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Your robot has quite limited degrees of freedom, and the interactive marker in MoveIt's rviz plugin doesn't play well with robots like that. The first thing you can try is ticking the Allow Approximate IK Solutions option in the rviz control panel for MoveIt. It's on the "Context" tab. I have a similar arm and ticking that box allowed me to control it interactively, although not particularly well.

If you haven't already, generating an IKFast plugin for your MoveIt config will probably help a lot. Follow the tutorial, and choose the TranslationDirection5D IK type when generating the IK solver. Aside from making planning significantly faster, this solver will be able to more reliably provide a solution than the default IK in MoveIt. My arm's reachable space improved when I generated an IKFast plugin.

Finally, a trick that some people use is to add a fake link and joint at the end effector. This gives MoveIt a little more freedom in planning. You have to add it to your URDF and to your MoveIt config (both as the end point of the planning group for the arm and set it as a passive joint, since there is nothing to control it). See the turtlebot_arm packages for an example. The fake link and joint are defined on line 25 of the URDF, and you can run the MoveIt setup assistant and point it at the turtlebot_arm_moveit_config package to load in the turtlebot_arm MoveIt config and see how it is set up.