Novel Approximations to the (Un)forced Pendulum–Cart System: Ansatz and KBM Methods

Abstract

In the present investigation, some novel analytical approximations to both unforced and forced pendulum–cart system oscillators are obtained. In our investigation, two accurate and effective approaches, namely, the ansatz method with equilibrium point and the Krylov–Bogoliubov–Mitropolsky (KBM) method, are implemented for analyzing pendulum–cart problems.The obtained results are compared with the Runge–Kutta (RK4) numerical approximation. The obtained approximations using both ansatz and KBM methods show good coincidence with RK4 numerical approximation. In addition, the global maximum error is estimated as compared to RK4 numerical approximation.

1. Introduction

The study of the mechanism of nonlinear oscillating systems is one of the most important and recent studies at the present time, which concerns many researchers, especially in the engineering field and in the field of automated systems and robotics [1,2,3,4,5,6,7,8,9,10,11]. Nonlinear control theory (NLCT) has received a lot of attention due to its technical importance and impact on numerous sectors of applications. Robotics, for example, is a popular application of NLCT. The inverted pendulum is significant and effective for modeling and designing the robotics control system. A cart–inverted pendulum system is one of the most effective models used in robotic systems. The pendulum–cart system (PCS) is a nonlinear, underactuated system with unstable zero dynamics that must be regulated to maintain its unstable equilibrium [12,13,14,15,16]. The most popular approach for swinging up an inverted pendulum is energy control, which involves regulating the system’s energy rather than its velocity and position. In the literature and based on nonlinear analysis, intelligent control techniques have been proposed for controlling the nonlinear system [17,18,19,20,21].

There are many nonlinear evolution equations that govern pendulum oscillations, for instance, Duffing-type oscillators, Helmholtz-type oscillators, and many other oscillators related to the Duffing and Helmholtz oscillators [1,2,3,4,5,6]. On the other side, a benchmark example for underactuated mechanical systems is the PCS, which is highly nonlinear in nature and chosen as a typical experimental device in the control field. The PCS setup is suitable for testing various types of control theories and building robotics systems. The Segway is a two-wheeled balancing vehicle used for transport systems that can move forward by shifting the weight forward and can move backward by shifting the weight backward. Motivated by the potential applications of the pendulum–cart in robotic systems and many other applications, in this paper, we give an analytical solution to the PCS as shown in Figure 1 [12]. The differential equations of motion for the PCS are given by [12,13]:

$$\left\{\begin{array}{c}{R}_1=\overline{m}{\ddot{x}}_C+ml(\ddot{\theta }cos\theta -{\dot{\theta }}^{2}sin\theta )-F=0,\\ R_2={\ddot{x}}_{C}lcos\theta +\ddot{\theta }{l}^2+glsin\theta =0,\end{array}\right.$$

(1)

which is subjected to the following initial conditions (ICs):

$$\left\{\begin{array}{c}\theta \left(0\right)={\theta }_0\&\dot{\theta }\left(0\right)=0,\\ x\left(0\right)=x_0\&\dot{x}\left(0\right)={\dot{x}}_0.\end{array}\right.$$

(2)

Here, $\theta \equiv \theta \left(t\right)$, $x\equiv x\left(t\right)$, and for simplicity we use $\overline{m}=(M+m)$, where m and M denote the cart and pole mass, respectively, while l represents the distance from the center of rod axis rotation to the center of rod mass, g indicates the gravitational acceleration, and F expresses the excitation force. In this system, we assume that the excitation force F is constant. Numerous authors have used several approaches for solving and analyzing different types of equations of motion. For instance, the ansatz method, the Krylov–Bogoliubov–Mitropolsky (KBM) method, and He’s homotopy method were devoted to solving and analyzing the quadratically damped Duffing oscillator [22]. In addition, the nonlinear damped oscillations including the Morse oscillator and a nonlinear harmonic oscillator have been analyzed via the KBM method [23]. Moreover, the generalized KBM method was extended for investigating strong nonlinear oscillators with slowly varying parameters [24]. The (un)forced Duffing–Van der Pol oscillators have been investigated via the KBM method, He’s homotopy method, and He’s frequency–amplitude formulation [25]. The ansatz method is considered one of the most effective methods in finding many approximate analytic solutions to many evolution equations including both ordinary and partial differential equations [26,27,28]. For example, the ansatz method was used to find some approximate solutions to the families of Duffing-type equations, Helmholtz–Duffing-type equations, damped nonplanar Korteweg–de-Vries-type equations, Kawahara-type equations, and Muhammad equations, which have great credit for modeling many nonlinear phenomena that arise in many branches of science in general physics and plasma physics in particular [6,26,27,28,29,30]. Accordingly, the main objective of our investigation is to analyze the PCS (1) in order to find some analytical approximations using both the ansatz and KBM methods. For analyzing the PCS (1) via the ansatz method, we should determine the equilibrium point, while the KBM method does not need to find the equilibrium point.

This investigation is organized in the following fashion: The suggested method for analyzing the pendulum–cart system problem is discussed in detail in Section 1. The obtained approximations are discussed and verified by numerical examples in Section 2. The most important results are summarized in Section 3.

2. Mathematical Approaches for Analyzing System (1)

In this section, we try to find some approximations to system (1) using two different approaches. In the first approach, the ansatz method is employed for analyzing system (1), while in the second approach, the KBM method is applied to find an approximation to system (1).

2.1. First Approach: Ansatz Method

It is clear that system (1) is a coupled system in two independent variables $\left(x_C,\theta \right)$ and for solving this system, we first eliminate ${\ddot{x}}_C$ from system (1), which leads to

$$\mathbb{R}\equiv \left[lm{cos}^2\theta -l\overline{m}\right]\ddot{\theta }-\left[cos\theta \left(F+lm{\dot{\theta }}^{2}sin\theta \right)+g\overline{m}sin\theta \right]=0.$$

(3)

Now, it is seen that Equation (3) is an ordinary differential equation (ode) in the variable $\theta$. Using a Chebyshev approximation, the following approximation to the trigonometric functions is utilized for $\left|\theta \right|\le \pi /3$:

$$\begin{array}{cc}\hfill sin\theta & \approx \theta -\frac{{\theta }^3}{6}+\frac{{\theta }^5}{124},\hfill \end{array}$$

(4)

$$\begin{array}{cc}\hfill cos\theta & \approx 1-\frac{{\theta }^2}{2}+\frac{{\theta }^4}{25}.\hfill \end{array}$$

(5)

According to the approximate formulas for $sin\theta$ and $cos\theta$ given in Equations (4) and (5), the approximate form of Equation (3) reads

$$\begin{array}{c}\left[{\left(\frac{{\theta }^4}{25}-\frac{{\theta }^2}{2}+1\right)}^{2}lm-l\overline{m}\right]\ddot{\theta }=\hfill \\ \left(\frac{{\theta }^4}{25}-\frac{{\theta }^2}{2}+1\right)\left[F+\left(\frac{{\theta }^5}{124}-\frac{{\theta }^3}{6}+\theta \right){\dot{\theta }}^{2}lm\right]+g\left(\frac{{\theta }^5}{124}-\frac{{\theta }^3}{6}+\theta \right)\overline{m}.\hfill \end{array}$$

(6)

Now, for deriving an approximation to Equation (6), the following ansatz is presented:

$$\left\{\begin{array}{c}\theta =d+acos\left(wt\right),\\ a={\theta }_0-d,\end{array}\right.$$

(7)

where $\theta \equiv \theta \left(t\right)$ and d indicates the equilibrium point. To determine the value of the equilibrium point d, the following MATHEMATICA command is utilized to solve system (1) at $\dot{\theta }=0$:

$$\mathrm{Solve}\left[\left\{R_1==0,R_2==0\right\},\left\{x_C^{\prime \prime }\left[t\right],{\theta }^{\prime \prime }\left[t\right]\right\}\right]//.{\theta }^{\prime }\left[t\right]\to 0$$

(8)

which leads to

$$\begin{array}{cc}\hfill {\theta }^{\prime \prime }\left[t\right]& =-\frac{Fcos\left(\theta \right)+g\overline{m}sin\left(\theta \right)}{l\left(\overline{m}-mcos{\left(\theta \right)}^2\right)},\hfill \end{array}$$

(9)

$$\begin{array}{cc}\hfill x_C^{\prime \prime }\left[t\right]& =\frac{F+gmcos\left(\theta \right)sin\left(\theta \right)}{\overline{m}-mcos{\left(\theta \right)}^2}.\hfill \end{array}$$

(10)

Note that $\left(x_C^{\prime \prime }\left[t\right],{\theta }^{\prime \prime }\left[t\right]\right)\equiv \left({\ddot{x}}_C,\ddot{\theta }\right)$. Now, solving Equation (8) or (9) using the following MATHEMATICA command,

$${\left(\mathrm{Solve}\left[-\frac{Fcos\left[\theta \left[t\right]\right]+g\overline{m}sin\left[\theta \left[t\right]\right]}{l\left(\overline{m}-mcos{\left[\theta \left[t\right]\right]}^2\right)}==0,\theta \left[t\right]\right]//.C\left[j\_\right]:\to 0\right)}^1//\mathrm{FullSimplify}$$

(11)

the equilibrium point for $F\ne 0$ is obtained as

$$d\equiv \theta \left[t\right]={tan}^{-1}\left[\frac{g\overline{m}}{\sqrt{F^2+g^2{\overline{m}}^2}},-\frac{F}{\sqrt{F^2+g^2{\overline{m}}^2}}\right],$$

(12)

which is equivalent to

$$d={tan}^{-1}\left(\frac{g\overline{m}}{F}\right).$$

(13)

Note that for $F=0$, we have $d=0$.

Inserting the ansatz (7) into Equation (6) yields

$$\mathbb{R}=\frac{-a}{119040000}Scos\left(wt\right)+\cdots ,$$

(14)

with

$$S=\left(\begin{array}{c}1575a^{10}lm{w}^2+75600a^8{d}^{2}lm{w}^2+24094a^{8}lm{w}^2+\\ 378000a^6{d}^{4}lm{w}^2+826980a^6{d}^{2}lm{w}^2-\\ 1382000a^{6}lm{w}^2+403200a^4{d}^{6}lm{w}^2+\\ 2760800a^4{d}^{4}lm{w}^2-25088000a^4{d}^{2}lm{w}^2+\\ 600000a^{4}gm+600000a^{4}gM+14632000a^{4}lm{w}^2+\\ 86400a^2{d}^{8}lm{w}^2+1770944a^2{d}^{6}lm{w}^2-\\ 34016000a^2{d}^{4}lm{w}^2+7200000a^2{d}^{2}gm+\\ 7200000a^2{d}^{2}gM+117254400a^2{d}^{2}lm{w}^2+\\ 14284800a^{2}d{F}_x-14880000a^{2}gm-14880000a^{2}gM-\\ 59520000a^{2}lm{w}^2+190464d^{8}lm{w}^2-4761600d^{6}lm{w}^2+\\ 4800000d^{4}gm+4800000d^{4}gM+39283200d^{4}lm{w}^2+\\ 19046400d^3{F}_x-59520000d^{2}gm-59520000d^{2}gM-\\ 119040000d^{2}lm{w}^2-119040000d{F}_x+119040000gm+\\ 119040000gM-119040000lM{w}^2\end{array}\right).$$

Note that for vanishing S, we obtain an algebraic equation which can be utilized in determining the value of w as follows:

$$w^2=\frac{S_1}{S_2},$$

(15)

with

$$S_1=4800\left(\begin{array}{c}125a^{4}gm+125a^{4}gM+1500a^2{d}^{2}gm+1500a^2{d}^{2}gM\\ +2976a^{2}dF-3100a^{2}gm-3100a^{2}gM+1000d^{4}gm\\ +1000d^{4}gM+3968d^{3}F-12400d^{2}gm-12400d^{2}gM\\ -24800dF+24800gm+24800gM\end{array}\right),$$

and

$$S_2=l\left(\begin{array}{c}-1575a^{10}m-75600a^8{d}^{2}m-24094a^{8}m-378000a^6{d}^{4}m\\ -826980a^6{d}^{2}m+1382000a^{6}m-403200a^4{d}^{6}m\\ -2760800a^4{d}^{4}m+25088000a^4{d}^{2}m-14632000a^{4}m\\ -86400a^2{d}^{8}m-1770944a^2{d}^{6}m+34016000a^2{d}^{4}m\\ -117254400a^2{d}^{2}m+59520000a^{2}m-190464d^{8}m\\ +4761600d^{6}m-39283200d^{4}m+119040000d^{2}m+119040000M\end{array}\right).$$

On the other side, for $F=0$, the value of w reads

$$w^2=-\frac{120000\left(5a^4-124a^2+992\right)g\overline{m}}{a^2\left(\begin{array}{c}1575a^8+24094a^6-1382000a^4\\ +14632000a^2-59520000\end{array}\right)lm-119040000lM}.$$

(16)

Furthermore, the period of the oscillator is given by

$$T=\frac{2\pi }{w}.$$

(17)

It remains to find the expression for $x_C$. To this end, the first equation in system (1) is utilized as

$${\ddot{x}}_C=\frac{1}{\overline{m}}\left[F+lm{\dot{\theta }}^{2}sin\theta -lm\ddot{\theta }cos\theta \right].$$

(18)

Using the approximate formulas for $sin\theta$ and $cos\theta$ given in Equations (4) and (5) in Equation (18), we obtain

$${\ddot{x}}_C=\frac{lm}{\overline{m}}\left[\frac{F}{lm}+{\dot{\theta }}^2\left(\frac{{\theta }^5}{124}-\frac{{\theta }^3}{6}+\theta \right)-\ddot{\theta }\left(\frac{{\theta }^4}{25}-\frac{{\theta }^2}{2}+\theta \right)\right].$$

(19)

The solution of Equation (19) under the ICs $x\left(0\right)=x_0$ and $x^{\prime }\left(0\right)={\dot{x}}_0$ reads

$$x_C={\dot{x}}_{0}t+\frac{lm}{297600\overline{m}}\left(S_3{w}^2+148800\frac{F}{lm}\right)t^2+\sum _{j=1}^7{W}_{j}cos\left(jwt\right),$$

(20)

with

$$\begin{array}{c}{W}_1=\frac{alm\left(375a^6+6000a^4{d}^2+2480a^4+6000a^2{d}^4+32736a^2{d}^2-74400a^2+23808d^4-297600d^2+595200\right)}{595200\overline{m}},\hfill \\ W_2=-\frac{a^{2}dlm\left(375a^4+23808a^2-1200d^4+48608d^2-297600\right)}{1190400(m+M)},\hfill \\ W_3=\frac{a^{3}lm\left(75a^4+3000a^2{d}^2-13640a^2+6000d^4-110112d^2+223200\right)}{5356800\overline{m}},\hfill \\ W_4=\frac{a^{4}dlm\left(125a^2+500d^2-4092\right)}{793600\overline{m}},\hfill \\ W_5=\frac{a^{5}lm\left(225a^2+3000d^2-7688\right)}{14880000\overline{m}},\hfill \\ W_6=\frac{5a^{6}dlm}{142848\overline{m}},\hfill \\ W_7=\frac{a^{7}lm}{388864\overline{m}},\hfill \\ S_3=\left(375a^{6}d+1500a^4{d}^3-372a^{4}d+600a^2{d}^5-496a^2{d}^3\right).\hfill \end{array}$$

(21)

For $F=0$, solution (20) reduces to

$$\begin{array}{c}{x}_C\left(t\right)=\frac{30675a^{7}lm+370636a^{5}lm-8544375a^{3}lm+51266250alm+25633125\left(F_x{t}^2+2(m+M)\left(t{\dot{x}}_0+x_0\right)\right)}{51266250\overline{m}}\hfill \\ -\frac{\left(75a^6+496a^4-14880a^2+119040\right)alm}{119040\overline{m}}cos\left(wt\right)+\frac{\left(15a^4-2728a^2+44640\right)a^{3}lm}{1071360\overline{m}}cos\left(3wt\right)\hfill \\ +\frac{\left(225a^2-7688\right)a^{5}lm}{14880000\overline{m}}cos\left(5wt\right)+\frac{a^{7}lm}{388864\overline{m}}cos\left(7wt\right).\hfill \end{array}$$

(22)

Now, for examining the accuracy of the obtained approximations, the following data in the absence (existence) of the excitation force are considered: $M=2$ kg, $m=1$ kg, $l=1$ m, $g=9.8$ m/s${}^2$, ${\theta }_0={20}^{\circ }$, ${\dot{\theta }}_0=0$, $x_0=1$ m, and ${\dot{x}}_0=0$, while the force is given by

$$\left\{\begin{array}{c}F=0,\\ F=0.01\mathrm{N},\end{array}\right.$$

(23)

Based on the mentioned data, the i.v.p. for the unforced case to be solved reads

$$\left\{\begin{array}{c}\ddot{\theta }cos\theta -{\dot{\theta }}^2-sin\theta +3\ddot{x}=0.01,\\ \ddot{\theta }+9.8sin\theta +\ddot{x}cos\theta =0,\end{array}\right.$$

(24)

and

$$\left\{\begin{array}{c}\theta \left(0\right)=20{}^{\circ },\dot{\theta }\left(0\right)=0,\\ x\left(0\right)=1,\dot{x}\left(0\right)=0.\end{array}\right.$$

(25)

The approximate solution based on the relations (7) and (20) to this problem reads

$$\theta =0.349066cos\left(3.74987t\right),$$

(26)

and

$$\begin{array}{c}\hfill x=1.114-0.11459cos\left(3.74987t\right)+0.000586335cos\left(11.2496t\right)\\ \hfill -8.9\times {10}^{-7}cos\left(18.7494t\right)+5.41\times {10}^{-10}cos\left(26.2491t\right).\end{array}$$

(27)

Using the same mentioned data for the forced case ($F\ne 0$), we have

$$\left\{\begin{array}{c}\ddot{\theta }cos\left(\theta \right)-{\dot{\theta }}^2-sin\left(\theta \right)+3\ddot{x}=0.01,\\ \ddot{\theta }+9.8sin\left(\theta \right)+\ddot{x}cos\left(\theta \right)=0,\end{array}\right.$$

(28)

with

$$\left\{\begin{array}{c}\theta \left(0\right)=20{}^{\circ },\dot{\theta }\left(0\right)=0,\\ x\left(0\right)=1,\dot{x}\left(0\right)=0.\end{array}\right.$$

(29)

The numerical values to the approximate solutions of this problem read

$$\theta \left(t\right)=0.349406cos\left(3.74972t\right)-0.000340136,$$

(30)

and

$$\begin{array}{cc}\hfill x_C\left(t\right)& =1.11411+0.00166669t^2-0.114699cos\left(3.74972t\right)\hfill \\ \hfill & -3.4265836921198773\times {10}^{-6}cos\left(7.49943t\right)\hfill \\ \hfill & +0.000588042cos\left(11.2491t\right)+8.68089\times {10}^{-9}cos\left(14.9989t\right)\hfill \\ \hfill & -8.936889\times {10}^{-7}cos\left(18.7486t\right)+5.44996\times {10}^{-10}cos\left(26.248t\right).\hfill \end{array}$$

(31)

The global maximum distance error $L_g$ in the whole domain is calculated for the unforced and forced cases as shown in

Table 1. The global maximum distance error $L_g$ for the ansatz method.

F	${\left.{\mathit{L}}_{\mathit{g}}\right|}_{\left(\mathit{\theta }\right)}$	${\left.{\mathit{L}}_{\mathit{g}}\right|}_{\left({\mathit{x}}_{\mathit{C}}\right)}$
0	$0.00232$	$0.00076$
$0.01$	$0.002314$	$0.0007565$

.

Both solutions (26) and (27) are presented in Figure 2. In Figure 2a, the angle profile $\theta$ according to solution (26) is considered, while the profile of position x is introduced in Figure 2b. Moreover, the forced case is introduced in Figure 3, in which the angle profile $\theta$ according to solution (30) is considered, while the profile of position x given in (31) is displayed. One can see from Figure 2 and Figure 3 as well as the global maximum error given in Table 1 that there is complete congruence between the analytical and numerical approximations, which confirms the high accuracy and effectiveness of the obtained approximations.

2.2. Second Approach: KBM Method

From Equation (3), we obtain

$$\ddot{\theta }\equiv {\theta }^{\prime \prime }\left(t\right)=\frac{Fcos\left(\theta \right)+gmsin\left(\theta \right)+gMsin\left(\theta \right)+lmsin\left(\theta \right)cos\left(\theta \right){\dot{\theta }}^2}{l\left(m{cos}^2\left(\theta \right)-\overline{m}\right)}.$$

(32)

Note here that F can take any value for (in)dependent time, i.e., F may be a constant value or $F=F\left(t\right)$. This property is not fulfilled in the above approach, in which the excitation force must be constant because equilibrium is considered only for autonomous systems.

Using series for the right-hand side of Equation (32) yields

$$\frac{Fcos\left(\theta \right)+gmsin\left(\theta \right)+gMsin\left(\theta \right)+lmsin\left(\theta \right)cos\left(\theta \right){\dot{\theta }}^2}{l\left(m{cos}^2\left(\theta \right)-\overline{m}\right)}=\sum _{j=0}^5{Z}_j{\theta }^j,$$

(33)

with

$$\begin{array}{cc}\hfill Z_0& =\frac{F}{l\left(m-\overline{m}\right)},\hfill \\ \hfill Z_1& =\frac{g\overline{m}}{l\left(m-\overline{m}\right)}+\frac{m{\dot{\theta }}^2}{\left(m-\overline{m}\right)},\hfill \\ \hfill Z_2& =\frac{F\left(m+\overline{m}\right)}{2l{\left(m-\overline{m}\right)}^2},\hfill \\ \hfill Z_3& =\frac{\left(2l{m}^2{\dot{\theta }}^2+5gm\overline{m}+4lm{\dot{\theta }}^2\overline{m}+g{\overline{m}}^2\right)}{6l{\left(m-\overline{m}\right)}^2},\hfill \\ \hfill Z_4& =\frac{F\left(5m^2+18m\overline{m}+{\overline{m}}^2\right)}{24l{\left(m-\overline{m}\right)}^3},\hfill \\ \hfill Z_5& =\frac{\left(16l{m}^3{\dot{\theta }}^2+61g{m}^2\overline{m}+88l{m}^2{\dot{\theta }}^2\overline{m}+58gm{\overline{m}}^2+16lm{\dot{\theta }}^2{\overline{m}}^2+g{\overline{m}}^3\right)}{120l{\left(m-\overline{m}\right)}^3}.\hfill \end{array}$$

Then, Equation (32) can be written in the following compact form:

$$\mathbb{R}=\ddot{\theta }+\frac{g\overline{m}}{lM}\theta +R\left(\theta ,\dot{\theta },F\right),$$

(34)

with

$$R\left(\theta ,\dot{\theta },F\right)=\frac{m{\dot{\theta }}^2\theta }{m-\overline{m}}+Z_0+Z_2{\theta }^2+Z_3{\theta }^3+Z_4{\theta }^4+Z_5{\theta }^5.$$

In order to apply the KBM method, we rewrite problem (34) in the following form of a p-problem (here, $p<<1$) to enable us to solve this equation:

$$\left\{\begin{array}{c}\ddot{\theta }+\frac{g\overline{m}}{lM}\theta +pR\left(\theta ,\dot{\theta },F\right)=0,\\ \theta \left(0\right)={\theta }_0\mathrm{and}{\theta }^{\prime }\left(0\right)={\dot{\theta }}_0.\end{array}\right.$$

(35)

For applying the KBM method on problem (35), we assume the solution in the following ansatz form:

$$\theta \equiv {\theta }_p\left(t\right)=acos\left(\psi \right)+\sum _{n=1}^{N-1}{p}^n{u}_n\left(a,\psi \right)+O\left(p^N\right),$$

(36)

where each $u_n$ is a periodic function of $\psi$ and both $a\equiv a\left(t\right)$ and $\psi \equiv \psi \left(t\right)$ are defined by

$$\frac{da}{dt}=\sum _{n=1}^{N-1}{p}^n{A}_n\left(a\right)+O\left(p^N\right),$$

(37)

and

$$\frac{d\psi }{dt}={\omega }_0+\sum _{n=1}^{N-1}{p}^n{\psi }_n\left(a\right)+O\left(p^N\right),$$

(38)

where ${\omega }_0^2=g\overline{m}/\left(lM\right).$

We define the KBM homotopy as follows:

$$H(\theta ,p)=\ddot{\theta }+{\omega }_0^2\theta +pR\left(\theta ,\dot{\theta },F\right).$$

(39)

The approximate analytical solution is obtained at $p=1$.

Now, to uniquely determine the values of the quantities $A_n$ and ${\psi }_n$, it is required that no $u_n$ contains secularity terms. For sake of simplicity, we assume that $N=1$, and applying the KBM yields

$${\theta }_p\left(t\right)=acos\left(\psi \right)-\frac{p}{46080g\overline{m}{M}^3}\left(Y_0+Y_2{a}^2+Y_3{a}^3+Y_4{a}^4+Y_5{a}^5+Y_7{a}^7\right),$$

(40)

with

$$\begin{array}{cc}\hfill Y_0& =46080M^{3}F,\hfill \\ \hfill Y_2& =3840M^{2}F(2m+M)(cos\left(2\psi \right)-3),\hfill \\ \hfill Y_3& =240g{M}^2\overline{m}(12m+M)cos\left(3\psi \right),\hfill \\ \hfill Y_4& =-16MF\left(24m^2+20mM+M^2\right)\left(20cos\left(2\psi \right)+cos\left(4\psi \right)-45\right),\hfill \\ \hfill Y_5& =-gM\overline{m}\left(\begin{array}{c}15\left(144m^2+76mM+M^2\right)cos\left(3\psi \right)\\ +\left(240m^2+140mM+M^2\right)cos\left(5\psi \right)\end{array}\right),\hfill \\ \hfill Y_7& =gm\overline{m}\left(15m^2+15mM+2M^2\right)\left(6cos\left(3\psi \right)+6cos\left(5\psi \right)+cos\left(7\psi \right)\right),\hfill \end{array}$$

while the values of $\left(\dot{a},\dot{\psi }\right)$ read

$$\begin{array}{c}\dot{a}=0,\hfill \\ \dot{\psi }=\sqrt{\frac{g\overline{m}}{lM}}+p\sqrt{\frac{g\overline{m}}{lM}}\frac{m{a}^6\left(15m^2+15mM+2M^2\right)+M{a}^4\left(96m^2+44mM+M^2\right)-24M^2{a}^2(4m+M)}{384M^3}.\hfill \end{array}$$

(41)

We assume that $a=A=$ constant, while the value of $\psi$ reads

$$\psi =\frac{\lambda }{384M^3}\sqrt{\frac{g\overline{m}}{lM}}\left(\begin{array}{c}{A}^{6}m\left(15m^2+15mM+2M^2\right)+\\ A^{4}M\left(96m^2+44mM+M^2\right)-\\ 24A^2{M}^2(4m+M)+384M^3\end{array}\right)t+B.$$

(42)

Here, the numbers $\lambda$ and p are free/optimal parameters which are chosen to obtain the least residual error as possible, and of which the default values read $\lambda =p=1$. The value of integration constant B can be obtained from the ICs.

In order to find the value of $x_C\left(t\right)$, we solve the second equation in system (1), i.e., $R_2=0$, for ${\ddot{x}}_C:$

$${\ddot{x}}_{C}lcos\theta +\ddot{\theta }{l}^2+glsin\theta =0,$$

(43)

We obtain

$${\ddot{x}}_C=-\frac{\ddot{\theta }{l}^2+glsin\theta }{lcos\theta }.$$

(44)

Now, applying the Chebyshev approximation on the RHS of Equation (44) yields

$${\ddot{x}}_C=-\frac{864g{\theta }^3}{49{\pi }^3}-\frac{18636g\theta }{6125\pi }-\frac{34}{39}l{\theta }^2\ddot{\theta }-\frac{35}{37}l\ddot{\theta }.$$

(45)

By solving the ode (45) using RK4 or MATHEMATICA command “NDSolve” with the help of ICs $x\left(0\right)=x_0$ and $x^{\prime }\left(0\right)={\dot{x}}_0$, we can obtain the expression for $x_C$.

For numerical examples, the comparison between the numerical approximation using the RK4 method and the obtained approximations (40) and (45) is presented in Figure 4, Figure 5 and Figure 6 for different values of the excitation force F as follows:

$$\left\{\begin{array}{c}F=0,\\ F=0.01\mathrm{N},\\ F=0.1cos\left(0.2t\right)\mathrm{N},\end{array}\right.$$

(46)

while the other parameters have the same values given in Figure 2 and Figure 3.

The global maximum error $L_g$ is estimated for the three cases of the excitation force given in Equation (46), as illustrated in

Table 2. The global maximum distance error $L_g$ for the KBM method.

F	${\left.{\mathit{L}}_{\mathit{g}}\right|}_{\left(\mathit{\theta }\right)}$	${\left.{\mathit{L}}_{\mathit{g}}\right|}_{\left({\mathit{x}}_{\mathit{C}}\right)}$
0	$0.041186$	$0.0112186$
$0.01$ N	$0.0155981$	$0.0126209$
$0.1cos\left(0.2t\right)\mathrm{N}$	$0.0752997$	$0.458301$

.

3. Conclusions

The equations of motion for unforced and forced pendulum–cart system oscillators have been solved and analyzed analytically using two accurate and effective approaches. In the beginning, the ansatz method was applied for deriving some effective and accurate approximations in terms of the trigonometric functions. Moreover, the KBM method was carried out to find some approximations to the problem under consideration. In the first approach, it was assumed that the excitation force F must be constant because equilibrium is considered only for autonomous systems. However, in the second approach, the excitation force F could take any value for dependent or independent time, i.e., F could be a constant value or $F=F\left(t\right)$. This feature is not fulfilled in the first approach, in which the excitation force must be constant, because equilibrium is considered only for autonomous systems. Furthermore, the analytical approximations were compared with the numerical approximations using the RK4 numerical approach. In addition, the global maximum distance error with respect to the RK4 numerical approximation was estimated. It was observed that the obtained approximations are characterized by high accuracy and convergence for long periods of time. Moreover, it was found that the accuracy of the obtained approximations using the first approach (ansatz method) is better than that of the second approach (KBM method).

The obtained results can help many authors in studying the mechanism of robotics motion and in investigating many nonlinear oscillations in plasma physics and several fields of science.