Dynamic Modeling and Adaptive Control of Cable-Driven Redundant Manipulator

Abstract

A cable-driven redundant manipulator (CDRM) characterized by redundant degrees of freedom and a lightweight, slender design can perform tasks in confined and restricted spaces efficiently. However, the complex multistage coupling between drive cables and passive joints in CDRM leads to a challenging dynamic model with difficult parameter identification, complicating the efforts to achieve accurate modeling and control. To address these challenges, this paper proposes a dynamic modeling and adaptive control approach tailored for CDRM systems. A multilevel kinematic model of the cable-driven redundant manipulator is presented, and a screw theory is employed to represent the cable tension and cable contact forces as spatial wrenches, which are equivalently mapped to joint torque using the principle of virtual work. This approach simplifies the mapping process while maintaining the integrity of the dynamic model. A recursive method is used to compute cable tension section-by-section for enhancing the efficiency of inverse dynamics calculations and meeting the high-frequency demands of the controller, thereby avoiding large matrix operations. An adaptive control method is proposed building on this foundation, which involves the design of a dynamic parameter adaptive controller in the joint space to simplify the linearization process of the dynamic equations along with a closed-loop controller that incorporates motor parameters in the driving space. This approach improves the control accuracy and dynamic performance of the CDRM under dynamic uncertainties. The accuracy and computational efficiency of the dynamic model are validated through simulations, and the effectiveness of the proposed control method is demonstrated through control tests. This paper presents a dynamic modeling and adaptive control approach for CDRM to enhance accuracy and performance under dynamic uncertainties.

Introduction

With the ongoing advancements in robotics technology, robots have become increasingly capable of replacing manual labor across a growing number of industrial applications. However, traditional robots are limited by their degrees of freedom, configuration, and size, which makes it difficult to achieve high-precision control and flexible obstacle avoidance in narrow spaces. In recent decades, redundant manipulators with large aspect ratios have been widely studied. These manipulators resemble snakes [1] and elephant trunks [2, 3] in appearance and move freely in complex spaces [4, 5].

The cable-driven redundant manipulator (CDRM) is a redundant manipulator driven by cables, and it has a redundant number of degrees of freedom [6, 7]. The driving motor for each joint is positioned behind the base, and each joint is actuated by cables. This design achieves a lightweight structure with a large aspect ratio for the manipulator, while also ensuring that the manipulator can operate underwater or in environments with electromagnetic interference [8]. Therefore, the CDRM can perform detection and maintenance tasks in the complex environments of nuclear power facilities [9, 10] and holds significant potential for applications in space-based on-orbit operations [11, 12] and laparoscopic inspection in medical scenarios [13]. However, the motion and force of each joint are highly coupled with the cable because of the large degree of freedom and complex internal structure of the CDRM and the control of the manipulator is very challenging for complex and limited application scenarios. Therefore, establishing an accurate and efficient dynamic model to improve the control accuracy of the CDRM is essential.

Traditional methods of dynamic modeling include the Newton Euler method [14, 15], principle of virtual work [16], Euler–Lagrange formalism [17, 18], etc. Based on the above methods, researchers conducted several research studies on the dynamic modeling of CDRM. Lau et al. [19, 20] proposed a general dynamic modeling method by deriving the Jacobian matrix and the cable arrangement matrix. Although this approach incorporates cable arrangement into the dynamic model, it fails to consider coupling effects between the cables and joints. The computational efficiency diminishes with an increase in the number of degrees of freedom (DOF), which poses challenges for its application in real-time control. Xu et al. [21] used the Newton Euler method for dynamic modeling, which improved calculation efficiency and considered the cable friction and coupling effect. Xu et al. [22] improved the Newton Euler method based on link eigenvectors and eigenpoints of the manipulator, simplifying the modeling process. These methods can efficiently calculate the dynamic model; however, the cable tension of the CDRM affects all joints through which the cable passes; the system has transient coupling. During motion, the tension on each cable exhibits significant nonlinearity, increasing the difficulty of designing controllers for the CDRM. Some researchers proposed an equivalent mapping of the cable tension because multiple motion space mapping relationships exist among cables, joints, and the end-effector.

Gu [23] mapped the cable tension in the driving space to equivalent torques in the joint space based on the principle of virtual work and designed a sliding mode controller with dynamic feedforward compensation. Chen et al. [24] mapped the cable tension to the equivalent torque in joint space, demonstrating that this method can reduce the transient coupling of the system. Further, they devised a compliance control method in joint space. Peng et al. [25] developed a trajectory tracking framework by integrating dynamic feedforward control with proportional-derivative (PD) control. The framework employs cable tension and end-effector pose deviation as optimization criteria for the real-time adjustment of PD gains. However, the overall mapping matrix becomes complex because of the multi-stage coupling between the cables and joints in the CDRM, which hinders the efficiency of inverse dynamics calculations.

In addition, the aforementioned control methods employ dynamic feedforward. However, the identification of dynamic parameters becomes challenging because of the complex structure and redundant actuation characteristics of the CDRM, which leads to dynamic uncertainties. These uncertainties reduce the accuracy of dynamic feedforward control, which affects the performance of the controller. In response to uncertain dynamics, robust control [26] and adaptive control theories are often proposed for enhancing control performance. He et al. [27] developed a dynamic parameter identification algorithm for humanoid robots and designed an adaptive control law. Yang et al. [28] designed a finite-time convergent adaptive controller for uncertain dynamic and kinematic and validated it on the Baxter robot. Ji et al. [29] designed an adaptive synchronization control method for cable-driven parallel robots, which regulates the coordinated motion of cables while mitigating uncertainties in dynamic and kinematic parameters. Linearizing dynamic equations becomes particularly challenging because of the redundant actuation characteristics of the CDRM where multiple solutions for the driving cable tensions exist. Consequently, the methods outlined above cannot be directly applied to CDRM.

To solve the above-mentioned problems, this paper employs screw theory for dynamic modeling. This approach succinctly and uniformly describes the motion of spatial mechanisms and addresses related problems [30, 31]. In addition, an efficient method is proposed for calculating the dynamic inverse solution, and based on this method, an adaptive control method tailored for the CDRM is proposed. The main contributions of this paper are as follows:

1. The mapping between the multilevel coupled joint space and cable space is simplified by utilizing screw theory and the principle of virtual work, which reduces the complexity of dynamic modeling while preserving the integrity and accuracy of the system.

2. A method based on the recursive computation of cable tension using equivalent torque is proposed, which eliminates the large matrix calculations induced by the high degrees of freedom in the robotic arm, thereby significantly enhancing the computational efficiency of inverse dynamics.

3. An adaptive control strategy, which involves a dynamic parameter adaptive controller in joint space, is introduced to simplify the linearization of dynamic equations. In addition, a closed-loop controller is designed in the driving space to compensate for motor parameters, which improves control accuracy and dynamic performance in the presence of dynamic uncertainties.

The remainder of this paper is structured as follows: Section 2 describes the structure of the CDRM and derives the kinematic model of the CDRM. Section 3 uses the screw theory to establish the dynamic model and deduce the recursive method. In Section 4, an adaptive control method for dynamic parameters is designed and its stability is analyzed. In Section 5, the CDRM is modeled within the Simulink simulation environment to validate the accuracy and computational efficiency of the dynamic modeling method proposed in this paper. Further, a comparative control test is designed to demonstrate the effectiveness of the adaptive control method. Finally, the conclusion of the study is summarized in Section 6.

Structural Design and Kinematic Modeling of the CDRM

Mechanical Design of the CDRM

The cable-driven redundant manipulator proposed in this paper is shown in Figure 1. This manipulator has a driving base and a manipulator body. The manipulator body is composed of $N$ links, where each link is connected by a universal joint. The universal joint has two rotating joints, and the manipulator has $2N$ DOF as a whole. The cables are controlled by motors in the driving base, which use discs to actuate the universal joint, thereby enabling motion in both the pitch and yaw directions. The proximal disc is located at the proximal end of links, which is close to the universal joint. The distal disc is located at the distal end of links, which is close to the next universal joint, as indicated in Figure 1. Using three cables to drive the 2-DOF universal joint is necessary because the cable can only bear tension and cannot bear pressure; the three cables at each joint are distributed at 120° to each other.

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Figure 1

CDRM model

Kinematic Model of the CDRM

The CDRM is driven by the cable, and therefore, any change in the length of the cable can cause changes in the joint angle of the universal joint, which makes each link reach the target pose. Therefore, there are multilevel kinematic mapping relationships between Cartesian, joint, cable, and driving spaces, as shown in Figure 2.

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Figure 2

Multilevel kinematic mapping relationship

Joint-Cartesian Kinematic

This paper establishes the kinematic mapping relationship between joint space and end Cartesian space based on screw theory. Figure 3 shows the screw axis of each rotating joint. For the convenience of calculation, all screw axes are established in the base coordinate system $\{s\}$, which is located in the center of the first universal joint. The screw axis formula under the initial configuration is [31]

1

$${\mathit{S}}_{\mathit{n}}=\left[\begin{array}{c}{\mathit{\omega }}_n\\ {\mathit{v}}_n\\ \end{array}\right]\mathbf{=}\left[\begin{array}{c}{\mathit{\omega }}_n\\ -{\mathit{\omega }}_n\times {\mathit{q}}_n\mathbf{+}{h}_n{\mathit{\omega }}_n\\ \end{array}\right],$$where ${\mathit{\omega }}_n$, ${\mathit{q}}_n$, and $h_n$ represent the unit rotation axis, coordinate of the rotation axis in the base coordinate system $\{s\}$, and screw pitch, respectively. Further, $h_n$ is zero because the joint in the manipulator are pure rotation pairs.

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Figure 3

Screw axis of the CDRM

The initial pose of the end-effector is defined as ${\mathit{T}}_{\mathit{End}}(0)$. According to the product of exponentials (POE) formula, the pose of the end-effector can be calculated as

2

$$\mathit{T}_{_{\mathit{End}}^{}}^{}{\mathbf{=}\mathit{e}}^{[{\mathit{S}}_1]{\mathit{\theta }}_1}{\mathit{e}}^{[{\mathit{S}}_2]{\mathit{\theta }}_2}\cdots {\mathit{e}}^{[{\mathit{S}}_{2N}]{\mathit{\theta }}_{2N}}\mathit{T}_{_{\mathit{End}}^{}}^{}(0),$$where $[{\mathit{S}}_{\mathbf{1}}],[{\mathit{S}}_{\mathbf{2}}],\cdots [{\mathit{S}}_{\mathbf{2}\mathit{N}}]$ represent the auxiliary skew symmetric matrix of ${\mathit{S}}_{\mathbf{1}}{\mathbf{,}\mathit{S}}_{\mathbf{2}}\mathbf{,}\cdots {\mathit{S}}_{\mathbf{2}\mathit{N}}$, and ${\theta }_1,{\theta }_2,\cdots {\theta }_{2N}$ represent the joint angle values of each rotating joint.

The mapping relationship between the spatial twist of the end-effector ${\mathit{V}}_{\mathit{End}}$ and joint angular velocity $\dot{\mathit{\theta }}$ is

3

$$\begin{array}{c}{\mathit{V}}_{\mathit{End}}={\dot{\mathit{T}}}_{\mathit{End}}{\mathit{T}}_{\mathit{End}}^{-1}=[{\mathit{J}}_{s,1},\cdots ,{\mathit{J}}_{s,2N}]\left[\begin{array}{c}{\dot{\theta }}_1\\ ⋮\\ {\dot{\theta }}_{2N}\\ \end{array}\right]={\mathit{J}}_s\dot{\mathit{\theta }},\\ {\mathit{J}}_{s,1}={\mathit{S}}_1,\cdots ,{\mathit{J}}_{s,2N}=\mathit{Ad}({\mathit{e}}^{[{\mathit{S}}_1]{\mathit{\theta }}_1}{\mathit{e}}^{[{\mathit{S}}_2]{\mathit{\theta }}_2}\cdots {\mathit{e}}^{[{\mathit{S}}_{2N}]{\mathit{\theta }}_{2N}}){\mathit{S}}_{2N},\\ \end{array}$$where ${\mathit{J}}_s$ represents the spatial Jacobian matrix, and each column of ${\mathit{J}}_s$ represents the screw axis of the rotating joint under the current configuration. Further, $\mathit{Ad}(\bullet )$ represents the adjoint representation, converts ${\mathit{e}}^{[{\mathit{S}}_1]{\mathit{\theta }}_1}\cdots {\mathit{e}}^{[{\mathit{S}}_{2n-1}]{\mathit{\theta }}_{2n-1}}$ to a 6 × 6 matrix, and is used to transform the screw axis from the initial configuration to the current configuration.

Joint-Cable Kinematic

There is a mapping relationship between the joint and cable spaces. The drive cable of the distal universal joint passes through all proximal universal joints because of the structural characteristics of the CDRM, which results in a high degree of coupling in the cable motion. Therefore, the cable length is obtained by summing the cable length at each universal joint it passes through with the length of the link.

The cable length at each universal joint it passes through is determined by the joint angle and position of the disc holes, as indicated in Figure 4. $h_j$ represents the distance between the proximal disc of the $n$th link and the distal disc of the $(n-1)$th link in the initial configuration; ${\mathit{p}}_{n,i}^l$ represents the position of the $i$th hole at the proximal disc of the $n$th link; and ${\mathit{p}}_{n-1,i}^h$ represents the position of the $i$th hole at the distal disc of the $(n-1)$th link.

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Figure 4

Cables at each universal joint

The distribution of disc holes is shown in Figure 5 because to each universal joint being driven by three cables with each cable spaced ${120}^{\circ }$ apart. The corresponding ${\phi }_i$ is defined as

4

$${\phi }_i=\left\{\begin{array}{c}\frac{(i-1)\pi }{18},\text{if}1\le i\le N,\\ \frac{(i-N-1)\pi }{18}+\frac{2\pi }{3},\text{if}N<i\le 2N,\\ \frac{(i-2N-1)\pi }{18}+\frac{4\pi }{3},\text{if 2}N<i\le 3N.\\ \end{array}\right)$$

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Figure 5

Distribution of disc holes

The position of each disk hole in the base coordinate system $\{s\}$ can be calculated as

5

$$\begin{array}{c}{\mathit{p}}_{n,i}^l{\mathbf{=}\mathit{T}}_{s,2n}{\left[\begin{array}{cccc}rsin({\phi }_i)& \frac{h_j}{2}& rcos({\phi }_i)& 1\end{array}\right]}^{\text{T}},\\ {\mathit{p}}_{n-1,i}^h{\mathbf{=}\mathit{T}}_{s,2\left(n-1\right)}{\left[\begin{array}{cccc}rsin({\phi }_i)& \frac{h_j}{2}+h_{\mathit{link}}& rcos({\phi }_i)& 1\end{array}\right]}^{\text{T}},\\ \end{array}$$where ${\mathit{T}}_{s,2n}$, ${\mathit{T}}_{s,2(n-1)}$, $r$, and $h_{\mathit{link}}$ represent the homogeneous transformation matrix between the $2n$th joint coordinate system $\{2n\}$, homogeneous transformation matrix between the $2(n-1)$th joint coordinate system $\{2(n-1)\}$ and base coordinate system $\{s\}$, radius of the disc, and length of link, respectively.

Figure 4 shows that the cable length formula at each universal joint is

6

$$l_{n,i}\mathbf{=}∥{\mathit{p}}_{n,i}^l-{\mathit{p}}_{n-1,i}^h∥,$$where $l_{n,i}$ represents the length of the $i$th cable in the $n$th universal joint. The total cable length formula is

7

$$\begin{array}{c}{l}_i=\sum _{j=1}^{m_i}(l_{j,i}+h_{\mathit{link}}),\\ \mathrm{\Delta }{l}_i=l_i-l_i(0),\\ \end{array}$$where $l_i$, $l_i(0)$, $\mathrm{\Delta }{l}_i$, and $m_i$ represent the length of the $i$th cable, initial length of the $i$th cable, change of cable length, and number of universal joints that the $i$th cable passes through, respectively.

The mapping relationship between the cable velocity and the joint angular velocity is obtained by the differential equation

8

$$\begin{array}{c}\dot{\mathit{l}}={\mathit{J}}_l\dot{\mathit{\theta }},\\ {\mathit{J}}_l=\frac{\partial \mathit{l}}{\partial \mathbf{\theta }},\\ \end{array}$$where $\dot{\mathit{l}}$, $\dot{\mathit{\theta }}$, and ${\mathit{J}}_l$ represent the cable velocity, joint angular velocity, and Jacobian matrix of the cable velocity and joint angular velocity, respectively.

Cable-Driving Kinematic

The cables are actuated by motors positioned at the base, which result in the following mapping relationship between the cable space and driving space.

9

$$\begin{array}{c}\dot{\mathit{l}}\mathbf{=}\mathit{W}\dot{\mathit{q}},\\ \ddot{\mathit{l}}\mathbf{=}\mathit{W}\ddot{\mathit{q}},\\ \end{array}$$where $\mathit{W}$, $\ddot{\mathit{q}}$, and $\dot{\mathit{q}}$ represent the coefficient of motion transmission between the cables and motors, motor angular acceleration, and motor angular velocity, respectively.

Dynamic Modeling of the CDRM

Establishing an accurate dynamic model is an important part of controlling CDRM. However, the CDRM has many DOF and high coupling. The driving force of each cable can affect the movement of multiple universal joints, which results in transient coupling in the system, improving the design difficulty of the controller. The efficiency of the dynamic solution is low, which is difficult to meet the control frequency requirements of the controller.

According to the principle of virtual work, cable tensions can be equivalently mapped to torques acting on the universal joint, which reduces transient coupling in the system and simplifies control complexity. In this work, screw theory is used to model and simplify equivalent mapping. Building on this model, a recursive method is employed to solve the inverse dynamics, which improves computational efficiency.

Cable Tension-Equivalent Torque Mapping

Figure 6 presents a schematic of the cable tension and joint equivalent torque in the $n$ th universal joint. There are $N$ universal joints and $2N$ rotating joints. As shown in Figure 6, each universal joint and link are driven by $3(N-n+1)$ cables. The $n$th, $(n+N)$th, $(n+2N)$th cable passes through the distal disc of the $(n-1)$th link, which is fixed to the proximal disc of the $n$th link and drives the joint through the tension on the disc. The remaining $3(N-n)$ cables pass through the proximal and distal discs of the $n$th link into the $(n+1)$th universal joint and drive the universal joint through the contact force with the hole in discs.

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Figure 6

Schematic of cable tension

According to the virtual work principle, the power generated by the equivalent torque of the joint is equal to the power generated by cable tension.

10

$${\mathit{\tau }}^{\text{T}}{\dot{\mathit{\theta }}\mathbf{=}\mathit{f}}^{\text{T}}\dot{\mathit{l}},$$where $\mathit{\tau }$, $\mathit{f}$, and $\dot{\mathit{l}}$ represent the equivalent torque, value of cable tension, and cable velocity, respectively.

Substituting Eq. (8) into Eq. (10) yields

11

$${\mathit{\tau }\mathbf{=}\mathit{J}}_l^{\text{T}}\mathit{f}.$$

Eq. (8) shows that ${\mathit{J}}_l$ needs to be obtained by differential calculation, which is inefficient and not conducive to improving the efficiency of inverse dynamics.

In this paper, screw theory is employed to represent the cable tension and contact force on the CDRM as a spatial wrench. The wrench power and joint torque power are equivalently mapped. Therefore, only the spatial twist of the CDRM and spatial Jacobian matrix in Eq. (3) are required, without differential calculation. For the convenience of calculation, all coordinates and vectors are established relative to the base coordinate system $\{s\}$ of the CDRM.

Let $f_{n,i}$ represent the cable tension of the $i$th cable at the $n$th universal joint. As shown in Figure 6, if the $i$th cable is fixed at the proximal disc of the $n$th link, i.e., $i=n,(n+N),(n+2N)$, the cable tension vector ${\mathit{F}}_{n,i}$ is

12

$${\mathit{F}}_{n,i}=f_{n,i}\frac{{\mathit{p}}_{n-1,i}^h-{\mathit{p}}_{n,i}^l}{\left.∥\left({\mathit{p}}_{n-1,i}^h-{\mathit{p}}_{n,i}^l\right.∥\right)}.$$

Given that the force vector and coordinates of the force vector are known, the six-dimensional spatial wrench of the cable on the CDRM can be calculated as

13

$${\overline{\mathit{F}}}_{n,i}\mathbf{=}\left[\begin{array}{c}{\mathit{m}}_{n,i}\\ {\mathit{F}}_{n,i}\\ \end{array}\right]\mathbf{=}\left[\begin{array}{c}{\mathit{p}}_{n,i}^l\times {\mathit{F}}_{n,i}\\ {\mathit{F}}_{n,i}\\ \end{array}\right].$$

According to the virtual work principle, the power generated by the equivalent torque of the joint is equal to the power generated by the spatial wrench as

14

$$\mathit{\tau }_{_{n,i}^{}}^{\text{T}}{\dot{\mathit{\theta }}\mathbf{=}\overline{\mathit{F}}}_{n,i}^{\text{T}}{\mathit{V}}_n,$$where ${\mathit{V}}_n$ represents the spatial twist in the base coordinate system $\{s\}$ and ${\tau }_{n,i}$ represents the equivalent torque generated by the $i$th cable in the $n$th universal joint.

${\mathit{V}}_n$ can be calculated by joint velocity $\dot{\mathit{\theta }}$ and spatial Jacobian matrix ${\mathit{J}}_s$ as

15

$${\mathit{V}}_n{\mathbf{=}\mathit{J}}_{s,2n}\dot{\mathit{\theta }},$$where ${\mathit{J}}_{s,2n}$ represents the first $2n$ columns of the spatial Jacobian matrix because the cable tension at the $n$th universal joint is only relevant for the first $2n$ rotating joints.

Substituting Eq. (15) into Eq. (14) yields

16

$$\begin{array}{c}{\mathit{\tau }}_{n,i}{\mathbf{=}\mathit{J}}_{s,2n}^{\text{T}}{\overline{\mathit{F}}}_{n,i},\\ {\mathit{\tau }}_{n,i}\mathbf{=}({\tau }_{n,i,2n},{\tau }_{n,i,2n-1},{\tau }_{n,i,2n-2}\cdots {\tau }_{n,i,1}),\\ \end{array}$$where ${\tau }_{n,i,2n}$ and ${\tau }_{n,i,2n-1}$ represent the equivalent torque acting on the $2n$th and $(2n-1)$th rotating joint in the $n$th universal joint, and ${\tau }_{n,i,2n-2}\cdots {\tau }_{n,i,1}$ represent the equivalent torque acting on the 1st to $(n-1)$th universal joint.

If the $i$th cable passes through the distal disc of the $n$th link as shown in Figure 7, the cable contact force vector ${\mathit{F}}_{n,i}^{\mathit{ch}}$ is

17

$$\begin{array}{c}{\mathit{F}}_{n,i}^{\mathit{ch}}\mathbf{=}{f}_{linkn,i}\frac{{\mathit{p}}_{n,i}^l-{\mathit{p}}_{n,i}^h}{\left.∥\left({\mathit{p}}_{n,i}^l-{\mathit{p}}_{n,i}^h\right.∥\right)}-f_{n+1,i}\frac{{\mathit{p}}_{n,i}^h-{\mathit{p}}_{n+1,i}^l}{\left.∥\left({\mathit{p}}_{n+1,i}^l-{\mathit{p}}_{n,i}^h\right.∥\right)},\\ f_{linkn,i}={\epsilon }_{n,i}^h{f}_{n+1,i},\\ \end{array}$$where $f_{linkn,i}$ represents the cable tension of the $i$th cable in the $n$th link and $f_{n,i}$ represents the cable tension of the $i$th cable in the $n$th universal joint. $\epsilon _{_{n,i}^{}}^h$ represents the coefficient of cable tension loss attributed to friction after the cable passes through the $i$th disc hole of the distal disc of the $n$th link, which can be calculated by the frictional contact model in Ref [22]. Let $\mu$ and $\psi$ represent the saturated viscous friction coefficient at the disc hole and angle of cable tension at the disc hole, respectively. It can be calculated based on the direction of the cable tension vector.

18

$$\psi _{_{_{n,i}^{}}^{}}^h={cos}^{-1}\left(\frac{{\mathit{p}}_{l,n,i}-{\mathit{p}}_{h,n,i}}{\left.∥\left({\mathit{p}}_{h,n,i}-{\mathit{p}}_{l,n,i}\right.∥\right)}.\frac{{\mathit{p}}_{h,n,i}-{\mathit{p}}_{l,n+1,i}}{\left.∥\left({\mathit{p}}_{l,n+1,i}-{\mathit{p}}_{h,n,i}\right.∥\right)}\right).$$

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Figure 7

Schematic of the cable contact force

On this basis, $\epsilon _{_{n,i}^{}}^h$ can be calculated as

19

$$\epsilon _{_{n,i}^{}}^h=\text{sgn}(v_{n,i}^h)e_{}^{\mu \psi _{_{_{n,i}^{}}^{}}^h},$$where $\text{sgn}(\bullet )$ and $v_{n,i}^h$ represent the sign function and velocity of the cables at the disk hole, respectively.

If the $i$th cable passes through the proximal disc of the $n$th link, the cable contact force vector ${\mathit{F}}_{n,i}^{\mathit{cl}}$ is

20

$$\begin{array}{c}{\mathit{F}}_{n,i}^{\mathit{cl}}=f_{n,i}\frac{{\mathit{p}}_{n-1,i}^h-{\mathit{p}}_{n,i}^l}{\left.∥\left({\mathit{p}}_{n,i}^l-{\mathit{p}}_{n-1,i}^h\right.∥\right)}-f_{linkn,i}\frac{{\mathit{p}}_{n,i}^l-{\mathit{p}}_{n,i}^h}{\left.∥\left({\mathit{p}}_{n,i}^h-{\mathit{p}}_{n,i}^l\right.∥\right)},\\ f_{n,i}={\epsilon }_{n,i}^l{f}_{linkn,i},\\ \end{array}$$where ${\epsilon }_{n,i}^l$ represents the coefficient of cable tension loss caused by friction after the cable passes through the $i$th disc hole of the proximal disc of the $n$th link.

Based on the contact force vectors, the wrench of the contact force can be calculated as

21

$${\overline{\mathit{F}}}_{n,i}^{\mathit{ch}}\mathbf{=}\left[\begin{array}{c}{\mathit{m}}_{n,i}^{\mathit{ch}}\\ {\mathit{F}}_{n,i}^{\mathit{ch}}\\ \end{array}\right]\mathbf{=}\left[\begin{array}{c}{\mathit{p}}_{n,i}^h\times {\mathit{F}}_{n,i}^{\mathit{ch}}\\ {\mathit{F}}_{n,i}^{\mathit{ch}}\\ \end{array}\right],$$

22

$${\overline{\mathit{F}}}_{n,i}^{\mathit{cl}}\mathbf{=}\left[\begin{array}{c}{\mathit{m}}_{n,i}^{\mathit{cl}}\\ {\mathit{F}}_{n,i}^{\mathit{cl}}\\ \end{array}\right]\mathbf{=}\left[\begin{array}{c}{\mathit{p}}_{n,i}^l\times {\mathit{F}}_{n,i}^{\mathit{cl}}\\ {\mathit{F}}_{n,i}^{\mathit{cl}}\\ \end{array}\right].$$

According to the principle of virtual work, the equivalent joint torque generated by the wrench of contact force can be calculated as

23

$${\mathit{\tau }}_{n,i}^{\mathit{ch}}{\mathbf{=}\mathit{J}}_{s,2n}^{\text{T}}{\overline{\mathit{F}}}_{n,i}^{\mathit{ch}},$$

24

$${\mathit{\tau }}_{n,i}^{\mathit{cl}}{\mathbf{=}\mathit{J}}_{s,2n}^{\text{T}}{\overline{\mathit{F}}}_{n,i}^{\mathit{cl}}.$$

Based on this mapping relationship between the cable tension and equivalent torque at the joints, the manipulator can be assumed to be a traditional joint motor-driven manipulator, and the equivalent torque required at each joint can be calculated using the Newton–Euler recursive method.

First, the twist ${\mathit{v}}_i$ and accelerations ${\dot{\mathit{v}}}_i$ at each joint coordinate system can be calculated by forward recursion based on $(\mathit{\theta },\dot{\mathit{\theta }},\ddot{\mathit{\theta }})$.

25

$$\begin{array}{c}\text{for}i=1\to N:\\ {\mathit{A}}_i={\mathit{Ad}}_{T_{i,0}}{\mathbf{(}\mathit{S}}_i\mathbf{)},\\ {\mathit{v}}_i={\mathit{Ad}}_{{\mathit{T}}_{\mathit{i}\mathbf{,}\mathit{i}\mathbf{-}\mathbf{1}}}{\mathbf{(}\mathit{v}}_{i-1}\mathbf{)}+{\mathit{A}}_i{\dot{\theta }}_i,\\ {\dot{\mathit{v}}}_i={\mathit{Ad}}_{T_{i,i-1}}{\mathbf{(}\dot{\mathit{v}}}_{i-1}\mathbf{)}+{\mathit{ad}}_{v_i}{\mathbf{(}\mathit{A}}_i\mathbf{)}{\dot{\theta }}_i+{\mathit{A}}_i{\ddot{\theta }}_i,\\ \end{array}$$where $\mathit{ad}$ represents the lie bracket.

The equivalent torque required for each joint ${\overline{\tau }}_i$ is then calculated using the backward recursion method.

26

$$\begin{array}{c}\text{for}i=N\to 1:\\ {\mathit{F}}_i={\mathit{Ad}}_{T_{i+1,i}}^{\text{T}}({\mathit{F}}_{i+1})+{\mathit{G}}_i{\dot{\mathit{v}}}_i-{\mathit{ad}}_{v_i}^{\text{T}}({\mathit{G}}_i{\mathit{v}}_i),\\ {\overline{\tau }}_i={\mathit{F}}_i^{\text{T}}{\mathit{A}}_i,\\ \end{array}$$where ${\mathit{G}}_i$ represents the inertia matrix.

In addition to the driving torque calculated above, when the manipulator moves, it needs to overcome the friction at each joint. Finally, the closed dynamic equation is obtained as

27

$$\begin{array}{c}\overline{\mathit{\tau }}=\mathit{M}(\mathit{\theta })\ddot{\mathit{\theta }}\mathbf{+}\mathit{C}(\mathit{\theta }\mathbf{,}\dot{\mathit{\theta }})\dot{\mathit{\theta }}\mathbf{+}\mathit{G}(\mathit{\theta })+{\mathit{F}}_{\mathit{jv}}\dot{\mathit{\theta }},\\ \overline{\mathit{\tau }}=\left(\begin{array}{c}{\overline{\tau }}_1\\ {\overline{\tau }}_2\\ ⋮\\ {\overline{\tau }}_{2N}\\ \end{array}\right),\\ \end{array}$$where $\overline{\mathit{\tau }}$, $\mathit{\theta }={({\theta }_1,{\theta }_2,{\theta }_3,...,{\theta }_{2N})}^{\text{T}}$, $\dot{\mathit{\theta }}={({\dot{\theta }}_1,{\dot{\theta }}_2,{\dot{\theta }}_3,...,{\dot{\theta }}_{2N})}^{\text{T}}$, $\ddot{\mathit{\theta }}={({\ddot{\theta }}_1,{\ddot{\theta }}_2,{\ddot{\theta }}_3,...,{\ddot{\theta }}_{2N})}^{\text{T}}$, $\mathit{M}(\mathit{\theta })$, $\mathit{C}(\mathit{\theta }\mathbf{,}\dot{\mathit{\theta }})$, $\mathit{G}(\mathit{\theta })$, and ${\mathit{F}}_{\mathit{jv}}$ represent the equivalent torque required by the manipulator, joint angle vector, joint velocity vector, joint acceleration vector, inertia matrix of the manipulator, matrix of the Koch force and centrifugal force of the manipulator, gravity vector of the manipulator, and viscous friction vector of the joints, respectively.

Substituting Eq. (16), Eq. (21), and Eq. (22) into Eq. (25) yields the following dynamic equations for the CDRM.

28

$$\overline{\mathit{\tau }}\mathbf{=}\mathit{M}(\mathit{\theta })\ddot{\mathit{\theta }}\mathbf{+}\mathit{C}(\mathit{\theta }\mathbf{,}\dot{\mathit{\theta }})\dot{\mathit{\theta }}\mathbf{+}\mathit{G}(\mathit{\theta })+{\mathit{F}}_{\mathit{jv}}\dot{\mathit{\theta }}={\mathit{J}}_t^{\text{T}}\mathit{F},$$where ${{\mathit{J}}_t}^{\text{T}}$ and $\mathit{F}$ represent the mapping matrix between the cable tension and the joint equivalent torque and cable tension vector, respectively.

Recursive Method for Solving Inverse Dynamics

The inverse dynamics can be computed either by solving an optimization problem or by utilizing the pseudoinverse of ${\mathit{J}}_t^{\text{T}}$ along with its null space because of the actuation redundancy in CDRMs. Eqs. (16), (21), and (22) indicate that the cable tension and contact forces at the $n$th link generate equivalent torques that affect all joints from the 1st to the $n$th universal joint, making ${\mathit{J}}_t$ in Eq. (26) very complex and reducing the efficiency of solving the inverse dynamics. This paper employs a method of recursively calculating the cable tension section by section from the end to enhance computational efficiency and avoid large matrix calculations, as illustrated in Algorithm 1.

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Algorithm 1

Recursive method

When $n=N$, only the cable tension acts on the $n$th link, and there is no cable contact force; therefore, the equivalent torque of the $n$th universal joint can be provided by the three cables fixed to the proximal disk of the $n$th link, which can be listed as

29

$$\begin{array}{c}\left(\begin{array}{c}{\overline{\tau }}_{2N-1}\\ {\overline{\tau }}_{2N}\\ \end{array}\right)={\mathit{J}}_{s,2N-1\sim 2N}^{\text{T}}\left({\overline{\mathit{F}}}_{N,N}+{\overline{\mathit{F}}}_{N,2N}+{\overline{\mathit{F}}}_{N,3N}\right),\\ \left(\begin{array}{c}{\overline{\tau }}_{2N-1}\\ {\overline{\tau }}_{2N}\\ \end{array}\right)={\mathit{D}}_N\left(\begin{array}{c}{f}_{N,N}\\ f_{N,2N}\\ f_{N,3N}\\ \end{array}\right),\\ \end{array}$$where ${\overline{\tau }}_{2N-1},{\overline{\tau }}_{2N}$ represents the equivalent torque of the $N$th universal joint, ${\mathit{J}}_{s,2N-1\sim 2N}$ represents columns $2N-1$ and $2N$ of the spatial velocity Jacobian matrix ${\mathit{J}}_s$.

The definition of ${\mathit{D}}_N$ is

30

$$\begin{array}{c}\hfill {\mathit{D}}_N{\mathbf{=}\mathit{J}}_{s,2N-1\sim 2N}^{\text{T}}\left(\begin{array}{cc}\begin{array}{cc}{\mathit{D}}_{N,1}& {\mathit{D}}_{N,2}\end{array}& {\mathit{D}}_{N,3}\end{array}\right),\\ \hfill {\mathit{D}}_{N,i}=\left(\begin{array}{cc}& {\mathit{P}}_{N-1,iN}^l\times \frac{{\mathit{P}}_{N-1,iN}^h-{\mathit{P}}_{N,iN}^l}{\left.∥\left({\mathit{P}}_{\mathit{N}\mathbf{,}\mathit{i}\mathit{N}}^{\mathit{l}}\mathbf{-}{\mathit{P}}_{\mathit{N}\mathbf{-}\mathbf{1}\mathbf{,}\mathit{i}\mathit{N}}^{\mathit{h}}\right.∥\right)}\hfill \\ \hfill & \frac{{\mathit{P}}_{N-1,iN}^h-{\mathit{P}}_{N,iN}^l}{\left.∥\left({\mathit{P}}_{\mathit{N}\mathbf{,}\mathit{i}\mathit{N}}^{\mathit{l}}\mathbf{-}{\mathit{P}}_{\mathit{N}\mathbf{-}\mathbf{1}\mathbf{,}\mathit{i}\mathit{N}}^{\mathit{h}}\right.∥\right)}\hfill \\ \hfill \end{array}\right).\\ \hfill \end{array}$$

There are multiple solutions for $f_{N,3N},f_{N,2N},f_{N,N}$ as indicated in Eq. (27), because of the actuation redundant characteristic of the CDRM. The following quadratic programming problem can be formulated to determine the unique optimal solution for the cable tensions by minimizing the sum of the squares of the cable tensions as the optimization objective and using the limit values of the cable tensions along with Eq. (29) as constraints.

31

$$\begin{array}{c}\underset{f}{min}\frac{1}{2}{\mathit{f}}^{\text{T}}\mathit{f},\\ \text{subject to}:\\ \left(\begin{array}{c}{\overline{\tau }}_{2N-1}\\ {\overline{\tau }}_{2N}\\ \end{array}\right)={\mathit{D}}_N\left(\begin{array}{c}{f}_{N,3N}\\ f_{N,2N}\\ f_{N,N}\\ \end{array}\right),\\ f_{max}\ge f\ge f_{min}.\end{array}$$

The spatial wrench of cable ${\overline{\mathit{F}}}_{N,N},{\overline{\mathit{F}}}_{N,2N},{\overline{\mathit{F}}}_{N,3N}$ can be obtained according to $f_{N,3N},f_{N,2N},f_{N,N}$. Substitute ${\overline{\mathit{F}}}_{N,N},{\overline{\mathit{F}}}_{N,2N},{\overline{\mathit{F}}}_{N,3N}$ into Eq. (16) as

32

$${\mathit{\tau }}_N{\mathbf{=}\mathit{J}}_{s,2N}^{\text{T}}{\mathbf{(}\overline{\mathit{F}}}_{N,N},{\overline{\mathit{F}}}_{N,2N},{\overline{\mathit{F}}}_{N,3N}\mathbf{)},$$where ${\mathit{\tau }}_N$ represents a $2N\times 1$ column vector, which indicates that ${\overline{\mathit{F}}}_{N,3N},{\overline{\mathit{F}}}_{N,2N},{\overline{\mathit{F}}}_{N,N}$ produces equivalent torque on all joints of the CDRM.

When $n<N$, the equivalent torque at the $n$th universal joint is provided by the cable tension and cable contact force at the $n$th to $N$th links. Therefore, for each forward recurrence of the calculation of the universal joint, the equivalent torque generated by the cable tension of the $(n+1)$th links and cable contact force of the cable passing through the $n$th link can be removed from $\overline{\mathit{\tau }}$ before calculating the cable tension at the $n$th universal joint.

From Eq. (16), the equivalent torque generated by the cable tension of the $(n+1)$ th links is

33

$${\mathit{\tau }}_{n+1}=\mathit{J}_{_{s,2(n+1)}^{}}^{\text{T}}({\overline{\mathit{F}}}_{n+1,n+1+2N}+{\overline{\mathit{F}}}_{n+1,n+1+N}+{\overline{\mathit{F}}}_{n+1,n+1}).$$

According to Eq. (25) and Eq. (26), the equivalent torque generated by the cable contact force of the $n$th link is

34

$$\begin{array}{c}{\mathit{\tau }}_i^{\mathit{ch}}{\mathbf{=}\mathit{J}}_{s,2n}^{\text{T}}\left({\overline{\mathit{F}}}_{n,i+2N}^{\mathit{ch}}{\mathbf{+}\overline{\mathit{F}}}_{n,i+N}^{\mathit{ch}}{\mathbf{+}\overline{\mathit{F}}}_{n,i}^{\mathit{ch}}\right),(n<i\le N),\\ {\mathit{\tau }}_i^{\mathit{cl}}{\mathbf{=}\mathit{J}}_{s,2n}^{\text{T}}\left({\overline{\mathit{F}}}_{n,i+2N}^{\mathit{cl}}{\mathbf{+}\overline{\mathit{F}}}_{n,i+N}^{\mathit{cl}}{\mathbf{+}\overline{\mathit{F}}}_{n,i}^{\mathit{cl}}\right)\mathbf{,}(n<i\le N).\\ \end{array}$$

Subtract the above torque from $\overline{\mathit{\tau }}$. For the ease of calculation, augment ${\mathit{\tau }}_{\mathit{i}}{\mathbf{,}\mathit{\tau }}_{\mathit{i}}^{\mathit{ch}}{\mathbf{,}\mathit{\tau }}_{\mathit{i}}^{\mathit{cl}}$ to $2N$ rows, with the augmented portion being 0.

35

$${\mathit{\tau }}_{\mathit{update}}=\left(\begin{array}{c}{\tau }_{n+1,1}+\sum _{i=n+1}^N{\tau }_{i,1}^{\mathit{ch}}+{\tau }_{i,1}^{\mathit{cl}}\\ {\tau }_{n+1,2}+\sum _{i=n+1}^N{\tau }_{i,2}^{\mathit{ch}}+{\tau }_{i,2}^{\mathit{cl}}\\ ⋮\\ {\tau }_{n+1,2N}+\sum _{i=n+1}^N{\tau }_{i,2N}^{\mathit{ch}}+{\tau }_{i,2N}^{\mathit{cl}}\\ \end{array}\right),$$

36

$$\overline{\mathit{\tau }}\mathbf{=}\overline{\mathit{\tau }}-{\mathit{\tau }}_{\mathit{update}}.$$

In the corrected $\overline{\mathit{\tau }}$, the equivalent torque of the $n$th universal joint is provided entirely by the three cables fixed to the proximal disk of the $n$th link, which can be listed as

37

$$\left(\begin{array}{c}{\overline{\tau }}_{2n}\\ {\overline{\tau }}_{2n}\\ \end{array}\right)={\mathit{J}}_{s,2(n-1)\sim 2n}^{\text{T}}\left({\overline{\mathit{F}}}_{n,n+2N}+{\overline{\mathit{F}}}_{n,n+N}+{\overline{\mathit{F}}}_{n,n}\right).$$

The unique optimal solution for the cable tension $f_{n,n+2N},f_{n,n+N},f_{n,n}$ is solved by quadratic programming according to Eqs. (29)–(31) in the same way.

Adaptive Control Method Based on Dynamic Feedforward

The mapping relationship between the cable tension and joint states is non-unique because of the actuation redundancy of CDRM, and this can lead to multiple solutions for the cable tension. In addition, the complexity of the CDRM structure significantly increases the difficulty of parameter identification, thereby leading to a reduction in the accuracy of dynamic feedforward. This paper formulates the dynamic equations in joint space to address the aforementioned issues, where the mapping between joint states and equivalent torques is unique, as illustrated in Eq. (27). This approach simplifies the linearization of the dynamic equations.

Based on the linearized form, a dynamic parameter adaptation law is developed using the regression matrix. This method enables real-time correction of dynamic parameters based on joint motion errors, which enhances control accuracy. Given that the cable forces in CDRM are generated by motors, it is essential to design a controller in the driving space to compensate for motor inertia, friction, and motion errors. This approach facilitates dual-space closed-loop control of the CDRM. The control block diagram is shown in Figure 8.

[See PDF for image]

Figure 8

Adaptive control block diagram

Adaptive Control Method

Considering the dynamic equation in joint space, the linear expression of Eq. (27) can be written as

38

$$\overline{\mathit{\tau }}=\mathit{M}(\mathit{\theta })\ddot{\mathit{\theta }}\mathbf{+}\mathit{C}(\mathit{\theta },\dot{\mathit{\theta }})\dot{\mathit{\theta }}\mathbf{+}\mathit{G}(\mathit{\theta })+{\mathit{F}}_{\mathit{jv}}{\dot{\mathit{\theta }}\mathbf{=}\mathit{Y}}_r(\ddot{\mathit{\theta }},\dot{\mathit{\theta }},\mathit{\theta }){\Phi }_r,$$where ${\mathit{Y}}_r\mathbf{(}\ddot{\mathit{\theta }},\dot{\mathit{\theta }},\mathit{\theta }\mathbf{)}$ represents the dynamic parameter regression matrix and ${\Phi }_r$ represents the identifiable dynamic parameters.

In the proposed control law, the following error variables are introduced.

39

$${\mathit{e}}_{\theta }{\mathbf{=}\mathit{\theta }}_d-\mathit{\theta },{\mathit{e}}_s{\mathbf{=}\dot{\mathit{\theta }}}_r-\dot{\mathit{\theta }},$$where ${\mathit{\theta }}_{\mathit{d}}$ represents the desired joint position and ${\dot{\mathit{\theta }}}_{\mathit{d}}$ represents the desired joint velocity. ${\dot{\mathit{\theta }}}_{\mathit{r}}$ can be defined as

40

$${\dot{\mathit{\theta }}}_{\mathit{r}}{\mathbf{=}\dot{\mathit{\theta }}}_{\mathit{d}}+{\mathit{\lambda }}_1{\mathit{e}}_{\mathit{\theta }},$$where ${\lambda }_1$ represents a given constant. Substituting Eq. (39) into Eq. (40) yields

41

$${\mathit{e}}_{\mathit{s}}{\mathbf{=}\dot{\mathit{e}}}_{\mathit{\theta }}+{\mathit{\lambda }}_1{\mathit{e}}_{\mathit{\theta }}.$$

Substituting Eq. (41) into Eq. (38) yields

42

$$\begin{array}{c}-\mathit{M}(\mathit{\theta }){\dot{\mathit{e}}}_s-(\mathit{C}(\mathit{\theta }\mathbf{,}\dot{\mathit{\theta }})+{\mathit{F}}_{\mathit{jv}}){\mathit{e}}_s\mathbf{=}\overline{\mathit{\tau }}-{\mathit{Y}}_r{\mathbf{(}\ddot{\mathit{\theta }}}_r,{\dot{\mathit{\theta }}}_r,\dot{\mathit{\theta }},\mathit{\theta }){\Phi }_r,\\ {\mathit{Y}}_r{\mathbf{(}\ddot{\mathit{\theta }}}_r,{\dot{\mathit{\theta }}}_r,\dot{\mathit{\theta }},\mathit{\theta }\mathbf{)}{\Phi }_r=\mathit{M}(\mathit{\theta }){\ddot{\mathit{\theta }}}_r+\mathit{C}(\mathit{\theta }\mathbf{,}\dot{\mathit{\theta }}){\dot{\mathit{\theta }}}_r+\mathit{G}(\mathit{\theta })+{\mathit{F}}_{\mathit{jv}}{\dot{\mathit{\theta }}}_r.\\ \end{array}$$

Based on Eq. (42), the control law in joint space can be designed as

43

$${\overline{\mathit{\tau }}\mathbf{=}\mathit{K}}_1{\mathit{e}}_s{\mathbf{+}\mathit{Y}}_r{\mathbf{(}\ddot{\mathit{\theta }}}_r,{\dot{\mathit{\theta }}}_r,\dot{\mathit{\theta }},\mathit{\theta }\mathbf{)}{\widehat{\Phi }}_r,$$where ${\mathit{K}}_1$, ${\mathit{Y}}_r$, ${\widehat{\Phi }}_r$, and ${\mathit{K}}_1{\mathit{e}}_s$ represent the given control gain, dynamic parameter regression matrix, estimated dynamic parameters, and error feedback term, respectively.

Substituting Eq. (43) into Eq. (42) yields the closed-loop dynamics in the joint space as

44

$$-\mathit{M}(\mathit{\theta }){\dot{\mathit{e}}}_{\mathit{s}}-\mathit{C}(\mathit{\theta }\mathbf{,}\dot{\mathit{\theta }}){\mathit{e}}_s-{\mathit{F}}_{\mathit{jv}}{\mathit{e}}_s{\mathbf{=}\mathit{K}}_1{\mathit{e}}_s{\mathbf{+}\mathit{Y}}_r{\mathbf{(}\ddot{\mathit{\theta }}}_r,{\dot{\mathit{\theta }}}_r,\dot{\mathit{\theta }},\mathit{\theta }){\tilde{\Phi }}_r,$$where ${\tilde{\Phi }}_r$ represents the error in the dynamic parameters.

The estimated ${\widehat{\Phi }}_r$ can be updated according to the following adaptive law

45

$${\dot{\widehat{\Phi }}}_r{\mathbf{=}\mathit{K}}_d{\mathit{Y}}_r^{\text{T}}{\mathbf{(}\ddot{\mathit{\theta }}}_r,{\dot{\mathit{\theta }}}_r,\dot{\mathit{\theta }},{\mathit{\theta }\mathbf{)}\mathit{e}}_s,$$where ${\mathit{K}}_d$ represents the gain for adaptive law.

In Eq. (43), $\overline{\mathit{\tau }}$ represents the equivalent torque output of the controller. The required cable tensions $\widehat{\mathit{f}}$ in driving space can be computed by substituting $\overline{\mathit{\tau }}$ into the recursive inverse solution algorithm described earlier.

Considering motor inertia and the friction in the motion control of CDRM is essential because the cables are driven by motors. The dynamic equation in the driving space is formulated as

46

$${\mathit{\tau }}_m={\mathit{I}}_m{\ddot{\mathit{q}}\mathbf{+}\mathit{F}}_v\dot{\mathit{q}}\mathbf{+}\mathit{N}\mathit{f},$$where ${\mathit{\tau }}_m$, ${\mathit{I}}_m$, ${\mathit{F}}_v$, $\ddot{\mathit{q}}$, $\dot{\mathit{q}}$, and $\mathit{N}$ represent the motor driving torque, motor inertia, motor friction coefficient, motor angular acceleration, motor angular velocity, and mapping coefficient between the motor torque and cable tension, respectively.

The following error variables are introduced in the driving space.

47

$${\mathit{e}}_q{\mathbf{=}\mathit{q}}_d-\mathit{q},{\mathit{e}}_{\mathit{qs}}{\mathbf{=}\dot{\mathit{q}}}_r-\dot{\mathit{q}},$$where ${\dot{\mathit{q}}}_{\mathit{r}}$ is defined as

48

$${\dot{\mathit{q}}}_r{\mathbf{=}\dot{\mathit{q}}}_d{\mathbf{+}\mathit{\lambda }}_2{\mathit{e}}_q,$$where ${\mathit{\lambda }}_{\mathbf{2}}$ represents a given constant vector. Substituting Eq. (48) into Eq. (47) yields

49

$${\mathit{e}}_{\mathit{qs}}{\mathbf{=}\dot{\mathit{e}}}_q{\mathbf{+}\mathit{\lambda }}_2{\mathit{e}}_q.$$

Based on this, the control law in the driving space is designed as

50

$${\mathit{\tau }}_m{\mathbf{=}\mathit{K}}_2{\mathit{e}}_{\mathit{qs}}{\mathbf{+}\widehat{\mathit{I}}}_m{\ddot{\mathit{q}}}_r{\mathbf{+}\widehat{\mathit{F}}}_v{\dot{\mathit{q}}}_r\mathbf{+}\mathit{N}\widehat{\mathit{f}},$$where ${\mathit{K}}_2$, ${\widehat{\mathit{I}}}_m$, and ${\widehat{\mathit{F}}}_v$ represent the feedback gain, estimated motor inertia, and estimated motor friction coefficient, respectively.

Substituting control law Eq. (50) into Eq. (46) yields the closed-loop dynamics in the driving space.

51

$$\begin{array}{c}-{\mathit{K}}_2{\mathit{e}}_{\mathit{qs}}{\mathbf{=}\widehat{\mathit{I}}}_m{\ddot{\mathit{q}}}_r-{\mathit{I}}_m{\ddot{\mathit{q}}\mathbf{+}\widehat{\mathit{F}}}_v{\dot{\mathit{q}}}_r-{\mathit{F}}_v\dot{\mathit{q}}\mathbf{+}\mathit{N}\mathbf{(}\widehat{\mathit{f}}-\mathit{f}\mathbf{)},\\ {\widehat{\mathit{I}}}_m{\dot{\mathit{e}}}_{\mathit{qs}}{\mathbf{+}\mathbf{(}\widehat{\mathit{I}}}_m-{\mathit{I}}_m{\mathbf{)}\ddot{\mathit{q}}\mathbf{+}\widehat{\mathit{F}}}_v{\mathit{e}}_{\mathit{qs}}{\mathbf{+}\mathbf{(}\widehat{\mathit{F}}}_v-{\mathit{F}}_v{\mathbf{)}\dot{\mathit{q}}\mathbf{+}\mathit{K}}_2{\mathit{e}}_{\mathit{qs}}\mathbf{+}\mathit{N}\mathbf{(}\widehat{\mathit{f}}-\mathit{f}\mathbf{)}\mathbf{=}\mathbf{0}.\\ \end{array}$$

Let $\stackrel{\mathbf{~}}{\mathit{f}}\mathbf{=}\widehat{\mathit{f}}-\mathit{f}$, ${\mathbf{\Delta }}_1{\mathbf{=}\mathbf{(}\widehat{\mathit{I}}}_m-{\mathit{I}}_m\mathbf{)}\ddot{\mathit{q}}$, and ${\mathbf{\Delta }}_{\mathbf{2}}{\mathbf{=}\mathbf{(}\widehat{\mathit{F}}}_v-{\mathit{F}}_v\mathbf{)}\dot{\mathit{q}}$ because the motor inertia and friction can be estimated with reasonable accuracy based on motor parameters. Therefore, ${\mathbf{\Delta }}_1,{\mathbf{\Delta }}_2$ is considered a bounded term.

In summary, the complete control law of the adaptive control method proposed in this paper is

52

$$\begin{array}{c}{\overline{\mathit{\tau }}\mathbf{=}\mathit{K}}_1{\mathit{e}}_s{\mathbf{+}\mathit{Y}}_r{\mathbf{(}\ddot{\mathit{\theta }}}_r,{\dot{\mathit{\theta }}}_r,\dot{\mathit{\theta }},\mathit{\theta }){\widehat{\Phi }}_r,{\dot{\widehat{\Phi }}}_r{\mathbf{=}\mathit{K}}_d{\mathit{Y}}_r^{\text{T}}{\mathbf{(}\ddot{\mathit{\theta }}}_r,{\dot{\mathit{\theta }}}_r,\dot{\mathit{\theta }},{\mathit{\theta }\mathbf{)}\mathit{e}}_s,\\ \widehat{\mathit{f}}=inversedynamics\mathbf{(}\overline{\tau }),\\ {\mathit{\tau }}_m{\mathbf{=}\mathit{K}}_2{\mathit{e}}_{\mathit{qs}}{\mathbf{+}\widehat{\mathit{I}}}_m{\ddot{\mathit{q}}}_r{\mathbf{+}\widehat{\mathit{F}}}_v{\dot{\mathit{q}}}_r\mathbf{+}\mathit{N}\widehat{\mathit{f}}.\\ \end{array}$$

Stability Analysis

This section conducts a stability analysis of the proposed adaptive control method. Define the following positive definite Lyapunov candidate function $V_1$

53

$$V_1\mathbf{=}\frac{1}{2}{\mathit{e}}_s^{\text{T}}\mathit{M}(\mathit{\theta }){\mathit{e}}_s\mathbf{+}\frac{1}{2}{\tilde{\Phi }}_r^{\text{T}}{\mathit{K}}_d^{-1}{\tilde{\Phi }}_r.$$

Differentiating $V_1$ with respect to time yields

54

$${\dot{V}}_1{\mathbf{=}\mathit{e}}_s^{\text{T}}\mathit{M}(\mathit{\theta }){\dot{\mathit{e}}}_s\mathbf{+}\frac{1}{2}{\mathit{e}}_s^{\text{T}}\dot{\mathit{M}}(\mathit{\theta }){\mathit{e}}_s{+\tilde{\Phi }}_r^{\text{T}}{\mathit{K}}_d^{-1}{\dot{\tilde{\Phi }}}_r.$$

Substituting Eq. (44) into Eq. (54) yields

55

$${\dot{V}}_1{\mathbf{=}\mathit{e}}_s^{\text{T}}\mathbf{(}-{\mathit{Y}}_r{\tilde{\Phi }}_r-\mathbf{(}{K}_1+{\mathit{F}}_{\mathit{jv}}){\mathit{e}}_s\mathbf{)}\mathbf{+}\frac{1}{2}{\mathit{e}}_s^{\text{T}}\mathbf{(}\dot{\mathit{M}}(\mathbf{\theta })-\mathbf{2}C(\mathit{\theta },\dot{\mathit{\theta }}){\dot{\mathit{e}}}_s+{\tilde{\Phi }}_r^{\text{T}}{\mathit{K}}_d^{-1}{\dot{\tilde{\Phi }}}_r.$$

Since $\mathit{M}(\mathit{\theta })$ is a positive definite symmetric matrix and $\dot{\mathit{M}}(\mathit{\theta })-\mathbf{2}\mathit{C}(\mathit{\theta },\dot{\mathit{\theta }})$ is a skew-symmetric matrix, Eq. (55) can be simplified to

56

$${\dot{V}}_1{\mathbf{=}\mathit{e}}_s^{\text{T}}\mathbf{(}-{\mathit{Y}}_r{\tilde{\mathrm{\Phi }}}_r-({\mathit{K}}_1+{\mathit{F}}_{\mathit{jv}}){\mathit{e}}_s{)+\tilde{\mathrm{\Phi }}}_r^{\text{T}}{\mathit{K}}_d^{-1}{\dot{\tilde{\mathrm{\Phi }}}}_r.$$

Substituting the adaptive law Eq. (45) into Eq. (56) yields

57

$$\begin{array}{c}{\dot{V}}_1\mathbf{=}-{\mathit{e}}_s^{\text{T}}({\mathit{K}}_1+{\mathit{F}}_{\mathit{jv}}){\mathit{e}}_s-{\mathit{e}}_s^{\text{T}}{\mathit{Y}}_r{\tilde{\mathrm{\Phi }}}_r{+\tilde{\mathrm{\Phi }}}_r^{\text{T}}{\mathit{Y}}_r^{\text{T}}{\mathit{e}}_s,\\ {\dot{V}}_1\mathbf{=}-{\mathit{e}}_s^{\text{T}}({\mathit{K}}_1+{\mathit{F}}_{\mathit{jv}}){\mathit{e}}_s.\\ \end{array}$$

From Eq. (57), ${\mathit{K}}_1$ and ${\mathit{F}}_{\mathit{jv}}$ are positive matrices, and it is evident that ${\dot{V}}_1$ remains consistently less than 0, which implies that both joint motion errors and dynamic feedforward errors will approach 0 under the adaptive control law in the joint space. Thus, $\tilde{\mathit{f}}$ is bounded and tends towards 0.

For the control law in the driving space, define the following Lyapunov candidate function $V_2$ as

58

$$V_2\mathbf{=}\frac{1}{2}{\mathit{e}}_{\mathit{qs}}^{\text{T}}{\widehat{\mathit{I}}}_m{\mathit{e}}_{\mathit{qs}}.$$

Differentiating $V_2$ with respect to time yields

59

$${\dot{V}}_2={\mathit{e}}_{\mathit{qs}}^{\text{T}}{\widehat{\mathit{I}}}_m{\dot{\mathit{e}}}_{\mathit{qs}}.$$

Substituting Eq. (51) into Eq. (59) yields

60

$${\dot{V}}_2=-{\mathit{e}}_{\mathit{qs}}^{\text{T}}{\widehat{\mathit{F}}}_v{\mathit{e}}_{\mathit{qs}}-{\mathit{e}}_{\mathit{qs}}^{\text{T}}{\mathit{K}}_2{\mathit{e}}_{\mathit{qs}}-{\mathit{e}}_{\mathit{qs}}^{\text{T}}{\mathbf{(}\mathbf{\Delta }}_1{\mathbf{+}\mathbf{\Delta }}_2\mathbf{+}\mathit{N}\stackrel{\mathbf{~}}{\mathit{f}}\mathbf{)}$$

Since $-{\mathit{e}}_{\mathit{qs}}^{\text{T}}{\widehat{\mathit{F}}}_v{\mathit{e}}_{\mathit{qs}}$, $-{\mathit{e}}_{\mathit{qs}}^{\text{T}}{\mathit{K}}_2{\mathit{e}}_{\mathit{qs}}$ is consistently less than 0 and ${\mathbf{\Delta }}_1{\mathbf{+}\mathbf{\Delta }}_2\mathbf{+}\mathit{N}\stackrel{\mathbf{~}}{\mathit{f}}$ is bounded, choosing a sufficiently large feedback gain ${\mathit{K}}_2$ ensures that ${\dot{V}}_2$ remains less than or equal to 0, guaranteeing system stability.

Simulation and Analysis

This paper establishes the simulation model of the 8 DOF CDRM in Simulink for verification to verify the validity and accuracy of the proposed dynamics modeling and adaptive control method. The parameters of the CDRM are shown in Table 1. The upper and lower tension bounds in the optimization problem are set as follows to maintain cable pre-tension, enhance the stability of the CDRM, and prevent cable fatigue due to excessive tension.

61

$$\begin{array}{c}{f}_{max}=250\text{N},\\ f_{min}=20\text{N}.\\ \end{array}$$

Table 1. Parameters of the CDRM

Item	Value	Unit
Body size	$\mathrm{\Phi }45\times 750$	$\text{mm}$
DOF	8	DOF
Link mass	$0.25$	$\text{kg}$
Link inertia	$I_{\mathit{xx}}=-4\text{.98}\times {10}^{-4},I_{\mathit{xy}}=-8\text{.3}\times {10}^{-7},I_{\mathit{xz}}=-1\text{.4}\times {10}^{-7},I_{\mathit{yy}}=1\text{.08}\times {10}^{-4},I_{\mathit{yz}}=8\text{.3}\times {10}^{-7},I_{\mathit{zz}}=4\text{.98}\times {10}^{-4}$	$\text{kg}·{\text{m}}^2$
Link centroid distance	$x=-1.68\times {10}^{-5},y=-0\text{.0775},z=-1.68\times {10}^{-5}$	$\text{m}$
Joint mass	0.004	$\text{kg}$
Joint inertia	$I_{\mathit{xx}}=3.4\times {10}^{-7},I_{\mathit{xy}}=0,I_{\mathit{xz}}=0,I_{\mathit{yy}}=6.7\times {10}^{-7},I_{\mathit{yz}}=0,I_{\mathit{zz}}=3.4\times {10}^{-7}$	$\text{kg}·{\text{m}}^2$
Joint centroid distance	$x=0,y=0,z=0$	$\text{m}$

The simulation validation include three parts: The first part involves establishing the simulation model to verify the accuracy of the dynamic model; the second part focuses on testing the elapsed time of the program to assess the efficiency of the dynamic computations; and the third part evaluates the performance of the controller to confirm that the adaptive control method enhances control performance.

Validation of the Dynamic Model

In the simulation process, the preset joint motion trajectories are input into the simulation solver of Simulink to solve the cable tension $f(s)$, and the same joint motion trajectories imported into the dynamics model established above to calculate the cable tension $f(c)$. The dynamic model was validated by comparing $f(s)$ and $f(c)$.

A quintic polynomial is used to interpolate the joint motion trajectory.

62

$$\left\{\begin{array}{c}{\theta }_i=0.25t^3-0.1875t^3+0.0375t^5,i=1,3,5,7,\\ {\theta }_i=0.0358+0.0803t^3-0.0602t^4+0.012t^5,i=2,\\ {\theta }_i=0.125t^3-0.0938t^4+0.0188t^5,i=4,6,8.\\ \end{array}\right)$$

The variation in cable tension during the motion is shown in Figure 9, where the red curve is the cable tension $f(s)$ obtained by the simulation solver and the black curve is the cable tension $f(c)$ calculated by the dynamics model. Figure 9 shows that the simulation value of cable tension is consistent with the cable tension calculated based on the dynamic model (maximum error within 0.1% and minimum value of cable tension is 20 N) because the CDRM has a drive redundancy characteristic. In the motion process, there is always one drive cable in each joint against the motion, and the goal of the quadratic planning is to minimize the sum of the squares of the cable tension such that the cable tension against the motion tends to be limited to the minimum value of 20 N of the cable tension in Eq. (59). The accuracy of the dynamic modeling method proposed in this paper can be verified.

[See PDF for image]

Figure 9

Calculation and simulation results of the cable tension

Validation of the Dynamics Computational Efficiency

The computer configuration used for the test is a 64-bit computer with a 3.10 GHz Core i5-12500H processor and 16 GB ram. The operating system environment is Ubuntu 20.04. We use C++ to implement the dynamic model and evaluate the computation time for dynamic models with varying DOFs. The time consumptions of dynamics models with 8 to 24 DOF are presented in Table 2, which implies that the recursive method proposed in this paper is computationally very efficient and can satisfy the high-frequency requirements of the controller.

Table 2. Time consumptions of the dynamics model

DOF	$8$	$12$	$24$
Min time	$0\text{.017 ms}$	$0\text{.028 ms}$	$0\text{.083 ms}$
Ave. time	$0\text{.023 ms}$	$0\text{.036 ms}$	$0\text{.088 ms}$
Max time	$0\text{.025 ms}$	$0\text{.043 ms}$	$0\text{.096 ms}$

Validation of the Adaptive Control Method

This section provides a simulation validation of the previously described adaptive control method. The initial value of ${\widehat{\Phi }}_r$ is set to 0 to demonstrate the effectiveness of the adaptive control method [28].

The design of the control gain for the adaptive control method is as follows: ${\mathit{K}}_1\in {\mathbb{R}}^8,$${\mathit{K}}_1=\text{diag}(0.1,0.05,0.04,0.02,0.04,0.02,0.04,0.02),$${\mathit{\lambda }}_1\in {\mathbb{R}}^8,{\mathit{\lambda }}_1=\text{diag}(5,10,5,10,5,10,5,10),$${\mathit{K}}_{\mathit{d}}\in {\mathbb{R}}^{80},{\mathit{K}}_{\mathit{d}}=0.02{\mathit{I}}_{80\times 80}$, where ${\mathit{I}}_{80\times 80}$ is an id-entity matrix,${\mathit{K}}_2\in {\mathbb{R}}^{12},{\mathit{K}}_2=\text{diag}(0.1,0.1,0.1,0.1,0.1,0.1,0.02,0.02,0.02,0.02,0.02,0.02),$${\mathit{\lambda }}_2\in {\mathbb{R}}^{12},{\mathit{\lambda }}_2=\text{diag}(20,20,20,20,20,20,200,200,200,200,200,200).$

Comparing the traditional feedback control method with the adaptive control method proposed in this paper reveals that the control law for the traditional feedback control method is

63

$$\begin{array}{c}{\overline{\mathit{\tau }}\mathbf{=}\mathit{K}}_1{\mathit{e}}_s\mathbf{+}\int \mathit{KI}{\mathit{e}}_{\theta }\text{d}t,\\ \widehat{f}=inversedynamics(\overline{\tau }),\\ {\mathit{\tau }}_m{\mathbf{=}\mathit{K}}_2{\mathit{e}}_{\mathit{qs}}{\mathbf{+}\widehat{\mathit{I}}}_m{\ddot{\mathit{q}}}_r{\mathbf{+}\widehat{\mathit{F}}}_v{\dot{\mathit{q}}}_r\mathbf{+}\mathit{N}\widehat{\mathit{f}},\\ \end{array}$$where ${\mathit{K}}_1\in {\mathbb{R}}^8$ and ${\mathit{K}}_2\in {\mathbb{R}}^{12}$ represent the same feedback gain used in the adaptive control method, $\mathit{KI}\in {\mathbb{R}}^8,$$\mathit{KI}=\text{diag}(0.06,0.03,0.02,0.02,0.02,0.02,0.01,0.01).$

The proposed adaptive and traditional feedback methods both operate with a control frequency of 1000 Hz.

Figure 10 illustrates a comparison of joint motion errors between the adaptive and traditional methods. In this test, initial dynamic parameters were set to zero, which leads to significant errors in both methods during the initial trajectory tracking phase. However, the adaptive method demonstrated smoother error convergence across all joints, while the traditional method exhibited oscillations in error reduction, thereby leading to instability in the motion of the CDRM, particularly in joints ${\theta }_1,{\theta }_2,{\theta }_3,{\theta }_4,{\theta }_5$, and ${\theta }_6$. These discrepancies can be attributed to the inherent nonlinear dynamics of the CDRM during motion. The traditional method, which employs a PID feedback control law, linearly compensates for the driving torque based on motion errors; however, it fails to fully address the nonlinear demands of the CDRM. Despite this and owing to the relatively low mass and inertia of the CDRM and the use of quintic spline interpolation for trajectory planning, motion errors in the traditional method gradually decrease to a small range. However, these errors remain larger compared to those observed with the adaptive method.

[See PDF for image]

Figure 10

Comparison of joint motion errors between the adaptive method and traditional method (no load)

A load motion test was conducted in which a 0.5 kg payload was added to the last link of the CDRM to further validate the robustness of the adaptive method, thereby simulating a scenario where the manipulator carries a working tool. Figure 11 shows that the motion errors of both the adaptive and traditional methods under the additional payload are presented.

[See PDF for image]

Figure 11

Comparison of joint motion errors between the adaptive and traditional methods (0.5 kg load)

In the traditional method, the convergence process of motion errors in joints ${\theta }_1,{\theta }_2,{\theta }_3,{\theta }_4,{\theta }_5$, and ${\theta }_6$ exhibited larger and more prolonged oscillations. This is attributed to increased nonlinear dynamics of the CDRM caused by the added payload, which degraded the control performance of the traditional method based on linear feedback. In joints ${\theta }_1$ and ${\theta }_3$, significant motion error oscillations were observed between 9 and 10 s, with the absolute value of the maximum error exceeding $5\times {10}^{-3}\text{rad}$ and showing a divergent trend.

In contrast, the adaptive method incorporates an adaptive law for dynamic parameters, thereby enabling real-time adjustment of dynamic parameters based on motion errors and the dynamic regression matrix. This approach effectively addresses the nonlinear dynamic requirements of the CDRM. Unlike the integral term in PID feedback control that accumulates compensation linearly, the motion errors in the adaptive method continue to converge rapidly and stably even with the added payload at the end-effector, thereby resulting in superior control performance for the CDRM.

Maintaining sufficient cable pre-tension while minimizing fluctuations in the tension is essential to ensure the stability of the CDRM during motion. Figure 12 presents the variation in cable tension during motion for both the adaptive and traditional methods. In the traditional method, significant tension fluctuations are observed, with the tension in the 4, 7, and 8th cables falling below 20 N within the interval highlighted by the blue box. This results in the CDRM failing to meet the minimum pre-tension requirement, which introduces a risk of instability. In contrast, the adaptive method, following the convergence of dynamic parameters, utilizes dynamic feedforward to provide more accurate driving torques while minimizing error feedback in the driving space. Consequently, cable tension variations in the adaptive method are smoother, and upon dynamic parameter convergence, the minimum pre-tension requirement is met consistently.

[See PDF for image]

Figure 12

Distribution of driving torques in the adaptive control method

In conclusion, compared to traditional control methods, the adaptive control method exhibits superior control precision and robustness. The variation in cable tension is smoother, and it effectively ensures that the CDRM meets the pre-tension requirements, which enhance motion stability.

Conclusions

1. The multilevel kinematic model of CDRM is derived, and dynamics modeling is improved by adopting the screw theory and principle of virtual work to establish the mapping relationship between the cable tension and contact force with the equivalent torque; this simplifies the model under the premise of guaranteeing modeling integrity.

2. The recursive method is employed to calculate inverse dynamics, thereby avoiding large matrix operations and enhancing computational efficiency. The simulation results confirm the accuracy of the dynamic model and high computational efficiency of the recursive method.

3. An adaptive control method is designed for the dynamic uncertainty of CDRM. The control tests demonstrate that the adaptive control method enhances both control accuracy and dynamic performance in the presence of dynamic uncertainty.

4. In the applications, the CDRM is subject to kinematic errors arising from manufacturing and assembly tolerances, as well as parameter inaccuracies in transmission modules. Performing kinematic parameter calibration and transmission module parameter identification is necessary to ensure control precision.

In future research, we plan to incorporate the elastic deformation and stress of the cable into the kinematics and dynamics model of the CDRM and design more robust controllers to address the complexities and constraints of the working environment.

Acknowledgements

Not applicable

Authors' Contributions

Qinchuan Li was in charge of the whole trial; Zihao Wang wrote the manuscript; Haifeng Zhang and Tengfei Tang assisted with structure and language of the manuscript. All authors read and approved the final manuscript.

Funding

Supported by National Natural Science Foundation of China (Grant No. 52405040), Research Project of State Key Laboratory of Mechanical System and Vibration (Grant No. MSV202514).

Data availability

The datasets used or analyzed during the current study are available from the corresponding author on reasonable request

Declarations

Competing interests

The authors declare no competing financial interests.

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