Page 176 - Flexible Robotics in Medicine

162 Chapter 6

In order for tip-based control to be implemented, equations must be found to relate the
0 1 0 1
x L 1
inputs @ A to the outputs @ t A .
y
z θ

Substituting (6.4) into (6.2),

x 1 y 5 R 2 Rcosφ
p ffiffiffiffiffiffiffiffiffiffiffiffiffiffi
2
2

1 p ffiffiffiffiffiffiffiffiffiffiffiffiffiffi
21
2
φ 5 cos 1 2 x 1 y 2 (6.6)
R
Substituting (6.6) into (6.3),

1 p ffiffiffiffiffiffiffiffiffiffiffiffiffiffi

2
z 5 L 1 1 Rsin cos 21 1 2 x 1 y 2 (6.7)
R

From trigonometric identities, this simplifies to

s ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
2
1 p ffiffiffiffiffiffiffiffiffiffiffiffiffi

2
L 1 5 z 2 R 1 2 12 x 1y 2
R
Similarly, substituting (6.6) into (6.1)
1 p ffiffiffiffiffiffiffiffiffiffiffiffiffiffi
2
t 5 Rcos 21 1 2 x 1 y 2 (6.8)
R

However, as cos 21 is a many-to-one mapping, it may result in errors when implemented
programmatically. A better alternative is to use the arctan2(y, x) function.

From (6.8),

t 1 p ffiffiffiffiffiffiffiffiffiffiffiffiffiffi

2
2
cos 5 1 2 x 1 y ; let RHS 5 A
R R
t ffiffiffiffiffiffiffiffiffiffiffiffiffi
p
sin 56 1 2 A 2
R
t t

21
t 5 Rtan 2 sin ; cos (6.9)
R R