗 32
x=linspace(-2,2);plot(x,x); hold on; plot(x,sqrt(4-x.^2))plot(x,-sqrt(4-x.^2))
軟體實作與計算實驗 33
function F = myfun(x) F(1) = x(1).^2 + x(2)^2-4; F(2) = x(1) - x(2);return
軟體實作與計算實驗 34
Least square of nonlinear functions
i
ixxxxf 2
21,),(min
21
軟體實作與計算實驗 35
function demo_lsq()x0= ones(1,2)*2;x = lsqnonlin(@myfun,x0)y=myfun(x);sum(y.^2)returnfunction F = myfun(x) F(1) = x(1).^2 + x(2)^2 -4; F(2) = x(1) - x(2);return
Demo_lsq
軟體實作與計算實驗 36
012),(
0242),(22
212
2
2
2
1211
21
xxxxf
xxxxf
軟體實作與計算實驗 37
x=linspace(-5,5);plot(x,sqrt(24-2*x.^2),'r');hold on; plot(x,-sqrt(24-2*x.^2),'r')plot(x,sqrt(12+x.^2),'b')plot(x,-sqrt(12+x.^2),'b')
軟體實作與計算實驗 38
Demo_lsq_2c
source code
軟體實作與計算實驗 39
223
212
11
180
180
1802
PP
PP
P
)sin()sin()sin(
)cos()cos()cos(
332211
332211
PlPlPly
PlPlPlx
t
t
t
t
yPlPlPlF
xPlPlPlF
)sin()sin()sin(),(
)cos()cos()cos(),(
332211212
332211211
軟體實作與計算實驗 40
Nonlinear systems
)sin()sin(
)cos()cos(
2211
2211
PlPly
PlPlx
t
t
t
t
yPlPlPPF
xPlPlPPF
)sin()sin(),(
)cos()cos(),(
2211212
2211211
軟體實作與計算實驗 41
)sin()sin(2
)cos()cos(2
21
21
PP
PP
2)sin()sin(),(
2)cos()cos(),(
21212
21211
PPPPF
PPPPF
軟體實作與計算實驗 42
Exercise
Write a malab function to solve the following nonlinear system
2)sin()sin(),(
2)cos()cos(),(
21212
21211
PPPPF
PPPPF
軟體實作與計算實驗 43
軟體實作與計算實驗 44
puma3d.m
gen_path.m
PUMA Robot Matlab/Servo To Go software
MATLAB Central File Exchange - 3D Puma Robot Demo
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