3.1. Forward dynamics formulation

The dynamic model of the cable is modified to accommodate the relative motions of their exit points with respect to the global frame of reference $\boldsymbol{{o}}$ - $X_0Y_0Z_0$ . Due to these additional DoF, the configuration of cable $C_k$ is updated to:

(7) \begin{align} \boldsymbol{{q}}_j = \left [\boldsymbol{{b}}_k^\top, q_1,\dots,q_{3{n_{\textrm{e}}}}\right ]^\top \in{\mathbb{R}^{3 ({n_{\textrm{e}}}+1)}},\; j=k=1,\dots,{n_{\textrm{k}}}. \end{align}

In Eq. (7), the symbol $\boldsymbol{{b}}_k$ denotes the position of the exit point of cable $C_k$ , whereas $q_\iota, \iota =1,\dots,3{n_{\textrm{e}}},$ with $n_{\textrm{e}}$ being the number of elements, represent the configuration of the cable. The number of cables is denoted by $n_{\textrm{k}}$ . Further changes to the formulation of dynamics based on the nature of inputs provided to the analyses are outlined below.

3.1.1. Inputs in the form of forces applied at the exit points of cables

The first kind of inputs considered in the following is the forces exerted by the driving mechanisms on the cables. In such cases, assuming no other external forces are acting on the cables, the vector of external generalised forces of cable $C_k$ has the following structure:

(8) \begin{align} \boldsymbol{\tau }_k = \left [\boldsymbol{\lambda }_i^\top, 0, \dots, 0\right ]^\top \in{\mathbb{R}^{3 ({n_{\textrm{e}}}+1)}},\; k=1,\dots,{n_{\textrm{k}}},\; i={n_{\textrm{k}}}+ k. \end{align}

The force $\boldsymbol{\lambda }_{{n_{\textrm{k}}}+ k}$ in Eq. (8) acts at the exit point of cable $C_k$ .

Alternatively, the generalised forces internal to the actuators of the driving mechanisms, denoted by $\boldsymbol{\tau }_i$ , could be specified as inputs to the analyses when there is no provision for direct measurement of $\boldsymbol{\lambda }_i$ . In such scenarios, as depicted in Fig. 3, every mechanism $\textrm{D}_k,\,k=1,\dots,{n_{\textrm{k}}}$ , is treated as a separate sub-system of the CDPR to compute the required restraining forces at the corresponding exit point.

Figure 3. Driving mechanisms, $\textrm{D}_k,\,k=1,\dots,{n_{\textrm{k}}}$ , as separate sub-systems of a Type II CDPR. The symbol $\boldsymbol{\tau }_i, i={n_{\textrm{k}}}+k,$ denotes the internal actuator forces of the $k$ th driving mechanism and $\boldsymbol{\lambda }_i$ represents the reaction forces at the exit point $\boldsymbol{{b}}_k$ of the cable $C_k$ .

At first, the connectivity between the sub-systems $\textrm{D}_k$ and $C_k$ is ensured by introducing the action-reaction pair of forces, denoted by $\boldsymbol{\lambda }_i$ . Subsequently, $\boldsymbol{\lambda }_i$ are computed from the imposition of additional constraint equations, $\boldsymbol{\eta }_i=\textbf{0}$ , where:

(9) \begin{align} \boldsymbol{\eta }_i = \boldsymbol{{b}}_k -{\boldsymbol{{e}}}_k,\; k=1,\dots,{n_{\textrm{k}}},\,i={n_{\textrm{k}}}+k. \end{align}

In Eq. (9), the location of the leading end of cable $C_k$ is denoted by $\boldsymbol{{b}}_k$ and its attachment point on $\textrm{D}_k$ by ${\boldsymbol{{e}}}_k$ .

In summary, the above formulation can handle inputs either in the form of forces directly applied on the exit points of cables or those that are internal to the actuators of the driving mechanisms. However, there can be many situations where the inputs are available in the form of motions of the exit points rather than the forces acting on them. One instance of such a scenario can be where the exit points of the cables are being carried by individual aerial vehicles, such as quadrotors or helicopters – instead of incorporating their dynamics into the overall model, one can simply track their motions and use those as kinematic inputs to the overall system. Analyses of such systems are discussed next.

3.1.2. Inputs in the form of trajectories tracked by the exit points of cables

The second kind of inputs to the analyses are the trajectories of the exit points of cables. As shown in Fig. 4, the point $\boldsymbol{{b}}_k$ of cable $C_k$ is considered to follow the input trajectory ${\boldsymbol{{e}}}_k(t)$ . Consequently, the inputs at some of the joints will be kinematic while the rest are dynamic in nature. Therefore, simulation of the CDPR requires a hybrid approach, that is, both the FD and ID analyses as classified in [Reference Featherstone30], pp. $171$ – $172$ . Such problems of CDPRs are addressed for the first time in the present work, to the best of the authors’ knowledge.

Figure 4. Representative trajectories $e_k(t)$ of the exit points $\boldsymbol{{b}}_k$ of cables $C_k$ in Type II CDPRs. The reaction forces acting at the exit point $\boldsymbol{{b}}_k$ are denoted by $\boldsymbol{\lambda }_{{n_{\textrm{k}}}+k}$ .

In the present work, the hybrid dynamic analysis of the robot is circumvented by introducing the following constraint equations:

(10) \begin{align} \boldsymbol{\eta }_i(\boldsymbol{{b}}_k, t) \,:\!=\, \boldsymbol{{b}}_k -{\boldsymbol{{e}}}_k(t) = 0,\; k=1,\dots,{n_{\textrm{k}}},\, i={n_{\textrm{k}}}+k. \end{align}

Clearly, Eq. (10) represents rheonomic constraints,Footnote ³ that is, they depend explicitly on time. Therefore, the associated Lagrange multipliers, denoted by $\boldsymbol{\lambda }_i$ , are also functions of time. Specifically, differentiating $\boldsymbol{\eta }_i$ twice results in the following expression:

(11) \begin{align}{\frac{\textrm{d} \boldsymbol{\eta }_i}{\textrm{d} t}} &={\frac{\textrm{d} }{\textrm{d} t}} \left ({\frac{\partial \boldsymbol{\eta }_i}{\partial t }} + \sum _{j=1}^v \boldsymbol{{J}}_{\boldsymbol{\eta }\boldsymbol{{q}}_{i,j}}{\dot{\boldsymbol{{q}}}}_j\right ) \nonumber \\ &={\frac{\partial ^2 \boldsymbol{\eta }_i}{\partial t^2}} + \sum _{j=1}^v \left ({\frac{\partial }{\partial q_j }} \left ({\frac{\partial \boldsymbol{\eta }_i}{\partial t }}\right ){\dot{\boldsymbol{{q}}}}_j + \boldsymbol{{J}}_{\boldsymbol{\eta }\boldsymbol{{q}}_{i,j}}{\ddot{\boldsymbol{{q}}}}_j +{\frac{\textrm{d} }{\textrm{d} t}}\left (\boldsymbol{{J}}_{\boldsymbol{\eta }\boldsymbol{{q}}_{i,j}}\right ){\dot{\boldsymbol{{q}}}}_j \right ) . \end{align}

Equation (11) may be simplified significantly by taking into cognisance the following.

1. 1. The variations of $\boldsymbol{\eta }_i$ w.r.t. time and the configuration variables, $\boldsymbol{{q}}_j$ , are mutually independent, hence ${\frac{\partial }{\partial q_j }}\left ({\frac{\partial \boldsymbol{\eta }_i}{\partial t }}\right ) ={\textbf{0}}$ .

2. 2. The matrix $\boldsymbol{{J}}_{\boldsymbol{\eta }\boldsymbol{{q}}_{i,j}}$ is of constant nature, since its entries are either ones or zeros. Therefore, ${\frac{\textrm{d} }{\textrm{d} t}}\left (\boldsymbol{{J}}_{\boldsymbol{\eta }\boldsymbol{{q}}_{i,j}}\right ) ={\textbf{0}}$ .

With these, Eq. (11) reduces to:

(12) \begin{align} &{\frac{\textrm{d}^2 \boldsymbol{\eta }_i}{\textrm{d} t^2}} ={\frac{\partial ^2 \boldsymbol{\eta }_i}{\partial t^2}} + \sum _{j=1}^v \boldsymbol{{J}}_{\boldsymbol{\eta }\boldsymbol{{q}}_{i,j}}{\ddot{\boldsymbol{{q}}}}_j ={\textbf{0}},\; i={n_{\textrm{k}}}+1,\dots,2{n_{\textrm{k}}}. \end{align}

The key difference between the above form of the loop-closure equation and that associated with the scleronomic types (as in Eq. (5)) is the additional term $\frac{\partial ^2 \boldsymbol{\eta }_i}{\partial t^2}$ , accounting for the explicit dependence of $\boldsymbol{\eta }_i$ on time. Consequently, the expression of the residual constraint accelerations $\boldsymbol{\psi }_i$ (see Eq. (15)) is modified to:

(13) \begin{align} \boldsymbol{\psi }_i ={\frac{\partial ^2 \boldsymbol{\eta }_i}{\partial t^2}} + \sum _{j=1}^v \boldsymbol{{J}}_{\boldsymbol{\eta }\boldsymbol{{q}}_{i,j}} \boldsymbol{{a}}_j,\; i={n_{\textrm{k}}}+1,\dots,2{n_{\textrm{k}}}. \end{align}

In Eq. (13), the symbol $\boldsymbol{{a}}_j \in{\mathbb{R}^{n_j}}$ denotes the unconstrained joint accelerations of the $j$ th sub-system and it is given by:

(14) \begin{align} \boldsymbol{{a}}_j ={{\boldsymbol{{M}}}_j}^{-1} \boldsymbol{{f}}_j,\;j=1,\dots,v, \end{align}

where $\boldsymbol{{f}}_j$ is the cumulative effect of the generalised forces devoid of the inertial and constrained forces acting on the $j$ th sub-system, as in Eq. (2). Finally, the Lagrange multipliers can be computed using the same expressions reported in ref. [Reference Mamidi and Bandyopadhyay2], that is:

(15) \begin{align} \sum _{s=1}^{u} \boldsymbol{{A}}_{i, s} \boldsymbol{\lambda }_s + \boldsymbol{\psi }_i = \textbf{0},\;i=1,\dots,u,\textrm{ and }u= 2{n_{\textrm{k}}}, \end{align}

where $\boldsymbol{{A}}_{i, s} \in{\mathbb{R}^{m_i \times m_i}}$ is the sub-matrix of the generalised inertia constraint matrix, mentioned in ref. [Reference Koul, Shah, Saha and Manivannan31]. Effectively, by virtue of the generalisations mentioned above, the modified RSSLM approach can simulate the dynamics of Type II CDPRs even when the trajectories of the driving mechanisms are provided as inputs.

4.1. Architecture of a 4-4 cable-driven parallel robot

A $4$ - $4$ CDPR consists of four cables $C_1,\dots,C_4$ connected to the MP at $\boldsymbol{\xi }_1,\dots,\boldsymbol{\xi }_4,$ and to the driving mechanisms at $\boldsymbol{{b}}_1,\dots,\boldsymbol{{b}}_4$ . It is decomposed into four cables and an MP, as shown in Fig. 5. The length, width and height of the MP are represented by ${l_{\textrm{m}}},{w_{\textrm{m}}},{h_{\textrm{m}}},$ respectively. Since the local frame of reference $\boldsymbol{\xi }_{\textrm{c}}$ - $X_aY_aZ_a$ is considered identical to the fixed one $\boldsymbol{{o}}$ - $X_0Y_0Z_0$ at the initial configuration, the ZYZ convention of Euler angles used for representing its orientation in the previous works, for example, [Reference Mamidi and Bandyopadhyay2], will lead to a parametric singularity (see, e.g., [Reference Shah, Saha and Dutt32], pp. $51$ – $52$ ). Therefore, the XYZ convention is used in this case to circumvent the singularity, and the corresponding DH parameters are listed in Table I.

Figure 5. Notional decomposition of the $4$ - $4$ cable-driven parallel robot into four cables and a moving platform.

The action-reaction forces at the disassembled joints are determined via the computation of the Lagrange multipliers $\boldsymbol{\lambda }_1,\dots,\boldsymbol{\lambda }_4$ associated with the loop-closure constraint functions in Eq. (3). The dimension of the configuration space of every cable is given by ${n_{\textrm{k}}}=3({n_{\textrm{e}}}+1), k=1,\dots,4,$ and hence, that of the $4$ - $4$ CDPR is $n=12{n_{\textrm{e}}}+18$ . The number of constraint functions is $m=24$ , that is, three per each end of the cable.

4.2. Pick-and-place operation by the 4-4 cable-driven parallel robot

The numerical values of the architecture and inertia parameters of the robot are listed in Table II. Further, the values of the various tolerances used in the simulation are the same as the ones reported in Table C.6 of [Reference Mamidi and Bandyopadhyay2]. The unstrained lengths of the cables are:

(16) \begin{align} l_k=0.75 \textrm{ m}, k=1,\dots,4. \end{align}

Since these unstrained lengths are small and do not vary with time, only five MRFEs are used for modelling each cable; thereby, ${n_{\textrm{e}}}=5$ . Subsequently, the coefficients of stiffness of the SDEs are determined using the expressions reported in ref. [Reference Mamidi and Bandyopadhyay2]. Their numerical values are given by:

(17) \begin{align} &{s_{\textrm{a}}} = 1.16 \times 10^7\,\textrm{N/m},\;{s_{\textrm{t}}} ={s_{\textrm{l}}}= 11.64\,\textrm{Nm/rad}. \end{align}

In Eq. (17), the subscripts a, t and l denote the values associated with deflections in the axial, transverse and lateral directions, respectively. Likewise, the calculated damping coefficients of cables are as follows:

(18) \begin{align}{d_{\textrm{a}}} = 2.40 \times 10^{5}\,\textrm{Ns/m},\;{d_{\textrm{t}}}={d_{\textrm{l}}}= 0.24\,\textrm{Nms/rad}. \end{align}

The initial pose of the robot is shown in Fig. 6. As mentioned before, the initial configuration of the MP is selected such that it rests on the ground with its local frame of reference $\boldsymbol{\xi }_{\textrm{c}}$ - $X_aY_aZ_a$ oriented in the same direction as the fixed frame of reference $\boldsymbol{{o}}$ - $X_0Y_0Z_0$ . Therefore, its position and orientation are given by:

(19) \begin{align} \boldsymbol{{q}}_5 = \left [0.70, 1.00, 0.10, 0, 1.57, 0\right ]^\top . \end{align}

Next, the initial configurations of the cables are chosen to be taut and oriented along the vertical direction. Accordingly, their numerical values are obtained as follows:

(20) \begin{align} &\boldsymbol{{q}}_1=\left [0.40, 0.60, 0.95, 0, 3.14, 0, \dots \right ]^\top, \boldsymbol{{q}}_2=\left [1.00, 0.60, 0.95, 0, 3.14, 0, \dots \right ]^\top, \nonumber \\ &\boldsymbol{{q}}_3=\left [0.40, 1.40, 0.95, 0, 3.14, 0, \dots \right ]^\top, \boldsymbol{{q}}_4=\left [1.00, 1.40, 0.95, 0, 3.14, 0, \dots \right ]^\top . \end{align}

Finally, the trajectories of the exit points of cables are planned as described below.

Figure 6. Initial configuration of the $4$ - $4$ cable-driven parallel robot with its moving platform resting on the ground, that is, the plane $X_0Y_0$ .

At first, the path followed by every point $\boldsymbol{{b}}_k, k=1,\dots,4,$ is divided into four line segments. As depicted in Fig. 7, the first segment, $L_1$ , is meant to vertically lift the MP from its initial location; the second, $L_2$ , is meant to spatially ascend it; the third, $L_3$ , is meant to spatially descend it; and the last, $L_4$ , is meant to vertically lower it to the desired destination on the ground. Next, the trajectories along each segment are defined via cubic polynomials of time, assuming that the points $\boldsymbol{{b}}_k$ start and end with zero velocities.

Figure 7. Specified identical path followed by the exit points of the cables $\boldsymbol{{b}}_k$ , $k=1,\dots,4$ . It comprises four line segments, $L_1$ to $L_4$ .

The input trajectories of the exit points are represented using the difference $\Delta{\boldsymbol{{e}}}_k(t) ={\boldsymbol{{e}}}_k(t)-{\boldsymbol{{e}}}_k(0),\,k=1,\dots,4$ . The components of the vector $\Delta{\boldsymbol{{e}}}_k=\left [\Delta e_{k\textrm{x}}, \Delta e_{k\textrm{y}}, \Delta e_{k\textrm{z}}\right ]^\top$ are defined as the following piece-wise functions:

(21) \begin{align} \Delta e_{k\textrm{x}} & = \Delta e_{k\textrm{y}} = \left \{ \begin{array}{l@{\quad}l} 0, & 0 \leq t \leq 2,\\[5pt] \dfrac{(t-2)^2(7-2t)}{2\sqrt{2}}, & 2 \lt t \leq 3,\\[15pt] \dfrac{82 + t (t(21 - 2 t) -72)}{2\sqrt{2}}, & 3 \lt t \leq 4,\\[9pt] \dfrac{1}{\sqrt{2}}, & 4 \lt t \leq 6. \end{array}\right ., \nonumber \\[10pt] \Delta e_{k\textrm{z}} &= \left \{ \begin{array}{l@{\quad}l} \dfrac{t^2(3-t)}{4}, & 0 \leq t \leq 2,\\[9pt] \dfrac{2-\sqrt{3} (t-2)^2 (2t-7)}{2}, & 2 \lt t \leq 3,\\[9pt] \dfrac{2+\sqrt{3}+\sqrt{3} (t-3)^2 (2t-9)}{2}, & 3 \lt t \leq 4,\\[9pt] \dfrac{(t-6)^2 (t-3)}{4}, & 4 \lt t \leq 6; \end{array}\right .\;k=1,\dots,4. \end{align}

The variations of $\Delta{\boldsymbol{{e}}}_k$ with time are depicted in Fig. 8. A video file named “motionOf44 cdprN5_ipTrjExPts.mp4” depicts the motion of the robot for the specified input.

Figure 8. Input trajectory of the exit points of the cables $\Delta{\boldsymbol{{e}}}_k ={\boldsymbol{{e}}}_k(t) -{\boldsymbol{{e}}}_k(0)$ , $k=1,\dots,4$ . The legends $\Delta e_{k\textrm{x}}$ , $\Delta e_{k\textrm{y}}$ , $\Delta e_{k\textrm{z}}$ represent the vector components of $\Delta{\boldsymbol{{e}}}_k$ . Due to the symmetry in the chosen path, $\Delta e_{k\textrm{x}}$ is identical to $\Delta e_{k\textrm{y}}$ .

The displacements of the MP for the input trajectories of $\boldsymbol{{b}}_k$ are shown in Fig. 9, using the difference $\Delta \boldsymbol{{q}}_5 (t)= \boldsymbol{{q}}_5 (t)-\boldsymbol{{q}}_5(0)$ . In addition, the variations of its instantaneous velocities and accelerations with time are depicted in Figs. 10a, 10b and Figs. 10c, 10d, respectively. Apart from the small yaw motion of the MP initiated at the end of vertical lift, that is, at $t=2$ s, there is no significant change in the orientation of the MP (see Fig. 9b).

Figure 9. Variation in the configuration of the moving platform of the $4$ - $4$ cable-driven parallel robot, $\Delta \boldsymbol{{q}}_5(t) = \boldsymbol{{q}}_5(t)-\boldsymbol{{q}}_5(0)$ , for the inputs given in Eq. (21).

Figure 10. Variations in the linear, angular velocities $\dot{\boldsymbol{{q}}}_5(t)$ and accelerations $\ddot{\boldsymbol{{q}}}_5(t)$ of the moving platform of the $4$ - $4$ cable-driven parallel robot, corresponding to the inputs given in Eq. (21).

Further, the variations of the linear displacements of the MP (shown in Fig. 9a) are similar to those of the exit points of the cables (see Fig. 8). Ideally, these variations would be the same if the dynamics of the robot were ignored. Therefore, an error measure $\delta{\boldsymbol{\xi }_{\textrm{c}}}(t)$ used to quantify such effects is defined as:

(22) \begin{align} \delta{\boldsymbol{\xi }_{\textrm{c}}}(t) = \Delta{\boldsymbol{\xi }_{\textrm{c}}}(t)- \Delta{\boldsymbol{{e}}}_k(t). \end{align}

The changes in $\delta{\boldsymbol{\xi }_{\textrm{c}}} (t)$ with time are depicted in Fig. 11. As is evident from the results, a kinematic estimation of the displacements of the MP will be erroneous. In this case, the maximum deviations are given by ${\| \delta{\boldsymbol{\xi }_{\textrm{c}}}\|}_\infty = \left [4.46, 3.30, 0.16\right ]^\top \times 10^{-2}$ m, which are approximately $\left [10, 8, 0\right ]^\top \%$ relative error in the respective directions. Similarly, the errors in estimating the linear velocities are ${\| \delta{\dot{\boldsymbol{\xi }}_{\textrm{c}}}\|}_\infty =\left [0.34, 0.32, 0.02\right ]^\top$ m/s and the linear accelerations are ${\| \delta{\ddot{\boldsymbol{\xi }}_{\textrm{c}}}\|}_\infty =\left [3.37, 4.90, 0.32\right ]^\top$ m/s ${}^2$ . Therefore, the dynamics of the cables and the MP should be included in the analyses for better accuracy.

Figure 11. Variations in the error $\delta{\boldsymbol{\xi }_{\textrm{c}}} = \Delta{\boldsymbol{\xi }_{\textrm{c}}}- \Delta \boldsymbol{{b}}_k(t)$ with time. The scalar components of $\delta{\boldsymbol{\xi }_{\textrm{c}}}$ are denoted by $\delta \xi _{\textrm{x}}$ , $\delta \xi _{\textrm{y}}$ and $\delta \xi _{\textrm{z}}$ .

The magnitudes of the axial extensions of the cables are minimal ( $O\left (10^{-6}\right )$ m) throughout the simulation of the robot. The changes in the constraint forces at the anchoring points of the cables $\boldsymbol{{c}}_k$ are shown in Fig. 12, and those corresponding to the exit points $\boldsymbol{{b}}_k$ are depicted in Fig. 13. All the cables pull the MP vertically for the entire duration of the simulation because ${\lambda }_{i_{\textrm{z}}}(t) \lt 0,\;\forall i=1,\dots,4,$ and ${\lambda }_{i_{\textrm{z}}}(t) \gt 0,\;\forall i=5,\dots,8$ . The former indicates that the cables counter the weight and vertical inertial forces of the MP, while the latter implies that the driving mechanisms are being pulled during the operation. Hence, these reaction forces complement each other, as seen by the change in sign of the respective plots. Also, the abrupt change in $|{\lambda }_{i_{\textrm{z}}}|$ in the vicinity of $t=6$ s is due to the impact of the MP on the ground.

Figure 12. Variations in the values of components of the reaction forces $\boldsymbol{\lambda }_i,\,i=1,\dots,4,$ with time, corresponding to the simulation of the $4$ - $4$ cable-driven parallel robot for the inputs given in Eq. (21).

Figure 13. Variations in the values of components of the reaction forces $\boldsymbol{\lambda }_i,\,i=5,\dots,8,$ with time, corresponding to the simulation of the $4$ - $4$ cable-driven parallel robot for the inputs given in Eq. (21).

The computational timeFootnote ⁴ taken to perform the simulation with the input trajectories ${\boldsymbol{{e}}}_k(t)$ of the points $\boldsymbol{{b}}_k$ in Eq. (21) is $2261$ s (approximately, $37$ minutes). Similar to the results presented in ref. [Reference Mamidi and Bandyopadhyay15], the solver increases the temporal resolution to accurately determine the robot’s response to the transitions, as the exit points switch segments in their paths. However, as depicted in Fig. 14, the majority of the time is spent in capturing the dynamics of the robot while spatially ascending and descending the MP. This could be attributed to the initiation and the rapid variations in transverse and lateral deflections of cables, which is also evident from the MP’s displacements shown in Fig. 9 from $t=2$ s to $5$ s.

Figure 14. Variations in the step size $\Delta t$ used by the solver ode15s at every instance of the simulation of the $4$ - $4$ CDPR for the input trajectories of the exit points of the cables $\boldsymbol{{b}}_k$ in Eq. (21). The sustenance of smaller steps can be seen during the spatial ascend and descend of the MP, that is, segments $L_2$ and $L_3$ in Fig. 7.

4.2.1. Validation of the simulation results

The satisfaction of the imposed kinematic constraints is verified by employing the error measures $e_1$ and $e_2$ defined in Appendix E. The variations of errors $e_1,\,e_2$ of the constraints responsible for the connectivity between the cables and the MP are depicted in Fig. 15. Similarly, those responsible for tracking the input trajectory of the exit points of cables are shown in Fig. 16. In both cases, the residues are within respective desirable limits for the entire duration of the simulation. Hence, the obtained results are confirmed to be valid.

Figure 15. Variations in the errors $e_1$ and $e_2$ with time, corresponding to the constraint functions associated with the connectivity between the cables and the MP. The definitions of $e_1$ and $e_2$ are given in Eqs. (E1) and (E2), respectively.

Figure 16. Variations in the errors $e_1$ and $e_2$ with time, corresponding to the constraint functions responsible for incorporating the input trajectories of the exit points of cables. The definitions of $e_1$ and $e_2$ are given in Eqs. (E1) and (E2), respectively.

4.3. Dynamic response of the robot in the event of mid-flight failure in the actuation

In this example, the driving mechanisms connected to the $4$ - $4$ CDPR are chosen to be four identical quadcopters. First, the dynamic model of the quadcopter is described. Next, the temporal changes in the forces internal to the quadcopters are devised. Finally, the response of the robot starting from the configuration shown in Fig. 6 is presented.

The architecture of the quadcopter that drives the cable $C_k$ is shown in Fig. 17. A local frame of reference, ${\boldsymbol{{e}}}_k$ - $X_{\textrm{m}_k}Y_{\textrm{m}_k}Z_{\textrm{m}_k}$ , is attached at its centre of mass. The orientation of this frame with respect to the inertial frame of reference $\boldsymbol{{o}}$ - $X_0Y_0Z_0$ is represented using the XYZ convention of Euler angles ${\alpha }_k,\beta _k,\gamma _k$ . Therefore, the configuration of the $k$ th quadcopter is defined by $\boldsymbol{{q}}_j=\left [\boldsymbol{{e}}_k^\top,{\alpha }_k,\beta _k,\gamma _k\right ]^\top$ , $j=k+5$ . The DH parameters required for modelling these sub-systems are listed in Table I.

Figure 17. Schematic of the model of quadcopter. The thrust forces and reactive moments of the $k$ th quadcopter are denoted by $f_{1_k},\dots,f_{4_k}$ and $m_{1_k},\dots,m_{4_k}$ , respectively. The reaction forces from the $k$ th cable is denoted by $\boldsymbol{\lambda }_{{n_{\textrm{k}}}+k}$ .

Further, the thrust forces and reactive moments of the rotors are directed along the $Z_{\textrm{m}_k}$ -axis at all times. The magnitudes of the former are denoted by $f_{1_k}, \dots, f_{4_k}$ , while those of the latter are represented by $m_{1_k}, \dots, m_{4_k}$ . Moreover, the ratio of these moments to the forces, denoted by $c_{\textrm{tm}}$ , is constant for specific propeller designs (see, e.g., [Reference Burggräf, Pérez Martínez, Roth and Wagner33]). Therefore, only four of these eight magnitudes can be varied independently. Furthermore, drag forces are neglectedFootnote ⁵ in the present work.

Accordingly, the equations of motion of the quadcopters can be expressed in the form of Eq. (1). Subsequently, the external forces, $\boldsymbol{\tau }_j$ , are computed using the expression:

(23) \begin{align} &\boldsymbol{\tau }_j = \left [f_{\textrm{t}_k}\sin{\beta _k}, -f_{\textrm{t}_k}\sin{\alpha _k}\cos{\beta _k}, f_{\textrm{t}_k}\cos{\alpha _k}\cos{\beta _k}, \left(f_{2_k}-f_{4_{k}}\right)r_k, \left(f_{3_k}-f_{1_{k}}\right)r_k,m_{\textrm{t}_k}\right ]^\top, \nonumber \\[3pt] &\textrm{where } f_{\textrm{t}_k}= f_{1_k}+\dots +f_{4_k},\,m_{\textrm{t}_k} = m_{1_k}+\dots +m_{4_k},\;k=1,\dots,4,\;j=k+5. \end{align}

In Eq. (23), the symbol $r_k$ denotes the offset of the rotors’ axes of rotation from the point $\boldsymbol{{e}}_k$ . Since there are no SDEs associated with these sub-systems, the vector $\boldsymbol{\tau }_j^{\textrm{s}}$ is null. In addition, the sub-matrices of the constraint Jacobian matrix ${\boldsymbol{{J}}}_{\boldsymbol{\eta }\boldsymbol{{q}}}$ associated with the constraint functions in Eq. (9) are constant. Therefore, there is no need for numerical estimation of these matrices, and their entries are either ones or zeroes. Furthermore, the sparsity of the GIMs ${\boldsymbol{{M}}}_j,\,j=6,\dots,9,$ is taken into account while computing their inverses. The numerical values of the architecture and inertia parameters of the quadcopters used in the simulation are listed in Table III.

The input actuator forces of the quadcopters are planned as follows. First, the yaw motion of quadcopters is prevented by imposing the following conditions on the reactive moments of the rotors:

(24) \begin{align} m_{1_k}=-m_{3_k},\,m_{2_k}=-m_{4_k},\;k=1,\dots,4. \end{align}

Next, the changes in the thrust forces for the four quadcopters are considered as:

(25) \begin{align} &f_{1_k} = f_{2_k} = f_{3_k} = f_{4_k} = f_k,\;k=1,\dots,4,\nonumber \\[10pt] &f_k = \left \{ \begin{array}{l l} \dfrac{5}{2}\sin{\left (\dfrac{\pi }{2} t\right )}, & 0 \lt t \leq \dfrac{1}{2},\\[10pt] \dfrac{5}{2}, & \dfrac{1}{2}\lt t \leq 3,\\[7pt] 0, & k=1, t \gt 3, \\[5pt] \dfrac{5}{2}, & k \neq 1, t \gt 3. \end{array}\right . \end{align}

Their variations with time are shown in Fig. 18. Finally, the corresponding values of the reactive moments are computed using the constant $c_{\textrm{tf}}$ . A video file named “motionOf44cdprN5_ipForces_vertInitCond.mp4” depicts the motion of the robot for the specified input when started from the configuration shown in Fig. 6.

Figure 18. Temporal variations of the input thrust forces of the quadcopters, $f_{1_k} = f_{2_k} = f_{3_k} = f_{4_k} = f_k$ , $k=1,\dots,4$ .

The difference $\Delta \boldsymbol{{q}}_5 = \boldsymbol{{q}}_5 (t)-\boldsymbol{{q}}_5(0)$ is used for studying the linear and angular displacements of the MP due to the input forces mentioned above. The variations of $\Delta \boldsymbol{{q}}_5$ with time are depicted in Fig. 19. Initially, the collective forces exerted by the quadcopters are not sufficient to lift the MP. However, after $t=0.48$ s, the MP starts to move up vertically, that is, $\Delta \xi _{\textrm{z}} \gt 0$ (see Fig. 19a). Subsequently, the MP accelerates in that direction with no significant change in its orientation. Even though the actuators of the first quadcopter are shut off at $t=3$ s to simulate its failure, the MP continues to ascend due to inertia of motion. However, its orientation no longer remains the same as the initial one (see Fig. 19b). Eventually, due to insufficient support from the remaining three quadcopters, the MP starts to descend at $t=4.21$ s. The corresponding changes in the velocities and accelerations of the MP are depicted in Figs. 20a, 20b and Figs. 20c, 20d, respectively.

Figure 19. Variation in the configuration of the moving platform of the $4$ - $4$ cable-driven parallel robot, $\Delta \boldsymbol{{q}}_5(t) = \boldsymbol{{q}}_5(t)-\boldsymbol{{q}}_5(0)$ , for the inputs given in Eq. (25).

Figure 20. Variations in the linear, angular velocities $\dot{\boldsymbol{{q}}}_5(t)$ and accelerations $\ddot{\boldsymbol{{q}}}_5(t)$ of the moving platform of the $4$ - $4$ cable-driven parallel robot, corresponding to the inputs given in Eq. (25).

Further, the variations in the reaction forces at the trailing ends of the cables are shown in Fig. 21. Similarly, the changes in those at the exit points of the cables are depicted in Fig. 22. As is evident from the plots, the variations of reaction forces in different cables remain the same until $t=3$ s. Moreover, the cables start to pull the MP vertically, that is, ${\lambda }_{i_z}\lt 0,\,i=1,\dots,4,$ only after the applied actuator forces reach a specific limit at $t=0.31$ s. Also, due to the inertia of rest of the MP, slightly larger forces are required to initiate the lift. Hence, an abrupt decrease in the magnitude of reaction forces $\left |{\lambda }_{i_z}\right |$ can be seen after $t=0.48$ s. Furthermore, the first cable becomes non-supportive and acts as additional load on the remaining quadcopters after $t=3$ s.

Figure 21. Variations in the values of components of the reaction forces $\boldsymbol{\lambda }_i,\,i=1,\dots,4,$ with time, corresponding to the simulation of the $4$ - $4$ cable-driven parallel robot for the inputs given in Eq. (25).

Figure 22. Variations in the values of components of the reaction forces $\boldsymbol{\lambda }_i,\,i=5,\dots,8,$ with time, corresponding to the simulation of the $4$ - $4$ cable-driven parallel robot for the inputs given in Eq. (25).

Figure 23. Variations in the time steps $\Delta t$ of the solver ode15s at every instance of the simulation of the $4$ - $4$ CDPR for the changes in the actuator forces of the quadcopters in Eq. (25). The sustenance of smaller steps can be seen after the failure of the first quadcopter after $t=3$ s. Such a failure renders the cable to be non-supportive and leads to disruptions in the motions of the remaining cables and quadcopters.

There is no observable difference in the orientation of the quadcopters. However, the changes in their linear displacements are as depicted in Fig. 24. Although the actuator inputs of the first quadcopter cease to exist after $t=3$ s, its motion is governed by the reaction forces $-\boldsymbol{\lambda }_5$ , shown in Fig. 22a.

Figure 24. Variations in the linear displacements, $\Delta \boldsymbol{{b}}_k = \boldsymbol{{b}}_k(t)-\boldsymbol{{b}}_k(0)$ , of the quadcopters, $\textrm{D}_k$ , $k=1,\dots,4,$ for the inputs specified in Eq. (25).

Similar to the results presented in Section 4.2, the deflections in the cables due to their axial compliance are minimal throughout the simulation. The computational time taken to obtain the response of $4$ - $4$ CDPR for the inputs given in Eq. (25) is $1.38 \times 10^{4}$ s. It takes only $7.39$ s to obtain results for $t\in [0, 3)$ s. Therefore, as seen in Fig. 23, most of the time is spent obtaining the results for $t \in [3, 6]$ s, where the cables undergo transverse and lateral displacements.

Furthermore, even with the same input forces as in the previous one, when the simulation is initiated using a different configuration of cables, such as the one shown in Fig. 25, it took only $121$ s of computational time. This difference can be attributed to gravity assisting the cables to deflect in the transverse direction when the quadcopters fail to apply sufficient pulling forces on the cables. In the former, due to the alignment of cables with the direction of gravity, that is, the vertical axis $\boldsymbol{{b}}_o$ - $Z_0$ (shown in Fig. 6), both gravity and the quadcopters induce compressive loads. A video file named “motionOf44cdprN5_ipForces_incInitCond.mp4” depicts the motion of the robot for the specified input when started from the configuration shown in Fig. 25.

Figure 25. Initial configuration of the $4$ - $4$ cable-driven parallel robot with its moving platform resting on the ground, that is, the plane $X_0Y_0$ . Also, in contrast to the configuration depicted in Fig. 6, the cables are not aligned along the direction of gravity, the axis $Z_0$ .

4.3.1. Validation of the simulation results

The values of the errors $e_1$ and $e_2$ defined in Appendix E, are computed for the entire duration of the simulation. The variations in these errors associated with the constraint functions $\boldsymbol{\eta }_i,\,i=1,\dots,4,$ are shown in Fig. 26, and those related to $\boldsymbol{\eta }_i,\,i=5,\dots,8,$ are depicted in Fig. 27. In both cases, the errors are within desirable limits. Therefore, the connectivity of the cables with the MP and the quadcopters is verified to be intact during the simulation of the robot.

Figure 26. Variations in the errors $e_1$ and $e_2$ with time, corresponding to the constraint functions associated with the connectivity between the cables and the MP. The definitions of $e_1$ and $e_2$ are given in Eqs. (E1) and (E2), respectively.

Figure 27. Variations in the errors $e_1$ and $e_2$ with time, corresponding to the constraint functions responsible for incorporating the input trajectories of the forces of the quadcopters. The definitions of $e_1$ and $e_2$ are given in Eqs. (E1) and (E2), respectively.

4.4. Pick-and-place operation with sinusoidal feed rates of the cables

The simulation is initiated from the configuration of the robot depicted in Fig. 25. In addition to the input trajectories of the exit points defined in Eq. (21), the input feed rates of the cables are provided using the following expression:

(26) \begin{align} \dot{l}_k = \frac{{\omega _{\textrm{c}}}}{10} \cos{\left ({\omega _{\textrm{c}}}t + (k-1)\frac{\pi }{10}\right )},\;k=1,\dots,4,\;{\omega _{\textrm{c}}}=\frac{\pi }{5}. \end{align}

The associated input trajectories of the cables’ lengths are depicted in Fig. 28. Accordingly, the masses, moments of inertia and locations of the centres-of-mass of rigid links of MRFEs are updated at every time instance. The stiffness and damping coefficients are recomputed. The necessary changes to the formulation of dynamics noted in ref. [Reference Mamidi and Bandyopadhyay15] are introduced.

Figure 28. Variations in the input unstrained lengths of cables of the 4-4 CDPR, $l_k,\,k=1,\dots,4$ , associated with the feed rates given in Eq. (26).

The changes in the position and orientation of the MP are represented using the difference $\Delta \boldsymbol{{q}}_5$ in Figs. 29a, 29b, respectively. The associated variations in the velocity and acceleration of the MP are shown in Figs. 30a, 30b and Figs. 30c, 30d, respectively. As opposed to the results presented in Section 4.2, significant changes in the MP’s angular displacements can be observed in this case due to the time-varying lengths of cables.

Figure 29. Variations in the configuration of the moving platform of the 4-4 cable-driven parallel robot, $\Delta \boldsymbol{{q}}_5= \boldsymbol{{q}}_5(t)- \boldsymbol{{q}}_5(0)$ , with time for the inputs given in Eqs. (21) and (26).

Figure 30. Variations in the linear, angular velocities $\dot{\boldsymbol{{q}}}_5(t)$ and accelerations $\ddot{\boldsymbol{{q}}}_5(t)$ of the moving platform of the $4$ - $4$ cable-driven parallel robot, corresponding to the inputs given in Eqs. (21) and (26).

Figure 31. Variations in the errors $e_1$ and $e_2$ with time, corresponding to the constraint functions associated with the connectivity between the cables and the MP. The definitions of $e_1$ and $e_2$ are mentioned in Eq. (E1) and Eq. (E2), respectively.

As is the case with the previous case studies, the axial extensions of the cables are minimal for the entire duration of the simulation. The variations in the forces exerted by the MP on the cables at their anchoring points are shown in Fig. 32. The forces responsible for the desired motion of the exit points of cables are depicted in Fig. 34. Evidently, all the cables pull the MP during the operation of the robot. However, the load is not shared uniformly amongst the cables due to the temporal changes in their lengths. A depiction of the response of the CDPR to simultaneous application of feeding of cables and the movement of their exit points is included as a video file named “motionOf44cdprN5_diffMoA.mp4.”

Figure 32. Variations in the values of components of the reaction forces $\boldsymbol{\lambda }_i,\,i=1,\dots,4,$ with time, corresponding to the simulation of the $4$ - $4$ cable-driven parallel robot for the inputs given in Eqs. (25) and (26).

Figure 33. Variations in the errors $e_1$ and $e_2$ with time, corresponding to the constraint functions that ensure the trajectories of the exit points of the cables to be the same as the inputs. The definitions of $e_1$ and $e_2$ are mentioned in Eq. (E1) and Eq. (E2), respectively.

Figure 34. Variations in the values of components of the reaction forces $\boldsymbol{\lambda }_i,\,i=5,\dots,8,$ with time, corresponding to the simulation of the $4$ - $4$ cable-driven parallel robot for the inputs given in Eqs. (25) and (26).