unicycle robot

8/13/2019 Unicycle Robot

1/45


2/45


3/45


4/45


5/45


6/45


7/45


8/45

τ

α

ẋ

rw

rf

I fg

I wg


9/45

x

α

δα

τ · δα

α

α

δx

τ rw

· δx

x

δW = τ rw

· δx + τ · δα


10/45

L = T − V

d

dt

∂L

∂ ẋ

−

∂L

∂x

=

τ

rwd

dt

∂L

∂ α̇

−

∂L

∂α

= τ

T = 1

2

mw ẋ

2

+ 1

2 mf

ẋ2 + 2ẋrf α̇ cos α + r

2f α̇

2

+ 1

2

I wg ẋ

2

r2w

+

1

2

I fg α̇

2

V = mf grf cos α

d

dt

ẋ

α̇

=

1ac−b2 cos2 α bcα̇2 sin α − b2g sin α cos α + τ c−brw cosαrw 1

ac−b2 cos2 α

bga sin α − b2α̇2 sin α cos α + τ

arw−b cosα

rw

a =mf + mw + I wg

r2w

b =mf rf

c =mf r2

f + I fg


11/45


12/45

a

b

c

d

dt

ẋ

α̇

α

=

0 0 −b2g

ac−b2

0 0

abg

ac−b2

0 1 0

ẋ

α̇

α

+

c−brwrw(ac−b2)

arw−brw(ac−b2)

0

τ

mf = 24 kg

mw = 1 kg

rf = 0.68 m

rw = 0.225 m


13/45


14/45

J =

ˆ ∞

0

xT Qx + uT Ru

dt

u = K x

Q =

10 0 00 0.1 0

0 0 0.01

R =0.1

K = −10 −22.3 −144.9


15/45

F e e d b a c k G a i n

Vec to r

L i n e a r i s e d U n i c y c l e

Dy n a mic s

-

u TL o o p b r o k e n

h e r e

G = −

s2 (a1k1 + a2k2) + s (a3k3) + b1k1

s3 + ds

a1 = 1.23 a2 = −1.69 a3 = −1.69 b1 = −16.53 d = −96.69

k1 = −10 k2 = −22.3 k3 = −144.9

0.5rads−1

10rads−1

10dB

9.8rads−1

−1


16/45

−40

−30

−20

−10

0

10

20

30

40

50

60

M a g n i t u d e ( d B )

10−2

10−1

100

101

102

103

−270

−225

−180

−135

−90

P h a s e ( d e g )

20rads−1

3Hz

3Hz

40o

6dB

10rads−1

s+as+b


17/45

0

5

10

15

20

25

30

35

40

M a g n i t u d e ( d B

)

10−3

10−2

10−1

100

101

102

103

−90

−60

−30

0

P h a s e ( d e g )

Bode Diagram

Frequency (rad/sec)

s+10s+0.1

0.85


18/45

−40

−20

0

20

40

60

80

100

120

M a g n i t u d e ( d B )

10−3

10−2

10−1

100

101

102

103

−360

−315

−270

−225

−180

−135

−90

P h a s e ( d e g )

Original Bode diagram

Compensated Bode diagram


19/45

6o

15o

1 ms−1


20/45

0 50 100 150 200 250 300 350 400 450 500−1

0

1

2

3

4

5

6

7

Time (e−2 s)

Translational Velocity (m.s−1)

Angular Velocity (rad.s−1)

Angular Position (rad)

Torque input (N.m)

6os−1

6.5Nm

7.5Nm


21/45

0 100 200 300 400 500 600−1

0

1

2

3

4

5

6

7

8

Time (e−2 s)

Translational Velocity (m.s−1)

Angular Velocity (rad.s−1)

Angular Position (rad)

Torque input (N.m)


22/45

1 ms

I n p u t / O u t p u t

M o d u l e sF P G A f i l e Hos t f i l e

L o c a t e d o n C o m p a c t R i o L o c a t e d o n l a p t o p

W i F iH a r d w i r e d


23/45

Inpu t / Ou tpu t

M o d u l e s

F P G A f i l e

L o c a t e d o n C o m p a c t R i o

L o c a t e d o n l a p t o p

W i F i

H a r d w i r e dHos t f i l eT ime-c r i t i ca l

con t ro l loop

Hos t f i l e

( o n F P G A ) (on m ic rocon t ro l le r )

L o w - f r e q u e n c y p o l l i n g


24/45

V olts

rads−1


25/45

m s−1


26/45

i ( t ) X O R i ( t -1 )

R e a d i n n e r l o o p

va lue, i ( t )

R e a d t i m e r a n d

c o n v e r t t o m s - 1R e s t a r t t i m e r

R e t u r n s p e e d

R e a d o u t e r l o o p

va l ue , o ( t )

i ( t ) X O R i ( t -1 )

A N D

o( t )

i ( t ) A N D o ( t )

S peed i s i n the fo rw a rd

d i rec t i on (pos i t i ve )

S p e e d i s i n t h e b a c k w a r d

d i rec t i on (nega t i ve )

T r ue

Fa l se

T r u e

T r u e

Fa l se

Fa l se


27/45

µ

µs

µs


28/45

1000 1200 1400 1600 1800 2000 2200−20

−15

−10

−5

0

5

10

15

20

25

PWM Duty Cycle (us)

T o r q u e ( N m )

measured data

Interpolated Data

Interpolated Data

Crude Approximation


29/45

1mm

2∗(25kg∗9.8ms−2)1mm∗10mm

= 49MP a


30/45


31/45


32/45

1.2mm

σmax =

Momentmax×ymaxI


33/45


34/45

0 1 2 3 4 5 6−0.3

−0.2

−0.1

0

0.1

0.2

0.3

0.4

Time (s)

A n g u l a r P o

s i t i o n ( r a d i a n s )


35/45

0 1 2 3 4 5 6−1

−0.5

0

0.5

1

1.5

Time (s)

A n g u l a r V e l o c i t y ( r a d i a n s s −

1 )

0 1 2 3 4 5 6−3

−2

−1

0

1

2

3

Time (s)

T r a

n s l a t i o n a l V e l o c i t y ( m s −

1 )


36/45

0 1 2 3 4 5 6−100

−80

−60

−40

−20

0

20

40

Time (s)

T o r q u e ( N

m )

30


37/45

0 1 2 3 4 5 6−4

−3

−2

−1

0

1

2

3

Time (seconds)

A m p l i t u d e s ( t o r q u e n o t t o s c a l e , o n l y f o r c o m p a r i s o n )

Torque data

Torque interpolated

Translational Velocity data

Translational Velocity interpolated

Angular Velocity data

Angular Velocity interpolated

Angular Position data

Angular Position interpolated


38/45

−5 −4 −3 −2 −1 0 1 2 3 4 5−10

−8

−6

−4

−2

0

2

4

6

8

10

nyquist locus

relay describing function

−180o


39/45

−5 −4 −3 −2 −1 0 1 2 3 4 5−10

−8

−6

−4

−2

0

2

4

6

8

10

original nyquist

nyquist with proportional gain


∼ 3.14 rads−1

s+1s+12


40/45

−5 −4 −3 −2 −1 0 1 2 3 4 5−10

−8

−6

−4

−2

0

2

4

6

8

10

original nyquist

nyquist with phase lead



41/45

15o


42/45

Q(state, action) = R(state, action) + γ.max[Q(next state, all actions)]

γ

+/ − 5o R = −1

R = 0

R = −1

δ ẋ

ẋ

δ ẋ Aδx + Bδu


43/45

A =∂f

∂x (xe, ue)

B =

∂f

∂u (xe, ue)

ẋ = f (x, u)

ẍ = 1

ac−

b2

cos2

α bcα̇2 sin α −

b2g sin2α

2

+ T c − brw cos α

rw

α̈ =

1

ac − b2 cos2 α

bga sin α −

b2α̇2 sin2α

2 + T

arw − b cos α

rw

α̇ = α̇

(xe, ue) = (0, 0)

∂f 1∂x1

(xe, ue) = ∂ ̈x

∂ ẋ (xe, ue) =0

∂f 1∂x2

(xe, ue) = ∂ ̈x

∂ α̇ (xe, ue) =0

∂f 1∂x3

(xe, ue) = ∂ ̈x

∂α (xe, ue) = −

b2g

ac − b2

α

∂f 2∂x

1

(xe, ue) = ∂ ̈α

∂ ẋ (xe, ue) =0

∂f 2∂x2

(xe, ue) = ∂ ̈α

∂ α̇ (xe, ue) =0

∂f 2∂x3

(xe, ue) = ∂ ̈α

∂α (xe, ue) =

abg

ac − b2

α

∂f 3∂x2

(xe, ue) = ∂ α̇

∂ α̇ (xe, ue) =1


44/45

d

dt

ẋ

α̇

α

=

0 0

−b2g

ac−b2

0 0

abg

ac−b2 0 1 0

ẋ

α̇

α

+

c−brwrw(ac−b2)

arw−brw(ac−b2)

0

τ


45/45

unicycle robot

Documents