A collaborative path planning method for mobile cable-driven parallel robots in a constrained environment with considering kinematic stability

In this work, we focus on a MCDPR composed of a classical redundantly constrained CDPR with eight cables (\(m=8\)) and a six DoF end-effector (\(n=6\)) mounted on four mobile bases (\(p=4\)), as shown in Fig. 2. The origin of the global coordinate system is denoted as \(O_{0}\) and the local coordinate system located at the center of the end-effector is expressed as \(O_{e}\). The jth mobile base, denoted as \({\mathscr {M}}_j\), \(j = 1,\ldots ,4\) carries two cables (\(m_{j}=2\)) and has three wheels (\(c=3\)) including one caster wheel and two driven wheels. The local coordinate system attached to \({\mathscr {M}}_j\) is denoted as \(O_{pj}\). The characteristics of the non-holonomic two-wheeled differential drive mechanism implicate that \({\mathscr {M}}_j\) is the planar robot, it can generate two DoF translational motions and one DoF rotational motion.

The ith cable vector attached onto \({\mathscr {M}}_j\) is named as \({\varvec{l}}_{ij}\), \(i=1,\ldots m_{j}\). As a result, the cable exit point and the anchor point are denoted as \(A_{ij}\) and \(B_{ij}\), respectively. According to the characteristics of parallel robots, selecting the ith closed-loop vector branch, \({\varvec{l}}_{ij}\) can be expressed bywhere \({\varvec{a}}_{ij}\) represents the position vector of \(A_{ij}\) in \(O_{0}\) coordinate system. \({\varvec{P}}=[{}^{0}{x},{}^{0}{{y}},{}^{0}{{z}}]^T\) is a three-dimensional position vector of the end-effector, and the rotation of the end-effector is not considered. \({}^{e}{\varvec{b}}_{ij}\) denotes the position vector of \(B_{ij}\) in the moving coordinate system.

Let \({\varvec{u}}_{ij}={\varvec{l}}_{ij}/\Vert {\varvec{l}}_{ij}\Vert \) be the unit vector pointing from \(B_{ij}\) to \(A_{ij}\) of the cable \({\varvec{l}}_{ij}\). The MCDPR is designed that \(A_{ij}\) lies on the axis \({}^{pj}z\) of \({\mathscr {M}}_j\), the rotation motion of \({\mathscr {M}}_j\) has no impact on \({\varvec{u}}_{ij}\). Hence, for a certain pose of the end-effector, \({\varvec{u}}_{ij}\) can be obtained by determining the position of \(O_{pj}\) in \(^{0}x{^{0}y}\) plane. The position of the mobile base is controlled by sending the velocity commands to two wheels, and the pose of the end-effector can be adjusted by changing the cable length.

To specify the MCDPR path planning problem more precisely, let \({\varvec{P}}_{k}\), \(k=1,\ldots ,N\), be the position vector of the end-effector at the kth time step. Correspondingly, the state of mobile bases \({\mathscr {M}}_j\) at the kth time step is denoted as \({\varvec{m}}_{j,k}=[{}^{0}{{x}_{{{j,k}}}}, {}^{0}{{y}_{{{j,k}}}}]\). Therefore, the state of four mobile bases and MCDPR at the kth time step is denoted as \({\varvec{M}}_{k}=[{\varvec{m}}_{1,k}^T , {\varvec{m}}_{2,k}^T , {\varvec{m}}_{3,k}^T ,{\varvec{m}}_{4,k}^T]^T \) and \({{{\varvec{X}}}_{k}}=[{\varvec{P}}_{k}^{T} , {\varvec{M}}_{k}^{T}]^T\), respectively.

One of the basic requirements for the feasible paths of MCDPR is to ensure mobile bases without colliding with each other or obstacles in the environment. For simplification purposes, the mobile bases and obstacles are modeled as cylindrical shaped structures. Therefore, for each time step, the position of the jth, \(j=1,\ldots ,4\), the mobile base needs to satisfy constraints as follows:where \(L_m\) represents the safety distance between two mobile bases and \(r_m\) is the radius of the mobile base. \({\varvec{o}}_{q}\) denotes the position of the qth obstacle on \(^{0}{x}^{0}{y} \) plane with radius \(r_o\). \({\varvec{d}}_{k}={\varvec{m}}_{j,k+1}-{\varvec{m}}_{j,k}\) is the directional vector of the mobile base in two adjacent states, and \(\beta _{\textrm{max}}\) is the maximum turning angle. Equation (2) aims to keep safety distance between each two mobile bases, (3) ensure that the mobile base does not collide with obstacles, and (4) prevents the abrupt direction change of the mobile base.

Since the mobile base carried CDPR is composed of multiple cables and the interference of cables/obstacles and end-effector/obstacles should be considered in the path planning of MCDPR. In this paper, the cables between \(A_{i,j}\) and \(B_{i,j}\) are simplified as line segments, and the end-effector is modeled as a cube. Let \(\varvec{c_k}\) and \(\varvec{b_k}\) denotes the two closest point between CDPR and obstacles at kth time step. It should be noted that \(\varvec{c_k}\) is either on the end-effector or cables, and \(\varvec{b_k}\) is on the obstacle. Therefore, to prevent the collision between CDPR and obstacles, let \(L_c\) be the minimum acceptable distance between CDPR and obstacles. We have the following equation:In addition, we consider the carried CDPR have limited workspace and cable length. For each configuration of mobile bases \({\varvec{M}}_{k}\), we can identify a corresponding workspace \({\mathcal {W}}_k\). Thus, the following constraints are conducted:where \(l_{\textrm{min}}\) and \(l_{\textrm{max}}\) denote the minimum and maximum length of cable, respectively. \(h_l\) represent two constant heights of exit point \(A_{i,j}\) in the \(O_{0}\) coordinate system. \(h_1\) and \(h_{2}\) represents the height of \(A_{1,j}\) and \(A_{2,j}\), \(j=1,\ldots ,4\), respectively.

In summary, MCDPR path planning is a problem that is concerned about finding feasible paths for the end-effector and multiple mobile bases from their given start locations to a sufficiently small area of their target locations \({\varvec{M}}_{goal}=[{\varvec{m}}_{goal,1}^T , {\varvec{m}}_{goal,2}^T , {\varvec{m}}_{goal,3}^T ,{\varvec{m}}_{goal,4}^T]^T \) while satisfying the constraints defined in (2)–(7) and optimizing the objective function.

Due to the kinematical redundancy of the MCDPR, ensuring excellent kinematics performance is the basic premise for studying the MCDPR path planning problem. We consider the classical CDPR carried by mobile bases, the velocity of the end-effector \(\dot{{\varvec{P}}}=[{}^{0}{\dot{x}}_{e}, {}^{0}{\dot{y}}_{e} , {}^{0}{\dot{z}}_{e}]^T\) due to the motion of cables can be expressed aswhere \(\varvec{{\dot{l}}}_{j}=[\varvec{{\dot{l}}}_{1j},\ldots ,\varvec{{\dot{l}}_{m_{j}j}} ]^T\) indicates the speed vector of the cables mounted on the mobile base \({\mathscr {M}}_j\), and \({\varvec{J}}_{j}\) is expressed aswhere \({\varvec{J}}_{j}\) is associated with the cable actuation wrench on \({\mathscr {M}}_j\). According to (8) and (9), the kinematic Jacobian matrix of the carried CDPR can be defined by \({\varvec{J}}=[{\varvec{J}}_{1},\ldots ,{\varvec{J}}_{j}]^T\), which maps the end-effector velocity to the actuated cable velocities. Thus, kinematic performance is defined as the sensitivity of kinematic Jacobian matrix \({\varvec{J}}\), which can be obtained by the condition number \(\kappa \), denoted aswhere \({\lambda }_{\textrm{max}}\) and \({\lambda }_{\textrm{min}}\) are the maximum and minimum eigenvalues of square matrix \({\varvec{J}}^{T}{\varvec{J}}\). The condition number of kinematic Jacobian matrix \(\kappa \) also describes the dexterity of the robot. Since \(\kappa \in [1,\infty )\), we normalized \(\kappa \) to define the kinematics performance index \(\gamma _{k} \in [0,1] \) expressed asThus, for guaranteeing the better kinematics performance of MCDPR, the larger \(\gamma _{k}\) should remain.

One major drawback of MCDPR is that the structure of the mobile base combined with the lifting column is susceptible to cable tension. The mobile base could be vibrated or tipped if the cable tension beyond a specific range. These unstable mobile base could increase the position error of the end-effector and lead to task failure. To prevent the MCDPR becoming unstable, the stability performance is defined in this section.

The free body diagram of jth mobile base is illustrated in Fig. 3a; it can be observed that the stability of the mobile base is influenced by both cable tension and its gravity. As shown in Fig. 3b, we define the wheel contact points of the mobile base \({\mathscr {M}}_{j} \) as \(C_{nj}\), \(n=1,2,3\), in a counter-clockwise direction, and the position vector of nth wheel contact point denotes as \({\varvec{c}}_{nj}=[{\varvec{c}}_{nj,x},{\varvec{c}}_{nj,y},{\varvec{c}}_{nj,z}]^T\). The left and right driven wheels of the mobile base are defined based on the forward direction (i.e., \(C_{2j}\) and \(C_{3j}\)). According to the Zero-Moment Point (ZMP) method, the static equilibrium of the mobile base relies on the moments generated at the boundaries of the wheel contact points [22]. The boundary between the two consecutive contact \((C_{nj},C_{n+1j})\) is denoted as \({\mathscr {L}}_{C_{nj}}\) of the directional vector \({\varvec{u}}_{C_{nj}}\). Thus, the moment \({\varvec{m}}_{C_{nj}}\) generated at the instant when \({\mathscr {M}}_{j} \) is about to tipping can be formulated aswhere \({\varvec{g}}_{j}=[\varvec{g_{j,x}},\varvec{g_{j,y}},\varvec{g_{j,z}}]^T\) is the position vector of \({\mathscr {M}}_{j} \)’s center of mass. \({\varvec{w}}_{gj}\) denotes the weight vector of \({\mathscr {M}}_{j} \). \({t}_{ij},i=1,\ldots ,m_{j}\), represents the cable tension associated \({\mathscr {M}}_{j}\). The first and second term of (12) denote the moment generated by the gravity and cable tension, respectively. To prevent negative tension and deformation of the cable, \({t}_{ij}\) is bounded between a minimum tension \(t_{\textrm{min}}\) and \(t_{\textrm{max}}\). Accordingly, the tipping constrains for \({\mathscr {M}}_{j} \) is to ensure the negative \({\varvec{m}}_{C_{nj}}\). It means that each boundary \({\mathscr {L}}_{C_{nj}}\) requires to satisfy the following constraints:The feasible interval of the cables tension of \({\mathscr {M}}_{j}\) could form an area called feasible tension polygon (FTP), denoted by \({\mathscr {F}}_{j} \). For the FTP without considering the tipping condition, \({\mathscr {F}}_{j}\) is a square with side length \(\Delta t=t_{\textrm{max}}-t_{\textrm{min}} \) shown in Fig. 4a. However, according to (12) and (13), the tipping condition is the function of cable tension which causes the linear constraints to cut the original FTP shown in Fig. 4b. Since the cable tension has infinite solutions, we define a riskiest feasible tension polygon \({\mathscr {F}}_{r}=\text {min}({\mathscr {F}}_{r}),j=1,\ldots ,4 \). The stability performance of MCDPR, \(\gamma _{s}\), can be defined as

where \({\mathscr {F}}_{r}\) can be obtained by getting the point of intersection (i.e., \({{\textbf {v}}}_1,{{\textbf {v}}}_2,{{\textbf {v}}}_3\) in Fig. 4b). It can be seen from (14) that \(\gamma _{s} \in [0,1]\), \(\gamma _{s}=1\) indicates the tipping constraints has no effects on the original \({\mathscr {F}}_{j}\) so that MCDPR is at the stability performance. In contrast, when \(\gamma _{s}=0\), the MCDPR will be tipping due to a cable tension solution that satisfies the static equilibrium does not exist.

Due to the high-dimensional state space and various coupled constraints of mobiles, the adaptive goal-biased (RRT) is proposed to generate feasible paths for mobile bases. The proposed RRT-based method is to find a set of feasible paths by building a tree-structured graph \({\mathcal {T}}({\mathcal {V}},{\mathcal {E}})\). The vertices \({\mathcal {V}}\) represents possible collision-free states of mobile bases \({\varvec{M}}_k\), and the edge \({\mathcal {E}}\) denotes the feasible transitions of mobile bases between two adjacent states.

Algorithm 1 summarizes the proposed approach, if the initial and goal state of mobile bases are feasible, the root tree is initialized with the initial state \({\varvec{M}}_{init}\), then the initial state is assigned to the first vertices \({\varvec{M}}_{new}\). Some important attributes are also initialized to vertices, \( {\varvec{M}}_{new}.\text {Goal\_arrived}\) indicates the arrival status of four mobile bases. In proposed method, once \({\mathscr {M}}_j\) reaches its target location, it is stays fixed and does not move, and the jth element of \( {\varvec{M}}_{new}.\text {Goal\_arrived}\) is changed to 1. Therefore, the algorithm proceeds with all elements in \( {\varvec{M}}_{new}.\text {Goal\_arrived}\) are not equal to 1.

Lines 5–31 of Algorithm 1 describe the execution process of the proposed method. For dealing with the high-dimensional state space of mobile bases and increase efficiency, the adaptive goal-biased sampling is designed (line 6 of Algorithm 1) to generate random state \({\varvec{M}}_{rand}=[{\varvec{m}}_{rand,1}^T , {\varvec{m}}_{rand,2}^T , {\varvec{m}}_{rand,3}^T ,{\varvec{m}}_{rand,4}^T]^T \) and its specific steps is given in Algorithm 2. The developed adaptive sampling strategy can guide the tree towards the goal region and the volume of the sampling space is reduced by adaptively adjusting the sampling strategy in a subspace for each mobile base. The \({\varvec{M}}_{partent}\) and \({\varvec{M}}_{goal}\) are the inputs of Algorithm 2. The elements in \( {\varvec{M}}_{partent}.\text {Goal\_extended}\) is checked to decide whether to assign the goal of jth mobile base \({\varvec{m}}_{goal,j}\) to \({\varvec{m}}_{rand,j}\) or not. If the element in parent vertices \({\varvec{M}}_{parent}\) fails to expand to its goal point directly, the adaptive sampling method is implemented in lines 5–11. GoalCircle(\({\varvec{m}}_{goal,j}\), \(r_{gc,j}\)) represents uniformly sampling in a circular area centered on the \({\varvec{m}}_{goal,j}\) with radius \(r_{gc,j}=\Vert {\varvec{m}}_{goal,j}-{\varvec{m}}_{r,j} \Vert \), and \({\varvec{m}}_{r,j}\) denotes a sub-vertices with a maximum expansion failure rate. Uniform(\({\mathcal {W}}_{m}\)) represents uniformly sampling in the workspace for mobile bases \({\mathcal {W}}_{m}\). The choice of which method for sampling depends on the \({\varvec{M}}_{parent}.\text {Failure\_rate}\).

To increase the robustness of the algorithm, the current number of vertices is stored in each iteration function (line 25 of Algorithm 1) once one of the mobile bases reach the goal point. In the beginning of the proposed method, \(V_j=0\), the function Nearest( \({\varvec{M}}_{rand}\),\({\mathcal {T}}\),\(V_j\)) return a vertices \({\varvec{M}}_{nearest}=[{\varvec{m}}_{nearest,1}^T , {\varvec{m}}_{nearest,2}^T , {\varvec{m}}_{nearest,3}^T ,{\varvec{m}}_{nearest,4}^T]^T \) closest to \({\varvec{M}}_{rand}\) by searching the vertices of entire tree \(\mathcal {T.V}\). However, after the one mobile base reaches the goal point, the search range is changed to \(\mathcal {T.V}[V_j\ldots ]\) to prevent the car from moving after reaching the target.

After \({\varvec{M}}_{nearest}\) is obtained, a new vertices \({\varvec{M}}_{new}=[{\varvec{m}}_{new,1}^T , {\varvec{m}}_{new,2}^T , {\varvec{m}}_{new,3}^T ,{\varvec{m}}_{new,4}^T]^T\) is generated using Steer(\({\varvec{M}}_{rand}\),\({\varvec{M}}_{nearest}\),\({\varvec{M}}_{parent}\)) function (line 8 of Algorithm 1), shown in Algorithm 3. The input \({\varvec{M}}_{parent}\) is used to determine if the mobile base has reached the goal point. Moreover, the inputs \(M_{rand}\) and \({\varvec{M}}_{nearest}\) is used to obtain a new node \({\varvec{M}}_{new}\). It is noted that the step size \(\eta \) depends on the state of \({\varvec{M}}_{parent}.\text {Goal\_arrived}[j]\), indicates \({\mathscr {M}}_j\) is fixed after arriving the goal point and returned variables \({\varvec{M}}_{new}\) have four components corresponding to the four mobile bases.

For each \({\varvec{M}}_{new}\), its feasibility is checked by using (2), (3), and (4) in “MCDPR parameterization and problem formulation” section. If \({\varvec{M}}_{new}\) is feasible according to previous constraints, an optimization strategy similar to RRT* is applied. The neighbor vertices of \({\varvec{M}}_{new}\) is obtained with a given radius \(r_c\), and if the edge from \({\varvec{M}}_{near}\) to \({\varvec{M}}_{new}\) is feasible, \({\varvec{M}}_{parent}\) is replaced by \({\varvec{M}}_{near}\) with a better cost (lines 12–20 of Algorithm 1). However, if \({\varvec{M}}_{new}\) is not feasible, some properties of \({\varvec{M}}_{nearest}\) and \({\varvec{M}}_{new}\) are updated to inspire the next iteration process (lines 21–30 of Algorithm ). In this work, the cost function is defined as total path length and \(\text {Dist}({\varvec{M}}_{new},{\varvec{M}}_{nearest})\) denotes the distance between two nodes:

Once the RRT tree \({\mathcal {T}}\) is constructed in the last section, the path for mobile bases \(^p{\varvec{M}}_{path}=[^p{\varvec{m}}_{path,1}^T , ^p{\varvec{m}}_{path,2}^T , ^p{\varvec{m}}_{path,3}^T, ^p{\varvec{m}}_{path,4}^T]^T, p=1,\ldots ,Q\), \({\varvec{M}}_{path}=[{\varvec{M}}_1, {\varvec{M}}_2,\ldots ,{\varvec{M}}_Q]^T\)can be generated by consecutively finding \({\varvec{M}}_{parent}\) from the end of the tree, and Q denotes the number of vertices on the path. It should be noted that the first vertices on the path \(^1{\varvec{M}}_{path}\) \({\varvec{M}}_{path,1}\) represents the initial state of mobile bases \({\varvec{M}}_{init}\), and the last vertices \(^Q{\varvec{M}}_{path}\) \({\varvec{M}}_{path,Q}\) is the goal state \({\varvec{M}}_{goal}\).

For each state on the path, \({\varvec{M}}_{path,p},p=1,\ldots ,Q\), it corresponds to a specific CDPR structure in which the position of pulleys is determined. Therefore, a global performance index \(\gamma \) with considering the stability and kinematic performance of MCDPR is formulated to generate the path \({\varvec{P}}_{path}=[{\varvec{P}}_{1},{\varvec{P}}_{2},\ldots ,{\varvec{P}}_{Q}]^T \)for the end-effector, namelyTo obtain \({\varvec{P}}_{path}\) with (16), a grid-based search method is developed in Algorithm 4. The pth element of \({\varvec{P}}_{path}\), \({\varvec{P}}_{path,p}\) is generated by maximizing \(\gamma \) in the mobile bases configuration \({\varvec{M}}_{path,p}\). According the definition in “Kinematics performance” and “Stability performance” sections, \(\gamma \) can be considered as a function of the end-effector position. Therefore, as shown in Fig. 5, we can obtain the distribution map of \(\gamma \) after gridding \({\mathcal {C}}_p\), and the cell with the maximum \(\gamma \) will be recorded. Hence, for each cell \(c_n\), it corresponds to a specific end-effector position and its feasibility is checked using (5)–and (7), which is considered in line 6 of Algorithm 4.

According to Algorithm 1 and , the final paths of the mobile bases and the end-effector are the sequences of straight line segments. Thus, the cubic splines [23] is adopted to smooth paths of the robot, whose geometric is constructed of piecewise third-order polynomials through each specified position point obtained from the proposed method. In this case, we define a single cubic spline for smoothing each independent path of the mobile base and the end-effector by connecting path points. Thus, the continuous motion profiles of the MCDPR can be guaranteed. Moreover, the boundary conditions are also imposed in which the velocities and accelerations at initial and final position points are null.

In this section, the proposed MCDPR path planning method is simulated in a cluttered environment. As shown in Fig. 6, a dynamic model of our MCDPR prototype is developed using CoppeliaSim (formerly known as V-REP) robot simulator software [24], to verify our proposed method. One should note that the end-effector is modeled as a point with a mass of 0.4 kg. We define 10 cylindrical obstacles with random positions in the environment. These obstacles have an identical radius and the height is 0.4 m. The dark blue obstacle is denoted as a target obstacle, where the MCDPR needs to move to this obstacle for performing a task action, i.e., picking and/or releasing an object. Therefore, goal points for each mobile base and the end-effector are defined around the target obstacle.

The proposed method is initialized using parameters given in Table 1, and the tree grows with a constant step size \(\eta =0.2\) m. Thus, a constant timestep \(\Delta T\) is required to transition the MCDPR state \({\varvec{X}}_{k+1}\) from the previous state \({\varvec{X}}_{k}\). In this work, \(\Delta T=0.2\) s is used to obtain high accuracy. Accordingly, the mass center of the mobile base can be directly obtained from the CoppeliaSim given by \([0.12, 0.34, 0.36]^T\) with respect to \(O_pj\) coordinate system. The simulation result is depicted in Fig. 7, and it is found that a collision-free path is generated for each mobile base to reach their goal points, and a path of the end-effector is also obtained by maximizing \(\gamma \). It should be noted that the safe region around obstacles is defined and its radius is given by \(r_m+r_o\). As a consequence, the mobile bases can be considered as a point to perform collision detection.

The velocity profiles of the end-effector and the mobile bases are shown in Fig. 8. As we can see, the continuous velocity profiles are guaranteed due to the post-processing. One should note that the mobile bases do not arrive the goal points at the same time. Thus, the total trajectory time 63 s due to \({\mathscr {M}}_{3}\) being the last one to reach its goal point. Moreover, Fig. 9 illustrates the three distance changes given by (2), (3), and (5). On account of the sampling characteristics of the proposed algorithm, the nodes will be retained only if they satisfy the given constraints. As a result, it can be found that MCDPR satisfies the distance constraints defined by these equations at each timestep, the collision-free trajectory is thus fulfilled.

In this section, the resulting MCDPR trajectories obtained from the last section is simulated in CoppeliaSim environment to verify the proposed method and the corresponding experiment is carried out. The software simulation framework is depicted in Fig. 10. For controlling the mobile bases, the resulting MCDPR trajectories are converted into the rotational velocities of the mobile base’s wheels using its kinematic model [25]. As shown in Fig. 11, due to the cubic splines are adopted, the continuous rotational velocities of wheels can be obtained, in which \(v_l\) and \(v_r\) represent the rotational velocities of the left and right wheels, respectively. For controlling the end-effector, we utilize (1) and (8) to adjust the length and speed of cables shown in Fig. 12. The obtained wheels’ rotational velocities are sent to CoppeliaSim to control the MCDPR model. The state of MCDPR can be directly obtained from the simulator, it will be compared with the desired path to obtain the error in evaluating the simulation results.

The clips of the simulation in CoppeliaSim are shown in Fig. 13, it can be seen that the MCDPR can follow the resulting paths and avoid the collision. In addition, the actual position of the MCDPR in CoppeliaSim is obtained to compute the error with the desired paths. As shown in Fig. 14, the maximum error about the mobile base and the end-effector is approximately 2.8 cm and 4.1 cm, respectively. The planned path is validated experimentally using a second scenario one obstacle as shown in Fig. 15, in which NVIDIA Jetson Nano\(^{\textrm{TM}}\) Developer Kit is adopted as a control computer to operate our MCDPR system. These results verify that the proposed method is feasible and effective.

The proposed path planning method for MCDPRs is compared with other existing methods in this section. Due to the random characteristics of the proposed path planning algorithm, a batch evaluation with 1000 simulations is conducted, and the average value is used to evaluate the proposed method. These simulations use the identical environment and parameters defined in the previous simulation. All the simulations are performed using \(\copyright \)MATLAB with CPU computations on an Intel ^®i7-9750 CPU@2.60 GHz and 32 GB RAM Windows 10 system. The results of the batch evaluation are presented in Table 2.

Comparing the proposed method with the Iterative Algorithm and Direct Transcription, our method reduced the time costs by 93.3% and 84.7%, respectively. This is due to goal points being pre-defined for each mobile base and the end-effector. In addition, our adaptive sampling method reduces the time cost by 76.6% compared to the conventional goal-biased sampling method in which the sampling probability of the goal point is given as 0.5. Thus, the adaptive goal-biased sampling strategy can be deployed to improve the efficiency of our method. Moreover, Iterative Algorithm and Direct Transcription require continuous iterative processes for optimization, it may become computationally more expensive with more complex environments. On the contrary, the proposed method does not need to perform an iterative search, but rather finds a suitable solution by adaptive sampling the state space. One should note that the Iterative Algorithm and Direct Transcription lead mobile bases and the end-effector to reach goal points at the same time. However, in our method, once one mobile base reaches the goal point, it will be fixed to wait for all mobile bases to reach goal points. Finally, we can observe that the average global performance index \(\gamma \) of the total path is improved by 31.1% and 14.4% with respect to the Iterative Algorithm and Direct Transcription. The average global performance index \(\gamma \) of the proposed method with and without adaptive sampling have almost identical value, which is due to the grid-based search algorithm performed by both all of them. Hence, the proposed method can bring greater stability and kinematic performance for MCDPRs during the motion of the end-effector.

Despite the better performance in the evaluation results, the path quality of the proposed method, i.e., lengths of the MCDPR path, may not be as good as Iterative Algorithm or Direct Transcription. However, the proposed method has higher efficiency and better comprehensive performance, these provide the possibility of real-time implementation. The path quality can be somewhat mitigated by performing additional sampling. Moreover, the post-processing using the cubic splines add additional discrete points between the two consecutive nodes, this might not satisfy all constraints. This problem may be resolved using a smaller step size \(\eta \) and increasing the safety region of obstacles.