State-Extended MPC for Trajectory Tracking and Optimal Obstacle Avoidance in Multi-Point Suspension Systems

Abstract

Ground-based three-dimensional motion testing of space manipulators typically relies on active suspension-based gravity compensation systems. The design of such systems faces two fundamental challenges: first, how multiple suspension winch units can precisely track the dynamic trajectories of the corresponding suspension interfaces on the manipulator; and second, how to achieve optimal collision avoidance among the suspension mechanisms themselves during the tracking process. To address these challenges, this paper presents a multi-point suspension system endowed with kinematic redundancy for the trajectory tracking task, thereby ensuring precise tracking of the manipulator’s complex three-dimensional motions. The key innovation of this work lies in formulating the internal collision avoidance constraints as safety distance functions and integrating them into the system states. These are then combined with the trajectory-tracking states to construct a unified state-extended system model that exhibits typical underactuated characteristics. For this model, and under the concurrent influence of external disturbances from both the manipulator’s motion and the proximity to collision boundaries, a dedicated Model Predictive Controller (MPC) is designed. The results demonstrate that the proposed controller can generate an optimal coordinated collision-avoidance motion plan for the suspension winch units while maintaining precise trajectory tracking, thereby effectively solving the coordinated motion-planning problem for such complex underactuated systems. The proposed MPC achieves maximum tracking errors of 0.64 mm (X) and 0.13 mm (Z)—substantially lower than the 1.3 mm and 1.9 mm results listed in the comparative scheme—while delivering optimal collision avoidance, which is only suboptimally realized in the baseline.

1. Introduction

1.1. The Challenge of Gravity Compensation for Space Manipulators

Space manipulators, serving as critical end-effectors for on-orbit servicing of spacecraft, have their performance and reliability directly determine the success of key missions such as space operations and maintenance. Due to the fundamental difference between the space microgravity environment and terrestrial conditions, it is imperative to conduct high-fidelity, ground-based three-dimensional motion testing prior to launch to comprehensively validate their motion accuracy and dynamic characteristics. The prerequisite for such testing is ground-based microgravity simulation via gravity compensation, which aims to counteract the influence of Earth’s gravity on the manipulator’s joint torques, thereby replicating its true motion state in space during ground experiments.

To achieve this objective, three primary categories of ground-based microgravity simulation and gravity compensation techniques have been developed: the air-floating method, the hardware-in-the-loop simulation method, and the cable-suspension method.

The air-floating method counteracts gravity and minimizes friction through gas lubrication, enabling near-weightless planar motion. Despite its low friction and simple setup, this approach is inherently limited to two-dimensional compensation. Studies confirm that air-bearing platforms cannot simulate the three-dimensional attitudes or spatial trajectories of space manipulators [1], making them unsuitable for validating complex operational tasks.

The hardware-in-the-loop simulation embeds physical hardware into a digital loop to compensate gravity within a simulated environment. It offers advantages for rapid algorithm verification and cost reduction, but the architecture cannot fully replicate the complete manipulator. Research notes that it only achieves limited microgravity simulation under specific configurations and relies heavily on model accuracy [2], focusing more on functional testing while lacking the full structural coupling and nonlinear dynamics of a physical system. Thus, the generalizability of the results remains constrained.

In contrast, the cable-suspension method applies compensation forces through multiple cables to directly counteract joint gravitational torques. Theoretically, it supports full-DOF spatial motion while preserving the complete physical structure and dynamics, making it the leading solution for high-fidelity ground-based verification [3]. Its core strength lies in delivering substantially higher test credibility compared to the other two methods.

However, as space manipulator missions grow in complexity, ground-based testing systems must support larger workspaces and more agile motion. This requirement has driven the development of multi-point suspension systems with redundant degrees of freedom, which in turn reveals a critical yet largely overlooked issue: the need to optimally avoid dynamic collisions among multiple suspension mechanisms. Existing studies on suspension systems mainly focus on single-point tracking accuracy or synchronized multi-point following control, without incorporating collision avoidance between suspension winch units as a core constraint for co-optimization with trajectory tracking. For example, Li C.Z. developed a fuzzy PID-based gravity compensation system for a single cable-suspended manipulator, targeting pose stability and compensation accuracy [4]. While effective for single-point or overall performance, this work implicitly assumes fixed or non-interfering suspension points and does not plan motion among the suspension units themselves.

During large-scale, highly dynamic coordinated motions, collision risks—both between suspension winch units and between units and surrounding structures—rise sharply. Such collisions risk physical damage, test interruption, and destabilization of gravity compensation, severely compromising test safety and validity.

Thus, a key challenge lies in enabling precise tracking and effective gravity compensation while fundamentally solving the optimal coordinated collision-avoidance problem for the suspension mechanisms themselves. Addressing this challenge is essential to advance ground-testing capabilities for space manipulators and to overcome current technical bottlenecks. Motivated by this gap, this paper investigates the coordinated control of trajectory tracking and optimal collision avoidance in multi-point suspension systems.

1.2. MPC Development and Applications

Model predictive control (MPC) is widely adopted for its explicit constraint handling and multi-objective optimization capabilities across diverse fields, from process industries to high-performance mechatronic systems [5]. Its rolling-horizon and feedback-correction mechanisms are particularly advantageous in trajectory tracking. For instance, Yulong You et al. developed an adaptive MPC scheme with a dynamic prediction model and adaptive weight tuning to improve tracking accuracy and robustness against model mismatch and disturbances [6]. In UAV control, MPC advances focus on tracking precision, robustness, and fault tolerance: Mohamad Amin Najafqolian et al. combined linear MPC based on convex quadratic programming (QP) for position control with deep network-based explicit MPC for attitude control, achieving excellent real-time tracking in hardware-in-the-loop tests [7]; Fei Dong et al. employed a deep network-based MPC to sustain stable hovering tracking of moving targets [8]; and Akram Eltrabyly et al. proposed a nonlinear MPC-based active fault-tolerant framework that maintained reliable tracking under simultaneous actuator failures and measurement noise [9]. In autonomous driving, MPC is central to path tracking and motion planning—Hengyang Wang et al. used a fuzzy adaptive algorithm to dynamically adjust MPC weights, co-optimizing tracking accuracy, steering smoothness, and lateral stability [10]. Applications extend to complex dynamic scenarios, such as high-speed overtaking trajectory planning in structured environments [11] and nonlinear MPC for challenging maneuvers like dynamic drifting [12].

Beyond high-precision tracking, the greater strength of MPC lies in its capacity to cooperatively handle multiple interdependent objectives and complex constraints within a single optimization problem. This is evident in vehicle motion control, where research extends beyond path tracking to active safety and stability under extreme conditions. For example, Ahmed Hassan et al. integrated a nonlinear autoregressive exogenous (NARX) neural network with MPC to improve lateral stability while reducing computational load [13]. For multi-axle steer-by-wire/brake-by-wire vehicles, time-varying nonlinear MPC coordinates tire forces and steering angles to suppress load transfer and prevent rollover [14]. In four-wheel-drive electric vehicles, MPC reconstructs and redistributes torque during motor failures to maintain stability [15]. Similarly, in constrained scenarios like automatic parking, Hai Zhao et al. used nonlinear MPC with state-dependent switching to decompose parking phases and handle constraints, enhancing solving efficiency and parking smoothness [16]. This multi-objective co-optimization paradigm also inspires other domains: Andrea Tagliabue et al. “compressed” a robust MPC policy into an efficient deep network via imitation learning for UAVs to cope with unknown disturbances [17], and Drew Hanover et al. designed a hybrid adaptive nonlinear MPC to strengthen quadrotor trajectory tracking under model uncertainty and external disturbances [18].

The selection of MPC for this study is driven by its fundamental capacity for multi-objective, constraint-based optimization over a predictive horizon—a necessity for systems where trajectory tracking must be concurrently optimized with dynamic collision avoidance. While recent advanced feedback controllers offer specific enhancements within the PID paradigm, they fall short in this integrated context due to inherent architectural limitations that are fundamentally distinct from the MPC framework. Sigmoid PID [19] and Gudermannian PID [20] controllers introduce smooth nonlinear gain mappings to improve transient response and robustness, yet their decision-making remains purely reactive, confined to instantaneous error signals without any mechanism to enforce future constraints or anticipate impending violations. BELBIC-PID controllers [21] provide data-driven adaptive regulation through biomimetic learning, but operate as feedback regulators that optimize tracking based on current and past measurements—lacking predictive optimization over a receding horizon and systematic constraint encoding. Similarly, Softsign fractional-order PID controllers [22] combine fractional calculus with smooth nonlinear damping to enhance setpoint regulation and disturbance rejection, yet their architecture remains rooted in the single-objective regulation paradigm, offering no framework for coordinating multiple competing objectives under evolving spatial constraints. In stark contrast, MPC uniquely incorporates a dynamic model to predict future states and proactively optimize control actions within a defined horizon, directly encoding state and input constraints—including time-varying collision boundaries—into the online optimization problem. This predictive, optimization-centric paradigm is therefore not merely an enhancement of feedback regulation, but a fundamentally different control methodology—one that is uniquely suited to address the coupled “tracking-and-avoidance” challenge in our kinematically redundant and underactuated suspension system.

After mapping the collision risk between suspension winch units into system states, the resulting multi-point coordination model exhibits typical underactuated characteristics—the control input dimension is lower than that of the coordinated states. MPC has shown strong potential in handling such systems. Research confirms that through careful design of objectives and constraints, MPC can generate safe and feasible control laws even for complex underactuated systems such as tilt-rotor vertical take-off and landing aircraft [23]. This offers a solid theoretical and methodological foundation for applying MPC to the coordinated motion planning of the underactuated multi-point suspension system in this work.

The theoretical scope and application scenarios of MPC continue to expand. In aerospace, it has been used for high-precision coupled attitude and orbital control during spacecraft rendezvous [24]. At the forefront of control theory, novel MPC architectures are emerging: for instance, event-triggered predictive control integrated with a classic PID structure has been applied to resource-constrained nonlinear cyber–physical systems [25]; meanwhile, networked MPC that accounts for bounded dynamic variables and their time-varying dependencies improves the handling of dynamic uncertainties [26]. Furthermore, to alleviate the online computational burden of MPC, researchers are exploring the use of tools such as neural networks to replace part of the online optimization with lightweight approximate models, thereby significantly boosting computational efficiency while preserving control performance [27].

This paper adopts the Model Predictive Control method to design a state-extended MPC controller. Departing from conventional approaches in suspension system research, which separate or simplify the trajectory tracking and mechanism collision avoidance problems, this work dynamically transforms the collision avoidance conditions between suspension winch units into internal system states. These are then integrated with the trajectory-tracking states, forming a unified, extended system model with underactuated characteristics. The existing literature contains limited research on such internal dynamic collision risks in multi-point suspension systems, let alone studies that co-optimize them with high-precision trajectory tracking within a single MPC framework. Therefore, this paper needs to systematically model this integrated “trajectory-avoidance” state-extended system and investigate the design of a corresponding, efficient, and reliable predictive control algorithm. The goal is to achieve safe and precise coordinated motion for the complex, underactuated multi-point suspension system.

1.3. Contributions of This Work

This study takes a 7-DOF space manipulator as the research object and establishes a multi-point active suspension gravity compensation system. It focuses on addressing the horizontal trajectory tracking of three fixed suspension interfaces on the manipulator body and the coordinated collision avoidance among the suspension winch units during three-dimensional motion.

To enhance the tracking accuracy for the manipulator’s complex motion trajectories and expand its operational workspace in ground testing, this paper designs a multi-point suspension system with kinematically redundant degrees of freedom. This design allows the suspension winch units to perform “crossing” motions in the horizontal plane, thereby significantly improving system flexibility. However, this redundancy, introduced to enhance tracking performance, simultaneously creates complex collision risks among the suspension mechanisms. To address this issue, the core innovation of this work is as follows: it transforms the internal collision avoidance constraints of the multi-point system into system states and integrates them with the traditional trajectory-tracking states, thereby constructing a unified state-extended system model with typical underactuated characteristics. This model innovatively integrates the two traditionally separate optimization objectives—“tracking” and “avoidance”—into a single framework for description and solution.

For this integrated, complex underactuated system, which is simultaneously subjected to two types of external disturbances—the manipulator’s motion and dynamic collision boundaries—this paper designs a dedicated MPC. This controller can, within each sampling period, simultaneously solve for control commands that ensure both high-precision trajectory tracking and optimal coordinated collision avoidance among the suspension winch units through rolling-horizon optimization. This effectively resolves the integrated motion-planning problem for such systems.

The contributions of this paper are summarized as follows:

• We propose a novel state-extension framework that explicitly models collision-avoidance constraints as internal system states, thereby integrating trajectory tracking and optimal collision avoidance into a unified underactuated MPC model for co-optimization within a single predictive horizon.
• We design and experimentally validate a tailored Model Predictive Controller for the proposed state-extended system, which simultaneously addresses kinematic redundancy, external disturbances, and dynamic collision constraints, achieving real-time and synchronized optimization of high-precision tracking and proactive safety.
• We evaluate the proposed method on a multi-point active suspension platform with redundant degrees of freedom, and the results demonstrate its capability to actively avoid inter-unit collisions while maintaining tracking accuracy.

The remainder of this paper is organized as follows: Section 2 establishes the continuous-time state-space equations that integrate trajectory tracking and optimal collision avoidance. Section 3 details the system discretization method and the design process of the receding-horizon optimal controller. Section 4 describes the construction of the experimental platform and presents an analysis of the experimental results. Finally, Section 5 provides conclusions and discusses potential future research directions.

2. Trajectory Tracking Modeling for a Multi-Point Suspension System with Collision Avoidance

2.1. Multi-Suspension Winch Unit Design with Redundant Degrees of Freedom

A three-degree-of-freedom (3-DOF) active gravity compensation suspension device can be used for ground-based three-dimensional motion testing of space manipulators. This system can actively follow the 3-DOF spatial motion of the manipulator while maintaining the preset suspension tension during the tracking process, thereby providing ground-based microgravity environment simulation for the space manipulator. In this paper, a 3-DOF active gravity compensation suspension device is defined as a single suspension winch unit. Each suspension winch unit includes a wire rope connected to the manipulator, through which an adjustable upward vertical force is applied to the manipulator. For the 7-DOF manipulator structure, as shown in Figure 1, three such suspension winch units, labeled 1, 3, and 5 in the figure, can be employed to achieve gravity compensation.

Each suspension winch unit is connected to the suspension interface on the manipulator via a wire rope and a dedicated suspension fixture. This fixture ensures that the wire rope always passes through the center of mass of the corresponding manipulator segment during its rotation. The intended effect is to apply a vertical, upward, and controllable suspension force at the suspension point, partially balancing the manipulator’s gravity.

The ultimate objective of gravity compensation is to maintain the torque induced by the manipulator’s own gravity on its joints within an acceptable range throughout its three-dimensional motion. To achieve this, the suspension winch units must remain directly above three specified positions on the manipulator during its motion to apply the required vertical upward forces. This study does not focus on the constant tension output of the suspension units, the three-point gravity compensation principle for the 7-DOF manipulator, or the 3D trajectory planning of the manipulator itself. Instead, it focuses on the motion control method for the suspension winch units that optimally tracks known trajectories of the manipulator’s suspension interfaces within the $x$-$z$ plane while also achieving optimal, active collision avoidance among the suspension unit structures.

As shown in the top view of Figure 2, coordinate system A serves as the global reference frame. Three X-axis Bridge Beams are arranged sequentially from left to right on their guides and can move independently along the x-direction. Mounted on each bridge beam is a Z-axis slider that moves along the z-direction and remains parallel to the x-axis. Suspended beneath each slider is a suspension winch unit that is capable of sliding along the slider. These winch units are connected via wire ropes to three target points on the manipulator to provide vertical upward forces.

The design of the Z-axis sliders provides the three suspension winch units with two control inputs in the x-direction. Its primary purpose is to resolve the motion constraint imposed by the inability of the bridge beams to “cross over” one another. Specifically, once the system is assembled, the three bridge beams maintain a fixed left-to-right order and cannot swap positions. If the x-direction motion of the suspension units relied solely on the bridge beams, the units themselves would be unable to “cross” each other along the x-axis. This would severely limit the flexibility of the space manipulator during three-dimensional motion testing. The introduction of the Z-axis sliders provides the suspension units with an additional degree of freedom in the x-direction. Each suspension unit can now generate a relative offset along the x-axis with respect to its supporting bridge beam, enabling “crossing” motions between the units. As illustrated in Figure 2, Suspension Winch Unit 5 is positioned to the left of Suspension Winch Unit 3, while Bridge Beam 5 remains to the right of Bridge Beam 3, demonstrating this crossing capability.

As shown in Figure 2, the three suspension winch units work in coordination with the space manipulator. By tracking three fixed points on the manipulator and maintaining positions directly above them, the units provide gravity compensation. However, while the Z-axis sliders expand the manipulator’s operational workspace, they also introduce collision risks. The three-point system presents four potential collision points, as illustrated in Figure 3, including risks between sliders (guide–guide collision) and between sliders and suspension units (guide–unit collision). The collision risk increases significantly when two adjacent sliders are too close in the z-direction; conversely, the risk is negligible when the z-direction distance is large.

The first type of collision depends on the x-direction distance between two adjacent X-axis Bridge Beams and the z-direction distance between their corresponding adjacent suspension winch units.

The second type of collision depends on the x-direction distance between one X-axis bridge beam and the suspension winch unit of an adjacent beam and the z-direction distance between the corresponding adjacent suspension winch units.

There are two risk points each for the first and second types of collision.

2.1.1. Trajectory-Tracking Status

The three yellow X-axis Bridge Beams can move along the x-direction of the global frame. Let $x_{a1}\left(t-\tau \right)$, $x_{a3}\left(t-\tau \right)$ and $x_{a5}\left(t-\tau \right)$ represent the global x-coordinates of the three bridge beams. The three red Z-axis sliders can move along the z-direction of the global frame on their respective bridge beams. Let $z_{a1}\left(t-\tau \right)$, $z_{a3}\left(t-\tau \right)$ and $z_{a5}\left(t-\tau \right)$ represent the global z-coordinates of the three sliders. Each Z-axis slider carries a suspension winch unit. Let $x_{b1}\left(t-\tau \right)$, $x_{b3}\left(t-\tau \right)$ and $x_{b5}\left(t-\tau \right)$ represent the x-coordinates of these suspension winch units within their respective slider coordinate frames. The origin of each slider frame is located at the intersection point of the slider and its bridge beam, with only one x-axis aligned along the slider. Since the x-axis of each slider frame is always parallel to the x-axis of the global frame, the global x-coordinate of a bridge beam and the x-coordinate of its suspension winch unit within the slider frame can be directly added. This yields the global x-coordinate of the suspension winch unit. Thus, the global xz-coordinates of the three suspension winch units are expressed as follows:

$$\left\{\begin{array}{l}{x}_{a1}\left(t-\tau \right)+x_{b1}\left(t-\tau \right)\\ z_{a1}\left(t-\tau \right)\\ x_{a3}\left(t-\tau \right)+x_{b3}\left(t-\tau \right)\\ z_{a3}\left(t-\tau \right)\\ x_{a5}\left(t-\tau \right)+x_{b5}\left(t-\tau \right)\\ z_{a5}\left(t-\tau \right)\end{array}\right.$$

(1)

These six coordinate values represent the six system states for tracking the robotic arm’s trajectory. The X-axis bridge beam, Z-axis slider, and suspension winch unit are all equipped with position drive systems. At time $t$, the system inputs are $x_{a1}\left(t\right)$, $x_{a3}\left(t\right)$, $x_{a5}\left(t\right)$, $x_{b1}\left(t\right)$, $x_{b3}\left(t\right)$, $x_{b5}\left(t\right)$, $z_{a1}\left(t\right)$, $z_{a3}\left(t\right)$, $z_{a5}\left(t\right)$. $\tau$ denotes the system time delay. Due to system lag, inputs prior to time $\tau$ are considered the current actual positions. It can be observed that the system has a total of six states and nine inputs.

$x_{c1}\left(t\right)$, $x_{c3}\left(t\right)$, $x_{c5}\left(t\right)$, $z_{c1}\left(t\right)$, $z_{c3}\left(t\right)$, $z_{c5}\left(t\right)$ are the coordinates, in the global coordinate system, of the suspension interface points on the space robotic arm. The objective of trajectory tracking in this paper is to control the suspension winch unit with redundant degrees of freedom to follow the trajectory of the suspension interface points on the robotic arm.

2.1.2. Optimal Collision Avoidance State

The x-coordinates of the three suspension winch units within the Z-axis slider coordinate system not only provide the suspension winch units with a richer range of motion but also serve as the basis for achieving collision avoidance. The prerequisite for collision avoidance is that it must not interfere with the suspension winch unit’s tracking of the robotic arm trajectory. This requires ensuring that the x-coordinate of the suspension winch unit in the global coordinate system remains unchanged; obstacle avoidance is achieved by adjusting the x-coordinate of the X-axis Bridge Beam and the x-coordinate of the suspension winch unit relative to the Z-axis slider.

Therefore, for collision protection, it is desirable for the suspension winch unit to be positioned as close as possible to the origin of the Z-axis slider coordinate system. This is because, in this configuration, the suspension winch unit can achieve maximum travel in both the positive and negative x-directions, providing the greatest margin for collision avoidance. Accordingly, three system states are defined, which are the coordinates $x_{b1}\left(t-\tau \right)$, $x_{b3}\left(t-\tau \right)$ and $x_{b5}\left(t-\tau \right)$ of the three suspension winch units in the Z-axis slider coordinate system. The optimal goal for collision avoidance is for all three of these states to be zero.

The safety distance is defined as the x-direction distance between two X-axis Bridge Beams that corresponds to just avoiding a collision when two Z-axis sliders move toward each other at a relative velocity. The collision protection boundary is a curve drawn along the z-direction based on this safety distance. A collision is considered to occur when the system’s physical region enters the area inside this boundary; otherwise, it is safe. It is important to note that when the z-distance between two Z-axis sliders is sufficiently large, the collision risk becomes negligible. However, when they approach at different velocities, the slope of the collision boundary changes dynamically as shown in Figure 4a,b. Therefore, the collision boundary is a function that varies with the motion state.

Assuming two X-axis Bridge Beams remain stationary, if two Z-axis sliders move towards each other at a relative velocity $v_z$ and the Bridge Beams take no avoidance action, a collision will occur. When the Z-axis sliders approach within a certain distance, the Bridge Beams can initiate avoidance by moving apart at their maximum separation speed $v_{x(\max )}$, thereby preventing a collision. By substituting the actual relative velocity $v_z$ and the maximum separation speed $v_{x(\max )}$, a linear collision boundary can be calculated. This boundary varies with the motion of the Z-axis sliders and has a slope that changes in real time.

While a triangular collision boundary aligns with physical intuition, it contains non-differentiable points. To address this, this paper defines the function $f$ as follows:

$$f\left(za\left(31\right),k_{31}\right)=G·e^{-\beta \left({\displaystyle \frac{k_{31}^2}{G^2}}{\left(za\left(31\right)\right)}^2\right)}$$

(2)

Here, to simplify the expression, the following abbreviations are introduced:

$$xa\left(31\right)=x_{a3}\left(t-\tau \right)-x_{a1}\left(t-\tau \right)$$

(3)

$$xat\left(31\right)=x_{a3}\left(t\right)-x_{a1}\left(t\right)$$

(4)

$$f\left(31\right)=f\left(za\left(31\right),k_{31}\right)$$

(5)

Similarly, the variable $x$ can be replaced with $z$, and the indices labeled “31” can be replaced with “53”, to represent combinations of other suspension winch units or directions.

$G=1000$ and $\beta =2$ are parameters of the function, while $za\left(31\right)$ and $k_{31}={\displaystyle \frac{v_{x(\max )}}{v_z}}$ are the independent variables of the function. In fact, $k_{31}$ is a function of the real-time approaching velocity $v_z$ of the Z-axis sliders. This function is also related to the maximum allowable value of $v_{x(\max )}$ the maximum horizontal avoidance speed. When the relative velocity approaches zero, $k_{31}$ tends towards infinity. When the relative velocity reaches the maximum value limited by the system, $k_{31}$ equals 1. As $k_{31}$ changes, the shape of the curve for function varies together with the collision boundary. The function is used to describe the escapable distance between the two collision parties during the motion of the Z-axis sliders. The relationship between its curve and the collision boundary is illustrated in Figure 5.

As shown in Figure 5, the linear collision boundary contains non-differentiable points, whereas the curve of function $f$ can envelope this boundary while providing a safety margin. Therefore, this paper adopts the $f$ function as the collision protection boundary for calculating the safety distance.

Parameter $G$ is called fundamental collision distance, representing the safe passage distance constant in the x-direction for the Z-axis sliders, which is determined by the system structure. Parameter $\beta$ is called the shape factor, and indicates the degree to which the function envelops the collision boundary. A smaller value of $\beta$ results in a larger margin between the function curve and the collision boundary, meaning the boundary is more comprehensively enveloped. When $\beta$ exceeds 2.45, the function no longer fully envelops the collision boundary. A value of $\beta =2$ is used throughout this paper. The influence of different $\beta$ parameters on the function is illustrated in Figure 5.

The system presents two types of potential collision points, defined as follows:

First type of collision (involving inter-beam collision avoidance):

Occurs between X-axis Bridge Beam 3 and X-axis Bridge Beam 1:

$$x_{a3}\left(t-\tau \right)-\left(x_{a1}\left(t-\tau \right)+f\left(za\left(31\right)\right)\right)<0$$

(6)

Occurs between X-axis Bridge Beam 5 and X-axis Bridge Beam 3:

$$x_{a5}\left(t-\tau \right)-\left(x_{a3}\left(t-\tau \right)+f\left(za\left(53\right)\right)\right)<0$$

(7)

Collisions occur when these two distances are less than 0. Structurally, these two safety distances have a trade-off relationship. Therefore, to ensure both potential collision risks are as far from the threshold as possible, we can set these two safety distances being equal as the optimal objective for collision avoidance. If a collision still cannot be avoided when the two safety distances are equal, then in any other scenario at least one potential collision is certain to occur.

The two collision points for the second type of collision are as follows:

Between suspension winch unit No. 3 and X-axis Bridge Beam No. 1:

$$x_{a3}\left(t-\tau \right)+x_{b3}\left(t-\tau \right)-\left(x_{a1}\left(t-\tau \right)+f\left(za\left(31\right)\right)\right)<0$$

(8)

Occurs between X-axis Bridge Beam 5 and X-axis Bridge Beam 3:

$$x_{a5}\left(t-\tau \right)-\left(x_{a3}\left(t-\tau \right)+x_{b3}\left(t-\tau \right)+f\left(za\left(53\right)\right)\right)<0$$

(9)

Similarly, setting the two distances equal serves as the optimal objective for collision avoidance. These four inequalities function as safety protections during operation to prevent collisions. As schematically illustrated in Figure 4c, conditions $a1=a2$ and $b1=b2$ represents the optimal collision avoidance state.

This paper defines the difference between the two safety distances of the first collision type as one system state, and the difference between the two safety distances of the second collision type as another system state. The goal of optimal collision avoidance is to drive both of these state values toward zero, thereby balancing the risks associated with obstacle avoidance.

In the subsequent sections of this paper, we will conduct detailed modeling of the system. The primary variables utilized at different Model Stages are summarized in

Table 1. Overview of variables across model stages.

Model Stage	Symbol	Description
Original Continuous-time System Equation	$A_0$	State Matrix
	$B_0$	Control Matrix
State-Extended Continuous-time Equation	$A$	State Matrix
	$B$	Control Matrix
	$E$	Internal Disturbance Matrix
	$x\left(t\right)$	State Vector
	$u\left(t\right)$	Control Input Vector
Continuous-time Tracking Error Equation	$A$	State Matrix
	$B$	Control Matrix
	$E$	Internal Disturbance Matrix
	$F$	Trajectory Disturbance Matrix
	$\tilde{x}$	State Vector
	$\tilde{u}$	Control Input Vector
Discrete-time Error Equation	$A_d$	State Matrix
	$B_d$	Control Matrix
	$d_k$	Disturbance
	${\tilde{x}}_k$	State Vector
	${\tilde{u}}_k$	Control Input Vector
State-Extended Discrete-time Error Equation	$\tilde{A}$	State Matrix
	$\tilde{B}$	Control Matrix
	${\tilde{d}}_k$	Disturbance
	${\lambda }_k$	State Vector
	$\Delta {\tilde{u}}_k$	Control Input Vector
Time-expanded Prediction System Equation	$V$	State Matrix
	$W$	Control Matrix
	$Z$	Disturbance Matrix
	${\lambda }_k$	State Vector
	$\Delta {\hat{u}}_k$	Control Input Vector
	${\tilde{d}}_k$	Disturbance Vector

.

2.1.3. Integrated System State Equations

Building upon the previous analysis of trajectory-tracking states and collision avoidance states, this section integrates both into a unified system state equation, providing a mathematical model foundation for subsequent controller design.

The system state vector at time $t$ is defined as follows:

$$x\left(t\right)=\left[\begin{array}{l}{x}_{a1}\left(t-\tau \right)+x_{b1}\left(t-\tau \right)\\ z_{a1}\left(t-\tau \right)\\ x_{a3}\left(t-\tau \right)+x_{b3}\left(t-\tau \right)\\ z_{a3}\left(t-\tau \right)\\ x_{a5}\left(t-\tau \right)+x_{b5}\left(t-\tau \right)\\ z_{a5}\left(t-\tau \right)\\ x_{b1}\left(t-\tau \right)\\ x_{b3}\left(t-\tau \right)\\ x_{b5}\left(t-\tau \right)\\ xa\left(31\right)-xa\left(53\right)-f\left(31\right)-f\left(53\right)\\ xa\left(31\right)-xa\left(53\right)+2x_{b3}\left(t-\tau \right)-f\left(31\right)-f\left(53\right)\end{array}\right]$$

(10)

The system state comprises a total of 11 components, with their physical meanings as follows:

Components 1–6 correspond to the $x$ and $z$ coordinates of the three suspension winch units in the global coordinate system, which are used to describe the trajectory tracking state.

Components 7–9 represent the $x$ coordinates of the suspension winch units within the Z-axis slider coordinate system, reflecting the offset of the suspension winch units relative to the Z-axis sliders.

Components 10–11 are, respectively, the difference in collision safety distances between the Z-axis slider and the Z-axis slider, and the difference in collision safety distances between the Z-axis slider and the suspension winch unit.

When the values of components 7–11 are zero, the system achieves the optimal collision avoidance state, where all collision risks are balanced.

The integrated system has 11 state variables, while the control input dimension is nine. Therefore, this system belongs to the category of underactuated control systems.

2.2. Establishment of Continuous State Transition Equations for Under-Driven Systems

This section establishes a continuous-time state transition model for the system based on the previously integrated system state equations. By introducing the Padé approximation method to handle the system time delay, the time derivative of the state vector is derived, ultimately yielding a linearized continuous-time state equation. This lays the foundation for subsequent discretization and controller design.

The theoretical basis of this approximation stems from comparing the Taylor expansion of signal $s_r\left(t\right)$ with the Padé approximation. Specifically, the Taylor expansion of $s_r\left(t\right)$ at point $t-\tau$ is:

$$s_r\left(t\right)=s_r\left(t-\tau \right)+{\dot{s}}_r\left(t-\tau \right)\tau +R_r\left(t\right)$$

(11)

where $R_r\left(t\right)$ denotes the higher-order Lagrange remainder term. The corresponding Padé approximation $R_{1,1}\left(t\right)$ of order [1, 1] satisfies

$$s_r\left(t-\tau \right)+{\dot{s}}_r\left(t-\tau \right)\tau -R_{1,1}\left(t\right)\approx 0$$

(12)

This implies that the error between the two is primarily confined to the residual term $R_r\left(t\right)$. Through derivation, the approximate relationship for the time-delay link is ultimately obtained:

$${\dot{s}}_r\left(t-\tau \right)=-{\displaystyle \frac{1}{\tau }}{s}_r\left(t-\tau \right)+{\displaystyle \frac{1}{\tau }}{s}_r\left(t\right)$$

(13)

This relationship forms the basis of the linear state equation. In this system, the delay constant $\tau =0.01$ is small and fixed, and the planned trajectory of the robotic arm is smooth, making the influence of higher-order errors represented by the residual term $R_r\left(t\right)$ controllable within the system’s operating frequency band. More importantly, the robustness of MPC does not entirely depend on the absolute accuracy of the predictive model. Its rolling optimization and feedback correction mechanism form a closed-loop system: optimization is continuously performed at each time step based on the latest measured state, enabling proactive compensation for prediction errors arising from model approximation. The high-performance experimental results presented in Section 4 directly demonstrate the effectiveness of this approximate model under the conditions of this study and the engineering robustness of the closed-loop system. For systems with larger delays or more severe dynamics, adopting higher-order approximations or precise delay models will be necessary directions for improvement.

The Pade approximation is a numerical method that is commonly used to handle time-delay systems, which approximates the time-delay term $e^{-\tau s}$ using a rational function. Here, $\tau$ is the time-delay constant. The first-order Pade approximation form is

$$e^{-\tau s}\approx {\displaystyle \frac{1-\frac{\tau s}{2}}{1+\frac{\tau s}{2}}}$$

(14)

In the time domain, this can be transformed into a differential equation form to approximate the derivatives of the state variables. This section employs a first-order Padé approximation to derive the system state $x\left(t\right)$, simplifying the treatment of the time-delay term.

Taking the first component $x_{a1}\left(t-\tau \right)+x_{b1}\left(t-\tau \right)$ of the state vector $x\left(t\right)$ as an example, the Padé approximation is applied to obtain its derivative. According to the first-order approximation, the state derivative can be expressed as follows:

$${\dot{x}}_1\left(t\right)\approx -{\displaystyle \frac{1}{\tau }}\left(x_{a1}\left(t-\tau \right)+x_{b1}\left(t-\tau \right)\right)+{\displaystyle \frac{1}{\tau }}\left(x_{a1}\left(t\right)+x_{b1}\left(t\right)\right)$$

(15)

Similarly, the derivatives for other state components can be obtained term by term. Furthermore, differentiating the function $f$ yields

$$f^{\prime }\left(31\right)=\left({\displaystyle \frac{4k_{31}^2}{-G^2}}·\left(z\left(31\right)\right){\left(z\left(31\right)\right)}^{\prime }+{\displaystyle \frac{4z\left(31\right)}{-G^2}}·k_{31}·{k_{31}}^{\prime }\right)·f\left(31\right)$$

(16)

Combining all components, the time derivative of the system state $x\left(t\right)$ can be written as follows:

$$\begin{array}{l}\dot{x}\left(t\right)=\\ \left[\begin{array}{l}-{\displaystyle \frac{1}{\tau }}\left(x_{a1}\left(t-\tau \right)+x_{b1}\left(t-\tau \right)\right)+{\displaystyle \frac{1}{\tau }}\left(x_{a1}\left(t\right)+x_{b1}\left(t\right)\right)\\ -{\displaystyle \frac{1}{\tau }}{z}_{a1}\left(t-\tau \right)+{\displaystyle \frac{1}{\tau }}{z}_{a1}\left(t\right)\\ -{\displaystyle \frac{1}{\tau }}\left(x_{a3}\left(t-\tau \right)+x_{b3}\left(t-\tau \right)\right)+{\displaystyle \frac{1}{\tau }}\left(x_{a3}\left(t\right)+x_{b3}\left(t\right)\right)\\ -{\displaystyle \frac{1}{\tau }}{z}_{a3}\left(t-\tau \right)+{\displaystyle \frac{1}{\tau }}{z}_{a3}\left(t\right)\\ -{\displaystyle \frac{1}{\tau }}\left(x_{a5}\left(t-\tau \right)+x_{b5}\left(t-\tau \right)\right)+{\displaystyle \frac{1}{\tau }}\left(x_{a5}\left(t\right)+x_{b5}\left(t\right)\right)\\ -{\displaystyle \frac{1}{\tau }}{z}_{a5}\left(t-\tau \right)+{\displaystyle \frac{1}{\tau }}{z}_{a5}\left(t\right)\\ -{\displaystyle \frac{1}{\tau }}{x}_{b1}\left(t-\tau \right)+{\displaystyle \frac{1}{\tau }}{x}_{b1}\left(t\right)\\ -{\displaystyle \frac{1}{\tau }}{x}_{b3}\left(t-\tau \right)+{\displaystyle \frac{1}{\tau }}{x}_{b3}\left(t\right)\\ -{\displaystyle \frac{1}{\tau }}{x}_{b5}\left(t-\tau \right)+{\displaystyle \frac{1}{\tau }}{x}_{b5}\left(t\right)\\ {\displaystyle \frac{1}{\tau }}\left(xa\left(53\right)-xa\left(31\right)\right)+{\displaystyle \frac{1}{\tau }}\left(xat\left(31\right)-xat\left(53\right)\right)-f^{\prime }\left(31\right)-f^{\prime }\left(31\right)\\ {\displaystyle \frac{1}{\tau }}\left(xa\left(53\right)-xa\left(31\right)+2x_{b3}\left(t-\tau \right)\right)+{\displaystyle \frac{1}{\tau }}\left(xat\left(31\right)-xat\left(53\right)+2x_{b3}\left(t\right)\right)-f^{\prime }\left(31\right)-f^{\prime }\left(31\right)\end{array}\right]\end{array}$$

(17)

Organizing the derivative expressions above into standard state-space form, the input vector is defined as follows:

$$u\left(t\right)=\left[\begin{array}{l}{x}_{a1}\left(t\right)\\ x_{b1}\left(t\right)\\ z_{a1}\left(t\right)\\ x_{a3}\left(t\right)\\ x_{b3}\left(t\right)\\ z_{a3}\left(t\right)\\ x_{a5}\left(t\right)\\ x_{b5}\left(t\right)\\ z_{a5}\left(t\right)\end{array}\right]$$

(18)

Then, the continuous-time state equation is

$$\dot{x}\left(t\right)=A·x\left(t\right)+B·u\left(t\right)+E$$

(19)

where $A$ is the system matrix:

$$A=\left[\begin{array}{cc}-{\displaystyle \frac{1}{\tau }}{I}_{9\times 9}& 0_{9\times 2}\\ S& 0_{2\times 2}\end{array}\right]$$

(20)

The submatrix $S$ is

$$S=\left[\begin{array}{lllllllll}{\displaystyle \frac{1}{\tau }}\hfill & 0\hfill & {\displaystyle \frac{-2}{\tau }}\hfill & 0\hfill & {\displaystyle \frac{1}{\tau }}\hfill & 0\hfill & {\displaystyle \frac{-1}{\tau }}\hfill & {\displaystyle \frac{2}{\tau }}\hfill & {\displaystyle \frac{-1}{\tau }}\hfill \\ {\displaystyle \frac{1}{\tau }}\hfill & 0\hfill & {\displaystyle \frac{-2}{\tau }}\hfill & 0\hfill & {\displaystyle \frac{1}{\tau }}\hfill & 0\hfill & {\displaystyle \frac{-1}{\tau }}\hfill & 0\hfill & {\displaystyle \frac{-1}{\tau }}\hfill \end{array}\right]$$

(21)

$B$ is the input matrix, composed of block matrices:

$$B=\left[\begin{array}{c}{I}_3\otimes B_1\\ I_3\otimes B_2\\ B_3\end{array}\right]$$

(22)

where

$$B_1=\left[\begin{array}{lll}{\displaystyle \frac{1}{\tau }}\hfill & {\displaystyle \frac{1}{\tau }}\hfill & 0\hfill \\ 0\hfill & 0\hfill & {\displaystyle \frac{1}{\tau }}\hfill \end{array}\right],B_2=\left[\begin{array}{ccc}0& {\displaystyle \frac{1}{\tau }}& 0\end{array}\right],B_3=\left[\begin{array}{lllllllll}-{\displaystyle \frac{1}{\tau }}\hfill & 0\hfill & 0\hfill & {\displaystyle \frac{2}{\tau }}\hfill & 0\hfill & 0\hfill & -{\displaystyle \frac{1}{\tau }}\hfill & 0\hfill & 0\hfill \\ -{\displaystyle \frac{1}{\tau }}\hfill & 0\hfill & 0\hfill & {\displaystyle \frac{2}{\tau }}\hfill & {\displaystyle \frac{2}{\tau }}\hfill & 0\hfill & -{\displaystyle \frac{1}{\tau }}\hfill & 0\hfill & 0\hfill \end{array}\right]$$

(23)

$E$ is the disturbance term, which includes the derivative of the function f and other unmodeled dynamics:

$$E=\left[\begin{array}{l}{0}_{9\times 1}\\ -f^{\prime }\left(z31\right)-f^{\prime }\left(z53\right)\\ -f^{\prime }\left(z31\right)-f^{\prime }\left(z53\right)\end{array}\right]$$

(24)

Here, it is assumed that the derivative of the function $f$ varies slowly over time and is determined by solving the current state within this period. By treating the derivative of $f$ as part of the disturbance term $E$, a linear expression of the system is obtained [28].

For the extended system, construct its controllability matrix

$$Q_c=\left[\begin{array}{cccc}B& AB& \cdots & A^{10}B\end{array}\right]$$

(25)

By calculation, we obtain

$$rank\left(Q_c\right)=11,$$

(26)

which is equal to the dimension of the extended system, indicating that the extended system is controllable. Removing the last five components of the state variables yields the state equation of the system before extension. We write this equation as

$${\dot{x}}_0\left(t\right)=A_0·x_0\left(t\right)+B_0·u\left(t\right)$$

(27)

where

$$A_0=-{\displaystyle \frac{1}{\tau }}\left[I_{6\times 6}\right], B_0=\left[I_3\otimes B_1\right]$$

(28)

The controllability matrix of the original system before extension is

$$Q_{c0}=\left[\begin{array}{cccc}{B}_0& A_0{B}_0& \cdots & A_0^5{B}_0\end{array}\right]$$

(29)

By calculation, we obtain:

$$rank\left(Q_{c0}\right)=6,$$

(30)

The dimension of the original system is six, so the original system is also controllable. The state extension performed in this paper for optimal collision does not affect the controllability of the system.

This section linearizes the nonlinear time-delay system using the Padé approximation, resulting in a concise continuous-time state equation. The disturbance term $E$ is handled independently to enhance the model’s robustness against dynamic uncertainties. This formulation provides a foundation for subsequent discretization and MPC design. Due to the system having a lower number of control inputs than state variables, it exhibits typical underactuated characteristics.

2.3. Establishment of the Continuous Error Model

Next, we discuss how the system model tracks a known state trajectory, $x_P\left(t\right)$. This trajectory corresponds to the path of the three Suspension Interface points on the robotic arm as it executes a task in three-dimensional space. The system input trajectory corresponding to this state trajectory, $u_P\left(t\right)$, is also known. Linearizing the system at any time $t$ yields

$$\dot{x}={\dot{x}}_P+A·\left(x-x_P\right)+B·\left(u-u_P\right)+E+F$$

(31)

These are the target trajectory states, where the first six components aim to track the robotic arm’s path, and components 7–11 are all 0. This is because states 7–9 represent the coordinates of the suspension winch units within the Z-axis slider coordinate frame, which are not directly related to the trajectory and are optimally set to 0 during operation. The two components, 10–11, represent the difference in safety protection distances involving the function ff, and their optimal state is also 0.

$$x_p\left(t\right)=\left[\begin{array}{l}{x}_{c1}\left(t\right)\\ z_{c1}\left(t\right)\\ x_{c3}\left(t\right)\\ z_{c3}\left(t\right)\\ x_{c5}\left(t\right)\\ z_{c5}\left(t\right)\\ 0_{5\times 1}\end{array}\right]$$

(32)

To quantify the tracking deviation, the error state $\tilde{x}$ and error input $\tilde{u}$ and the derivative of the error state are defined:

$$\left\{\begin{array}{l}\tilde{x}=\left(x-x_P\right)\\ \tilde{u}=\left(u-u_P\right)\\ \dot{\tilde{x}}=\left(\dot{x}-{\dot{x}}_P\right)\end{array}\right.$$

(33)

For the continuous-time error system:

$$\dot{\tilde{x}}\left(t\right)=A\tilde{x}\left(t\right)+B\tilde{u}\left(t\right)+E+F$$

(34)

Expanding the state of the error system yields

$$x\left(t\right)-x_P\left(t\right)=\left[\begin{array}{l}{x}_{a1}\left(t-\tau \right)+x_{b1}\left(t-\tau \right)-x_{c1}\left(t\right)\\ z_{a1}\left(t-\tau \right)-z_{c1}\left(t\right)\\ x_{a3}\left(t-\tau \right)+x_{b3}\left(t-\tau \right)-x_{c3}\left(t\right)\\ z_{a3}\left(t-\tau \right)-z_{c3}\left(t\right)\\ x_{a5}\left(t-\tau \right)+x_{b5}\left(t-\tau \right)-x_{c5}\left(t\right)\\ z_{a5}\left(t-\tau \right)-z_{c5}\left(t\right)\\ x_{b1}\left(t-\tau \right)\\ x_{b3}\left(t-\tau \right)\\ x_{b5}\left(t-\tau \right)\\ xa\left(31\right)-xa\left(53\right)-f\left(za\left(31\right)\right)-f\left(za\left(53\right)\right)\\ xa\left(31\right)-xa\left(53\right)+2x_{b3}\left(t-\tau \right)-f\left(za\left(31\right)\right)-f\left(za\left(53\right)\right)\end{array}\right]$$

(35)

This expression clearly reflects the trajectory-tracking deviation and the collision avoidance state.

Taking the derivative of the system error state gives

$$\begin{array}{l}\dot{x}\left(t\right)-{\dot{x}}_p\left(t\right)=\\ \left[\begin{array}{l}-{\displaystyle \frac{1}{\tau }}\left(x_{a1}\left(t-\tau \right)+x_{b1}\left(t-\tau \right)-x_{c1}\left(t\right)\right)+{\displaystyle \frac{1}{\tau }}\left(x_{a1}\left(t\right)+x_{b1}\left(t\right)-x_{c1}\left(t\right)\right)-{\dot{x}}_{c1}\left(t\right)\\ -{\displaystyle \frac{1}{\tau }}\left(z_{a1}\left(t-\tau \right)-z_{c1}\left(t\right)\right)+{\displaystyle \frac{1}{\tau }}\left(z_{a1}\left(t\right)-z_{c1}\left(t\right)\right)-{\dot{z}}_{c1}\left(t\right)\\ -{\displaystyle \frac{1}{\tau }}\left(x_{a3}\left(t-\tau \right)+x_{b3}\left(t-\tau \right)-x_{c3}\left(t\right)\right)+{\displaystyle \frac{1}{\tau }}\left(x_{a3}\left(t\right)+x_{b3}\left(t\right)-x_{c3}\left(t\right)\right)-{\dot{x}}_{c3}\left(t\right)\\ -{\displaystyle \frac{1}{\tau }}\left(z_{a3}\left(t-\tau \right)-z_{c3}\left(t\right)\right)+{\displaystyle \frac{1}{\tau }}\left(z_{a3}\left(t\right)-z_{c3}\left(t\right)\right)-{\dot{z}}_{c3}\left(t\right)\\ -{\displaystyle \frac{1}{\tau }}\left(x_{a5}\left(t-\tau \right)+x_{b5}\left(t-\tau \right)-x_{c5}\left(t\right)\right)+{\displaystyle \frac{1}{\tau }}\left(x_{a5}\left(t\right)+x_{b5}\left(t\right)-x_{c5}\left(t\right)\right)-{\dot{x}}_{c5}\left(t\right)\\ -{\displaystyle \frac{1}{\tau }}\left(z_{a5}\left(t-\tau \right)-z_{c3}\left(t\right)\right)+{\displaystyle \frac{1}{\tau }}\left(z_{a5}\left(t\right)-z_{c3}\left(t\right)\right)-{\dot{z}}_{c3}\left(t\right)\\ -{\displaystyle \frac{1}{\tau }}{x}_{b1}\left(t-\tau \right)+{\displaystyle \frac{1}{\tau }}{x}_{b1}\left(t\right)\\ -{\displaystyle \frac{1}{\tau }}{x}_{b3}\left(t-\tau \right)+{\displaystyle \frac{1}{\tau }}{x}_{b3}\left(t\right)\\ -{\displaystyle \frac{1}{\tau }}{x}_{b5}\left(t-\tau \right)+{\displaystyle \frac{1}{\tau }}{x}_{b5}\left(t\right)\\ {\displaystyle \frac{1}{\tau }}\left(xa\left(53\right)-xa\left(31\right)\right)+{\displaystyle \frac{1}{\tau }}\left(xat\left(31\right)-xat\left(53\right)\right)-f^{\prime }\left(31\right)-f^{\prime }\left(53\right)\\ {\displaystyle \frac{1}{\tau }}\left(xa\left(53\right)-xa\left(31\right)+2x_{b3}\left(t-\tau \right)\right)+{\displaystyle \frac{1}{\tau }}\left(xat\left(31\right)-xat\left(53\right)+2x_{b3}\left(t\right)\right)-f^{\prime }\left(31\right)-f^{\prime }\left(53\right)\end{array}\right]\end{array}$$

(36)

After organization, the input for the error model is

$$u-u_P=\left[\begin{array}{l}{x}_{a1}\left(t\right)-\left(x_{c1}\left(t\right)+\tau {\dot{x}}_{c1}\left(t\right)\right)\\ x_{b1}\left(t\right)\\ z_{a1}\left(t\right)-\left(z_{c1}\left(t\right)+\tau {\dot{z}}_{c1}\left(t\right)\right)\\ x_{a3}\left(t\right)-\left(x_{c3}\left(t\right)+\tau {\dot{x}}_{c3}\left(t\right)\right)\\ x_{b3}\left(t\right)\\ z_{a3}\left(t\right)-\left(z_{c3}\left(t\right)+\tau {\dot{z}}_{c3}\left(t\right)\right)\\ x_{a5}\left(t\right)-\left(x_{c5}\left(t\right)+\tau {\dot{x}}_{c5}\left(t\right)\right)\\ x_{b5}\left(t\right)\\ z_{a5}\left(t\right)-\left(z_{c5}\left(t\right)+\tau {\dot{z}}_{c5}\left(t\right)\right)\end{array}\right]$$

(37)

Therefore, the system input corresponding to the robotic arm trajectory is derived as

$$u_p\left(t\right)=\left[\begin{array}{l}{x}_{c1}\left(t\right)+\tau {\dot{x}}_{c1}\left(t\right)\\ 0\\ z_{c1}\left(t\right)+\tau {\dot{z}}_{c1}\left(t\right)\\ x_{c3}\left(t\right)+\tau {\dot{x}}_{c3}\left(t\right)\\ 0\\ z_{c3}\left(t\right)+\tau {\dot{z}}_{c3}\left(t\right)\\ x_{c5}\left(t\right)+\tau {\dot{x}}_{c5}\left(t\right)\\ 0\\ z_{c5}\left(t\right)+\tau {\dot{z}}_{c5}\left(t\right)\end{array}\right]$$

(38)

The physical meaning of the optimal input is consistent with our defined system states. If the system state lags behind the input, then the ideal input is the lead of the target trajectory.

A new disturbance term $F$ is introduced during the organization process:

$$F=\left[\begin{array}{c}{0}_{9\times 1}\\ -2{\dot{x}}_{c3}\left(t\right)+{\dot{x}}_{c1}\left(t\right)+{\dot{x}}_{c5}\left(t\right)\\ -2{\dot{x}}_{c3}\left(t\right)+{\dot{x}}_{c1}\left(t\right)+{\dot{x}}_{c5}\left(t\right)\end{array}\right]$$

(39)

This term arises from the influence of the target trajectory and reflects the dynamic disturbance of the robotic arm motion on the collision boundary.

3. Controller Design for Optimal Collision Avoidance in a Multi-Suspension Winch Unit System

Based on the system model established in the previous chapters, this chapter designs an MPC specifically for the multi-suspension winch unit system. By discretizing the continuous system, constructing a prediction model, and solving a receding horizon optimization problem, coordinated control of trajectory tracking and collision avoidance is achieved. The content of this section is divided into three parts: deriving the discrete error model of the system; establishing a future time-domain prediction model with disturbance terms; and designing the MPC objective function and constraint-handling strategy to complete the controller framework.

3.1. Establishment of Discrete Error Models

For a sampling time $T$, under the Zero-Order Hold (ZOH) assumption, the continuous-time system matrices can be precisely discretized into the following form:

$${\tilde{x}}_{k+1}=A_d{\tilde{x}}_k+B_d{\tilde{u}}_k+d_k$$

(40)

where

$$A_d=e^{A_{c}T}, B_d={\displaystyle {\int }_0^T{e}^{A_{c}T}d\tau }·B_c, d_k=\left({\displaystyle {\int }_0^T{e}^{A_{c}T}d\tau }\right)·\left(E+F\right)$$

(41)

The discrete parameters are solved via augmented matrix exponential:

$$M=\left[\begin{array}{ccc}{A}_d& B_d& d_k\\ 0& 0& 0\\ 0& 0& 0\end{array}\right]·T, e^M=\left[\begin{array}{ccc}{\varphi }_{11}& {\varphi }_{12}& {\varphi }_{13}\\ 0& I& 0\\ 0& 0& I\end{array}\right]$$

(42)

Solving yields:

$$A_d={\varphi }_{11},B_d={\varphi }_{12},d_k={\varphi }_{13}$$

(43)

The discrete error equation describes the state error and input error of the system while tracking the trajectory, serving as the foundation for designing the MPC controller in the following sections.

3.2. Future Time-Domain Prediction Models with Disturbance Terms

Using MPC control requires predicting the system’s behavior over the future $n$ steps based on the current time $k$ and solving for the optimal sequence within this horizon. To formulate the optimal objective function linearly, new system states ${\lambda }_k$ and system inputs $\Delta {\tilde{u}}_k$ are defined:

$${\lambda }_k={{\left[{{\tilde{x}}_k}^T,{{\tilde{u}}_{k-1}}^T\right]}^T}_{\left(20*1\right)}$$

(44)

$$\Delta {\tilde{u}}_k=\left({\tilde{u}}_k-{\tilde{u}}_{k-1}\right)$$

(45)

The expanded system state equation is written as

$${\lambda }_{k+1}=\tilde{A}{\lambda }_k+\tilde{B}\Delta {\tilde{u}}_k+{\tilde{d}}_k$$

(46)

Here, the new system state matrix $\tilde{A}$, input matrix $\tilde{B}$, disturbance term ${\tilde{d}}_k$, and output matrix $\tilde{C}$ are defined as

$$\tilde{A}={\left[\begin{array}{cc}A& B\\ 0_{9*11}& I_{9*9}\end{array}\right]}_{\left(20*20\right)}, \tilde{B}={\left[\begin{array}{c}B\\ I_{9*9}\end{array}\right]}_{\left(20*9\right)}, {\tilde{d}}_k={\left[\begin{array}{c}{d}_k\\ 0\end{array}\right]}_{\left(20*1\right)}, \tilde{C}={\left[\begin{array}{cc}{I}_{\left(11*11\right)}& 0_{\left(11*9\right)}\end{array}\right]}_{\left(11*20\right)}$$

(47)

The output equation is defined as

$$y_k=\tilde{C}{\lambda }_k$$

(48)

Within the prediction horizon $n$ of the model predictive control (from current time $k$ to $k+n-1$), it is assumed that the disturbance term remains constant at its current value, i.e.,

$$d_{k+i\left|k\right.}=d_k, i=0,1,\cdots ,N-1$$

(49)

where $d_{k+i\left|k\right.}$ represents the predicted disturbance value for the $i$-th step starting at time $k$. This assumption holds when disturbances change slowly or the sampling period is small.

For the receding horizon starting at time $k$:

Step 1 state equation:

$${\lambda }_{k+1}=\tilde{A}{\lambda }_k+\tilde{B}\Delta {\tilde{u}}_k+{\tilde{d}}_k$$

(50)

Step 2 state equation:

$${\lambda }_{k+2}=\tilde{A}{\lambda }_{k+1}+\tilde{B}\Delta {\tilde{u}}_{k+1}+{\tilde{d}}_{k+1}$$

(51)

$${\lambda }_{k+2}=\tilde{A}\left(\tilde{A}{\lambda }_k+\tilde{B}\Delta {\tilde{u}}_k+{\tilde{d}}_k\right)+\tilde{B}\Delta {\tilde{u}}_{k+1}+{\tilde{d}}_k$$

(52)

$${\lambda }_{k+2}={\tilde{A}}^2{\lambda }_k+\tilde{A}\tilde{B}\Delta {\tilde{u}}_k+\tilde{B}\Delta {\tilde{u}}_{k+1}+\left(\tilde{A}+I\right){\tilde{d}}_k$$

(53)

Step n state equation:

$${\lambda }_{k+n}={\tilde{A}}^n{\lambda }_k+{\displaystyle \sum _{i=0}^{n-1}{\tilde{A}}^{n-1-i}\tilde{B}\Delta {\tilde{u}}_{k+i}}+{\displaystyle \sum _{i=0}^{n-1}{\tilde{A}}^{n-1-i}{\tilde{d}}_k}$$

(54)

Step n output equation:

$$y_{k+n}=\tilde{C}{\tilde{A}}^n{\lambda }_k+{\displaystyle \sum _{i=0}^{n-1}\tilde{C}{\tilde{A}}^{n-1-i}\tilde{B}\Delta {\tilde{u}}_{k+i}}+{\displaystyle \sum _{i=0}^{n-1}\tilde{C}{\tilde{A}}^{n-1-i}{\tilde{d}}_k}$$

(55)

To construct the optimal function over $n$ steps, the system inputs and outputs for the future $n$ steps are concatenated. We redefine the output and input sequences for the future $n$ steps as

$${\hat{y}}_k={\left[\begin{array}{l}{y}_{k+1}\\ y_{k+2}\\ ⋮\\ y_{k+n}\end{array}\right]}_{\left(11n*1\right)}, \Delta {\hat{u}}_k={\left[\begin{array}{l}\Delta {\tilde{u}}_k\\ \Delta {\tilde{u}}_{k+1}\\ ⋮\\ \Delta {\tilde{u}}_{k+n-1}\end{array}\right]}_{\left(9n*1\right)}$$

(56)

And the coefficient matrices $V$, $W$ and $Z$:

$$V={\left[\begin{array}{l}\tilde{C}\tilde{A}\\ \tilde{C}{\tilde{A}}^2\\ ⋮\\ \tilde{C}{\tilde{A}}^n\end{array}\right]}_{\left(11n*20\right)}, W={\left[\begin{array}{cccc}\tilde{C}\tilde{B}& 0_{\left(11*9\right)}& \cdots & 0_{\left(11*9\right)}\\ \tilde{C}\tilde{A}{\tilde{B}}^2& \tilde{C}\tilde{B}& \cdots & 0_{\left(11*9\right)}\\ ⋮& \cdots & \ddots & \cdots \\ \tilde{C}{\tilde{A}}^{n-1}\tilde{B}& \tilde{C}{\tilde{A}}^{n-2}\tilde{B}& \cdots & \tilde{C}\tilde{B}\end{array}\right]}_{\left(11n*9n\right)}, Z={\left[\begin{array}{c}\tilde{C}{I}_{20*20}\\ \tilde{C}\tilde{A}+\tilde{C}{I}_{20*20}\\ ⋮\\ {\displaystyle \sum _{i=0}^{n-1}\tilde{C}{\tilde{A}}_k^{n-1-i}}\end{array}\right]}_{\left(11n*20\right)}$$

(57)

Thus, the modified output equation for the future $n$ steps is obtained. The physical meaning of the outputs over the future $n$ steps is the trajectory tracking state deviation and input deviation over that period.

$${\hat{y}}_k=V·{\lambda }_k+W·\Delta {\hat{u}}_k+Z·{\tilde{d}}_k$$

(58)

This is a linear output equation based on time $k$, projecting forward $n$ steps.

3.3. Solving the MPC Optimal Objective

Based on the future time-domain prediction model established in Section 3.2, the core task of the MPC controller is to find an optimal control input sequence within the prediction horizon that enables the system to both accurately track the target trajectory and ensure the smoothness and physical feasibility of the control actions. This objective is achieved by constructing a quadratic objective function—the quadratic form not only possesses convexity, guaranteeing the uniqueness of the global optimal solution, but also allows for flexible prioritization of different optimization goals through weighting matrices, catering to the system’s dual requirements for trajectory tracking accuracy and control input constraints.

First, define the core objective function at time $k$:

$$J_k={\displaystyle \sum _{i=1}^n{‖y_{k+i}‖}^{2}Q}+{\displaystyle \sum _{i=0}^{n-1}{‖\Delta {\tilde{u}}_{k+i}‖}^{2}R}$$

(59)

Here, $n$ is the prediction horizon length, representing the number of steps the controller optimizes forward in a receding manner; $y_{k+i}$ denotes the predicted system output for the future $i$-th step (i.e., at time $k+i$) based on the state information at time $k$, with its physical meaning being the combination of trajectory-tracking state deviation and input deviation at that moment; $\Delta {\tilde{u}}_{k+i}$ is the control increment for the future $i$-th step, i.e., the difference between the current control input and the previous control input. $Q$ and $R$ are the output deviation weighting matrix and the control increment weighting matrix, respectively, both being positive definite symmetric matrices. The diagonal elements of $Q$ are non-negative and are used to adjust the weights of different output deviation terms—if priority is given to trajectory tracking accuracy, the elements corresponding to the trajectory deviation dimensions in $Q$ can be increased. The diagonal elements of $R$ are also non-negative and are used to constrain the magnitude of control increments, preventing abrupt changes in control inputs that could cause system oscillations or exceed hardware drive capabilities (such as motor torque limits), thereby ensuring the smoothness of control actions.

Equation (63) inherently represents a trade-off between tracking accuracy optimization and control smoothness constraint. The first term ${\displaystyle \sum _{i=1}^n{‖y_{k+i}‖}^{2}Q}$ denotes the weighted cumulative sum of output deviations across all time steps within the prediction horizon, with the objective of minimizing this portion to drive the system output as close as possible to the target trajectory. The second term ${\displaystyle \sum _{i=0}^{n-1}{‖\Delta {\tilde{u}}_{k+i}‖}^{2}R}$ denotes the weighted cumulative sum of control increments across all time steps within the prediction horizon, with the objective of minimizing this portion to avoid abrupt changes in the control inputs, thereby protecting the actuators and enhancing system stability.

Remark 1. 

The standard form of the QP problem is given by

$$J={\displaystyle \frac{1}{2}}{x}^{T}Hx+f^{T}x+c$$

(60)

The standard form can include equality constraints, inequality constraints, and boundary constraints. In this paper, Equation (62) represents the system dynamic equality constraint. Since the system output appears only as quadratic terms in the objective function, substituting it does not introduce new constraints, but merely alters the coefficients of the objective function. Therefore, substituting Equation (62) is equivalent to implicitly incorporating the equality constraint into the objective function, thereby reducing the number of constraints in the QP problem and simplifying the solution. Subsequently, only the inequality constraints and boundary constraints based on the physical limitations of the actuators need to be specified to proceed with the optimal solution.

To transform the objective function into the standard QP form suitable for direct solving, the linear output equation (projected forward $n$ steps from time $k$) is substituted into the optimal solution objective.

$$J_k={‖{\hat{y}}_k‖}^{2}Q+{‖\Delta {\hat{u}}_k‖}^{2}R$$

(61)

Here, $Q$ and $R$ are constant matrices. The values of ${\hat{y}}_k$ and $\Delta {\hat{u}}_k$ that minimize this objective function represent the optimal trajectory tracking over the future $n$ steps. From this solution, the optimal input at time $k$ can be extracted.

Substitute the predictive model into the above equation:

$$J_k={‖V·{\lambda }_k+W·\Delta {\hat{u}}_k+\Theta ·{\tilde{d}}_k‖}^{2}Q+{‖\Delta {\hat{u}}_k‖}^{2}R$$

(62)

Express the weighted norm as a matrix quadratic form:

$$J_k={\left(V·{\lambda }_k+W·\Delta {\hat{u}}_k+\Theta ·{\tilde{d}}_k\right)}^{T}Q\left(V·{\lambda }_k+W·\Delta {\hat{u}}_k+\Theta ·{\tilde{d}}_k\right)+\Delta {\hat{u}}_k^{T}R\Delta {\hat{u}}_k$$

(63)

To match the quadratic form standard form of QP, let

$$\left\{\begin{array}{l}H=2W^{T}QW+2R\\ f^T=2W^{T}Q\left(V{\lambda }_k+\Theta {\tilde{d}}_k\right)\\ c={\left(V{\lambda }_k+\Theta {\tilde{d}}_k\right)}^{T}Q\left(V{\lambda }_k+\Theta {\tilde{d}}_k\right)\end{array}\right.$$

(64)

we obtain

$$J_k={\displaystyle \frac{1}{2}}\Delta {\stackrel{⌢}{u}}_k^T\left(2W^{T}QW+2R\right)\Delta {\stackrel{⌢}{u}}_k+2W^{T}Q\left(V{\lambda }_k+\Theta {\tilde{d}}_k\right)\Delta {\stackrel{⌢}{u}}_k+{\left(V{\lambda }_k+\Theta {\tilde{d}}_k\right)}^{T}Q\left(V{\lambda }_k+\Theta {\tilde{d}}_k\right)$$

(65)

The design of constraints essentially involves control input constraints and control increment constraints, with continuous definitions expressed in terms of values at time $k$. First, the actual control input vector at time $k$ is

$$\left(u_{k\left(\min \right)}-u_{P\left(k\right)}\right)\le {\tilde{u}}_k\le \left(u_{k\left(\max \right)}-u_{P\left(k\right)}\right)$$

(66)

The control input deviation is

$${\tilde{u}}_k=\left(u_k-u_{P\left(k\right)}\right)$$

(67)

The increment of the control input deviation is

$$\Delta {\tilde{u}}_k=\left({\tilde{u}}_k-{\tilde{u}}_{k-1}\right)$$

(68)

Therefore, based on the maximum and minimum values of the input, the change in input per cycle has maximum and minimum limits. The upper and lower bounds of the control input deviation are determined by the physical limits of the actual control input:

$$\left(u_{k\left(\min \right)}-u_{P\left(k\right)}\right)\le {\tilde{u}}_k\le \left(u_{k\left(\max \right)}-u_{P\left(k\right)}\right)$$

(69)

Abbreviated as

$${\tilde{u}}_{k\left(\min \right)}\le {\tilde{u}}_k\le {\tilde{u}}_{k\left(\max \right)}$$

(70)

Expressing the increment of the control input deviation in terms of the original control input:

$$\Delta {\tilde{u}}_k=\left(u_k-u_{P\left(k\right)}\right)-\left(u_{k-1}-u_{P\left(k-1\right)}\right)$$

(71)

Rearranging and combining yields

$$\Delta {\tilde{u}}_k=\left(u_k-u_{k-1}\right)-\left(u_{P\left(k\right)}-u_{P\left(k-1\right)}\right)$$

(72)

Thus, the constraints on the deviation increment are jointly determined by the actual control increment constraints and the change in the reference input:

$$\left(\Delta u_{k\left(\min \right)}-\Delta u_{P\left(k\right)}\right)\le \Delta {\tilde{u}}_k\le \left(\Delta u_{k\left(\max \right)}-\Delta u_{P\left(k\right)}\right)$$

(73)

Abbreviated as

$$\Delta {\tilde{u}}_{k\left(\min \right)}\le \Delta {\tilde{u}}_k\le \Delta {\tilde{u}}_{k\left(\max \right)}$$

(74)

Deriving according to the standard form of inequality constraints. The control deviations over the future $n$ steps are expressed in terms of the current deviation and the increment sequence. First, at time $k$:

$${\tilde{u}}_k={\tilde{u}}_{k-1}+\Delta {\tilde{u}}_k$$

(75)

At time $k+1$:

$${\tilde{u}}_{k+1}={\tilde{u}}_{k-1}+\Delta {\tilde{u}}_k+\Delta {\tilde{u}}_{k+1}$$

(76)

And so forth:

$${\tilde{u}}_{k+n-1}={\tilde{u}}_{k-1}+\Delta {\tilde{u}}_k+\Delta {\tilde{u}}_{k+1}+\cdots +\Delta {\tilde{u}}_{k+n-1}$$

(77)

Writing the above n relationships in matrix form:

$$\left[\begin{array}{l}{\tilde{u}}_k\\ {\tilde{u}}_{k+1}\\ ⋮\\ {\tilde{u}}_{k+n-1}\end{array}\right]=\left[\begin{array}{l}{\tilde{u}}_{k-1}\\ {\tilde{u}}_{k-1}\\ ⋮\\ {\tilde{u}}_{k-1}\end{array}\right]+\left[\begin{array}{cccc}I& 0& 0& 0\\ I& I& 0& 0\\ ⋮& ⋮& \ddots & ⋮\\ I& I& \dots & I\end{array}\right]\left[\begin{array}{l}\Delta {\tilde{u}}_k\\ \Delta {\tilde{u}}_{k+1}\\ ⋮\\ \Delta {\tilde{u}}_{k+n-1}\end{array}\right]$$

(78)

The boundary constraints in standard quadratic form are obtained:

$$\left[\begin{array}{c}\Delta {\tilde{u}}_{k\left(\min \right)}\\ ⋮\\ \Delta {\tilde{u}}_{\left(k+n-1\right)\left(\min \right)}\end{array}\right]\le \Delta {\hat{u}}_k\le \left[\begin{array}{c}\Delta {\tilde{u}}_{k\left(\max \right)}\\ ⋮\\ \Delta {\tilde{u}}_{\left(k+n-1\right)\left(\max \right)}\end{array}\right]$$

(79)

At this point, the standard quadratic form can be solved using QP to obtain $\Delta {\hat{u}}_k$, and then $u_k$ is calculated according to

$$\Delta {\hat{u}}_k\Rightarrow \Delta {\tilde{u}}_k\Rightarrow {\tilde{u}}_k\Rightarrow u_k$$

(80)

In this section, we discussed the controller design for robotic arm trajectory tracking with optimal collision avoidance and derived an MPC-based method for solving the optimal control input.

4. Test Results

This section validates the effectiveness of the state-extended model predictive control method proposed in this paper. First, the hardware platform is introduced. Subsequently, the operational results of robotic arm trajectory tracking and suspension winch unit obstacle avoidance are presented, followed by a quantitative analysis combining graphical and tabular data. Finally, the data is summarized, addressing the innovations highlighted in the introduction, to verify the performance of the proposed method.

4.1. Test System

The optimal control method proposed in this paper can be applied to the experimental system illustrated in Figure 6. This system serves as the physical implementation of the research concept outlined in Section 1.3. Its core lies in constructing a multi-point suspension hardware architecture with kinematically redundant degrees of freedom, aimed at serving the dual control objectives of “trajectory tracking” and “coordinated collision avoidance.”

As shown in Figure 6, the experimental system primarily consists of three independently driven “X-axis Bridge Beam–Z-axis slider–suspension winch unit” assemblies, a space robotic arm body, a central controller, and a measurement system. The three X-axis Bridge Beams are installed in parallel on an overhead track and can move independently along the $x$-axis of the global coordinate system. A Z-axis slider, capable of moving along the bridge beam’s $z$-axis, is suspended beneath each X-axis Bridge Beam. Each Z-axis slider is equipped with a suspension winch unit that can slide along a guide rail (i.e., the $x$-axis direction). The suspension winch units are connected via wire ropes to the three Suspension Interface points on the robotic arm links. This redundant DOF design—where a suspension winch unit can be driven either by its X-axis Bridge Beam or can slide relative to its Z-axis slider—forms the key physical foundation for achieving the “crossing” motion between suspension winch units mentioned in the introduction, thereby expanding the robotic arm’s test workspace.

The system’s core control objective is to make the three suspension winch units precisely track the motion trajectories of their corresponding target points on the robotic arm within the horizontal plane ($x$-$z$ plane). During tracking, collisions between suspension mechanism components (between X-axis Bridge Beams, and between X-axis Bridge Beams and suspension winch units) are actively avoided by coordinating the motions of the Bridge Beams and the winch units. The experimental platform integrates high-precision encoders and LiDAR to acquire real-time information on suspension winch unit positions, robotic arm target point positions, and potential collision risks, feeding this data back to the central controller. Following the state-extended MPC algorithm designed in Section 3 of this paper, the controller performs online receding-horizon optimization to solve for the optimal control sequence over the future horizon that balances tracking accuracy and collision avoidance safety. This sequence then drives the physical actuators (Bridge Beam motors, guide rail motors) to accomplish the tracking and obstacle avoidance tasks.

4.2. Robotic Arm Motion Trajectory

To comprehensively evaluate the trajectory tracking and coordinated collision avoidance performance of the multi-suspension winch unit system under complex, dynamic scenarios, this paper plans and executes a comprehensive motion trajectory for a 7-DOF space robotic arm. This trajectory encompasses significant pose variations and multi-joint coordinated movements. Its design aims to fully engage the system’s tracking capabilities and collision avoidance requirements. The corresponding translational and rotational motions of the robotic arm’s base coordinate system (Frame B) relative to the global coordinate system (Frame A), as well as the angle variations of its seven joints, are detailed in

Table 2. Robotic arm motion trajectory table.

Time	Joint 1	Joint 2	Joint 3	Joint 4	Joint 5	Joint 6	Joint 7	Bx	Bz	By
0	0	0	90	180	−30	−90	0	0	0	0
40	−180	0	90	180	−30	−90	0	0	0	0
30	−180	0	90	114.98	−30	−90	0	0	0	0
30	−180	0	0	114.98	−30	−90	0	0	0	0
20	−120	−23.18	−48.81	114.98	−30	−90	0	0	0	0
20	−120	−23.18	−48.81	114.98	11.02	−90	0	0	0	0
10	−120	−23.18	−48.81	114.98	11.02	−62.64	0	0	0	0
10	−120	−23.18	−48.81	114.98	11.02	−62.64	−153.69	0	0	0
40	−120	−97.81	−35.33	109.85	−6.99	−48.07	−114.39	0	−120	−49.31
18	−120	−115.77	−26.55	72.67	12.8	−41.44	−105.95	0	−140	−71.5
15	−120	−145.15	−42.77	76.46	3.64	−45.57	−84.4	0	−200	−90
20	−180	−150.03	−29.24	83.89	−54.66	−30.03	0	0	−200	−90
30	−270	−172.45	−36.12	90.16	−84.26	−83.47	93.79	0	−200	−90
13	−270	−144.96	−45.67	107.02	−92.47	−75.3	98.71	−390	−330	−113.88
12	−270	−119	−44.27	104.34	−92.57	−68.53	103.12	−750	−450	−135.92
10	−270	−97.49	−45.68	106.81	−94.44	−65.58	105.2	−1150	−350	−154.28
14	−270	−68.03	−54.39	123.03	−100.55	−71.1	101.4	−1470	−210	−180

.

Table 2 fully describes a continuous motion trajectory with a duration of approximately 300 s. In it, the first column “Time” indicates the duration (in seconds) taken for the robotic arm to move from the previous configuration to the configuration in the current row. Throughout the motion, the robotic arm’s base undergoes significant displacement (Bx changes from 0 mm to −1470 mm, Bz from 0 mm to −210 mm) and rotation (Bry changes from 0° to −180°). Simultaneously, the seven joints move coordinately, producing a wide range of angular variations—for instance, Joint 1 moves from 0° to −270°, and Joint 2 from 0° to −68.03°. This coupling of base displacement with multi-joint motion causes the robotic arm’s end-effector and the three fixed target points on its links to generate a highly dynamic, nonlinear trajectory in three-dimensional space, encompassing long-distance translation, attitude adjustments, and complex paths. This poses a severe challenge to the tracking accuracy and coordination capability of the ground-based suspension winch unit system.

To visually demonstrate the spatiotemporal characteristics of the tracking targets, Figure 7, Figure 8 and Figure 9 present the $x$-direction trajectories, $z$-direction trajectories, and $x$-$z$ plane trajectories of the three target points in the global coordinate system, respectively. From Figure 7 and Figure 8, it can be seen that the $x$ and $z$ coordinates of the target points vary over time, and the motions of the three points are both interrelated and independent, effectively simulating the non-uniform motion of different parts of a robotic arm in real tasks. Figure 7 reveals that the $x$-trajectories of the suspension winch units exhibit “crossing” phenomena. This trajectory can verify the performance of the redundant degrees of freedom when facing complex paths. The $x$-$z$ plane trajectories in Figure 9 further illustrate the complexity of the motion: the three trajectories interweave, cover a wide area, and the relative distances and orientations between any two target points continuously change dynamically during the motion. This means that the three suspension winch units responsible for tracking them must not only accomplish their respective independent rapid tracking tasks but also frequently adjust their relative positions in the process, thereby continuously facing potential mutual approach and collision risks.

$x_P\left(t\right)$, $u_P\left(t\right)$ constitute an artificially designed and manually validated trajectory. This trajectory encompasses potential collision risks while simultaneously ensuring collision avoidance feasibility, meaning that no physically unavoidable collisions are present. During actual operation, the method proposed in this paper tracks this trajectory as its target, while seeking optimal system inputs for collision avoidance in real time. Since the reference trajectory is feasible, at any time step $k$ the following feasible solution exists.

$$\Delta {\hat{u}}_k^p=\left[\begin{array}{l}{\tilde{u}}_{k\left|k\right.}^p-{\tilde{u}}_{k-1}\\ {\tilde{u}}_{k\left|k+1\right.}^p-{\tilde{u}}_{k\left|k\right.}^p\\ ⋮\\ {\tilde{u}}_{k\left|k+n-1\right.}^p-{\tilde{u}}_{k\left|k+n-2\right.}^p\end{array}\right]$$

(81)

Therefore, while tracking this trajectory, the method proposed in this paper is recursively feasible.

Define the cost function

$$V*\left({\lambda }_k\right)=\underset{\Delta \hat{u}}{\min }\left({‖{\hat{y}}_k‖}^{2}Q+{‖\Delta {\hat{u}}_k‖}^{2}R\right)$$

(82)

There exists a constant $M>0$, such that

$$0<V*\left({\lambda }_k\right)<M$$

(83)

Its upper bound, $M$, can be obtained via $J_k^p$, and its lower bound is guaranteed by non-negativity. Therefore, under the condition of tracking this trajectory, the algorithm proposed in this paper is stable. The following experimental results also validate this conclusion.

In summary, the verification in this work uses a pre-planned trajectory as the tracking target. This trajectory requires collision avoidance and admits a feasible collision-avoidance solution, thereby validating the proposed trajectory tracking and optimal collision avoidance algorithm based on state-extended MPC.

4.3. System Performance

The system parameters, collision boundary function parameters, MPC weighting parameters, and MPC prediction parameters used in the experiments of this study are summarized in the

Table 3. Key parameter settings for the controller.

Parameter Category	Parameter Name	Symbol	Data
System Dimension	State Dimension	$n_x$	11
	Control input dimensions	$n_u$	9
Collision Boundary Parameters	Fundamental Collision Distance	$G$	1000 mm
	Shape Factor	$\beta$	2
	Maximum horizontal avoidance speed	$v_{x(\max )}$	500 mm/s
MPC Weight Matrix	State Weight Matrix	$Q$	Suspension Winch Unit offset state: 0.01
			Tracking and Safety Distance Status: 0.1
	Control Input Weight Matrix	$R$	10 (for Input Component: 1, 3, 4, 6, 7, 9)
			1 (for Input Component: 2, 5, 8)
Time parameter	Time domain prediction	$N_{pred}$	10
	Control Cycle	$T$	0.01 s
	System Delay	$\tau$	0.01 s

below.

$$\left\{\begin{array}{l}Q\left(n,n\right)=0.1,n=1\cdots 6,10,11\\ Q\left(n,n\right)=0.01,n=7,8,9\\ others=0\end{array}\right.$$

(84)

$$\left\{\begin{array}{l}R\left(n,n\right)=10,n=1,3,4,6,7,9\\ R\left(n,n\right)=1,n=2,5,8\\ others=0\end{array}\right.$$

(85)

The experimental trajectory comprised 33,184 steps, with a time interval of 0.01 s between adjacent trajectory points and a control cycle of 0.01 s, resulting in a total trajectory duration of 331.8 s. Based on these parameters, the controller predicts 10 steps ahead, solves for the optimal solution, and selects the first control input as the current command. The actual computation time per cycle is 4.477 milliseconds. This computation time occupies 44.8% of the control cycle, indicating that the computational load is within an acceptable range.

4.3.1. Trajectory Tracking Performance Analysis

The system’s tracking performance is evaluated based on the deviation between the positions of the suspension winch units and the desired positions of the robotic arm target points. Figure 10 and Figure 11, respectively, show the real-time tracking trajectories (solid lines) and the desired trajectories (dashed lines) of the three suspension winch units in the $x$-direction and $z$-direction. It can be visually observed that the actual trajectories closely coincide with the desired trajectories throughout the entire highly dynamic motion. Figure 12 further illustrates this tracking effect in the $x$-$z$ plane, where all three suspension winch unit trajectories tightly follow their respective complex, interwoven desired paths without noticeable lag or deviation.

A quantitative analysis of the tracking error is presented in Figure 13 and Figure 14. Throughout the experiment, the tracking errors for all suspension winch units in both the $x$-direction and $z$-direction were strictly confined within a range of ±1 mm. This result directly addresses the first core objective raised in the Introduction and Section 1.3. It demonstrates that the designed MPC controller can achieve high-precision tracking of dynamic trajectories for multiple target points, even in the presence of time delays and external disturbances. This meets the stringent accuracy requirements for tracking in ground-based gravity compensation testing of space robotic arms.

To validate the effectiveness of the proposed method, a comparative experiment was conducted using a standard PID controller under the same reference trajectory. The tracking errors of the target in the $x$ and $z$ directions are shown in Figure 15 and Figure 16. It can be observed that the trajectory-tracking errors of the PID-based approach are significantly larger than those achieved using the proposed method.

As can be seen from the figures above, the proposed method achieves excellent tracking performance for the target trajectory. The maximum tracking error in the X-direction is 0.64 mm with a maximum standard deviation of 0.32 mm, while in the Z-direction, the maximum tracking error is 0.13 mm with a maximum standard deviation of 0.29 mm. In comparison, the PID controller yields substantially larger tracking errors, with a maximum of 1.3 mm in the X-direction and 1.9 mm in the Z-direction, and corresponding maximum standard deviations of 0.47 mm and 0.84 mm, respectively. The specific tracking performance for the three trajectories is detailed in the

Table 4. Quantitative analysis of trajectory-tracking performance.

Trajectory	Maximum Deviation (mm)		Standard Deviation (mm)
	Proposed MPC	Comparative PID	Proposed MPC	Comparative PID
$x_{c1}\left(t\right)$	0.319	0.552	0.161	0.172
$x_{c3}\left(t\right)$	0.640	0.752	0.322	0.185
$x_{c5}\left(t\right)$	0.354	1.306	0.161	0.469
$z_{c1}\left(t\right)$	0.131	1.561	0.028	0.607
$z_{c3}\left(t\right)$	0.115	1.778	0.028	0.682
$z_{c5}\left(t\right)$	0.113	1.941	0.029	0.838

below.

4.3.2. Collision Avoidance State Analysis

The innovation of this paper lies in transforming the collision avoidance problem into a state optimization objective. Figure 17 shows state components 7–9, which are the offsets of each suspension winch unit within the Z-axis slider coordinate frame. Ideally, these offsets should be as close to zero as possible, indicating that the suspension winch unit is positioned at the center of its motion range, thereby reserving maximum space for avoidance maneuvers in any direction. The experimental results show that throughout the entire motion, the three offset states consistently oscillate slightly near zero, with their absolute values maintained at a low level. This proves that the controller successfully regulates the system into a “ready” state with sufficient avoidance redundancy while accomplishing the tracking task.

The core verification of collision avoidance performance is reflected in state components 10–11, namely the differences between the two types of collision safety distances, as shown in Figure 18. The optimal collision avoidance objective is to stabilize both state values at zero, meaning the system balances all potential collision risks rather than addressing one at the expense of another. Experimental data indicate that throughout the entire process, where robotic arm motion causes dynamic changes in collision risk, both safety distance differences are effectively stabilized near zero. This result strongly confirms the effectiveness of the state-extended model and the MPC framework: through online optimization, the controller actively and coordinately adjusts the motion of each suspension winch unit and X-axis Bridge Beam, successfully equalizing all collision risks and maintaining them above safety thresholds. This achieves the “optimal coordinated collision avoidance” proposed in the introduction.

4.3.3. Comparative Analysis of Collision Avoidance States

In previous experiments, the proposed method successfully obtained an optimal collision avoidance solution while performing trajectory tracking. To more clearly analyze the successful implementation of this optimal collision avoidance, we conducted a comparative test. In this comparative test, we employed another collision-avoidance strategy to control the system’s avoidance maneuvers. Specifically, instead of maintaining the system in a state of maximum collision avoidance margin at all times, this alternative strategy activates the actuators for avoidance only when a collision is imminent. This approach also ensures safe operation during the experiment. We have plotted the collision state curve from the comparative test, which corresponds to the curve obtained using our proposed method.

As can be seen from the Figure 19 and Figure 20, the state variables of the comparative group exhibit significant and abrupt changes. States 7–9 reflect the coordinates of the suspension winch unit on the Z-axis slider. In the comparative group, the suspension winch unit moves to its coordinate limit of 700 mm. This indicates that even when the unit reaches its extreme position on the slider, it still cannot fully meet the requirements for trajectory tracking. Consequently, the unit remains at its limit on the slider, forcing the X-axis Bridge Beam to compensate for the remaining tracking error. In this scenario, if a collision requiring avoidance occurs, at least one of the involved parties is already in a limiting state. Collision avoidance can then only be achieved by adjusting the position of the other suspension winch unit on its corresponding slider.

In contrast, the proposed method maintains the suspension winch unit near the origin of the Z-axis slider, with a maximum deviation of no more than 3 mm. This demonstrates that when a potential collision is imminent, the unit retains ample travel distance on the slider to execute the avoidance maneuver effectively.

Similarly, States 10–11 reflect the difference in safety margins between two potentially conflicting collision constraints. A larger difference implies that one side has a substantial safety margin while the other has a very small one. In the comparative experiment, this difference reached 2000 mm. However, under our proposed method, the difference is kept below 1 mm.

While the comparative system can still operate safely, the proposed method achieves more than just safety. It actively seeks the optimal solution for collision avoidance, enabling the system to maintain the best possible safety margins at all times to handle potential collisions proactively.

4.3.4. Control Input and System Stability

The time-varying curves of the system’s nine control inputs ($x$-direction acceleration of the three X-axis Bridge Beams, $z$-direction acceleration of the three Z-axis sliders, and $x$-direction acceleration of the three suspension winch units on the Z-axis sliders) are shown in Figure 21, Figure 22 and Figure 23. All control inputs are smooth and continuous, and none exceed the physical limits of the actuators. The smoothness of the input signals indicates a stable optimization solution process, with no high-frequency chattering or abrupt changes. Furthermore, all eleven system states remain bounded and convergent throughout the entire experiment, with no observed divergence in the system.

It is worth noting that the fundamental safety protection mechanisms remained active throughout our experiments. Therefore, while performing trajectory tracking and optimal collision avoidance, the system is guaranteed to operate safely even when confronted with sudden trajectory changes or infeasible paths—even if an optimal collision-avoidance solution might not be found under such conditions. System safety protection serves as the foundational layer of the overall control architecture.

5. Conclusions

This paper addresses the core challenge in ground-based three-dimensional motion testing of a 7-DOF space robotic arm using an active multi-point suspension gravity compensation system—the coordinated realization of high-precision trajectory tracking and optimal cooperative collision avoidance. Systematic research on modeling, control, and experimentation was conducted. The designed controller ultimately achieved both optimal control objectives. The main conclusions are as follows:

An active multi-point suspension gravity compensation system with kinematically redundant degrees of freedom was successfully designed and implemented, providing the hardware foundation for coordinated trajectory tracking and collision avoidance. By endowing each suspension winch unit with an x-direction offset degree of freedom relative to its X-axis Bridge Beam, the system enables “crossing” motion between suspension winch units. This not only expands the ground test workspace for the space robotic arm but also reserves sufficient motion margin for collision avoidance. It resolves the inherent conflict between “tracking range” and “avoidance safety” in traditional multi-point suspension systems, validating the rationality and effectiveness of the redundant DOF design.

A modeling method that integrates collision avoidance conditions with trajectory-tracking states was proposed, constructing a unified underactuated state-extended system model. Innovatively, this paper transforms inter-point collision avoidance constraints into five system states (the offsets of suspension winch units relative to Z-axis sliders, and the differences between two types of collision safety distances) and integrates them with the six trajectory-tracking states into an 11-dimensional state vector. This unifies the traditionally separated objectives of “tracking” and “avoidance” within a single modeling framework. This model accurately describes the system’s underactuated characteristics (11 states, nine control inputs) and, through the application of Padé approximation to handle time delay and the linearization of the error model, lays the groundwork for controller design.

An MPC suitable for complex underactuated systems was designed, achieving simultaneous optimization of trajectory tracking and optimal collision avoidance. Addressing two types of external disturbances—robotic arm motion and collision boundary changes—the controller generates optimal control commands in each sampling cycle (0.01 s) through receding horizon optimization and QP solution. The experimental results verify the controller’s excellent performance: the trajectory tracking error of multiple suspension winch units for three target points on the robotic arm remains stable within ±1 mm; the collision avoidance-related states (States 7–11) consistently approach zero, achieving balanced avoidance of collision risks; the system control inputs are smooth without saturation, and all 11 states show no divergence. This effectively solves the cooperative motion-planning problem for complex underactuated systems.

In summary, through hardware innovation, breakthroughs in modeling methodology, and controller optimization, this paper successfully resolves the core contradiction between “high-precision tracking” and “optimal collision avoidance” in active multi-point suspension gravity compensation systems for ground-based testing of space robotic arms.

Future research can explore expanding the number of suspension points and increasing redundant degrees of freedom. This paper focuses on a three-point suspension system. However, the complex structure of space robotic arms in practical engineering may require more suspension points for gravity compensation. Future work needs to investigate systems with four or more suspension points, studying the nonlinear growth patterns of collision avoidance states and establishing a modular, scalable state-extended model to address the problems of intensified coupling and a sharp increase in collision risk points among multiple suspension units. Furthermore, the online computational load of the MPC controller increases significantly with the expansion of the prediction horizon and state dimensions. Future research could integrate artificial intelligence technologies such as deep learning and reinforcement learning to streamline the computational process of the MPC’s receding horizon optimization.