Joints

Every joint connects exactly two links.

A joint:

• provides freedoms to allow one rigid body to move relative to another.
• provides constraints on the possible motions of the two rigid bodies it connects.

$dof\ (of\ joint) = dof\ (of\ rigid\ body) - number\ of\ constraints\ (of\ joint)$

Typical Joints

Configuration Space

Configuration: a specification of the positions of all points of the robot.

Configuration space (C-space):

• The space of all configurations.
• The n-dimensional space containing all possible configurations of the robot.

The configuration of a robot is represented by a point in its C-space.

Degrees of Freedom (dof)

Degrees of freedom: smallest number of real-valued coordinates needed to represent its configuration. It’s the dimension of the C-space or minimum number of real-valued numbers needed to represent the configuration.

Degree of freedom depends on:

• Number of links
• Number of joints
• Types of joints
• Distribution of joints within mechanism

It does not depend on length of links.

Rigid body (number of degrees of freedom)

• 2D (planar bodies): 3
• 3D (spatial bodies): 6

Degrees of freedom in 3D:

• x
• y
• z
• roll
• pitch
• yaw

Grübler’s Formula

• $N$: Number of links (ground is also a link)
• $J$: Number of joints
• $m$: Degrees of freedom of rigid body (3 in 2D, 6 in 3D)
• $f_i$: Number of freedoms of joint $i$
• $c_i$: Number of constraints of joint $i$
• where $\forall i;f_i + c_i = m$

The formula holds only if all constraints for the joints are independent.

Literature

Notes taken from:

Modern Robotics: Mechanics, Planning, and Control by Kevin M. Lynch and Frank C. Park, Cambridge University Press, 2017