Equilibrium stability assessment is a primary issue in continuum robots (CRs). The possible stable-to-unstable transitions that CRs may admit complicate the use of CRs in tasks where safety and human–robot interactions are mandatory. In this context, metrics measuring the distance from instability are essential but rarely developed. Existing metrics are frequently based on the evaluation of matrices involving mixed units, thus resulting in unit-dependent metrics. Moreover, the physical meaning of existing metric is hard to interpretate. This article proposes to use the magnitude of a force that brings instability to the CR equilibrium as a measure of the distance to instability. The major advantages of this metric are the intrinsic physical meaning, the practical interpretation of the results, and the well-defined unit of the measurements. The proposed metric (named directional critical load index) is based on the linearization of the eigenvalues of the reduced Hessian matrix of the total potential energy, which can be achieved regardless of the employed discretization technique. Three different case studies illustrate and demonstrate the main results of this article.

This article studies the trajectory planning for underactuated cable-driven parallel robots (CDPRs) in the case of rest-to-rest motions, when both the motion time and the path geometry are prescribed. For underactuated manipulators, it is possible to prescribe a control law only for a subset of the generalized coordinates of the system. However, if an arbitrary trajectory is prescribed for a suitable subset of these coordinates, the constraint deficiency on the end-effector leads to the impossibility of bringing the system at rest in a prescribed time. In addition, the behavior of the system may not be stable, that is, unbounded oscillatory motions of the end-effector may arise. In this article, we propose a novel trajectory-planning technique that allows the end effector to track a constrained geometric path in a specified time, and allows it to transition between stable static poses. The design of such a motion is based on the solution of a boundary value problem, aimed at a finding solution to the differential equations of motion with constraints on position and velocity at start and end times. To prove the effectiveness of such a method, the trajectory planning of a six-degrees-of-freedom spatial CDPR suspended by three cables is investigated. Trajectories of a reference point on the moving platform are designed so as to ensure that the assigned path is tracked accurately, and the system is brought to a static condition in a prescribed time. Experimental validation is presented and discussed.