In kinematics of mechanisms we derive the constraint equations depending on the architecture and then analyse the singularities of the mechanisms by deriving their Jacobian matrices.
For example:
$\qquad {\bf f}=\{f_1, f_2,\cdots,f_n\}^\top=\bf 0$, where
$\qquad f_i=f_i({\bf x, y})=0$, and
$\qquad \bf {x,y}$ are vectors of dimensions $n$, ${\bf x}=\{x_1, x_2, \cdots,x_n\}^\top$ and ${\bf y}=\{y_1,y_2,\cdots,y_n\}^\top$.
On differentiating $\bf f$:
$\qquad\frac{\partial \bf f}{\partial \bf x}\dot{\bf x}+\frac{\partial \bf f}{\partial \bf y}\dot{\bf y}={\bf 0}$
The singularities are given by the Jacobian matrices losing rank. There are two Jacobians which can be calculated:
- $\frac{\partial \bf f}{\partial \bf x}$
- $\frac{\partial \bf f}{\partial \bf y}$
The singularity conditions can be derived by taking the determinant of the Jacobian matrices. The theory is based on Implicit function theorem and is widely accepted in the community.
Recently, we tried a slightly different approach where the initial set of constraint equations, $\bf f$ is reduced to a single equation by elimination all but one variables from $\bf y$, which is aided by using trigonometric identities and resultants.
$\qquad f_u=f_u({\bf x},y_n)$
Then two approaches are used to find the singularity condition given by $\frac{\partial \bf f}{\partial \bf y}$ in the original set of equations:
A1. Finding Jacobian with respect to single variable $y_n$, i.e., $\frac{\partial f_u}{\partial y_n}$.
A2. Finding the condition for $f_u$ to have a double root, i.e., finding the partial derivative and then elimination the variable $y_n$ from the equation $f_u=0$ and $\frac{\partial f_u}{\partial y_n}=0$.
The advantage of using A2 is that the condition no longer depends on $\bf y$.
From simulations, I have found that the approach A2 gives extra solutions to the original Jacobian, $\frac{\partial \bf f}{\partial \bf y}$, derived from initial set of equations, ${\bf f}$. I believe this is true because there might have been extra roots added during the elimination procedure (for example: taking squares of $\cos$ and $\sin$ and adding them).
Questions:
Is there any other reason from Mathematics perspective that $\det\frac{\partial \bf f}{\partial \bf y}\neq \frac{\partial f_u}{\partial y_n}$?
Can it be proved mathematically that A2 is not necessary and sufficient?
Please feel free to comment if further clarification is required.