DOF discussion of quadruped

Formula

With basic mechanism formula

\[\sigma = 6m - \sum_{i=1}^n p_i\]

where

\(\sigma\), DOF \(m\), number of movable parts (links) \(p_i\), number of degree constrained by \(i\)-th kinematic pair(joint or contact). \(n\), total number of kinematic pairs

In special, rotation joint, \(p_i = 5\) point contact, \(p_i = 3\)

Case study

DOF of quadruped base body with 4 legs on the ground.

\(m = 13\) number of rotation joint(\(p_i = 5\)), \(n_1 = 12\) number of point contact(\(p_i = 3\)), \(n_2 = 4\)

\[\begin{split}\begin{equation} \begin{split} \sigma &= 6 \cdot 13 - 5 \cdot 12 - 3 \cdot4\\ &=6 \end{split} \end{equation}\end{split}\]

DOF of quadruped base body with 3 legs on the ground.

\(m = 10\) number of rotation joint(\(p_i = 5\)), \(n_1 = 9\) number of point contact(\(p_i = 3\)), \(n_2 = 3\)

\[\begin{split}\begin{equation} \begin{split} \sigma &= 6 \cdot 10 - 5 \cdot 9 - 3 \cdot3\\ &=6 \end{split} \end{equation}\end{split}\]

DOF of quadruped base body with 2 legs on the ground.

\(m = 7\) number of rotation joint(\(p_i = 5\)), \(n_1 = 6\) number of point contact(\(p_i = 3\)), \(n_2 = 2\)

\[\begin{split}\begin{equation} \begin{split} \sigma &= 6 \cdot 7 - 5 \cdot 6 - 3 \cdot2\\ &=6 \end{split} \end{equation}\end{split}\]

DOF of quadruped base body with 1 legs on the ground.

\(m = 4\) number of rotation joint(\(p_i = 5\)), \(n_1 = 3\) number of point contact(\(p_i = 3\)), \(n_2 = 1\)

\[\begin{split}\begin{equation} \begin{split} \sigma &= 6 \cdot 4 - 5 \cdot 3 - 3 \cdot1\\ &=6 \end{split} \end{equation}\end{split}\]

DOF of quadruped base body with 0 legs on the ground.

\(m = 1\) number of rotation joint(\(p_i = 5\)), \(n_1 = 0\) number of point contact(\(p_i = 3\)), \(n_2 = 0\)

\[\begin{split}\begin{equation} \begin{split} \sigma &= 6 \cdot 1 - 5 \cdot 0 - 3 \cdot0\\ &=6 \end{split} \end{equation}\end{split}\]

Is it over-actuated? Is it under-actuated?

Quadruped with 0 leg on the ground: under-actuated Quadruped with 1 leg on the ground: under-actuated Quadruped with 2 cross legs on the ground: under-actuated

There are 6 actuator by two Ab/ad Quadruped with 3 legs on the ground: over-actuated but NOT redundancy Think 3 legs case as 2 cross legs with 1 more leg. After 2 cross legs configuration is determined, the orientation and position of base body is determined. Then the location that connect to 3rd leg is determined. If ground is in reachable domain, \(\ast\)for certain ground contact point, the configuration of 3rd leg has to be determined.

So, in this case, even we have 9 actuator with 6DOF, actuators are not independent and there isn’t redundancy.

Quadruped with 4 legs on the ground: over-actuated but NOT redundancy Same reason like 3 legs

\(\ast\) Notice: certain ground contact point, in the background of stand phase, the contact point is fixed and we are controlling the orientation or position of base body.

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