Reachable Position with endeffector robotic arm

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Hey there,

I have a 7dof robotic Arm. All the 7 Joints are rotational and have upper and lower joint angle limits. Now assume my endeffector 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).

I know this is a quite general robotics question. If this is not the right place to ask it. I would be delighted if you could point me to a different forum where I can ask this question.