Degrees of Freedom (Mobility)
Kinematics and Machine Dynamics · Chapter 2
Chapter overview
- Count the motions of a free rigid body.
- Translate joints into constraints.
- Estimate mobility for planar and spatial mechanisms.
- Recognize dependent constraints and singular configurations.
What mobility measures
The mobility M is the number of independent configuration parameters needed to locate every link relative to ground.
A free planar rigid body needs (x,y,\theta): three parameters.
A free spatial rigid body needs three translations and three orientation parameters: six parameters locally.
Ground is a link, but contributes no motion parameters.
Links, coordinates, and constraints
Four links include ground. Before connecting the moving links, there are 3(4-1)=9 planar coordinates.
Joint freedoms in space
| Joint | Relative freedom f | Independent constraints 6-f |
|---|---|---|
| Revolute: rotation about one axis | 1 | 5 |
| Prismatic: translation along one axis | 1 | 5 |
| Helical: coupled rotation and translation | 1 | 5 |
| Cylindrical: independent slide and rotation | 2 | 4 |
| Spherical: three rotations | 3 | 3 |
More joint types
A planar joint allows two translations and one rotation: f=3 in space.
A universal joint allows two independent relative rotations: f=2.
A planar cam or gear contact without an imposed no-slip condition usually removes one of the three relative planar freedoms.
Rolling without slip adds a tangential velocity constraint; state that assumption explicitly.
Deriving the planar count
For n_L links including ground and n_J joints:
M_{\mathrm{count}}=3(n_L-1)-\sum_{i=1}^{n_J}(3-f_i).
Equivalently,
\boxed{M_{\mathrm{count}}=3(n_L-n_J-1)+\sum_i f_i.}
This equals mobility when the counted constraints are independent at a regular configuration.
Lower and higher pairs
Let J_1 count one-freedom planar joints, such as pins and sliders, and J_2 count two-freedom planar contacts.
\boxed{M_{\mathrm{count}}=3(n_L-1)-2J_1-J_2.}
A pin joining k separate links counts as k-1 binary pin joints.
Count a rigid assembly as one link; count all connections to ground.
Worked example: a four-bar
There are four links including ground and four revolute joints.
M=3(4-1)-2(4)=1.
One input angle locally determines the other link positions on a chosen assembly branch.
The count does not identify the branch, allowable input range, or singular positions.
Worked example: a slider-crank
Four links; three revolute joints and one prismatic joint:
M=3(4-1)-2(3+1)=1.
Worked example: a cam and follower
Three links: ground, cam, and translating follower.
Two lower pairs: cam-to-ground revolute and follower-to-ground prismatic.
One higher pair: cam-to-follower contact.
M=3(3-1)-2(2)-1=1.
The contact must remain closed for this model to apply.
Spatial mobility
\boxed{M_{\mathrm{count}}=6(n_L-1)-\sum_i(6-f_i).}
For an open serial chain of six revolute joints, n_L=7 and n_J=6:
M=6(7-1)-5(6)=6.
Six joint freedoms do not guarantee six independent end-effector velocity directions at every posture.
Constraint rank is the deciding quantity
Write the constraints as g(q)=0, with N unconstrained coordinates.
J(q)\dot q=0,\qquad J=\frac{\partial g}{\partial q}.
At a regular configuration,
\boxed{M=N-\operatorname{rank}J.}
Counting equations works only when their gradients are independent.
Redundant constraints: a physical example
A rigid door rotates about a hinge axis. Adding a second perfectly coaxial hinge does not remove its rotation.
Treating both hinges as independent spatial revolute joints gives
M_{\mathrm{count}}=6-5-5=-4.
The actual mobility is one because several constraints are dependent. Negative counts are a reason to inspect geometry, not negative physical freedom.
Singular configurations
At a singular posture, J can lose rank and the instantaneous nullspace can grow.
An extra instantaneous velocity direction need not extend to a finite motion: higher-order constraints can still prevent it.
Distinguish:
- mobility of the regular mechanism;
- instantaneous mobility at a particular posture;
- number of independent actuator inputs.
A reliable counting procedure
- Mark ground and identify rigid links.
- Identify every joint and its allowed relative motion.
- Choose planar or spatial coordinates consistently.
- Compute the generic count.
- Inspect special geometry, contact conditions, and singularities.
References and next chapter
Satici, Kinematics and Machine Dynamics, Chapter 2, pp. 5–6; Lecture01 mobility slides.
The compiled chapter follows Wilson and Sadler, Kinematics and Dynamics of Machinery.
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