THREE-DIMENSIONAL KINEMATICS AND KINETICS - Biomechanics and Motor Control of Human Movement - page 178

z

z ′

z ″′

″′

x

z ″

x ″

x ′

x

y

y ′

y ″′

y ″

Figure 7.1 Cardan sequence of three rotations about the x , y , z axes. The first rotation,
θ 1 , about the x axis to get x , y , z ; the second rotation, θ 2 , about the new y axis to get
x , y , z , and the final rotation, θ 3 , about the new z axis to get the desired x , y , z .

point will have coordinates x 1 , y 1 , z 1 in the x , y , z axis system. Based on the

rotation θ 1 :

x 1 = x 0
y 1 = y 0 cos θ 1 + z 0 sin θ 1

z 1 =− y 0 sin θ 1 + z 0 cos θ 1

Using the shorthand notation c 1 = cos θ 1 and s 1 = sin θ 1 , in matrix notation

this may now be written as:
⎡ ⎤ ⎡ ⎤ ⎡ ⎤ ⎡ ⎤

x 1
y 1
z 1

10 0
0

x 0
y 0
z 0

x 0
y 0
z 0

⎣ ⎦ = ⎣ ⎦ ⎣ ⎦ = ⎣ ⎦

c 1 s 1 [ 1 ] (7.1)

0 − s 1 c 1

After the second rotation θ 2 about y , this point will have coordinates x 2 , y 2 , z 2
in the x , y , z axis system.
⎡ ⎤ ⎡ ⎤ ⎡ ⎤ ⎡ ⎤

x 2
y 2
z 2

c 2 0 − s 2
01 0
s 2

x 1
y 1
z 1

x 1
y 1
z 1

⎣ ⎦ = ⎣ ⎦ ⎣ ⎦ = ⎣ ⎦

[ 2 ] (7.2)

0 c 2

Finally, the third rotation θ 3 about z yields the coordinates x 3 , y 3 , z 3 in the

x , y , z axis system.

⎡ ⎤ ⎡ ⎤ ⎡ ⎤ ⎡ ⎤

x 3
y 3
z 3

c 3 s 3 0 x 2

y 2
z 2

x 2
y 2
z 2

⎣ ⎦ = ⎣ ⎦ ⎣ ⎦ = ⎣ ⎦

− s 3 c 3 0

001

[ 3 ]

(7.3)