Intelligent Control of a Space Manipulator Ground Unfold Experiment System with Lagging Compensation

Abstract

In ground testing of space manipulators, gravity compensation is a critical testing requirement. The objective of this paper was to design a space manipulator gravity compensation test platform for ground tests and solve the problems of force control oscillation and precision degradation caused by the execution lag encountered in the development process. An intelligent PID controller was designed for this active-suspension gravity compensation experimental mechanism of a space manipulator on the ground, and a specially designed second-order method was used to solve the problem of the execution lag in this mechanism. The intelligent controller was developed based on adaptive dynamic programming and redesigned to improve its transient performance. The simulation was carried out, and its results were compared with the results on a real machine to demonstrate the effectiveness of this set of experimental controllers. This paper compares in detail the results of the designed method on system input and output and shows the effectiveness of this method in dealing with the execution lag of the mechanism. In conclusion, in this work, we successfully designed and implemented an intelligent PID controller for an active-suspension gravity compensation experimental mechanism of a space manipulator on the ground, and the experimental results demonstrate the effectiveness of the proposed method.

1. Introduction

Space manipulators are a crucial component of space exploration missions due to their ability to perform a wide range of tasks in space. Many studies have focused on various aspects of space manipulator design, such as dynamic modeling, kinematics and dynamics analysis, and control strategies. Semiautonomous teleoperation based on learning from demonstration is an effective method of remote operation of space manipulators, particularly in scenarios with limited communication and repeated operation problems [1]. The control of space manipulators has its own characteristics and testing methods due to the unique working environment of space. In this regard, a hybrid mapping method using an exoskeleton device and surface electromyography sensors has been proposed for manipulator teleoperation [2]. A non-equidistant fractional-order accumulation (NEFA) model of a visual-based space manipulator that uses a modified extrapolation method to predict the trajectory of a motion target floating in outer space has been presented [3]. Collectively, these studies demonstrate the importance of space manipulators for space exploration and highlight ongoing research efforts to improve their capabilities and performance. To further enhance the performance of space manipulators, ground-based experiments and testing are essential. These tests are typically conducted in a controlled environment and allow researchers to simulate and evaluate different scenarios, such as the effects of gravity and other environmental factors on the manipulator’s performance. One area of focus in ground-based testing of space manipulators is gravity compensation. This involves developing techniques to offset the effects of gravity on the manipulator and maintain its position and orientation in space. The success of space missions relies heavily on the performance and reliability of space manipulators, and ground-based testing plays a critical role in ensuring their success. Due to the fact that space manipulator design work is carried out in a zero-gravity environment, the weight of the manipulator itself will have a significant impact on its movement during ground testing. The torque generated by gravity on the joints of the manipulator may even exceed the mechanical load range, making it impossible for the manipulator to perform certain actions. Several studies have contributed to the development of effective test systems for space manipulators and improve our understanding of their control mechanisms. To achieve efficient assembly, time-optimal trajectory planning based on the genetic algorithm is utilized, and the feasibility of the proposed method is verified through preliminary ground experiments of assembling submirror modules of space telescopes using the KUKA LWR iiwa-7 [4]. A novel active gravity compensation concept has been proposed to simulate space on-orbit microgravity environments for free-flying robots [5] on the ground. Dynamic requirements for designing ground manipulator platforms that can be utilized to verify the performance of the controllers developed for space manipulators are realized [6]. New projects have also been implemented for environmental testing of space robotic arms [7]. Passive gravity compensation and active gravity compensation are the two main methods used in ground-based testing of space manipulators. Passive gravity compensation is achieved by using pulleys and weights to balance the gravity of the manipulator. The advantage of this method is that it provides stable static gravity balance, but its disadvantage is that the additional inertia generated by the weights during the movement of the manipulator can affect the testing results. In contrast, active gravity compensation is achieved by using specially designed actuators to actively control the force applied to the manipulator and maintain a balance between the gravity and the applied force, which is measured by force sensors. This method has the advantage of not generating additional inertia during the movement of the manipulator, and the testing system can achieve good gravity balance within the required accuracy range. Although the effect of active gravity compensation may be slightly inferior to that of passive gravity compensation in static gravity balance, it still has a very high static balance capability and is an effective method. In the design of an active gravity compensation space manipulator test bed, the precision and response speed of the control system can be improved through improved mechanism design and control algorithms to continuously improve the gravity compensation effect of the manipulator on the ground.

The test system designed in this paper used active gravity compensation. In order to achieve good gravity compensation, we designed an intelligent controller and used ADP to adjust the parameters. Finally, the coefficients were fine-tuned based on actual performance. Adaptive dynamic programming (ADP) is a data-driven adaptive optimal control approach. Researchers have proposed novel event-triggered control approaches to solve the adaptive optimal output regulation problem for a class of linear discrete-time systems [8] and an event-triggered robust optimal control approach for large-scale systems with both parametric and dynamic uncertainties through robust adaptive dynamic programming, policy iteration, and small gain techniques [9]. Additionally, a reinforcement learning-based adaptive optimal control approach has been proposed to learn the optimal control gain of the model to solve traffic congestion [10]. These approaches have been extended to solve robust optimal output regulation problems of a class of partially linear systems under both dynamic uncertainties and denial-of-service attacks [11]. Furthermore, an adaptive tire cornering stiffness strategy and a trajectory-tracking autonomous steering control strategy have been proposed for intelligent vehicle trajectory tracking on different road surfaces, including when affected by inclement weather such as rain and snow [12]. ADP has been developed both in theory and practice, leading to a growing interest in its potential applications in various control systems. The problem of attitude control for hypersonic vehicles in the presence of actuator faults and unknown uncertainties, including modeling uncertainties and aerodynamic parameter uncertainties, has been investigated using the integral sliding mode technique [13]. ADP has shown significant potential in the aerospace field, including investigating fixed-time attitude coordinated control problems for multi-spacecraft systems with unknown external disturbances [14]. The design of intelligent controllers has been widely applied in various fields. A new approach for energy management in photovoltaic (PV)/battery/fuel cell (FC) systems has been presented, which compensates for uncertainties using a proposed deep-learning type-2 fuzzy logic compensator (T2FLC) and optimization rules based on the immersion and invariance (I&I) theorem, resulting in acceptable voltage/energy regulation performance under different disturbances and without relying on mathematical models [15]. A novel observer-based fuzzy control method for chaotic systems was presented in this study utilizing a generalized type-2 fuzzy logic system to approximate uncertainties and adjust parameters through robustness investigation for improved control performance [16]. The vehicle dynamic behavior under the influence and effects of an antiroll bar mechanism is understood through a basic vehicle dynamic modeling with four DOFs on a half-car model, which shows how external forces on the front antiroll bar can control the handling and ride comfort of the car [17]. A comparison of performance for an active antiroll bar (ARB) system using two types of control strategy has been analyzed, where the LQG control strategy has been investigated, and a novel LQG CNF fusion control method has been developed to improve the active antiroll bar system performance for vehicle ride comfort and handling [18]. An observer-based predictive control approach has been proposed to actively compensate random communication constraints in the feedback channel of each agent and between agents in the cooperative output tracking control problem of linear heterogeneous multiagent systems with random communication constraints [19]. Additionally, leader-following consensus of distributed multiagent systems (MAS) under sampled-data control has been investigated [20]. Decentralized tracking control (DTC) problems for input-constrained unknown nonlinear interconnected systems have been solved using event-triggered adaptive dynamic programming [21]. The dynamic event-triggered optimal control problem of discrete-time (DT) nonlinear Markov jump systems (MJSs) has been investigated using policy iteration (PI) adaptive dynamic programming (ADP) algorithms [22]. Their successful experience has given us confidence in designing ground test equipment for space manipulators.

We present a control system design method for test equipment that uses a suspension method to balance the gravity of the manipulator. Dynamic compensation of the manipulator’s gravity in real time during its movement puts high demands on the test system’s dynamic response ability. Suitable control parameters become key to the success of the test system development. Intelligent controller design can improve controller performance under certain conditions. For example, the H∞ control of linear discrete-time systems can employ off-policy reinforcement learning to solve the game algebraic Riccati equation online using measured data along the system trajectories [23]. Another approach based on a Q-learning algorithm has been proposed to solve the infinite-horizon linear quadratic tracker (LQT) for unknown discrete-time systems in a causal manner [24]. During the actual debugging process, we discovered that the actuator’s lag caused output shaking, which is a common problem in controller design. Scholars often use adaptive methods to solve it. For instance, a novel adaptive nonsingular terminal sliding mode (TSM) control procedure has been proposed for the speedy and finite time stabilization of nonlinear cyber-physical systems (CPSs) dealing with unwanted disturbances, actuator cyberattacks, and time-varying delays [25]. A time-delayed adaptive dynamic programming framework for optimal energy control has also been utilized to reconstruct the state utilizing more comprehensive historical data [26]. These adaptive control methods were applied to enhance the performance of our test system. The papers referenced in this paragraph are only a few examples of the extensive literature available on the topic of adaptive control, which has been demonstrated to have vast potential in improving the performance of complex control systems. In this paper, we designed a ground test system and presented a gravity compensation method for space manipulators, which allows a space manipulator to perform motion tests in the ground gravity environment. We used the adaptive dynamic programming (ADP) method to adjust the controller parameters and proposed a compensation method for the system’s lagging. We present our own solution to improve the controller’s vibration problem. Our approach involves using ADP to adjust the intelligent parameters of the controller and fine-tune them based on the actual operating effect.

The remainder of this paper is as follows. We formulate the problem and the intelligent controller design in Section 2. Then, the simulation results and the results from the real equipment are compared in Section 3. The algorithm in our solution for solving the lagging is discussed in Section 4. Finally, the conclusions are contained in Section 5.

2. Problem Formulation

2.1. Model Building

In this article, we designed a set of mechanisms composed of three identical parts. Its function was to apply the set value of vertical upward force to the given three points in the space through a steel wire rope, and it could adapt to the three-dimensional space movement of the given three points maintaining the vertical direction and tension of the steel wire rope at the set value. The three components of this mechanism operated independently, and each part could apply force to a given point. We called this part the “lifting point”. The structure of a single lifting point and the definition of the coordinate system used in the article are shown in Figure 1.

The Y direction of the coordinate system is the vertical upward direction, and the XZ direction is in the horizontal plane. There were three pulleys in the lifting point named ABC. Pulley A was a fixed pulley with a sensor above it to measure the tension of the steel wire rope. Pulley B was a movable pulley that could actively move in the Y direction as an actuator, causing the retraction and release of the steel wire rope in the Y direction. The C pulley was an actuator that could wind the steel wire rope, which could also control the retraction and release of the steel wire rope in the Y direction. Between points N and M there was a part where the steel wire rope extended out of the lifting point. Point M was connected with the point to which we applied the force in the space so that the force could be applied to point M through the steel wire. There was an inclination measuring mechanism at point N, which could measure the inclination of the steel wire rope along the X and Z directions. These two inclination angles were α and γ. Maintaining these two inclination angles at zero, it could be considered that the steel wire rope was vertical and the force applied to point M was also vertical. A spring was connected in series to the steel wire rope between points M and N, and the total distance between points M and N was l.

To describe this system, the coordinates of the given point in space are expressed as ${\left({\mathrm{x}}_2,{\mathrm{y}}_2,{\mathrm{z}}_2\right)}^{\mathrm{T}}$, the coordinates of M in the natural state are ${\left({\mathrm{x}}_1\left(\mathrm{t}-\tau \right),{\mathrm{y}}_1\left(\mathrm{t}-\tau \right),{\mathrm{z}}_1\left(\mathrm{t}-\tau \right)\right)}^{\mathrm{T}}$, ${\left({\mathrm{x}}_1,{\mathrm{y}}_1,{\mathrm{z}}_1\right)}^{\mathrm{T}}$ is the input of the system, and ${\left(\alpha ,\mathrm{F},\gamma \right)}^{\mathrm{T}}$ is the output of the system. The relationship between them can be expressed by Equation (1):

$$\left\{\begin{array}{l}\mathrm{l}\cos \alpha \left(\mathrm{t}\right)\cos \gamma \left(\mathrm{t}\right)={\mathrm{y}}_{\max }-{\mathrm{y}}_2\left(\mathrm{t}-\tau \right)\hfill \\ \mathrm{l}\sin \alpha \left(\mathrm{t}\right)={\mathrm{x}}_2\left(\mathrm{t}\right)-{\mathrm{x}}_1\left(\mathrm{t}-\tau \right)\hfill \\ \frac{\mathrm{F}\left(\mathrm{t}\right)}{\mathrm{k}}=-{\mathrm{y}}_2\left(\mathrm{t}\right)+{\mathrm{y}}_1\left(\mathrm{t}-\tau \right)\hfill \\ \mathrm{l}\cos \alpha \left(\mathrm{t}\right)\sin \gamma \left(\mathrm{t}\right)={\mathrm{z}}_2\left(\mathrm{t}\right)-{\mathrm{z}}_1\left(\mathrm{t}-\tau \right)\hfill \end{array}\right.$$

(1)

where k is the coefficient in Hooke’s law, which is the ratio of force to displacement. In this model, this is the inherent parameter of the system.

We considered carrying out linearization when the wire rope was nearly in a vertical state with the condition of small inclination. Under this condition, $\sin \left(\alpha \right)$ approximately equals $\alpha$, and $\cos \left(\alpha \right)$ approximately equals 1. We linearized Equation (1) and sorted it out to obtain Equation (2):

$$\left\{\begin{array}{l}\mathit{\alpha }\left(\mathrm{t}\right)=\frac{1}{\mathrm{l}}{\mathrm{x}}_2\left(\mathrm{t}\right)-\frac{1}{\mathrm{l}}{\mathrm{x}}_1\left(\mathrm{t}-\tau \right)\hfill \\ {F}\left(\mathrm{t}\right)=-\mathrm{k}{\mathrm{y}}_2\left(\mathrm{t}\right)+\mathrm{k}{\mathrm{y}}_1\left(\mathrm{t}-\tau \right)\hfill \\ \mathit{\gamma }\left(\mathrm{t}\right)=\frac{1}{\mathrm{l}}{\mathrm{z}}_2\left(\mathrm{t}\right)-\frac{1}{\mathrm{l}}{\mathrm{z}}_1\left(\mathrm{t}-\tau \right)\hfill \end{array}\right.$$

(2)

Equation (2) is the mathematical model of an independent lifting point.

In the modern control theory, we usually use the letter “x” to represent the state of the system. In this article, the letter “x” had already been used to represent the coordinates of points. Therefore, in order to not use the same letter twice, we used the letter “s” to represent the state of the system, while other commonly used letters remained unchanged. We set the system status to the coordinate of the suspension point with delay; the system input was the coordinate of the suspension point, and the system delay was τ. We expressed the mathematical model of the system with Equation (3):

$$\left\{\begin{array}{l}{{\mathbf{s}}^{\prime }}_{(\mathrm{n})}=\mathrm{A}{\mathbf{s}}_{(\mathrm{n})}+\mathrm{B}{\mathbf{u}}_{(\mathrm{n}-\tau )}\\ {\mathbf{y}}_{(\mathrm{n})}=\mathrm{C}{\mathbf{s}}_{(\mathrm{n})}+\mathrm{D}{\mathbf{u}}_{(\mathrm{n}-\tau )}+\mathrm{E}\end{array}\right.$$

(3)

where

$${\mathbf{s}}_{(\mathrm{n})}=\left[\begin{array}{l}{\mathrm{x}}_{1(\mathrm{n}-\tau )}\\ {\mathrm{y}}_{1(\mathrm{n}-\tau )}\\ {\mathrm{z}}_{1(\mathrm{n}-\tau )}\end{array}\right],{\mathbf{u}}_{(\mathrm{n})}=\left[\begin{array}{l}{\mathrm{x}}_{1(\mathrm{n})}\\ {\mathrm{y}}_{\mathrm{s}1(\mathrm{n})}\\ {\mathrm{y}}_{\mathrm{b}1(\mathrm{n})}\\ {\mathrm{z}}_{1(\mathrm{n})}\end{array}\right], {\mathbf{y}}_{(\mathrm{n})}=\left[\begin{array}{l}{\alpha }_{(\mathrm{n})}\\ {\mathrm{F}}_{(\mathrm{n})}\\ {\gamma }_{(\mathrm{n})}\end{array}\right], {\mathrm{E}}_{(\mathrm{n})}=\left[\begin{array}{l}\frac{1}{\mathrm{l}}{\mathrm{x}}_{2(\mathrm{n})}\\ {-\mathrm{ky}}_{2(\mathrm{n})}\\ \frac{1}{\mathrm{l}}{\mathrm{z}}_{2(\mathrm{n})}\end{array}\right]$$

Since there were two actuators in the Y direction of the lifting point, we used ${\mathrm{y}}_{\mathrm{s}}$ and ${\mathrm{y}}_{\mathrm{b}}$ to represent the inputs of these two actuators, and we obtained Equation (4):

$${\mathrm{y}}_{1(\mathrm{n})}={\mathrm{y}}_{\mathrm{s}1(\mathrm{n})}+{\mathrm{y}}_{\mathrm{b}1(\mathrm{n})}$$

(4)

The values of matrices A, B, C, and D were as follows:

$$\mathrm{A}=\left[\begin{array}{ccc}-\frac{1}{\tau }& 0& 0\\ 0& -\frac{1}{\tau }& 0\\ 0& 0& -\frac{1}{\tau }\end{array}\right],\mathrm{B}=\left[\begin{array}{cccc}\frac{1}{\tau }& 0& 0& 0\\ 0& \frac{1}{\tau }& \frac{1}{\tau }& 0\\ 0& 0& 0& \frac{1}{\tau }\end{array}\right]\mathrm{C}=\left[\begin{array}{ccc}-\frac{1}{\mathrm{l}}& 0& 0\\ 0& \mathrm{k}& 0\\ 0& 0& -\frac{1}{\mathrm{l}}\end{array}\right]\mathrm{D}={\left[0\right]}_{3\ast 3}$$

Here, we obtained the mathematical model of one lifting point of the system. This system had three lifting points in total, with the same mathematical model.

2.2. Controller Design

It is checkable that System (3) is with input delay. The design of controllers was not the same as in delay-free systems. With this issue in mind, we first developed a controller based on adaptive dynamic programming [11,27,28] via online data. Since the online data could be inaccurate due to the influence of measurement noise, we modified the learned controller to improve the performance of the closed-loop system. The goal of controller design was to design the inputs of the four actuators of each lifting point to maintain the output of the system at the desired value. The input of the system satisfied the relationship given in Equation (5), with the corner mark zero representing the initial state:

$$\left\{\begin{array}{l}{\mathrm{x}}_{1(\mathrm{n})}={\displaystyle \sum _0^{\mathrm{n}}{}^{\mathrm{l}}\mathrm{v}_{\mathrm{x}1(\mathrm{n})}\mathrm{d}\mathrm{t}}+{\mathrm{x}}_{\mathrm{s}(0)}\hfill \\ {\mathrm{y}}_{\mathrm{s}(\mathrm{n})}={\displaystyle \sum _0^{\mathrm{n}}{}^{\mathrm{l}}\mathrm{v}_{\mathrm{y}\mathrm{s}(\mathrm{n})}\mathrm{d}\mathrm{t}}+{\mathrm{y}}_{\mathrm{s}(0)}\hfill \\ {\mathrm{y}}_{\mathrm{b}(\mathrm{n})}={\displaystyle \sum _0^{\mathrm{n}}{}^{\mathrm{l}}\mathrm{v}_{\mathrm{y}\mathrm{b}(\mathrm{n})}\mathrm{d}\mathrm{t}}+{\mathrm{y}}_{\mathrm{b}(0)}\hfill \\ {\mathrm{z}}_{1(\mathrm{n})}={\displaystyle \sum _0^{\mathrm{n}}{}^{\mathrm{l}}\mathrm{v}_{\mathrm{z}1(\mathrm{n})}\mathrm{d}\mathrm{t}}+{\mathrm{z}}_{10}\hfill \end{array}\right.$$

(5)

We could control the input of the four controllers by adjusting the speed commands of the four actuators. Before giving the expression of speed commands, we describe the four methods used in this paper.

Method 1, saturation:

The saturation here was to limit the amplitude of a variable and the change of the variable with time at the same time [29]. When l was added to the upper left corner of the variable, this expression was used to represent the value of the variable after limiting, as in Equation (6).

$${}^{\mathrm{l}(\mathrm{v},\mathrm{a})}\mathrm{v}_{(\mathrm{n})}=\left\{\begin{array}{cc}{\mathrm{v}}_{(\mathrm{n})}\hfill & {\mathrm{v}}_{(\mathrm{n})}\le {\mathrm{v}}_{\max }, \left|{\dot{\mathrm{v}}}_{(\mathrm{n})}\right|\le {\mathrm{a}}_{\max }\hfill \\ {\mathrm{a}}_{\max }\mathrm{d}\mathrm{t}+{}^{\mathrm{l}}\mathrm{v}_{(\mathrm{n}-1)}\hfill & {\mathrm{v}}_{(\mathrm{n})}\le {\mathrm{v}}_{\max },\left({\mathrm{v}}_{(\mathrm{n})}-{}^{\mathrm{l}}\mathrm{v}_{(\mathrm{n}-1)}\right)/\mathrm{d}\mathrm{t}>{\mathrm{a}}_{\max }\hfill \\ -{\mathrm{a}}_{\max }\mathrm{d}\mathrm{t}+{}^{\mathrm{l}}\mathrm{v}_{(\mathrm{n}-1)}\hfill & {\mathrm{v}}_{(\mathrm{n})}\le {\mathrm{v}}_{\max },\left({\mathrm{v}}_{(\mathrm{n})}-{}^{\mathrm{l}}\mathrm{v}_{(\mathrm{n}-1)}\right)/\mathrm{d}\mathrm{t}<-{\mathrm{a}}_{\max }\hfill \\ {\mathrm{v}}_{\max }\hfill & {\mathrm{v}}_{(\mathrm{n})}>{\mathrm{v}}_{\max }\hfill \end{array}\right.$$

(6)

Method 2, piecewise constant function:

The coefficient of a variable is a function of that variable [30]. We added a piecewise constant function with a hysteresis-like structure at the boundary and then limited the amplitude of the result according to method 1 to obtain this function. The expression of the function is shown in Equation (7):

$$\mathrm{K}=\left\{\begin{array}{ll}{\mathrm{K}}_1\hfill & \mathrm{e}<{\mathrm{e}}_1-{\mathrm{d}}_1\hfill \\ {\mathrm{K}}_1,{\mathrm{K}}_2\hfill & {\mathrm{e}}_1-{\mathrm{d}}_1\le \mathrm{e}\le {\mathrm{e}}_1+{\mathrm{d}}_1\hfill \\ {\mathrm{K}}_2\hfill & {\mathrm{e}}_1+{\mathrm{d}}_1<\mathrm{e}<{\mathrm{e}}_2-{\mathrm{d}}_2\hfill \\ {\mathrm{K}}_2,{\mathrm{K}}_3\hfill & {\mathrm{e}}_2-{\mathrm{d}}_2\le \mathrm{e}\le {\mathrm{e}}_2+{\mathrm{d}}_2\hfill \\ ⋮\hfill & ⋮\hfill \end{array}\right.,\mathrm{K}(\mathrm{e})={}^{\mathrm{l}}\mathrm{K}$$

(7)

where ${\mathrm{K}}_1$ is the control gain learned by the adaptive dynamic programing method. In the following text, the coefficients processed according to the function defined here were used in the later PID controller coefficients and other coefficients, unless otherwise specified. We set a width at each switching point. When an independent variable changed near the switching point plus width, the output could be switched only when the independent variable increased. When an independent variable changed near the decreasing width of the switching point, the output was switched only when the independent variable decreased. This is shown in Figure 2.

In this way, when an independent variable was the original data collected by the sensor with noise, frequent switching of the function output could be avoided.

Method 3, filtering:

We added the letter “f” to the upper left corner of a filtered variable to represent it as Equation (8). The filters used in this paper were Butterworth second-order low-pass filters [31].

$${}^{\mathrm{f}}\mathrm{e}_{(\mathrm{n})}=\mathrm{f}\left({\mathrm{e}}_{(\mathrm{n})},{\mathrm{e}}_{(\mathrm{n}-1)},{\mathrm{e}}_{(\mathrm{n}-2)},{}^{\mathrm{f}}\mathrm{e}_{(\mathrm{n}-1)},{}^{\mathrm{f}}\mathrm{e}_{(\mathrm{n}-2)}\right)$$

(8)

Method 4, lagging suppression:

This method is used to suppress the system input and output oscillation caused by lagging. In the system described in this paper, it showed a very good effect. We used a simple lagging system model to illustrate this algorithm.

As shown in Figure 3, the input of the system is $\mathrm{p}$, which is processed by the algorithm into $\mathrm{q}$. Then, $\mathrm{q}$ is the new input for the lagging model, and the output of the model is $\mathrm{r}$. The measured value of $\mathrm{r}$ is fed back to the algorithm. Here, the input and the output of the model are required to be the same physical quantity.

The algorithm has two expressions: first-order and second-order. For the first-order algorithm, the algorithm output $\mathrm{q}$ is a function of the system input $\mathrm{p}$ and the model output $\mathrm{r}$, as shown in Equation (9):

$${\mathrm{q}}_{(\mathrm{n})}=\mathrm{f}\left(\mathrm{p},\mathrm{r}\right)$$

(9)

The first-order function expression is shown in Equation (10):

$$\mathrm{f}\left(\mathrm{x},\mathrm{r}\right)=\left(1-\mathrm{m}\right){\mathrm{p}}_{(\mathrm{n})}+\mathrm{m}{\mathrm{r}}_{(\mathrm{n}-1)}$$

(10)

where $\mathrm{m}$ is the constant coefficient between zero and one, $0<\mathrm{m}<1$; $\mathrm{m}$ represents the proportional relationship between the model output and the system input in the algorithm output. A larger $\mathrm{m}$ represents a larger proportion of the model output. When $\mathrm{m}$ is between zero and one, the total gain of the algorithm can be maintained at 1.

The essence of lag compensation is to make the calculated control input of the system closer to the measured control input. When the calculation only involves the current cycle’s computed value and measured value, it is called first-order compensation. When the computation and the measured values of the previous control cycle are introduced, it is called second-order compensation. For the second-order algorithm, the algorithm output $\mathrm{q}$ is a function of the system input $\mathrm{p}$, the model output $\mathrm{r}$, and the algorithm output $\mathrm{q}$, as shown in Equation (11):

$${\mathrm{q}}_{(\mathrm{n})}=\mathrm{f}\left(\mathrm{p},\mathrm{q},\mathrm{r}\right)$$

(11)

The second-order function expression is shown in Equation (12):

$$\mathrm{f}\left(\mathrm{p},\mathrm{q},\mathrm{r}\right)=\left(1-\mathrm{m}\right)\left(\left(1-\mathrm{v}\right){\mathrm{x}}_{(\mathrm{n})}+\mathrm{v}{\mathrm{y}}_{(\mathrm{n}-1)}\right)+\mathrm{m}\left(\left(1-\mathrm{w}\right){\mathrm{z}}_{(\mathrm{n}-1)}+\mathrm{w}{\mathrm{z}}_{(\mathrm{n}-2)}\right)$$

(12)

where $\mathrm{m}$, $\mathrm{v}$, and $\mathrm{w}$ are the constant coefficients between zero and one, keeping the total gain at 1.

Next, we discuss the speed commands of the four actuators in one lifting point.

The system in this paper had three output variables, and we defined three errors that corresponded to the difference between the set values of the three output variables and the measured values of the sensors. The error is described in Equation (13):

$$\left\{\begin{array}{l}{\mathrm{e}}_{\alpha }=\alpha -{\alpha }^{*}\hfill \\ {\mathrm{e}}_{\mathrm{F}}={\mathrm{F}}^{*}-\mathrm{F}\hfill \\ {\mathrm{e}}_{\gamma }=\gamma -{\gamma }^{*}\hfill \end{array}\right.$$

(13)

For ${\mathrm{x}}_1$, we designed a negative proportional feedback controller for the inclination α as Equation (14):

$${\mathrm{v}}_{\mathrm{x}1}={\mathrm{K}}_{\mathrm{p}}({}^{\mathrm{f}}\mathrm{e}_{\alpha (\mathrm{n})})\cdot {}^{\mathrm{f}}\mathrm{e}_{\alpha (\mathrm{n})}$$

(14)

In Equation (14), the coefficient ${\mathrm{K}}_{\mathrm{p}}$ was set according to the function given in method 2 so that the actuator was difficult to start for adjustment when the error was very small, and a low-pass filter with a cutoff frequency of 2 Hz was to be used for ${\mathrm{e}}_{\alpha (\mathrm{n})}$.

There were two actuators in the Y direction, ${\mathrm{y}}_{\mathrm{s}}$ and ${\mathrm{y}}_{\mathrm{b}}$; ${\mathrm{y}}_{\mathrm{s}}$ was characterized by a shorter range of motion but could adapt to a greater acceleration; In contrast, ${\mathrm{y}}_{\mathrm{b}}$ was characterized by a long range of motion but could adapt to a smaller acceleration. When designing the controller, we fed forwards part of the input of ${\mathrm{y}}_{\mathrm{b}}$ to ${\mathrm{y}}_{\mathrm{s}}$, and the existence of this part of the input could prevent the motion range of ${\mathrm{y}}_{\mathrm{s}}$ from being excessively large.

The expression of speed input ${\mathrm{v}}_{\mathrm{y}\mathrm{s}}$ is shown in Equation (15):

$${\mathrm{v}}_{\mathrm{y}\mathrm{s}(\mathrm{n})}=\left(1-\mathrm{k}\right)\left(\left(1-\mathrm{a}\right){\mathrm{v}}_{\mathrm{y}\mathrm{s}\mathrm{c}\mathrm{m}\mathrm{d}(\mathrm{n})}+\mathrm{a}{}^{\mathrm{l}}\mathrm{v}_{\mathrm{ys}(\mathrm{n}-1)}\right)+\mathrm{k}\left(\left(1-\mathrm{b}\right){\mathrm{v}}_{\mathrm{y}\mathrm{s}\mathrm{m}\mathrm{e}\mathrm{s}(\mathrm{n}-1)}+\mathrm{b}{\mathrm{v}}_{\mathrm{y}\mathrm{s}\mathrm{m}\mathrm{e}\mathrm{s}(\mathrm{n}-2)}\right)$$

(15)

where ${\mathrm{v}}_{\mathrm{y}\mathrm{s}\mathrm{c}\mathrm{m}\mathrm{d}}$ represents the speed command, ${\mathrm{v}}_{\mathrm{y}\mathrm{s}}$ represents the speed input processed by method 4, and ${\mathrm{v}}_{\mathrm{y}\mathrm{s}\mathrm{m}\mathrm{e}\mathrm{s}}$ represents the measured value of the actuator speed. The speed command consists of two parts, the filtered PID controller output and ${\mathrm{v}}_{\mathrm{y}\mathrm{b}\mathrm{s}}$, as in Equation (16):

$${\mathrm{v}}_{\mathrm{y}\mathrm{s}\mathrm{c}\mathrm{m}\mathrm{d}(\mathrm{n})}={}^{\mathrm{f}}\left({\mathrm{K}}_{\mathrm{p}}({\mathrm{e}}_{\mathrm{F}(\mathrm{n})}){\mathrm{e}}_{\mathrm{F}(\mathrm{n})}+{}^{\mathrm{l}}\left({\mathrm{K}}_{\mathrm{i}}({\mathrm{e}}_{\mathrm{F}(\mathrm{n})}){\displaystyle \sum _0^{\mathrm{n}}{\mathrm{e}}_{\mathrm{F}(\mathrm{n})}\mathrm{d}\mathrm{t}}\right)+{}^{\mathrm{l}}\left({\mathrm{K}}_{\mathrm{d}}({\mathrm{e}}_{\mathrm{F}(\mathrm{n})})\frac{{}^{\mathrm{f}}\left({\mathrm{e}}_{\mathrm{F}(\mathrm{n})}-{\mathrm{e}}_{\mathrm{F}(\mathrm{n}-1)}\right)}{\mathrm{d}\mathrm{t}}\right)\right)+{\mathrm{K}}_{\mathrm{F}\mathrm{F}}{}^{\mathrm{l}}\mathrm{v}_{\mathrm{y}\mathrm{b}\mathrm{s}(\mathrm{n})}$$

(16)

The PID coefficients in Equation (16) are a function of the force error F according to method 2, and the variation in the coefficients is shown in Figure 4. The output amplitude of the integral action and the differential action are limited by method 1. The amplitude limiting function of the integral can keep the integral within the controllable range and avoid excessive integral output, which requires a long time of reverse error to clear. The input of the differential action was filtered by method 3 to reduce the noise impact caused by the difference.

The existence of ${\mathrm{v}}_{\mathrm{y}\mathrm{b}\mathrm{s}}$ was used to adjust ${\mathrm{v}}_{\mathrm{y}\mathrm{s}\mathrm{c}\mathrm{m}\mathrm{d}}$, which could keep ${\mathrm{v}}_{\mathrm{y}\mathrm{s}\mathrm{c}\mathrm{m}\mathrm{d}}$ within the effective range; see Equation (17):

$${\mathrm{v}}_{\mathrm{y}\mathrm{b}\mathrm{s}(\mathrm{n})}={\mathrm{K}}_{\mathrm{b}\mathrm{s}}({\mathrm{y}}_{\mathrm{s}(\mathrm{n}-1)}){\mathrm{y}}_{\mathrm{s}(\mathrm{n}-1)}$$

(17)

The expression of speed input ${\mathrm{v}}_{\mathrm{y}\mathrm{b}}$ is shown in Equation (18):

$${\mathrm{v}}_{\mathrm{y}\mathrm{b}(\mathrm{n})}=\left(1-\mathrm{k}\right)\left(\left(1-\mathrm{a}\right){\mathrm{v}}_{\mathrm{y}\mathrm{b}\mathrm{c}\mathrm{m}\mathrm{d}(\mathrm{n})}+\mathrm{a}{\mathrm{v}}_{\mathrm{y}\mathrm{b}(\mathrm{n}-1)}\right)+\mathrm{k}\left(\left(1-\mathrm{b}\right){\mathrm{v}}_{\mathrm{y}\mathrm{b}\mathrm{m}\mathrm{e}\mathrm{s}(\mathrm{n}-1)}+\mathrm{b}{\mathrm{v}}_{\mathrm{y}\mathrm{b}\mathrm{m}\mathrm{e}\mathrm{s}(\mathrm{n}-2)}\right)$$

(18)

Similarly to ${\mathrm{v}}_{\mathrm{y}\mathrm{s}}$, the speed command consists of two parts, the filtered proportional feedback controller output and ${\mathrm{v}}_{\mathrm{y}\mathrm{b}\mathrm{s}}$, as in Equation (19):

$${\mathrm{v}}_{\mathrm{y}\mathrm{b}\mathrm{c}\mathrm{m}\mathrm{d}(\mathrm{n})}={}^{\mathrm{f}}\left({\mathrm{K}}_{\mathrm{p}}({\mathrm{e}}_{\mathrm{F}(\mathrm{n})}){\mathrm{e}}_{\mathrm{F}(\mathrm{n})}\right)+{}^{\mathrm{l}}\mathrm{v}_{\mathrm{y}\mathrm{b}\mathrm{s}(\mathrm{n})}$$

(19)

${\mathrm{v}}_{\mathrm{y}\mathrm{b}\mathrm{s}}$ exists in both ${\mathrm{v}}_{\mathrm{y}\mathrm{s}}$ and ${\mathrm{v}}_{\mathrm{y}\mathrm{b}}$.

For ${\mathrm{z}}_1$, we designed the same negative proportional feedback controller for the inclination γ as Equation (20):

$${\mathrm{v}}_{\mathrm{z}1(\mathrm{n})}={\mathrm{K}}_{\mathrm{p}}({}^{\mathrm{f}}\mathrm{e}_{\gamma (\mathrm{n})})\cdot {}^{\mathrm{f}}\mathrm{e}_{\gamma (\mathrm{n})}$$

(20)

3. Results

Here, we conducted simulation and actual testing of the experimental system proposed in this paper. The experimental system we designed had a total of three lifting points, each of which could achieve three-dimensional motion. Each lifting point was connected to the tested space manipulator with a steel wire rope, providing vertical upward tension to the manipulator. In the simulation and actual testing in the paper, in order to make a clear comparison, we used a swinging rod as the load of the experimental system to study the tension exerted by a lifting point on the rod during its movement. We hoped that the lifting point could apply a vertical upward force of 300 N to the moving rod. In order to maintain the magnitude and direction of this force, the experimental system must have been able to adapt to the three-dimensional motion of the rod. We judged the operation of the experimental system by measuring the magnitude of the tensile force in the results and the inclination angle of the steel wire rope and the vertical direction.

For the model and the controller given in Section 2, we conducted verification by simulation and physical experiment. Since the structure of the three lifting points of the system was the same, the data results of only one lifting point are given here. For both verifications, we set the output as shown in

Table 1. Output values of the verifications.

	α	F	γ
Value	0	300	0

.

3.1. Simulation

3.1.1. Simulation Model

We built the simulation system according to the system structure given in Figure 5.

The simulation model simulated the motion process, as shown in Figure 6. The upper part of Figure 6 is the lifting point of the test system, which can move horizontally and retract the steel wire rope. At the bottom of Figure 6, there is a rod that can swing with a single degree of freedom, which was used to simulate the motion of the tested space robotic arm. The lifting point was connected to the rod with a steel wire rope to apply tension to the rod. When the rod moved, the lifting point needed to make an active follow-up motion to maintain the verticality of the wire rope while maintaining the tension on the wire rope at the set value. When the swing direction of the rod did not coincide with the X- or Z-direction of the lifting point, the motion of the rod and the wire rope connection point in the lifting point coordinate system had a three-dimensional trajectory diagram as shown in Figure 7. The lifting point of the experimental equipment could follow this three-dimensional trajectory. The motion of the point was simulated.

3.1.2. Simulation Results

We took the motion data of two cycles as the result. The three-dimensional input of the system is shown in Figure 7.

Figure 8 shows the speed inputs of the four actuators and the curves of the three outputs of the simulation system. The tension for the active gravity balance was set to 300 N.

Figure 8a shows that the green curve starts to change before the other two curves in each cycle, and ${\mathrm{v}}_{\mathrm{y}\mathrm{s}}$ reflects quickly. When ${\mathrm{v}}_{\mathrm{y}\mathrm{s}}$ works, ${\mathrm{v}}_{\mathrm{y}\mathrm{b}}$ starts increasing and ${\mathrm{v}}_{\mathrm{y}\mathrm{s}}$ drops. In Figure 8b, these two speed inputs are similar to intermittent startup, which is consistent with the coefficient change logic we set. Figure 8c,d are the output curves of the system. It can be seen that the outputs of the system fluctuated near the setting values, and the system performed well.

We compared the control methods used by other scholars in the active gravity balance test bench under the same operating conditions [6]. The simulation results show that the control method proposed in this paper had a better control effect on the experimental system in this paper. This indicates that the control method selected in this article was more suitable for this control system. The tension for active gravity balance was also set to 300 N. The specific results are shown in Figure 9.

It can be seen that when other control methods were applied in this system, the measured force error of the active gravity balance reached ca. 20 N, which is greater than the 10 N state achieved by the control method in this article. The inclination error between the steel wire rope and the vertical direction was also slightly greater than the simulation results of the control method in this paper.

3.2. Results on the Experiment System

3.2.1. Real System

The actual state of the system is shown in Figure 9. Figure 9a shows the three lifting points, and Figure 10b shows the detail of one lifting point.

3.2.2. Three-Dimensional Real Results

We discuss the motion data of two cycles. The three-dimensional input of the real system is shown in Figure 11.

Figure 12 shows the speed inputs of the four actuators and the curves of the three outputs of the real system.

It can be seen that both the simulation and actual measurement results show that when the tested test system worked, it could adapt to the vertical movement speed of the load approaching at 80 mm/s and the horizontal movement speed of 15 mm/s. Within this speed range, the force applied by the lifting point fluctuates between 290 N and 310 N, and the absolute value of inclination angle between the steel wire rope and the vertical direction keeps less than 0.6°. The performance of the real system was very similar to that of the simulation system.

4. Discussion

In the previous two chapters, we discussed the system model, the controller, and the operation results. In the design of the Y-direction controller, we used method 4 to reduce the lagging effect. In this chapter, we show the performance of this method in this system.

We simplified the structure of the system, keeping only the actuator ${\mathrm{y}}_{\mathrm{b}}$, and the movement of the point was limited to the vertical direction. The performance is shown in Figure 13 in contrast.

It can be seen from the curve that when method 4 was used, the fluctuation amplitude of the system input and the output curve decreased significantly. Detailed results for 200 ms are shown in Figure 14.

It can be seen that there was an obvious and stable lagging relationship between the system input speed and the measured speed, and the system output F fluctuated by approximately ±10 N before method 4 was applied. When method 4 was applied, the input speed of the system was almost consistent with the measured speed without obvious lagging, and the fluctuation of F was also suppressed to approximately ±2 N.

The lag compensation method proposed in this paper can compensate for the execution lag in this system. The particularity of this compensation method lies in directly processing the deviation between the calculated input and the measured input of the controller, making the calculated input closer to the measured input, thereby reducing the oscillation effect caused by the lag.

5. Conclusions

In this paper, our goal was to design ground test equipment with active-suspension gravity compensation for space manipulators. The approach proposed in this paper can be applied in ground gravity compensation requirements of space manipulators and other space equipment. To achieve this, we designed an intelligent controller that implements 3 degrees of freedom of active-suspension gravity compensation. The controller was tuned using the ADP method, and the parameters were fine-tuned based on the actual operating conditions. To improve control accuracy and solve the execution lag problem found during the actual testing, we used four methods mentioned in the paper. The designed controller can be used in other similar controller design requirements, and the protective measures in our controller can prevent safety issues during the initial debugging. When the controller is applied in a new environment, its control parameters need to be retuned according to the actual situation. For the lag compensation method proposed by us, it should satisfy two conditions when applied to other control systems. First, the input of the system should be measurable, and second, the controller of the system should be able to adapt to the disturbance brought by lag compensation. The essence of this lag compensation is to mix the controller inputs calculated from the current control cycle and the previous control cycle and the measured controller inputs in a certain proportion as the controller input for the current cycle. In the system described in this paper, the lag time is approximately four control cycles, and our compensation method can effectively suppress the control oscillation caused by this lag. The application scope of this lag compensation method can be studied further. In the paper, we conducted a simulation of a complete application of the system, and the simulation results were close to the actual results, indicating that our intelligent controller can achieve our control goals and complete the control requirements of the ground test equipment for active-suspension gravity compensation of a space manipulator. With the assistance of this system, space manipulators can be tested in a ground environment, which facilitates the study of motion characteristics of space manipulators by researchers. In actual use, our test equipment has successfully assisted a space manipulator in completing ground deployment tests, with stable operation.