Force Balance of a Spatial Metamorphic 6R Closed-Chain Linkage With Specific Kinematic Conditions

For deployable, reconfigurable, and metamorphic linkages force balance is an important property to bear the static and dynamic forces caused by the mass of each element, either due to gravity or due to inertia when moving. Force balance refers to ‘shaking force balance’ meaning that for all motion of the linkage, no dynamic reaction forces are exerted to its base and to the surroundings preventing base vibrations. This typically is an important feature in high-speed and high-precision robotic manipulation tasks. Force balance also refers to ‘static balancing with solely mass’. Since the center of mass of a force balanced linkage is stationary for all motion, a force balanced linkage is also statically balanced which means that the linkage is not affected by gravity and remains stationary in any pose without the need of actuators or brakes. This property is important for the design of, among others, large moving structures such as mobile platforms and deployable structures used in architecture.

In this paper the force balance of a spatial metamorphic plano-spherical 6R closed linkage with plane symmetry is investigated. It is shown that because of its specific kinematic conditions — being pantograph relations in the projected plane — advantageous balance solutions can be found for each of the four reconfiguration modes. For specific geometric conditions on the link design also solutions are found where the links balance one another without the need of any countermass. It is also shown how an advantageous compromise force balance solution for multiple modes is found. For this purpose an approximate force balance solution was investigated. The results are compared with the general force balance solutions of a general spatial 6R closed linkage, which are also shown and explained. All balance solutions are illustrated to have realistic interpretations.