A Survey of Planar Underactuated Mechanical System

Abstract

Planar underactuated mechanical systems have been a popular research issue in the area of mechanical systems and nonlinear control. This paper reviews the current research status of control methods for a class of planar underactuated manipulator (PUM) systems containing a single passive joint. Firstly, the general dynamics model and kinematics model of the PUM are given, and its control characteristics are introduced; secondly, according to the distribution position characteristics of the passive joints, the PUM is classified into the passive first joint system, the passive last joint system, and the passive intermediate joint system, and the analysis and discussion are carried out in respect to the existing intelligent control methods. Finally, in response to the above discussion, we provide a brief theoretical analysis and summarize the challenges faced by PUM, i.e., uncertainty and robustness of the system, unified control methods and research on underactuated systems with uncontrollable multi-passive joints; at the same time, the practical applications have certain limitations that need to be implemented subsequently, i.e., anti-jamming, multi-planar underactuated robotic arm co-control and spatial underactuated robotic arm system development. Aiming at the above challenges and problems in the control of PUM systems, we elaborate on them in points, and put forward the research directions and related ideas for future work, taking into account the contributions of the current work.

1. Introduction

During our daily life, we often encounter mechanical systems according to which “less control over more”, such as ships in the water [1,2], two-wheeled balance bikes on the ground [3,4] and air vehicles in the air [5,6]. This “less control over more” behavior is called underactuated characteristic. An underactuated system is characterized by having a lesser number of independent control inputs compared to its degrees of freedom [7]. Underactuated systems represent a common category of nonlinear systems [8,9,10], which do not have corresponding control inputs for certain state quantities in the system’s bitmap space due to the decrease in the number of input excitations [11], which results in underactuated systems generally having nonholonomic constraints. Therefore, most of the underactuated mechanical systems belong to the category of nonholonomic systems, so the study of underactuated mechanical systems can provide new solutions for nonholonomic systems. Because the number of actuators of the planar underactuated manipulator is less than that of the fully actuated manipulator, the volume and weight of the underactuated manipulator are smaller, which can provide new ideas for people to reduce the cost and energy consumption of the mechanical system and improve the compactness of the structure. Undriven systems are receiving more and more attention and gradually become the focus of research in the field of robotics. At the same time, it also means that the study of such systems is of great significance and broad application prospects for the progress of the control theory of nonholonomic systems and non-linear control theory.

Underactuated system control faces difficult problems such as high nonlinearity [12], parameter ingestion [13], multi-objective control requirements [14] and limited control volume [15], which might significantly impact system performance instability and complicate system control. To address these control problems, researchers have put forward sliding mode control methods [16], fuzzy control methods [17], neural network control [18,19] methods and adaptive control methods [20], effectively solving the system control problems, and promoting them to many fields widely used to promote their development [21,22,23]. The planar underactuated system is a specific application of underactuated system in two-dimensional plane; compared with the underactuated system in three-dimensional space, the planar underactuated system is simpler in structure, but has the same complexity and challenge of the underactuated system.

Table 1. The benefits and drawbacks of planar underactuated mechanical systems.

Advantages	Disadvantages
Lower cost	Increased control complexity
Simple structure	Performance constraint
Small size	Precision and stability challenges
Low energy consumption	Poor adaptability to environmental changes
Easy to maintain and repair	Increased difficulty in design and optimization

lists the benefits and drawbacks of underactuated mechanical systems.

The planar underactuated mechanical system is not limited by gravity [24,25], and such systems are commonly used in microgravity conditions such as space and deep ocean [26,27]. Moreover, due to the complexity and diversity of microgravity working environments such as space and deep ocean, the systems are difficult to be directly manipulated by human beings [28,29]. The research on control techniques for planar underactuated mechanical systems is significant for space exploration and deep-sea exploration. When a planar full-drive mechanical system fails during operation, causing the full-drive system to become an underactuated system, if the drive device is difficult to repair promptly, the control strategy of the planar underactuated system can be employed to guarantee the uninterrupted functioning of the system, which greatly improves the reliability of the system movement. As a result, the control approach of gravity-constrained vertical underactuated systems cannot achieve the complex and varied control objectives of such systems, and system control is extremely challenging.

The study of stabilization control of planar underactuated systems is of greater strategic significance for aerospace and deep ocean exploration engineering, which is currently under development [30,31,32]. However, planar underactuated mechanical systems have very complex nonlinear properties, and any accessible point is a balance in the system, and balance the linear approximation model is uncontrollable [33]. Therefore, the control objectives of such systems are complex and diverse and cannot be realized by the control method of gravity-constrained vertical underactuated systems [34], and the system control is very difficult.

Planar underactuated mechanical system control technology is also developing and becoming more and more popular in the industry as a result of the ongoing advancements in intelligent and automation technologies [35]. Researchers are working to develop more efficient and stable control algorithms to meet the market demand for high-precision, high-efficiency mechanical systems [36]. As a new type of mechanical equipment, the market demand for planar underactuated mechanical systems is also growing [37]. The planar underactuated mechanical system can be used for attitude control of spacecraft, orbit adjustment of satellites, etc. Its unique underactuated characteristics make the system able to maintain stability in microgravity environments and realize precise control of the target [38]. In deep-sea exploration, planar underactuated mechanical systems can be used for motion control of underwater robots and positioning of underwater equipment [39]. In the field of industrial automation, planar underactuated mechanical systems can be used for material handling and assembly of automatic production lines [40]. Classes of underactuated mechanical systems are also being investigated, such as underactuated bionic fish systems [41], unmanned aircraft systems [42,43], one-wheeled mobile robots [44], exoskeleton robotic arms [45] and inverted pendulum robotic vehicles [46], etc., which need to be carried out for research, development and application in real-world engineering and promote the development of underactuated mechanical systems.

At present, for planar manipulator systems containing a single passive joint, the proposed control method is designed according to the characteristics of each specific system, and by further analyzing the constraint relationship between the passive and active joints, the following problems exist:

(1) More intuitive inter-joint constraint relations such as angular constraint relations, angular velocity constraint relations and chain structure are obtained through each specific model, and the stability control strategy and trajectory tracking control strategy designed in this way are not universally applicable.

(2) In addition, these existing control techniques are used to design controllers to achieve effective control of the system according to a nominal model, and do not consider the parameter absorption of the system and external disturbances, and the performance indices such as stability and robustness also need to be optimized.

(3) Most of the published research results only use numerical simulation to achieve the system control objectives, without specific practical experiments to verify the designed control strategy.

The control methods for underactuated manipulators mainly include partial feedback linearization, trajectory planning and tracking control, sliding mode control, backstepping method and so on. For multi-link underactuated manipulators, the model reduction method is often used for processing. Underactuated mechanical systems typically require more complex algorithms to control due to their complex nonlinear properties. Moreover, the nonlinearity of the system, parameter perturbation, multi-objective control requirements and limited control variables may increase the difficulty of control. In partial feedback linearization, the internal dynamics may become a self-made system without input control, and its stability is difficult to guarantee. If the internal dynamics are unstable, it can result in an unstable whole system. Different control methods may only be applicable to specific underactuated systems or specific dynamic models. For complex underactuated systems or unknown dynamic models, it may be necessary to develop new control methods or conduct more in-depth research.

At present, there are few review articles on planar underactuated manipulator (PUM) systems with different passive joint positions. This paper systematically reviews the current research status of planar manipulator systems containing a single underactuated joint, focuses on analyzing the main control methods of different kinds of planar underactuated manipulator systems and describes the problems to be further solved and prospective research avenues.

The rest of this paper is structured as follows. Section 2 describes the modeling and control property analysis of the PUM. Section 3 describes current control techniques for different types of PUM systems. Section 4 describes the challenges of current research on PUM systems. Section 5 contains concluding remarks and future research.

2. Modeling of Planar Underactuated Manipulator

This section introduces the PUM system in terms of modeling and analysis, passive joint positions and control characteristics. These elements are the basis for exploring and analyzing the PUM system and the theoretical basis for designing the control strategy.

2.1. Modeling and Analysis

The physical structure of the planar n-DoF active-active-⋯-active (AA⋯A) manipulator is shown in Figure 1, the blue part indicates a section of manipulator. The parameters of the ith $(i=1,2,3,\dots ,n)$ link include the angle $q_i$, the mass $m_i$, the length $L_i$, the distance from its joint to its center of mass $L_{ci}$, the moment of inertia $J_i$ and the end-point coordinate is $(x,y)$.

The above system is referred to as a full drive system; however, the system is downgraded to an underactuated system if one of the active linkages in the full drive system is damaged, at which point the drive joints are unable to provide drive and it becomes an underdrive system. The planar ${\mathrm{A}}^m{\mathrm{PA}}^n$ system is a multilink underactuated system containing a single driven joint and the planar ${\mathrm{A}}^m{\mathrm{PA}}^n$ system travels in the level plane without considering gravity, i.e., the kinetic potential energy is zero. In order to analyze the planar underactuated system’s own characteristics more deeply, the ${\mathrm{A}}^m{\mathrm{PA}}^n$ system, as the most typical generalized planar underactuated robotic arm system model, is modeled and analyzed dynamically as well as kinematically.

2.1.1. Dynamic Model

The majority of the literature is on the basis of dynamic modeling by the Euler–Lagrange method. The dynamics model of the planar ${\mathrm{A}}^m{\mathrm{PA}}^n$ system is developed according to the Euler–Lagrange method, which allow us to visualize the mathematical relationship between the individual linkage angles of the planar ${\mathrm{A}}^m{\mathrm{PA}}^n$ system and its active joint moments. The physical structure of the planar ${\mathrm{A}}^m{\mathrm{PA}}^n$ manipulator is shown in Figure 2. The parameters of the ith $(i=1,2,\dots m,m+1,\dots ,m+n,m+n+1)$ link include the angle $q_i$, the mass $m_i$, the length $L_i$, the distance from its joint to its center of mass $L_{ci}$ and the moment of inertia $J_i$.

Firstly, the Lagrangian function $L\left(q,\dot{q}\right)$ of the system is determined as

$$L(q,\dot{q})=K(q,\dot{q})-P\left(q\right)$$

(1)

where $q={[q_1{q}_2\dots q_{m+1}\dots q_{m+n+1}]}^T\in {\mathbb{R}}^{(m+n+1)\times 1}$ is the angle vector of all $m+n+1$ links in the system, and $\dot{q}={[{\dot{q}}_1,{\dot{q}}_2,\dots {\dot{q}}_n,{\dot{q}}_{m+n+1}]}^T\in {\mathbb{R}}^{(m+n+1)\times 1}$ is the angular velocity vector of all $m+n+1$ links in the system. Because the system moves in the horizontal plane, the potential energy $P\left(q\right)$ is zero, and the specific expression of kinetic energy $K\left(q,\dot{q}\right)$ is

$$K(q,\dot{q})={\displaystyle \frac{1}{2}}{\dot{q}}^{T}M\left(q\right)\dot{q}$$

(2)

where $M\left(q\right)\in {\mathbb{R}}^{(m+n+1)\times (m+n+1)}$.

The Lagrangian equation is defined as

$${\displaystyle \frac{d}{dt}}{\displaystyle \frac{\partial L(q,\dot{q})}{\partial \dot{q}}}-{\displaystyle \frac{\partial L(q,\dot{q})}{\partial q}}=\tau$$

(3)

where $\tau =[{\tau }_1,{\tau }_2,\dots {\tau }_m,0\dots {\tau }_{m+n+1}]\in {\mathbb{R}}^{(m+n+1)\times 1}$.

The system Lagrangian function is derived for the angle as well as the angular velocity, respectively, and then replaced in Equation (3), i.e.,

$$\begin{array}{c}\hfill M\left(q\right)\ddot{q}+\dot{M}\left(q\right)\dot{q}-{\displaystyle \frac{1}{2}}{\displaystyle \frac{\partial }{\partial q}}\left({\dot{q}}^{T}M\left(q\right)\dot{q}\right)=\tau \end{array}$$

(4)

where $\ddot{q}={[{\ddot{q}}_1,{\ddot{q}}_2,\dots {\ddot{q}}_m,{\ddot{q}}_{m+1},\dots {\ddot{q}}_{m+n+1}]}^T\in {\mathbb{R}}^{(m+n+1)\times 1}$ is the angular acceleration vector of the system.

The dynamic model can be derived as

$$M\left(q\right)\ddot{q}+H(q,\dot{q})=\tau$$

(5)

where $q={[q_1,q_2,\dots ,q_{m+n+1}]}^T$, $\dot{q}={[{\dot{q}}_1,{\dot{q}}_2,\dots ,{\dot{q}}_{m+n+1}]}^T$, and $\ddot{q}={[{\ddot{q}}_1,{\ddot{q}}_2,\dots ,{\ddot{q}}_{m+n+1}]}^T$ are the vector of angle, angular velocity and angular acceleration, respectively; $M\left(q\right)\in {\mathbb{R}}^{m+n+1\times m+n+1}$ is the inertia matrix and is symmetric and positive definite, and $H\left(q,\dot{q}\right)\in {\mathbb{R}}^{m+n+1\times 1}$ contains the Coriolis and centrifugal forces.

Lemma 1.

A necessary and sufficient condition for the partial integrability of the underactuated joint dynamics constraint in the system is as follows:

(i) The gravity term in the underactuated joint dynamics constraint is constant;

(ii) The underactuated joint variables do not appear in the inertia matrix.

2.1.2. Kinematic Model

The specific content of the position control of the planar ${\mathrm{A}}^m{\mathrm{PA}}^n$ system is to design a suitable strategy according to the equation of motion state of the planar ${\mathrm{A}}^m{\mathrm{PA}}^n$ system to realize the control of the system endpoints. This requires obtaining the geometric relationship between the system endpoint coordinates position coordinates and the length and angle of each link of the system.

To obtain this geometric relationship, it is necessary to compute an expression for the coordinates of the endpoints of the system. According to the coordinate transformation method, the kinematic model of the planar ${\mathrm{A}}^m{\mathrm{PA}}^n$ system is derived, and the kinematic model is utilized to obtain specific mathematical expressions between the system endpoint coordinate positions and the lengths and angles of the system links.

The structural sketch of the planar ${\mathrm{A}}^m{\mathrm{PA}}^n$ manipulator is shown in Figure 3. ${\sum }_0$ is the base coordinate system, ${\sum }_i(i=1,2\dots m+n+1)$ is the kinetic coordinate system established on the ith connecting rod, and ${\sum }_{m+n+2}$ is the kinetic coordinate system established at the end point.

From Figure 3, the chi-square transformation matrix ${}^{0}T_1$ of the moving coordinate system ${\sum }_1$ with respect to the base coordinate system ${\sum }_0$ is

$${}^{0}T_1=\left[\begin{array}{ccc}-sin{q}_1& -cos{q}_1& 0\\ cos{q}_1& -sin{q}_1& 0\\ 0& 0& 1\end{array}\right]$$

(6)

The chi-square transformation matrix ${}^{m+n+1}T_{m+n+2}$ of the moving coordinate system ${\sum }_{m+n+2}$ with respect to the moving coordinate system ${\sum }_{m+n+1}$ is

$${}^{m+n+1}T_{m+n+2}=\left[\begin{array}{ccc}1& 0& L_{m+n+1}\\ 0& 1& 0\\ 0& 0& 1\end{array}\right]$$

(7)

The chi-square transformation matrix ${}^{i-1}T_i$ with respect to the dynamic coordinate system $\sum {}_i(i=1,2\dots n+1)$ is

$${}^{i-1}T_i=\left[\begin{array}{ccccc}cos{q}_i& & -sin{q}_i& & L_{i-1}\\ sin{q}_i& & cos{q}_i& & 0\\ 0& & 0& & 1\end{array}\right]$$

(8)

Then, the chi-square transformation matrix ${\sum }_{m+n+2}$ of the moving coordinate system ${}^{0}T_{m+n+2}$ with respect to the base coordinate system ${\sum }_0$ is

$$\begin{array}{c}\hfill {}^{0}T_{m+n+2}={}^{0}T_1{}^{1}T_2\dots {}^{m+n}T_{m+n+1}{}^{m+n+1}T_{m+n+2}\\ \hfill =\left[\begin{array}{ccc}-A& -B& -C\\ \\ B& -A& D\\ \\ 0& 0& 1\end{array}\right]\end{array}$$

(9)

where $A=sin\left({\sum }_{i=1}^{m+n+1}{q}_i\right)$, $B=cos\left({\sum }_{i=1}^{m+n+1}{q}_i\right)$, $C={\sum }_{i=1}^{m+n+1}\left(L_{i}sin\left({\sum }_{j=1}^i{q}_j\right)\right)$, $D={\sum }_{i=1}^{m+n+1}\left(L_{i}cos\left({\sum }_{j=1}^i{q}_j\right)\right)$.

According to the above equation, the coordinates of the end point of the planar ${\mathrm{A}}^m{\mathrm{PA}}^n$ system underactuated robotic arm can be obtained as follows:

$$\left\{\begin{array}{c}x=-\sum _{i=1}^{m+n+1}\left(L_{i}sin\left(\sum _{j=1}^i{q}_j\right)\right)\hfill \\ y=\sum _{i=1}^{m+n+1}\left(L_{i}cos\left(\sum _{j=1}^i{q}_j\right)\right)\hfill \\ \theta =q_1+q_2+\dots q_{m+n+1}\hfill \end{array}\right.$$

(10)

where $\theta$ is angle of position.

2.2. Position of the Passive Joint

When one of the active joints in the planar n-DoF manipulator is missing or damaged, such system becomes the underactuated manipulator. As in the ${\mathrm{A}}^m{\mathrm{PA}}^n$ model analyzed above, according to the position of the passive joint, we divide the planar underactuated manipulator into the following three configurations:

• Configuration 1: The planar ${\mathrm{PA}}^n$ manipulator, $\tau ={\left[0,{\tau }_2,{\tau }_3,\dots ,{\tau }_n\right]}^T$.
• Configuration 2: The planar ${\mathrm{A}}^m{\mathrm{PA}}^n$ manipulator, $\tau ={\left[{\tau }_1,\dots ,{\tau }_m,0,\dots ,{\tau }_n\right]}^T$.
• Configuration 3: The planar ${\mathrm{A}}^m$P manipulator, $\tau ={\left[{\tau }_1,{\tau }_2,\dots ,{\tau }_m,0\right]}^T$.
  ${\tau }_i(i=1,2,3,\dots )$ denotes the applied torque at the i-th joint and $\tau$ is the applied torque vector.

2.3. Control Properties

For the control of PUM systems, they are categorized as position control, position and attitude control and trajectory tracing control.

Position control is defined as the design of a suitable controller that controls the system from some initial position to some target position and keeps the system in that state to maintain stability.

The positional and attitude control includes position control, and considering that the end point arrives at the target position, the system linkage also arrives at the target pose angle. Therefore, positional and pose control of the system is more complex and difficult than the position control.

The trajectory tracking control consists of trajectory planning and tracking control, planning appropriate motion trajectory for driven joint, which enables the manipulator to track the trajectory to arrive the target state, and underactuated joint state could only be passively controlled and changed by the driven joints. Therefore, the difficulty of trajectory tracking control is in planning the trajectory for the driven joint and designing the relevant control method.

3. Control Techniques

The complexity of underactuated systems is caused by their underactuated characteristics, which require more degrees of freedom to be actuated with fewer actuators and for which many conventional nonlinear control approaches are inapplicable. Recently, feedback-based controlling laws, energetic attenuation, adaptive, sliding mode, fuzzy, and other control methods have emerged to improve control and stabilization accuracy. This section will comprehensively analyze the corresponding control methods and their research status from the control object.

3.1. The Passive First Joint

Planar manipulators with underactuated first joints can be categorized into complete and first-order non-complete systems. First-order nonholonomic planar manipulators systems are mainly classified into two categories [47]: planar three-link underactuated manipulators (referred to as planar PAA systems) and planar multilink underactuated manipulators (referred to as planar ${\mathrm{PA}}^n(n>2)$ systems), which are characterized by first-order nonholonomic properties and angular velocity constraints relationships [48]. This subsection will organize and introduce its control methods in detail.

3.1.1. Acrobot

The physical structure of the above-mentioned manipulator system is illustrated in Figure 4, where $m_i(i=1,2)$ denotes the mass of the i-th link, $L_i$ is the length of the i-th link, $l_i$ represents the distance from the i-joint to the center of mass of the i-link, J denotes the inertia of the i-th link with respect to the center of mass, $q_i$ denotes the angle of rotation of the connecting rod, and ${\tau }_2$ is the control force acting on the second connecting link.

Let $q={[q_1,q_2]}^T$, the kinetic equation of Acrobot satisfy

$$M_{11}\left(q\right){\ddot{q}}_1+M_{12}\left(q\right){\ddot{q}}_2+H_1(q,\dot{q})=0$$

(11)

$$M_{21}\left(q\right){\ddot{q}}_1+M_{22}\left(q\right){\ddot{q}}_2+H_2(q,\dot{q})={\tau }_2$$

(12)

where

$$\left\{\begin{array}{c}{M}_{11}\left(q\right)=y_1+y_2+2y_{3}cos{q}_2\hfill \\ M_{12}\left(q\right)=M_{21}=y_2+y_{3}cos{q}_2\hfill \\ M_{22}=y_2\hfill \\ H_1(q,\dot{q})=-y_3(2{\dot{q}}_1{\dot{q}}_2+{\dot{q}}_2^2)sin{q}_2\hfill \\ H_2(q,\dot{q})=y_3{\dot{q}}_1^{2}sin{q}_2\hfill \end{array}\right.$$

(13)

The structural parameters $y_i(i=1,2,3)$ are

$$\left\{\begin{array}{c}{y}_1=m_1{l}_1^2+m_2{l}_1^2+J_1\hfill \\ y_2=m_2{l}_2^2+J_2\hfill \\ y_3=m_2{l}_1{l}_2\hfill \end{array}\right.$$

(14)

The equation on the right side of (11) equals zero, which imposes a constraint on the state variables of the underactuated planar acrobot. This section explores the interconnections among the state variables through the integration of (11), forming the foundation for our control strategy derivation. It is important to note that lacks a gravity term, and the inertia matrix, denoted as $M\left(q\right)$, does not include the underactuated variable $q_1$. Consequently, (11) is partially integrable. Furthermore, it can be readily verified that $M_1=\left[M_{11}{M}_{12}\right]$ is involutive, indicating that (11) is completely integrable. This completeness allows for the derivation of relationships between the angular velocities and the angles as expressed.

The Acrobot would be a classical holonomic system having perspective constraints. The angle between the angles of the actively and underactuated link has a certain mathematical relationship by integrating the underactuated equation of the planar Acrobot. Based on this angular constraint relationship, controllers can be designed for attractive links to achieve the controlling objectives in the systems. Zhang et al. [49] suggested the trajectory track control approach after investigating the stability of the underactuated two-link gymnastics robotics Acrobot. By integrating the dynamic equations with Acrobot, Lai et al. [50] obtained the constraints on the angle and acceleration of the links of the chain, and relevant control strategies are suggested for its kinematic status constrain.

3.1.2. PAA Manipulator System

In the planar PAA manipulator system, the corresponding system equations for the first underactuated link can only be integralized once, and certain mathematical constraints are found to exist between the angle velocities in second and third actuated chains and the angular velocity of the first underactuated chain, and this angular velocity constraint is referred to as a first-order nonholonomic characteristic [51]. The physical structure of the planar PAA manipulator system is shown in Figure 5.

Let $q={[q_1,q_2,q_3]}^T$, Based on the dynamic Equation (5), the limitation imposed on the passive joint is

$$M_{11}{\ddot{q}}_1+M_{12}{\ddot{q}}_2+M_{13}{\ddot{q}}_3+H_1=0$$

(15)

The relationships governing state constraints for an underactuated system can be derived through quadrature, provided that the integrability conditions are met. This principle serves as the basis for the control strategy. The integrability conditions pertinent to a planar three-link PAA underactuated system are outlined in Lemmas 2 and 3.

Lemma 2.

The condition stated in (15) is considered partially integrable only if the gravitational torque described in (15) remains constant, and the variable associated with the passive joint is absent from the inertia matrix, denoted as $M\left(q\right)$.

Lemma 3.

Constraint (15) can be considered fully integrable solely under the condition that the system demonstrates partial integrability.

For the control for the systems, Lai et al. [24] proposed a strategy based on angular velocity constraints between the planar PAA system connecting links and angular constraint relationship of planar chapel Acrobot, and two Lyapunov functions were constructed to design controllers for the actuated connecting links, and the angular control of each connecting link was implemented at the same time. This methodology offers innovative insights and strategies for the regulation of planar PAA systems. Zhang et al. [52] designed an intelligent optimization algorithm, and based on it, a segmentation control approach is introduced to ultimately realize the control objectives of the planar PAA system. Zhang et al. [53] suggested a switching controlling strategy to solve position-attitude control problems, and designed the main controller and auxiliary controller for alternating switching control to realize the position-attitude controlling objectives. Gao et al. [54] proposed an innovative two-phase switch control scheme aimed at the PAA under-actuated manipulator, based on enhanced optimized particle swarms.

Furthermore, for this 3-DoF PUM, Lai et al. [55] propose a rapid and efficient continuous state feedback control approach utilizing intelligent optimization for a planar three-link PAA underactuated system characterized by first-order nonholonomic constraints. To address this issue, it is essential to analyze the first-order non-global properties of planar PAA systems, based on which the system is degraded for planar type Acrobot by controlling the angular velocities of the two driving connecting rods in the system to a certain linear relationship; then, the system has the holonomic characteristics and uses the holonomic characteristics for fast control of planar PAA systems [56]. Considering the uncertainty in the system, Zhang et al. [57] formulate a series of rapid terminal sliding mode controllers aimed at following designated trajectories, while also establishing adaptive laws to ensure the stability and convergence of the tracking system.

Although these studies have made significant progress in the control of planar PAA systems, there are still some knowledge gaps and issues that require further research. For example, how to further enhance the control precision and robustness of the system, especially when facing complex environments and uncertainties; how to optimize the control algorithms to reduce the computational complexity and resource consumption; and how to apply the existing research results to a wider range of domains and scenarios to achieve a wider range of application values. In order to solve these problems, we propose a classification of future research work: first, in-depth study of the dynamic characteristics and control mechanisms of planar PAA systems, and exploration of more advanced control methods and algorithms; second, strengthening interdisciplinary cooperation, and applying advanced technologies such as machine learning to the control of planar PAA systems; and third, carrying out experimental research and application validation, and continuously optimizing and improving the control strategies and methods through practical data and feedback control strategies and methods.

3.1.3. ${\mathrm{PA}}^n(n>2)$ Manipulator System

The planar ${\mathrm{PA}}^n(n>2)$ manipulator has the same first-order nonholonomic characteristic; that is, a certain mathematical restraint among the angular velocities of the actuated links and the first underactuated link.

For the planar PA system (as shown in Figure 6a), it is actually the same structural model as the previously mentioned Acrobot, which essentially consists of two linkages, with the first linkage joint being the passive joint and the second linkage joint being the active joint. An inverse kinematics approach is employed to determine the desired angle of the two segments, an active flexible link controller is designed to stabilize it at the target angle and suppress its vibration and the controller’s parameters are optimized using a genetic algorithm [58].

For the planar PAAA system (as shown in Figure 6b), the literature [59] proposed a switching control method with three phases; according to the designed connection sequence of the three drive linkages, the system was sequentially downgraded to three planar virtual Acrobot; to achieve the overall system control goal, the angle of motion constraints of the actual systems and the angular constraint connection of the planar virtual Acrobot are taken into consideration. However, there is jitter and instability during the switching process, and intelligent switching strategies are needed to enhance the stability and continuity of the system. Drawing from the previous research, to achieve the position control of such systems, Lai et al. [60] proposed a new idea of reduced-order control, which simplifies the planar ${\mathrm{PA}}^n$ system by analyzing only the first active linkage and the second passive linkage. According to the angle constraint relationship, two Lyapunov functions were constructed to design controllers for the driving linkage, and the angle control of each linkage is implemented at the same time.

For planar ${\mathrm{PA}}^n$ systems (as shown in Figure 6c), Wang et al. [61] suggested a segmented controlling method based on a combination of particle swarm optimization algorithms or genetic algorithms, and bring the first underactuated linkage to its target angle while the drive lever is being controlled.

On the basis of realizing the positioning control in a planar ${\mathrm{PA}}^n$ system and ensuring system stabilization control, Wang et al. [62] suggested a successive control methods, which only requires design of one controller to attain the control target of the system. Considering the existence of uncertain parameters perturbation and external disruptions within the system, Wang et al. [63] suggested a robust controller strategy with adaptive tracking control to accomplish position control of planar ${\mathrm{PA}}^n$ systems. Wang et al. [64] offered a model-reduction and internet smart computation-based control approach for a planar n-link underactuated manipulator with a passive first joint. The goal of the control approach is to achieve the goal of moving the end-point from the beginning state to the position that is desired.

We also consider a special case regarding the planar PAPA systems (as shown in Figure 6d), for which Tafrishi et al. [65] suggested an inverse dynamics model independent of singularities in the inertial coupling for an underactuated manipulator. The subject of the study is a 4-DoF mass-rotating underactuated manipulator that has two passively and two active couplings, where singularity-free inverse dynamics are enabled by assuming a positive deterministic condition on the inertia matrix.

The present study primarily emphasizes the aspect of control of planar ${\mathrm{PA}}^n$ systems with single or a few connecting rods, and it is still a challenge to analyze the global stability of multilink systems, which can be explored for the dynamics of multilink systems, and a more comprehensive stability analysis model can be established to provide a theoretical basis for the control of the system. Existing control methods are mainly based on the system model under ideal conditions, but in practical applications, the system may face complex environmental disturbances and uncertainties, which requires researchers to carry out extensive studies on the system control strategy in complex environments, and to study the plane underactuated mechanical system physically, and to introduce external devices such as sensors, in order to promote the adaptability and robustness of the system. Meanwhile, the real-time and energy consumption optimization of the system control strategy is also an important performance evaluation index, and more efficient control algorithms and energy-saving strategies, such as the use of event-triggered control, need to be explored to improve the real-time and energy efficiency of the system.

For the planar underactuated manipulator system with passive joints, the planar Acrobot is a complete system with angle constraints, and the angle of the passive joint can be obtained directly from the angle of the active joint. Therefore, the control method of the system is relatively simple, and the control of passive joints can be realized by simple trajectory tracking. For the planar PAA system and planar ${\mathrm{PA}}^n(n>2)$ system, the system can be classified as a first-order nonholonomic system that is subject to constraints on angular velocity. The angle relationship needs to be obtained by integrating the underactuated equation. The control method is complex, and the conventional trajectory tracking control is not enough to achieve its control objectives. For such systems, some researchers use intelligent optimization algorithms, state feedback, switching control, model degradation and other methods to study. The results of these studies show that these methods are effective.

3.2. The Passive Last Joint

In this section, we will explore the case wherein the end joints cannot be driven. Planar manipulator systems in which the end joints are underactuated are divided into three main categories: (1) Planar Pendubot (first joint: active, second joint: passive); (2) Planar AAP systems; and (3) Planar ${\mathrm{A}}^m$P$(m>2)$ systems.

3.2.1. Pendubot

The Lagrangian equation can be utilized to articulate the dynamic equations pertinent to the Pendubot system.

$$\left[\begin{array}{cc}{M}_{11}& M_{12}\\ M_{12}& M_{22}\end{array}\right]\left[\begin{array}{c}{\ddot{q}}_1\\ {\ddot{q}}_2\end{array}\right]+\left[\begin{array}{c}{H}_1\\ H_2\end{array}\right]=\tau$$

(16)

The elements of the matrix are

$$\left\{\begin{array}{c}{M}_{11}=y_1+y_2+2y_{3}cos{q}_2\hfill \\ M_{12}=y_2+y_{3}cos{q}_2\hfill \\ M_{22}=y_2\hfill \\ H_1=-y_3+\left(2{\dot{q}}_1{\dot{q}}_2+{\dot{q}}_2^2\right)sin{q}_2\hfill \\ H_2=y_3{\dot{q}}_1^{2}sin{q}_2\hfill \end{array}\right.$$

(17)

where

$$\left\{\begin{array}{c}{y}_1=m_1{l_1}^2+m_2{L_1}^2+J_1\hfill \\ y_2=m_2{l_2}^2+J_2\hfill \\ y_3=m_2{L}_1{l}_2\hfill \end{array}\right.$$

(18)

The constraint equation for the planar Pendubot can be derived as indicated in (17).

$$M_{12}{\ddot{q}}_1+M_{22}{\ddot{q}}_2+H_2=0$$

(19)

The constraint represented by Equation (19) cannot be integrated to derive a relationship for velocity constraints, leading to the classification of the planar Pendubot as a second-order nonholonomic system. Consequently, when employing traditional methods to maneuver the system towards the desired angles, the passive link frequently exhibits a non-zero angular velocity. To address this challenge, the nilpotent approximation was suggested for controlling the planar Pendubot; however, this approach is iterative in nature, potentially resulting in prolonged timeframes to meet the control objectives. Therefore, there is a necessity for an alternative and more effective control strategy to achieve the desired outcomes for the planar Pendubot.

The planar Arcobot and the planar Pendubot are the simplest manipulator structures with two degrees of freedom and are very representative. Taking these two underactuated manipulator as objects, the following experimental simulation analysis is carried out.

In the first control phase, a uniform trajectory is planned for both manipulator models as follows:

$$S_1\left(t\right)=\left\{\begin{array}{cc}{q}_{a0}+{\tilde{\mathit{q}}}_a\left[{\displaystyle \frac{t}{t_1}}-{\displaystyle \frac{sin\left(\frac{2\pi t}{t_1}\right)}{2\pi }}\right],\hfill & 0⩽t<t_1\hfill \\ \\ q_{at},\hfill & t=t_1\hfill \end{array}\right.$$

(20)

where ${\tilde{q}}_a=q_{a\mathrm{f}}-q_{a0}$; $q_{a0}$ is the initial angle of the driving connecting rod; $q_{a\mathrm{f}}$ is the target angle of driving connecting rod; $t_1$ is the end time of the first stage. In this study, $t_1=1$ s. It can be seen from Equation (20) that, when moving along this trajectory, the driving link will start from its initial position $q_{a0}$ and arrive at the target state $q_{a\mathrm{f}}$ at $t_1$. This enables the achievement of the control objective for the system’s driving connection.

The trajectory tracking control is designed as follows:

$${\tau }_a=\left(\begin{array}{c}-P_1{\mu }_1^a-f_a-D_1{\dot{\mu }}_1^a\end{array}\right)g_{aa}^{-1}$$

(21)

$P_1$ and $D_1$ are the control coefficients of the PD controller, $g_{aa}$ and $f_a$ are the corresponding parameters of the driving linkage in the matrix $M\left(q\right)$ and $f\left(x\right)$, respectively. ${\mu }_1^a=x_a-S_1\left(t\right)$, ${\mathcal{X}}_a$ is the corresponding state of the driving link.

In the second-order control phase, the oscillatory trajectory is formulated with adjustable parameters in the following manner:

$$S_2\left(t\right)=\left\{\begin{array}{cc}{q}_{a\mathrm{f}},\hfill & t=t_1\hfill \\ q_{a\mathrm{f}}+Us\left(t\right),\hfill & t_1<t<t_{\mathrm{f}}\hfill \\ q_{a\mathrm{f}},\hfill & t⩾t_{\mathrm{f}}\hfill \end{array}\right.$$

(22)

where $U={\displaystyle \frac{1}{2}}\left(tanh\alpha -tanh\beta \right)$; $s\left(t\right)=A_{m}sin\left(wt\right)$; $\alpha =\left(\begin{array}{c}8t-8t_1-4t_{\mathrm{r}}\end{array}\right)/t_{\mathrm{r}}$; $t_{\mathrm{r}}$ and $t_{\mathrm{d}}$ are two constants of the rise time and fall time of the pulse function, respectively; $\beta =\left(\begin{array}{c}8t-8t_{\mathrm{f}}-4t_{\mathrm{d}}\end{array}\right)/t_{\mathrm{d}}$. When $t\in \left(\begin{array}{cc}{t}_1,& t_{\mathrm{f}}\end{array}\right)$ (the conclusion time of stage 2 control), $A_m(m=1,2)$ and w are the amplitude and frequency of $S_{\mathrm{t}}$, respectively. $A_m$, w and $t_{\mathrm{f}}$ are adjustable trajectory parameters.

According to the underactuated connecting rod target state design evaluation function is

$$h_1=\left|\left[\begin{array}{c}{q}_p\left(t_{\mathrm{f}}\right)\end{array}\right]-q_{p\mathrm{f}}\right|+\left|\begin{array}{c}{\dot{q}}_p\left(t_{\mathrm{f}}\right)\end{array}\right|$$

(23)

where $q_p\left(t_{\mathrm{f}}\right)$ and ${\dot{q}}_p\left(t_{\mathrm{f}}\right)$ are the angle and angular velocity of the underactuated linkage at the moment of $t_{\mathrm{f}}$, respectively; and $q_{p\mathrm{f}}$ is the target angle of the underactuated linkage, which is optimized using the differential evolutionary algorithm to optimize the parameters $A_m$, w and $t_{\mathrm{f}}$.

The trajectory tracking controller of the second stage is designed as

$${\tau }_a=\left(\begin{array}{c}-P_2{\mu }_2^a-f_a-D_2{x}_{a+2}\end{array}\right)g_{aa}^{-1}$$

(24)

where $P_2$ and $D_2$ are the control coefficients of the controller, ${\mu }_2^a={\mathbf{x}}_a-S_2\left(t\right)$.

The model parameters are selected as follows: $m_1=2$ kg, $m_2=1$ kg, $L_1=L_2=1$ m, $J_1=0.167$, $J_2=0.083$, Parameters of controllers (21) and (24) are set to $P_1=P_2=1$ and $D_1=D_2=1.8$. The parameters of the algorithm are as follows: variation factor $p_m=0.3$, $p_c=0.7$, iteration number $G_{max}=200$, error range of evaluation function ${\epsilon }_1=1\times {10}^{-4}$. Other simulation parameters are set to $t_1=t_{\mathrm{r}}=t_{\mathrm{d}}=1 \mathrm{s}$.

To assess the feasibility of the proposed strategy, a set of parameters was selected to build the Sinmulink model of the system, and a set of feasibility verification simulations was carried out with Matlab 2024a software. The initial state and target state parameters of the two connecting rods are set as $\left[\begin{array}{cccc}{q}_{10}& q_{20}& q_{1\mathrm{d}}& q_{2\mathrm{d}}\end{array}\right]=\left[\begin{array}{cccc}0& 0& -0.847& 6.481\end{array}\right]$rad and $\left[\begin{array}{c}{\dot{q}}_{10},{\dot{q}}_{20},{\dot{q}}_{1\mathrm{d}},{\dot{q}}_{2\mathrm{d}}\end{array}\right]=\left[\begin{array}{cccc}0& 0& 0& 0\end{array}\right]\mathrm{rad}/\mathrm{s}$.

The optimized trajectory parameters are obtained by intelligent optimization algorithm. The plane Acrobot is $A_1=0.22\mathrm{rad}$, $w=2.76\mathrm{rad}/\mathrm{s}$ and $t_{\mathrm{f}}=2.23$ s. The plane Pendubot is $A_2=0.90$ rad, $w=0.56$ rad/s and $t_{\mathrm{f}}=3.43$ s. The simulation results are shown in Figure 7 and Figure 8 to control the stability of the target state eventually.

The effects of different initial state and target state parameters and target state parameters on the simulation results are discussed, and a set of simulation experiments are conducted as follows. The initial parameters of the system are selected as $\left[\begin{array}{cccc}{q}_{10}& q_{20}& q_{1\mathrm{d}}& q_{2\mathrm{d}}\end{array}\right]=\left[\begin{array}{cccc}0& 0& -0.517& 2.419\end{array}\right]$ and $\left[\begin{array}{c}{\dot{q}}_{10},{\dot{q}}_{20},{\dot{q}}_{1\mathrm{d}},{\dot{q}}_{2\mathrm{d}}\end{array}\right]=\left[\begin{array}{cccc}0& 0& 0& 0\end{array}\right]$ rad/s.

The optimized trajectory parameters are obtained by algorithm. The plane Acrobot is $A_1=0.07\mathrm{rad}$, $w=1.76\mathrm{rad}/\mathrm{s}$ and $t_{\mathrm{f}}=4.55$ $\mathrm{s}$. The plane Pendubot is $A_2=0.26$ rad, $w=3.42\mathrm{rad}/\mathrm{s}$ and $t_{\mathrm{f}}=3.72$ $\mathrm{s}$.

The simulation results of the second group are shown in Figure 9 and Figure 10. The system finally realizes the stable control of the target state.

Through two sets of simulation experiments, it can be seen that the proposed control strategy remains applicable when different initial state and target state parameters are changed for simulation, which proves the efficacy of the designed control strategy.

The Pendubot is a planar 2-DoF underactuated manipulator with the passive second joint [66]. The physical structure of the Pendubot manipulator is shown in Figure 11. Although the planar Pendubot linear approximation model is uncontrollable, the system has the nilpotent approximation property [67]. To exploit this property, A.D. Luca et al. [66,68] design an open-loop iterative strategy to realize the stabilization system control. Based on the reaserch of A.D. Luca, He et al. [69] proposed a parametric polynomial periodic input control strategy with general applicability, but the factors that determine the accuracy and even the stability of the controller is complex, and difficult to be used for real-time control. Suzuki et al. [70] proposed a nonlinear control strategy with consistent asymptotic stability by using nilpotent approximation technique and parameterized periodic input control method.

Suzuki et al. [71] used averaging to simplify the planar Pendubot model, and the periodic control inputs is applied to the active joint and the underactuated joint converges to target value. Alfredo et al. [72] suggested control strategy for tracking discretely planned desired trajectory for Pendubot. According to the system control objective trajectory of the driving linkage, optimization algorithm should be used to optimize the parameters of the desired trajectory, and feedback control should be used to regulate the tracking error value of the model.Wu et al. [73] proposed a Pendubot controlling approach based on Fourier transform and intelligence optimization. A time-dependent controller corresponding to the harmonic term frequency is designed, which achieves a controlling goal for moving these systems around from an initial point to a destination point.

Researchers achieve precise control of Pendubot through feedback control of state observer. For examples, Zhang et al. [74] proposed an Equivalent Input Disturbance bases system to reduce effect of velocity noise on the system control performance. Ramírez-Neria et al. [75] proposed an active disturbance rejection control strategy based on extended state observer to achieve the trajectory tracking controller of the Pendubot system with effective disturbance suppression. Xin et al. [76] provided an analytical solution that, when applied to the appropriate closed-loop system, minimizes the real component of the main pole of the linearized model around the downward equilibrium point of all Pendubot while preserving stiffness.

Harandi et al. [77] presented a specific construction of an ideal inertial matrix to simulate the partial differential equations associated with kinetic and potential energy shaping based on interconnections and damping allocation passivity, and the method was applied to a range of Pendubot systems. Turrisi et al. [78] proposed an iterative method for feedback nonlinearization with an active and passive degree of freedom based on an online updated model, estimating the perturbations caused by model uncertainties in active and passive degrees for freedom during the learning process. Parulski et al. [79] proposed the strategy based on partial feedback linearizations, and specifically investigated the properties of closed-loop systems and zero dynamics.

Wei et al. [80] addressed the matching and mismatching uncertainty of Pendubot by using an orthogonal decomposition method to downgrade the control task to equal constraints on the state of the system, and proposed an adaptive robust control method based on constraints that are more fragmented, so that the mismatched portion disappears after the decomposition. Wei et al. [81] proposed an internal stabilization controller based on output feedback to stabilize Pendubot in an unsteady vertically elevated position, based on a computational algorithm with various results from the “polynomial matrix method”.

Examining plants with two symmetrical real zeros around the origin, a two-origin zero, and two combination fictitious zeros in a single approach, Xin et al. [82] proposed a more general parameterization of the second-order strongly enhancing controller, with results applicable to the Pendubot, and verified the theoretical conclusions through simulations. The investigation focused on the minimum order of a firmly stabilizing control system for a two-chain planar robot proceeding in a horizontal plane.

To address the position control problem for an under-excited uncertain dual flex-joint horizontal cantilever with only one actuator, Yan et al. [83] established a dynamic model of position double flexible jointed. Its characteristics were analyzed and an adaptation neurocorrection-poststepping strategy was proposed to achieve positional control of positional biflexible joints, which can effectively deal within the impact from parametric perturbations, models’ uncertainties and outside disturbances, and relaxes restrictions that traditional control methods.

Aiming at the unstable unforced equilibrium position problem encountered in point-to-point tracking control, Gulan et al. [84] proposed a Pendubot unified control scheme founded on nonlinear moving horizontal estimation and nonlinear model predictive control, which employs a Gaussian–Newton real-time iterative scheme to be implemented on such a fast, underactuated mechatronic system, to present an efficient solution to the problem of obtaining underlying nonlinear optimization through sequential quadratic programming.

Although there have been many theoretical research results, experimental validation and practical applications are still essential. Future research could focus more on the testing and application of Pendubot in the laboratory or real industrial environment to verify its feasibility and practicality.

3.2.2. AAP Manipulator

The physical structure of the AAP manipulator system is shown in Figure 12. The dynamics of the planar AAP type robot is modeled as follows.

$$\left[\begin{array}{ccc}{M}_{11}& M_{12}& M_{13}\\ M_{21}& M_{22}& M_{23}\\ M_{31}& M_{32}& M_{33}\end{array}\right]\left[\begin{array}{c}{\ddot{q}}_1\\ {\ddot{q}}_2\\ {\ddot{q}}_3\end{array}\right]+\left[\begin{array}{c}{H}_1\\ H_2\\ H_3\end{array}\right]=\left[\begin{array}{c}{\tau }_1\\ {\tau }_2\\ 0\end{array}\right]$$

(25)

For a planar AAP type manipulator, the underactuated characteristic part of (28) is

$$M_{31}{\ddot{q}}_1+M_{32}{\ddot{q}}_2+M_{33}{\ddot{q}}_3+H_3=0$$

(26)

According to Equation (26), the constraint relations for the underactuated linkage are as follows:

$$\left\{\begin{array}{c}{\dot{q}}_3=-{\int }_0^T{\displaystyle \frac{M_{31}{\ddot{q}}_1+M_{32}{\ddot{q}}_2+M_{33}{\ddot{q}}_3}{M_{33}}}\mathrm{d}t-{\dot{q}}_{30}\hfill \\ q_3={\int }_0^T{\dot{q}}_3\mathrm{d}t+q_{30}\hfill \end{array}\right.$$

(27)

The constraint relationship between the connecting rods cannot be obtained directly with Equation (26). However, when the trajectory of the actuated linkage is given, the state of the underactuated linkage is calculated indirectly through Equation (26). Therefore, using the relationship between the underactuated linkage and the actuated linkage, Equation (26), it is feasible to indirectly achieve a control objective for the underactuated linkage while directly controlling the actuated linkage to achieve a control objective.

About the planar AAP manipulator, the literature [85] uses differential geometry theory to research the controllability problem of the three-linkages mechanical system when passive joints are in different positions, and proves that the AAP system is a second-order nonholonomic system and satisfies the STLC condition. Recently, many scholars have made use of various methods such as differential geometry theory, nonlinear feedback control law, fuzzy control strategy and so on, to explore the control problem of the AAP manipulator in depth.

Arai et al. [86] performed a detailed analysis for the controlled planar AAP system with underactuated end joints, designed a trajectory from arbitrary primary positions which are adjusted to desired positions based on the second-order canonical chained standard type of the system, and asymptotically stabilize the system through nonlinear feedback control law. Luca et al. [87] based themselves on the second-order canonical chained standard type of the system to calculate the impact coordinates of third linkages, designed a smooth desired trajectory according to impact coordinates and designed the feedback stability controller to enable the driven link to move around the necessary trajectory, ultimately realizing the stability control of the AAP system. Shiroma et al. [88] designed the feedback controller for stabilized controls of planar AAP systems, in view of which they analyzed the dynamic coupling features of underactuated manipulators systems [89], and used this characteristic in positions controlling the underactuated manipulator. Meanwhile, they also focused on the nonholonomic constraints of the planar AAP system [90], in view of which they studied the position control of this system [91]; they also studied the motion planning issue [92] and trajectory tracking issue [93,94] of the planar AAP system.

Liu et al. [95] designed segmented controlling method for positions controlling problems of a planar AAP system. The whole control process is organized in two phases: the first stage is based on a dynamics model to realize the position regulation for the first active linkage control, and the second phase realizes the control of the second and third passive joints without considering the motion of the first joints. This stage adopts the fuzzy control strategy, which considers the motion of the system as the extension or folding of underactuated joints and the rotation of active joints. In this stage, the fuzzy control strategy is used, and the motion of the systems is regarded as the synthesis of the extension or folding of underactuated joints and the rotation of active joints.

Urrea et al. [96] designed and implemented an original method for controlled oscillation compensation for a 3-DOF underactuated manipulator to reduce the impact of passive joints to the whole performances of this manipulator. Wu et al. [97] designed three error models to increase the precision of movement calibration of the 3-DOF underactuated manipulator, which solved the problems in terms of models intricacy, precision of verification, noise uncertain impact, parameters identifiability, etc.

Huang et al. [98] suggested a generalized monitoring scheme for a planar 3-DoF underactuated manipulator with a single passive joint, based on track control and trajectories forecasting. Two trajectories were created for each active joint in order to achieve the control goals. The one-stage trajectory was created using the active links’ initial and target angles, and the second stage trajectory was created using the constraints that exist between the passive and active joints. By following the intended paths, the sliding mode variable structure controller and the differential evolutionary algorithm make sure that every link eventually reaches the target angle.

Flexible jointed manipulators have also been of great interest. However, the characteristic that some states of single-input multi-degree-of-freedom flexible joint manipulators cannot be directly controlled by a single input makes their position control and vibration suppression difficult to be solved. To address this problem, Yan et al. [99] suggested a framework-based dual-loop adaptive neural control of sliding modes approach. The objective is to adjust the actuator angle so that it aligns with the time-dependent reference actuator angle that is dynamically generated, for which a dual closed-loop controller with a torque controller in the inner loop was built. In order to reduce the vibration of the two links, the outer loop’s design is predicated on the dynamic coupling connection.

3.2.3. ${\mathrm{A}}^m$P$(m>2)$ Manipulator Systems

Furthermore, owing to the swift advancements in science and technology, the manipulator must be capable of performing certain intricate tasks that are beyond the scope of a 2-DOF manipulator. Planar ${\mathrm{A}}^m$P$(m>2)$ systems have second-order nonholonomic characteristics, and this class of systems satisfies the STLC [100] and is fully controllable [101]. The physical structure of planar ${\mathrm{A}}^m$P manipulator systems is seen in Figure 13. Luca [102] suggested trajectory planning and feedback control strategy for planar ${\mathrm{A}}^m$P systems. The AAP system is converted to a second-order chain reaction standard type, while effect coordinates of the last link are calculated and the smooth desired trajectory is computed based on the impact coordinates. Eventually, the stabilizing control of the entire system is achieved by the movement of a feedback stabilizing controller along the intended trajectory.

For the planar AAAP system position control problem, Luca et al. [103] used the linked reaction between accelerations of joints in order the control motion of passive joints, and proposed a bi-modal fuzzy PID control algorithm to realize the trajectory tracking at the end of manipulator and attitude controlling end rod.

Swarm planar underactuated robots can coordinate among themselves to do a job within a certain time limit, as well as ignoring gravity, being energy-efficient, and being lightweight. Huang et al. [104] proposed a control strategy for such robots through motion planning and intelligent algorithms, and optimized the parameters using differential evolutionary algorithm.

The planar underactuated robotic arm system with underactuated joints located at the end joints has second-order non-complete characteristics with angular acceleration constraint relations, and the angular relations need to be obtained after quadratic integration. For this kind of research object, researchers have proposed two-stage control, adaptive sliding mode control, fuzzy control and other methods to solve their control problems, and there are also researchers who use the oscillation compensation method to improve the calibration accuracy of the system and eliminate the influence of noise uncertainty. These methods can theoretically realize the control of planar underactuated robotic arm systems with underactuated joints located at the end joints, but the complexity of the control methods is high, and the requirements for hardware resources are also high when applied to the actual hardware system. The above control strategys are mainly designed for planar four-degrees-of-freedom manipulator as the object of research, and there is no analysis of the general characteristics of the planar manipulators with underactuated end joints in terms of position control and working attitude, whose scope of application has certain limits. Meanwhile, for the manipulator in actual working conditions, it often has clear requirements about its control speed. Therefore, the position control of the planar ${\mathrm{A}}^n$P system needs in-depth research.

3.3. The Passive Middle Joint

The planar manipulator with underactuated intermediate joints has second-order nonholonomic characteristic and satisfies the STLC condition [105,106]. Since the underactuated joints are uncertain in the middle of the manipulator [107], then this class of system can be classified into various cases as the system’s number degrees in freedom increases.

3.3.1. APA Manipulator Systems

The physical configuration of the planar APA manipulator system is illustrated in Figure 14. The dynamical equations of the system are established according to the Euler–Lagrange method as

$$M_{21}{\ddot{q}}_1+M_{22}{\ddot{q}}_2+M_{23}{\ddot{q}}_3+H_2=0$$

(28)

Since the planar manipulator with underactuated intermediate joints has second-order non-complete characteristics and its linear approximation model lacks controllability, it is also difficult to directly obtain the mathematical constraint relationship between the states of the actuated and underactuated joints, which makes it difficult to further design the stabilization control strategy. The existing position control methods are mainly to control the first driven joint at the target angle, and downgrade the original system to a first-order non-complete system for stable control. So far, no effective control strategy has been proposed to realize the position control of this type of system.

The planar APA manipulator [108] does not have the angular velocity constraint relation as planar PAA manipulator, nor can it be transformed into the chain structure of a planar AAP manipulator, resulting in the inability to structure a controller to realize stability control of this type of system based on the above mathematical constraints. Luca [109] analyed various types of planar underactuated manipulators, and pointed out that the planar APA system’s remains a challenging and open research topic. Bhave et al. [110] proposed an terminal sliding mode approach for the position control of such systems; however, this method alters the underactuated nature of the system by immobilizing its passive joint once the passive joint reaches the target position. Xiong et al. [111] introduced a technique that utilizes energy dissipation for the planar APA manipulator system. Li et al. [112] proposed a two-stage drift suppression control method. The algorithm for the system solution gives both the dynamic model and the desired angle. And a differential evolutionary algorithm is used to find suitable parameters for the parametric trajectory. Huang et al. [113] transformed the APA manipulator systems into a two-dimensional suppositional Pendubot and a two-dimensional suppositional Acrobot. By controlling the virtual system separately, the overall control goal is finally achieved.

3.3.2. ${\mathrm{APA}}^n(n>2)$ Manipulator Systems

For the planar APAA system, a phased control strategy grounded in model degradation was proposed by the literature [114], and the PSO algorithm is employed to determine the desired angle for each linkage within the system. The system development of the controller is aimed at guiding the system from the starting position to the desired target position. Xiong et al. [115] introduced a stabilizing control strategy via energy decay. According to the energy attenuation approach, they manage the initial connection to the target, while the planar APAA system is simplified into a PAA system. And the control of the PAA system is divided into two phases. Furthermore, in order to simplify the control process and achieve the control objectives faster, Huang et al. [116] introduced a technique that utilizes underactuated constraints and intelligent optimization for the planar APAA system, designing the trajectory with adjustable parameters calculated by intelligent optimization.Controllers are engineered to monitor the specified trajectory of active joints, so that active and underactuated links reach target state synchronously under the underactuated constraints.

For the planar APAAA system, Xiong et al. [117] introduced a position control method that relies on the model degradation. By sustaining the initial state of the first rod, the original has been degraded to a first-order nonholonomic system; secondly, the system is degraded into three planar virtual Acrobot systems with holonomic constraints by the angle of the drive lever which is controlled in segments, and based on the angle constraint relationship of the planar virtual Acrobot systems, using the PSO algorithm to obtain the desired angle that corresponds to the desired position. The controller is developed in phases utilizing the Lyapunov formula to realize the system control objectives.

For planar ${\mathrm{APA}}^n$ systems, Xiong et al. [118] introduced a segmented position control approach. The overall control is segmented into n − 2 stages. In every control phase, the initial joint is regulated to be stationary and immobile. According to the angular velocity constraints between the second underactuated linkage and the other actuated linkages, it is necessary to use an intelligent optimization algorithm to obtain the desired angle that aligns with the intended position of system endpoint.According to the control objectives in the n − 2 stages, the controllers are sequentially structured to meet the control purpose in each stage, so as to realize the overall control purpose for the whole system.

3.3.3. ${\mathrm{A}}^m{\mathrm{PA}}^n(m\ge 1,n\ge 1,m+n\ge 2)$ Manipulator Systems

For the planar AAPA system [119], a control method of model degradation was proposed [120], which sequentially degrades the original system into a two-dimensional suppositional AAP system and a two-dimensional suppositional Acrobot system.

Using the chain structure of the two-dimensional suppositional AAP system and the angular constraint relationship of the two-dimensional suppositional Acrobot to stabilize and control the two suppositional systems to attain the control purpose for the entire system.

The model structure of the planar ${\mathrm{A}}^m{\mathrm{PA}}^n(m\ge 1,n\ge 1,m+n\ge 2)$ manipulator system is shown in Figure 15. Wang et al. [121] proposed a differential evolutionary algorithm incorporating the geometric relationship of PUM to transform the positional attitude control into angular control. Wu et al. [122] introduced a position control approach that leverages bidirectional motion planning alongside intelligent optimization techniques. Additionally, all trajectory tracking controllers are made to direct the manipulator’s end effector from its initial position to a predetermined target point using the sliding mode control approach. This method is applied to planar underactuated manipulators with second-order nonholonomic restrictions, where the passive links can be positioned in different locations except the first joint.

A planar manipulator with underactuated intermediate joints has second-order noncomplete characteristics and satisfies the STLC condition. However, due to the uncertainty of the underactuated joints in the middle of the robotic arm, the complexity of the system control increases as the number of joint increases. The control methods for this type of system are similar to those for the planar underactuated robotic arm system with end joints as underactuated joints, model degradation, fuzzy control and adaptive control are used for the research, but most researchers mainly conduct theoretical studies and simulations for validation, and rarely apply them to actual physical systems.

4. Current Challenges of PUM

In the previous section, the control methods of planar underdriven robotic arm systems according to different classifications were discussed, which have been difficult to study due to their complex dynamics models, structural characteristics and integral properties. And due to the system defects, not all of the above methods can be applied to the real working environment. Meanwhile, for the various special cases of planar underdriven robotic arm, it is necessary to analyze and design the characteristics. In this section, the above problems will be discussed from both theoretical analysis and practical applications.

4.1. Theoretical Analysis

4.1.1. Research on Uncertainty and Robustness of System Model

When modeling underactuated robots [69], we usually simplify the actual system model and ignore the high-order terms for the convenience of analysis and calculation, which leads to great differences between the model and the practical characteristics of systems. In addition, there are always measurement errors in system parameters [57,63], some of which are even difficult to measure directly, so modeling errors will significantly reduce the control performance of the system. Except for modeling errors, most existing research results do not fully consider the influence of various external disturbances [123], and often compensate for them roughly or ignore them. Therefore, it is urgent to research the robustness of the system to deal with the influence of model error and external disturbance on control performance.

4.1.2. Research on Unified Control Method

Due to the different degrees of freedom and passive joint positions, each underactuated mechanical system has different coupling characteristics and different dynamic models, which makes the analysis and control of the system very complicated. The majority of the research deals with the stabilization control of individual underactuated mechanical systems, and only a few studies address specific types of PUM. For example, the literature [124] proposed a unification controlling method applicable to 2-DOF Acrobot and Pendubot and considered the case where the primary speed is nonzero or an external perturbation is applied to the passive joints. The literature [98] based in trajectory mapping and tracing controls is proposed to 3-DOF planar underactuated mechanical systems that have a single actuated linkage. However, it is hard to extend these control techniques to other types of PUM. It is a difficult task to research a unified control strategy that can be suitable for all underactuated mechanical systems.

4.1.3. Research on Multi Passive Joints Underactuated System

Almost all current studies only discuss underactuated mechanical systems PUMs where the PUM system has only one passive joint. For the study of PUM with multiple passive joints, there is no mature control theory yet. Although a vibration control approach for PUM with two passive joints has been proposed in the literature [125], the authors only designed a dynamic controller to control the system in a vertical plane with a certain tilt angle, and static equilibrium stabilization was not achieved. The literature [126] proposed a stable control approach for the PUM containing multiple passive joints, and took the planar four-link underactuated mechanical system as the research object. However, only the stability of the control approach is proved, and a stable control strategy is not found. The research on underactuated mechanical systems with multiple passive joints needs to be deepened.

4.2. Practical Applications

4.2.1. Anti-Interference

At present, the research on the underactuated manipulator is primarily designed to control the controller and realize its effective control based upon nominal modeling, without considering the parameters perturbation and external interference of the system. In practical application scenarios, underactuated manipulators will encounter a lot of interference uncertainties [127]. Among them, the modeling, observation and suppression of interference are three important links. It is necessary to study appropriate models to describe the influence of various interference sources on the manipulator, such as force disturbance model, friction model, sensor noise model and so on. For different types of interference, it is necessary to design corresponding interference observation and estimation methods. For the suppression and compensation of interference, it is important to investigate the anti-interference control algorithms to reduce their influence of interference on the motion of the manipulator, which usually includes the design of robust control algorithm, adaptive control algorithm, predictive control algorithm and so on. In a word, in future research work, it is necessary to analyze or carry out robust control from the perspective of the robustness of control methods, so as to realize more perfect motion control research on the system.

4.2.2. Multi-Underactuated Manipulator Coordination

Due to its nonlinear characteristics, the control of underactuated manipulators is relatively difficult. Current research has focused on the study of a single underactuated manipulator. In practical application scenarios, due to the high complexity of work environment and tasks, the cooperation of multiple underactuated manipulators has higher efficiency [128]. In the coordination of multi-underactuated manipulators, it is necessary to consider four aspects: task allocation and planning, communication and coordination, obstacle avoidance and collision detection, action synchronization and cooperative control. When multiple underactuated manipulators work together, it is necessary to design appropriate task allocation and planning algorithms to determine the tasks and actions of each underactuated manipulator. It usually involves task optimization, path planning, resource allocation and other issues. In the aspect of communication and coordination, it is necessary to design an effective communication protocol and coordination mechanism to realize the information exchange between underactuated manipulators. It generally includes distributed control algorithms, collaborative decision-making algorithms, synchronization and communication strategies, etc. To ensure safety and coordination between manipulators, in addition, the manipulators’ collisions and obstacle avoidance must be taken into account. To achieve action synchronization and cooperative control between multiple manipulators, it is necessary to design appropriate control algorithms and strategies to achieve action synchronization and coordination between manipulators. Generally, it involves timing control, synchronous control, cooperative control and other technologies. In the future, multi-agent systems composed of multiple homogeneous or heterogeneous underactuated manipulator systems will be an important research hotspot.

4.2.3. Space Underactuated Manipulator

The PUM system is an underactuated system without gravity constraint. Such systems mainly exist in microgravity environments such as space and deep sea, while the working environment of manipulator systems is mostly in three-dimensional space [129]. Consequently, the three-dimensional underactuated manipulator requires an extension of the planar underactuated manipulator’s control mechanism. When it is extended to three-dimensional space, it is necessary to reconsider its dynamic and kinematic models and constraint analysis, and design corresponding stability control methods. For the study of stabilisation of PUM in three dimensions, this is a crucial application.

5. Concluding Remarks and Future Research

Underactuated mechanical systems are far from being adequately researched, the results about them are far from being widely recognized and universally valued, their potential application areas have not been fully explored, and specific engineering application systems are still very much lacking. Since general methods are only applicable to achieve the control objectives of a particular system, a mastery of general control strategies is required to achieve general control with an underactuated system. Specific analysis was carried out according to the structure of different structured systems. For the advancement of theory study of PUM systems and its combination with actual operation application, further research is to be carried out in the following aspects.

(1) The suggested control strategies for the PUM system with a single passive joint are created using the system’s unique properties, and further analyzed on the basis of constraints between active and passive joints, such as the angular constraints, angular velocity constraints and the chain structure. These control methods are not designed by directly utilizing the underactuated coupling constraint relationship and do not have universal applicability. How to combine intelligent optimization algorithms to explore a unified control strategy using only the underactuated coupling relationship is an urgent problem to explore urgently for a PUM system containing a single underactuated joint.

(2) The current domestic and international research status for planar underactuated manipulator systems only considers the system containing single underactuated joints, and the common control methods are usually designed for a certain type of system. In contrast, control strategy of planar underactuated manipulator systems containing multiple passive joints is still an open problem, and no effective control method has been proposed, so there is a need for an in-depth study of the system, so as to propose a more effective control strategy.

(3) The current research content is mainly for a single plane underactuated manipulator system, while the actual engineering applications, the need for two or more manipulators to work together, by more than one homogenized or heterogeneous underactuated manipulator system composed of multi-intelligent systems, will be an important research hotspot in the future.

(4) One type of underactuated system that is not limited by gravity is the planar underactuated manipulator system, and this kind of system mainly exists in microgravity environments such as space and sea, while most of the working environment of manipulator systems is inside the three-dimensional space. Therefore, research on space-underactuated manipulators is also necessary.

(5) The current research work is still mainly carried out for the analog simulation; the design of the controller, control strategy and anti-disturbance methods are only verified in the numerical analog simulation, while in the actual working environment, the mechanical system often has to encounter a variety of disturbances, such as the system controller’s own small perturbations, the error between the models and the outside world for the interference of the mechanical system, which is an issue that requires a realistic solution. In reality, the disturbance suppression of planar underdriven mechanical systems will be an important research hotspot in the future.

(6) The current research results are based on the nominal model for designing the controller and realizing its effective control, without considering the parameter uptake and external interference of systems. Therefore, in order to carry out more intensive research on the controlling of systems in future research work, it is required to examine the control technique from the standpoint of robustness or robust control.

(7) From the current domestic and international research on planar underdriven manipulator systems, researchers through simulation use idealized parameter data or optimization algorithms to determine the structural parameters of the mechanism, as the size of the manipulator cannot be characterized and does not have practical engineering significance; at the same time, the weight needs to be considered, and the majority of PUM systems are set to be in a gravity-free state (i.e., weight is not analyzed) or in a microgravity state. In practical engineering, weight needs to be strictly considered for special planar underdriven systems, such as inverted pendulums and cranes, etc. It is especially important to build a bridge between the idealized controller design and the controller design in practical engineering.

(8) Most of the published research results only use numerical simulation; therefore, to test the validity of designed control strategies without specific experimental validation, it is necessary to develop a system of a planar underactuated manipulator and experimentally verify the proposed control strategy to unify theory and practice.