Compliance Control of a Cable-Driven Space Manipulator Based on Force–Position Hybrid Drive Mode

Abstract

Multi-cable cooperative control is essential for cable-driven space manipulators to achieve in-orbit services such as fault spacecraft maintenance, fuel injection, on-orbit assembly, and orbital garbage removal. To prevent the cables from becoming slack or excessively tight, the force in each cable must be distributed appropriately. The force distribution among different cables requires real-time adjustments; otherwise, the system may become unstable. This paper proposes a compliance control method based on the force–position hybrid drive mode to address the challenges of multi-cable cooperative control. Firstly, the mapping relationship between the cable space and the joint space of the cable-driven space manipulator is established. Then, the force mapping relationship for this structure is derived. The control scheme categorizes the cables into two types: active-side cables and antagonistic-side cables. Position control and force control are implemented separately, significantly reducing the computational requirements and enhancing the overall performance of the control system. Finally, the feasibility of the proposed algorithm is demonstrated through simulations and compared with the PID control method. When tracking the same trajectory, the proposed method reduces the tracking error by 49.14% and the maximum force by 58.58% compared to the PID control method, effectively addressing the problem of force distribution in multi-rope coordinated control.

1. Introduction

With advancements in space technology, the construction of super-large spacecraft and space structure platforms has become a critical focus for future development, such as the construction of space solar power stations [1], large communication antennae [2] and space telescopes [3,4]. Utilizing space manipulators to assemble large modular structures [5] in-orbit offers a feasible, stable, and cost-effective method for constructing space structure platforms. Considering the harshness and complexity of the space environment, the space manipulator is of great significance in applications in the aviation industry [6,7].

Since the 1990s, extensive research has been conducted worldwide on space manipulator technology [8]. Prominent examples of large robotic arms used for on-orbit assembly, maintenance, and operations include the Mobile Servicing System (MSS) [9] of the International Space Station, the Japanese Experiment Module Remote Manipulator System (JEMRMS) [10], and the European Robotic Arm (ERA) [11]. These manipulators adopt traditional modular rotary joints combined with carbon fiber composite tubes, with modular joints driving the arm segments. While this structure offers high operational precision and strong torque output, it falls short of achieving the 100-m range required for assembling large space facilities. Taking the Canadian arm-2, for example [12], as the longest known space manipulator, it measures 17.6 m and weighs 1800 kg. Such designs tend to be large and heavy, with joint mass accounting for 85% to 90% of the total arm mass. This makes it challenging for the manipulator to balance the requirements for long reach, high rigidity, lightweight construction, and a high deployment-to-storage ratio [13].

In order to achieve future tasks and applications [14], it is necessary to significantly increase the operating length and flexibility of the manipulator while reducing the weight of the manipulator itself. NASA proposed a tendon-actuated lightweight in-space manipulator [15]. The structure combines the drive joint with the rotating hinge through a unique cable drive form to achieve a ${360}^{\circ }$ rotation between the lightweight arm rods, and combines the boom to achieve the arm of force between the tendon cable and the joint rotation axis [16]. This innovative design achieves greater joint torque with reduced weight, facilitating longer reach and more efficient performance in extended-range operations.

Compared with traditional manipulators, space manipulators do not need a high speed when performing tasks [17], but they need to maintain the stability and flexibility of the operating force during contact operation, which is important for completing the task [18].The compliant control of the manipulator means that the manipulator can adjust the end pose according to the size of the end force while performing tasks according to a specific trajectory. Impedance/admittance control, first proposed by Hogan [19], is one of the primary methods for achieving compliance control. Through the design of the control system, a spring–damper–mass second-order system is formed between the end effector of the manipulator and the environment to realize the mutual transformation of force and position. How to blend the active compliance from impedance control with the physical compliance from cable transmission and consider the non-negligible cable elongation is one of the difficulties in the compliance control of cable-driven manipulators. Bao et al. [20] proposed the impedance control of a rope-driven manipulator in the force–position hybrid control method. The force and torque at the operating point are estimated by detecting the load on the drive rope without the need for a dedicated sensor at the operating point. The forces and torques at the operational point are estimated by monitoring loads along their drive cable without the need for a dedicated transducer at the operational point. Li et al. [21] proposed a compliance blending framework with Cartesian space algebraic super-positioning, using well-described approaches for both impedance control and cable motion compensation. Su et al. [22] realized coordinated variable impedance control for multi-segment cable-driven manipulators. They established a dynamic model based on a pseudo-rigid-body model. The model estimated the magnitude of external forces using a momentum observer to achieve impedance control in the operational space. Data simulation and experiments prove the feasibility of the proposed method. He et al. [23] proposed a variable impedance control rate, which is based on the construction of a new Lyapunov function. Then, the operation space variable impedance control for a single segment cable-driven continuum manipulator is realized by the aid of a pseudo-rigid-body model. Some numerical simulations have also demonstrated the stability of the variable impedance control system. Mazare M, et al. [24] proposed variable impedance control in the configuration space for a modular robot manipulator in the presence of model uncertainties and external forces to solve the problem of force and position coupling, and designed the adaptive back-stepping sliding mode controller to guard against uncertainties. Liang et al. [25] proposed finite-time observer-based variable impedance control of cable-driven continuum manipulators. In order to realize the closed-loop variable impedance controller, a finite-time observer is designed to estimate the acceleration feedbacks, which can avoid the difficulty of directly sensing the interaction forces and shows excellent robust stability in noisy environments. Zhang et al. [26] designed a convolutional dynamic-jerk-planning algorithm for impedance control of variable-stiffness cable-driven manipulators, achieving effective motion planning and stiffness control of the cable-driven manipulators. Li et al. [27] proposed a force-sensing algorithm and compliance control framework for a cable-driven redundant manipulator without a six-axis force/torque sensor. The experimental results indicated that the force-sensing accuracy exceeded 95%, whereas the compliance controller demonstrated outstanding compliant behavior in human–robot interaction tasks. Li et al. [28] proposed a hybrid impedance control method for realizing robust impedance control and friction compensation for cable-driven parallel robots simultaneously.

The optimal control of the tension is also a pivotal concern in the impedance control of the cable-driven manipulator. Zhou et al. [29] proposed an adaptive adjustment inertia weight particle swarm optimization algorithm for the multi-objective optimal design of cable-driven parallel robots. Sun et al. [30] proposed a compliant trajectory tracking method for a portable cable-driven robot. A real-time force redistribution algorithm was introduced into its auxiliary tracking controller to ensure that cable tensions remained continuous, thereby avoiding sudden discrete jumps. The results demonstrated the feasibility of the force redistribution without compromising the trajectory tracking performance. Chen et al. [31] proposed an impedance control method based on cable tension optimization for the cable-driven snake-like manipulator. This method achieves both stiffness and load capacity while maintaining good compliance. However, the current optimization algorithms are mainly for cable-driven parallel robots and cable-driven snake-like manipulators with limited attention given to the cable-driven manipulator of the structure proposed in this paper.

In this paper, compliance control based on the force–position hybrid drive mode is proposed to solve the problem of the force distribution of multi-cable cooperative control for a space manipulator composed of drive cables, a rotary joint, links, and a spreader. The rest of this article is organized as follows. Section 2 establishes the mapping relationship between the cable space and the joint space of the manipulator and the force mapping relationship. In Section 3, a compliance control framework based on the force–position hybrid drive mode under the admittance model is proposed. Important simulations and prototype experiments are conducted in Section 4. Finally, Section 5 concludes the entire paper.

2. Modeling

2.1. Description

As shown in Figure 1, the cable-driven space manipulator system consists of a satellite platform and a cable-driven manipulator with n degrees of freedom.

To facilitate subsequent modeling, the system is simplified. The manipulator in Figure 1 is composed of multiple single-joint modules shown in Figure 2. Each joint module is driven by four motors, which control four cables to enable motion. The system is composed of five main components: a cable-driven control device, links, a spreader, drive cables, and a symmetrical rotary joint.The link and spreader form the primary structure of the manipulator. The length of the link determines the operational range. The drive cables connect the link, spreader, and rotary joints, allowing for the removal of the large high-torque motor gearbox traditionally located at the joint axis in conventional manipulators.

The cable-driven control device serves as the power source for the entire manipulator. It drives the cables by rotating drums via motors, enabling joint movement throughout the manipulator. Each cable is individually controlled by a dedicated motor. A newly designed symmetrical rotary mechanism is implemented at the joints, featuring two rotational axes, which allows for the 360-degree rotation of the joints.

2.2. Joint-to-Cable Kinematics

To facilitate subsequent modeling, the simplified model of the multiple single-joint module is shown in Figure 3. $L^i$ is the distance from the center of the drum to the center of the joint rotation axis. $H^i$ and $D^i$ are the single-sided length and width of the spreader, respectively. $S^i$ is the distance from the rotational axis to the corresponding fixed end of the spreader. ${\phi }^i$ represents the angle between $S^i$ and the centerline of the spreader. These variables are all determined by the mechanical dimensions of the manipulator itself. $l^i={[l_1^i,l_2^i,l_3^i,l_4^i]}^{\mathrm{T}}$ is the length of the cable for the i-th joint. ${\theta }^i={[{\theta }_1^i,{\theta }_2^i]}^{\mathrm{T}}$ is the i-th joint rotation angle of the manipulator.

The following assumptions are made during the modeling process of this paper:

• Assume that the manipulator only moves in the $x-o-y$ plane, and the small motion in the z-axis direction is neglected.
• Assume that there is no gravity during the movement of the manipulator.
• Assume that the tendons on both sides of the suspender are fixed in the same horizontal plane, ignoring the position deviation of the tendon’s fixed position.

Based on the geometric relationships in Figure 3, the formulas for calculating $S^i$ and ${\phi }^i$ are as follows:

$$\left\{\begin{array}{c}{S}^i=\sqrt{{D^i}^2+{H^i}^2}\hfill \\ {\phi }^i=arctan\left({\displaystyle \frac{H^i}{D^i}}\right)\hfill \end{array}\right.$$

(1)

From this, a mapping relationship can be established between the length of cables and the angle of rotation of the joint:

$$\left\{\begin{array}{c}{l}_1^i=\sqrt{{S^i}^2+{L^i}^2+2S^i{L}^{i}cos({\phi }^i+{\theta }_1^i)}\hfill \\ l_2^i=\sqrt{{S^i}^2+{L^i}^2+2S^i{L}^{i}cos({\phi }^i-{\theta }_1^i)}\hfill \\ l_3^i=\sqrt{{S^i}^2+{L^i}^2+2S^i{L}^{i}cos({\phi }^i+{\theta }_2^i)}\hfill \\ l_4^i=\sqrt{{S^i}^2+{L^i}^2+2S^i{L}^{i}cos({\phi }^i-{\theta }_2^i)}\hfill \end{array}\right.$$

(2)

Differentiating both sides of Equation (2) yields

$${[{\dot{l}}_1^i,{\dot{l}}_2^i,{\dot{l}}_3^i,{\dot{l}}_4^i]}^{\mathrm{T}}=J_q^i{[{\dot{\theta }}_1^i,{\dot{\theta }}_2^i]}^{\mathrm{T},}$$

(3)

where the Jacobian matrix of the joint module can be expressed as

$$J_q^i=\left[\begin{array}{c}{\displaystyle \frac{-S^i{L}^{i}sin({\phi }^i+{\theta }_1^i)}{l_1^i}}\\ {\displaystyle \frac{S^i{L}^{i}sin({\phi }^i-{\theta }_1^i)}{l_2^i}}\\ 0\\ 0\end{array}\begin{array}{c}0\\ 0\\ {\displaystyle \frac{-S^i{L}^{i}sin({\phi }^i+{\theta }_2^i)}{l_3^i}}\\ {\displaystyle \frac{S^i{L}^{i}sin({\phi }^i-{\theta }_2^i)}{l_4^i}}\end{array}\right]$$

(4)

2.3. Force Mapping

The manipulator achieves joint rotation through cables. One side of the cable is connected to a motor, which adjusts the cable length through rotation; the other side is connected to the spreader, pulling it to rotate around the axis. Based on the principle of virtual work, the relationship between the joint torque ${\tau }^i$ and cable force $f^i$ can be established as follows:

$${\tau }^i={J_q^i}^{\mathrm{T}}{f}^i$$

(5)

The specific expression for ${J_q^i}^{\mathrm{T}}$ is given by Equation (4). When the deformation of the cable is small, it can be treated as being linear elastic, and its stress state can be modeled using Hooke’s law:

$$f^i=K_c^i\Delta l^i$$

(6)

In Equation (6), $K_c^i$ represents the cable stiffness, expressed as follows:

$$K_c^i={\displaystyle \frac{ES}{\Delta l^i}}$$

(7)

In Equation (7), E represents the Young’s modulus of the cable, S represents the cross-sectional area of the cable, and $\Delta l^i$ represents the cable’s deformation. Equations (5)–(7) establish the relationship between the cable force and joint torque.

Based on the principle of virtual work, the relationship between the joint torque ${\tau }^i$ and end force $F^i$ can also be established as follows:

$${\tau }^i={J_m^i}^{\mathrm{T}}{F}^i$$

(8)

where $F^i={\left[\begin{array}{cc}{F}_x^i& F_y^i\end{array}\right]}^T$ is the end force, $J_m^i$ is the Jacobian matrix relating the joint angles to the end Cartesian space, and its specific expression is as follows:

$$J_m^i=\left[\begin{array}{cc}-2D^{i}sin{\theta }_1^i-L^{i}sin({\theta }_1^i+{\theta }_2^i)& -L^{i}sin({\theta }_1^i+{\theta }_2^i)\\ 2D^{i}cos{\theta }_1^i+L^{i}cos({\theta }_1^i+{\theta }_2^i)& L^{i}cos({\theta }_1^i+{\theta }_2^i)\end{array}\right]$$

(9)

Equations (8) and (9) establish the relationship between the end force and joint torque.

The relationship between the motor output torque ${\tau }_m^i$ and the controlled cable force is as follows:

$$r{f}^i={\tau }_m^i\ast N\ast {\eta }_1$$

(10)

where r is the radius of the motor reducer output shaft, N is the reduction ratio, and ${\eta }_1$ is the transmission efficiency of the motor reducer. The relationship between the motor output torque and the motor control current $I^i$ can be expressed as follows:

$${\tau }_m^i=K_i\ast I^i-\epsilon$$

(11)

where $K_i$ is the current torque constant, and $\epsilon$ is a nonlinear term that includes inertial forces, damping forces, and other factors. Therefore, the relationship between the current and cable tension can be derived as follows:

$$f^i={\displaystyle \frac{K_i\ast N\ast {\eta }_1}{r}}{I}^i-{\displaystyle \frac{N\ast {\eta }_1}{r}}\epsilon$$

(12)

Equations (10)–(12) describe the mapping between the cable force and the motor current.

3. Compliance Control

Due to the nature of cables as weakly constrained flexible transmission media, friction on pulleys and guide wheels can result in tension loss, which negatively impacts the positional accuracy and force transmission efficiency. When transmitting motion and force through cables, it is essential to consider both the cable length and tension.

The definitions of the cable impedance coefficients are as follows: let $K_l$ be the stiffness coefficient, $B_l$ be the damping coefficient, and $M_l$ be the mass coefficient. The cable admittance model can be described as follows:

$$f-f_d=K_l(l-l_d)+B_l(\dot{l}-{\dot{l}}_d)+M_l(\ddot{l}-{\ddot{l}}_d)$$

(13)

where $f_d$ and f represent the desired cable force and actual cable force, respectively. $l_d$ and l represent the desired cable length and the actual cable length, respectively. ${\dot{l}}_d$ and $\dot{l}$ represent the desired cable velocity and the actual cable velocity, respectively. ${\ddot{l}}_d$ and $\ddot{l}$ represent the desired cable acceleration and the actual cable acceleration, respectively.

The number of pulses per motor revolution is denoted as n. The relationship between the motor pulse count $q_c$ and the motor angle q is as follows:

$$q=q_c{\displaystyle \frac{360}{n}}$$

(14)

The relationship between the motor pulse count $q_c$ and the cable length l is given by

$$l={\displaystyle \frac{q}{N}}{\displaystyle \frac{\pi }{180}}r=q_c{\displaystyle \frac{360}{nN}}{\displaystyle \frac{\pi }{180}}r=q_c{\displaystyle \frac{2\pi r}{nN}}$$

(15)

combing Equation (10) and Equation (8), we can obtain

$$f_d-f={\displaystyle \frac{2\pi r}{nN}}(K_l(q_{cd}-q_c)+B_l({\dot{q}}_{cd}-{\dot{q}}_c)+M_l({\ddot{q}}_{cd}-{\ddot{q}}_c))$$

(16)

Equations (13)–(16) establish the mapping between the motor rotation angle and cable force.

According to the following motor dynamics equation, the motor output torque $\tau$, motor angular velocity $\dot{q}$, and motor angular acceleration $\ddot{q}$ are planned to determine the required motor current:

$$K_{i}i=J_m\ddot{q}+b_m\dot{q}+\tau$$

(17)

where $J_m$ is the moment of inertia, and $b_m$ is the viscous damping coefficient. The control voltage $V_m$ is calculated based on the required current, current change rate, and motor rotational speed:

$$L_m{\displaystyle \frac{di}{dt}}+R_{m}i+K_e\dot{q}=V_m$$

(18)

where $L_m$ is the armature inductance, $R_m$ is the armature resistance, and $K_e$ is the back electromotive force (EMF) constant.

In summary, the control scheme for the manipulator is shown in Figure 4. The cable on the side that retracts during manipulator movement is defined as the active-side cable, while the side that releases the cable is the antagonistic-side cable. When the direction of the manipulator’s movement changes, the retraction and release states of the cables on both sides are adjusted, and the control mode for each side is switched accordingly.

In the control process, the desired control variable is the joint angle, while the controllable inputs are the motor angle, speed, and current. The ideal control process involves position control for the active-side cables and tension control for the antagonistic-side cables. The active-side cables enable precise cable length control, ensuring joint angle accuracy, while the antagonistic-side cables behave similarly to a spring. When the antagonistic-side cables are stretched, they act like a stretched spring, increasing the cable tension. If there is an overshoot in the angle, the active cable will release some length, reducing the tension in the antagonistic cable, thus achieving a spring-like effect. In essence, one side maintains position control, while the other maintains tension control, keeping the cable taut at all times.

4. Simulation and Experiment

4.1. Simulation Environment

As shown in Figure 5, we use Simscape multibody on the Matlab platform to build the manipulator, which allows us to define structural parameters such as dimensions, mass, and density, as well as to configure rotational joints and add external sensors. The specific physical parameters of the manipulator are shown in

Table 1. Physical structure parameter table of the manipulator.

Physicial Structure	Parameters
Manipulator Mass	$M_L$ = 5.804 kg
Spreader Mass	$M_S$ = 1.368 kg
Manipulator Length	L = 1.952 m
Spreader Height	h = 0.37 m
Motor Reducer Radius	r = 0.03 m

, with the simulation model parameters set according to the physical model.

By cooperating with other components of the Simulink, we built a manipulator simulation system in Matlab, as shown in Figure 6.

4.2. Trajectory Tracking Simulation

The initial state of the manipulator is $\mathsf{\Theta }={[0,0]}^{\mathrm{T}}$ and $\dot{\mathsf{\Theta }}={[0,0]}^{\mathrm{T}}$. The expected trajectory of the joint is ${\mathsf{\Theta }}_d={[{\displaystyle \frac{\pi }{6}}sin\left({\displaystyle \frac{\pi t}{5}}\right),{\displaystyle \frac{\pi }{6}}sin\left({\displaystyle \frac{\pi t}{5}}\right)]}^{\mathrm{T}}$. The simulation results are shown below.

We applied the PID control method to track the set trajectory for this object. The following presents a comparison of the proposed admittance control (AC) method with the PID control method, focusing on the joint angle error and the force in cables 1 and 3.

Figure 7 depicts the tracking trajectory and error curves. Figure 8 displays the variation in the cable length and tension, showing that the cable tension remains above 0 N when following the upper trajectory. Figure 9 displays the joint angular velocity and the joint torque. In Figure 10, the maximum tracking error for the PID control method is $19.71\times {10}^{-3}$ rad, while the maximum tracking error for the AC method is $10.02\times {10}^{-3}$ rad, representing a reduction of 49.14%. In Figure 11, the maximum force for the PID control method is 1251.2 N, while the maximum force for the AC method is 518.3 N, a decrease of 58.58%. These results demonstrate that the control method proposed in this paper outperforms the PID control method.

4.3. Zero-Force Drag Simulation

When the joint flexibility is high, the manipulator can easily respond to external forces, achieving the effect of zero-force drag. To evaluate this, we conduct a zero-force drag simulation, applying a specified drag torque:

$$\tau =\left\{\begin{array}{cc}0& 0<\mathrm{t}\le 1\\ 37.5(t-1)& 1<\mathrm{t}\le 2\\ 37.5& 2<\mathrm{t}\le 3\\ -37.5(t-4)& 3<\mathrm{t}\le 4\\ 0& 4<\mathrm{t}\le 6\\ -37.5(t-6)& 6<\mathrm{t}\le 7\\ -37.5& 7<\mathrm{t}\le 8\\ 37.5(t-9)& 8<\mathrm{t}\le 9\\ 0& \mathrm{t}\ge 9\end{array}\right.$$

(19)

The simulation results are as follows:

Figure 12 illustrates the given drag torque and the real-time torque of the manipulator joint. Figure 13 presents the trajectory and tracking error of the manipulator during the dragging process and the maximum tracking error is $3.16\times {10}^{-3}$ rad. Figure 14 depicts the variations in the cable length and tension of the manipulator during the tracking process and the maximum cable force is 368.8 N. These results demonstrate that the compliance control method proposed in this chapter can adjust the joint torque in real time to counteract external forces and enable the manipulator’s dragging functionality.

4.4. Experiment

We constructed a prototype of the manipulator system (Figure 15), installing rotary encoders on the joint rotation axes to measure joint angles and tension sensors on each cable to provide real-time feedback on cable tension. Based on this prototype, we conducted a zero-force dragging experiment to validate the effectiveness of the control algorithm.

We set the desired torque to 0 Nm and applied a joint admittance model in the algorithm to convert the difference between the desired joint torque and the actual contact torque into a joint angle offset. Then, through the joint module’s mapping relationship between the joint angle, cable length, and motor angle, the joint angle offset was converted into a motor angle control variable, achieving zero-force dragging.

Figure 16 depicts the estimated joint torque and joint angle increment. Figure 17 depicts the motor position and the cable force. Figure 18 displays the motor current and the motor velocity. The experimental results indicate that within a certain range of joint contact forces, the control system can effectively estimate the joint torque and convert it into a joint offset. During the zero-force dragging process, multiple cables remained taut, demonstrating the feasibility of the algorithm.

5. Conclusions

In this paper, a kinematic modeling method and a compliance control method are proposed for the cable-driven space manipulator. This provides an innovative solution to solve the contradiction between the position accuracy and motion stability of traditional control methods. By leveraging a force–position hybrid drive mode, the proposed method achieves joint torque perception through compliance control and dynamically adjusts the force output. Additionally, an admittance model enables the manipulator to adapt to external environmental changes, striking a balance between force flexibility and position accuracy.

The simulation and experimental results demonstrate that the designed compliance control method effectively maintains a constant preset force in the antagonistic cables, preventing manipulator instability caused by slack or excessive tension. This ensures stable force output during complex operations. This method also allows for real-time adjustments to the manipulator’s force output in response to changes in the assembly environment, facilitating precise docking and assembly tasks. By reducing the risk of rigid impact-induced component damage, the approach significantly enhances the reliability of complex space operations, making it particularly suitable for assembling precision spacecraft.

The current research focuses on simulations and experiments with a single-degree-of-freedom module. The contact friction and inelastic deformation of the actual cables are not considered in the simulation environment, which will be incorporated in future modeling efforts. In complex tasks, the compliant control of multi-degree-of-freedom manipulators becomes particularly critical. However, the simulation system and experiments presented in this paper have not yet validated the performance of the multi-degree-of-freedom system. Future research and experiments will be conducted to assess the applicability of the compliant control method in more realistic task environments.