Equilibrium Analysis-Based Trajectory Generation to Suppress Oscillatory Motion of Under-Constrained Cable-Driven Parallel Robot

Abstract

Cable-driven parallel robots (CDPRs) are among parallel robots, and they use several cables instead of rigid links to manipulate their end-effector (EE) or payload. Since the EE or payload of the CDPR is moved and controlled by adjusting the length of the light and flexible several cables, the mass and inertia of the moving parts among the mass of the entire CDPR system are quite small, making high-speed operations of the CDPR possible and reducing the energy consumption of the CDPR system. Among the CDPRs, the types of using fewer cables than the degrees of freedom (DoFs) of the EE are called under-constrained cable-driven parallel robots (UCCDPRs). Compared to fully-constrained and over-constrained CDPRs, UCCDPRs have a simpler structure, which results in low cost, and are free from problems caused by interference between cables or collisions between cables and objects existing in the workspace. Thus, UCCDPRs have been utilized in various application fields. However, an UCCDPR has several fundamental drawbacks brought about by structural distinction, even if disadvantages caused by cable properties such as flexibility, sagging, and unidirectional force transmission are excluded. Since fewer cables than the DoFs of the EE are used to control the EE, some of the DoFs of the EE are constrained by the cables but not the others. This means that the EE can be still moveable even if the lengths of all cables are fixed and taut. Even with the control of the cables and the gravity pull of the EE, the generalized coordinates of the EE are not uniquely determined. Due to these characteristics, unintended oscillatory motions and vibrations of the EE are caused. This paper proposes a method to prevent unwanted oscillatory motions and suppress the residual vibrations of the EE in general UCCDPRs. First, a methodology to find the stable equilibrium configuration in the workspace of the general UCCDPRs is proposed, and then an equilibrium-based trajectory (EQBT) generator is designed. EQBT generator makes it possible to obtain a feasible cable length that allows the EE to track a given trajectory. Even if the orientation trajectory is not given and only the position trajectory is given, the cable length to follow the given position trajectory of the EE can be obtained through the EQBT generator based on the equilibrium configuration corresponding to the given position trajectory. In addition, the proposed method allows a theoretical frequency analysis at all positions in the workspace by linearizing the internal dynamics of the UCCDPR around an equilibrium point obtained from the proposed method. Thus, it is possible to apply an input-shaping algorithm to generic UCCDPRs to suppress the residual vibrations of the EE. The effectiveness and performance of the proposed method were proved through computer simulations and hardware experiments.