Bryant angles

Lesson 8-B: Bryant Angles x-y-z Convention of Euler Angles

• •

Bryant angles are the x-y-z convention of the Euler angles The x-y-z frame is rotated three times: first about the x-axis by an angle new y-axis by an angle
2 1

; then about the

; then about the newest z-axis by an angle

3

. If the three angles frame.

are chosen correctly, then the rotated frame will coincide with the z 1 2 2 1

z

z
3

3

x

1

x

2

x

3

•

The transformation matrix is found by considering three planar transformation matrices cos 2 0 sin 2 cos 3 sin 3 0 1 0 0

D = 0 cos 0 sin

1 1

sin cos

1 1

C=

0 sin

1
2

0
2

B = sin 0

3

0 cos

cos 0

3

0 1

•

The transformation matrix A is the product of these three planar transformation matrices cos 2 0 sin 2 cos 3 sin 3 0 1 0 0

A = DCB = 0 cos 0 sin

1 1

sin cos

1 1

0 sin c 1c s 1c

1
2

0
2

sin 0 s
3 3

3

0 cos c 2s
3

cos 0

3

0 1

c 2c 3 A = c 1s 3 + s 1s 2 c s 1s
3

2 2 2

3 3

3

s 1s 2 s

s 1c c 1c

c 1s 2 c

3 + c 1s 2 s

where: c cos and s sin

• • • • •

Note that the resulting transformation matrix, similar to the z-x-z convention, is highly nonlinear in terms of the three angles This process does not tell us how to chose the value for each angle! If the angles are not chosen correctly, following the rotations, the x-y-z frame will not coincide with the frame! We have the same problem of “singularity” as in the z-x-z convention!

Inverse Problem Assume that the values of the nine direction cosines; i.e., all the nine elements of the transformation matrix, are known. How do we determine the three Bryant angles? We equate some of the direction cosines with the entries of the transformation matrix A:

c 2c c 1s s 1s
3 3

3 3 3

c 2s c 1c s 1c
3 3

3 3 3

s

2 2 2

a11 = a21 a31 sin =

a12 a22 a32

a13 a23 a33

+ s 1s 2 c c 1s 2 c sin s 1s 2 s + c 1s 2 s sin =