A major advantage of reconfigurability is the possibility of modifying the workspace shape and the robot performances within it. Not only obstacles or cable collisions can be avoided (Gagliardini et al., 2016a) but robot key performances, such as the maximum payload or energy absorption, can be optimized. Indeed, the performances of a CDPR can be evaluated through several global or local indices that have been proposed in the literature (see, for example (Gouttefarde et al., 2007; Duan and Duan, 2014) and the references therein). By reconfiguring the exit-points of a R-CDPR, not only global but also local properties can be altered considerably to meet some desired requirements.

The capability of a TR-CSPR of balancing an external action (e.g., payload or environmental forces) is a relevant example of performance that can be optimized by reconfiguration. The WEC index proposed in (Boschetti and Trevisani, 2018) provides an evaluation of the maximum external wrench that the EE of a CDPR can bear along a given direction of interest and it is first applied here to TR-CSPRs.

The WEC is a local performance index that computes the maximum wrench component the cables can exert on the EE along a prescribed direction while maintaining assigned wrench components on the remaining directions and taking explicitly into account the lower ( t _ , generally greater than zero) and upper ( t ¯ , usually depending on cable and motor physical properties) cable tension limits.

For translational CDPRs, just forces rather than combined forces and torques (wrenches) are considered applied to the EE. Then, the general formulation presented in (Boschetti and Trevisani, 2018) can be applied straightforwardly to TR-CSPRs actuated by n cables converging in the EE, however, reconfigurability makes the optimization problem non-linear: the computation of the WEC index imposes the solution of a non-linear optimization problem where cable tensions and the Cartesian coordinates of the movable exit-points are the unknowns. Let identify the spatial directions of the cable forces by the unit vectors u i , with i = 1 , … , n . Under the hypotheses of massless and inextensible cables, the i-th cable force is directed as the vector associated to the i-th cable, which connects the Cartesian position p ∈ R 3 of the EE to the Cartesian position a i ∈ R 3 of the i-th exit-point; therefore, its direction is defined by u i = a i − p a i − p . The matrix collecting vectors u i is referred to as the structure matrix S : S a = u 1 a 1 u 2 a 2 ⋯ u n a n ∈ R 3 × n (1) Where a is the set of Cartesian coordinates of the reconfigurable exit-points and, in the general case, a = a 1 , … , a n . For sake of clarity, in Eq. 1, and in all the equations below, the dependance of vectors, and matrices from the sole unknowns of the problem is explicitly shown.

Through S a and by gathering the cable forces t i into vector t , the n-dimensional cable tension space is related to the three-dimensional space of the EE. If the equilibrium of the EE is considered, in addition to the force exerted by the cables ( S a t ), the EE is subject to the global external force denoted by w e , which usually comprises the gravitational force w g and other external forces w f : w e = w g + w f (2)

Let w ∼ a , t denote the total force acting on the EE: w ∼ a , t ≜ S a t + w e (3)

It can be recast as: w ∼ a , t ≜ W a t ∼ (4) where the structure matrix S a is augmented with vector w e yielding to the wrench matrix W a = S a w e , which multiplies the vector of the generalized cable forces t ∼ = t T 1 T . W a can be also interpreted as the matrix that collects the three row vectors, namely W a = w a x T w a y T w a z T T , that projects t ∼ onto the axes of the global reference frame. Each row vector can be chosen to evaluate the force exertion capability of the TR-CSPR along a Cartesian direction. Following a similar reasoning, by means of a proper rotation matrix R, it is possible to evaluate the force exertion capability along any arbitrary direction d: R W a = w a d T w a o 1 T w a o 2 T T (5) When the rotation matrix is employed, two additional orthogonal directions come along with d: o 1 and o 2 . These directions, taken together, define a new Cartesian reference frame on which the total wrench w ∼ a , t is projected: R w ∼ a = w ∼ d a , t w ∼ o 1 a , t w ∼ o 2 a , t T . Hence, the computation of the WEC _d is carried out by solving the following optimization problem: W E C d ≜ max t , a w ∼ d a , t = w a d T t ∼ s . t . w a o 1 T w a o 2 T t ∼ = w ∼ o 1 a , t w ∼ o 2 a , t = w ∼ r a i ∈ Σ i i = 1 , … , n t – < t i < t ¯ i = 1 , … , n (6) which consists in finding a and t that maximize the force exerted along the direction of interest d ( w a d T t ∼ ), while assigning given values w ∼ r (typically, but not necessarily, null values) to the forces along the directions o 1 ; o 2 , keeping the Cartesian coordinates of a i within the set of allowable configurations Σ i and keeping the cable tensions within t _ ; t ¯ . The optimization problem is non-linear because of reconfigurability which makes W a function of the unknowns a. Additionally, such non-linear terms also lead to a non-convex cost function and to non-convex constraints representing the force equilibria along the directions o 1 ; o 2 . In contrast, computing the WEC for fixed positions of the exit-points, as proposed in (Boschetti and Trevisani, 2018), leads to a linear and convex optimization problem whose solution is straightforward.

In this section, the general formulation of Eq. 6 is particularized for a three-cable TR-CSPR with a single reconfigurable exit-point. The TR-CSPR is shown in Figure 1I in a sample configuration, that is here discussed in detail: the EE, depicted as a black dot, has a mass m e e = 3 kg (assumed comprehensive of the payload) and it is placed at p = 1.4 0 1 T m; the fixed exit-points are placed at a 1 = 2 1 2 T m; a 2 = 2 − 1 2 T m while the reconfigurable exit-point is initially placed at a 3 = 0 0 3 T m (denoted as the nominal position). The Cartesian coordinates of the reconfigurable exit-point a 3 = a 3 x a 3 y a 3 z T can be independently changed within a rectangular cuboid Σ 3 defined through the respective allowable ranges [-1, 1.1] m [-1, 1] m and [1.2, 3] m. The components of a 3 are the unknowns of the problem together with the three cable tensions collected in t. The lower and upper cable tension limits are set to t _ = 0.1 N and t ¯ = 100 N. In Figure 1I all the exit-points are depicted in red, the reconfigurable one is in its nominal position and Σ 3 is shaded in blue. The global Cartesian reference frame Oxyz is shown too.

As a reasonable choice for this robot architecture, attention is paid to the maximum upward force that the cables can exert on the EE, the so-called WEC _z , while keeping null force components along the horizontal directions ( w ∼ r = 0 0 T N) and bounded cable tensions (with reference to the notation adopted in sub-Section 2.1.1, in this case; d = z , o 1 = x ; o 2 = y ). The following non-linear optimization problem is therefore stated: W E C z ≜ max t , a 3 w ∼ z a 3 , t = w a 3 z T t ∼ s . t . w a 3 x T w a 3 y T t ∼ = w ∼ x a 3 , t w ∼ y a 3 , t = w ∼ r a 3 ∈ Σ 3 t – < t i < t ¯ i = 1 , … , 3 (7)

The non-convex optimization problem formulated in Eq. 7 imposes the use of global optimization routines to get rid of the presence of several local minima and hence to find the actual optimal solution (Richiedei et al., 2019; Belotti et al., 2020). In this work, the standard GlobalSearch routine of Matlab is used to generate the set of initial guesses through an efficient and reliable scatter-search algorithm, while the gradient-based, fmincon interior-point solver is used as the local solver.

The analysis of such a sample position leads to W E C z = 163.2 N, by exploiting the optimal reconfiguration shown in Figure 1II, Figure 1III and Figure 1IV, where all the cable tensions reach the upper bound t 1 = 100 N, t 2 = 100 N, and t 3 = 100 N. The optimal position of the reconfigurable exit-point is painted in green, while its nominal position is painted in grey, the fixed exit-points are shown in red. Additionally, Figure 1II shows a top view of the TR-CSPR with the boundaries of Σ 3 sketched in dashed line, Figure 1III shows a side view, and finally, Figure 1IV reports a three-dimensional view, with Σ 3 shaded in blue. The green arrow applied to the EE represents w ∼ a 3 , t , which has only an upward component.

Just to show how the optimal reconfiguration depends on the direction of interest, and hence on the task features, at the same position p of the EE the WEC is now computed considering another arbitrary direction d = − 0.88 0.34 0.32 T . The computation of WEC _d allows determining the optimal configuration assuring the exertion of the maximum force along direction d while maintaining null force components along o 1 and o 2 ( w ∼ r = 0 0 T N). In this case such maximum force is equal to 90.3 N and the optimal configuration of the exit-point is depicted in Figure 1V; Figure 1VI and Figure 1VII with apparent meaning. Cable tensions take the following values t 1 = 27 N, t 2 = 0.4 N, and t 3 = 100 N.

In general, by repeating the computation of the WEC _d at all the positions where equilibrium can be maintained by the EE, it is possible to perform a thorough task-related analysis of the performance achievable within the whole workspace. For example, the WEC _z analysis discussed above can be repeated in the Statically Feasible Workspace (SFW) of the TR-CSPR to analyze how the robot load-carrying capability changes as the EE changes position. The SFW is defined as the set of the EE poses for which static equilibrium against gravity can be obtained using a limited range of cable tensions (Trevisani, 2013). The results of such an analysis are shown in Figure 2: the three coordinates of the EE p = p x p y p z T change within the ranges [-1, 2] m [-1, 1] m and [0, 3] m with the step size 0.2 m in the x, y, and z directions, respectively. The investigated positions of the EE are depicted and painted in accordance with the values taken by the WEC _z ; for sake of clarity, Σ 3 is shown too. The predicted performance within the SFW can be immediately recognized by the color of the dots.

The result of this analysis can be exploited to preliminary assess whether a task can be carried out in a part of interest of the workspace. For example, the possibility of moving quasi-statically a load can be easily verified through WEC _z : if the EE passes through positions where the WEC _z takes values greater than the load, there exists at least one configuration of the TR-CSPR which assures static equilibrium.

In the end, the WEC-based analysis appears useful to carry out preliminary investigations of task-related performances throughout the workspace, while it is not suitable to manage continuous exit-point reconfigurations for two reasons: firstly, the non-linear nature of the WEC formulation imposes a strong computational effort which is not compatible with real-time implementations; secondly, the optimal configurations computed in adjacent positions of the EE can differ significantly and could lead to discontinuous reconfigurations during the execution of a continuous displacement of the EE.