Admissible spring configurations for statically balanced planar articulated manipulators have been investigated in previous studies. However, in these spring configurations, springs are only identified by the connection between links. The attachment angles and distance for springs to be properly installed remain unaddressed. In this study, a method to determine attachment angles and distance for springs is developed to ensure all the springs are acting for the benefit of static balancing. Here the gravitational and elastic potential energies are represented in stiffness matrix form, it is shown that term by term compatibility exists between the first row of gravitational stiffness matrix and the first row of the elastic stiffness matrix. In accordance with these compatibility conditions, the admissible spring attachment angles are found to ensure all the ground-connected springs are acting for the benefit of gravity balancing. And the remained components below the first row of the elastic stiffness matrix are offset by the non-ground-connected springs. In accordance with the compatibility between the remained components and the elastic stiffness matrix of non-ground-connected springs, the spring attachment angles to ensure all the non-ground-connected springs acting for the benefit of elastic balancing are found. The determination of the admissible spring configurations is revisited in addition to the connection between links, the attachment angles of springs are also specified. The admissible spring configurations of statically balanced planar articulated 3- and 4-link manipulators are derived. A 4-link planar manipulator is used as an example for illustration.
With the use of springs, a method to balance the constant forces in arbitrary directions on a planar serial manipulator is developed in this study. Gravity balancing has been discussed a lot in the past. However, manipulators usually bear forces from various directions rather than only a fixed one as gravity. For instance, an industrial manipulator would bear forces from everywhere during the working process. Therefore, a method to balance these forces in arbitrary directions with springs is proposed. Based on the representation of energy, spring energy is the function of springs’ attachment points. Two spring systems with different attachment angles are needed to balance respectively forces in arbitrary directions and gravity. The spring installations of the above systems on 3-DoF manipulators are proposed. Finally, a resistive force-balanced manipulator with/without gravity balance in the grinding process is shown. In sum, this paper for the first time develops the balancing method for forces in arbitrary directions, expanding the spring balance theory to a broader application.
Abstract. This paper proposes a spring efficiency assessment of a statically spring-balanced planar serial manipulator. The admissible spring configurations for the static balancing of planar serial manipulators without auxiliary links have been determined in the past. Gravity is balanced by the spring configuration systematically; however, the spring configuration also contains counter-effects between springs. Conceptually, with fewer counter-effects between springs, there is less burden on the spring system, which means that the springs are used more efficiently, and accordingly, the system would be safer, and its service life would be longer. In this study, the spring energy is represented in a quadratic form. The coefficients in a quadratic form represent the change in elastic energy with the relative position between links, which is named “elastic pseudo-stiffness”. Compared to the quadratic form of gravity energy, those elastic pseudo-stiffnesses for static balancing are regarded as positive contributions of a spring, while those that contain counter-effects are seen as negative ones. Spring efficiency is defined as the ratio of the elastic pseudo-stiffnesses, which has positive contributions for balancing to total elastic pseudo-stiffnesses. To use springs efficiently, the counter-effects, which are functions of spring parameters, need to be decreased, including spring stiffness and the attachment location of springs on links. A method to use spring efficiently by adjusting spring parameters is developed. Furthermore, it is found that, for a spring attached between adjacent links, the spring efficiency is 100 %, and the spring efficiency decreases while the number of joints over which the spring spans increases. In a spring manipulator system, the efficiency is negatively correlated to the payload. As an example, an efficiency assessment on a 3 degrees of freedom (DOF) manipulator is shown at the end.
Serially connected statically balanced manipulators with springs have been used in many applications. However, a portion of the torques caused by springs countering each other lead to an imbalance in gravitational torques and, therefore, are deemed as waste torques for springs to achieve static balance. In this paper, the torque contribution of a typical spring is classified as gravity-balancing torque and counter-torque based on the accumulated joint angle of the gravitational torque. Then, the internal counter-torque is defined as the sum of the magnitude of the terms of these counter-torques at each joint. Through the adjustment of spring attachment parameters, the internal counter-torque can be minimized with preferable spring attachment parameters while maintaining a static-balancing condition. A typical four-link manipulator with a preselected spring configuration is shown as an illustrative example. The results show that there are 28% and 50% reductions in the internal counter-torque at joints 2 and 3, respectively, through the adjustment of spring attachment parameters. Hence, the waste torques in statically balanced serially connected manipulators are reduced to the lowest quantity.
Force analysis with regard to serial connected manipulators are discussed thoroughly in the past. However, force analysis of statically balanced manipulator using springs has not been widely addressed because spring forces and motions do not share an immediate association. In this paper, spring forces are represented as accumulative joint angles of links crossed by springs and attached angles/lengths of springs. Torque equilibrium equations regarding the pre-connected joint of a typical link as contributed by gravity force and spring force can be inwardly formed link by link from the end link. Compatibility with the same accumulative joint angle can be formulated under static balance conditions. Hence, spring attachment parameters such as spring stiffness and attachment lengths, are constrained by given link properties and spring attachment angles. Thus, spring forces can be determined by a chosen set of stiffness and attached lengths of springs, and the joint reaction force can then be determined. Example figures of 3-DOF manipulators show that joint reaction forces are reduced by 22.6%, 40.1% and 75.7% at joint 1, 2 and 3 than those without springs. It is found that besides balancing gravity, the statically balanced manipulator is with lower joint reaction forces. Hence, the manipulator can be more lightweight by using compact joints and links with the same material. Furthermore, the static and dynamic performance of the manipulator can be improved by the effect of reduced joint reaction forces as well.
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