I have a 7 DOF robotic Arm. All the 7 Joints are rotational and have upper and lower joint angle limits. Now assume my end effector has a certain Pose $x = [X, Y, Z, a, b, c]$.
My Question now is how to calculate the maximal reachable $X$ Position if all other Parameters $(Y, Z, a, b)$ remain the same assuming certain $q_{lower}$ and $q_{upper}$ joint limits.
I know how to do that for differential kinematics: $dx_{max} = J * dq_{max}$ ($J$ is Jacobian Matrix and $dq_{max}$ is the max. angle velocity of the joints).