SME
Kinematics of Mechanisms · Problem type
Mobility of a spatial chain (Kutzbach)
Use this for “A robot arm has 7 links (with its base) and 6 revolute joints. What is its mobility?”
Chart
M = F − C₁ − C₂ − C₃
36 − 30 − 0 − 0 = +6
The flows in add and the flows out take away: M = +6.
This is an example. Type your own values to start a new problem.
M
=
6
×
(
n
−
1)
−
5
×
f₁
−
4
×
f₂
−
3
×
f₃
F
Link freedoms
worked out (not typed)
C₁
One-freedom loss
worked out (not typed)
C₂
Two-freedom loss
worked out (not typed)
C₃
Three-freedom loss
worked out (not typed)
Formulas
M
=
6
×
(
n
−
1)
−
5
×
f₁
−
4
×
f₂
−
3
×
f₃
+6 = 6 × (7 − 1) − 5 × 6 − 4 × 0 − 3 × 0
F
=
6
×
(
n
−
1)
36 = 6 × (7 − 1)
C₁
=
5
×
f₁
30 = 5 × 6
C₂
=
4
×
f₂
0 = 4 × 0
C₃
=
3
×
f₃
0 = 3 × 0
Assumptions
•
Rigid links in space: each free link has 6 freedoms, 3 moves and 3 turns.
•
Joints are named by the freedoms they allow: revolute or prismatic 1, cylindrical 2, spherical 3.
•
The base counts as one of the n links and never moves.
•
Redundant constraints (parallel axes, for one) would make the count wrong.
Step-by-step
WE KNOW
n = 7, f₁ = 6, f₂ = 0, f₃ = 0
FIND
mobility (M), link freedoms (F), one-freedom loss (C₁), two-freedom loss (C₂), three-freedom loss (C₃)
Step 1 · Find mobility (M)
Use M = 6 × (n − 1) − 5 × f₁ − 4 × f₂ − 3 × f₃
Every link but the base has 6 freedoms; a joint that allows k of them takes away 6 − k.
M
=
6
×
(7
−
1)
−
5
×
6
−
4
×
0
−
3
×
0
M
=
6
×
6
−
5
×
6
−
4
×
0
−
3
×
0
M
=
36
−
30
M
=
+6
Step 2 · Find link freedoms (F)
Use F = 6 × (n − 1)
The freedoms of the moving links before any joint: 6 for each link but the base.
F
=
6
×
(7
−
1)
F
=
6
×
6
F
=
36
Step 3 · Find one-freedom loss (C₁)
Use C₁ = 5 × f₁
Each revolute or prismatic joint takes away 5 freedoms.
C₁
=
5
×
6
C₁
=
30
Step 4 · Find two-freedom loss (C₂)
Use C₂ = 4 × f₂
Each cylindrical joint takes away 4 freedoms.
C₂
=
4
×
0
C₂
=
0
Step 5 · Find three-freedom loss (C₃)
Use C₃ = 3 × f₃
Each spherical joint takes away 3 freedoms.
C₃
=
3
×
0
C₃
=
0
Check
Put the numbers back in. Both sides should match.
+6 = 6 × (7 − 1) − 5 × 6 − 4 × 0 − 3 × 0 ✓
36 = 6 × (7 − 1) ✓
30 = 5 × 6 ✓
0 = 4 × 0 ✓
0 = 3 × 0 ✓
Topic check
See what you know: 5 questions on this topic
New questions each time. Pass to mark it complete.