Chapter 2 Math Fundamentalsalonzo/books/math3.pdf · Chapter 2 Math Fundamentals Part 3 2.6.1...

Chapter 2 Math Fundamentals

Part 3 2.6.1 Kinematic Models of Video Cameras 2.6.2 Kinematic Models of Laser Rangefinders

Mobile Robotics - Prof Alonzo Kelly, CMU RI 1

Outline • 2.6.1 Kinematic Models of Video Cameras • 2.6.2 Kinematic Models of Laser Rangefinders


Outline • 2.6.1 Kinematic Models of Video Cameras

– Perspective Projection

• 2.6.2 Kinematic Models of Laser Rangefinders


Video Cameras • Image formation in cameras follows the perspective

projection. • It is nonlinear. • Unique in two ways:

– reduces the dimension of the input vector by one – it requires a post normalization step to re-establish a unity scale

factor.

5 Mobile Robotics - Prof Alonzo Kelly, CMU RI

xi

yi

zi

wi

1 0 0 00 0 0 00 0 1 0

0 1f--- 0 1

xs

ys

zs

1

=

6

Video Cameras • By similar triangles:

• As a homogeneous transform:

zs

ys

xs

f

(xi,yi) (xs,ys,zs)

xixsf

ys f+-------------

xs1 ys f⁄+--------------------= =

z izsf

ys f+-------------

zs1 ys f⁄+--------------------= =

yi 0=

The nonlinearity culprit

ys

xs

zs

How can you tell this not invertible? Mobile Robotics - Prof Alonzo Kelly, CMU RI



8

Laser Rangefinders • Two kinds

– Scanning devices use actuated mirrors to steer the beam.

– Flash ladars which work like cameras.

• For the former, model what happens when a unit vector is reflected off of all of the mirrors involved.


Configurations

9 9 Mobile Robotics - Prof Alonzo Kelly, CMU RI


– The Reflection Operator – Kinematics of the Azimuth Scanner – Summary

10 10

Mobile Robotics - Prof Alonzo Kelly, CMU RI

Contrast with Robot Kinematics

• Robots -> fundamental operator is a rotation • Mirrors -> fundamental operator is a reflection

11

Link φ u ψ w 0 0 0 ψ1 0 1 0 L1 ψ2 0 2 0 L2 ψ3 0 3 0 L3 0 0

x0,1

y0,1

ψ1

x2

y2

ψ2 x3

y3

ψ3

y4

x4

L1 L2 L3

0

1 2 3

T40

c1 s1– 0 0

s1 c 1 0 0

0 0 1 00 0 0 1

c2 s2– 0 L1

s2 c2 0 0

0 0 1 00 0 0 1

c 3 s3– 0 L2

s3 c3 0 0

0 0 1 00 0 0 1

1 0 0 L3

0 1 0 00 0 1 00 0 0 1

=



Reflection Operator

• Subtract twice the projection of incident ray onto mirror normal.

12


Reflection Operator

Ref n̂( ) I 2 n̂ n̂⊗( )–1 2nxnx– 2n– xny 2n– xnz

2n– ynx 1 2n yny– 2n– ynz

2n– znx 2n– zny 1 2nznz–

= =

Outer Product

Matrix Reflection Operator (Householder Transform)

13

Mirror Gain

• When mirror rotates through angle θ

• Beam rotate through angle of 2θ

14

input

output

2θ θ

input output


Box 2.6 Kinematic Modelling of Rangefinders • 1: Choose coordinates fixed to sensor

housing. • 2: Express beam leaving laser diode as a

unit vector. • 3: Express normal of mirror 1 in terms of

its rotation angle. • 4: Reflect the beam off mirror 1 • 5: Express normal of mirror 2 in terms of

its rotation angle. • 6: Reflect result of step 4 off mirror 2 • 7: Result is the orientation of the beam

expressed in terms of the mirror articulation angles.

15

input

output

mirror 1

mirror 2





Scanning Mechanisms Azimuth

• Enter along x • Reflect around “z” • Reflect around “y” • Leave along “y”

Elevation

• Enter along z • Reflect around “-x” • Reflect around “xy” • Leave along “y”



Azimuth Scanner

ys

xs Polygon mirror

Nodding mirror

18


Azimuth Scanner

v̂p Ref n̂P( ) v̂m=

v̂m 0 1– 0T=

v̂p 0 sψ– cψT=

19


Azimuth Scanner

α2--- π

4--- θ

2---–=put:

v̂p 0 sψ– cψT=

n̂n sα2--- 0 c

α2---–

T=

v̂n

0sψ–

cψ

2cψcα2---

sα2---

0

cα2---–

+

2cψcα2--- sα

2---

sψ–

cψ 2cψcα2--- cα

2---

–

cψsαsψ–

cψcα–

cψs π2--- θ–

sψ–

cψc π2--- θ–

–

= = = =

v̂n cψcθ[ ] sψ[ ]– cψsθ[ ]–T=

20


Azimuth Scanner

• Equivalent to a rotation about y by θ and then a rotation about the new z axis by -ψ.

Rcosψcosθ

vs

xs

ys

zs

RcψcθRsψ–

R– cψsθ

= =

21


Inverse Kinematics

vs

xs

ys

zs

RcψcθRsψ–

R– cψsθ

= =

Rψθ

xs2 ys

2 zs2+ +

ys– xs2 zs

2+⁄( )atan

z– s xs⁄( )atan

=

22


Image of Flat Terrain

Setup:

23


Image of Flat Terrain

R h cψsθβ( )⁄=

Image Plane

Ground Plane

hyperbola

xg h tθβ⁄=yg htψ sθβ⁄–=

zg 0=

24


Resolution • Linearize fwd kinematics:

• Jacobian Determinant:

• Approximation:

dxg dygh ψsec( )2

sθβ( )3----------------------- dψdθ=

J R3

h------≈

Laser spot size / spacing Grows with

Cube of Range

dxg

dyg

0 h–sθβ( )2

----------------

h ψsec( ) 2–sθ β

-------------------------- htψcθβsθβ( )2

-------------------

dψdθ

×=





Summary

• Video cameras are modeled by a perspective projection.

• Laser rangefinder models are nonlinear and cannot be represented by a constant homogeneous transform - like a camera.

• However, our mechanism modeling rules apply perfectly (see text) and one can also use a reflection operator to model them.

27

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