Abstract

The movable anchor points make reconfigurable cable-driven parallel robots (RCDPRs) advantageous over conventional cable-driven parallel robots with fixed anchor points, but the movable anchor points also introduce an inherent problem—reconfiguration planning. Scholars have proposed reconfiguration planning approaches for RCDPRs, taking into account the statics and kinematics of RCDPRs. However, a real-time reconfiguration planning approach that considers the dynamics of an RCDPR and is computationally efficient enough to be integrated into the RCDPR's dynamic controller is still not available in the literature yet. This paper develops a real-time reconfiguration planning approach for RCDPRs. A novel reconfiguration value function is defined to reflect the “value” of an RCDPR configuration and provide a reference index for the reconfiguration planning of an RCDPR. And then, the developed approach conducts reconfiguration planning based on the value of RCDPR configurations. The developed approach is computationally efficient, reducing the reconfiguration planning time by more than 93%, compared to single iteration of a box-constrained optimization-based reconfiguration planning approach. Such a high efficiency allows the developed approach to be integrated into an RCDPR's dynamic controller that usually runs with a high frequency. Integrating reconfiguration planning and dynamic control enhances the control performance of the RCDPR. To verify the effectiveness of the developed approach and the integration of reconfiguration planning and dynamic control for RCDPRs, a case study of an RCDPR with seven cables and four movable anchor points is conducted.

1 Introduction

A cable-driven parallel robot (CDPR) has two platforms—the base and the moving platform. Anchor points on both platforms are paired and connected through cables [1]. A CDPR usually has a large workspace, low inertia, and high payload-to-weight ratio. CDPRs are suitable for several applications such as material handling and instrumentation [24], but the design and control of CDPRs are challenging. A notable challenge for CDPRs is the collision issue, which includes cable-to-cable collision, cable-to-platform collision, and cable-to-object collision. A reconfigurable cable-driven parallel robot (RCDPR) permits the movement of its anchor points, providing an efficient way to avoid collisions [5]. The movement of anchor points can be achieved by mounting anchor points on mobile bases (e.g., sliding bars [58], mobile robots [9], and aerial robots [10], as shown in Fig. 1). With movable anchor points, RCDPRs can be applied to a wider range of applications than CDPRs. For example, with anchor points mounted on aerial robots, an RCDPR can have different workspaces, depending on the reconfiguration of the aerial robots. This benefit makes the RCDPR suitable for aerial manipulation tasks, such as fire-fighting in high-rise buildings [11], as shown in Fig. 1(c).

Fig. 1
RCDPRs with movable anchor points mounted on (a) sliding bars [5], (b) mobile robots [9], and (c) aerial robots [10]
Fig. 1
RCDPRs with movable anchor points mounted on (a) sliding bars [5], (b) mobile robots [9], and (c) aerial robots [10]
Close modal

Although RCDPRs have the advantage in collision avoidance and workspace, RCDPRs suffer an inherent problem—reconfiguration planning. To address this problem of RCDPRs, researchers have made a lot of efforts. Zanotto et al. [12] addressed the reconfiguration planning of a planar RCDPR based on the idea of decoupling the pose of the moving platform and the disposition of cables to maximize a local performance index in a wide portion of the workspace of the RCDPR. Zhou et al. [13] studied RCDPRs whose anchor points on the base are mounted on mobile bases and discussed two reconfiguration planning approaches in the case of trajectory tracking. The first reconfiguration planning approach is to perform a global optimization to find the optimal configuration and then move the anchor points to reach the optimal configuration. The second reconfiguration planning approach is to perform a faster and more possible local search and then move the anchor points. It was shown that the latter approach is acceptable in the case of trajectory tracking, though it might lead to a local optimal issue. Nguyen et al. [14] developed a reconfiguration planning approach based on solving two sub-optimization problems for RCDPRs. Bounds on the reconfiguration parameter are defined in the first problem and the reconfiguration planning of an RCDPR is transformed into a box-constrained optimization problem. The second problem is to solve the box-constrained optimization problem utilizing gradient-based optimization algorithms. The computation time of the reconfiguration planning approach can be reduced by limiting the number of iterations of an optimization algorithm. Gagliardini et al. [5] solved the problem of optimizing the sequence of reconfigurations that allow the moving platform of an RCDPR to follow a prescribed path in the case of a discrete set of possible configurations and developed a discrete reconfiguration planning approach for RCDPRs. To determine the configuration of an RCDPR that satisfies a given workspace requirement per second, a reconfiguration planning approach consisting of two separate phases was proposed in Ref. [15]. In the first phase, the RCDPR moves as a whole to make the geometrical center of the workspace of the RCDPR and that of the required workspace close to each other. In the second phase, the anchor points of the RCDPR are adjusted to achieve the entire required workspace.

An RCDPR needs to track a dynamic trajectory in real-time in several applications (e.g., fire-fighting in high-rise buildings as shown in Fig. 1(c)). For an RCDPR applied to fire-fighting in high-rise buildings, the moving platform needs to track a dynamic trajectory depending on fire spreading, which has the feature of time and space. Dynamic controllers (e.g., proportional–integral–derivative (PID) controllers [16], optimal controllers [17], sliding mode controllers [18], adaptive controllers [19], and robust controllers [20]) have been widely used to control CDPRs to track dynamic time-sensitive trajectories. Dynamic controllers have been applied to RCDPRs with anchor points mounted on aerial robots in Refs. [10,21] as well. For an RCDPR that is required to track a dynamic trajectory in real-time, the control performance of the RCDPR depends on not only the dynamic controller of the RCDPR but also the reconfiguration planning approach of the RCDPR. To optimize the control performance of an RCDPR in tracking a dynamic trajectory, the integration of reconfiguration planning and dynamic control is needed.

The existing reconfiguration planning approaches [13,14] consider only the statics and kinematics of RCDPRs [21] and cannot be applied to the dynamic control of RCDPRs. Moreover, although the existing reconfiguration planning approaches for RCDPRs [13,14] can improve their computation efficiency by limiting the number of iterations of an optimization algorithm, they still cannot be integrated into an RCDPR's dynamic controller that usually runs with a frequency of more than 100 Hz (e.g., 250 Hz in Ref. [22] and 500 Hz in Ref. [18]). To address the fast trajectory planning of multiple aerial robots, nonlinear constraints and cost functions were cast into a global cost function [23]. A gradient-based algorithm was then used to determine target trajectories quickly. To deal with the real-time motion planning of robotic systems with nonlinear dynamics, Singh et al. proposed a framework consisting of an offline phase and an online phase in Ref. [24]. The artificial potential field was used in Ref. [25] to achieve real-time path planning of CDPRs in dynamic environments. This paper applies several real-time planning techniques [2325] originally developed for general robotic systems (e.g., robotic arms, mobile robots, and aerial robots) to RCDPRs and develops a real-time reconfiguration planning approach for RCDPRs. The developed real-time reconfiguration planning approach is computationally efficient enough to be integrated into the dynamic controller of an RCDPR running with a frequency of 100 Hz. This paper achieves three contributions that are as follows:

  • This paper develops a real-time reconfiguration planning approach. The approach is computationally efficient, reducing the reconfiguration planning time by more than 93%, compared to single iteration of a box-constrained optimization-based reconfiguration planning approach. The high efficiency allows the approach to be integrated into a dynamic controller that runs with a frequency of 100 Hz.

  • A novel reconfiguration value function is proposed based on the cost functions and the constraint functions of the reconfiguration planning of an RCDPR to reflect the “value” of an RCDPR configuration. The value of an RCDPR configuration can be applied to several applications, such as an analysis of the configurations of an RCDPR and the reconfiguring planning of an RCDPR. In this paper, the value of RCDPR configurations is used to achieve the real-time reconfiguration planning of an RCDPR.

  • The developed real-time reconfiguration planning approach is integrated into the dynamic controller of an RCDPR. The integration of real-time reconfiguration planning and dynamic control enhances the control performance of the RCDPR in tracking a dynamic trajectory.

The remainder of the paper is organized as follows. Section 2 presents the preliminaries of the present paper. In Sec. 3, a reconfiguration value function is proposed and a real-time reconfiguration planning approach is developed for RCDPRs. The integration of reconfiguration planning and the dynamic control for RCDPRs is presented in Sec. 4. A case study is presented to demonstrate and validate the developed reconfiguration planning approach and the integration of the reconfiguration planning and dynamic control of an RCDPR in Sec. 5. Finally, this paper is summarized in Sec. 6.

2 Preliminaries

This section introduces the preliminaries of the present paper in detail, including a problem statement, notations of a CDPR, and the dynamic control of a CDPR.

2.1 Problem Statement.

The control performance of a CDPR depends on its dynamic controller which usually runs with high frequencies (e.g., 250 Hz in Ref. [22] and 500 Hz in Ref. [18]). The control performance of an RCDPR depends on not only its dynamic controller but also its reconfiguration planning approach. To optimize the control performance of an RCDPR in tracking a dynamic trajectory in applications such as fire-fighting for high-rise buildings, as shown in Fig. 1(c), the integration of real-time reconfiguration planning and dynamic control is needed.

Since the trajectory to be tracked is time-sensitive and dynamic, the reconfiguration planning of the RCDPR must work synchronously with the movement of the moving platform, rather than running ahead of the movement of the moving platform. Thus, reconfiguration planning approaches for RCDPRs based on a determined trajectory in Ref. [14] or a required workspace in Ref. [15] cannot be applied when the trajectory is time-sensitive and dynamic. Besides, due to their relatively low computation efficiency, the existing reconfiguration planning approaches cannot be integrated into an RCDPR's dynamic controller that usually runs with a frequency of more than 100 Hz.

To enhance the control performance of an RCDPR tracking a dynamic trajectory, this paper addresses two problems: (1) how to improve the computation efficiency of reconfiguration planning such that reconfiguration planning can be integrated into the dynamic controller of the RCDPR? and (2) how to integrate reconfiguration planning and dynamic control of the RCDPR? It should be noted that, since the trajectory to be tracked is time-sensitive and dynamic, the trajectory may change before the achievement of the globally optimal configuration of an RCDPR. Thus, a locally optimal configuration is acceptable for the RCDPR in tracking a dynamic trajectory according to [13].

2.2 Notations.

Notations of a CDPR with p degrees-of-freedom (DOFs) and n cables are shown in Fig. 2. ui(i = 1, 2, …, n) is the unit vector along the ith cable. li(i = 1, 2, …, n) is the length of the ith cable. Ai and Bi(i = 1, 2, …, n) are the anchor points of the ith cable on the base and the moving platform, respectively. The positions of the anchor points Ai and Bi are represented by vectors ai and bi(i = 1, 2, …, n), respectively. The base frame is represented by Fb. The moving platform frame Fm is attached to the moving platform of the CDPR.

Fig. 2
Notations of a CDPR with p DOFs and n cables
Fig. 2
Notations of a CDPR with p DOFs and n cables
Close modal

2.3 Dynamic Control of Cable-Driven Parallel Robots.

The dynamic control of CDPRs has been intensively investigated by several researchers. Different controllers (e.g, PID controllers [16], optimal controllers [17], sliding mode controllers [18], adaptive controllers [19], and robust controllers [20]) have been applied to the dynamic control of CDPRs. The dynamics of a CDPR can be expressed as [16]
M(x)x¨+C(x,x˙)x˙+G(x)=w
(1)
where x, x˙, and x¨ represent the pose, velocity, and acceleration of the moving platform of the CDPR, respectively. M(x) and C(x,x˙) denote the mass matrix and the Coriolis/centripetal matrix, respectively, G(x) is the gravity vector, and w is the wrench applied on the moving platform of the CDPR by cable tensions. The tension of cables required to generate a certain wrench on the moving platform can be obtained by solving the inverse dynamics problem [16]
w=JTτ
(2)
where J represents the Jacobian matrix [26]. τ is a cable tension vector. The steps of the dynamic control of a CDPR [27] are summarized in Fig. 3. In Fig. 3, a dynamic controller in the task space designed based on the dynamics of the CDPR expressed in Eq. (1) determines a wrench to be applied on the moving platform (i.e., a target wrench). Based on the target wrench, cable tensions to be applied by actuators (i.e., target cable tensions) are calculated by solving the inverse dynamics problem in Eq. (2). Actuators apply the target cable tensions then.
Fig. 3
Flow diagram of the dynamic control of a CDPR
Fig. 3
Flow diagram of the dynamic control of a CDPR
Close modal

3 Real-Time Reconfiguration Planning Approach for Reconfigurable Cable-Driven Parallel Robots

This section develops a real-time reconfiguration planning approach for RCDPRs based on several techniques [2325] originally developed for the real-time planning of general robotic systems (e.g., robotic arms, mobile robots, and aerial robots), aiming to improve the computation efficiency of reconfiguration planning such that reconfiguration planning can be integrated into the dynamic controller of an RCDPR. The developed real-time reconfiguration planning approach achieves three major advantages: (1) the proposed reconfiguration planning approach includes an offline phase and an online phase. The offline phase undertakes the computation that can be done offline, reducing the computation time of reconfiguration planning in the online phase, (2) nonlinear constraints of the reconfiguration planning of an RCDPR are transformed to cost functions to reduce the computation time, and (3) a reconfiguration value function is proposed for an RCDPR to reflect the value of an RCDPR configuration. A neural network (NN) is applied to approximate the reconfiguration value function. The approximated reconfiguration value function, rather than the cost function of an optimization algorithm, is applied to the reconfiguration planning of an RCDPR.

The flow diagram of the proposed reconfiguration planning approach is shown in Fig. 4. The steps presented in Fig. 4 are provided in detail below.

Fig. 4
Flow diagram of the proposed real-time reconfiguration planning approach for RCDPRs
Fig. 4
Flow diagram of the proposed real-time reconfiguration planning approach for RCDPRs
Close modal

3.1 Offline Phase of Real-Time Reconfiguration Planning Approach.

An offline phase and an online phase [24] are designed for the proposed reconfiguration planning approach to reduce the computation time of reconfiguration planning in the online phase. The offline phase aims to address the calculations that are computationally expressive and can be dealt with offline. These calculations include but are not limited to the calculation of the size of the available wrench set [28], the transformation of nonlinear constraints, and matrix inversion. In this way, the computation time of reconfiguration planning in the online phase can be reduced. The offline phase includes steps as below.

3.1.1 Constant and Variable Parameters.

To conduct the reconfiguration planning of an RCDPR, one needs to define the constant and variable parameters of the RCDPR for the proposed reconfiguration planning approach. For the convenience of demonstration, it is assumed that the anchor points on the moving platform are fixed while the anchor points are mounted on movable bases. The constant parameters of the RCDPR usually include as follows:

  • The number of cables, n.

  • The mass matrix of the moving platform, M.

  • The position of anchor points is fixed on the moving platform, bi, in the moving platform frame.

The variable parameters of the RCDPR include

• The position of anchor points on the movable bases, ai.

3.1.2 Cost Function.

To optimize the performance of an RCDPR based on the proposed reconfiguration planning approach, one may need to define a set of cost functions. The cost functions can be different from case to case depending on the task to be performed by the RCDPR or on the user preferences [5]. In the present paper, a cost function is designed based on the available wrench set of an RCDPR, aiming to enlarge the feasible wrench that can be used for the dynamic control of the RCDPR.

  • Available wrench set

For the dynamic control of an RCDPR at a pose, a large available wrench set is preferred. However, since a cable can only be pulled and sustain a limited magnitude of tension, an RCDPR can only generate a bounded set of wrenches by cables. The Available Wrench set of a CDPR, denoted as AW, can be defined as [28]
AW={w=Jτ|τminττmax}
(3)
where τmax is the vector of maximum cable tensions, and τmin is the vector of minimum cable tensions.
In this study, both the isotropy [29] and the size of the AW of an RCDPR are taken into account. An available wrench cost function based on the AW of an RCDPR at a pose is defined as
fAW(x,a)=μrrAW(AW(x,a))
(4)
where rAW is defined as the maximum feasible isotropic wrench of an RCDPR with a given configuration at a certain pose for a given characteristic length [30]. The maximum feasible isotropic wrench refers to the maximum wrench an RCDPR with a given configuration at a certain pose can provide in any arbitrary direction in the wrench space [8]. The determination of rAW can be referred to [11,28]. μr is an available wrench weight coefficient.

3.1.3 Constraint Function.

Reconfiguration planning of an RCDPR has to satisfy several constraints that can be expressed by inequality constraint functions or equality constraint functions. The inequality constraint functions are nonlinear constraints, resulting in a long computation time for a reconfiguration planning approach [23]. Thus, the proposed reconfiguration planning approach transforms the nonlinear constraints into cost functions according to [23] to reduce the computation time. The constraint functions of an RCDPR considered in this study are introduced as follows:

  • Cable interference

An RCDPR suffers cable collision issues. If two or more cables collide, the geometric and dynamic models of the RCDPR are not valid anymore and the cables can be damaged. In order to ensure that cables do not interfere, one can investigate the distances between cables according to [31]. The distance between the ith cable and the jth cable, denoted as dij, should satisfy
dijd*
(5)
where d* is the minimum distance between two cables. The proposed reconfiguration planning approach transforms the nonlinear constraint presented in Eq. (5) into a cable interference cost function according to [23]. The cable interference cost function of the ith cable and the jth cable is defined based on Eq. (5) as
fijI=e(dijd*μd)
(6)
where μd is a cable distance penalty coefficient.
The cables of an RCDPR can connect to the same anchor point. If two or more cables connect to the same anchor point, one cannot ensure that cables do not interfere by investigating the distances between cables. Alternatively, one can ensure that cables do not interfere by investigating the angular differences between cables that connect to the same anchor point. The angular difference between the ith cable and the jth cable, denoted as θij, should satisfy
θijθ*
(7)
where θ* is the minimum angular difference between two cables. The cable interference cost function of the ith cable and the jth cable is defined as
fijI=e(θijθ*μθ)
(8)
where μθ is a cable angular difference penalty coefficient.
Based on the cable interference cost function of the ith and the jth cables defined in Eqs. (6) and (8), the cable interference cost function of the RCDPR is
fI(x,a)=ijfijI
(9)

The number of fijI to be accumulated is Cn2=n!/2!(n2)! [5].

  • Feasible position of movable anchor points

The movable anchor points of an RCDPR can only move within certain feasible regions usually. Let
aiAi
(10)
where Ai is a simply connected set of feasible positions of ai. The proposed reconfiguration planning approach transforms the nonlinear constraint in Eq. (10) into a feasible position cost function according to [23]. The feasible position cost function of the ith anchor point is defined as
fiA={e(minaiAiμa)aiAie(minaiAiμa)aiAi
(11)
where Ai is a set of positions of the bound of Ai. minaiAi represents the minimum distance between ai and the bound of Ai. μa is a feasible position penalty coefficient. The feasible position cost function of the RCDPR is
fA(x,a)=infiA
(12)

3.1.4 Reconfiguration Value Function.

A state value function demonstrates the value of a state of a system in reinforcement learning [32]. Inspired by the concept of the state value function, this paper proposes a reconfiguration value function for the reconfiguration planning of RCDPRs. The reconfiguration value function of an RCDPR is defined based on the cost functions and constraint functions of the RCDPR and reflects the value of an RCDPR configuration. In this way, the reconfiguration planning of an RCDPR can be transformed into the optimization of the reconfiguration value function of the RCDPR. In this study, the reconfiguration value function, denoted as V(x, a), is defined as
V(x,a)=fAW(x,a)+fI(x,a)+fA(x,a)
(13)
By substituting Eqs. (4), (9), and (12) into Eq. (13), one has
V(x,a)=μrrAW(AW(x,a))+ijfijI+infiA
(14)

One may apply a general optimization algorithm to optimize V(x, a) defined in Eq. (13) and then achieve the reconfiguration planning of an RCDPR. However, the reconfiguration planning based on a general optimization algorithm can be very slow, especially when the RCDPR has a large number of cables. The slow reconfiguration planning is caused by the calculation of fI(x, a) shown in Eq. (13). According to Eq. (9), fI(x, a) is the sum of fijI(ij), the number of which is n!/2!(n − 2)!. To reduce the computation time of reconfiguration planning, the proposed reconfiguration planning approach calculates the numerical approximation of V(x, a) (i.e., approximated reconfiguration value function) in the offline phase. The dimension of the approximated reconfiguration value function depends on the number of DOFs and the number of the movable anchor points of an RCDPR, not the number of pairs of cables of the RCDPR. It should be noted that the number of movable anchor points of an RCDPR is far less than the number of pairs of cables of the RCDPR if the number of cables of the RCDPR is large. The approximated reconfiguration value function, rather than the cost function of a general optimization algorithm, is applied to the reconfiguration planning of an RCDPR then.

To approximate V(x, a), an NN is utilized in the proposed reconfiguration planning approach. An NN is a well-known universal approximator [33]. An NN has been used to approximate the forward kinematics of a CDPR in Ref. [34]. An NN was used in parallel with a controller to compensate for model uncertainties in controlling a CDPR in Ref. [35]. In our earlier study [27], an NN was used to approximate the inverse dynamics of a CDPR. The proposed reconfiguration planning approach applies an NN to approximate V(x, a) and achieves an approximated reconfiguration value function, denoted as VNN(x, a), as
VNN(x,a)V(x,a)
(15)

To approximate V(x, a) based on an NN, one can use an NN with a proper size to approximate a proper number of randomly selected data samples of V(x, a). Specifically, if one uses a fully-connected NN (i.e., a feedforward NN), abbreviated as FC, to approximate V(x, a), an FC with a single hidden layer that includes a small number of units (e.g., 16 or 32) can be used at the beginning. Then, the FC is used to approximate an increasing number of randomly selected data samples of V(x, a), until the approximation error of the FC no longer decreases. If the final approximation error of the FC is acceptable, one can integrate the VNN(x, a) represented by the achieved FC into the proposed reconfiguration planning approach. If the final approximation error of the FC is not acceptable, one should increase the size of the FC (i.e., increase the number of layers or the number of units of a layer) and repeat the above-mentioned steps, until the final approximation error of an achieved FC is acceptable.

It should be noted that, except for reducing the computation time of reconfiguration planning, utilizing VNN(x, a) can address cost functions that are hard to achieve analytical formulations (e.g., a cost function based on the volume of the workspace of an RCDPR) based on interval analysis [36] or Monte Carlo method.

3.2 Online Phase of Real-Time Reconfiguration Planning Approach.

In the online phase of the proposed reconfiguration planning approach, an RCDPR determines the target movement of movable anchor points based on VNN(x, a) achieved in the offline phase. Then, the RCDPR moves the movable anchor points according to the determined target movement.

3.2.1 Determination of the Target Movement of Movable Anchor Points.

Reconfiguration planning of an RCDPR can be regarded as the redesign of a CDPR. Lou et al. have shown that gradient-based algorithms (e.g., sequential quadratic programming) are with higher computational efficiency than probabilistic algorithms (e.g., differential evolution, particle swarm optimization, and genetic algorithm) in optimizing the mechanical design of parallel robots [37]. In [23], a gradient-based algorithm has been applied to fast trajectory planning of multiple aerial robots that are movable bases of RCDPRs in Refs. [10,21].

In the proposed reconfiguration planning approach, the gradient of VNN(x, a) with respect to the position of movable anchor points is used to determine the target movement of movable anchor points. The target movement of movable anchor points, denoted as Δa*, can be expressed as
Δa*={α[VNN(x,a)a1,VNN(x,a)a2,,VNN(x,a)an]T[VNN(x,a)a1,VNN(x,a)a2,,VNN(x,a)an]T2,[VNN(x,a)a1,VNN(x,a)a2,,VNN(x,a)an]T200,[VNN(x,a)a1,VNN(x,a)a2,,VNN(x,a)an]T2=0
(16)
where α is the step size of the target movement of movable anchor points. One can set the value of α to adjust the overall magnitude of the target movement of movable anchor points at a certain step. In Eq. (16), the gradient of VNN(x, a) determines the directions of the target movement of movable anchor points and the proportion of the magnitudes of the target movement of different movable anchor points.

Once the target movement of movable anchor points is determined based on Eq. (16), RCDPR needs to control the movable anchor points to perform the target movement. An optimized configuration can be achieved by the RCDPR then.

4 Integration of Real-Time Reconfiguration Planning and Dynamic Control of a Reconfigurable Cable-Driven Parallel Robot

This section proposes an approach to integrate the reconfiguration planning and dynamic control of an RCDPR. According to the proposed reconfiguration planning approach, an approximated reconfiguration value function can be calculated in the offline phase. If the total time used for reconfiguration planning with the approximated reconfiguration value function and the calculation of the dynamic control law in each step is less than the cycle time of the dynamic control loop, the proposed real-time reconfiguration planning approach can be integrated into the dynamic controller of the RCDPR.

The flow diagram of the integration of the reconfiguration planning and dynamic control of an RCDPR is shown in Fig. 5. There are two loops—a fast dynamic control loop (e.g., 100 Hz in the case study of this paper) and a slow reconfiguration planning loop (e.g., 10 Hz in the case study of this paper). To integrate the slow reconfiguration planning loop into the fast dynamic control loop, the sum of the computation time of dynamic control law and that of the reconfiguration planning should be less than the period of the fast dynamic control loop. In the case study of this paper, the computation of the dynamic control law takes less than 0.0001 s and the computation time of the reconfiguration planning takes less than 0.0042 s. Thus, the slow reconfiguration planning loop can be integrated into the fast dynamic control loop in the case study. One can also see that the reconfiguration planning loop can be as fast as the dynamic control loop if only the computation time of reconfiguration planning is concerned. However, a fast reconfiguration planning loop tends to provide micro target movements at a high frequency for moveable anchor points. It can be hard for the movable anchor points to achieve the micro target movements. The micro target movements may also result in the vibration of the movable anchor points, making the RCDPR unstable. To enhance the stability of the RCDPR, the frequency of the slow reconfiguration planning loop is set to be one-tenth of the frequency of the fast dynamic control loop in the case study of this paper. It should be noticed that the ratio doesn't have to be one-tenth.

Fig. 5
Flow diagram of the integration of real-time reconfiguration planning and dynamic control of an RCDPR
Fig. 5
Flow diagram of the integration of real-time reconfiguration planning and dynamic control of an RCDPR
Close modal

The fast dynamic control loop of an RCDPR shown in Fig. 5 is the same as the dynamic control loop of a CDPR shown in Fig. 3. Thus, this section focuses on the slow reconfiguration planning loop, rather than the fast dynamic control loop which is discussed in Sec. 2. The steps of implementing the slow reconfiguration planning loop are explained as follows:

  1. At the beginning of a slow reconfiguration planning loop, the current positions of the movable anchor points and the current pose of the moving platform of an RCDPR are assumed to be determined by measurement devices such as motion capture systems and inertial measurement units. The target movement of movable anchor points is calculated based on the gradient of the approximated reconfiguration value function with respect to the positions of the movable anchor points according to Eq. (16).

  2. The RCDPR executes the calculated target movement of the movable anchor points and moves the movable anchor points to their new positions.

  3. The RCDPR updates the Jacobian matrix in the fast dynamic control loop based on the new positions of the movable anchor points and the new pose of the moving platform.

5 Case Study

To evaluate the efficiency of the proposed real-time reconfiguration planning approach in reducing the computation time of reconfiguration planning of an RCDPR and the effectiveness of the integration of reconfiguration planning and dynamic control in improving the control performance of an RCDPR, this section presents a case study of an RCDPR with a three DOF point-mass moving platform driven by seven cables, as shown in Fig. 6. The redundancy of actuation provides the movable aerial robots of the RCDPR with large degrees-of-freedom in reconfiguration planning while maintaining the RCDPR fully constrained. The RCDPR in the case study is required to simultaneously track a dynamic trajectory based on a dynamic controller that runs with a frequency of 100 Hz, according to the dynamic controllers for CDPRs in practice [18,22], and improve the control performance via reconfiguration based on the proposed real-time reconfiguration planning approach integrated into the 100 Hz dynamic control loop of the dynamic controller. The case study is conducted in a Gazebo simulator [38] that has been applied to several studies of CDPRs [39,40] and aerial robots [4144]. The Gazebo simulator runs at a frequency of 1000 Hz on a computer with an Intel i9-10900K CPU, an NVIDIA RTX3080Ti GPU, and a 64-Gigabyte memory.

Fig. 6
An RCDPR with a point-mass moving platform driven by four movable aerial robots and three mobile robots with fixed positions
Fig. 6
An RCDPR with a point-mass moving platform driven by four movable aerial robots and three mobile robots with fixed positions
Close modal

5.1 Setups of a Reconfiguration Cable-Driven Parallel Robot and the Integration of Real-Time Reconfiguration Planning and Dynamic Control.

In the case study, an RCDPR with a 0.1 kg point-mass moving platform is driven by four movable aerial robots and three mobile robots with fixed positions, as shown in Fig. 6. Since mobile robots usually move much slower than aerial robots and can be limited by terrain, the three mobile robots are assumed to have fixed positions according to [11]. Inspired by [10], it is assumed that every aerial robot or mobile robot drives a non-extensible cable by a pulley and can adjust the tension of the cable. The robots apply a resultant force on the point-mass moving platform cooperatively to control the moving platform. Every aerial robot is controlled by its motion controller to achieve a target movement obtained by a reconfiguration planning approach while maintaining a constant altitude. Every mobile robot holds a constant position. Based on the demonstration presented above, one can find that the RCDPR discussed in the case study has three DOFs and four movable anchor points on the base. The movable anchor points of the RCDPR are fixed at certain positions, working as a reference. If the movable anchor points of the RCDPR are fixed, the RCDPR is regarded as a CDPR. The position of the anchor points on the base of the RCDPR and CDPR is listed in Table 1.

Table 1

Position of anchor points of RCDPR and CDPR in the base frame (unit: meter)

Anchor point on the basePosition of an anchor point of RCDPRPosition of an anchor point of CDPR
A1(variable, variable, 8.00)(−2.50, 2.50, 8.00)
A2(variable, variable, 8.00)(−2.50, −2.50, 8.00)
A3(variable, variable, 8.00)(2.50, −2.50, 8.00)
A4(variable, variable, 8.00)(2.50, 2.50, 8.00)
A5(2.50, 1.45, 0.00)(2.50, 1.45, 0.00)
A6(−2.50, 1.45, 0.00)(−2.50, 1.45, 0.00)
A7(0.00, −2.90, 0.00)(0.00, −2.90, 0.00)
Anchor point on the basePosition of an anchor point of RCDPRPosition of an anchor point of CDPR
A1(variable, variable, 8.00)(−2.50, 2.50, 8.00)
A2(variable, variable, 8.00)(−2.50, −2.50, 8.00)
A3(variable, variable, 8.00)(2.50, −2.50, 8.00)
A4(variable, variable, 8.00)(2.50, 2.50, 8.00)
A5(2.50, 1.45, 0.00)(2.50, 1.45, 0.00)
A6(−2.50, 1.45, 0.00)(−2.50, 1.45, 0.00)
A7(0.00, −2.90, 0.00)(0.00, −2.90, 0.00)

The dynamic control of the RCDPR is addressed based on the inverse dynamics of the moving platform [10,21] and a PID controller in task space [16,27]. The gains of the PID controller are presented in Table 2. The tension distribution of cables is determined by optimizing the maximum tension of cables based on interior-point nonlinear programming, inspired by [4547]. It should be noted that, in the case study, the tension of a cable ranges from 0.00 N to 5.00 N. When a cable sustains a certain magnitude of tension, the corresponding aerial robot that can generate a total thrust of 50.00 N at most will apply an additional thrust to its control thrust, compensating for the disturbance caused by the tension of the cable. Thus, the change in configuration of the RCDPR caused by the tension of cables is negligible in the case study.

Table 2

Gains of the PID controller applied to the dynamic control of RCDPR and CDPR

ProportionalIntegralDerivative
x-direction0.100.001.80
y-direction0.050.001.80
z-direction0.120.000062.80
ProportionalIntegralDerivative
x-direction0.100.001.80
y-direction0.050.001.80
z-direction0.120.000062.80

The parameters of cable interference cost function (i.e., μθ and θ*) and feasible position cost function (i.e., μa) are designed to construct “barriers” that guide the reconfiguration planning of an RCDPR to satisfy the cable interference constraint and the feasible position constraint. Then, the parameters of available wrench cost function (i.e., μr), cable interference cost function (i.e., μθ and θ*), and feasible position cost function (i.e., μa) are adjusted collaboratively. The collaborative adjustment of the parameters aims to make the available wrench cost function dominant if the constraints are far from being violated and to make the cable interference cost function or the feasible position cost function dominant if a constraint is close to being violated. The value of the parameters of the cost functions of reconfiguration planning in the case study is presented in Table 3. Since the cables of the RCDPR included in the case study connect to the same anchor point on the moving platform, fI(x, a) is determined based on Eqs. (8) and (9). Therefore, one should define μθ and θ*, and doesn't have to define μd and d*. Based on the parameters of the cost functions of reconfiguration planning presented in Table 3, V(x, a) can be determined according to Eqs. (4), (9), (12), and (13).

Table 3

Parameters of cost functions and constraint functions for reconfiguration planning

ParameterValue
μθ0.02
μa0.03
μr3.00
θ*5.00 deg
ParameterValue
μθ0.02
μa0.03
μr3.00
θ*5.00 deg

In the case study, the moving platform of the RCDPR has three DOFs and the vector x of V(x, a) has three elements as a result. Let x = [x, y, z]T. Every movable anchor point of the RCDPR has two DOFs in a horizontal plane and the vector a of V(x, a) has eight elements as a consequence. Let a = [a1x, a1y, a2x, a2y, a3x, a3y, a4x, a4y]T. Based on the definitions of x and a, the feasible position of the moving platform and movable anchor points are presented in Table 4.

Table 4

Parameters of the feasible position of the moving platform and movable anchor points (unit: meter)

ParameterRange
x[−3.00, 3.00]
y[−3.00, 3.00]
z[0.00, 5.00]
a1x[−5.00, 0.00]
a1y[0.00, 5.00]
a2x[−5.00, 0.00]
a2y[−5.00, 0.00]
a3x[0.00, 5.00]
a3y[−5.00, 0.00]
a4x[0.00, 5.00]
a4y[0.00, 5.00]
ParameterRange
x[−3.00, 3.00]
y[−3.00, 3.00]
z[0.00, 5.00]
a1x[−5.00, 0.00]
a1y[0.00, 5.00]
a2x[−5.00, 0.00]
a2y[−5.00, 0.00]
a3x[0.00, 5.00]
a3y[−5.00, 0.00]
a4x[0.00, 5.00]
a4y[0.00, 5.00]

5.2 Determination of an Approximated Reconfiguration Value Function.

According to the proposed real-time reconfiguration planning approach, to obtain VNN(x, a), an NN is applied to approximate V(x, a) based on a set of 100,000 randomly selected data samples of V(x, a). A data sample is a tuple consisting of x, a, and V(x, a). To approximate V(x, a), the input of the NN, which is a combination of (x, a), has 11 elements. The output of the NN is the approximated value of V(x, a). The parameters of the NN used to approximate V(x, a) are shown in Table 5. The architecture of the NN used to approximate V(x, a) is presented in Fig. 7. The NN is an FC with the ReLU activation function. The input of the FC includes x and a. The output of the FC is the value of VNN(x, a). The FC has three hidden layers and each hidden layer has 256 units. Given a specific configuration of an RCDPR, the FC can output the value of VNN(x, a) based on the inputted x and a determined by the configuration.

Fig. 7
Architecture of the NN used to approximate the reconfiguration value function
Fig. 7
Architecture of the NN used to approximate the reconfiguration value function
Close modal
Table 5

Parameters of the NN used to approximate the reconfiguration value function

Number of unitsValue
Input layer11
The first hidden layer256
The second hidden layer256
The third hidden layer256
Output layer1
Number of unitsValue
Input layer11
The first hidden layer256
The second hidden layer256
The third hidden layer256
Output layer1
To verify the accuracy of the achieved VNN(x, a), the approximation error of the achieved VNN(x, a) is investigated based on a set of 10,000 randomly selected combinations of (x, a). For a combination of (x, a), the approximation error of VNN(x, a) with respect to V(x, a) is defined as
ΔV=VNN(x,a)V(x,a)2max(V(x,a))min(V(x,a))
(17)
where max(V(x, a)) and min(V(x, a)) represent the maximum value of V(x, a) and the minimum value of V(x, a) determined by the 10,000 randomly selected combinations of (x, a), respectively. *2 denotes the two-norm of *. The approximation errors of the achieved VNN(x, a) are listed in Table 6. According to Table 6, the mean approximation error of the achieved VNN(x, a) is 2.70%. The mean approximation error is the mean of the approximation errors of VNN(x, a) calculated according to Eq. (17) based on the 10,000 randomly selected combinations of (x, a). This suggests that for the set of 10,000 randomly selected combinations of (x, a), the mean difference between the values of V(x, a) and the corresponding values of VNN(x, a) is 2.70%. The maximum approximation error of the achieved VNN(x, a) is 50.28%. The maximum approximation error is the maximum of the approximation errors of VNN(x, a) calculated according to Eq. (17) based on the 10,000 randomly selected combinations of (x, a). This suggests that for the set of 10,000 randomly selected combinations of (x, a), the maximum difference between the values of V(x, a) and the corresponding values of VNN(x, a) is 50.28%. V(x, a) includes several exponential cable interference cost functions and feasible position cost functions, as shown in Eqs. (8) and (11). Thus, V(x, a) is sensitive to the position of movable anchor points, when the movable anchor points are close to the bound of the set of feasible positions or when a pair of cables is close to each other. In view of this, the authors think VNN(x, a) with a mean approximation error of 2.70% and a maximum approximation error of 50.28% is acceptable and thus try to conduct reconfiguration planning for the RCDPR based on the achieved VNN(x, a). If the performance of reconfiguration planning of the RCDPR based on the achieved VNN(x, a) is not acceptable for certain applications, according to [33], one can use an NN with a larger size than the NN shown in Table 5 and more data samples to obtain VNN(x, a) with higher accuracy than the achieved VNN(x, a). Then one can improve the performance of reconfiguration planning of the RCDPR based on the VNN(x, a) with higher accuracy.
Table 6

Approximation error of the approximated reconfiguration value function

Approximation errorValue
Mean approximation error2.70%
Maximum approximation error50.28%
Approximation errorValue
Mean approximation error2.70%
Maximum approximation error50.28%

5.3 Trajectory Tracking Test of the Reconfiguration Cable-Driven Parallel Robot Based on the Integration of Real-Time Reconfiguration Planning and Dynamic Control.

The RCDPR shown in Fig. 6 is controlled to track two dynamic time-sensitive trajectories to evaluate the effectiveness of the proposed real-time reconfiguration planning approach and the integration of reconfiguration planning and dynamic control of an RCDPR. A CDPR with anchor points shown in Table 1 is controlled to track the two trajectories as well, working as a reference. The first target trajectory, as shown in Fig. 8, is within the wrench feasible workspace of the CDPR. The second target trajectory, as shown in Fig. 9, is not entirely contained in the wrench feasible workspace of the CDPR. To track the second target trajectory, the RCDPR has to change the configuration. For the RCDPR, the integration of the proposed reconfiguration planning approach and a PID controller with gains shown in Table 2 is applied. The reconfiguration planning approach adopts the abovementioned VNN(x, a) and the step size of the target movement of movable anchor points α = 0.05 m. For the CDPR, only a PID controller with gains shown in Table 2 is applied.

Fig. 8
Trajectory of (a) RCDPR and (b) CDPR in tracking the first target trajectory
Fig. 8
Trajectory of (a) RCDPR and (b) CDPR in tracking the first target trajectory
Close modal
Fig. 9
Trajectory of (a) RCDPR and (b) CDPR in tracking the second target trajectory
Fig. 9
Trajectory of (a) RCDPR and (b) CDPR in tracking the second target trajectory
Close modal

In a trajectory tracking test, the RCDPR and CDPR need to (1) stabilize the moving platform at (0.00, 0.00, 4.00) (unit: meter) from 0 s to 30 s; (2) track the first or the second target trajectory from 30 s to 60 s; and (3) stabilize the moving platform at (0.00, 0.00, 4.00) (unit: meter) from 60 s to 90 s. The trajectories of the RCDPR and CDPR in tracking the first target trajectory are presented in Fig. 8. The trajectories of the RCDPR and CDPR in tracking the second target trajectory are presented in Fig. 9. Figure 8 shows that both the RCDPR and CDPR can track the first target trajectory, though the performance of the CDPR is worse than the performance of the RCDPR. One can see from Fig. 9(a) that the RCDPR can track the second target trajectory. However, according to Fig. 9(b), one can see that when the CDPR is controlled to track the second target trajectory, the CDPR is out of control. Figure 9 suggests that with the proposed real-time reconfiguration planning approach, the RCDPR can achieve a larger workspace than the CDPR can achieve.

The trajectories of the four movable anchor points of the RCDPR in tracking the first target trajectory and in tracking the second target trajectory are presented in Fig. 10. It is shown that the four movable anchor points of the RCDPR can move within the feasible regions defined in Table 4 and can change the configuration of the RCDPR when the moving platform of the RCDPR is controlled to track the first and the second target trajectories. This suggests that the integration of reconfiguration planning and dynamic control of an RCDPR can optimize the configuration of the RCDPR while controlling the RCDPR and does not violate the constraint of the feasible regions.

Fig. 10
Trajectory of the movable anchor points of the RCDPR in tracking (a) the first target trajectory and (b) the second target trajectory
Fig. 10
Trajectory of the movable anchor points of the RCDPR in tracking (a) the first target trajectory and (b) the second target trajectory
Close modal

To further analyze the performance of the proposed real-time reconfiguration planning approach and the integration of reconfiguration planning and dynamic control of an RCDPR, this paper investigates the computation time of reconfiguration planning, the trajectory tracking error of the RCDPR and CDPR, the maximum feasible isotropic wrench of the RCDPR and CDPR, the cable interference cost function of the RCDPR, and the feasible position cost function of the RCDPR in the case study.

  • Computation time of reconfiguration planning approaches

To evaluate the performance of the proposed reconfiguration planning approach in reducing the computation time of reconfiguration planning, the computation time of the proposed reconfiguration planning approach and that of single iteration of a box-constrained optimization-based reconfiguration planning approach [14] are investigated based on the trajectory of the RCDPR shown in Fig. 8(a). Based on the sequential least square quadratic programming (SLSQP) algorithm that has been applied to the reconfiguration planning of an RCDPR in Ref. [15], the box-constrained optimization-based reconfiguration planning approach addresses the following optimization problem that is based on Eqs. (4), (7), and (10)
minafAW(x,a)subjectto{θijθ*(i,j=1,2,,7andij)aiAi(i=1,2,3,4)
(18)

By addressing the optimization problem defined in Eq. (18), the box-constrained optimization-based reconfiguration planning approach can optimize the available wrench set of the RCDPR with cable interference constraint functions and feasible position constraint functions can be satisfied. It should be noticed that the proposed reconfiguration planning approach addresses the same reconfiguration planning problem in the case study.

The computation time of the reconfiguration planning approaches is presented in Fig. 11 and Table 7. According to Fig. 11 and Table 7, if SLSQP is applied to the reconfiguration planning of the RCDPR, it takes 0.0367 s on average and 0.0560 s at most to perform single iteration. It is shown in Sec. 4 that to integrate the reconfiguration planning loop into the dynamic control loop of an RCDPR, the sum of the computation time of a dynamic control law and that of the reconfiguration planning should be less than the period of the dynamic control loop. The maximum computation time of single iteration of SLSQP suggests that the dynamic controller of the RCDPR could not run with a frequency of more than 1/0.0560 s ≈ 18 Hz, if reconfiguration planning based on SLSQP was integrated into the dynamic control loop of the RCDPR. The limitation of the frequency of the dynamic controller of the RCDPR limits the control performance of the RCDPR. Thus, reconfiguration planning based on SLSQP is not integrated into the dynamic controller of the RCDPR in the case study.

Fig. 11
Computation time of reconfiguration planning approaches
Fig. 11
Computation time of reconfiguration planning approaches
Close modal
Table 7

Computation time of reconfiguration planning approaches (unit: second)

Mean computation timeMaximum computation time
Single iteration of SLSQP0.03670.0560
The proposed reconfiguration planning approach0.00140.0036
Ratio of the proposed reconfiguration planning approach to single iteration of SLSQP3.81%6.43%
Mean computation timeMaximum computation time
Single iteration of SLSQP0.03670.0560
The proposed reconfiguration planning approach0.00140.0036
Ratio of the proposed reconfiguration planning approach to single iteration of SLSQP3.81%6.43%

According to Fig. 11 and Table 7, the proposed reconfiguration planning approach takes 0.0014 s on average and 0.0036 s at most to determine the target movement of movable anchor points. The computation time of reconfiguration planning is reduced by more than 93% based on the proposed reconfiguration planning approach, compared to single iteration of SLSQP. If the proposed reconfiguration planning approach is integrated into the dynamic controller of the RCDPR, the frequency of the dynamic controller can be higher than 100 Hz. Based on the analysis of the computation time of the reconfiguration planning approaches, the effectiveness of the proposed reconfiguration planning approach in reducing the computation time of reconfiguration planning and the feasibility of integrating the proposed reconfiguration planning approach into a dynamic controller that runs with a frequency of 100 Hz is verified.

  • Trajectory tracking error

To investigate the control performance of the integration of reconfiguration planning and dynamic control of an RCDPR, the trajectory tracking error of the RCDPR and CDPR in trajectory tracking tests are researched. The trajectory tracking error, denoted as Δx, is defined as the two-norm of the difference between the position of the moving platform and the target position
Δx=xx*2
(19)
where the position of the moving platform x is observable in the Gazebo simulator. The trajectory tracking error of the RCDPR and CDPR in trajectory tracking tests are listed in Table 8. According to the mean trajectory tracking error and the maximum trajectory tracking error in tracking the first target trajectory, one can find that with the proposed reconfiguration planning approach integrated into the dynamic controller of the RCDPR, the RCDPR achieves a smaller trajectory tracking error than the CDPR, if a target trajectory is within the wrench feasible workspace of the CDPR. Based on the mean trajectory tracking error and the maximum trajectory tracking error in tracking the second target trajectory, one can tell that the RCDPR can track a target trajectory that is not entirely contained in the wrench feasible workspace of the CDPR. The analysis of the trajectory tracking error of the RCDPR and CDPR in trajectory tracking tests shows that by real-time reconfiguration, the RCDPR can achieve better control performance than the CDPR.
  • Available wrench set

Table 8

Trajectory tracking error of RCDPR and CDPR (unit: meter)

Mean trajectory tracking error in tracking the first target trajectoryMaximum trajectory tracking error in tracking the first target trajectoryMean trajectory tracking error in tracking the second target trajectoryMaximum trajectory tracking error in tracking the second target trajectory
CDPR0.06360.1300Not applicableNot applicable
RCDPR with reconfiguration planning0.03900.09600.08390.2299
Mean trajectory tracking error in tracking the first target trajectoryMaximum trajectory tracking error in tracking the first target trajectoryMean trajectory tracking error in tracking the second target trajectoryMaximum trajectory tracking error in tracking the second target trajectory
CDPR0.06360.1300Not applicableNot applicable
RCDPR with reconfiguration planning0.03900.09600.08390.2299

To analyze the reasons for the difference in control performance between RCDPR and CDPR in trajectory tracking tests, the available wrench set, denoted as AW, of the RCDPR and CDPR in trajectory tracking tests is investigated. In the available wrench cost function of reconfiguration planning defined in Eq. (4), the isotropy and the size of AW of an RCDPR are reflected by the maximum feasible isotropic wrench rAW. The rAW of the RCDPR and CDPR in trajectory tracking tests is studied to analyze the available wrench set of the RCDPR and CDPR.

The rAW of the RCDPR and CDPR in tracking the first target trajectory and in tracking the second target trajectory is presented in Figs. 12 and 13, respectively. The mean of rAW and the minimum rAW of the RCDPR and CDPR are listed in Table 9. According to the mean of rAW and the minimum rAW of the RCDPR and CDPR in tracking the first target trajectory shown in Table 9, if a target trajectory is within the wrench feasible workspace of the CDPR, the RCDPR with the integration of reconfiguration planning and dynamic control can achieve a larger rAW than the CDPR can achieve. The larger rAW tends to enable the RCDPR to perform a larger resultant force if it is necessary and achieve a better control performance as a result.

Fig. 12
rAW of (a) RCDPR and (b) CDPR in tracking the first target trajectory
Fig. 12
rAW of (a) RCDPR and (b) CDPR in tracking the first target trajectory
Close modal
Fig. 13
rAW of (a) RCDPR and (b) CDPR in tracking the second target trajectory
Fig. 13
rAW of (a) RCDPR and (b) CDPR in tracking the second target trajectory
Close modal
Table 9

rAW of RCDPR and CDPR (unit: N)

Mean of rAW in tracking the first target trajectoryMinimum rAW in tracking the first target trajectoryMean of rAW in tracking the second target trajectoryMinimum rAW in tracking the second target trajectory
CDPR3.93312.97922.07340.0000
RCDPR with reconfiguration planning5.21694.52162.81860.4592
Mean of rAW in tracking the first target trajectoryMinimum rAW in tracking the first target trajectoryMean of rAW in tracking the second target trajectoryMinimum rAW in tracking the second target trajectory
CDPR3.93312.97922.07340.0000
RCDPR with reconfiguration planning5.21694.52162.81860.4592

According to Table 9, the mean of rAW and the minimum rAW of the RCDPR and CDPR in tracking the second target trajectory show that if a target trajectory is not entirely contained in the wrench feasible workspace of the CDPR, the CDPR moves out of the wrench feasible workspace eventually. The rAW of the CDPR becomes zero and the CDPR is out of control. By reconfiguration, the RCDPR with the integration of reconfiguration planning and dynamic control can always achieve a non-zero rAW in tracking the second target trajectory. Thus, the RCDPR can track the second target trajectory.

  • Cable interference cost function

The cable interference cost functions of the RCDPR in tracking the first target trajectory and the second target trajectory are presented in Fig. 14. The mean cable interference cost function and the maximum cable interference cost function of the RCDPR in tracking the target trajectories are listed in Table 10. The mean cable interference cost function and the maximum cable interference cost function of the RCDPR are positive numbers smaller than the resolution of a variable of type double, both in tracking the first target trajectory and in tracking the second target trajectory. A variable of type double has a floating-point precision of up to 15 digits. Numerically, the mean cable interference cost function and the maximum cable interference cost function of the RCDPR are zero. This suggests that every pair of cables have not been close to each other when the RCDPR is tracking the first target trajectory or tracking the second target trajectory. This phenomenon accords with the position of the movable anchor points of the RCDPR in tracking the target trajectories shown in Fig. 10.

  • Feasible position cost function

Fig. 14
Cable interference cost function of the RCDPR in tracking: (a) the first target trajectory and (b) the second target trajectory
Fig. 14
Cable interference cost function of the RCDPR in tracking: (a) the first target trajectory and (b) the second target trajectory
Close modal
Table 10

Cable interference cost function of the RCDPR in tracking the target trajectories

Mean cable interference cost functionMaximum cable interference cost function
In tracking the first target trajectory0.00000.0000
In tracking the second target trajectory0.00000.0000
Mean cable interference cost functionMaximum cable interference cost function
In tracking the first target trajectory0.00000.0000
In tracking the second target trajectory0.00000.0000

The feasible position cost functions of the RCDPR in tracking the first target trajectory and the second target trajectory are presented in Fig. 15. The mean feasible position cost function and the maximum feasible position cost function of the RCDPR in tracking the target trajectories are listed in Table 11. One can see from Fig. 15(a) and Table 11 that in tracking the first target trajectory, the value of the feasible position cost function of the RCDPR is about zero. The value of the feasible position cost function suggests that when the RCDPR is tracking the first target trajectory, movable anchor points have not been close to the bound of the set of feasible positions. One can also observe this phenomenon in Fig. 10(a).

Fig. 15
Feasible position cost function of the RCDPR in tracking: (a) the first target trajectory and (b) the second target trajectory
Fig. 15
Feasible position cost function of the RCDPR in tracking: (a) the first target trajectory and (b) the second target trajectory
Close modal
Table 11

Feasible position cost function of the RCDPR in tracking target trajectories

Mean feasible position cost functionMaximum feasible position cost function
In tracking the first target trajectory0.00470.0484
In tracking the second target trajectory0.04202.5446
Mean feasible position cost functionMaximum feasible position cost function
In tracking the first target trajectory0.00470.0484
In tracking the second target trajectory0.04202.5446

According to Fig. 15(b), the value of the feasible position cost function of the RCDPR in tracking the second target trajectory jumps to about 2.50 at 37 s and converges to about zero at 45 s. This suggests that when the RCDPR is tracking the second target trajectory, at least one of the movable anchor points moves close to the bound of the set of feasible positions from 37 s to 45 s. As shown in Fig. 10(b), the moveable anchor point A1 once moves close to the bound of its feasible positions and moves away from the bound of its feasible positions eventually. This validates the effectiveness of the feasible position cost function of the RCDPR in ensuring that moveable anchor points are within the bound of the set of feasible positions.

6 Conclusion

This paper developed a real-time reconfiguration planning approach that can dramatically reduce the computation time of reconfiguration planning, compared to single iteration of a conventional box-constrained optimization-based reconfiguration planning approach, for the dynamic control of RCDPRs. To reduce the computation time of reconfiguration planning, techniques used for real-time planning of robotic systems were applied to the reconfiguration planning of RCDPRs. A novel reconfiguration value function based on cost functions and constraint functions was proposed for RCDPRs to reflect the value of an RCDPR configuration. Based on optimizing an approximation of the reconfiguration value function, rather than the cost function of an optimization algorithm, the computation time of reconfiguration planning was dramatically reduced. With reduced computation time of reconfiguration planning, the developed reconfiguration planning approach was integrated into an RCDPR's dynamic controller that runs with a frequency of 100 Hz, and the integration of reconfiguration planning and dynamic control was achieved.

A case study of an RCDPR with seven cables and four movable anchor points was conducted to verify the effectiveness of the developed real-time reconfiguration planning approach and the integration of reconfiguration planning and dynamic control for RCDPRs. The case study shows that the computation time of reconfiguration planning according to the developed reconfiguration planning approach is reduced by more than 93%, compared to single iteration of a box-constrained optimization-based reconfiguration planning approach. Based on the integration of reconfiguration planning and dynamic control, the RCDPR can perform tasks that a CDPR cannot perform or achieve better control performance than the CDPR.

In the future, the authors plan to develop an RCDPR prototype that includes moveable anchor points mounted on movable aerial robots and fixed anchor points on the ground and conduct trajectory tracking tests based on the RCDPR prototype to correlate the results of the case study in simulation.

Acknowledgment

The authors would like to thank the editors and reviewers for their valuable comments and suggestions. This work was partially supported by the Guangdong Basic and Applied Basic Research Foundation for Young Scientists (Grant No. 2021A1515110021), Shenzhen Science and Technology Program (Grant No. RCBS20210609103819024), and the Research Foundation for Advanced Talents, Harbin Institute of Technology Shenzhen (Grant No. CA11409019).

Conflict of Interest

There are no conflicts of interest.

Data Availability Statement

The datasets generated and supporting the findings of this article are obtainable from the corresponding author upon reasonable request.

Nomenclature

     
  • n =

    number of cables

  •  
  • p =

    number of degrees-of-freedom of the moving platform

  •  
  • w =

    wrench applied on the moving platform by cables

  •  
  • x =

    pose of the moving platform

  •  
  • V =

    reconfiguration value function

  •  
  • C =

    Coriolis/centripetal matrix of the moving platform

  •  
  • G =

    gravity vector of the moving platform

  •  
  • J =

    Jacobian matrix

  •  
  • M =

    mass matrix of the moving platform

  •  
  • dij =

    distance between the ith and the jth cables

  •  
  • li =

    length of the ith cable

  •  
  • ai =

    position of the ith anchor point on the base

  •  
  • bi =

    position of the ith anchor point on the moving platform

  •  
  • ui =

    unit vector of the ith cable

  •  
  • d* =

    minimum distance between two cables

  •  
  • fAW =

    available wrench cost function

  •  
  • fA =

    feasible position cost function

  •  
  • fI =

    cable interference cost function

  •  
  • rAW =

    maximum feasible isotropic wrench

  •  
  • fiA =

    feasible position cost function of the ith anchor point

  •  
  • fijI =

    cable interference cost function of the ith cable and the jth cable

  •  
  • Ai =

    the ith anchor point on the base

  •  
  • Bi =

    the ith anchor point on the moving platform

  •  
  • Fb =

    base frame

  •  
  • Fm =

    moving platform frame

  •  
  • VNN =

    approximated reconfiguration value function

  •  
  • Ai =

    set of feasible positions of the anchor point ai

  •  
  • Ai* =

    set of positions of the bound of Ai

  •  
  • α =

    step size of a gradient-based algorithm

  •  
  • Δa* =

    target movement of movable anchor points

  •  
  • ΔV =

    approximation error of the approximated reconfiguration value function

  •  
  • Δx =

    tracking error of the moving platform

  •  
  • θij =

    angular difference between the ith and the jth cables

  •  
  • θ* =

    minimum angular difference between two cables connected to the same anchor point

  •  
  • μa =

    feasible position penalty coefficient

  •  
  • μd =

    cable distance penalty coefficient

  •  
  • μr =

    available wrench weight coefficient

  •  
  • μθ =

    cable angular difference penalty coefficient

  •  
  • τ =

    vector of cable tensions

  •  
  • τmin =

    vector of minimum cable tensions

  •  
  • τmax =

    vector of maximum cable tensions

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