2.1 Non-reconfigurable CDPM

The CDPM faces some constraints because cables can pull the end effector, not push it [Reference Lahouar, Ottaviano, Zeghoul, Romdhane and Ceccarelli24]. Merlet and Daney [Reference Merlet and Daney30] indicates that collision between the cables and the end effector may limit the workspace. Nguyen and Gouttefarde [Reference Nguyen and Gouttefarde22] discussed that a collision may occur in some cases: (1) between cables, (2) between cables and the end effector (self-collision), (3) between the end effector and the environment, and (4) between cables and objects in the environment. In these situations, designing an algorithm to determine optimal trajectory motion between two collision-free configurations is important because the CDPM cables may collide with other cables, human, or static objects in the workspace.

Kowalczyk et al. [Reference Kowalczyk, Michałek and Kozłowski31] presented a vector field orientation control algorithm for the end effector in an environment with obstacles in order to track the desired trajectory. Also, the local artificial potential function (APF) [Reference Ge and Cui32] is used as a collision avoidance method that surrounds obstacles. So, a robot can be repulsed when it is close to the boundaries of obstacles. In the artificial potential field method, by defining the functions of artificial attraction and repulsion potential, the robot is attracted to the target and repelled by obstacles. Lyapunov function candidate is used to analyze stability. Lahouar et al. [Reference Lahouar, Ottaviano, Zeghoul, Romdhane and Ceccarelli24] worked on collision-free path planning for four-cable-driven parallel robot in order to avoid collisions between cables and obstacle and between cables.

There are some approaches in order to solve the path planning problem such as moving obstacles include APF methods [Reference Malone, Chiang, Lesser, Oishi and Tapia33], sample-based methods [Reference Otte and Frazzoli34], geometry-based methods [Reference Yang, Alvarez and Bruggemann35], and velocity obstacle-based methods [Reference Chakravarthy and Ghose36]. Bak et al. [Reference Bak, Hwang, Yoon, Park and Park37] presented a modified goal-biased RRT algorithm and GJK algorithm to find the distance between the robot and fixed objects and solve the cable collision problem. Zhang et al. [Reference Zhang, Shang and Cong38] proposed an improved RRT algorithm with a cost function in order to guarantee the obstacle avoidance and finding the optimal path. Bordalba et al. [Reference Bordalba, Porta and Ros39] proposed a method in order to find a collision-free path between two points while keeping the cables in tension and simultaneously adhering to the actuators and joints force capabilities. They validated their presented method by experimental data on a specific CDPM. Also, in the proposed method, positions and velocities of two points are needed to compute a collision-free path. A new method presented by Wischnitzer et al. [Reference Wischnitzer, Shvalb and Shoham40] allows collisions between cables by expanding the workspace compared to free-collision workspace mechanisms. The presented method formulates the inverse kinematics of 6-DOF redundant CDPM by considering two colliding cables while keeping a feasible and positive WCW. They presented theoretical and experimental results and expansion workspace compared with a free-collision one. However, the vibration problem due to colliding cables, especially in high-speed applications, is not studied. Also, the friction at the contact point between cables is not considered. Aref and Taghirad [Reference Aref and Taghirad41] studied the collisions between cables, cable to the body of 6-DOF cable-driven manipulator. The whole workspace could be indicated based on designing a full force feasible mechanism, and then the workspace boundaries are recognized by a collision-free algorithm. Analysis of this approach takes less computational time and is practical in real-time application, while the collision between cables and obstacles inside the CDPM workspace is not considered, whereas the effect of the collision avoidance method in the workspace is discussed in our presented paper. Pinto et al. [Reference Pinto, Moreira, Lima, Sousa and Costa42] proposed SPIDERobot as 4-DOF-driven mechanism for pick and place in industrial applications. A new approach is introduced based on visually locating the robot’s position and the position of obstacles to optimize the robot’s trajectory by visual interpretation of the workspace. This method can be helpful for collision avoidance between cables and the environment. Simulated models indicate that this approach is valid for cable mechanisms under constrain. However, this approach is not applied for reconfigurable, fully constrained CDPM. The authors do not present an analysis of the wrench or feasible workspace.

Wang et al. [Reference Wang, Zi, Qian and Zhang43] presented the WCW of planar 3-DOF cable robot without collision. This research finds the area where the end effector and obstacles are not colliding. The study did not evaluate workspace and the performance of the method on different spatial geometric configurations (motorized reel location) of the CDPM. Blanchet and Merlet [Reference Blanchet and Merlet44] investigated interference detection between the cable–cable and cable–objects for a 6-DOF freedom cable-driven robot by two algorithms based on interval analysis. The authors proposed a non-crossed cable model. The crossed cable configuration has a larger workspace and higher torques than the non-crossed cable configuration.

Otis et al. [Reference Otis, Perreault, Nguyen-Dang, Lambert, Gouttefarde, Laurendeau and Gosselin45] proposed a method to manage cable interference between two 6-DOF foot platforms in a cable-driven locomotion interface. This method estimates the cable interference geometrically for any constrained trajectory. Then, they proposed an algorithm that could determine which cable could be released from an active actuation state while keeping cables in tension to maintain the trajectory. The problem of tension discontinuity is solved by presenting a collision prediction scheme applied to redundant actuators. Also, the limitation of the workspace as the main issue due to folding two cables on each other is discussed. This workspace is limited due to one cable release of its actuation state. A sudden increase in tension of other cables may cause mechanical vibration and instability.

More recently, Meziane et al. [Reference Meziane, Cardou and Otis46] proposed a new approach in order to avoid collision between cables by using an admittance controller when a robot is in physical interaction with a human. Collision detection between two cables can cause changing the trajectory of the end effector. While moving the end effector toward a collision, the user feels a virtual force and pushes the end effector to increase the distance between the two cables but in the inverse direction of the current motion to avoid this collision.

These previously studied algorithms implemented with fixed configurations on CDPMs affected the performance of the system to avoid collision with obstacles. Also, these algorithms for complex tasks and cluttered environments are not very reliable because the geometric configurations of CDPMs are limited. Indeed, reconfiguration is introduced by relocating the positions of the attachment points on base (motorized reel location) to allow the initial desired trajectory.

2.2 Geometric reconfiguration of the base

Variable-structure CDPMs can be defined by allowing collisions between cables and fixed objects in the workspace. Allowing collisions along the length of cables is discussed by Rushton and Khajepour [Reference Rushton and Khajepour47]. Attachment points of cables can be changed when a collision is recognized along the length of a cable and cause changes in the dynamic structure of the mechanism. One of the significant advantages of variable-structure cable robot (VSCM) is the ability to cover a non-convex workspace. In comparison, moving obstacle is not discussed in this paper. Geometric reconfiguration method, proposed by Youssef and Otis [Reference Youssef and Otis29], allows avoiding collision between cables by relocating attachment points on rails. However, collision avoidance method between cable and moving obstacles is not presented by authors. Anson et al. [Reference Anson, Alamdari and Krovi48] investigated to add a moving base for a planar 3-DOF mechanism driven by four cables. Using mobile base for the CDPM increases kinematic redundancy which maximizes the size and quality of the WCW, but it requires precision to control.

A Tension Factor index tests the quality of the WCW. Also, the WCW is analyzed by comparing two planar mobile base configurations and a traditional CDPM. Two configurations, rectangular base and circular base, are introduced. Each base is restricted from moving on its linear rail in the rectangular configuration, and an angle is made between each rail and the two adjacent rails. Also, the bases are restricted from moving along a circular rail in a circular configuration. The authors mentioned that the circular base has a better WCW where the end effector could reach any position and orientation. This research focuses on planar CDPM, and cable interference avoidance is not presented.

2.3 Geometric reconfiguration of the end effector

Barbazza et al. [Reference Barbazza, Oscari, Minto and Rosati49] proposed a method as online reconfiguration of attachment points on the end effector for a 3-DOF under-constrained suspended CDPM with four cables. The reconfigurable end effector is presented by changing attachment points’ position at the end effector to avoid collisions with obstacles. Also, when the end effector is far from obstacles, the attachment points on the end effector can be reconfigured to increase performance. The algorithm can be defined to optimize trajectory in some processes, such as pick and place. But in this research, attachment points on the base are fixed, and analysis workspace caused by online reconfiguration is not evaluated.

2.4 The perspective of this article

The initial trajectory in the haptic system or collaborative physical human–robot interaction should be unchanged because the trajectory comes from an operator. Still, the collision between cables and obstacles constrains the performance of CDPM and can be dangerous for user safety. This paper focuses on the collision avoidance method to improve the performance of CDPM using geometric reconfiguration with workspace analysis. Whereas the workspace between humans and robots is shared, previous research adapts the end effector’s trajectory to avoid collision between cables or between cables and the environment or release a cable from its actuation state to keep the same trajectory. Some methods presented in previous section are used to track predetermined trajectory. Also, collision between cables and cables with obstacles are considered, whereas these methods may limit the workspace.

Figure 1. Geometric configuration of the cable-driven parallel mechanism: (a) Arrangement of attachment points on the rails and (b) arrangement of attachment points on the end effector.

This paper presents a new method to detect and avoid collision between cables and human, while the trajectory of the end effector is unchanged and end effector tracks the trajectory inside the workspace. To perform this idea, a model of the interference between cables and human should be considered that is presented in Section 3.

3.1 Coordinate system and attachment points

In this study, a fixed global frame (X–Y–Z) can be seen in Fig. 1, at the bottom left corner, and the Z axis direction is vertically upward, and the X–Y axes can be found according to right-hand rule. Whereas the attachment points on the rails are fixed in conventional CDPM, to avoid collision, our approach moves up/down the cable attachment points on the rail (reel location) in order to increase the distance between the human with cables to the safest threshold as suggested in ref. [Reference Haddadin, Albu-Schäffer and Hirzinger50]. As it can be observed in Fig. 1, servo-actuated reels are connected to the moving platform with eight cables to get a fully constrained CDPM in the six DOFs. Each cable is connected to the end effector at points $\mathbf{B}_{i}$ . To relocate the attachment points on rails $\mathbf{A}_{i}$ , equations are obtained to find the desired cable length to reach a given end effector position and orientation. Suggested configuration is used to detect distance between cables and human and to obtain new position of attachment points on rails while keeping the end effector trajectory unchanged. In this research, the cables are considered massless and straight lines.

To verify the performance of the presented approach to avoid collision between cables and a human in the shared workspace, a human skeleton is inserted through a colliding distance with cables in simulation where the end effector performs its trajectories for its production tasks.

There are eight attachment points ( $\mathbf{B}_{i}$ ) on the end effector (rectangular prism shape). The dimensions of the end effector are 0.5 × 0.5 × 0.5 $\mathrm{m}^{3}$ . The location of attachment points ( A i) on the base and the locations of the attachment points ( $\mathbf{B}_{i}$ ) on the end effector with respect to the local frame ( $\varepsilon$ in meters) are observed respectively in Table I and Table II.

Table I. Initial positions of the eight attachment points (m).

Table II. Local positions of the eight attachment points (m).

The human limbs length and human height are presented in Table III.

Table III. Human limbs length (m).

Kinematic and driving constraints are discussed in Sections 3.2 and 3.3. These constraints should be considered to evaluate forward and inverse kinematics. Kinematic constraint restricts moving attachment points only in vertical direction.

3.2 Kinematics constraints

The generalized coordinates of the system or $\vec {\mathbf{q}}$ consisting of $\vec {\mathbf{q}}_{ac}$ and $\vec {\mathbf{q}}_{er}$ are presented in (1):

(a) $\vec {\mathbf{q}}_{ac}$ is defined as generalized coordinates of the attachment points on the rail.

(b) $\vec {\mathbf{q}}_{er}$ is defined as generalized coordinates of the attachment points on the end effector.

(1) \begin{align} \vec {\mathbf{q}}& =\left[\left({\vec {\mathbf{q}}_{er}}\right)^{T}, {\left({\vec {\mathbf{q}}_{ac}}\right)^{T}}^{T}\right] \nonumber \\ \vec{\mathbf{q}}_{er} & = [\textbf{X}_{p},\,\textbf{Y}_{p},\,\textbf{Z}_{p},\boldsymbol\alpha_{p},\,\boldsymbol\beta_{p},\,\boldsymbol\gamma_{p}], \nonumber \\ \vec{\mathbf{q}}_{ac} &= [\textbf{X}_{i},\,\textbf{Y}_{i},\,\textbf{Z}_{i}] \end{align}

In this project, the movement in vertical direction is limited for each attachment points. Kinematic constraints for moving in X and Y directions have been restricted as follows [Reference Youssef and Otis29]:

(2) \begin{equation} \left[\begin{array}{c} \mathbf{x}_{i}\\[5pt] \mathbf{y}_{i} \end{array}\right]=\left(\begin{array}{c} \mathbf{k}_{i}^{1}\\[5pt] \mathbf{k}_{i}^{2} \end{array}\right)i=1\;to\;n \end{equation}

Cartesian coordinates of attachment points on reels in the global coordinate system are indicated by $\mathbf{x}_{i}$ and $\mathbf{y}_{i}$ . $\mathbf{k}_{i}$ is a constant and indicates attachment points position with respect to the fixed frame.

3.3 Driving constraints

The driving constraints are introduced as defined motion trajectories. In this project, three groups of driving constraints are considered.

1. a. The first group of driving constraints, due to the vertical motion of the eight attachment points on the rails, is presented in (3) as suggested in ref. [Reference Youssef and Otis29]:

   (3) \begin{equation} \mathbf{z}_{i}=\mathbf{c}_{i}\left(t,\vec {\mathbf{q}}\right)\text{for }i=1\;\text{to}\;\mathrm{n}, \end{equation}
   where $\mathbf{z}_{i}$ is defined as Z coordinate of each reel on the rails and $\mathbf{c}_{i}\left(t,\vec {\mathbf{q}}\right)$ is a function to drive the attachment point vertically. Also, this function should depend on $t$ (time), $\vec {\mathbf{q}}$ (generalized coordinates), human limbs velocity, and sampling frequency for example.

   For this research work, this value is fixed as a constant. Therefore, each attachment point can move vertically 0.1 m up and down in simulation as explained in Section 3.6.

1. b. The second group of driving constraints can be introduced due to the extension and retraction of each cable attached to the end effector. It is indicated by changing the length of cables. The length of the cable can be defined by hypotenuse of a right-angle triangle by the vertices $\mathbf{A}_{1}$ , $\mathbf{B}_{1}$ , and $\mathbf{B}_{1Z}$ . As can be seen in Fig. 2, these equations of second group constraint are presented as follows:

(4) \begin{equation} \left({{\mathbf{B}_{x}}}\right)_{i}^{2}+\left({{\mathbf{B}_{y}}}\right)_{i}^{2}+\left({\mathbf{A}_{i}}-\left({{\mathbf{B}_{z}}}\right)_{i}\right)^{2}=\mathbf{p}_{i}^{2}\,\,\,\text{for }i=1\;\text{to}\;\mathrm{n} \end{equation}

1. c. The third group of driving constraints in this paper is workspace analysis. Workspace analysis of CDPM is the ability of the end effector to translate or orient under some constraints. Five workspace categories are introduced: static equilibrium workspace, WCW, feasible wrench workspace, dynamic workspace, and collision-free workspace [Reference Duan and Duan51]. The feasible wrench workspace is presented as all possible positions and orientations of the end effector while maintaining a positive tension among all the cables. Set of positive tension vector in all the cables maintain each set of external wrenches acting on the end effector. The feasible wrench workspace is presented as follows:

   (5) \begin{equation}\tilde{\mathrm{w}}\vec {\mathbf{t}}+\vec {\mathbf{w}_{J}}=\vec {\mathbf{0}_{6}}\quad \mathrm{t}_{min}\lt \vec {\mathbf{t}}\lt \mathrm{t}_{max}\;\text{ and }\tilde{\mathrm{w}}=\left[\begin{array}{c@{\quad}c@{\quad}c} \vec {u_{1}} & \ldots & \vec {u_{8}}\\[5pt] \vec {r_{1}}\vec {u_{1}} & \ldots & \vec {r_{8}}\vec {u_{8}} \end{array}\right] \end{equation}
   $\vec {\mathbf{w}_{J}}$ or external wrench acting on the end effector is weight of the end effector as 25 n in the Z direction. $\tilde{\mathrm{w}}$ and $\vec {\mathbf{t}}$ are structure matrix and tension in the cables, respectively. The two variables ( $\vec {r_{i}}$ and $\vec {u_{i}}$ ) in the structure matrix $\tilde{\mathrm{w}}$ are explicit functions in the position and orientation of the end effector as well as the attachments points position. The tension of each cable is positive value, and it is between minimum tension $\mathrm{t}_{min}$ = 20 and maximum tension $\mathrm{t}_{max}$ = 90 n [Reference Youssef and Otis29].

Figure 2. Second group of driving constraints [Reference Youssef and Otis29].

Finally, the total constraint equations can be introduced as Eqs. (2) to (5). The next section presents the algorithm to solve the forward and inverse kinematics using these constraints.

3.4 Forward and inverse kinematic of proposed reconfiguration theory

3.4.2 Inverse kinematic

Figure 3 shows the schematic for the end effector and attachment points A _i, where O is the fixed global frame and $\varepsilon$ is a local frame attached to the end effector. This figure shows cable i and human limb as two-lines segments.

Figure 3. Second group of driving constraints.

In the reconfigurable CDPM theory proposed in this paper, the inverse kinematics is defined as (5). Meanwhile, that variable inputs are introduced as the pose of the end effector and the attachment point’s location on the reels, and output is known as cable length. Regarding Fig. 3, the vector loop-closure equation or length of cables for cable i is introduced as follows:

(6) \begin{equation} \mathbf{p}_{i}=\| \vec {\mathbf{B}}_{i}-\vec {\mathbf{r}}_{Ai}\| \text{ for }i=1\;\text{to}\;\mathrm{n} \end{equation}

where (7) indicates the coordinates of point $\mathbf{B}_{i}$ as follows:

(7) \begin{equation} \vec {\mathbf{B}}_{i}=\vec {\mathbf{r}}_{ef}+\mathbf{R}\vec {\mathbf{r}}'_{\!\!i}\text{ for }i=1\;\text{to}\;\mathrm{n}\left(\text{number of cables}\right) \end{equation}

$\vec {\mathbf{r}}_{ef}$ is the position vector of the center of mass of the end effector and R is introduced as rotation matrix of moving platform frame with respect to local frame. $\vec {\mathbf{r}}'_{\!\!i}$ is the local coordinate frame attached to the end effector $\varepsilon$ . Shortest distance computation can be used to detect collision between human and cables that is discussed in next section.

Figure 4. Vector formulation of the distance between limb and cables.

3.5 Shortest distance computation as collision detection between cables and human

Some formulations are introduced to define the interference contact points [Reference Otis, Nguyen-Dang, Laurendeau and Gosselin53]. One method is based on determining a line perpendicular to two lines to define the shortest distance. Whereas interference happens when the cables are coplanar, it is necessary to add a condition to recognize that an intersection occurs inside or outside the cable lengths. The shortest distance between cables can be geometrically computed as discussed in refs. [Reference Meziane, Cardou and Otis46] and [Reference Otis, Nguyen-Dang, Laurendeau and Gosselin53]. Two non-dimensional variables $\mathbf{d}_{h}$ and $\mathbf{d}_{i}$ are introduced, where $\mathbf{d}_{i}\mathbf{p}_{i}$ and $\mathbf{d}_{h}\mathbf{p}_{h }$ are illustrated by red lines in Fig. 4. $\mathbf{A}_{i}\mathbf{B}_{i} \mathrm{and}$ $\mathbf{A}_{h}\mathbf{B}_{h}$ present the length of eight cables and human arm, respectively. $\mathbf{d}_{i}\mathbf{p}_{i}$ is distance between $\mathbf{B}_{i}$ and the line that represents the shortest distance between cable and human. $\mathbf{d}_{h}\mathbf{p}_{h}$ is distance between $\mathbf{B}_{h}$ and the line that represents the shortest distance between cable and human. This new approach is used to avoid cables being parallel at infinity. $\mathbf{d}_{hi}$ is introduced as a distance vector between cables and human as follows:

(8) \begin{align} \mathbf{d}_{hi} & =\mathbf{d}_{h}\mathbf{p}_{h}-\mathbf{d}_{i}\mathbf{p}_{i}-\mathbf{b}_{h}+\mathbf{b}_{i} \nonumber \\[5pt] & \quad 0\leq \mathbf{d}_{i}\leq 1\;and\;0\leq \mathbf{d}_{h}\leq 1 \end{align}

$\mathbf{d}_{h}$ and $\mathbf{d}_{i}$ are coefficients of two cable lengths. The maximum length of cables is considered, when $\mathbf{d}_{h}$ and $\mathbf{d}_{i}$ are in upper value or one. Nonlinear optimization method known as KKT conditions [Reference Taghirad and Bedoustani54, Reference Carricato and Merlet55] can guarantee the main goal to minimize the norm of $\mathbf{d}_{hi}$ in (8) as a cost function. Several multipliers can solve the optimization problem by equality and inequality constraints. But in this paper, inequality constraints are considered. So, the problem is introduced as follows:

Minimise, $\parallel \mathbf{d}_{hi}\parallel _{2}$

Subject to:

(9) \begin{align} & -\mathbf{d}_{h}\leq 0 \nonumber \\[5pt] & \mathbf{d}_{h}-1\leq 0 \nonumber \\[5pt] &-\mathbf{d}_{i}\leq 0 \nonumber \\[5pt]& \mathbf{d}_{i}-1 \leq 0 \end{align}

The optimal values of the coefficients $\mathbf{d}_{h}$ and $\mathbf{d}_{i}$ can help to find minimum distance. KKT theorem introduces the Lagrange multipliers, λ, with $\boldsymbol{\lambda }_{i}\geq 0$ and i = 1,…4, as each constraint [Reference Meziane, Cardou and Otis46]. The Lagrange function by inequality constraints is introduced as follows:

(10) \begin{align} \left(\mathbf{d}_{h},\mathbf{d}_{i},\boldsymbol{\lambda }\right)&=\mathrm{f}\left(\mathbf{d}_{h},\mathbf{d}_{i}\right)+\boldsymbol{\lambda }^{\mathrm{T}}\mathrm{h}\left(\mathbf{d}_{h},\mathbf{d}_{i}\right) \nonumber \\ \textbf{f}(\textbf{d}_{h},\textbf{d}_{i}) & = \textrm{min} \|\textbf{d}_{hi}\|_{2} = \textrm{min} \|\textbf{d}_{h}\textbf{p}_{h}-\textbf{d}_{i}\textbf{p}_{i}-\textbf{b}_{h}+\textbf{b}_{i}\|_{2}, \nonumber \\ \boldsymbol{\lambda}_{T} &= [\boldsymbol{\lambda}_{1}\;\;\boldsymbol{\lambda}_{2}\;\;\boldsymbol{\lambda}_{3}\;\;\boldsymbol{\lambda}_{4}], \nonumber \\ \textbf{h}(\textbf{d}_{h},\textbf{d}_{i}) & = \left[\begin{matrix} -\textbf{d}_{h} \\ \textbf{d}_{h}-1 \\ -\textbf{d}_{i} \\ \textbf{d}_{i}-1 \end{matrix}\right] \end{align}

Meziane et al. [Reference Meziane, Cardou and Otis46] introduced the simplified equation with all conditions and cases. Different $\mathbf{d}_{h}$ and $\mathbf{d}_{i}$ can be generated while Lagrange multipliers are activated and deactivated. However, $\boldsymbol{\lambda }_{1}$ and $\boldsymbol{\lambda }_{2}$ or $\boldsymbol{\lambda }_{3}$ and $\boldsymbol{\lambda }_{4}$ could not be activated simultaneously, since $\mathbf{d}_{h}$ (or $\mathbf{d}_{i}$ ) cannot accept two values. Some cases are not acceptable when $\mathbf{d}_{i}$ is at the upper limit of the inequality constraint. In this situation, the lengths of cables are in high position; therefore, there is no collision between human and cables. But when $\mathbf{d}_{i}$ is at the lower limit of the inequality constraint or free, the collision between the human and the cables could be occurring. Six cases are introduced as follows:

Case 1: $\mathbf{d}_{h}$ and $\mathbf{d}_{i}$ are found by the inequality constraints. Indeed, $0\leq \mathbf{d}_{h}\leq 1$ and $0\leq \mathbf{d}_{i}\leq 1$ . The Lagrange multipliers are free $\boldsymbol{\lambda }_{h}$ = 0. So, (10) can be observed as follows.

(11) \begin{equation} \left[\begin{array}{c@{\quad}c} {\mathbf{p}_{h}}^{T}\mathbf{p}_{h} & {-\mathbf{p}_{h}}^{T}\mathbf{p}_{i}\\[5pt] {-\mathbf{p}_{i}}^{T}\mathbf{p}_{h} & {\mathbf{p}_{i}}^{T}\mathbf{p}_{i} \end{array}\right] \left[\begin{array}{c} {\mathbf{d}_{h}}^{*}\\[5pt] {\mathbf{d}_{i}}^{*} \end{array}\right]=\left[\begin{array}{c} {\mathbf{p}_{h}}^{T}\mathbf{b}_{hi}\\[5pt] {-\mathbf{p}_{i}}^{T}\mathbf{b}_{hi} \end{array}\right] \end{equation}

while $\colon \mathbf{b}_{hi}=\mathbf{b}_{h}-\mathbf{b}_{i}$

Case 2: $\mathbf{d}_{i}$ is located at the lower limit of the inequality constraint and $\mathbf{d}_{h}$ is free. Therefore, $\boldsymbol{\lambda }_{3}=1$ is active, and $\boldsymbol{\lambda }_{1}=\boldsymbol{\lambda }_{2}=0$ deactivated as well $\boldsymbol{\lambda }_{4}=0$ .

(12) \begin{align} \begin{cases} {\mathbf{d}_{h}}^{\boldsymbol{*}}=\dfrac{{p_{h}}^{T}b_{hi}}{{p_{h}}^{T}p_{h}}\\[9pt] {\mathbf{d}_{i}}^{\boldsymbol{*}}=0 \end{cases} \end{align}

Case 3: $\mathbf{d}_{h}$ is the upper limit of the inequality constraint and $\mathbf{d}_{i}$ is free. So, $\boldsymbol{\lambda }_{2}$ = 1 is active and $\boldsymbol{\lambda }_{3}=\boldsymbol{\lambda }_{4}=0$ deactivated.

(13) \begin{align} \begin{cases} {\mathbf{d}_{h}}^{\boldsymbol{*}}=1\\[9pt] {\mathbf{d}_{i}}^{\boldsymbol{*}}=\dfrac{{p_{i}}^{T}p_{h}-{p_{i}}^{T}b_{hi}}{{p_{i}}^{T}p_{i}} \end{cases} \end{align}

Case 4: when $\mathbf{d}_{h}$ is located at the lower limit of the inequality constraint and $\mathbf{d}_{i}$ is free, so $\boldsymbol{\lambda }_{\mathbf{1}}$ = 1 is active and $\boldsymbol{\lambda }_{3}$ = $\boldsymbol{\lambda }_{4}$ = $\boldsymbol{\lambda }_{2}$ = 0 is deactivated.

(14) \begin{align} \begin{cases} {\mathbf{d}_{h}}^{\boldsymbol{*}}=0\\[9pt] {\mathbf{d}_{i}}^{\boldsymbol{*}}=\dfrac{-{p_{i}}^{T}b_{hi}}{{p_{i}}^{T}p_{i}} \end{cases} \end{align}

Case 5: $\mathbf{d}_{h}$ is at the upper limit of the inequality constraint and $\mathbf{d}_{i}$ is at the lower limit. Indeed $\boldsymbol{\lambda }_{\mathbf{1}}=1, \boldsymbol{\lambda }_{4}=1$ are active, and $\boldsymbol{\lambda }_{2}$ = $\boldsymbol{\lambda }_{3}$ = 0 is deactivated.

(15) \begin{equation} \begin{cases} {\mathbf{d}_{h}}^{\boldsymbol{*}}=1\\[5pt] {\mathbf{d}_{i}}^{\boldsymbol{*}}=0 \end{cases} \end{equation}

Case 6: $\mathbf{d}_{h}$ and $\mathbf{d}_{i}$ are located at the lower limit of the constraints. $\boldsymbol{\lambda }_{1}=\boldsymbol{\lambda }_{3}=1$ is activated and $\boldsymbol{\lambda }_{2}=\boldsymbol{\lambda }_{4}=0$ is deactivated.

(16) \begin{equation} \begin{cases} {\mathbf{d}_{h}}^{\boldsymbol{*}}=0\\[5pt] {\mathbf{d}_{i}}^{\boldsymbol{*}}=0 \end{cases} \end{equation}

Cable–human collision can be considered as two-line interference. A collision is detected when two lines get close to each other to a certain threshold value.

The next step is computation of $\mathbf{d}_{hi}$ for each value $\mathbf{d}_{h}$ and $\mathbf{d}_{i}$ . The distance $\mathbf{d}_{hi}$ are the distance between the two points $\mathbf{B}_{h}$ and $\mathbf{B}_{i}$ . The algorithm for the computation of the shortest distance between human and cables can be seen in Fig. 5. The first cable is selected, then coefficients for all six cases can be determined. In this case where the constraints are satisfied, the shortest distance according to (8) is then estimated. Hence, the computed shortest distance is compared with a threshold value such that the shortest distance value is increased by relocating vertically up and down the corresponding attachment point until the shortest distance between cables and human exceeds the threshold value, which means that the collision has been avoided, as it is discussed in later sections. The relocation theory of attachment points is discussed in the next section.

Figure 5. Collision avoidance algorithm.

3.6. Cable interference avoidance – relocation of corresponding attachment points

After solving the inverse kinematics and the computation of the shortest distance between human and cables, the attachment points on the rails are relocating to solve the collision between cables and human while maintaining the end effector trajectory.

The shortest distance computation between human and cables depends on position of the human arm and the cables that can be found according to position of the attachment points on reels and the attachment points on end effector. In case of a near collision between human and cable $i$ ( $i$ = 1 to 8), the reel moves up or down when $\mathbf{d}_{hi}$ is lower than threshold. Therefore, the position of the reel $i$ on the rail is displaced by a step as scalar $\Delta \mathrm{q}$ . Parameter $\Delta \mathrm{q}$ is fixed at 1.5% of the distance between two attachment points on the one rail in Z direction. In this research, parameter $\Delta \mathrm{q}$ is selected theoretically 0.1 m. So, the new location of the attachment point $\left(\vec {\mathbf{q}}_{{i_{z}}}\right)_{k+1}$ or $\left(\vec {\mathbf{q}}_{{i_{z}}}\right)_{new}$ is updated as follows: where the previous location of the attachment point $\left(\vec {\mathbf{q}}_{{i_{z}}}\right)_{initial}$ or $\left(\vec {\mathbf{q}}_{{i_{z}}}\right)_{k}$ moves as $\Delta \mathrm{q}$ on the rail:

(17) \begin{equation}\left(\vec {\mathbf{q}}_{{i_{z}}}\right)_{k+1}=\left(\vec {\mathbf{q}}_{{i_{z}}}\right)_{k}+\Delta \mathrm{q}, \left(\vec {\mathbf{q}}_{{i_{z}}}\right)_{k}=\left(z_{i}\right)_{k}\end{equation}

k: actual sampling time

k + 1: next sampling time

$z_{i}$ : position of attachment points in Z direction on the rail

\begin{align*} \Delta \mathrm{q}=\begin{cases} 0\text{ if }\text{there is not collision between human and cables}\\[5pt] 0.1\text{ if }\text{there is collision between human and cables } \end{cases} \end{align*}

This theory can be valid by constraining the maximum vertical location of a reel on the rail where the attachment point should not overpass. The location of the reel can move vertically by a servo linear actuator. This idea of reconfiguration to avoid collision between cables and human limb simulated in Matlab is presented in Section 4.

4.1 Results

The presented method can determine the shortest distance as collision detection. If the shortest distance is lower than tolerance and close to collision, the algorithm moves the attachment point on rails and updates the cable lengths. The results of the shortest distance between cables and human can be seen in Fig. 7 for three situations when human is close to cables 8, cable 2, and cable 6 for circular trajectory. The end effector tracks the circular trajectory (as the desired trajectory) presented by Youssef and Otis [Reference Youssef and Otis29]. To guarantee human safety, the threshold between human and cable is considered as 0.35 m.

Figure 7. Shortest distance between cables and human near cable 8 (circular trajectory).

As shown in Fig. 7 when human is near cable 8, all cables except cable 8 are at acceptable distance from human limb. The distance between human near cable 8 and cables are $\mathbf{d}_{{h_{i}}}.$ It can be observed that the distance between cable 8 and human is less than threshold, and the position of cable 8 needs to be modified to avoid cable–human collision. Therefore, Fig. 8 shows that there is no change in the position of reels 1 to 7. The reels 2,4, and 6 are located at 0.5 m, and the reels 1, 3, 5, and 7 are located at 7.5 m.

Figure 8. Attachment point location of cables 1 to 7 – human near cable 8.

Figure 9 demonstrates the shortest distance between cable 8 and human and the relocation of attachment point 8. When the shortest distance between human and cable 8 falls below the threshold, detected by an algorithm at around 10 s, the relocation of attachment points on reels could avoid collision between cable 8 and the human limb. So, the distance is increased until the distance between cable 8 and human reaches a reliable distance and above the threshold.

Figure 9. Circular trajectory: (a) shortest distance between cable 8 and human (human is placed near cable 8) and (b) attachment point location of cable 8.

To investigate the performance of the proposed method, the results have been obtained when human is close to cable 2 and cable 6. Figure 10(b-1) shows the distance between human and cable 2 when human is located close to cable 2 and Figure 10(a-1) indicates the distance between human and cable 6 when human is placed near cable 6. There are four points that distance between human and cable 6 is in threshold, and three threshold points in distance between human and cable 2 can be seen in Fig. 10(b-2).

Figure 10. Circular trajectory: (a-1) distance between human and cable 6 (human is located close to cable 6), (a-2) attachment point position of cable 6 on the rail, (b-1) distance between human and cable 2 (human is located close to cable 2), and (b-2) attachment point position of cable 2 on the rail.

Figure 11 indicates the length of cables for all three situations. According to Fig. 11, there are changes suddenly in length of cables because of close distance between cable 8 and skeleton placed near cable 8 in Fig. 11(a), between human and cable 6 in Fig. 11(b) when human model is located near cable 6 and between human and cable 2 in Fig. 11(c).

Figure 11. Length of cables (circular trajectory): (a) human near cable 8, (b) human near cable 6, and (c) human near cable 2.

4.2 The wrench feasible workspace

Firstly, cables tension distribution equation can be solved to map feasible workspace. The computational algorithm is used to map the reconfigurable workspace. Figure 12 indicates the initial (blue) and final (red) workspaces according to the reconfiguration of the mechanism due to circular motion of end effector and distance between human arm and cable 2, 6, and 8.

Figure 12. Circular trajectory (cables–human collision when human is near cable 8): (a) final and original workspaces 3D, (b) front view, (c) side view, and (d) top view.

As it can be observed in Fig. 12, final workspace is changed due to relocation of the attachment points on the rail, while trajectory of the end effector kept unchanged. Upper attachment points (1, 3, 5, and 7) are located Initially at 7.5 m. In Fig. 13, cable 8 was initially at 0.5 m and is moving to 1 m to avoid collision when human is located near cable 8. As can be seen in Fig. 12(a), the top part of the final workspace and the initial workspace are same. This can be clearly observed by comparing between the X–Z views Fig. 12(b). Meanwhile, there was no noteworthy change by comparing initial and final workspace the X–Y views (Fig. 12(d)) or the Y–Z views Fig. (12 c). The results for two other situations are observed in Figs. 13 and 14. The attachment point 2 in Fig. 13(b) is moved from 0.5 to 0.8 m. This movement could prove Fig. 12 where five points in threshold distance is observed. Also, Fig. 14(c) indicates initial and final workspaces when human is near cable 6; meanwhile, the attachment point 6 can be moved from 0.5 to 0.9 m in order to avoid collision between human and cable 6.

Figure 13. Circular trajectory (cables–human collision when human is near cable 2): (a) final and original workspaces 3D, (b) front view, (c) side view, and (d) top view.

Figure 14. Circular trajectory (cables–human collision when human is near cable 6): (a) final and original workspaces 3D, (b) front view, (c) side view, and (d) top view.

Figures 15, 16, and 17 indicate the difference between original and final workspaces and are presented by the boundary of some points of the end effector’s position in an original workspace that cannot be found in the final workspace. As can be seen in Figs. 15, 16, and 17, the final workspace is strictly smaller at the bottom part due to a change in the position of the attachment points on rails 8, 6, and 2. The position of the upper attachment points does not change in the initial and final workspace, and the final trajectory is inside the initial workspace.

Figure 15. Difference between original and final workspaces when human is near cable 8: (a) difference between final and original workspace 3D, (b) front view, (c) side view, and (d) top view.

Figure 16. Difference between original and final workspaces when human is near cable 2: (a) difference between final and original workspace 3D, (b) front view, (c) side view, and (d) top view.

Figure 17. Difference between original and final workspaces when human is near cable 6: (a) difference between final and original workspace 3D, (b) front view, (c) side view, and (d) top view.

4.3 Two humans inside the workspace

The risk of collision between humans and robots could be raised when many humans are in the workspace. The suggested algorithm can avoid collision between all humans and each cable. In addition of collision avoidance between one human and cables, collision avoidance between two humans located in the hybrid workspace generating up to three collision avoidances simultaneously using reel position relocalization strategy is discussed in this section. At the same time, the endeffector keeps its original trajectory setpoint. These two humans are in the workspace near cables 6 and 8. In this section, the first human #1 is located near cable 6 while both arms are moving, and the second one is located near cable 8 while only one arm of this human is moving. Figure 18 demonstrates the collision avoidance between the first arm of human #1 with cable 6 for 11 times. This algorithm avoids collision between left arm of human with cable 6 for 8 times.

Figure 18 (a) Distance between both first human arms and cable 6 and (b) distance between second human arm and cable 8.

Finally, as shown in Fig. 19, the algorithm moves attachment point 6 from 0.5 to 2.4 m. Meanwhile, moving attachment point 8 from 0.5 m to 1 m is indicated in Fig. 19 to avoid collision between the second human and cable 8. The first human moves both arms while one of the second human arms moves simultaneously through the workspace. Supplementary material, Video 2: Experiment 2 (Two humans in the workspace-Part 2) [Reference Khoshbin, Youssef, Meziane and Otis57] shows that attachment point 8 moves up (from 0.5 to 0.9 m) when the first human is located near cable 8. Meanwhile, the attachment point 6 moves up 1.9 m (from 0.5 to 2.4 m), while the end effector keeps its trajectory in the WCW.

Figure 19. Original and final workspaces.