From Measurement to Interpretation - Kinematic Analysis of Human Movement - page 45

cos αα − sin 0 1 0 0 cos γ 0 sin γ

⎡ ⎤⎡ ⎤⎡ ⎤

⎢ ⎥⎢ ⎥⎢ ⎥

i

− R =

1

sin αα cos 0 0 cos β − sin β 0 1 0

⎢ ⎥⎢ ⎥⎢ ⎥

⎢ ⎥⎢ ⎥⎢ ⎥

0 0 1 0 sin β cos β − sin γ 0 cos γ

⎣ ⎦⎣ ⎦⎣ ⎦

which gives the following analytical form, as a function of
the three angles:

cos αγ ⋅ cos -sin αβγ ⋅ sin ⋅ sin -sin α β αγ ⋅ cos cos ⋅ sin + sin αβ γ ⋅ sin ⋅ cos

⎡ ⎤

⎢ ⎥ ⎡⎤ i − 1

R = sin α ⋅ cos γ + cos α ⋅ sin βγ ⋅ sin cos α ⋅ cos β sin α ⋅ sin γ − cos α ⋅ sin βγ ⋅ cos = ⎣⎦ δ

⎢ ⎥

i ij

⎢ ⎥

-cos βγ ⋅ sin sin β cos β γ ⋅ cos

⎣ ⎦

where

ij represents the element in i th line and j th column of
the matrix.

δ

To obtain the values of these joint angles, at each instant
of the movement, we use the simplest elements of this
matrix from the following trigonometric functions:

⎛ ⎞

− δ − 1

α = tg

12

⎜ ⎝ ⎠

δ

22

( )

− 1

β = sin δ

32

⎛ ⎞

− δ

γ = tg

− 1 31

⎜ ⎝ ⎠

δ

33

However, it can be seen that in these expressions, the
cosine of angle appears in the denominator for the

calculation of angles

β

, which is a problem when it
becomes close to zero; in other words, when

and

α γ

β is close to 90°.

⎛ ⎞ − δ ⎛ −− (sin .cos ) α β ⎞

α = tg

− 1 12

= tg

− 1

⎜ ⎟ ⎜ ⎟

δ cos α .cos β

⎝ ⎠

⎝ ⎠

22