Switching Control of Planar PPR Underactuated Robot with External Interference and Non-Zero Initial Velocity

Abstract

Planar underactuated robots are mainly applied in the microgravity field, such as deep sea and deep space. The system modeling and stability control of planar underactuated robots are prerequisites to ensure successful task completion. The planar prismatic–prismatic–rotational (PPR) underactuated robot is one type of planar underactuated robot structure. In this paper, the dynamic system model of planar PPR underactuated robots is built, and the switching-control strategy is designed. In the first phase, an improved PD controller based on the linkage coupling of the PPR model is designed to adjust the state quantities of the three linkages and stabilize the first two linkages to the target state. This controller has certain robustness and rapidity. In the second phase, the PPR model is downgraded to the PR model, an open-loop iterative controller is designed, and the third underactuated link is stabilized to the target state through oscillation convergence. Finally, the effectiveness and applicability of the proposed strategy were verified through the comparison of setting torque interference and the simulation of the initial velocity of the link.

1. Introduction

With the continuous in-depth exploration of the deep sea and deep space, the role of robots in such environments is becoming increasingly important. In fact, in extreme environments, such as weightlessness or microgravity, people cannot directly perform manual operations, and these basic operation tasks must be realized indirectly by robots [1,2,3,4,5]. When a part of the robot fails or is destroyed, the robot has underactuated characteristics in a certain sense.

Analyzing the matching between the quantity of control inputs and the quantity of robot degrees of freedom, they can be divided into fully actuated robots with the quantity of control inputs equal to the quantity of degrees of freedom and underactuated robots with the quantity of control inputs less than the quantity of degrees of freedom [6,7,8,9,10]. The fully actuated robot has linearized control characteristics, and each control input can control the robot’s movement in one degree-of-freedom direction [11,12,13,14]. The number of control inputs of underactuated robots is insufficient. To complete a motion in the direction of more degrees of freedom, it is indispensable to conduct in-depth research and analysis on the complex nonlinear characteristics and specific conditions of underactuated robots. In the same practical engineering applications, underactuated robots only need a small number of controllers to complete the task, which has advantages such as energy saving, lightness, and flexibility compared with fully actuated robots under the same conditions [15,16,17,18,19,20].

More complex planar underactuated robots can be constructed based on prismatic linkage and rotational underactuated linkage, and their control strategies are more complicated. The prismatic–rotational (PR) underactuated robot is the simplest form of construction, consisting of these two most basic linkages. From the perspective of motion stability, Alaci et al. [21] proved that there are irregularity and strong instability in the motion process of PR underactuated robots. Ref. [22] measured the absolute position of the prism-actuated link rod and the relative angle of the rotating link rod and designed a nonlinear observer utilizing the lower-order theory to verify the asymptotic convergence of the observation error. Wu [23] eliminated the angular acceleration of the actuated link from the dynamics equation of a PR underactuated robot and developed a unified controller with a brake to address the point-to-point control challenge in the motion space.

Planar prismatic–prismatic–rotational (PPR) underactuated robots have a wider working range for extraveparular exploration in special environments. In Ref. [24], the Jacobian matrix of the planar PPR underactuated robot was obtained by using spiral theory, and a controller was crafted based on this matrix to achieve stability control. Ichida [25] converted the dynamic model of the planar PPR underactuated robot into an extended non-holonomic double-integral form and designed a controller to achieve the robot’s position control. In Ref. [26], the end effector of the planar PPR underactuated robot achieved the desired position with arbitrary attitude. Ref. [27] analyzed the passivity of a planar PPR underactuated robot, and a controller was developed utilizing the energy strategy to meet specific control objectives. In Ref. [28], a time-varying feedback controller was constructed using a direct method and backstepping technique to bring the planar PPR underactuated robot convergence to the target state.

Most of the above research on planar PPR underactuated robots is based on model construction and controller design, and some achievements have been realized. However, the control methods for this structure are relatively simple and need to be improved in terms of stability time and control performance indicators. The stability control methods for planar PPR underactuated robots need to be further expanded.

This article mainly conducts research in the following areas: (1) The introduction elaborates on the research significance of underactuation, summarizes the current research status of planar PPR underactuated robots, and extracts the directions that need to be improved or enhanced. (2) Section 2 establishes the model of the planar PPR underactuated robot, analyzes its underactuated characteristics, and divides the entire control process into two different phases. (3) In Section 3, the state-space expression is established, and an improved PD controller based on the linkage coupling of the planar PPR underactuated robot model itself is designed to control the first two driving linkages to stabilize to the target state. (4) In Section 4, the planar PPR underactuated robot is reduced to a planar PR model. An open-loop iterative controller is designed. The oscillation of the second driving link is utilized to stabilize the third underactuated link to the desired state, and ultimately all the links of the entire planar PPR robot are stabilized to the target state. (5) Section 5 is the simulation. Under the conditions of external torque interference and non-zero initial velocity of the links, two sets of comparative simulations with different state parameters were established, respectively, to verify the effectiveness and robustness of the proposed strategy. (6) Section 6 summarizes and generalizes the above work and clarifies the areas that need improvement in the future.

2. System Model and Analysis

2.1. Dynamic Model

The PPR underactuated robot moves in horizontal space without considering the gravitational potential energy, and the prismatic connecting rod is independently controlled by the actuated joint. The first and second joints of the robot are mutually orthogonal moving joints, the third joint is a rotating joint, the two moving joints are actuated joints, and the rotating joints are underactuated joints.

The translational motion of the prismatic link is controlled under the action of the control torque, and the rotating link is connected with the underactuated joint without a driving device, which is not directly controlled by any controller and can rotate freely. It is an important control strategy for the planar PPR underactuated robot to propel the rotating underactuated link indirectly to reach the target state by controlling the prism-actuated link motion.

The end point of the first actuated joint is taken as the origin, and a planar PPR underactuated robot model is established in the O-$xy$ plane, as shown in Figure 1. There are three links in the model $(r=1,2,3)$, and each link has a number of symbolic parameters, whose meanings and units are shown in

Table 1. Planar PPR underactuated robot model parameters.

Symbol	Significance	Unit
$m_r$	The mass	kg
$L_r$	The length of link	m
$q_{1,2}$	The displacement	m
$q_3$	The angle	rad
$J_r$	The moment of inertia	$\mathrm{kg}·{\mathrm{m}}^2$
$l_r$	The distance from joint to center of mass	m
${\tau }_r$	The torque	$\mathrm{N}·\mathrm{m}$

.

The dynamics model is established utilizing the Euler–Lagrange approach. Firstly, the Lagrange function $L(q,\dot{q})$ of the system is determined as

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

(1)

$q={\left[q_1{q}_2{q}_3\right]}^T\in {\mathbb{R}}^{3\times 1}$ is the state vector of the system link and $\dot{q}={\left[{\dot{q}}_1{\dot{q}}_2{\dot{q}}_3\right]}^T\in {\mathbb{R}}^{3\times 1}$ is the velocity vector of the system link. Since the system is moving in the horizontal plane, the potential energy $P\left(q\right)$ is zero, and the kinetic energy $K(q,\dot{q})$ is expressed as

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

(2)

The Lagrange 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)

By differentiating the Lagrange function of the system with the velocity quantity, respectively, for the state quantity and substituting it into Equation (3), we can obtain

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

(4)

In the above equation, $M\left(q\right)\in {\mathbb{R}}^{3\times 3}$ is a symmetric matrix with positive-definite characteristics, and $\tau =\left[{\tau }_1{\tau }_2{\tau }_3\right]\in {\mathbb{R}}^{3\times 1}$ is the control torque. $\ddot{q}={\left[{\ddot{q}}_1{\ddot{q}}_2{\ddot{q}}_3\right]}^T\in {\mathbb{R}}^{3\times 1}$ is the system acceleration vector, and $H(q,\dot{q})$ is the combined vector of Coriolis force and centrifugal force. In the dynamic Equation (4), we let

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

(5)

where $M\left(q\right)$ and $H(q,\dot{q})$ are expressed as follows:

$$M\left(q\right)=\left[\begin{array}{ccc}{M}_{11}& M_{12}& M_{13}\\ M_{12}& M_{22}& M_{23}\\ M_{13}& M_{23}& M_{33}\end{array}\right]=\left[\begin{array}{ccc}{m}_1+m_2+m_3& 0& -m_3{l}_{3}sin{q}_3\\ 0& m_2+m_3& m_3{l}_{3}cos{q}_3\\ -m_3{l}_{3}sin{q}_3& m_3{l}_{3}cos{q}_3& J_3+m_3{l}_3^2\end{array}\right]$$

(6)

$$H(q,\dot{q})=\left[\begin{array}{c}{H}_1\\ H_2\\ H_3\end{array}\right]=\left[\begin{array}{c}-m_3{l}_3{\dot{q}}_3^{2}cos{q}_3\\ -m_3{l}_3{\dot{q}}_3^{2}sin{q}_3\\ 0\end{array}\right]$$

(7)

$$\tau =\left[\begin{array}{c}{\tau }_1\\ {\tau }_2\\ 0\end{array}\right]$$

(8)

The nonlinearity of planar PPR underactuated robots is mainly reflected in the following aspects:

(1)

In terms of dynamic equations, the dynamic equation of the planar PPR underactuated robot is established based on the Euler–Lagrange equation. The inertia matrix $M\left(q\right)$ is a nonlinear function of the generalized coordinate q, and the Coriolis force and centrifugal force terms $H(q,\dot{q})$ are nonlinear functions of the generalized coordinates q and $\dot{q}$.

(2)

In terms of kinematics, the kinematics of the planar PPR underactuated robot are described by the homogeneous transformation matrix from the base coordinate system to the end-effector coordinate system. The trigonometric functions containing the joint angle in the homogeneous transformation matrix indicate that there is a nonlinear mapping relationship between the position and attitude of the end effector and the joint angle.

(3)

In terms of the relationship between control input and system response, the planar PPR underactuated robot has a non-minimum phase characteristic. This non-minimum phase characteristic leads to a complex nonlinear relationship between control input and system response. Furthermore, the actuators of underactuated robot systems usually have nonlinear characteristics, such as saturation and dead zones.

2.2. Analysis of Model Characteristics

There is an underactuated link in the planar PPR underactuated robot system, and the control of this link can only be realized by controlling the rest of the prism-actuated link so that the rotating underactuated link is indirectly controlled and adjusted and the stability control is achieved at the same time. Therefore, it is essential to understand the control properties of planar PPR underactuated robots to achieve stability control.

According to Equations (6)–(8), the system dynamics equation is reformulated as follows:

$$\left\{\begin{array}{c}(m_1+m_2+m_3){\ddot{q}}_1-m_3{l}_{3}sin{q}_3{\ddot{q}}_3-m_3{l}_3{\dot{q}}_3^{2}cos{q}_3={\tau }_1\hfill \\ (m_2+m_3){\ddot{q}}_2+m_3{l}_{3}cos{q}_3{\ddot{q}}_3-m_3{l}_3{\dot{q}}_3^{2}sin{q}_3={\tau }_2\hfill \\ (-m_3{l}_{3}sin{q}_3){\ddot{q}}_1+m_3{l}_{3}cos{q}_3{\ddot{q}}_2+(J_3+m_3{l}_3^2){\ddot{q}}_3=0\hfill \end{array}\right.$$

(9)

According to Equation (9), the underactuated equation of the planar PPR underactuated robot is as follows:

$$(-m_3{l}_3\sin q_3){\ddot{q}}_1+\left(m_3{l}_3\cos q_3\right){\ddot{q}}_2+(J_3+m_3{l}_3^2){\ddot{q}}_3=0$$

(10)

In order to analyze the integral properties of dynamic constraints of underactuated joints in planar PPR systems, the following lemma is given.

Lemma 1

([29]). The adequate and essential conditions for the integrability of the dynamic constraint part of the underactuated joint in the underactuated system are as follows:

(1)

The gravitational component within the dynamic constraint of the underactuated joint remains invariant;

(2)

Underactuated joint variables are absent from the inertia matrix.

The gravity term of the dynamics equation of a planar PPR system is 0, which satisfies condition (1) in Lemma 1. However, it does not satisfy condition (2) of Lemma 1. Thus, the constraint equations corresponding to the underactuated joints in the planar PPR system are not integrable. Thus, planar PPR robot is a class of systems with the non-integrability property.

Through Equations (9) and (10), we can obtain

$${\ddot{q}}_3=-{\displaystyle \frac{M_{31}{\ddot{q}}_1+M_{32}{\ddot{q}}_2+H_3}{M_{33}}}=-{\displaystyle \frac{(-m_3{l}_{3}sin{q}_3){\ddot{q}}_1+(m_3{l}_{3}cos{q}_3){\ddot{q}}_2}{J_3+m_3{l}_3^2}}$$

(11)

Through the integration of Equation (11), we can obtain

$$\left\{\begin{array}{c}{\dot{q}}_3=-{\int }_0^t{\ddot{q}}_3\mathrm{d}t+{\dot{q}}_{30}\hfill \\ q_3={\int }_0^t{\dot{q}}_3\mathrm{d}t+q_{30}\hfill \end{array}\right.$$

(12)

$q_{30}$ is the starting angle, and ${\dot{q}}_{30}$ is the starting angular velocity.

Equation (12) provides the constraint relationship between the actuated link and the underactuated link of the planar PPR system. Although Equation (12) is not integrable, according to this constraint relationship, we can indirectly control the underactuated link by controlling the actuated link.

2.3. Control Method

The specific switching-control process is shown in Figure 2. The PD controller is an automatic regulation device composed of two control units: proportional and differential. Its core principle is to respond to system deviations in real time through proportional action and predict the trend of deviation changes in combination with differential action, thereby correcting the dynamic characteristics of the system in advance.

P: Control signals are directly generated based on the current system deviation, which has a linear relationship with the deviation amount, enhancing the system response speed.

D: Predict future trends by calculating the rate of deviation change, generate advanced correction signals, and effectively suppress system oscillations.

The first control phase designs the PD controller for the prismatic actuated link, moves the actuated link from the starting position to the desired state, and ensures that the prismatic actuated link can reach the target state. Meanwhile, the third rotating underactuated link will be in a drift state and rotate freely.

In the second control phase, the first prism-actuated link is kept in the target state at all times, and the planar PPR underactuated robot model is reduced to a PR underactuated robot model. According to the nilpotent approximation characteristic of PR underactuated robot and the iterative contraction theory, the second prism-actuated link is made to move under the open-loop iterative command. The underactuated link is controlled indirectly and quickly leaves the drift state.

In each iteration period, the starting state and the ending state of the prismatic actuated link are consistent with the state before the start of the iteration, the angle of the rotating underactuated link gradually approaches the desired angle, and the angular speed gradually slows down and stabilizes at zero speed to achieve stability control.

3. The First-Phase Control

The control method of the first stage is as follows. Let the state variable x be

$$x=\left[\begin{array}{c}{x}_1\\ x_2\\ x_3\\ x_4\\ x_5\\ x_6\end{array}\right]=\left[\begin{array}{c}{q}_1\\ q_2\\ q_3\\ {\dot{q}}_1\\ {\dot{q}}_2\\ {\dot{q}}_3\end{array}\right]$$

(13)

the state-space expression $\dot{x}=f\left(x\right)+g\left(x\right)\tau$, and the specific expression is as follows:

$$\left[\begin{array}{c}{\dot{x}}_1\\ {\dot{x}}_2\\ {\dot{x}}_3\\ {\dot{x}}_4\\ {\dot{x}}_5\\ {\dot{x}}_6\end{array}\right]=\left[\begin{array}{c}{x}_4\\ x_5\\ x_6\\ f_1\\ f_2\\ f_3\end{array}\right]+\left[\begin{array}{ccc}0& 0& 0\\ 0& 0& 0\\ 0& 0& 0\\ g_{11}& g_{12}& g_{13}\\ g_{21}& g_{22}& g_{23}\\ g_{31}& g_{32}& g_{33}\end{array}\right]\left[\begin{array}{c}{\tau }_1\\ {\tau }_2\\ 0\end{array}\right]$$

(14)

some of the variables in Equation (14) are as follows:

$$\left\{\begin{array}{c}\left[\begin{array}{c}{f}_1\\ f_2\\ f_3\end{array}\right]=-M^{-1}\left(q\right)H(q,\dot{q})\hfill \\ g=\left[\begin{array}{ccc}{g}_{11}& g_{12}& g_{13}\\ g_{21}& g_{22}& g_{23}\\ g_{31}& g_{32}& g_{33}\end{array}\right]=M^{-1}\left(q\right)\left[\begin{array}{ccc}1& 0& 0\\ 0& 1& 0\\ 0& 0& 0\end{array}\right]\hfill \end{array}\right.$$

(15)

In the following, the corresponding controller is researched for the control target, and the Lyapunov function $V\left(x\right)$ is constructed

$$V\left(x\right)={\displaystyle \frac{1}{2}}[P_1{(x_1-x_{1\mathrm{d}})}^2+P_2{(x_2-x_{2\mathrm{d}})}^2+(x_4^2+x_5^2)]$$

(16)

$P_1$ and $P_2$ are positive constants; $x_{1\mathrm{d}}=q_{1\mathrm{d}}$ and $x_{2\mathrm{d}}=q_{2\mathrm{d}}$ are the target state quantities of the first and second actuated links, respectively.

Calculating the derivative of the Lyapunov function (16), we get

$$\begin{array}{cc}\hfill \dot{V}\left(x\right)& =x_4(P_1(x_1-x_{1\mathrm{d}})+f_1+g_{11}{\tau }_1+g_{12}{\tau }_2)\hfill \\ \hfill & +x_5(P_1(x_2-x_{2\mathrm{d}})+f_2+g_{21}{\tau }_1+g_{22}{\tau }_2)\hfill \end{array}$$

(17)

According to Equation (17), in order to ensure the system state convergence, the control law is

$$\left\{\begin{array}{c}{\tau }_1=(P_1(-x_1+x_{1\mathrm{d}})-f_1-D_1{x}_4-g_{12}{\tau }_2)g_{11}^{-1}\hfill \\ {\tau }_2=(P_2(-x_2+x_{2\mathrm{d}})-f_2-D_2{x}_5-g_{21}{\tau }_1)g_{22}^{-1}\hfill \end{array}\right.$$

(18)

$D_1>0$ and $D_2>0$ are parameters that can adjust the convergence rate of the system state. Since $M\left(q\right)$ and $M^{-1}\left(q\right)$ are positive-definite matrices, there are $g_{11}>0$ and $g_{22}>0$, which can avoid the occurrence of singular problems. Therefore, the control law ensures that the following convergence conditions are established:

$$\dot{V}\left(x\right)=-D_1{x}_4^2-D_2{x}_5^2\le 0$$

(19)

The following uses the Lasalle invariance principle [30] to prove that the control rate of Equation (18) can achieve the control objective of the first phase of the system. We first outline the definition of an invariant set.

Lemma 2. 

Let $x\left(t\right)$ be the solution of Equation (14). If there exists a set Ω such that, when $x\left(0\right)\in \mathrm{\Omega }$, $x\left(t\right)\in \mathrm{\Omega }$, and $\forall t\ge 0$, then Ω is called the invariant set of Equation (14).

Since $\dot{V}\left(x\right)\le 0$, then $\dot{V}\left(x\right)$ is bounded and the invariant set is characterized as

$$\mathrm{\Omega }=\{x\in {\mathbb{R}}^{6\times 1}\left|V\left(x\right)\le \lambda \right\}$$

(20)

$\lambda$ is the given small positive number.

Define the set as

$$\mathrm{\Phi }=\left\{\begin{array}{c}x\left(t\right)\in \mathrm{\Omega }|\dot{V}\left(x\right)=0\end{array}\right\}$$

(21)

let W be the greatest invariant set in $\mathrm{\Phi }$, and then $x\left(0\right)\in \mathrm{\Phi }$, $x\left(t\right)\in W$, $t\to \infty$.

When $\dot{V}\left(x\right)\equiv 0$, then $x_4\equiv 0$ and $x_5\equiv 0$, and it follows that ${\dot{x}}_4\equiv 0$ and ${\dot{x}}_5\equiv 0$. Then, it follows from Equation (14) that

$$\left\{\begin{array}{c}{f}_1=-g_{11}{\tau }_1-g_{12}{\tau }_2\hfill \\ f_2=-g_{21}{\tau }_1-g_{22}{\tau }_2\hfill \end{array}\right.$$

(22)

Substituting Equation (22) into Equation (17) yields $x_1=x_{1\mathrm{d}}$, $x_2=x_{2\mathrm{d}}$. Therefore, in this control phase, the maximum invariant set of the first and second actuated links of the system is

$$W=\{x\in {\mathbb{R}}^{6\times 1}\left|x_1=x_{1\mathrm{d}},x_2=x_{2\mathrm{d}},x_4=x_5=0\right\}$$

(23)

According to Lasalle’s principle [30], when Equation (21) reaches the goal of this stage, the states of the first and second actuated links are $q_1=q_{1\mathrm{d}}$, ${\dot{q}}_1=0$, $q_2=q_{2\mathrm{d}}$, and ${\dot{q}}_2=0$, respectively.

Therefore, the position adjustment control of the system is complete when the following conditions are met:

$$\left\{\begin{array}{c}\left|x_1-x_{1\mathrm{d}}\right|\le e_1\hfill \\ \left|x_2-x_{2\mathrm{d}}\right|\le e_1\hfill \\ \left|x_4\right|\le e_2\hfill \\ \left|x_5\right|\le e_2\hfill \end{array}\right.$$

(24)

and $e_1$ and $e_2$ are given small positive numbers.

According to Equation (9), the system underactuated equation can be obtained as

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

(25)

and, according to the analysis of Equations (23) and (25), ${\dot{x}}_6\equiv 0$ can be obtained when the first and second actuated links are controlled to the target angle, and then $x_6\equiv \epsilon$, and $\epsilon$ is a constant, indicating that the underactuated link rotates at a constant speed.

According to the above analysis, when the front two actuated links are controlled to the target angle, the underactuated links still rotate at a constant speed, and the first phase of control is completed.

4. The Second-Phase Control

4.1. Model Reduction

When the planar PPR system drives all links to maintain their respective target states, the underactuated links rotate freely. After the first-phase control, the planar PPR system will be degraded to a planar virtual PR system, and the second and third linkages of the planar PPR system are the actuated links and underactuated links of the planar virtual PR system, respectively. Meanwhile, the first actuated link will stop moving, and the control torque is applied alone to the second actuated link, thus driving the third underactuated link to reach the target state. The order-reduction model is shown in Figure 3.

Supplementary: After the end of the first phase, the first joint link has already stabilized to the target state and stopped moving, and the open-loop iterative control strategy acts solely on the second link. At this time, it enters the second phase, the PPR model is reduced to PR model, and the first link stops participating in the second phase.

The planar PR underactuated robot model after order reduction is as follows:

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

(26)

where $M{\left(q\right)}^{*}$ and $H^{*}(q,\dot{q})$ are as follows, and the control torque is ${\tau }_2^{*}$.

$$M{\left(q\right)}^{*}=\left[\begin{array}{cc}{m}_2+m_3& -m_3{l}_{3}sin{q}_3\\ -m_3{l}_{3}sin{q}_3& m_3{l}_3^3+J_3\end{array}\right]$$

(27)

$$H^{*}(q,\dot{q})=\left[\begin{array}{c}-m_3{l}_3{\dot{q}}_3^{2}cos{q}_3\\ 0\end{array}\right]$$

(28)

According to Equations (26)–(28), the underactuated part of the reduced-order dynamics equation of planar PR robot can be obtained as follows:

$$\left(\begin{array}{c}-m_3{l}_{3}sin{q}_3\end{array}\right){\ddot{q}}_2+\left(\begin{array}{c}{m}_3{l}_3^2+J_3\end{array}\right){\ddot{q}}_3=0$$

(29)

$${\ddot{q}}_3={\ddot{q}}_2{\displaystyle \frac{m_3{l}_3}{m_3{l}_3^2+J_3}}sin{q}_3$$

(30)

By combining Equation (30), a new state-space expression can be obtained as follows:

$$\dot{x}=h\left(x\right)+k\left(x\right)A\left(t\right)$$

(31)

where $A\left(t\right)$ represents the newly developed control input, and

$$\left\{\begin{array}{cc}h\left(x\right)=\hfill & {\left[\begin{array}{cccc}{\dot{q}}_2& {\dot{q}}_3& 0& 0\end{array}\right]}^{\mathrm{T}}\hfill \\ k\left(x\right)=\hfill & {\left[\begin{array}{cccc}0& 0& 1& Csin{q}_3\end{array}\right]}^{\mathrm{T}}\hfill \end{array}\right.$$

(32)

$$C={\displaystyle \frac{m_3{l}_3}{m_3{l}_3^2+J_3}}$$

(33)

Equation (31) has non-holonomic integral property, which makes it difficult to obtain the constraint relation directly through integration. Therefore, we construct the accessibility matrix in the initial state $x_0={[q_2^0,q_3^0,{\dot{q}}_2^0,{\dot{q}}_3^0]}^{\mathrm{T}}$ based on Equation (31) and calculate the local coordinates at $x_0$ and perform the privileged coordinate transformation [31]. The obtained privileged coordinate $z={[z_1,z_2,z_3,z_4]}^{\mathrm{T}}$ preserves the reachability of the original system. By processing the privileged coordinates with Taylor expansion method, a nilpotent approximation model is obtained as follows:

$$\dot{z}=\left[\begin{array}{c}1\\ 0\\ -z_2\\ 0\end{array}\right]-A\left(t\right)\left[\begin{array}{c}0\\ -1\\ 0\\ {\left({\dot{q}}_3^{\mathrm{I}}{z}_1\right)}^2/4Ccos{q}_3^{\mathrm{I}}+{\displaystyle \frac{1}{2}}{z}_3\end{array}\right]$$

(34)

at the beginning of each cycle, the angle and angular velocity of the underactuated link are $q_3^{\mathrm{I}}$ and ${\dot{q}}_3^{\mathrm{I}}$, respectively. Equation (34) has exactly the same characteristics as Equation (31). The planar PPR underactuated robot can be shrunk to a stable target state by designing and using the auxiliary control input $A\left(t\right)$.

4.2. Design of Controller

In this stage, the underactuated link is driven by the second prismatic actuated link based on Equation (34). We set $t=0$ as the initial time, and the actuated link is able to return to the desired state at zero speed. The auxiliary control input $A\left(t\right)$ should satisfy the following conditions:

$$\left\{\begin{array}{c}{\int }_0^{\mathrm{T}}A\left(t\right)\mathrm{d}t=0\hfill \\ {\int }_0^{\mathrm{T}}{\int }_0^{t}A\left(b\right)\mathrm{d}b\mathrm{d}t=0\hfill \end{array}\right.$$

(35)

According to Equations (34) and (35), $z_1\left(T\right)=T$, $z_2\left(T\right)=z_3\left(T\right)=0$, and the following error relation can be obtained:

$$\left\{\begin{array}{c}\mathrm{\Delta }{q}_3=q_3^{\mathrm{II}}-q_3^{\mathrm{I}}={\dot{q}}_3^{\mathrm{I}}{z}_1\left(T\right)={\dot{q}}_3^{\mathrm{I}}T\hfill \\ \mathrm{\Delta }{\dot{q}}_3={\dot{q}}_3^{\mathrm{II}}-{\dot{q}}_3^{\mathrm{I}}=C^{2}sin2q_3^{\mathrm{I}}\hfill \end{array}\right.$$

(36)

$C=m_3{l}_3/(m_3{l}_3^2+J_3)$, $q_3^{\mathrm{II}}$ and ${\dot{q}}_3^{\mathrm{II}}$ indicate the angle and angular velocity of the underactuated link at the end of each cycle, respectively. $q_3^{\mathrm{d}}$ represents the desired angle of the underactuated link.

By calculating Equations (34) and (36), it can be obtained as

$$\mathrm{\Delta }{\dot{q}}_3=-C^{2}sin{q}_3^{\mathrm{I}}cos{q}_3^{\mathrm{I}}{\int }_0^{\mathrm{T}}{z}_2^2\left(t\right)\mathrm{d}t+C{\left({\dot{q}}_3^{\mathrm{I}}\right)}^{2}sin{q}_3^{\mathrm{I}}{\int }_0^{\mathrm{T}}{z}_3\left(t\right)\mathrm{d}t$$

(37)

In light of the preceding analysis, the following auxiliary control inputs are designed:

$$A\left(t\right)=\left\{\begin{array}{cc}-Ecos(4\pi t/T),\hfill & t\in \left[0,{\displaystyle \frac{T}{2}}\right)\hfill \\ Ecos4\pi (t-{\displaystyle \frac{T}{2}})/T,\hfill & t\in \left[{\displaystyle \frac{T}{2}},T\right]\hfill \end{array}\right.$$

(38)

where E represents the highest absolute value observed in each cycle, and the cycle control input is illustrated in Figure 4.

According to Equations (35)–(38),

$$\mathrm{\Delta }{\dot{q}}_3={\displaystyle \frac{E^2{T}^3{C}^{2}sin2q_3^{\mathrm{I}}}{64{\pi }^2}}$$

(39)

Establish the subsequent relationships to guarantee that the underactuated link undergoes iterative contraction during each cycle:

$$\left\{\begin{array}{c}\left|q_3^{\mathrm{d}}-q_3^{\mathrm{II}}\right|\le {\eta }_1\left|q_3^{\mathrm{d}}-q_3^{\mathrm{I}}\right|\hfill \\ \left|{\dot{q}}_3^{\mathrm{II}}\right|={\eta }_2\left|{\dot{q}}_3^{\mathrm{I}}\right|\hfill \end{array}\right.$$

(40)

the convergence coefficients ${\eta }_1$ and ${\eta }_2$ are in the range $\left[0,1\right)$.

By combining Equations (36), (39) and (40), we can obtain

$$\left\{\begin{array}{c}T=(1-{\eta }_1)(q_3^{\mathrm{d}}-q_3^{\mathrm{I}})/{\dot{q}}_3^{\mathrm{I}}\hfill \\ E={\displaystyle \frac{8\pi }{CT}}\sqrt{{\dot{q}}_3^{\mathrm{I}}(1-{\eta }_2)/T\sin 2q_3^{\mathrm{I}}}\hfill \end{array}\right.$$

(41)

Equation (41) satisfies $0<\mathrm{T}$, and $\mathrm{E}<\infty$.

The simultaneous Equations (26), (29), (30) and (33) can be obtained

$${\tau }_2^{*}=(m_2+m_3-C{m}_3{l}_3{sin}^2{q}_3)A\left(t\right)-m_3{l}_3{\dot{q}}_3^{2}cos{q}_3$$

(42)

The actuated link achieves the position control goal in the first phase, and the underactuated link is in a free rotating state. At this time, the second-phase controller should be used to iteratively operate the actuated link by using the control torque, as in Equation (42).

When the designed supplementary control input $A\left(t\right)$ is applied to the aforementioned controller, i iteration cycles T, $t_{\mathrm{d}}=t_0+i(i=1,2,\dots ,n)$, must be experienced during the process from iteration start time $t_0$ to iteration end time $t_d$. At the end of each cycle, the angle contraction of the third underactuated link is closer to the target state, and the states of the actuated link and underactuated link at the final of each iteration cycle are regarded as the initial state of the next cycle, and the iteration continues until the virtual planar PR underactuated robot stabilizes to the target state after n iteration cycles.

5. Simulation

The planar PPR underactuated robot model established by using MATLAB/Simulink R2023b (23.2) is shown in Figure 5 below. When describing the planar PPR underactuated robot system as a series of complex mathematical expressions, the S-function is used to input the expressions in text form, and then the designed control method is written in the S-function and the program is debugged.

The functions of each module are as follows. First, create three new S-function scripts, namely SF-plant, SF-controller, and SF-judge. SF-plant represents the controlled physical system, which receives control signals as input and produces corresponding outputs according to its own dynamics. The function of the SF-controller module is to calculate the appropriate control signal based on the expected output and actual output of the system so that the output of the system tracks the expected output as much as possible. The SF-judge contains judgment conditions, which serve as the basis for the entire system to stabilize to the target state and stop. In the block diagram, tau is used as the control input, u as the control output, Clock as the simulation step-size time, Scope-input and Scope-output, respectively, monitor the waveforms of the control input and control output separately, and Scopes-x(1-6), respectively, monitor the waveform changes of each state parameter.

Set the PD controller parameters of the first phase as $P_1=P_2=0.6$, $D_1=D_2=1.7$. Set the parameters and convergence coefficient of the second-phase controller to ${\eta }_1={\eta }_2=0.7$. The specific parameters of the planar PPR underactuated robot model are shown in

Table 2. Parameter values of the planar PPR underactuated robot model.

Link r	${\mathit{m}}_{\mathit{r}}/\mathbf{kg}$	${\mathit{L}}_{\mathit{r}}/\mathbf{m}$	${\mathit{l}}_{\mathit{r}}/\mathbf{m}$	${\mathit{J}}_{\mathit{r}}/\mathbf{kg}·{\mathbf{m}}^2$
r = 1	1	1	0.5	0
r = 2	1	1	0.5	0
r = 3	0.5	0.5	0.25	1

.

5.1. Case 1

When the initial velocity of all the links of the planar PPR underactuated robot is 0, the initial state is set to the target state as follows:

$$\left\{\begin{array}{c}{x}_0={\left[\begin{array}{cccccc}0.7& 0.6& 0.8& 0& 0& 0\end{array}\right]}^{\mathrm{T}}\hfill \\ x_{\mathrm{d}}={\left[\begin{array}{cccccc}0.5& 0.35& 2.4& 0& 0& 0\end{array}\right]}^{\mathrm{T}}\hfill \end{array}\right.$$

(43)

Figure 6a demonstrates that the first prism-actuated link stabilizes to the desired state and stops moving, while the third underactuated rotating link stabilizes to the desired state with the second prism-actuated link. The velocity of the planar PPR underactuated robot link starts from 0, and, when the motion velocity of the first link converges to 0, the first prism-actuated link stabilizes and stops moving. At the moment of 20 s, under the open-loop iteration strategy, the second link oscillates and drives the third underactuated rotating link to eventually converge and stabilize to the target state. The control torque applied during the whole motion process remains at $[-1.0,1.0]\mathrm{N}·\mathrm{m}$ and eventually converges to 0.

Figure 6b shows that the control torque of the PPR system is subjected to the external interference of $0.01\mathrm{N}·\mathrm{m}$ from $10\mathrm{s}$ to $12\mathrm{s}$, and the initial velocity of the link is zero. The stability control can also be realized in the end. The first prism-actuated link stabilizes to the target state and stops moving. Even with external interference, the third underactuated link can still stably converge under the drive of the second prism-actuated link.

When the initial velocity of the third link of the planar PPR underactuated robot is $0.01\mathrm{rad}/\mathrm{s}$, the initial state is set, and the target state is as follows:

$$\left\{\begin{array}{c}{x}_0={\left[\begin{array}{cccccc}0.7& 0.6& 0.8& 0& 0& 0.01\end{array}\right]}^{\mathrm{T}}\hfill \\ x_{\mathrm{d}}={\left[\begin{array}{cccccc}0.5& 0.35& 2.4& 0& 0& 0\end{array}\right]}^{\mathrm{T}}\hfill \end{array}\right.$$

(44)

Figure 7a shows that the PPR system can also realize the stable control of three links under the proposed control strategy when the initial velocity of the third underactuated link is $0.01\mathrm{rad}/\mathrm{s}$ without interference.

Figure 7b shows that the control torque of the PPR system is subjected to external interference of $0.01\mathrm{N}·\mathrm{m}$ for $10\mathrm{s}$ to $12\mathrm{s}$, and the initial velocity of the third underactuated link is $0.01\mathrm{rad}/\mathrm{s}$. The whole system still achieves successful stability.

Table 3. Simulation comparison of case 1.

Simulation i	Initial Velocity	Interference	Stable Time
1	0	0	$t<35\mathrm{s}$
2	0	$0.01\mathrm{N}·\mathrm{m}$	$t<35\mathrm{s}$
3	${\dot{q}}_3=0.01\mathrm{rad}/\mathrm{s}$	0	$t<35\mathrm{s}$
4	${\dot{q}}_3=0.01\mathrm{rad}/\mathrm{s}$	$0.01\mathrm{N}·\mathrm{m}$	$t<35\mathrm{s}$

shows the comparative analysis of the four simulations in case 1.

5.2. Case 2

In order to verify the universality of the proposed control strategy, another set of parameters are established for simulation. When the initial velocity of all links of the planar PPR underactuated robot is 0, the initial state and target state are set as follows:

$$\left\{\begin{array}{c}{x}_0={\left[\begin{array}{cccccc}0.7& 0.6& 0.8& 0& 0& 0\end{array}\right]}^{\mathrm{T}}\hfill \\ x_{\mathrm{d}}={\left[\begin{array}{cccccc}2.1& 1.9& 1.8& 0& 0& 0\end{array}\right]}^{\mathrm{T}}\hfill \end{array}\right.$$

(45)

Figure 8a shows that the first prism-actuated link achieves stability in the first phase. At the moment of 23 s, under the open-loop iteration strategy, the second link oscillates and drives the third underactuated rotating link to finally converge and stabilize to the target state. The torque of the whole motion process is maintained at $[-5,5]\mathrm{N}·\mathrm{m}$ and eventually converges to 0.

Figure 8b shows that the control torque of the planar PPR underactuated robot is subjected to the external interference of $0.05\mathrm{N}·\mathrm{m}$ from $13\mathrm{s}$ to $15\mathrm{s}$, and the starting velocity of the link is 0. Even in the presence of external interference, the whole system can still converge stably.

When the starting velocity of the first link of the planar PPR underactuated robot is $0.02\mathrm{m}/\mathrm{s}$, the starting and desired states at this point are as follows:

$$\left\{\begin{array}{c}{x}_0={\left[\begin{array}{cccccc}0.7& 0.6& 0.8& 0.02& 0& 0\end{array}\right]}^{\mathrm{T}}\hfill \\ x_{\mathrm{d}}={\left[\begin{array}{cccccc}2.1& 1.9& 1.8& 0& 0& 0\end{array}\right]}^{\mathrm{T}}\hfill \end{array}\right.$$

(46)

Figure 9a shows that the PPR system can also realize the stable control of three links under the control strategy proposed in this paper under the condition that the PPR system is not disturbed and the initial velocity of the first underactuated link is $0.02\mathrm{m}/\mathrm{s}$.

Figure 9b shows that the control torque of the PPR system is subjected to external interference of $0.05\mathrm{N}·\mathrm{m}$ from $13\mathrm{s}$ to $15\mathrm{s}$, and the starting velocity of the third underactuated link is $0.02\mathrm{m}/\mathrm{s}$. The whole system still achieves successful stability.

Table 4. Simulation comparison of case 2.

Simulation i	Initial Velocity	Interference	Stable Time
1	0	0	$t<45\mathrm{s}$
2	0	$0.05\mathrm{N}·\mathrm{m}$	$t<45\mathrm{s}$
3	${\dot{q}}_1=0.02\mathrm{m}/\mathrm{s}$	0	$t<45\mathrm{s}$
4	${\dot{q}}_1=0.02\mathrm{m}/\mathrm{s}$	$0.05\mathrm{N}·\mathrm{m}$	$t<45\mathrm{s}$

shows the comparative analysis of the four simulations in case 2.

5.3. Comparison and Analysis

In case 1 and case 2, different comparative simulations are carried out, respectively, around the two points of whether there is interference and whether there is initial velocity. Case 1 and case 2 involve different target states, the external interference selected $0.01\mathrm{N}·\mathrm{m}$ and $0.05\mathrm{N}·\mathrm{m}$, the initial velocity of the link selected $0.01\mathrm{rad}/\mathrm{s}$ and $0.02\mathrm{m}/\mathrm{s}$, and the final stability time was controlled within 40 s and 50 s, respectively. The planar PPR underactuated robot system can always converge successfully and stabilize to the target state under the proposed switching control.

According to the latest literature [32] on the same type of planar PR underactuated robot, it is evident from the comparative analysis in

Table 5. Comparison with the PR underactuated robot of the same type.

	PR Underactuated Robot	PPR Underactuated Robot
Phase 1 control	Sliding mode control	Improved PD control
Phase 2 control	Iterative shrinkage	Iterative shrinkage
Stable time	40–50$\mathrm{s}$	35–45$\mathrm{s}$
Initial velocity	—	Quick response
Torque interference	—	Effective resistance

that this improved PD controller has certain advantages in stabilization time, with the average stabilization time shortened by about 10 s. Moreover, in the presence of initial velocity and external torque interference in the link, this control method can still respond quickly and suppress the interference error, and it has certain advantages in terms of performance indicators.

6. Conclusions

In this paper, a two-phase switching-control strategy is researched for the planar PPR underactuated robot, and the efficacy of the suggested control method is confirmed by the simulation results. Through the simulation test of controlling individual variables, it is demonstrated that the planar PPR underactuated robot can achieve stable control and has strong robustness under the conditions of interference and non-zero initial link velocity, respectively. This control strategy is innovative and extends the stability control strategy of the entire planar PPR underactuated robot. In the future, the research of planar PPR underactuated robots will be further optimized in terms of anti-interference and energy consumption, and reducing the stability time will be considered. The strategy proposed in this paper provides a theoretical basis for subsequent research.