**5. An application example**

We apply the aforementioned theory to a 3D crane system to understand the proposed methodology comprehensively.

## **5.1. Problem statement**

An overhead crane is a symbol of underactuated mechanical systems. Overhead cranes are typically used to transport cargo over short distances or to small areas, such as automotive factories and shipyards. We have investigated the nonlinear feedback control problem for a 3D overhead crane [8] with three actuators used to stabilize five outputs. The crane system, which is composed of four masses, is physically modeled in **Figure 4**. The distributed masses of the bridge are converted into a concentrated mass *mb*, which is placed at the center of the bridge. *ml* denotes the equivalent mass of the hoist mechanism, whereas *mt* and *mc* are the masses of the trolley and cargo, respectively. The system includes five degrees of freedom, which correspond to five generalized coordinates. *x*(*t*) is the trolley motion, *z*(*t*) is the bridge movement, and cargo position is characterized by three generalized coordinates (*l*, *θ*, and *φ*). Therefore, the generalized coordinates of the system are described by = . Additionally, the friction of cargo hoisting, as well as trolley and bridge motions, is linearly characterized by damping factors *br*, *bt* , and *bb*, respectively. The control signals *ub*, *ut* and *ul* correspondingly demonstrate the driving forces of trolley motion, bridge movement, and cargo lifting translation.

**Figure 4.** Physical modeling of a 3D overhead crane.

(28)

(29)

= ) *if the*

and *mc* are the

. The stability of the linear system (28)

with

as a 2(*n*–*m*)×2(*n*–*m*) Jacobian matrix of components ∂*fi*

252 Nonlinear Systems - Design, Analysis, Estimation and Control

linearization theorem [4].

*The equilibrium point* ( = ˙

**5. An application example**

methodology comprehensively.

**5.1. Problem statement**

bridge. *ml*

(28) *is unstable*.

*stable*.

*linearized system* (28) *is strictly stable*.

/∂*xj*

= ) *of the nonlinear system* (26) *is unstable if the linearized system*

can be analyzed by considering the positions of the eigenvalues of **A** or using several traditional techniques, such as the Routh-Hurwitz criterion [3], the root locus method, and so on. Thus, by investigating the stability of the linear system (28), we can understand the dynamic behavior of the nonlinear system (27), or equivalently, zero dynamics (26), according to Lyapunov's

*We cannot conclude the stability of the nonlinear system* (26) *if the linearized system* (28) *is marginally*

As we will see in the examples provided in Section 5, the analysis of system stability using the

We apply the aforementioned theory to a 3D crane system to understand the proposed

An overhead crane is a symbol of underactuated mechanical systems. Overhead cranes are typically used to transport cargo over short distances or to small areas, such as automotive factories and shipyards. We have investigated the nonlinear feedback control problem for a 3D overhead crane [8] with three actuators used to stabilize five outputs. The crane system, which is composed of four masses, is physically modeled in **Figure 4**. The distributed masses of the bridge are converted into a concentrated mass *mb*, which is placed at the center of the

denotes the equivalent mass of the hoist mechanism, whereas *mt*

masses of the trolley and cargo, respectively. The system includes five degrees of freedom, which correspond to five generalized coordinates. *x*(*t*) is the trolley motion, *z*(*t*) is the bridge movement, and cargo position is characterized by three generalized coordinates (*l*, *θ*, and *φ*). Therefore, the generalized coordinates of the system are described by = .

aforementioned theorem yields the constraint equations of the controller parameters.

*The nonlinear system* (26) *is asymptotically stable around the equilibrium point* ( = ˙

The main objective of this example is to design a controller for simultaneously conducting five tasks: (1) tracking the bridge, (2) moving the trolley to its destinations, (3) lifting/lowering the payload to the desired length of the cable, (4) keeping the cargo swing angles small during transportation, and (5) completely suppressing these swings at payload destinations.

By using Lagrange's equation to constitute the mathematical model, overhead crane dynamics can be represented by matrix equation (9) in which the component matrices are determined by the following formulas:

$$\mathbf{M}(\mathbf{q}) = \begin{bmatrix} m\_{11} & 0 & m\_{13} & m\_{14} & m\_{15} \\ 0 & m\_{22} & m\_{23} & 0 & m\_{25} \\ m\_{31} & m\_{32} & m\_{33} & 0 & 0 \\ m\_{41} & 0 & 0 & m\_{44} & 0 \\ m\_{51} & m\_{52} & 0 & 0 & m\_{55} \end{bmatrix}, \mathbf{C}(\mathbf{q}, \dot{\mathbf{q}}) = \begin{bmatrix} b\_{\flat} & 0 & c\_{13} & c\_{14} & c\_{15} \\ 0 & b\_{\flat} & c\_{23} & 0 & c\_{25} \\ 0 & 0 & b\_{\flat} & c\_{34} & c\_{35} \\ 0 & 0 & c\_{43} & c\_{44} & c\_{45} \\ 0 & 0 & c\_{55} & c\_{54} & c\_{55} \end{bmatrix},$$

$$\mathbf{F} = \begin{bmatrix} \boldsymbol{\mu}\_{b} & \boldsymbol{\mu}\_{\boldsymbol{\iota}} & \boldsymbol{\mu}\_{\boldsymbol{\iota}} & \mathbf{0} & \mathbf{0} \end{bmatrix}^{\boldsymbol{\tau}}, \; \mathbf{G} \begin{pmatrix} \mathbf{q} \\ \end{pmatrix} = \begin{bmatrix} \mathbf{0} & \mathbf{0} & \mathbf{g}\_{\boldsymbol{\mathfrak{s}}} & \mathbf{g}\_{\boldsymbol{\mathfrak{s}}} & \mathbf{g}\_{\boldsymbol{\mathfrak{s}}} \end{bmatrix}.$$

The coefficients of the **M**(**q**) matrix are given by

11 <sup>=</sup> <sup>+</sup> <sup>+</sup> , 13 <sup>=</sup> 31 <sup>=</sup> sincos, 14 <sup>=</sup> 41 <sup>=</sup> coscos,

15 <sup>=</sup> 51 <sup>=</sup> − sinsin, 22 <sup>=</sup> <sup>+</sup> , 23 <sup>=</sup> sin, 25 <sup>=</sup> 52 <sup>=</sup> cos, and 32 <sup>=</sup> sin, 33 <sup>=</sup> <sup>+</sup> , 44 <sup>=</sup> 2 cos 2, 55 <sup>=</sup> 2 . The coefficients of the , ˙ matrix are determined by 13 <sup>=</sup> coscos˙ − sinsin˙, 14 <sup>=</sup> coscos ˙ − cossin˙ − sincos˙, 15 <sup>=</sup> − cossin˙ − sinsin ˙ − sincos˙, 23 <sup>=</sup> cos˙, 25 <sup>=</sup> cos ˙ − sin˙, 34 <sup>=</sup> − cos 2˙, 35 <sup>=</sup> − ˙, 43 <sup>=</sup> cos 2˙, 44 <sup>=</sup> cos 2 ˙ − 2 cossin˙, and 45 <sup>=</sup> − 2 cossin˙, 53 <sup>=</sup> ˙, 54 <sup>=</sup> 2 cossin˙, 55 <sup>=</sup> ˙ .

The nonzero coefficients of the **G**(**q**) vector are given by

$$\mathbf{g}\_3 = -m\_c \mathbf{g} \cos \varphi \cos \theta,\,\mathbf{g}\_4 = m\_c \mathbf{g} l \sin \varphi \cos \theta,\,\mathbf{g}\_5 = m\_c \mathbf{g} l \cos \varphi \sin \theta$$

#### **5.2. Controller design**

The overhead crane is an underactuated system in which five output signals are driven by three actuators. Using the nonlinear feedback methodology, we construct a control law

$$\mathbf{F} = \begin{bmatrix} \mathbf{U} & \mathbf{0}\_{2 \times 1} \end{bmatrix}^{\mathrm{T}},\tag{30}$$

with <sup>=</sup> to drive the actuated states <sup>=</sup> to the desired destinations <sup>=</sup> and the actuated states (cargo swings) 2 <sup>=</sup> toward zero.

Applying the theory proposed in Sections 1−4, we determine the structure of the controller in Equation (24), where **K***ad* = diag(*Kad*1, *Kad*2, *Kad*3), **K***ap* = diag(*Kap*1, *Kap*2, *Kap*3), **K***ud* = diag(*Kud*1, *Kud*2), and **K***up* = diag(*Kup*1, *Kup*2) are the positive matrices of control gains, and = 1 <sup>0</sup> <sup>0</sup> 2 0 0 is a

weighting matrix.

#### **5.3. System stability**

15 <sup>=</sup> 51 <sup>=</sup> −

and 32 <sup>=</sup>

13 <sup>=</sup>

14 <sup>=</sup>

23 <sup>=</sup>

35 <sup>=</sup> −

45 <sup>=</sup> −

 2

**5.2. Controller design**

with <sup>=</sup>

weighting matrix.

<sup>=</sup>

15 <sup>=</sup> −

sinsin, 22 <sup>=</sup> <sup>+</sup>

, 44 <sup>=</sup>

˙ −

˙, 54 <sup>=</sup>

sin, 33 <sup>=</sup> <sup>+</sup>

254 Nonlinear Systems - Design, Analysis, Estimation and Control

coscos˙ −

˙ −

cossin˙ −

cos˙, 25 <sup>=</sup>

˙, 43 <sup>=</sup>

coscos

The coefficients of the , ˙ matrix are determined by

cos

The nonzero coefficients of the **G**(**q**) vector are given by

j q

cos

cossin˙, 53 <sup>=</sup>

sinsin˙,

cossin˙ −

sinsin

˙ −

2˙, 44 <sup>=</sup>

, 23 <sup>=</sup>

2, 55 <sup>=</sup>

cos 2˙,

cossin˙, 55 <sup>=</sup>

to drive the actuated states <sup>=</sup> to the desired destinations

cossin˙, and

˙ .

 jq

(30)

is a

 2 cos

sincos˙,

sin˙, 34 <sup>=</sup> −

cos 2 ˙ − 2

> 2

3 45 *g m g g m gl g m gl* = -= = *cc c* cos cos , sin cos , cos sin

The overhead crane is an underactuated system in which five output signals are driven by three actuators. Using the nonlinear feedback methodology, we construct a control law

and the actuated states (cargo swings) 2 <sup>=</sup> toward zero.

Applying the theory proposed in Sections 1−4, we determine the structure of the controller in Equation (24), where **K***ad* = diag(*Kad*1, *Kad*2, *Kad*3), **K***ap* = diag(*Kap*1, *Kap*2, *Kap*3), **K***ud* = diag(*Kud*1, *Kud*2),

and **K***up* = diag(*Kup*1, *Kup*2) are the positive matrices of control gains, and =

 j q

sincos˙,

sin, 25 <sup>=</sup> 52 <sup>=</sup>

 2 . cos,

As presented in Section 4, we analyze the local stability of the internal dynamics (25), or equivalently, the zero dynamics (26). Applying Equation (26) to a 3D overhead crane, the zero dynamics of the system is expanded as

$$l\_d l\_d \ddot{\boldsymbol{\phi}} - 2l\_d \tan \theta \dot{\theta} \dot{\boldsymbol{\phi}} - \alpha\_l K\_{\text{ul}} \frac{\cos \phi}{\cos \theta} \dot{\boldsymbol{\phi}} - \alpha\_l K\_{\text{ul}} \frac{\cos \phi}{\cos \theta} \boldsymbol{\varphi} + \mathbf{g} \frac{\sin \phi}{\cos \theta} = 0 \,, \tag{31}$$

$$\begin{pmatrix} l\_d \ddot{\theta} + l\_d \cos \theta \sin \theta \dot{\phi}^2 + \alpha\_l K\_{u\phi 1} \sin \varphi \sin \theta \dot{\phi} - \alpha\_2 K\_{u\phi 2} \cos \theta \dot{\theta} \\ + \alpha\_l K\_{u\rho 1} \sin \varphi \sin \theta \dot{\varphi} - \alpha\_2 K\_{u\rho 2} \cos \theta \dot{\theta} + \mathbf{g} \cos \varphi \sin \theta \end{pmatrix} = 0 \,\text{}\,\text{d}\,\tag{32}$$

The stability of the zero dynamics, which comprises Equations (31) and (32), is analyzed using Lyapunov's linearization theorem. First, we represent the zero dynamics in the first-order form by setting the four state variables as

$$z\_1 = \rho, z\_2 = \dot{\rho}, \; z\_3 = \theta, z\_4 = \dot{\theta}$$

Then, the zero dynamics exhibits the following state-space forms:

(33)

$$\dot{z}\_2 = \left( 2 \tan z\_3 z\_4 z\_2 + \frac{\alpha\_l K\_{\text{sub}}}{l\_d} \frac{\cos z\_1}{\cos z\_3} z\_2 + \frac{\alpha\_l K\_{\text{sub}}}{l\_d} \frac{\cos z\_1}{\cos z\_3} z\_1 - \frac{\text{g}}{l\_d} \frac{\sin z\_1}{\cos z\_3} \right) = f\left( \mathbf{z} \right), \tag{34}$$

$$
\dot{\mathbf{z}}\_3 = \mathbf{z}\_{4\prime} \tag{35}
$$

$$\dot{z}\_{4} = \begin{pmatrix} -\cos z\_{3}\sin z\_{3}z\_{2}^{2} - \frac{\alpha\_{1}K\_{\text{all}}}{l\_{d}}\sin z\_{1}\sin z\_{3}z\_{2} + \frac{\alpha\_{2}K\_{\text{all}}}{l\_{d}}\cos z\_{3}z\_{4} \\\\ -\frac{\alpha\_{1}K\_{\text{all}}}{l\_{d}}\sin z\_{1}\sin z\_{3}z\_{1} + \frac{\alpha\_{2}K\_{\text{all}}}{l\_{d}}\cos z\_{3}z\_{3} - \frac{\mathbf{g}}{l\_{d}}\cos z\_{1}\sin z\_{3} \end{pmatrix} = h\left(\mathbf{z}\right). \tag{36}$$

Using = 1 2 3 4 as the state vector, the nonlinear zero dynamics (33)–(36) are asymptotically stable around the equilibrium point **z** = **0** ( = ˙ = ) if the linearized system is strictly stable. Linearizing the zero dynamics around **z** = **0** leads to a linear system as follows:

$$
\dot{\mathbf{z}} = \mathbf{A} \mathbf{z} \,, \tag{37}
$$

where

$$\mathbf{A} = \begin{bmatrix} 0 & 1 & 0 & 0 \\ \frac{\partial f}{\partial z\_1} & \frac{\partial f}{\partial z\_2} & \frac{\partial f}{\partial z\_3} & \frac{\partial f}{\partial z\_4} \\ 0 & 0 & 0 & 1 \\ \frac{\partial h}{\partial z\_1} & \frac{\partial h}{\partial z\_2} & \frac{\partial h}{\partial z\_3} & \frac{\partial h}{\partial z\_4} \end{bmatrix}\_{\mathbf{x}=\mathbf{0}} = \begin{bmatrix} 0 & 1 & 0 & 0 \\ \frac{\alpha\_1 K\_{up1} - \mathbf{g}}{l\_d} & \frac{\alpha\_1 K\_{up1}}{l\_d} & 0 & 0 \\ 0 & 0 & 0 & 1 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & \frac{\alpha\_2 K\_{up2} - \mathbf{g}}{l\_d} & \frac{\alpha\_2 K\_{up2}}{l\_d} \end{bmatrix} \tag{38}$$

is a Jacobian matrix in which the characteristic polynomial exhibits the following form:

$$\begin{split} \left| \mathbf{A} - \mathbf{s} \mathbf{I}\_{4} \right| &= \mathbf{s}^{4} - \frac{\left( \alpha\_{t} K\_{\text{ul}1} + \alpha\_{2} K\_{\text{ul}2} \right)}{l\_{d}} \mathbf{s}^{3} + \left( \frac{\alpha\_{t} \alpha\_{2} K\_{\text{ul}1} K\_{\text{ul}2}}{l\_{d}^{2}} - \frac{\alpha\_{t} K\_{\text{up}1} + \alpha\_{2} K\_{\text{ug}2} - 2 \mathbf{g}}{l\_{d}} \right) \mathbf{s}^{2} \\ &+ \frac{\alpha\_{t} \alpha\_{2} \left( K\_{\text{ul}1} K\_{\text{up}2} + K\_{\text{up}1} K\_{\text{ul}2} \right) - \mathbf{g} \left( \alpha\_{t} K\_{\text{ul}1} + \alpha\_{2} K\_{\text{ul}2} \right)}{l\_{d}^{2}} \mathbf{s} \\ &+ \frac{\alpha\_{t} \alpha\_{2} K\_{\text{ul}1} K\_{\text{up}2} + \mathbf{g}^{2} - \mathbf{g} \left( \alpha\_{t} K\_{\text{up}1} + \alpha\_{2} K\_{\text{up}2} \right)}{l\_{d}^{2}} . \end{split} \tag{39}$$

The linearized system (37) is stable around the equilibrium point **z** = **0** if **A** is a Hurwitz matrix. On the basis of the Hurwitz's criterion and the results of the calculations, the constraint condition of the controller parameters is determined as

$$
\alpha\_1 K\_{\text{ud1}} + \alpha\_2 K\_{\text{ud2}} < 0 \,, \tag{40}
$$

$$
\alpha\_1 \alpha\_2 K\_{\text{u}\text{d}1} K\_{\text{u}\text{d}2} > l\_d \left( \alpha\_1 K\_{\text{u}\text{p}1} + \alpha\_2 K\_{\text{u}\text{p}2} - \text{2g} \right), \tag{41}
$$

$$\log \alpha\_2 \left( K\_{\text{udl}} K\_{\text{up}2} + K\_{\text{uql}} K\_{\text{udl}2} \right) > \text{g} \left( \alpha\_1 K\_{\text{udl}} + \alpha\_2 K\_{\text{udl}2} \right), \tag{42}$$

$$
\alpha\_1 \alpha\_2 K\_{up1} K\_{up2} + \mathbf{g}^2 > \mathbf{g} \left( \alpha\_1 K\_{up1} + \alpha\_2 K\_{up2} \right). \tag{43}
$$

Therefore, if Equations (40)−(43) among the control parameters are maintained, then the zero dynamics is stable around the equilibrium point **z** = **0**, which leads to the local stability of the internal dynamics (25).

### *5.4. Simulation and experiment*

(37)

(38)

(39)

(40)

(41)

(42)

(43)

where

256 Nonlinear Systems - Design, Analysis, Estimation and Control

is a Jacobian matrix in which the characteristic polynomial exhibits the following form:

The linearized system (37) is stable around the equilibrium point **z** = **0** if **A** is a Hurwitz matrix. On the basis of the Hurwitz's criterion and the results of the calculations, the constraint

Therefore, if Equations (40)−(43) among the control parameters are maintained, then the zero dynamics is stable around the equilibrium point **z** = **0**, which leads to the local stability of the

condition of the controller parameters is determined as

internal dynamics (25).

The overhead crane dynamics (9) driven by the control inputs (30) is numerically simulated in the case of a crane system that involves complicated operations. Accordingly, the trolley is forced to move from its initial position to the desired displacement at 0.4 m. The bridge is driven from its starting point to the desired location at 0.3 m, and the cargo is lifted with a cable length of 1–0.7 m of cable reference. These processes (lifting the cargo, moving the trolley, and driving the bridge) must be initiated simultaneously, with the cargo suspension cable initially perpendicular to the ground. The parameters used for the simulation are listed in **Table 1**.


**Table 1.** Crane system parameters.

**Figure 5.** Overhead crane system used for the experiments.

Additionally, an experimental study is conducted to verify the simulation results. **Figure 5** shows a laboratory crane system used for the experiment. In this system, three DC motors for the bridge motion, trolley movement, and cargo hoisting motion are used. Five incremental encoders are applied for measuring bridge and trolley motions, the movement of the cargo along the cable, and the two swing angles of the cargo.

Three-dimensional overhead crane is controlled by a target PC in which a control structure is built based on MATLAB/SIMULINK with an xPC target foundation. A host PC is linked to the target PC, and the crane system is connected to the target PC by two interface cards. The 6602 card sends PWM signals to the motor amplifiers and obtains feedback pulses from the encoders. The 6025E multifunction card is utilized for sending direction control signals to the motor amplifiers.

**Figure 6.** Bridge motion.

**Figure 7.** Trolley motion.

**Figures 6**–**18** describe both the simulation and the experiment results. **Figures 6**–**8** show the paths of the bridge motion, trolley movement, and payload lifting translation, respectively. All the responses approach asymptotically to the destinations. However, the simulation paths are smoother and achieve steady states earlier than the experiment ones. The bridge moves and stops accurately at the load endpoint after 4 s in the simulation and 6 s in the experiment. The trolley reaches its destination after 4.1 s in the simulation and 6.2 s in the experiment. The crane lifts the payload from an initial length (1 m) of cable to the desired length (0.7 m) of cable after 4.2 s.

**Figure 8.** Cargo hoisting motion.

**Figure 9.** Cargo swing angle *φ*.

card sends PWM signals to the motor amplifiers and obtains feedback pulses from the encoders. The 6025E multifunction card is utilized for sending direction control signals to the

**Figures 6**–**18** describe both the simulation and the experiment results. **Figures 6**–**8** show the paths of the bridge motion, trolley movement, and payload lifting translation, respectively. All the responses approach asymptotically to the destinations. However, the simulation paths are smoother and achieve steady states earlier than the experiment ones. The bridge moves and stops accurately at the load endpoint after 4 s in the simulation and 6 s in the experiment. The trolley reaches its destination after 4.1 s in the simulation and 6.2 s in the experiment. The crane lifts the payload from an initial length (1 m) of cable to the desired length (0.7 m) of cable after

motor amplifiers.

258 Nonlinear Systems - Design, Analysis, Estimation and Control

**Figure 6.** Bridge motion.

**Figure 7.** Trolley motion.

**Figure 8.** Cargo hoisting motion.

4.2 s.

**Figure 10.** Cargo swing angle *θ*.

**Figure 11.** Velocity of bridge motion.

**Figure 12.** Velocity of trolley motion.

**Figures 9** and **10** indicate the responses of the cargo swings. The payload swing angles are in a small boundary during the payload transportation: *φ*max = 2.2° and *θ*max = 2.9° for the simulation and *φ*max = 2.3° and *θ*max = 2.4° for the experiment. The simulated cargo swings are

completely vanished after short settling periods, *ts* = 4 s for *φ* and *ts* = 4.5 s for *θ*, within one vibration period. Slight steady-state errors remain in the experimental responses, which achieve the approximate steady state after over two oscillation periods.

**Figure 13.** Cargo hoisting velocity.

The velocity components depicted in **Figures 11**–**15** asymptotically approach to zero. The movements of the bridge and the trolley, as well as the lifting movement of the payload at transient states, composed of two phases, namely, the increasing and decreasing velocity periods. As indicated clearly in the simulated curves, the trolley speeds up within the first 1.7 s and slows down within the last 2.4 s. The cargo is then lifted with increasing speed within the first 0.7 s and with decreasing speed within the remaining 3.5 s.

**Figure 14.** Payload swing velocity ˙.

**Figure 15.** Payload swing velocity ˙.

**Figure 16.** Bridge moving force.

completely vanished after short settling periods, *ts* = 4 s for *φ* and *ts* = 4.5 s for *θ*, within one vibration period. Slight steady-state errors remain in the experimental responses, which

The velocity components depicted in **Figures 11**–**15** asymptotically approach to zero. The movements of the bridge and the trolley, as well as the lifting movement of the payload at transient states, composed of two phases, namely, the increasing and decreasing velocity periods. As indicated clearly in the simulated curves, the trolley speeds up within the first 1.7 s and slows down within the last 2.4 s. The cargo is then lifted with increasing speed within

achieve the approximate steady state after over two oscillation periods.

260 Nonlinear Systems - Design, Analysis, Estimation and Control

the first 0.7 s and with decreasing speed within the remaining 3.5 s.

**Figure 13.** Cargo hoisting velocity.

**Figure 14.** Payload swing velocity ˙.

**Figure 15.** Payload swing velocity ˙.

**Figure 17.** Trolley driving force.

The nonlinear control forces are illustrated in **Figures 16**–**18**. The simulation responses achieve steady states after 4, 4.1, and 4.2 s for the bridge moving, trolley moving, and cargo lifting forces, respectively.

$$\text{At steady states, } u\_t^{\text{SS}} = u\_b^{\text{SS}} = 0 \text{ N and } u\_l^{\text{SS}} = -m\_c g = -9.81 \times 0.85 = -8.34 \text{ N.}$$

Evidently, differences in responses still exist between the simulation and the experiment responses because the dynamic model and the realistic overhead crane do not match completely. Several nonlinearities that exist in practice, such as the cable flexibility, the backlash of the gear motors, and nonlinear frictions, are not considered in the system dynamics. If the mathematical model is close to a realistic system, then the results will certainly be accurate.

**Figure 18.** Payload hoisting force.
