Natural Oscillations of Underactuated Cable-Driven Parallel Robots

Abstract

Underactuated Cable-Driven Parallel Robots (CDPR) employ a number of cables smaller than the degrees of freedom (DoFs) of the end-effector (EE) that they control. As a consequence, the EE is underconstrained and preserves some freedoms even when all actuators are locked, which may lead to undesirable oscillations. This paper proposes a methodology for the computation of the EE natural oscillation frequencies, whose knowledge has proven to be convenient for control purposes. This procedure, based on the linearization of the system internal dynamics about equilibrium configurations, can be applied to a generic robot suspended by any number of cables comprised between 2 and 5. The kinematics, dynamics, stability and stiffness of the robot free motion are investigated in detail. The validity of the proposed method is demonstrated by experiments on 6-DoF prototypes actuated by 2, 3, and 4 cables. Additionally, in order to highlight the interest in a robotic context, this modelling strategy is applied to the trajectory planning of a 6-DoF 4-cable CDPR by means of a frequency-based method (multi-mode input shaping), and the latter is experimentally compared with traditional non-frequency-based motion planners.

A New Performance Index for Underactuated Cable-Driven Parallel Robots

“…where (•) denotes the skew-symmetric representation of a vector. Matrix K is referred to as Geometric Stiffness of the CDPR [18,19], because it depends on geometry, and it is fundamentally different from the so-called Passive Stiffness generated by cable deformations (not considered in this paper). From Eq.…”

“…Note that, if the equilibrium is stable, K ⊥ f is positive definite, thus invertible [18]. Then, the free-coordinate variation upon a variation of cable lengths is:…”

Design of a Planar Cable-Driven Parallel Robot for Non-Contact Tasks

“…In addition, the pulley swivel axes are equipped with rotary encoders to achieve redundant kinematic measures, and cable tensions are measured with shear beam load cells integrated into the proximal anchor systems. Like other planar CDPR that share the idea of parallelogram mechanisms [23,26], this system allows the platform to move on a defined plane, excluding low-amplitude oscillatory motions induced by external disturbances [27,28]. On the other hand, unlike the others, the proposed system shows three advantages: the use of a single cable loop, instead of two separate cables, ensures that the parallel cable segments are in tension when controlled by a single actuator, thus reducing the number of actuators to be used; the cable anchor system includes a measurement unit to continuously monitor cable tensions and cable angles with respect to the frame; the mounting arrangement of the swivel pulleys allows to estimate the cable lengths as distances between two precisely defined points, which are the intersections between the swivel axes and the work plane, without introducing geometrical approximations.…”

Equilibrium Configuration Analysis and Equilibrium-Based Trajectory Generation Method for Under-Constrained Cable-Driven Parallel Robot

“…Even with the control of the cables and the gravity pull of the EE, the generalized coordinates of the EE are not uniquely determined, causing unwanted oscillatory motions of the EE. This property unique to UCCDPRs raises a few challenging problems such as finding a stable equilibrium configuration [10], finding a feasible solution set of the forward and inverse kinematics [11], [12], [13], [14], [15], [16], [17], [18], and subduing unwanted oscillatory motions of the EE [19], [20], [21], [22], [23], [24], [25], [26], [27], [28], [29], [30], [31], [32], [33], [34], [35], [36].…”

“…And a rest-to-rest trajectory planning method was proposed by finding the solutions of the equations of motion of the UCCDPR satisfying given boundary value and constraints on position and velocity at start and end times [30]. Also, trajectory generation methods to suppress unwanted oscillations of EE were proposed by using an input shaping algorithm based on the analysis of natural frequencies [31], [32], [33], [34], [35], [36]. However, all the above-mentioned studies were conducted only on 2-and 3-cable UCCDPR, Gentry, Crane types, or generating simple linear trajectories for 4-cable systems.…”