Under-constrained Cable-driven Parallel Robots Research Articles

Consistent Solution Strategy for Static Equilibrium Workspace and Trajectory Planning of Under-Constrained Cable-Driven Parallel and Planar Hybrid Robots

This paper presents a consistent solution strategy for static equilibrium workspaces of different types of under-constrained robots. Considering the constraint conditions of cable force and taking the least squares error of the static equilibrium equation as the objective, the convex optimization solution is carried out, and the static equilibrium working space of the under-constrained system is obtained. A consistent solution strategy is applied to solve the static equilibrium workspaces of the cable-driven parallel and planar hybrid robots. The dynamic models are presented and introducing parameters that are applied to make the system stable for point-to-point movements. Based on this model, the traditional polynomial-based point-to-point trajectory planning algorithm is improved by adding unconstrained parameters to the kinematic law function. The constraints of the dynamics model are incorporated into the trajectory planning process to achieve point-to-point trajectory planning for the under-constrained cable-driven robots. Finally, under-constrained cable-driven parallel robots with three cables and planar hybrid robot with two cables are taken as examples to carry out numerical simulation. The final results show that the point-to-point trajectory planning algorithm introducing parameters is effective and feasible and can provide theoretical guidance for the design of subsequent under-constrained robots.

Forward Kinematics for Suspended Under-Actuated Cable-Driven Parallel Robots With Elastic Cables: A Neural Network Approach

Abstract Kinematic analysis of under-constrained cable-driven parallel robots (CDPR) has been a topic of interest because of the inherent coupling between the loop-closure and static equilibrium equations. The non-linearity of the problem is magnified with the addition of the coupling between the cable lengths and their tensions based on the elastic cable model. The paper proposes an unsupervised neural network algorithm to perform real-time forward geometrico-static analysis of such robots in a suspended configuration under the action of gravity. The formulation determines a non-linear function approximation to model the problem and proves to be efficient in solving consecutive and close waypoints in a path. The methodology is applied on a six-degree-of-freedom spatial under-constrained suspended CDPR. Specific comparison results in simulation and hardware to show the effectiveness of the proposed method in tracking a given path are illustrated. Finally, the degree of constraint satisfaction is presented against the results obtained from non-linear least-square optimization.

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

Under-constrained cable-driven parallel robots (UCCDPRs) manipulate the end-effector (EE) by employing fewer number of cables than the degree of freedom of the EE, which causes unwanted swaying motions and oscillations. These unwanted vibrations can be suppressed by using the regenerated EE trajectory through the input-shaper. However, in the case of the UCCDPR systems, generating the feasible EE trajectory and deriving the natural frequency for designing the input-shaper is not straightforward since finding the stable equilibrium configuration is challenging. This paper proposes a novel methodology to find the stable equilibrium configuration of the general UCCDPRs in the assigned EE position. With the proposed method, the EE trajectory can be reproduced as a vibration suppression trajectory through an input shaper designed based on the analysis of the natural frequencies, as well as generating feasible EE trajectories based on the equilibrium configuration. Also, 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 based on the equilibrium configuration corresponding to the given position trajectory. Computer simulations and hardware experiments were conducted to verify the effectiveness and performance of the proposed method.

Vibration Reduction of Flexible Rope-Driven Mobile Robot for Safe Façade Operation

In recent years, cable-driven-parallel robots (CDPRs) have been studied for façade operations. There are various types of CDPRs; however, underconstrained CDPRs are capable of wider operating in façade workspaces than overconstrained CDPRs. Therefore, in this article, a dual ascender robot (DAR) was used for façade operations. Herein, two suggestions for safe façade operations are presented. First, a flexible nylon fiber rope was modeled such that the vibration direction, natural frequency, and damping ratio of the DAR could be converted through a Jacobian matrix and modal decomposition from the rope model. Second, input shaping control was applied to reduce vibrations, based on the vibration model of a DAR using the rope model. Modal decomposition was verified using a verification experiment, and the effect of input shaping was evaluated by comparing the w/input shaping and w/o input shaping experiments. w/input shaping case was shown about 48% reducing robot vibration and about 35% shortening settling time compare with w/o input shaping case.

Experimental study on the kinematic control of a cable suspended parallel robot for object tracking purpose

In this paper, two novel methods are proposed and compared which allows computation of the position of the end-effector in under-constrained cable-driven parallel robots. In the first method, from the data collected from an inertial measurement unit attached on the end-effector, the forward kinematic problem is reduced to a linear system of equations in which the position of the end-effector can be readily obtained real-time which is a definite asset for control purpose. In what concern the second method, a LoLiMoT neural network is trained with data collected from the simulation environment to solve the forward kinematic problem. Based on the obtained position feedback, an experimental closed loop kinematic control is performed and the proposed method is validated based on the observation of a camera which is mounted on the end-effector. A simple low cost mechanical structure is designed and constructed to measure the cable tensions by use of bending Loadcells. The sensors were calibrated and their data were used to guarantee the existence of minimum force in cables during the end-effector’s motion. At last, an algorithm for tracking a moving object is proposed and implemented. The experimental results verify the efficiency of the proposed methods.

Parasitic Inclinations in Cable-Driven Parallel Robots using Cable Loops

Cable-Driven Parallel Robots (CDPRs) also noted as wire-driven robots are parallel manipulators with flexible cables instead of rigid links. A CDPR consists in a base frame, a Moving-Platform (MP) and a set of cables connecting in parallel the MP to the base frame. CDPRs are well-known for their advantages over the classical parallel robots in terms of large workspace, reconfigurability, large payload capacity and high dynamic performance. In spite of all the mentioned advantages, one of the main shortcomings of the CDPRs is their limited orientation workspace. The latter drawback is mainly due to cable interferences and collisions between cables and surrounding environment. Hence, a planar four-Degree-of-Freedom (DoF) under-constrained CDPR with an articulated MP is introduced and studied in this paper. The end-effector is articulated through a cable loop, which enables the robot to obtain a modular pose determination, namely orientation and positioning. As a result, the mechanism under study has an unlimited and singularity-free orientation workspace in addition to a large translational workspace. It should be noted that some unwanted rotational motions of the moving platform, namely, parasitic inclinations, arise due to the cable loop. Finally, those parasitic inclinations are modeled and assessed for the mechanism at hand.

Trajectory generation to suppress oscillations in under-constrained cable-driven parallel robots

Cable-driven parallel robots (CDPRs) have many advantages over conventional link-based robot manipulators in terms of acceleration due to their low inertia. This paper concerns about under-constrained CDPRs, which have a less number of cables than six, often used favorably due to their simpler structures. Since a smaller number of cables than 6 are employed, however, their payloads have extra degrees of motion freedom and exhibit swaying motions or oscillation. In this paper, a scheme to suppress unwanted oscillatory motions of the payload of a 4-cable-driven CDPR based on a Zero-vibration (ZV) input-shaping scheme is proposed. In this method, a motion in the 3-dimensional space is projected onto the independent motions on two vertical planes perpendicular to each other. On each of the vertical plane, the natural frequency of the CDPR is computed based on a 2-cable-driven planar CDPR model. The precise dynamic model of a planar CDPR is obtained in order to find the natural frequency, which depends on the payload position. The advantage of the proposed scheme is that it is possible to generate an oscillation-free trajectory based on a ZV input-shaping scheme despite the complexity in the dynamics of the CDPR and the difficulty in computing the natural frequencies of the CDPR, which is required in any ZV input-shaping scheme. To verify the effectiveness of the proposed method, a series of computer simulations and experiments were conducted for 3- dimensional motions with a 4-cable-driven CDPR. Their results showed that the motions of the CDPR with the proposed method exhibited a significant reduction in oscillations of the payload. However, when the payload moves near the edges of its workspace, the improvement in oscillation reduction diminished as expected due to the errors in model projection.

Solving the direct geometrico-static problem of underconstrained cable-driven parallel robots by interval analysis

This paper presents an efficient interval-analysis-based algorithm to solve the direct geometrico-static problem (DGP) of underconstrained cable-driven parallel robots (CDPRs). Solving the DGP for a generic CDPR consists of finding all possible equilibrium poses of the end effector for the given cable lengths. Since cables impose unilateral constraints, configurations with one or more slack cables may occur. When the number of taut cables is smaller than six the robot is underconstrained and the DGP solutions must be found considering loop-closure and mechanical equilibrium equations simultaneously. The presented algorithm can find all DGP solutions of a generic CDPR in a numerically robust and safe way. By using interval analysis the proposed procedure can directly search for real solutions with non-negative cable tensions and it can take advantage of the physical constraints of the robot. The implemented procedure is discussed in detail, and the testing and experimental validation on working prototypes are presented.

Direct Geometrico-static Problem of Underconstrained Cable-Driven Parallel Robots With <inline-formula> <tex-math notation="LaTeX">$n$</tex-math></inline-formula> Cables

This paper studies the direct geometrico-static problem of underconstrained cable-driven parallel robots supported by $n$ cables, with $n\le 6$ . The task consists in identifying the equilibrium poses of the end-effector when cable lengths are specified. The problem is challenging, because the end-effector preserves some freedoms after cable lengths are assigned by motors. Hence, kinematics and statics are coupled, and they must be tackled simultaneously. A general method is presented to model the problem by a set of algebraic equations, and a least-degree univariate polynomial in the corresponding ideal is numerically found for any value of $n$ . For the efficient computation of the solution set, a software package is developed, which implements an algorithm based on homotopy continuation. Distinctive features of the code are that it finds all problem solutions, including those with slack cables, and stability analysis is integrated in order to identify feasible configurations.

Direct Geometrico-Static Problem of Underconstrained Cable-Driven Parallel Robots With Three Cables

Abstract This paper studies the direct geometrico-static problem (DGP) of underconstrained cable-driven parallel robots (CDPRs) with three cables. The task consists in determining the end-effector pose and the cable tensile forces when the cable lengths are assigned. The problem is challenging, because kinematics and statics are coupled, and they must be tackled simultaneously. An effective elimination procedure is proposed and a least-degree univariate polynomial free of spurious factors is obtained in the ideal governing the problem. This is proven to admit 156 solutions in the complex field. Several approaches for the efficient computation of the complete solution set are presented, including an eigenproblem formulation and homotopy continuation.

Stability Analysis of Underconstrained Cable-Driven Parallel Robots

This paper studies cable-driven parallel robots with less than six cables, in crane configuration. A geometrico-static model is provided, and the stability of static equilibrium is assessed within the framework of a constrained optimization problem. The method relies on ordinary linear-algebra routines, and it may be very simply applied to the most general architectures. Several examples are provided, concerning robots with a number of cables that range from 2 to 4.

Real solutions of the direct geometrico-static problem of under-constrained cable-driven parallel robots with 3 cables: a numerical investigation

This paper addresses the direct geometrico-static problem of under-constrained cable-driven parallel robots with 3 cables. The task at hand consists in finding all equilibrium configurations of the end-effector when the cable lengths are assigned. This problem is known to admit 156 solutions in the complex field, but the upper bound on the number of real solutions is as yet an open issue. Finding this bound is the objective of the paper. For this purpose, three numerical approaches are developed, namely a continuation procedure adapted from an algorithm originally proposed by Dietmaier and two evolutionary techniques based on a genetic algorithm and particle swarm optimization. In all cases, a number of sets of robot parameters for which the direct geometrico-static problem provides at the most 54 real configurations is found. The coherence of the obtained results leads to conjecture that the achieved bound is tight. However, formal proof is yet to be discovered.

Computing the lowest equilibrium pose of a cable-suspended rigid body

We solve the problem of finding the lowest stable-equilibrium pose of a rigid body subjected to gravity and suspended in space by an arbitrary number of cables. Besides representing a contribution to fundamental rigid-body mechanics, this solution finds application in two areas of robotics research: underconstrained cable-driven parallel robots and cooperative towing. The proposed approach consists in globally minimizing the rigid-body potential energy. This is done by applying a branch-and-bound algorithm over the group of rotations, which is partitioned into boxes in the space of Euler-Rodrigues parameters. The lower bound on the objective is obtained through a semidefinite relaxation of the optimization problem, whereas the upper bound is obtained by solving the same problem for a fixed orientation. The resulting algorithm is applied to several examples drawn from the literature. The reported Matlab implementation converges to the lowest stable equilibrium pose generally in a few seconds for cable-robot applications. Interestingly, the proposed method is only mildly sensitive to the number of suspending cables, which is shown by solving an example with 1000 cables in two hours.