Page 81 - Introduction to Autonomous Mobile Robots

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π π 3 1 Chapter 3
sin --- – cos --- – l ------- – --- l–
3 3 2 2
J = 0 – cos π – l = 0 1 – l (3.32)
1f
π π
3 1
sin – --- – cos – --- l– – ------- --- l–
–
3 3 2 2
Once again, computing the value of ξ · requires calculating the inverse, J – 1 , as needed
I 1f
in equation (3.31). One approach would be to apply rote methods for calculating the inverse
of a 3x3 square matrix. A second approach would be to compute the contribution of each
Swedish wheel to chassis motion, as shown in section 3.2.2. We leave this process as an
exercise for the enthusiast. Once the inverse is obtained, ξ · can be isolated:
I

1 1
-------0 – -------
3 3
· – 1 1 2 1 ·
ξ = R θ() – --- --- – --- J ϕ (3.33)
I 2
3 3 3
1 1 1
–
– ----- ----- – ---- -
3l 3l 3l
Consider a specific omnidrive chassis with l = 1 and r = 1 for all wheels. The robot’s
local reference frame and global reference frame are aligned, so that θ = 0 . If wheels 1,
,
(
,
(
2, and 3 spin at speeds ϕ =( 1 4) ϕ = 1) ϕ = 2) , what is the resulting motion of the
2
3
whole robot? Using the equation above, the answer can be calculated readily:
1 1 2
-------0 – ------- -------
x · 100 3 3 100 4 3
· 1 2 1 4
·
ξ = y = 010 – --- --- – --- 010 1 = – --- (3.34)
I
3
3
3 3
θ · 001 001 2 7
---
--- –
– 1 1 - 1 – ---
-- –
3 3 3 3
x
So this robot will move instantaneously along the -axis with positive speed and along
y
the axis with negative speed while rotating clockwise. We can see from the above exam-
ples that robot motion can be predicted by combining the rolling constraints of individual
wheels.