I want to use a 6-DoF robot arm to reach a user-given point
$$P_{target} = (x, y, z)$$
with arbitrary orientation.
My robot arm is equipped with a laser pen and what I want to do is to aim at point $P_{target}$ along the axis perpendicular to the flange face of my robot arm with a predefined distance $D$.
I want to minimize the movement of my robot arm, that is, with given start pose
$$(x_i, y_i, z_i, q_{xi}, q_{yi}, q_{zi}, q_{wi})$$
and final pose
$$(x_f, y_f, z_f, q_{xf}, q_{yf}, q_{zf}, q_{wf})$$
the value
$${(x_f-x_i)}^2 + {(y_f - y_i)}^2 + {(z_f - z_i)}^2 + {(q_{xf} - q_{xi})}^2 + {(q_{yf} - q_{yi})}^2 + {(q_{zf} - q_{zi})}^2 + {(q_{wf} - q_{wi})}^2$$
can be optimized.
How could I list the equations and get the result
$$(q_{xf}, q_{yf}, q_{zf}, q_{wf})$$
?
Thanks a lot for any hint and help.
Hint.
A possible variational formulation assuming for simplicity final position without final orientation.
Supposing you have the end point dynamics as
$$ \ddot X = \Phi(X, \dot X, \theta) $$
where $\theta$ are the elbow actuators, you need to obtain
$$ \min_{\theta}\int_0^{t_f} ||\dot X|| dt\ \ \mbox{s. t.}\ \ \ \ddot X = \Phi(X, \dot X,\theta), \, X(t_f) = X_f $$
NOTE
We assumed minimum movement as minimum path covered distance.