Parallel robotic automated docking method for realizing space segment assembly

Abstract

Assembling large, heavy space segments presents a significant challenge in aerospace engine production. Rigid collisions often occur during the docking process, impacting the precision and quality of engine assembly. Traditional manual docking depends on workers’ experience to prevent collisions, but it is labor-intensive and low in productivity, making it impractical. Parallel robots, known for their high precision and heavy load capacity, are widely used in precision assembly under heavy load conditions. Therefore, automated docking methods using parallel robots capable of avoiding rigid collisions have emerged as an excellent solution to these issues. This paper presents a framework for easy implementation in practical production. The Stewart parallel robot facilitates automatic docking of heavy aerospace components without rigid collisions. Fractional-order variable damping admittance control is proposed, allowing the robot to dynamically adjust the assembly trajectory based on real-time interaction forces, thus preventing rigid collisions during docking. Additionally, adaptive robust sliding mode control is developed, enhancing the robot’s tracking accuracy for desired poses and making it suitable for high-precision assembly.

Introduction

Segment docking is crucial in assembling large aerospace equipment, directly affecting the accuracy and stability of aerospace products^1,2. Aerospace segments are typically large and heavy, requiring docking without rigid collisions to prevent damage to segments and internal seals^3,4. Traditional manual docking requires high precision from operators and is labor-intensive, rendering it impractical for most applications⁵. Parallel robots provide strong load capacity, precise position control, and low end-effector inertia, making them ideal for assembling large or heavy components^6,7. Thus, using parallel robots for efficient automated segment docking is crucial.

Assembling aerospace equipment requires high precision, often necessitating interference fits for segment assembly^8,9. Consequently, conventional automated docking methods, which involve robots following preplanned trajectories, often struggle to prevent rigid collisions during docking. Compliant control effectively helps avoid rigid collisions during docking. Robot compliance is its ability to respond to interactive forces from the environment¹⁰. In constrained environments, robots require high stiffness for positional accuracy and compliance to avoid collisions—an enduring challenge in research^11,12. To balance the demands of stiffness and compliance, researchers have investigated active compliant control technologies. Wei and Wang proposed adaptive hybrid force/position control for parallel robots in task execution¹³. They validated the adaptive controller’s performance through experiments in rigid and compliant environments. Kalani and Akbarzadeh addressed dynamic position and force control in chewing robots by developing kinematic and dynamic models for a 6RSS parallel robot. They proposed an impedance control scheme for compliant control in parallel robots¹⁴. Additionally, Xu et al. proposed an impedance-based robot-assisted iterative learning sliding mode control scheme to estimate invariant dynamic parameters. This significantly enhances robot flexibility^15,16. Admittance control suits scenarios that involve external forces and require flexibility, while impedance control is ideal for tasks needing precise force or position control. In precision assembly, the robot must adjust the applied force to ensure quality. Admittance control enables the robot to adapt to external forces by adjusting the position and speed of the end effector, making it especially suitable for segmented docking of parallel robots.

During the docking of the heavy space segment, the parallel robot frequently encounters unknown interaction forces, posing significant external disturbances to its closed-loop control system¹⁷. Additionally, the Coriolis and friction terms cannot be determined after carrying the space segment, rendering the system highly nonlinear and uncertain. Therefore, sliding mode control is the preferred method for such systems. Zhang and Qi et al. propose a hybrid adaptive iterative learning sliding mode control (AILS) approach to reduce the effects of periodic disturbances in the system and ensure a fast response¹⁸. Mostafa Nazari and Hossein Darvishi Nejad et al. proposed an adaptive time-varying sliding mode control (ATVSMC) for desalination systems that reduced the overshooting and stabilization times of bypass flow rates by about 3.5% and 7.6%, respectively¹⁹. Jiang Tao and Yan Yan et al. used adaptive control gain and sliding variables as inputs to reconstruct the hyperbolic tangent function, solving the trajectory tracking problem of mechanical systems with parameter uncertainties, external disturbances and actuator failures²⁰. Because of the existence of sliding surface in sliding mode control, the flutter problem is inevitable in sliding mode control. To solve this problem, Zhang and Ma et al. proposed a fractional-order sliding mode control method (FOSMC) to reduce the disadvantage of sliding mode control flutter problem to a certain extent²¹. Jing and Zhang et al. study an improved adaptive supertwisted sliding mode control (ASTSMC) for robot manipulators with input saturation to improve the convergence rate while avoiding singular perturbations²². Yanfen Song and Zijun Li et al. developed an optimized leader-follower consensus control for multi-robot manipulator systems by combining sliding mode control (SMC) and reinforcement learning (RL)²³. Fatma Abdelhedi and Rimjallouli Khlif et al. proposed a new fractional order sliding mode control (F-FOSMC) strategy based on fuzzy logic parameter tuning, which minimizes the required motion energy consumption while improving the accuracy and rapidness of the transient response^24,25.

To address the slow response and insufficient robustness of traditional motion control to unknown interaction forces during docking, this paper introduces an adjustment coefficient to traditional admittance control to enhance the robot’s compliance and integrates fractional-order control to accelerate its response to external forces. To enhance the robustness of motion control against unknown forces, an adaptive robust sliding mode control method is proposed to improve the tracking accuracy of the parallel robot. This method enables the parallel robot to adapt to external interaction forces and avoid rigid collisions, thereby facilitating high-quality automatic docking of space segments.

The remainder of the paper is structured as follows: Sect. 2 provides an overview of the practical application scenarios addressed. Section 3 describes the control algorithms used in the automated segment docking method. Section 4 presents the experimental results and discussions. Finally, Sect. 5 offers the conclusions drawn from the study.

Description of application scenarios

This paper proposes a framework in which a Stewart parallel robot adjusts its motion trajectory in real time using six-dimensional force feedback to prevent rigid collisions between spacecraft segments and accomplish automated docking tasks. As shown in Fig. 1, the system includes a fixed segment mounted on a support structure, a six-axis force sensor on the end effector, and a movable segment for docking. The active compliant control algorithm continuously adjusts the predefined assembly trajectory using real-time six-dimensional force information, providing corrective motion profiles for automated docking. Specifically, in the docking direction, where movement freedom is constrained, the robot follows the predetermined assembly trajectory without compliant control. In the other five degrees of freedom, the robot dynamically responds to interaction forces while executing the planned assembly trajectory.

As shown in Fig. 2, the automated segment docking system consists of components including the Stewart parallel robot, fixed support structure, movable segment, and fixed segment. The pivotal platform of the Stewart parallel robot controls and adjusts the posture of the movable segment. It includes an active platform, fixed platform, piston rods for each branch, servo motors, drivers, force sensors, and an industrial control computer. Each component of the docking system defines target positions and coordinate systems. As depicted in Figs. 1 and 2, axial positioning pin holes help determine the completion of the segment assembly docking task. An overview of coordinate systems and key definitions is provided as follows:

• (1) Earth coordinate system (reference), denoted as: ${O_0} - {X_0}{Y_0}{Z_0}$;

• (2) Static coordinate system (connected to the static platform of the parallel robot), denoted as: ${O_1} - {X_1}{Y_1}{Z_1}$;

• (3) Mobile coordinate system (connected to the mobile platform of the parallel robot), denoted as: ${O_2} - {X_2}{Y_2}{Z_2}$;

• (4) Axial positioning pin holes of the movable segment: ${H_1}$,${H_2}$,${H_3}$,${H_4}$,${H_5}$,${H_6}$;

• (5) Axial positioning pin holes of the fixed segment: ${G_1}$,${G_2}$,${G_3}$,${G_4}$,${G_5}$,${G_6}$.

Fig. 1

Schematic diagram of horizontal docking of segments.

Fig. 2

Schematic of the distribution of axial locating pin holes on the end faces of the compartments.

Equivalent alternative model of robots carrying multiple objects

The force information collected by the six-axis force sensor does not directly represent the interaction forces between the end faces of the spacecraft segments. To calculate the interaction forces between these end faces using the force information, all components on the end effector of the Stewart parallel robot must be modeled. Because the components on the end effector are tightly interconnected, the modeling approach treats them as a unified whole. This involves calculating the mass, center of mass, moment of inertia, and other physical characteristics. Ultimately, a cuboid mass block with equivalent physical properties replaces these components in the modeling process. The schematic representation of this equivalent model is shown in (Fig. 3).

Fig. 3

Schematic representation of an equivalent alternative model of all components on the end-effector.

All components (load-bearing tool, six-axis force sensor, movable segment) and the equivalent mass block have a mass ${{\text{m}}_{\text{B}}}$= 52.68 kg, with dimensions $lm=(l{m_x},l{m_y},l{m_z})$ and inertial properties $IB=(I{B_{xx}},I{B_{yy}},I{B_{zz}})$ as shown in (Table 1).

Table 1 Mass and geometric parameters of load-bearing props.

Conversion of the six-dimensional force transducer to a force system on the end face of the cabin section

During the docking process, the interaction forces between the end faces of the segments are irregularly distributed around the circumference. However, this complex force system can be simplified to act at the center of the circle. To correspond with the six degrees of freedom of the six-axis force sensor, this complex force system is simplified here into forces in three directions $({F_x},{F_y},{F_z})$and moments around these three directions $({T_X},{T_Y},{T_Z})$. A simplified schematic is shown in (Fig. 4).

Fig. 4

Schematic diagram of the interaction forces distributed circumferentially around the end face of the segment simplified towards the upper end of the six-dimensional force transducer.

The arbitrary force system on the segment’s end face is projected onto three coordinate axes. Each force along the coordinate axes is then simplified under specific conditions to converge at point O, resulting in a six-dimensional force representation of the arbitrary force system on the segment’s end face, denoted as O.This results in a six-dimensional spatial force system that aligns with the structure of a six-axis force sensor. Subsequently, applying the principle of force translation, the force system from the six-axis sensor is transformed into the spatial force system of the movable segment. Notably, when transforming moments, additional moments caused by translational forces occur only when the projection of the axis perpendicular to direction $OO^{\prime}$ is also perpendicular to the plane formed by the translational force direction. The following conditions must be satisfied:

$${T_{{k_a}}}={F_i} \cdot O{O^{\prime}_j} \ne 0 \Leftrightarrow {n_k} \times {n_i} \times {n_j}=0(i=x,y,z;j=x,y,z;k=x,y,z)$$

(1)

where: ${T_{{k_a}}}$represents the additional moment caused by translational forces,${F_i}$represents any force in force system O, $O{O^{\prime}_j}$represents the projection of $OO^{\prime}$in any direction of force system O,${n_k}$ and ${n_i}$respectively denote unit direction vectors of ${T_{{k_a}}}$,${F_i}$and$O{O^{\prime}_j}$.

The formula for transforming the spatial force system of the movable segment into the six-axis force sensor system can be derived from the method described above:

$${F^{\prime}_X}={F_X}$$

(2)

$${F^{\prime}_Y}={F_Y}$$

(3)

$${F^{\prime}_Z}={F_Z}$$

(4)

$${T^{\prime}_X}={T_X}+{F_Y} \cdot O{O^{\prime}_Z}+{F_Z} \cdot O{O^{\prime}_Y}$$

(5)

$${T^{\prime}_Y}={T_Y}+{F_X} \cdot O{O^{\prime}_Z}+{F_Z} \cdot O{O^{\prime}_X}$$

(6)

$${T^{\prime}_Z}={T_Z}+{F_X} \cdot O{O^{\prime}_Y}+{F_Y} \cdot O{O^{\prime}_X}$$

(7)

Control system design

This paper proposes fractional-order variable damping admittance control (FVDAC) and adaptive robust sliding mode control (ARSMC) to enable rapid handling of interaction forces and precise, stable dynamic motion control of robots for automated docking of aerospace segments. The proposed control schemes are illustrated in (Fig. 5).

FVDAC rapidly adjusts the robot’s motion in real time during docking to avoid rigid collisions between segments. During the docking process, this controller receives interaction forces and real-time position data from force sensors and the Stewart parallel platform. It adjusts the predefined assembly trajectory $({x_d},{\dot {x}_d},{\ddot {x}_d})$in Cartesian coordinates in real time, generating an ideal compliant motion plan. It incorporates a variable damping matrix that adjusts damping parameters in real time based on interaction forces, enhancing robot compliance.

To prevent unknown interaction forces during collaboration from impacting the motion accuracy of the Stewart parallel robot and, consequently, assembly precision, this paper proposes adaptive robust sliding mode control (ARSMC). This approach ensures precise and stable dynamic control of the robot. The controller generates stable control laws to maintain high consistency between the motion state of the robot end-effector $(x,\dot {x},\ddot {x})$and the motion plan designed by FVDAC. Additionally, a sigmoid function is used to adjust the sliding mode surface to suppress the jitter phenomenon common in sliding mode control.

Fig. 5

Block diagram of the control system, which consists of a fractional-order variable damping admittance control and an adaptive robust sliding mode control. The former produces a compliant trajectory.

Fractional-order variable damping admittance control

Admittance control endows the robot system with spring-like compliance and adaptability, allowing dynamic adjustment of the motion trajectory based on external forces²⁶, as illustrated in (Fig. 6). This section enhances traditional admittance control by introducing a variable damping matrix and fractional-order variable damping admittance control. These improvements enhance the flexibility and responsiveness of admittance control to interaction forces, enabling fully automated segmental docking without rigid collisions.

Fig. 6

Admittance control acting on a parallel platform.

The standard equation of classical admittance control is:

$${{\mathbf{M}}_D}({\mathbf{\ddot {x}}} - {{\mathbf{\ddot {x}}}_{\mathbf{c}}})+{{\mathbf{K}}_D}({\mathbf{\dot {x}}} - {{\mathbf{\dot {x}}}_c})+{{\mathbf{K}}_P}({\mathbf{x}} - {{\mathbf{x}}_c})={{\mathbf{Q}}_i}$$

(8)

Where:$({\mathbf{x,\dot {x},\ddot {x}}})$represents the robot’s motion state when there is no external force acting on it,${{\mathbf{M}}_D}$,${{\mathbf{K}}_D}$, and${{\mathbf{K}}_P}$refer to the inertia, damping, and stiffness matrices in cartesian co-ordinates,${{\mathbf{Q}}_i}$represents the inter-segment interaction forces computed after transformation.

The motion output of the robot under admittance control largely depends on the admittance parameters. Among the admittance parameters, stiffness governs response displacement, damping influences velocity, and inertia affects acceleration. Stiffness is set based on the ideal response displacement, and since large accelerations do not occur during docking, both stiffness and inertia remain fixed. Ideally, as interaction force increases, the robot should adjust its motion more rapidly, necessitating a reduction in damping. To implement this concept, we propose variable damping admittance control, which dynamically adjusts the damping matrix based on interaction forces between segments during docking. To implement this concept, we propose variable damping admittance control, which dynamically adjusts the damping matrix based on interaction forces between segments during docking. The proposed damping function is as follows:

$$\alpha =tanh({{\mathbf{Q}}_i}^{T}{{\mathbf{Q}}_i})$$

(9)

The hyperbolic tangent function is chosen for constructing adjustment coefficients for the following reasons:

• (1) Symmetry around the x = 0 axis: The hyperbolic tangent function is symmetric about the x = 0 axis, ensuring it affects forces equally in both positive and negative directions.

• (2) Range limitation [0,1]: The function’s output is constrained within the range [0,1]. When no force is applied, the function value is 0; as the internal force increases, the value approaches 1. This normalization facilitates subsequent adjustments to the damping matrix.

• (3) Steep slope near x = 0: The function exhibits a steep slope near x = 0, allowing for rapid changes in the function value when forces are applied.

In summary, these properties make the hyperbolic tangent function suitable for adjusting coefficients in variable damping impedance control, ensuring an effective and symmetric response to forces in all directions while providing rapid responsiveness to changes. Based on the above adjustment coefficient, the damping coefficient in the impedance control is adjusted as follows:

$${{\mathbf{K}}_D}(\alpha )=(1 - \alpha ){{\mathbf{K}}_{{D_c}}}$$

(10)

where: ${{\mathbf{K}}_{{D_c}}}$is a fixed damping matrix, ${{\mathbf{K}}_D}(\alpha )$is the damping matrix adjusted based on the value of variable$\alpha$.

Although the previously described variable damping strategy helps the robot achieve compliance during collaboration, the controller exhibits transient overshoot and slow tracking response. The inherent memory characteristic of fractional-order derivatives can improve the transient response of the control system, thereby enhancing the performance of the closed-loop system. Various mathematical definitions of fractional-order derivatives exist²⁴. This paper focuses on solving fractional-order differential equations using the Laplace transform, thus employing the Caputo definition. The Caputo derivative of any order is computed using Formula (11):

$$_{0}^{C}D_{t}^{p}f(t)\mathop =\limits^{\Delta } \frac{1}{{\Gamma (n - p)}}{\int_{0}^{t} {(t - \tau )} ^{n - p - 1}}{f^{(n)}}(\tau )d\tau ,(0 \leqslant n - 1<p<n,n \in N)$$

(11)

The Laplace transform corresponding to the Caputo derivative of any order is given by:

$$\begin{gathered} L\{ _{0}^{C}D_{t}^{p}f(t);s\} =L\{ _{0}^{{RL}}D_{t}^{{p - n}}{f^{(n)}}(t);s\} ={s^{p - n}}G(s) \hfill \\ ={s^{p - n}}({s^n}Q(s) - \sum\limits_{{k=0}}^{{n - 1}} {{s^{n - k - 1}}{x^{(k)}}_{e}(0)} ) \hfill \\ ={s^p}Q(s) - \sum\limits_{{k=0}}^{{n - 1}} {{s^{p - k - 1}}{x^{(k)}}_{e}(0)} ,(n - 1<p<n) \hfill \\ \end{gathered}$$

(12)

Replacing the integer order in Eq. (8) with a fractional order yields a fractional-order variable damped admittance control (FVDAC) on a single degree of freedom:

$${M_D}{}^{C}D_{t}^{\beta }({x_e})+{K_D}(\alpha ){}^{C}D_{t}^{\gamma }({x_e})={Q_i},(1<\beta <2,0<\gamma <1)$$

(13)

Simultaneous Laplace transformation of both sides of Eq. (13) yields:

$${M_D}{s^p}{X_e}(s)+{K_D}(\alpha ){s^q}{X_e}(s)={M_D}{s^{p - 1}}{x_e}(0)+{M_D}{s^{p - 2}}{\dot {x}_e}(0)+{K_D}(\alpha ){s^{q - 1}}{x_e}(0)+Q(s)$$

(14)

The collation leads to:

$$\begin{gathered} {X_e}(s)=\frac{{{M_D}{s^{p - 1}}{x_e}(0)+{M_D}{s^{p - 2}}{{\dot {x}}_e}(0)+{K_D}(\alpha ){s^{q - 1}}{x_e}(0)+Q(s)}}{{{M_D}{s^p}+{K_D}(\alpha ){s^q}}} \hfill \\ =\frac{{{M_D}{s^{p - 1}}{x_e}(0)}}{{{M_D}{s^p}+{K_D}(\alpha ){s^q}}}+\frac{{{M_D}{s^{p - 2}}{{\dot {x}}_e}(0)}}{{{M_D}{s^p}+{K_D}(\alpha ){s^q}}}+\frac{{{K_D}(\alpha ){s^{q - 1}}{x_e}(0)}}{{{M_D}{s^p}+{K_D}(\alpha ){s^q}}}+\frac{{Q(s)}}{{{M_D}{s^p}+{K_D}(\alpha ){s^q}}} \hfill \\ \end{gathered}$$

(15)

The Mittag-Leffler function is widely used in fractional-order calculus and differential equations because of its analytic properties in the complex domain, facilitating infinite series expansions that aid in solving fractional-order differential equations. The definition of the multivariate Mittag-Leffler function in the n-dimensional case is as follows:

$${E_{({a_1}, \cdots ,{a_n}),b}}({z_1}, \cdots ,{z_n})=\sum\limits_{{k=0}}^{\infty } {\sum\limits_{{{l_1}+ \cdots +{l_n}=k}} {\frac{{k!}}{{{l_1}!{\text{ }} \times \cdots \times {l_n}!}}} } \frac{{\mathop \prod \limits_{{i=1}}^{n} {z^l}_{{{i^i}}}}}{{\Gamma (b+\sum\limits_{{i=1}}^{n} {{a_i}{l_i}} )}}$$

(16)

where:$b>0,{a_i}>0,\left| {{z_i}} \right|<\infty ,{l_i}>0,i=, \cdots ,n$.

Specifically, for$n=1$, the unitary Mittag-Leffler function is commonly utilized:

$${E_{{a_1},b}}({z_1})=\sum\limits_{{k=0}}^{\infty } {} \frac{{{z^k}_{1}}}{{\Gamma (b+k{a_1})}}{\text{ }}{a_1},b>0,\left| {{z_1}} \right|<\infty$$

(17)

The Laplace transform of the Mittag-Leffler function is given by:

$$L\{ {t^{\alpha k+\beta - 1}}E_{{\alpha ,\beta }}^{{^{{(k)}}}}( \pm a{t^\alpha });s\} =\int_{0}^{\infty } {{e^{ - st}}{t^{\alpha k+\beta - 1}}} E_{{\alpha ,\beta }}^{{^{{(k)}}}}( \pm a{t^\alpha })dt=\frac{{k!{s^{\alpha - \beta }}}}{{{{({s^\alpha } \mp a)}^{k+1}}}},Re(s)>{\left| a \right|^{\frac{1}{a}}}$$

(18)

By using the Mittag-Leffler function to describe the above fractional-order differential equation, the following inverse transformation can be obtained from its Laplace transform:

$${L^{ - 1}}\{ \frac{{{M_D}{s^{p - 1}}{x_e}(0)}}{{{M_D}{s^p}+{K_D}(\alpha ){s^q}}};s\} =\frac{{{M_D}}}{{{K_D}(\alpha )}}{x_e}(0){t^{q - p}}{E_{q - p,q - p+1}}( - \frac{{{M_D}}}{{{K_D}(\alpha )}}{t^{q - p}})$$

(19)

$${L^{ - 1}}\{ \frac{{{K_D}(\alpha ){s^{q - 1}}{x_e}(0)}}{{{M_D}{s^p}+{K_D}(\alpha ){s^q}}};s\} =\frac{{{K_D}(\alpha )}}{{{M_D}}}{x_e}(0){t^{p - q}}{E_{p - q,p - q+1}}( - \frac{{{K_D}(\alpha )}}{{{M_D}}}{t^{q - p}})$$

(20)

$${L^{ - 1}}\{ \frac{{{K_D}(\alpha ){s^{q - 1}}{x_e}(0)}}{{{M_D}{s^p}+{K_D}(\alpha ){s^q}}};s\} =\frac{{{K_D}(\alpha )}}{{{M_D}}}{x_e}(0){t^{p - q}}{E_{p - q,p - q+1}}( - \frac{{{K_D}(\alpha )}}{{{M_D}}}{t^{p - q}})$$

(21)

$${L^{ - 1}}\{ \frac{{Q(s)}}{{{M_D}{s^p}+{K_D}(\alpha ){s^q}}};s\} ={Q_i} * (\frac{1}{{{K_D}(\alpha )}}{t^{q - 1}}{E_{q - p,q}}( - \frac{{{M_D}}}{{{K_D}(\alpha )}}{t^{q - p}}))$$

(22)

The analytical solution of the control system can then be obtained using the above equations:

$$\begin{gathered} {x_e}=\frac{{{M_D}}}{{{K_D}(\alpha )}}{x_e}(0){t^{q - p}}{E_{q - p,q - p+1}}( - \frac{{{M_D}}}{{{K_D}(\alpha )}}{t^{q - p}})+\frac{{{M_D}}}{{{K_D}(\alpha )}}{{\dot {x}}_e}(0){t^{q - p+1}}{E_{q - p,q - p+2}}( - \frac{{{M_D}}}{{{K_D}(\alpha )}}{t^{q - p}}) \hfill \\ \frac{{{K_D}(\alpha )}}{{{M_D}}}{x_e}(0){t^{p - q}}{E_{p - q,p - q+1}}( - \frac{{{K_D}(\alpha )}}{{{M_D}}}{t^{p - q}})+{Q_i} * (\frac{1}{{{K_D}(\alpha )}}{t^{q - 1}}{E_{q - p,q}}( - \frac{{{M_D}}}{{{K_D}(\alpha )}}{t^{q - p}}) \hfill \\ \end{gathered}$$

(23)

Fractional-order variable damping admittance control introduces parameters$\beta$ and$\gamma$, enhancing the adjustment of dynamic interactions between the robot and interaction forces, thereby increasing the flexibility of control strategy design. When the orders$\beta$and$\gamma$both take their minimum values, it becomes first-order damping control, and when the order$\beta$and$\gamma$both take their maximum values, it becomes second-order integer admittance. Thus, it is understood that as the order decreases, fractional-order characteristics gradually diminish inertial storage energy, while energy dissipation characteristics become dominant. The specific orders of the controller can be determined based on this characteristic.

Adaptive robust sliding mode control

Unknown interacting forces render the inertial and Coriolis terms in the dynamic model of the Stewart parallel robot uncertain²⁷, significantly disturbing the closed-loop system. Therefore, the robustness of motion control is crucial for the Stewart parallel robot to accurately track compliant trajectories generated by fractional-order variable damping admittance control, even in the presence of unknown interaction forces, thereby meeting stringent segment docking accuracy requirements. An expression for the Lagrangian dynamics equation is derived when an external force acts on the end effector of the Stewart parallel robot:

$${\mathbf{M}}({\mathbf{l}}){\mathbf{\ddot {l}}}+{\mathbf{C}}({\mathbf{l}},{\mathbf{\dot {l}}}){\mathbf{\dot {l}}}+{\mathbf{n}}({\mathbf{l}},{\mathbf{\dot {l}}})={{\mathbf{u}}_c} - {{\mathbf{J}}^T}({\mathbf{l}}){{\mathbf{h}}_e}$$

(24)

Where: ${\mathbf{M}}({\mathbf{l}})$represents the inertia matrix of the Stewart parallel robot, ${\mathbf{C}}({\mathbf{l}},{\mathbf{\dot {l}}})$is the Coriolis force term, ${\mathbf{n}}({\mathbf{l}},{\mathbf{\dot {l}}})$is an n-dimensional vector dependent on the system state (including nonlinear coupling terms such as centripetal force, friction force, and gravity), ${{\mathbf{h}}_e}$is an external force, ${{\mathbf{u}}_c}$is the control input term.

A common approach is to select the following inputs as described in²⁸:

$${\mathbf{y}}={\mathbf{\bar {J}(l)}}({\mathbf{\ddot {\tilde {x}}}}+{{\mathbf{K}}_{{D_t}}}{\mathbf{\dot {\tilde {x}}}}+{{\mathbf{K}}_{{P_t}}}{\mathbf{\tilde {x}}} - {\mathbf{\dot {J}}}({\mathbf{l}},{\mathbf{\dot {l}}}){\mathbf{\dot {l}}})$$

(25)

where: ${\mathbf{\tilde {C}}}({\mathbf{l}},{\mathbf{\dot {l}}})={\mathbf{\hat {C}}}({\mathbf{l}},{\mathbf{\dot {l}}}) - {\mathbf{C}}({\mathbf{l}},{\mathbf{\dot {l}}})$,${\mathbf{\tilde {n}}}({\mathbf{l}},{\mathbf{\dot {l}}})={\mathbf{\hat {n}}}({\mathbf{l}},{\mathbf{\dot {l}}}) - {\mathbf{n}}({\mathbf{l}},{\mathbf{\dot {l}}})$.

This control rate is applicable only in regular robot motion, and its robustness is difficult to ensure under external forces. Therefore, a more effective control approach is needed to maintain the robustness of the robot motion controller in such conditions.

For the one-dimensional case, the dynamical system in Eq. (24) can be represented by the following state equation:

$$\left\{ \begin{gathered} \frac{{d{x_1}}}{{dt}}={x_2} \hfill \\ M \cdot \frac{{d{x_2}}}{{dt}}=u(t)+\Delta \hfill \\ \end{gathered} \right.$$

(26)

Where: M is a constant greater than zero, representing the unknown inertia of the system;$\Delta = - (C(l,\dot {l})+{J^T}(l){h_e}+n(l,\dot {l}))$represents the sum of various types of uncertainties and disturbances, which can be reasonably considered as bounded, in other words: $\Delta \leqslant D$.

Design the following slide mold surface:

$$s=\dot {e}+c \cdot e={\dot {x}_d} - {x_2}+c \cdot e$$

(27)

Where:${x_d}$denotes the desired position signal, c is a constant greater than zero, and $e={x_d} - {x_1}$ is the tracking error of the position.

The derivation is obtained by multiplying both sides of the equation by M:

$$M \cdot \dot {s}=M({x_d} - {x_2}+c \cdot e)$$

(28)

Taking$\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{M}$to be an estimate of M, define the Li function as follows:

$$V=\frac{1}{2}M \cdot {s^2}+\frac{1}{{2\Upsilon }}{\tilde {M}^2}$$

(29)

Where:$\tilde {M}=M - \overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{M}$,$\Upsilon$is a constant always greater than zero.

Derivation of the above equation yields the following equation:

$$\begin{gathered} \dot {V}=M \cdot s\dot {s}+\frac{1}{\gamma }\tilde {M}{{\tilde {M}}^2} \hfill \\ {\text{ }}=s \cdot M({{\ddot {x}}_d} - {{\dot {x}}_2}+c \cdot \dot {e}) - \frac{1}{\gamma }\tilde {M}\dot {\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{M} } \hfill \\ {\text{ =}}s( - u - \Delta +M \cdot {{\ddot {x}}_d}+c \cdot M \cdot \dot {e}) - \frac{1}{\gamma }\tilde {M}\dot {\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{M} } \hfill \\ \end{gathered}$$

(30)

The design index converges to the law, then the control rate can be set to:

$$u=\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{M} ({\ddot {x}_d}+c \cdot \dot {e})+k \cdot s - \eta \cdot \operatorname{sgn} (s)$$

(31)

Where: k and$\eta$are a constant always greater than zero.

Substituting Eq. (31) into Eq. (30), becomes:

$$\begin{gathered} \dot {V}=s \cdot \tilde {M}({{\ddot {x}}_d}+c \cdot \dot {e}) - k \cdot {s^2} - \eta \cdot \left| s \right| - s \cdot \Delta - \frac{1}{\gamma }\tilde {M}\dot {\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{M} } \hfill \\ {\text{ =}} - k \cdot {s^2} - \eta \cdot \left| s \right| - s \cdot \Delta +\tilde {M}[s({{\ddot {x}}_d}+c \cdot \dot {e}) - \frac{1}{\gamma }\dot {\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{M} }] \hfill \\ \end{gathered}$$

(32)

For the system to be asymptotically stable, it is sufficient that the last term is less than or equal to zero. Thus, the adaptive rate can be designed as:

$$\dot {\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{M} }=s\gamma ({\ddot {x}_d}+c \cdot \dot {e})$$

(33)

Thus Eq. (32) becomes:

$$\dot {V}= - k \cdot {s^2} - \eta \cdot \left| s \right| - s \cdot \Delta \leqslant - k \cdot {s^2} \leqslant 0$$

(34)

V=0, if and only if $s=0$. According to the LaSalle invariance principle, s tends to 0 as t approaches infinity, indicating that the closed-loop system is asymptotically stable with a convergence rate dependent on k. It is worth noting that the above conditions show that${\text{V}}$is bounded as t tends to infinity, It is also shown that$\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{M}$is also bounded, but there is no guarantee that$\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{M}$will converge to M. Also, s will tend to zero as t tends to infinity, but there is no guarantee that$\tilde {M}$tends to zero.

In order to prevent$\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{M}$from being too large and causing the control input$u(t)$to be too large, and thus$\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{M} <0$, it is necessary to design the adaptive rate so that$\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{M}$varies in the range$[{M_{\hbox{min} }},{M_{\hbox{max} }}]$, an adaptive mapping algorithm can be used to correct the adaptive rate to using a projection function:

$$\dot {\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{M} }=proj(s\gamma ({\ddot {x}_d}+c \cdot \dot {e}))=\left\{ \begin{gathered} 0{\text{ , if }}\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{M} \geqslant {M_{\hbox{max} }}{\text{ }}and{\text{ }}s\gamma ({{\ddot {x}}_d}+c \cdot \dot {e})>0 \hfill \\ 0{\text{ , if }}\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{M} \leqslant {M_{\hbox{min} }}{\text{ }}and{\text{ }}s\gamma ({{\ddot {x}}_d}+c \cdot \dot {e})<0 \hfill \\ s\gamma ({{\ddot {x}}_d}+c \cdot \dot {e}){\text{ , otherwise}} \hfill \\ \end{gathered} \right.$$

(35)

The corrected adaptive rate can be used to ensure that$\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{M} \in [{M_{min}},{M_{\hbox{max} }}]$ is constant by making$\dot {\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{M} }=0$when$\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{M}$is about to exceed its range of variation.

Finally, within the sliding subspace, the ideal scenario is for the control signal to operate at an infinite frequency²⁹. In practice, however, this signal alternates at high frequencies, resulting in torque oscillations. A common method to suppress sliding mode control jitter is to use various continuous functions to replace the sign function in Eq. (31). In this study, the following sigmoid function is used for substitution, effectively suppressing jitter:

$$G(x)=\frac{{1 - {e^{ - ax}}}}{{1+{e^{ - ax}}}}$$

(36)

Where: a is a designable coefficient and the larger the value of a, the closer the sigmoid function approaches the sign function. The sigmoid function shares the same upper and lower bounds as the sign function, ensuring that the previously described stability reasoning remains valid after this substitution.

The fractional-order variable damping admittance control (FVDAC) and adaptive robust sliding mode control (ARSMC) discussed in this section establish relationships within the control system. VDAC, as an upper-level decision-making algorithm, dynamically adjusts pre-calibrated motion plans based on interaction forces during assembly. The ARSMC subsequently receives these motion plans, ensuring stable and precise execution by parallel robots and preventing rigid collisions between cabin segments.

Experimental verification

Experimental settings

Experiments were conducted on the Z-axis degree of freedom under maximum load using FVDAC and ARSMC. A docking experiment with a single cabin segment was then performed to demonstrate the feasibility of this docking strategy. The experiments utilized a Stewart parallel robot with a rated load capacity of 100 kg. The physical prototype is depicted in (Fig. 7).

Fig. 7

Automated docking platform for Stewart parallel robot segments.

The upper computer communicates with the lower computer via the UDP protocol using a LabVIEW program, with a sampling period of 5 ms. It reads the robot’s state in real time, applies a gravity compensation algorithm for external force perception from a six-dimensional force sensor, and generates motion plans using the FVDAC algorithm. The lower computer executes a motion control program to implement the calculated control rate. Following repeated commissioning, the parameters listed below were selected to achieve moderate compliant behavior in the parallel robot:

For FVDAC:${{\mathbf{M}}_D}=dia{g_6}\{ 5,5,5,0.6,0.6,0.6\}$,${{\mathbf{K}}_{{D_c}}}=dia{g_6}\{ 5,5,5,10,10,10\}$,${{\mathbf{K}}_{{P_c}}}=dia{g_6}\{ 8,8,8,8,8,8\}$, $\beta =1.2$,$\gamma =0.9$.

For ARSMC: $c=20$,$\eta =5$,$k=2$,$\Upsilon =100$,$a=10$.

According to reference³⁰, the initial sensor signals are subjected to zero drift compensation and real-time gravity compensation. A low-pass Butterworth filter is then applied to eliminate noise signals.

Variable damping admittance control experiment

Experiments were conducted to evaluate the effect of fractional order control on both Cartesian translational and rotational degrees of freedom. Figure 8 compares the response performances of traditional and fractional admittance control. Figure 5 illustrates the end-pose trajectory curve of the parallel robot under fractional and conventional admittance control after applying a force${F_Z}$of 20 N and a moment${T_Z}$of 2 N·m.

Fig. 8

Tracking curves for traditional and fractional admittance control. (a) Comparison of the robot’s response to external forces in the degree of freedom of translation along Z axis under two control algorithms. (b) Comparison of the response performance of the robot to external forces in the degree of freedom of rotation around the Z axis under two control algorithms.

The experiments were repeated five times, with Fig. 8a, b presenting the results of the first repetition. Table 2 shows the improvement rates in response performance of the control algorithm before and after changing the derivative order to fractional order.

Table 2 Rate of improvement in the response performance of the fractional order admittance control algorithm.

Experimental results indicate that, in the translational degree of freedom along the Z-axis, fractional order control increases average response speed by 51.16% and decreases average overshoot by 21.83% compared to traditional control with a 20 N force. Fractional order admittance control achieves 41.71% higher response speed and 4.39% higher overshoot compared to conventional control when applying 20 N·m torque in the rotational degree of freedom around the Z-axis. These results demonstrate that fractional order control enhances the robot’s response to external forces.

Adaptive robust sliding mode control experiment

Experiments validated the robustness of the Adaptive Robust Sliding Mode Control (ARSMC) against unknown interaction forces. In this experiment, the predefined assembly trajectory was removed, with the robot’s behavior entirely determined by the fractional-order variable damping admittance control (FVDAC). The tests included both simple inverse dynamics control Eq. (25) and ARSMC. The experiments required an upward force of 20 N along the Z-axis for 3 s, followed by a downward force of 20 N on the end effector of the parallel robot. The applied forces corresponded to the robot’s position variation curve shown in (Fig. 9).

Fig. 9

Comparison of tracking error between inverse dynamics control and ARMSC. (a): Forces applied along the Z-axis degrees of freedom during the experiment. (b): Tracking error of inverse dynamic control on expected pose under external force (c): tracking error of ARSMC on expected pose under external force.

After external force, the tracking error of the controllers increases over time. Around 5.3 s, both the inverse dynamics controller and the adaptive robust sliding mode controller (ARSMC) reach their peak tracking errors. The peak error for the inverse dynamics controller is 1.18 mm, while ARSMC achieves only 0.57 mm. Compared to the inverse dynamics controller, ARSMC shows smaller and more bounded tracking errors, whereas the former may lead to larger errors and unexpected robot movements during interactions.

The above experiments were repeated five times, and the peak tracking errors for the two control algorithms, along with the percentage reduction relative to each other, are presented in (Table 3):

Table 3 Peak tracking error and reduction percentage of inverse dynamics control and adaptive robust sliding mode control.

Analysis of experimental results indicates that greater forces along the Z-axis increase tracking errors. The average peak tracking error for the inverse dynamic controller is 1.20 mm, while for the adaptive robust sliding mode controller (ARSMC), it is only 0.59 mm. The peak tracking error of ARSMC is reduced by 50.83% compared to the inverse dynamics controller. The experiment demonstrates that ARSMC effectively enhances robot tracking performance under unknown external forces.

Segment automated docking experiment

Segment automated docking experiment further validates the advantages and practicality of the proposed variable damping admittance control and adaptive robust sliding mode control in real-world cabin docking scenarios. The experimental setup involved positioning the cabin segment model on the end effector of the Stewart robot. The docking process is divided into two stages, as illustrated in (Fig. 10):

Fig. 10

Stages in the segment docking process: (a) Robot in the starting position. (b) Robot approaches the stationary segment following the predefined assembly trajectory and begins contact. (c) Completion of the segment docking task.

Process 1: The robot moves the movable cabin segment toward the fixed cabin segment without contact, generating no interaction forces. Consequently, the robot’s motion trajectory aligns with the pre-calibrated assembly trajectory.

Process 2: The variable damping admittance control algorithm adjusts the pre-calibrated motion trajectory in real time based on the interaction forces, except for adjustments in the Y direction.

Figure 11 provides a comprehensive overview of the robot’s posture and interaction forces across all degrees of freedom throughout the entire experimental process.

Fig. 11

Interaction force versus robot position change curves during segment docking. (a): Interaction forces on the X, Y, and Z translational degrees of freedom. (b): Robot position changes in the X, Y, and Z translational degrees of freedom. (c): Interaction forces on the Rx, Ry, and Rz rotational degrees of freedom. (d): Robot attitude changes in the Rx, Ry, and Rz rotational degrees of freedom.

Figure 11a illustrates the following sequence of events along the Z-axis:

From 3.2 to 4.2 s, contact between the cabin segments generates a negative Z-axis interaction force, peaking at 5.2 N. Consequently, the robot moves approximately 3.6 mm downward along the Z-axis.

From 4.4 to 6.9 s, contact generates a positive Z-axis interaction force, peaking at 48.4 N. In response, the robot moves approximately 10.3 mm upward along the Z-axis.

This compliant behavior prevents rigid collisions between the cabin segments, protecting their components and achieving automated docking. During the subsequent docking process, no further interaction forces are observed along the Z-axis. Therefore, the robot’s position in this degree of freedom requires no further adjustment. Similar behavior is observed in the other four degrees of freedom, excluding the Y-axis.

Focusing on Figs. 10b and 11a, the Y direction serves as the docking direction. No interaction force is generated in this direction during the first 18.6 s; however, the X and Z directions adjust their compliant behavior according to the interaction forces until they reach 0. From 18.6 to 20 s, the robot begins to carry the movable module segment along the positive Y-axis, generating interaction force in the Y direction while the interaction forces in the X and Z directions remain 0. The same behavior is observed in the three rotational degrees of freedom (X, Y, Z), indicating that the robot uses a compliant response to align the movable segment with the stationary segment, successfully completing the docking task.

Figure 12 illustrates the effectiveness of the proposed Variable Damping Admittance Control in the X-axis degree of freedom during the docking process:

From 9.4 to 11.6s, a peak interaction force of 96.7 N occurs in the negative direction along the X-axis. During this interval, the adjustment coefficient of the control law rises from 0 to nearly 1 in response to the interaction force, then decreases back to 0 as the force diminishes. Similarly, the damping matrix starts at 5, decreases to nearly 0 due to the adjustment coefficient, and then returns to 5. A similar process occurs between 12.3 and 14.6s, demonstrating analogous variations.

Fig. 12

(a): Change in the adjustment coefficient$\alpha$following the interaction force along the X-axis. (b): Variation of the variable damping matrix${{\mathbf{K}}_D}$after the interaction force is generated in the X-axis.

This adaptive behavior results in lower damping values when interaction forces are present, enhancing the robot’s compliance. Conversely, in the absence of interaction forces, the variable damping admittance control assigns higher damping values to ensure precise motion control.

Conclusion

This paper proposes a novel automatic docking method that utilizes parallel robots for collision-free docking. First, a fractional-order variable-damped admittance control (FVDAC) is introduced, allowing the robot to dynamically adjust its motion trajectory based on segment interactions to prevent rigid collisions. Additionally, an adaptive robust sliding mode control (ARSMC) is developed to enhance the adaptability of Stewart parallel robots in high-precision assembly scenarios. This controller ensures accurate and stable motion control of the Stewart parallel robot despite unknown forces, satisfying the stringent docking accuracy requirements of space modules.

The study compares the tracking performance of conventional inverse dynamic control and adaptive robust sliding mode control under identical interaction conditions, along with the response performance of admittance control before and after introducing fractional-order control. Experimental results indicate that the peak tracking error of the proposed adaptive robust sliding mode control is reduced by an average of 50.83% compared to traditional inverse dynamic control. Following the introduction of fractional-order control, the response speed of admittance control to interaction forces increases by an average of 51.16% in translational degrees of freedom and 41.71% in rotational degrees of freedom. Finally, the effectiveness of the proposed variable damping matrix and the feasibility of the method are validated through piecewise automatic docking experiments. Results demonstrate that the variable damping matrix can dynamically adjust the damping values of admittance parameters in real time based on interaction forces. In summary, this method facilitates real-time trajectory adjustments of the robot under interaction forces, enabling it to complete the heavy aircraft cabin docking task without rigid collisions.