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 ∗Address all correspondence to this author. 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. INTRODUCTION Deployable, reconfigurable, and metamorphic linkages are useful for devices that need large flexibility and adaptability. For instance deployable structures are used in car roofs [1], in architecture [2], and in portable emergency shelters [3] where they need to be capable of being closed in a compact space and to extend over a large space when undeployed. Reconfigurable and metamorphic robot manipulators are capable of changing certain kinematic properties [4, 5] by which they are more flexible in carrying out different tasks or they are capable to perform more complicated tasks [6, 7]. One challenge in the design of deployable, reconfigurable, and metamorphic linkages is to bear the forces due to the mass and the motion of the mass of the elements, which can be significant already for small dimensions. Because of gravity these linkages may collapse if not properly and continuously actuated, which can happen especially at the instant moment of mobility change when moving from one configuration mode to another. For the linkage in motion the inertial forces can cause unwanted behavior such as base vibrations, especially at high speeds. Force balance then becomes an important aspect in the design of the linkage. A force balanced linkage is designed such that the common center of mass (CoM) of all elements is in a stationary point in 1 Copyright c ⃝ 2016 by ASME the base for all motion [8]. Therefore gravity has no effect on the motion of the linkage and on its performance, a property known as static balance. For instance force balanced movable canopies can be opened and closed in any position with minimal effort [9]. When in motion, a force balanced linkage does not exert any dynamic reaction forces (shaking forces) to its base, by which no base vibrations are induced. This is advantageous for low cycle times with high precision [10]. Force balance may also contribute to smoothen dynamic transitions through singularity positions as for unbalanced mechanisms gravity and inertial forces may easily cause uncontrollable dynamic effects. In the design of force balanced linkages it is challenging to obtain balance solutions that can be applied in practice since often significant additional mass, inertia, and complexity is needed [11, 12]. One approach to find applicable solutions is to investigate a linkage for specific kinematic or geometric conditions such that advantageous balance solutions are found [8, 13]. This approach has resulted in, among others, the successful design of a planar high-speed balanced parallel manipulator [10]. To the best of the authors’ knowledge this approach has not yet been applied to spatial linkages. In this paper the spatial metamorphic plano-spherical 6R closed linkage with plane symmetry that was presented in [14] is investigated for force balance. It is shown that advantageous force balance solutions can be found for each of the four reconfiguration modes because of the linkage’s specific kinematic conditions. This paper is organized as follows. First the geometry of the spatial plano-spherical 6R linkage is explained together with its metamorphic capabilities. Then, for comparison, the general force balance solutions for general spatial 6R closed chains are shown and explained. Subsequently the force balance solutions of the spatial plano-spherical 6R linkage are presented and discussed for each mode individually. Force balance solutions for multiple modes are discussed at the end, including approximate force balance solutions. GEOMETRY OF THE SPATIAL PLANO-SPHERICAL 6R LINKAGE WITH PLANE SYMMETRY The spatial plano-spherical 6R linkage is a plane symmetric metamorphic linkage consisting of a planar sublinkage and a spherical sublinkage. The geometry of the linkage was explained in [14] and is shown in Fig. 1. The six revolute pairs are A0, A1, A2, A3, A4, and A5 and the linkage is symmetric with respect to the plane through the rotational axis of A0 normal to the paper. The planar sublinkage is formed of the links A0A1 and A0A5 where joint A0 is the base pivot and the rotational axes of A1 and A5 are parallel to the rotational axis of A0 for all motion. These links have equal length l1 and have a mass m1 of which the CoM is located at a distance e1 from the axis through A0 as illustrated. Because of symmetry, these parameters are shown only for one a l 1